The Symbolic Basis of Physical Intuition
A Study of Two Symbol Systems in Physics Instruction
by
Bruce Lawrence Sherin
B.A. (Princeton University) 1985
M.A. (University of California, Berkeley) 1988
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Science and Mathematics Education
in the
GRADUATE DIVISION
of the
UNIVERSITY of CALIFORNIA, BERKELEY
Committee in charge:
Professor Andrea A. diSessa, Chair
Professor Rogers P. Hall
Professor Daniel S. Rokhsar
Professor Barbara Y. White
1996
1
The Symbolic Basis of Physical Intuition
A Study of Two Symbol Systems in Physics Instruction
© 1996
by
Bruce Lawrence Sherin
2
Abstract
The Symbolic Basis of Physical Intuition:
A Study of Two Symbol Systems in Physics Instruction
by
Bruce L. Sherin
Doctor of Philosophy in Science and Mathematics Education
University of California, Berkeley
Professor Andrea A. diSessa, Chair
This dissertation is a comparative study of the use of two symbol systems in physics
instruction. The first, algebraic notation, plays an important role as the language in which physicists
make precise and compact statements of physical laws and relations. Second, this research explores
the use of programming languages in physics instruction. Central to this endeavor is the notion that
programming languages can be elevated to the status of bona fide representational systems for
physics.
I undertook the cognitive project of characterizing the knowledge that would result from each
of these two instructional practices. To this end, I constructed a model of one aspect of the
knowledge associated with symbol use in physics. This model included two major types of
knowledge elements: (1) symbolic forms, which constitute a conceptual vocabulary in terms of
which physics expressions are understood, and (2) representational devices, which function as a set
of interpretive strategies. The model constitutes a partial theory of “meaningful symbol use” and
how it affects conceptual development.
The empirical basis of this work is a data corpus consisting of two parts, one which contains
videotapes of pairs of college students solving textbook physics problems using algebraic notation,
and one in which college students program computer simulations of various motions. The
videotapes in each half of the corpus were transcribed and analyzed in terms of the above model,
and the resulting analyses compared. This involved listing the specific symbolic forms and
representational devices employed by the students, as well as a measurement of the frequency of
use of the various knowledge elements.
A conclusion of this work is that algebra-physics can be characterized as a physics of balance and
equilibrium, and programming-physics a physics of processes and causation. More generally, this
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work provides a theoretical and empirical basis for understanding how the use of particular symbol
systems affects students’ conceptualization.
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Acknowledgements
This work, to a great extent, grew out of my interactions with my advisor, Andrea A. diSessa. He
shaped my views on thinking and learning generally, and also contributed directly to many of the
ideas in this thesis.
My views have also been strongly influenced by Rogers Hall and Barbara White. Rogers gave me
many new ways to think about representational artifacts—really, more than I could handle—and
Barbara kept me anchored to the world of physics instruction. Thanks also to Daniel Rokhsar, for
his physics-eye view.
Many other members of the EMST/SESAME community helped me along the way. Thanks
especially to the members of the Boxer Research Group, for listening to me talk about this work
for several years, and giving helpful feedback. Thanks also to the members of the Representational
Practices Group and the ThinkerTools Group for the added perspective provided by their own
research.
I could never have finished—or taken so long—without my friends to distract me. Thanks to the
Geebers for Geebing, the thing that they do best.
My parents have never been anything but supportive in my academic pursuits, a fact that I really
do appreciate. Thanks also to my grandparents, my sister Heidi, my parents-in-law, all my
brothers-in-law, sisters-in-law, nieces, nephews, and everybody else for generally being a mellow
bunch.
My wife Miriam, in a thousand ways, made this work possible. She did the last-minute tasks that I
couldn’t do, and kept me sane and happy. If she’s proud of me, then I’m satisfied.
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Chapter 1. Introduction and Major Themes
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26
319
64
9
6
33
fattened kids
fattened goat-heifers
pastured sheep
pastured he-goats
pastured ewes
pastured goats
weaned lambs…
Circa 2500 BCE, an inhabitant of Mesopotamia—probably a man—stands working on a clay
tablet, writing in cuneiform. He is recording information concerning livestock to be presented in
trade. The marks he makes are not a careful transcription of speech, but they are also not a
haphazard collection of information. The information is represented in a specific form, as a list.
This ancient man is making and using an external representation.
To a present day individual, the making of such a list is a mundane event. We make lists
before going shopping and to keep track of the tasks we need “to do.” But the timeless and
ubiquitous nature of written lists does not suggest that they are simple and unimportant
constructions; on the contrary, it implies that they have a power that transcends time and task.
Why are lists so useful that they are not only written on sheets of paper, they have even been
etched on clay tablets?
Some of the answers to this question are relatively straightforward. The list is useful to our
ancient Mesopotamian because he can use it later as an aid to memory. In general, the use of
written lists allows individuals to remember long lists of information with relative ease, as
compared with committing the same information to memory. In addition, lists, such as shopping
lists, can serve as resources during future action. While at the supermarket, we can procure the
items on our list one by one, checking them off as we go.
Furthermore, there is the striking fact that the list above is a real one: This particular list of
information was actually written by some inhabitant of Mesopotamia during the third
millennium, BCE (It is taken from Goody, 1977). Thus, the information has accomplished the
trick of traveling through time several millennia so that we can read it, a striking feat!
Some aspects of the power of lists and other external representations are somewhat less easy to
recognize and explain. In The Domestication of the Savage Mind, Jack Goody discusses the
important role played by lists during the early development of writing (Goody, 1977). According
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to Goody, writing—and lists, in particular—did not simply constitute a means of performing the
same tasks, such as remembering existing information, more easily and efficiently. The changes
introduced by written lists were not only changes in degree or extent. Rather, he argues, the
development of lists changed people’s practices as well as the very nature of knowledge and
understanding.
My concern here is to show that these written forms were not simply by-products of the interaction
between writing and say, the economy, filling some hitherto hidden ‘need,’ but that they represented a
significant change not only in the nature of transactions, but also in the ‘modes of thought’ that
accompanied them, at least if we interpret ‘modes of thought’ in terms of the formal, cognitive and
linguistic operations which this new technology of the intellect opened up. (p. 81)
The point is this: As our list-writer works, he must bend his thought and action to the
framework provided by the list. For this reason, the thoughts of our list-writer, during the
moments of composing the list, are in part shaped by the structure of the list. Thus, the product
of this activity is also in part determined by the list structure. Goody even argues that the use of
lists led to the development of scientific taxonomies and eventually to the development of science
as we know it.
Step forward to the 20th century. Lists and other external representations are still quite
popular. External representations are an important part of everyday activity, as well as of work in
almost all disciplines. Here we turn to the scientific disciplines; in particular, we turn to the
domain of physics. Within the discipline of physics, equations and related symbolic expressions
are an extremely important external representational form. Physicists write equations and
manipulate symbols in order to perform extended and complex computations. Furthermore,
physicists use equations as a means of making precise and compact expressions of physical laws,
which then appear in academic papers and in textbooks.
As with Goody’s lists, the implications of equation-use in physics extend to the subtle and
profound. Equation-use is so pervasive in physics that, for the novice physics student, it may be
hard to escape the intuition that equations ARE physics knowledge. And it may seem like “doing
physics” means nothing more than using and manipulating symbols. This may be more than a
foolhardy misconception, however. At least one physicist believes that there is some truth behind
this appearance. In The Character of Physical Law, physicist Richard Feynman, expresses a similar
intuition (Feynman, 1965):
The burden of the lecture is just to emphasize the fact that it is impossible to explain honestly the
beauties of the laws of nature in a way that people can feel, without their having some deep
understanding in mathematics. I am sorry, but this seems to be the case. (p. 39)
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Why not tell me in words instead of in symbols? Mathematics is just a language, and I want to be able
to translate the language. … But I do not think it is possible, because mathematics is not just another
language. Mathematics is a language plus reasoning; it is like a language plus logic. (p. 40)
The apparent enormous complexities of nature, with all its funny laws and rules, each of which has
been carefully explained to you, are really very closely interwoven. However, if you do not appreciate
the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one
to the other. (p. 41)
In these quotes, Feynman does not quite say that physics knowledge is nothing but equations,
but he is making a related claim. Here, he expresses the intuition that, in some manner, it is not
possible to fully understand physics without being an initiate into the physicist’s use of
mathematics and the associated symbols. Feynman tells his students that he cannot explain “in
words instead of symbols.” To understand physics, you must have the experience of living in the
tight net of physical relations expressed as equations, going from one to the other. You have to
write equations and solve them, derive one equation from another, and from this experience
develop a sense for the properties of the equations and the relations among them.
From Goody’s lists to Feynman’s equations, the power of external representations is manifest.
They allow us to easily remember long lists of facts for a duration that is essentially unlimited.
And they allow us to perform extended, context-free manipulations to compute results.
Furthermore, these authors point to the possibility of more subtle implications of external
representations for the nature of human knowledge. The first, an historical view, posits a role for
external representations in an historically developing body of knowledge. The second view boils
down to a physicist’s intuition that an appreciation of the beauties of physics is somehow tied up
with their expression in the language of mathematics.
Similar notions have appeared in many forms, and in a variety of literature. Jerome Bruner
spoke of external representations as a type of “cultural amplifier” (Bruner, 1966). Whorf
hypothesized that the vocabulary and syntax of a language shape the speaker’s view of reality
(Whorf, 1956). Lev Vygotsky was the father of a branch of psychology for which the
internalization of external signs plays a fundamental role in the development and character of
thought (Vygotsky, 1934/1986). And there is an extensive body of literature devoted to
determining the effect of literacy on thought (e.g., Goody & Watt, 1968; Scribner & Cole,
1981).
For each of the above researchers, knowledge and external representations are inextricably
related. But the diversity of views here will require some sorting, and the intuitions call for
expression in theoretically precise terms. Here, I will not adopt the claims or perspective of any of
8
these researchers, or add directly to any of their research programs. Unlike these anthropologists
and historians, I will not look across cultures or time for differences in thought or the nature of
knowledge associated with the use of various symbol systems. Instead, I will use the methods of
cognitive science to build a model of symbol use within a particular domain. Then, given this
model, I can begin to tentatively address some of the long-standing issues concerning external
representations and human knowledge.
A Tale of Two (Representational) Systems
The inquiry to be described here will be built around a comparative study of two symbol
systems within the domain of physics. The first of these is the traditional system of algebraic
notation. By “algebraic notation” I mean to include the full range of equations that physicists
write. Within the practice of physics, algebraic notation plays an important role as the language in
which physicists make precise and compact statements of physical laws and relations, and in which
they solve problems and derive new results. Here we will be particularly concerned with the use of
algebraic notation as it occurs in introductory physics instruction.
The use of the second representational system is novel: This research explores the use of
programming languages in physics instruction. Central to this endeavor is the notion that
programming languages can be elevated to the status of bona fide representational systems for
physics. This means treating programming as potentially having a similar status and performing a
similar function to algebraic notation in physics learning. It means that instead of representing
laws as equations, students will represent them as programs.
This proposal—to replace algebraic notation with a programming language—would certainly
constitute a significant change in the nature of physics and physics instruction. If programming
becomes a major part of the physics curriculum, then students will be engaged in very different
activities: They will be programming instead of solving traditional physics textbook problems.
And the knowledge that students take away from this new curriculum will be substantially
different, at least in some superficial respects. Students will learn programming and programming
algorithms, rather than some derivational and problem solving strategies that are particular to
textbook physics problems.
In addition, we must also keep in mind the more profound possibilities raised by the likes of
Goody and Feynman. Goody might tell us that we must expect to alter the nature of physics
knowledge in a more fundamental manner. And Feynman might worry that an approach to
9
physics that passes through programming may deprive students of an appreciation of the “beauty
of nature.”
These observations are more than interesting asides; they have serious implications for how the
comparative study at the heart of this inquiry must be understood. If external representations are
inextricably tied to the nature of physics knowledge, then it will not be reasonable to think of
programming as a new representation for doing the same old jobs. In Goody’s terms,
representations do not just fill some need, they also shape and determine the nature of the
knowledge. For this reason, it will not be precisely correct to ask whether programming or algebra
does certain jobs better. I cannot ask which of the systems better “represents” physics knowledge. I
cannot even really ask which of these representational systems will lead to better student learning
of physics. In my view, there is no physics that exists separate from these representational systems
and the associated tasks, that can be represented more accurately or taught better.
As an alternative, I thus propose to ask and answer the following questions: How does
“programming-physics” differ from “algebra-physics?” And: Is “programming-physics” a
respectable form of physics? These questions will provide the focus and anchor for this inquiry.
Two practices for introductory physics instruction
There is still another complication to be considered. I cannot assume that these symbol
systems, somehow defined prior to their application to physics, uniquely determine the associated
practices. In other words, for example, given any programming language, there are many possible
practices of programming-physics that one could build around that programming language. For
this study, I must therefore pick out a particular algebra-physics and a particular programmingphysics for study.
Specifying a practice of algebra-physics for study does not pose a significant difficulty. Since I
am interested in programming as an alternative for physics instruction, I will examine algebraphysics as it is practiced by students in traditional introductory physics courses. Because of the
great uniformity that exists across such traditional courses (which still make up most physics
instruction), this constitutes a relatively well-defined and uniform practice. Perhaps the most
prominent aspect of this practice is an emphasis on solving a certain class of problems. A problem
and solution, taken from the textbook by Tipler (1976), is shown in Figure Chapter 1. -1. This
problem is wholly typical of those that students solve during the first weeks of an introductory
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course. Note that it employs a few equations (notably v = vo + at and ∆x = vot + 12 at 2 ) that
students are expected to have learned during this period.
A ball is thrown upward with an initial velocity of 30 m/s. If its acceleration is 10 m/s2
downward, how long does it take to reach its highest point, and what is the distance to the
highest point?
v = vo + at
0 = 30 m s + −10 m s 2 t
(
t=
)
30 m s
= 3.0s
10 m s 2
(
)
∆x = vot + 12 at 2 = (30 m s)(3.0s ) + 12 −10 m s 2 (3.0s )2
= +90m − 45m = 45m
Figure Chapter 1. -1. A typical problem and solution (from Tipler, 1976).
For the case of programming, there is less to build on, and there is certainly no uniformly
accepted method of employing programming languages in physics instruction. In this work, I will
draw on a practice of programming-physics that was developed by the Boxer Research Group at
U.C. Berkeley, and was implemented in 6th grade and high school classrooms (diSessa, 1989;
Sherin, diSessa, and Hammer, 1993). For the purposes of this study, this practice was transported
into a laboratory setting, and it was employed with university students who had already
completed two semesters of introductory physics. The resulting complications will be discussed
later.
Unlike the problem-based approach typical of traditional introductory physics courses, the
courses developed by the Boxer Research Group were all simulation-based; rather than solving
standard physics textbook problems, the students spent their time programming simulations of
motion. Figure Chapter 1. -2 shows an example program, which simulates the motion of a
dropped ball. This simulation is written in the Boxer programming environment (diSessa,
Abelson, and Ploger, 1991). At first glance, the external representations that appear in this figure
probably seem very different than the equations in Figure Chapter 1. -1—and they are!
Nonetheless, this research will uncover a surprising degree and kind of similarity. The Boxer
programming language and the practice of programming-physics employed in this study will be
described in some detail in Chapter 8.
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Instructions
Make a realistic simulation of the motion of a ball
that is dropped from someone's hand.
Data
drop
setup
change
change
change
repeat
pos 0
vel 0
acc 1
23 tick
pos
vel
acc
253
23
1
Data
Data
Data
tick
Doit
change pos
pos + vel
Doit
change vel
vel + acc
Doit
fd vel
dot
Doit
Data
Data
Figure Chapter 1. -2. Simulation of a dropped ball.
Finally, I want to note that, just looking at Figure Chapter 1. -1 and Figure Chapter 1. -2, it is
evident that each of these practices involves more than a single external representation—they
involve more than just algebraic notation and programming statements. For example, both of the
above figures have textual elements, and the programming figure shows a representation of the
ball’s motion involving dots. Furthermore, even more external representations are involved in these
practices than appear in these figures. For instance, algebra-physics uses graphs and various types
of diagrams. Nonetheless, I have chosen to name these two practices after algebra and
programming because my attention will be focused tightly around these particular symbol
systems. In addition, I believe that these symbol systems play a critical role in constraining the
nature of their respective practices.
The advantageous properties of this study
Why is this particular comparative study the right place to explore the relation between
external representations and understanding? First, it is apparent that symbol use plays a substantial
and important role in the discipline of physics, both in instruction and in the work of practicing
experts. The process of learning physics traditionally involves a great deal of writing equations and
12
manipulating them to solve problems, and equations are the target form for physicists’ laws.
Because symbol use plays such an important role in physics, this study is pretty much guaranteed
to encounter some of the more interesting interactions between knowledge and external
representations. Second, although algebraic notation plays a central role in physics, it is not as
integrated in everyday activity as natural language; thus, its role should be somewhat easier to
fathom. The point is that describing the relationship between literacy and thought (Goody &
Watt, 1968), or thought and language (Vygotsky, 1934/1986) is probably a much harder
problem than describing the relationship between equations and physics understanding. We can
thus think of this project as a first and easier step toward these other, more ambitious
undertakings.
Third, the use of a comparative study has some benefits. I will be able to do more than
speculate about how physics built around an alternative symbol system might differ; I can actually
uncover some specific differences. It is also important that the two practices we are comparing are
not altogether different. If the two practices were too divergent, what type of comparisons might
we make? How, for example, would one compare physics and chemistry?
Finally, this study has interest beyond the more theoretical issues I have described. One of the
aims of this project is to explore the feasibility of a novel instructional approach for physics.
Preliminary research has found that the use of a programming language in physics instruction is a
promising alternative and that programming languages may even have certain advantages (as well
as certain disadvantages) over algebraic notation. The study described here will contribute to the
investigation of this novel instructional approach.
Overview of the Approach
Taken down to a single driving notion, this study can be seen as an exploration of the
possibility of replacing algebraic notation with a programming language in physics instruction.
But, when the question is viewed with a little care, such a study blows up to a much larger scale,
and it is evident that a number of subsidiary questions must be answered. First, we cannot possibly
understand the implications of replacing algebra with programming if we do not have some
notion of the role played by algebraic notation in traditional physics learning. So, already this
study has doubled in size; before even starting to look at the use of programming in physics, we
have to go back and understand certain aspects of traditional physics learning using algebraic
notation.
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And we have seen that it is probably not going to be good enough to think simply in terms of
replacing algebraic notation with a programming language. Such a switch may have far-reaching
effects on the nature of physics knowledge and understanding. So, we really cannot do anything
like a simple comparison, there is some tricky business about the relationship between knowledge
and external representations. This is a problem to overcome, but it is also a positive feature of this
inquiry: One of the outcomes of this study will be some insight into this complex relationship.
In order to compare algebra-physics and programming-physics I proceeded as follows. I began
by collecting a data corpus consisting of two parts, one which contains observations of students
engaged in the practice of algebra-physics and another in which students were engaged in
programming-physics. The question then becomes: How can we go about comparing these two
sets of observations?
The key to producing a useful comparison of algebra-physics and programming-physics based
on these observations was the construction of a model of some aspects of symbol use in physics.
This model, which I will preview below in a moment, is a model in the spirit of cognitive science; I
attempted to account for a certain class of human behavior (symbol use in physics) by
hypothesizing that people possess and employ certain elements of knowledge.
This model is the central result of this research and the key to getting at the goals of this
inquiry. Viewed at a certain level, this model will be general enough to describe both varieties of
symbol use under consideration in this study. Thus, it can provide the categories for
understanding and comparing the observations of algebra-physics and programming-physics in
the data corpus. In addition, once we have the model, we can look at it with more general issues in
mind.
In the remainder of this section I will briefly describe the data corpus and the model of
symbol use on which my analysis is based.
The data corpus, in brief
All of the subjects in this study were UC Berkeley students enrolled in “Physics 7C.” Physics
7C is a third semester introductory course intended primarily for engineering majors of all
persuasions. The fact that these students were in Physics 7C implies that they had already
completed two semesters of instruction, Physics 7A and 7B. Many also had some instruction
during high school.
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The subjects were divided into two distinct pools, an “algebra” pool and a “programming”
pool, with each pool consisting of 5 pairs of students. All experimental sessions were conducted in
a laboratory setting and these sessions were videotaped. Students in the algebra pool worked with
their partner at a blackboard to solve a pre-specified set of problems. Most of these problems were
fairly traditional textbook problems, but a few more unusual tasks were also included. The pairs
typically solved the problems in 5 1/2 hours spread over 5 sessions. The result was a total of 27
hours of videotape of these students using algebraic notation to solve physics problems. A subset
of this data, 10 hours, was selected for more focused analysis.
Programming pool subjects worked in pairs at computers. The students were first given about
four hours of training, usually requiring 2 two-hour sessions. Following the training, the students
were asked to program a set of simulations of physical motions. They typically required 4
additional two-hour sessions to complete these tasks. In sum, the result was a total of 53 hours of
videotape of students programming simulations. A subset of this data, 16 hours, was selected for
more focused analysis
All of the videotaped sessions selected for focused analysis were carefully transcribed and the
resulting transcripts were analyzed. The examples given in upcoming chapters are all taken from
these transcripts, and the model described is drawn from this analysis. However, I will not
describe the data corpus or my analysis techniques in more detail until Chapter 6. That chapter
will also include an account of the rationale for the above design.
Overview of the model: The Theory of Forms and Devices, in brief
As I mentioned above, the model will be a theory in the spirit of cognitive science, which
means it will posit knowledge elements to explain certain behaviors. The scope of the model will
be restricted in a number of senses. Most importantly, I will attempt to describe only a small
subset of the knowledge necessary to provide a complete account of symbol use in physics. This
subset is particularly important, I will argue, in the construction of novel symbolic expressions and
the interpretation of expressions, both in algebra-physics and programming-physics. Furthermore,
I will argue that this small subset of knowledge is precisely where our attention must be focused in
order to get at the relation between physics knowledge and external representations.
The model of symbol use will be built around two major theoretical constructs, what I call
“symbolic forms” and “representational devices.” I will now briefly discuss each of these
constructs. (A more complete discussion can be found in Chapters 3 and 4.)
15
On Symbolic Forms
An important contention of this work is that initiates in algebra-physics are capable of
recognizing a particular type of structure in physics expressions. More specifically, they are able to
see expressions in terms of a set of templates that I refer to as “symbol patterns,” each of which is
associated with a simple conceptual schema. The knowledge element involving this combination
of symbol pattern and schema I refer to as a “symbolic form” or just as a “form,” for short.
In a sense, symbolic forms constitute a conceptual vocabulary in terms of which physics
equations are written and understood. For example, one of the most common forms is what I call
“BALANCING .” In the conceptual schema associated with BALANCING , a situation is schematized as
involving two influences, such as two forces, in balance. Furthermore, the symbol pattern
associated with BALANCING involves two expressions separated by an equal sign:
=
BALANCING
For illustration, imagine an example in which a student is solving a problem involving a block
that has two opposing forces acting on it. Because the block is not moving, the student writes the
expression F1 = F2. Now, the question is, how precisely did the student know to write this
equation? The answer I will give to this question is that the BALANCING form has been engaged and,
when it is engaged, it pins down what the student needs to write at a certain level. In this case, it
specifies that the student needs to write two expressions separated by an equal sign, with each side
of the equation associated with one of the two influences in balance. Furthermore, this process can
also work in reverse; a student can look at an equation, like F1 = F2, and see it in terms of the
BALANCING
form.
Symbolic forms, in part, develop out of symbolic experience and then become a way of
experiencing the world of symbols. Because of the developed ability to be sensitive to the patterns
associated with forms, the symbolic world of experience is more structured and meaningful for
the physics initiate than one might first expect.
On Representational Devices
In general, a number of different symbolic forms may be recognized in any particular physics
expressions. The question then arises: What determines, at any particular time, the forms that a
student recognizes in an expression? The answer, I will attempt to show, is that initiates in physics
possess a repertoire of interpretive strategies that I call “representational devices.” Roughly,
16
representational devices set a stance within which symbolic forms are engaged. The term “device”
here is intended to call to mind “literary devices,” for reasons that I hope to make apparent in
Chapter 4.
I will argue that there are three major classes of representational devices. In one of these classes,
which I call Narrative devices, an equation is embedded in an imaginary process in which some
change occurs. For example, suppose that a student has derived the expression a=F/m, which gives
the acceleration of an object in terms of the mass and the force applied. The student can then
imagine a process, for example, in which the mass increases, and infer from the equation that the
acceleration must decrease. This would be an interpretation involving a Narrative device.
The other two classes of representational devices are what I call Static and Special Case
devices. In Static devices, an equation is seen as describing a moment in a motion. For example, an
equation may be seen as true only at the apex of a projectile’s trajectory. In Special Case devices, a
stance is set by considering some restricted set of cases to which the equation applies. For
example, a student may consider the behavior of an equation in an extreme or limiting case.
In sum, the Theory of Forms and Devices describes a collection of knowledge that allows
initiates in physics to see a certain type of meaningful structure in physics expressions. Two
specific types of knowledge are involved in this collection: symbolic forms, which correspond to a
set of meaningful structures that students can recognize in expressions, and representational
devices, which constitute a repertoire of interpretive stances. Primarily, it is by looking at the
forms and devices associated with algebra-physics and programming-physics that I will compare
these two practices.
In the remainder of this chapter, I will now work toward situating this work in relation to
broader issues and in relation to existing research. I will begin, in the next section, by attempting
to provide a definition of “external representation.” Then I will present a more in depth discussion
of a central issue that I have so far only touched upon: the relation between knowledge and
external representations. Finally, I will discuss how this work relates to existing research
concerning physics learning and instruction.
A Definition of ÒExternal RepresentationÓ
Much of what I say in this document will be specific to the external representational practices
described in the previous section. But I will also want to argue for some generality of my
17
observations, especially when these observations are framed in broader terms. However, in order to
make these claims of generality there is some additional work that must be done. If I want to say
that my observations apply more broadly, then I need to say something about the scope of
application. In particular, it would be helpful to have a definition of “external representation.”
So, what is an “external representation.” Let’s take a moment, first, to think about the type of
definition that is required here. My project is to build a model of some knowledge associated with
symbol use in physics. In essence, I am going to be providing a kind of description of a class of
human behaviors. Thus, the definition we need is a definition that characterizes this class of
behavior, that specifies it in contrast to other behaviors. Finding such a definition may very well be
difficult, since human activity is complicated and messy. In fact, there is no reason to believe that
there is a simple definition that lines up with our intuitions concerning what constitutes a
representation and still specifies a relatively circumscribable class of human activity.
An alternative approach, taken by a number of other researchers, is to define external
representation without explicit reference to behavior or cognition. They begin, instead, by
defining a “representational system.” For example, in a typical version of such a definition, a
representational system is defined in terms of correspondences between a “representing” and a
“represented world.” This type of definition of representation hooks up with our intuitive notion
of representation quite clearly—a representation is something that “stands for” something else.
Such correspondence definitions have their uses, and I will follow this line for a while in Chapter 7.
However, I believe that this is not the right type of definition for the present project. Since I am
building a model of symbol use, I need a definition that slices off a class of human activity.
Let’s take a stab at this type of definition. I will define an “external representational practice”
as a class of human activity that involves:
1.
A set of externally realizable marks. These marks will be organized, in some manner, either
in space or through time. They may be arrangements of objects, marks on paper, sounds, or
even smells.
2.
Conventions concerning the making of marks. There will be conventions—socially dictated
regularities in human behavior that are shared by many individuals—pertaining to how the
marks are organized in space and time, and how the making of marks is integrated with other
aspects of activity.
18
3.
Conventions concerning action associated with existing arrangements of marks. People who
have learned to participate in the practice will be sensitive to distinctions in the arrangement of
marks. This means that their action in the presence of the external representation—what they
say and do because of and with a particular collection of marks—will depend sensitively on
conventionally defined distinctions in the arrangement of the marks.
To see that this definition can actually do some work, let’s look at what it rules out:
•
A random pile of stones out in the middle of nowhere. This is not an instance of human
activity so it’s not even covered by the definition.
•
Some children make a haphazardly arranged pile of stones. This violates requirements (2) and
(3) of the definition. To the extent that there are conventions, they are not widely shared by
many individuals.
•
A table is set prior to a meal. In this case, requirement (2) is met since there are conventions
concerning the arrangement of the objects. However, requirement (3) is not met; fine
distinctions in the arrangement of objects are not important for future action.
•
The entrails of a goat are used to predict the future. Here, there is an arrangement of physical
marks that is interpreted, so requirement (3) is met. But the act of arranging the marks is not,
presumably, a normal component of the practice, so requirement (2) is not met and this is not
a representational practice.
Note that this is not really a technical definition and I am relying on the reader’s intuitions, to
a great extent, to apply it. Furthermore, the agreement of specific instances with the requirements
of the definition may be a matter of degree rather than clearly establishable fact. For example, a
sophisticated diner may know that the outermost utensils in a place setting are always supposed to
be used first. Thus, the arrangement of objects is significant for human action and dining becomes
a representational practice. My first response to this is that maybe this is not such a ridiculous
conclusion. To the extent that the arrangement of silverware communicates information from the
table setter to the diner, perhaps it is reasonable to think of this as a very simple representational
practice.
My second response is to again appeal to the type of definition that I am trying to create.
Since I am attempting to slice off a class of human behavior, there are certain to be some fuzzy
19
edges. The best I can really shoot for is a rough characterization of this class, something that gives
the reader a sense for the scope of the conclusions to be drawn here. The hope is that the
intuitively central cases (equations, graphs, writing) are clearly included by the definition, and that
intuitively excludable cases (the haphazard arrangement of stones) are excluded.
Thus, this definition does not specify properties of external representations that make them
absolutely unique among artifacts; most notably, I have not tried to argue that external
representations have the special property that they refer to or stand for ideas or other elements of
the environment. Instead, I have characterized a representational practice as involving the assigning
of significance to fine distinctions in the arrangements of elements of the environment.
Furthermore, the conventions for assigning this significance are shared among individuals. On
these dimensions, the difference between a representational practice and other activity is a matter
of degree; the most typical instances of representational practices involve a sensitivity to many and
numerous fine distinctions, with the associated conventions shared by many individuals.
Relations Between Knowledge and External Representations
I am working from the premise that replacing algebraic notation with programming must
change the nature of physics knowledge and understanding. But, to this point, I have only
proffered some vague notions and rough suggestions as to the manner in which knowledge and
external representations are related. I have Goody’s notion that lists shape knowledge and
Feynman’s intuition that an appreciation of physics is tied up with its mathematical expression.
But, more precisely, what can it mean to say that knowledge is inextricably representational? In
what ways might physics knowledge and external representations be related?
Here I will take some preliminary steps toward answering these questions. To do this, I will lay
out a very basic framework on which we can hang some of our intuitions concerning relations
between knowledge and external representations. However, I will not push too hard on these
intuitions and I certainly do not intend this discussion to be exhaustive in its stating of relations
between knowledge and external representation. My purpose is only to make a basic accounting of
some of these relations within a simple framework.
Direct versus indirect effects of external representations
In order to introduce the framework on which the rest of the discussion in this section
depends, I will begin with an important distinction between “direct” and “indirect” influences of
20
external representations on the knowledge of individuals. This distinction is illustrated nicely by
some competing viewpoints concerning the effects of literacy on thought. To begin, I turn to
Goody and Watt’s seminal article The Consequences of Literacy (Goody & Watt, 1968). In this
article, Goody and Watt’s agenda was to react against what they perceived to be the over-extreme
cultural relativism of the time. To respond to this relativism, they proposed that a number of
significant differences existed between “oral” and “literate” societies. Most importantly for our
concerns here, they presented a number of strong hypotheses concerning the effects of literacy on
human thought.
The arguments of Goody and Watt rest largely on the analysis of historical cases. For
example, they claim that the development of the specific type of alphabet used by the Ancient
Greeks was responsible for—or, at least, allowed—many of the important intellectual
achievements of the time. Included is the claim that the development of this type of literacy
allowed the development of a new sort of logic and even “scientific” thought.
Two researchers, Sylvia Scribner and Michael Cole, proposed to directly test these claims. In
The Psychology of Literacy, Scribner and Cole describe an anthropological research project
involving the Vai people of Northwestern Liberia (Scribner & Cole, 1981). For the purposes of
studying the consequences of literacy for individuals, the Vai provided an opportunity for a
potentially illuminating case study. The interesting fact about the Vai culture is that there are
three widely practiced forms of literacy, as well as widespread non-literacy. First, the Vai employ
an independently invented phonetic writing system—Vai script—that is not used elsewhere. No
formal institutional settings exist for the propagation of this script; rather, it is transmitted
informally among the Vai. In addition, there are Qur’anic schools where the Arabic script is
learned, and English writing is taught in the context of formal western-style schooling.
Scribner and Cole’s framing idea was that the Vai culture provided a natural laboratory in
which the effects of literacy could be studied separately from the effects of schooling. Their
method involved the administering of various psychological tests to individuals with differing
literacies and with differing schooling backgrounds. Through such a study, Scribner and Cole
reasoned, they could isolate the consequences of literacy for the mental abilities of individuals.
The results of this research, at least with regard to encompassing cognitive consequences, were
generally negative.
There is nothing in our findings that would lead us to speak of cognitive consequences of literacy with
the notion in mind that such consequences affect intellectual performance in all tasks to which the
human mind is put. Nothing in our data would support the statement quoted earlier that reading and
21
writing entail fundamental “cognitive restructurings” that control intellectual performance in all
domains (Scribner & Cole, 1988, p. 70).
Literacy, it seemed, did not lead Vai individuals to perform better on measures of general
cognitive ability, such as tests of logic or categorization tasks.
However, Scribner and Cole did find differences in more specific “skills” associated with each
of the individual forms of literacy. For example, it turns out that letter writing is one of the most
common uses of the Vai script. For this reason, Scribner and Cole hypothesized that Vai literacy
would require and foster the development of some specialized skills not required by oral
communication.
In writing, meaning is carried entirely by the text. An effective written communication requires
sensitivity to the informational needs of the reader and skill in use of elaborative linguistic techniques.
(Scribner & Cole, 1988, p. 66)
In fact, Scribner and Cole did find a correlation between Vai script literacy and communication
skills. Similarly, because the use of the Arabic script was associated with memorization tasks in
Qur’anic learning, they predicted that this Arabic literacy would be associated with increased
memory skills. This hypothesis was born out, though only in the types of memory tasks that were
similar to those found in the Qur’anic schools.
So Scribner and Cole’s results seem to indicate individual differences due to literacy at the
level of specific skills, but not at the level of more general cognitive capacities. Goody’s response to
these results was that, given the nature of the study performed, these results are not at all
surprising:
That is to say, we did not expect the ‘mastery of writing’ (of whatever form) to produce in itself an
immediate change in the intellectual operations of individuals … But if we are referring to an
operation like syllogistic reasoning, the expectation that ‘mastery of writing’ in itself would lead
directly to its adoption is patently absurd. The syllogism, as we know it, was a particular invention of a
particular place and time; there were forerunners of a sort in Mesopotamia just as there were
forerunners of Pythagoras’ Theorem; nor on a more general level is the process of deductive inference
unknown in oral societies. But we are talking about a particular kind of puzzle, ‘logic’, theorem, that
involves a graphic lay-out. In this sense the syllogism is consequent upon or implied in writing.
However its use as distinct from its invention does not demand a mastery of writing. Once invented, it
can be fed back into the activities of individual illiterates or even non-literates just as the same
individuals can be taught to operate the arithmetic table or, as Scribner and Cole point out, to decode
Rebus writing. (Goody, 1987, p. 221)
Goody (1987) believes that psychological tests that compare individuals in a society should not be
expected to uncover the more general effects predicted by Goody and Watt (1968). According to
Goody, we should expect the stronger consequences of literacy to be manifested only at the level
22
of societies, not at the level of individuals. Thus, we should not expect to see any strong
differences between Vai individuals due to differences in literacy.
To clarify this point, Goody makes a distinction between what he calls “mediated” and
“unmediated” effects of literacy (Goody, 1987). The idea here is that the consequences of literacy
for the thought processes of individuals need not arise through the direct use of writing by
individuals. Instead, the consequences of literacy may play themselves out at a society-wide level
and then be “fed back” into the thought of individuals. An individual need not even be literate to
be affected. Given this viewpoint, Goody and Watt’s original hypotheses need not be manifested
as differences between individuals in the Vai society.
There are many other extant criticisms of the hypotheses of Goody and Watt, such as the
claim that their arguments tend toward an inappropriate “technological determinism” (Street,
1984). But my purpose here is not to engage in a thorough analysis of particular hypotheses for the
consequences of literacy. Instead, my goal has been to use this debate as a context for illustrating
that hypotheses concerning the effects of external representations on individual knowledge may be
of two sorts: We can posit direct effects, which arise due to an individual’s own participation in a
representational practice, and indirect effects, which do not necessarily arise from actual use of the
representation. The main purpose of the simple framework shown in Figure Chapter 1. -4 is to
highlight this distinction.
Representational
Practice
Direct
Effects
Literate
Individual
External
Representation
"Products"
Indirect
Effects
Cultural
Mediation
Literate or
Non-Literate
Individual
Figure Chapter 1. -4. An individual’s knowledge can change through interaction with an external
representation. In addition, the products of this interactions may be “fed back” into the knowledge of literate
as well as non-literate individuals.
23
I will take a moment to walk the reader through Figure Chapter 1. -4. To begin, we have a
“literate” individual, defined as such by their ability to participate in a representational practice.
Following the structure provided by this practice, this individual interacts with an external
representation. Participation in this practice may have a direct (unmediated) affect on the
knowledge or thought processes of the literate individual. In addition, there may be some
outcome of this interaction, a “product” whose nature is shaped by the external representation and
the associated practice. This product may then be processed by society and ultimately fed back
into the knowledge of individual literates or non-literates. Also note that the products of
representational activity may feed back into a reshaping of the representational practice itself.
This simple framework will guide us in the discussion that follows. In the remainder of this
section I will go into slightly more detail concerning some particular notions regarding individual
knowledge and external representations. Where appropriate, I will relate these notions to the
framework presented in Figure Chapter 1. -4.
A Òmore powerful systemÓ
One stance that we can take is that the literate individual and external representation,
interacting as in Figure Chapter 1. -4, constitute a system whose properties we can study. The
point here, which has been made by many researchers, is that this system may in some ways be
“more powerful” than the individual acting alone. Before saying how this stance reflects on the
relationship between external representations and knowledge, I want to take a moment to explore
this view.
The “more powerful system” notion has many roots, one of which is in the work of Jerome
Bruner (Bruner, 1966). Bruner argues that cultures provide individuals with a number of “cultural
amplifiers,” artifacts and techniques that amplify the capacities of individuals, both physical and
mental. More recently Norman (1991), following Cole and Griffin (1980), argued that the
amplifier metaphor is somewhat misleading. Norman does allow that a person-artifact system can
indeed have enhanced capabilities when compared with an individual acting alone—in fact, this
observation is central for him—but he does not believe that this enhancement is accurately
described as a process of “amplification.” Instead, he argues that artifacts that enhance human
cognitive capabilities do so by changing the nature of the task that an individual must perform,
not by “amplifying” individual action.
24
One common way in which the system view is applied is to describe a person-artifact system
as having an increased memory capacity or increased knowledge base to draw on. Consider the
system consisting of a literate person together with a pencil and a sheet of paper. In a sense, this
system may be said to possess a “better memory” than the person acting alone.
In fact, at least one author, Merlin Donald, takes the system view, together with an emphasis
on memory-enhancing properties of external representations, as a central way to understand the
functioning of the modern human mind. In The Origins of the Modern Mind, Donald attempts to
lay out, in broad sweep, the evolution of the human mind (Donald, 1991). His story posits three
stages in this evolution. The first stage, associated with Homo Erectus, is the development of
“mimetic culture.” According to Donald, Homo Erectus was distinguished from apes in its ability
to “mime, or re-enact, events.” During the second stage, associated with the appearance of Homo
Sapiens, speech developed. In Donald’s scheme, both of these stages involved true biological
evolution in the sense that there were concomitant changes in human biology.
The third stage in Donald’s evolution of the mind is distinguished from these first two in that,
rather than involving any biological changes, it involved the development of what Donald calls
“external symbolic storage.” Donald’s external symbolic storage is precisely what we have been
calling external representations. For Donald, the “storage” feature gets the lion’s share of his
emphasis:
External symbolic storage must be regarded as a hardware change in human cognitive structure, albeit
a nonbiological hardware change. Its consequence for the cognitive architecture of humans was similar
to the consequence of providing the CPU of a computer with an external storage device, or more
accurately, with a link to a network (p. 17).
To be fair, Donald allows a somewhat more complex role for external representations than a
simple memory-aid story. Because his external symbolic storage supplements short-term as well as
long-term memory, its presence can alter the nature of short-term thinking processes. To Donald,
it is as if the very “architecture” of the mind has been augmented; he treats external
representations as comparable to an additional supply of memory that just happens to be
connected through the visual channel.
So what does the “more powerful system” view tell us about the relation between external
representations and the knowledge of individuals? The answer is that it does not tell us much. For
the simple reason that the focus of a system analysis is on the system and not on an individual, it
does not require many presumptions about how the thought processes of individuals must change.
However, the fact that the system view allows external representations to profoundly enhance
25
cognition, without requiring changes in the nature of thought processes, does have important
implications for this discussion. Because it can predict strong effects without positing any major,
or even permanent, changes in the thought processes of individuals, the system view frees us to
take the stance that there are no such important changes.
Incidentally, symbol use in physics is often understood through the lens of the “more powerful
system” view. The idea is that when equations are written on a sheet of paper, they can be
manipulated following a relatively simple set of rules. Thus computations and derivations, which
might be difficult or impossible otherwise, are made easy or, at least, possible. Again, the point is
that this does not necessitate any strong influence on the abilities of individuals in the absence of
external symbols. The person-representation system is capable of performing remarkable
computations, but these remarkable computations still only require the person to perform a
relatively simple task, the rule-based manipulation of symbols.
The Òresidual effectsÓ of symbol use
Restricting ourselves to the system view really begs the question of whether there are any
residual effects on individuals due to the use of external representations. So now let us consider
this question explicitly. Suppose a person is in interaction with an external representation as in
Figure Chapter 1. -4. What types of residual effects might there be due to the direct interaction
of person and representation? Here I consider a few possibilities.
Knowledge is adapted by and for symbol use
Recall that, as I explained above, Norman argued that a person-artifact system gets its
enhanced abilities by allowing the person to perform an alternative, easier task. The important
observation here is this: Although the artifact allows a person to perform a task that is somehow
easier, the individual may still need to learn or otherwise develop new capabilities in order to
perform this alternative task. Thus we see an opening for the possibility of residual effects. The
knowledge and capabilities of an individual may need to be changed and reorganized for the
specific tasks required by representational practices. In fact, the positive results found by Scribner
and Cole can be understood in this way. The representational practices associated with Vai script,
such as letter-writing, could require individuals to develop certain skills.
It is interesting to note that these required changes in what an individual must learn are not
always seen in a positive light. To illustrate, I begin with an analogy. Imagine that a person spends
a significant amount of time using some sort of physical support, such as a crutch. It is possible
26
that certain portions of this person’s physique will gradually be modified. Some muscles may
atrophy. Or, for example, if the person is using a wheelchair, the muscles in their arms might grow
stronger while the muscles in their legs grow weaker. In an analogous manner, Plato worried, in
the Phaedrus, that the use of writing would dampen a scholar’s ability to memorize information
(Plato, trans. 1871).
Similar observations can be made for the case of symbol use in physics. If students can learn to
remember and use certain equations, they may never need to learn the conceptual information
that is somehow embodied in these equations. They can know the equations and not their
meanings. In this way, the use of external representations could have a direct effect on the nature
of an individual’s understanding of physics, literally what an individual “knows.” A student may,
in a sense, just know enough to fill in the gaps between the equations and to use equations
effectively.
The point I am trying to make here is, I believe, relatively uncontroversial. If we spend time
engaged in a particular variety of symbol use, then our knowledge and capabilities will be adjusted
for the requirements of that variety of symbol use. We will learn what we need, and not learn what
we do not need. In this way, the knowledge of an individual may be adapted by and for symbol
use. Of course, in any particular case, we can argue about the type and degree of adaptations, but
the general point still holds.
A symbolic world of experience
Now I want to propose a second way to think about possible residual effects on individual
symbol users due to direct interaction. This viewpoint is not logically distinct from that presented
just above, but I believe that announcing it will help us to recognize and understand some
additional possible relations between knowledge and external representations. Here, I connect to
Feynman’s intuition that one must have a certain type of mathematical-symbolic experience in
order to appreciate the beauty of nature.
I summarize this viewpoint as follows. Start with the (admittedly vague) presumption that
individual knowledge derives from experience. For example, much of our physics-relevant
knowledge comes from our experiences living in the physical world. Before anybody takes their
first physics course, they know what a physical object is, just because they are a person living in the
world. They have held objects, kicked them, painted them, etc. Now here is the key move
associated with this viewpoint: We can simply take the experience of manipulating symbols to be a
27
new class of experience. Working with equations—reading them, writing them, manipulating
them, talking to other people about them—is a kind of experience that need not be any less
meaningful than our experience in the physical world. And this experience forms part of the basis
for our understanding of physics.
Instrumental Psychology and internalization
In speaking of a “symbolic world of experience,” I make it clear that the character of human
experience is shaped by humans. In part, our experience consists of a world of our own devising.
One theory of psychology, the “Instrumental Psychology” of Lev Vygotsky, puts this idea at its
center. For Vygotsky, it is essential that individuals are subject to stimuli not only from the
“natural environment,” they also actively modify their stimuli. Furthermore, the tools for
modifying and mastering the environment are inventions that have been perfected over the course
of human history, and which are somewhat specific to the culture in which the individual is
embedded.
Strikingly, for Vygotsky, not only is it the case that the knowledge of individuals is somehow
affected by stimuli of human devising, these stimuli happen to be the origins of the higher,
uniquely human forms of mental activity. External, social activity is internalized and gives human
thought its particular character. In addition, spoken language—a specific external
representation—is assigned a special role in this process. For Vygotsky, it is language that gives
order to the initially undifferentiated stream of infant thought.
In mastering external speech, the child starts from one word, then connects two or three words; a little
later, he advances from simple sentences to more complicated ones, and finally to coherent speech made
up of a series of such sentences; in other words, he proceeds from a part to the whole. In regard to
meaning, on the other hand, the first word of the child is a whole sentence. Semantically, the child
starts from the whole, from a meaningful complex, and only later begins to master the separate
semantic units, the meanings of words, and to divide his formerly undifferentiated thought into those
units. The external and the semantic aspects of speech develop in opposite directions—one from the
particular to the whole, from word to sentences, and the other from the whole to the particular, from
sentence to word. (Vygotsky, 1934/86, p. 219) [My emphasis.]
Vygotsky takes us close to what is perhaps the strongest form of “direct effect” one can
imagine, the view that thought is, in some sense, an internalized version of external symbolic
activity. For example, someone who adopts this most extreme of views might assert that human
thought is internalized speech. While Vygotsky does not quite propose this extreme view, he does
believe that the use of external signs is primarily responsible for the shaping of the highest levels of
individual thought.
28
Indirect effects of symbol use
In the preceding sub-section I discussed residual effects on the knowledge of individuals due
to direct interactions with external representations. Although they will not be of primary concern
in the rest of this work, I want to briefly consider some hypotheses concerning indirect effects. The
general form of these hypotheses is that the use of certain external representations shapes a
society’s knowledge products in a certain way, and these knowledge products are then internalized
by individuals. Again, these are indirect in the sense that the knowledge of an individual may be
affected even if they are not literate with the external representation in question.
One example of such an indirect effect I call “epistemic structuring.” To illustrate I turn once
again to Jack Goody. In The Domestication of the Savage Mind, Goody extends his discussion of
lists from purely administrative lists, such as the one with which I began this chapter, to lists
designed for a more academic purpose. In particular, he describes the “Onomasticon of
Amenope,” which was composed by an Egyptian around the end of the Second Millennium BCE.
Apparently, this document was the result of an attempt to make a truly exhaustive list of “all
things that exist.” It bears the title:
Beginning of the teaching for clearing the mind, for instruction of the ignorant and for learning all
things that exist: what Ptah created, what Thoth copied down, heaven with its affairs, earth and what is
in it, what the mountains belch forth, what is watered by the flood, all things upon which Re has shone
all that is grown on the back of the earth, excogitated by the scribe of the sacred books in the House of
Life, Amenope, son of Amenope. (p. 100)
For Goody, the fact that this document takes the form of a list is essential; the
representational form has the effect of shaping the ideas that are represented:
We can see here the dialectical effect of writing upon classification. On the one hand it sharpens the
outlines of the categories; one has to make a decision as to whether rain or dew is of the heavens or of
the earth. At the same time, it leads to questions about the nature of the classes through the very fact of
placing them together. How is the binary split into male and female related to the tripartite division
between man, stripling, and old man? (p. 102)
Collins and Ferguson (1993) would say that the list is functioning as an “epistemic form,” a
“generative framework” for knowledge. Goody’s point is that the mere fact of listing leads to
reflection and progress in elaborating the categories used for describing the world.
The idea here is that the “product” of the person interacting with the list is shaped by the
nature of the list structure. Then, once this “product” is produced, it can be fed back into the
knowledge of individuals, both literate and non-literate. Furthermore, after it is processed by
29
society, the product need not even be in the form of a list; distinctions can be taught to nonliterate as well as literate individuals, without restriction to the list structure.
It is important to note that these proposed indirect effects can vary greatly in how profoundly
they purport to alter the nature of individual cognition. For example, we could imagine taking
one of Amenope’s specific distinctions and teaching it to someone. This would not lead to a very
profound change in the nature of that individual’s thought. However, some of the indirect effects
proposed by researchers suggest more general changes in the nature of thinking. Above I
mentioned Goody and Watt’s hypothesis that the invention of the Greek alphabet allowed the
development of techniques for reasoning logically—techniques that could be learned by any
individual.
Similarly, other authors have argued that the appearance of literacy led to generally important
intellectual achievements. David Olson, for example, makes some of his own very strong
hypotheses concerning consequences associated with the development of writing (Olson, 1994).
According to Olson, an important feature of writing, in contrast to spoken language, is that it
must be interpreted without many of the contextual cues that are available during verbal
discourse. For this reason, it is much harder for writers to convey their intended “illocutionary
force.” Therefore, Olson argues, writing must contain additional clues that convey the author’s
stance to what is said. For example an author must tell the reader whether they mean their
statements to be taken literally or as metaphor; and an author must tell a reader whether they
intend to be presenting an hypothesis or a fact. Olson’s big leap is that, for this reason, the
development of writing had to be accompanied by developments in epistemology; the categories
available for describing knowledge had to be made explicit and refined. It is in this manner, he
argues, that categories like “fact” and “hypothesis” were born. With these new categories available,
the cognitive processes of individuals—even non-literate individuals—would be generally
enhanced.
Summary: Where I will stand
In this section I have briefly discussed some hypotheses concerning deep relations between
external representations and the knowledge of individuals. Centrally, I distinguished the direct
effects that arise from an individual’s use of an external representations with indirect effects, which
can even influence non-literates. In this project, it will be helpful to think of my focus as being on
30
direct, non-mediated effects of external representations. I will be interested in the knowledge and
abilities that must develop specifically for symbol use in physics.
Furthermore, I will lean heavily on the notion that physicists develop an intuition—they get a
feel for physics—through participation in what I have called “the symbolic world of experience.” I
will argue that, to know physics as physicists know it, one must be an initiate into the full world of
physics practice, including learning to write and manipulate symbols, and learning to say things
with and about equations in the manner of a physicist. Note that this view implicitly presumes
that experience with symbols is not any less meaningful or important than other types of
experience, such as daily experience interacting with objects in the physical world. In fact, a major
part of the work of this document will be to illustrate that, for physicists and even for mildly
advanced physics students, an equation is an element in a highly structured and meaningful world.
However, my own orientation stops somewhat short of strong versions of the internalization
view. I do not believe that symbol use is uniquely responsible for the development of human
thought. Nor do I believe that direct interaction with external representations tends to engender
broad changes in the character of the thought of individuals, such as a move toward more
“logical” or “scientific” thought. External representations are a meaningful and important element
of our experience, but activities with external representations are not unique in giving human
thought its character.
Finally, it should be noted that, throughout the discussion in this section, I have essentially
presumed that it is useful to speak of knowledge as something that is localized in the minds of
individuals. Though, to some, this may seem like an obvious and unproblematic move, some
researchers have questioned whether it is appropriate and scientifically productive. Researchers
such as Greeno and Moore (1993) and Hall (1996) take the stance that knowledge is most
profitably understood as a theoretical entity that spans person and elements of the environment.
Note that, since the environment may include external representations, this view provides a quite
difference sense in which knowledge and external representations may be related.
I will not adopt the stance that knowledge must be interactionally defined in this manner;
throughout this document, I will talk as if knowledge is something that can exist localized in
individuals. I will have somewhat more to say on this topic in Chapter 7, but I will make one
additional point here. The notion that knowledge must be interactionally defined is an element of
31
some versions of a more general viewpoint that is known under such names as “situated action.”1
Situated action, as compared to traditional cognitive science, tends to emphasize the influence of
social and historical factors on individual behavior, as well as the importance of the immediate
environment. In allowing that knowledge is, in some sense, dependent on specific practices,
including the use of cultural artifacts such as equations, I hope to be taking into account many of
the concerns that are central to situated action. Again, I will have more to say on this subject later.
Concerning Physics Learning and Instruction
A central goal of this inquiry is to contribute to research in physics learning and instruction.
We want to know what it means to understand physics, and why students do or do not succeed in
learning this topic. Of course, the emphasis here remains on symbol use, how the using of
equations contributes to and is part of physics understanding. In this section, I will describe how
the current project ties into work that has already been done.
Figure Chapter 1. -6. A typical misconception.
Intuitive knowledge and the learning of physics
In the early 1980’s, a number of studies appeared that seemed to indicate that physics
instruction was, stated simply, a disaster. Although students could solve many of the problems
that are typical of introductory instruction, such as the one shown in Figure Chapter 1. -1, they
apparently could not answer some very basic qualitative questions. For example, a typical question
asked students what happens when a ball is shot out of a spiral tube (Figure Chapter 1. -6). In
response, students frequently stated that the ball continues to move in a curved path after leaving
the tube (e.g., McCloskey, 1984). This answer is not correct; once the ball leaves the tube, it
1
See, for example, the special issue of the journal Cognitive Science that is devoted to this topic
(ÒSituated Action,Ó 1993).
32
moves in a straight line, continuing in the direction that it had at the moment it exited the tube.
These alternative answers were attributed to “misconceptions” or “alternative conceptions” by
researchers.2
The presence of misconceptions such as this one is shocking because it suggests that students
lack an understanding of some of the most fundamental aspects of Newtonian physics. For
example, Newton’s first law states that:
Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to
change that state by forces impressed thereon. (Newton, 1687/1995)
What this means is that, if you go out into outer space and throw a baseball, the baseball continues
forever, moving with the same speed and in the same direction that you threw it. It never slows
down and it never turns, unless it happens to come near to a planet or asteroid. It does not matter
how the ball is launched; the ball doesn’t care and it doesn’t remember. Even if it is launched from
a spiral tube, once the ball leaves the tube, it just continues in a straight line. Thus the spiral tube
misconception is in contradiction with one of the first things that students learn in their
introductory physics classes, Newton’s first law.
And such errors appear to be commonplace. Halloun and Hestenes (1985b), in a study that
employed a multiple choice test given to hundreds of university physics students, found that 44%
of these students exhibited the belief that a force is required to maintain motion. (Though
individual students tended to answer questions inconsistently.) Nearly half, it seems, did not learn
Newton’s first law!
These results led to the obvious question: What is the problem with physics instruction?
Researchers hypothesized that part of the answer was that students spent the vast majority of their
time engaged in solving problems. And it was possible to get good at solving these problems
without really understanding the content of the equations that you were using. I will return to this
hypothesis below.
But the over-emphasis on problem solving could not be the whole story. The fact that objects
continue moving in the absence of forces seems like a straightforward notion, even if students were
not given much time to mull it over. This fact even seems like something that can be memorized,
whether or not you believe it. What is so intrinsically difficult about the notion that objects just
2
For a variety of accounts of student difficulties see, for example: R. Cohen, Eylon, and Ganiel, 1983;
Halloun and Hestenes, 1985a & 1985b; McDermott, 1984; Trowbridge and McDermott, 1980 & 1981;
Viennot, 1979. Also, see Smith, diSessa, and Roschelle (1993) for a critique of the misconceptions
33
keep moving in a straight line? In some manner, it seemed, students were actually resistant to
learning physics.
What researchers came to realize was that students’ existing knowledge of the physical world
has a strong impact on the learning of formal physics. In fact, students start physics instruction
with quite a lot of knowledge about the physical world. Furthermore, this knowledge is based in
quite a number of years of experience, so it might well be resistant to change.
So, one of the major outcomes of this “disaster” research was the realization that students have
prior knowledge of the physical world that is relevant to physics learning. The study of this prior
knowledge—often called “intuitive physics” or “naive physics” knowledge—has become an
endeavor of its own. When it takes its simplest form, naive physics research simply catalogs
student’s alternative conceptions (e.g., Halloun & Hestenes, 1985b). Further research attempted
to go beyond mere lists of misconceptions to develop generalizations about the nature of student
beliefs. These generalizations were directed at answering two distinct questions:
1. Can intuitive physics be formalized?
2. How is intuitive physics actually realized as knowledge in people?
The first question takes student’s answers to misconception-style questions and searches for a
specification of some underlying physics that these answers reflect. To understand the nature of
this approach, imagine that it was applied to the statements of expert physicists. In that case, the
assumption is that this methodology would produce Newtonian physics (Newton’s three laws,
literally) as the formalization of a physics underlying expert answers.
However, in searching for a similar formalization of intuitive physics, researchers must make
many questionable assumptions. In particular, they must interpret students’ answers to questions
as reflecting more general beliefs. In so doing, they beg the question of whether a formalization
exists.
Furthermore—and this is the point of the second question—even once you have found a
formalization, you still do not know how this physics is realized as knowledge in people. For
example, even if we know that expert physics formalizes as Newton’s laws, we still know very little
about what knowledge experts actually have. Most of the first formalizers, such as Halloun and
Hestenes (1985b) and Viennot (1979), gloss this issue with statements like: “[students show] the
perspective.
34
belief that, under no net force, an object slows down” (Halloun & Hestenes, 1985b). What is left
often by such assertions is how such a belief is realized as knowledge. Do students actually possess
propositional knowledge of this form? Or is there some more fundamental type of knowledge
that causes students to behave, in certain circumstances, as if they have this particular belief?
Throughout this manuscript, I intend particularly to build on the work of one researcher who
is self-consciously directed at answering the second question. In his Toward an epistemology of
physics, Andrea diSessa sets out to describe a portion of intuitive physics knowledge that he calls
the “sense-of-mechanism” (diSessa, 1993). The idea is that elements of the sense-of-mechanism
form the base level of our intuitive explanations of physical phenomena. As an example, diSessa
asks us to think about what happens when a hand is placed over the nozzle of a vacuum cleaner.
When this is done, the pitch of the vacuum cleaner increases. According to diSessa, the way people
explain this phenomena is they say that, because your hand is getting in the way of its work, the
vacuum cleaner has to start working much harder. The point is that this explanation relies on a
certain primitive notion: Things have to work harder in the presence of increased resistance if they
want to produce the same result.
diSessa’s program involves the identification of these primitive pieces of knowledge which he
calls “phenomenological primitives” or just “p-prims,” for short. (The p-prim that is employed to
explain the vacuum cleaner phenomenon is known as “OHM’S P -PRIM .”) The word
“phenomenological” appears in the name of this knowledge element in part because they are
presumed to be abstracted from our experience. OHM’S P -PRIM , for example, is abstracted from
the many experiences we have in the physical world in which things have to work harder in the
presence of increased resistance. diSessa lists a variety of p-prims, including p-prims pertaining to
force and motion (e.g., FORCE AS MOVER , DYING AWAY ) and p-prims pertaining to constraint
phenomena (e.g., GUIDING , ABSTRACT BALANCE). Furthermore, during physics instruction, the senseof-mechanism is refined and adapted, and comes to play an important role in expert physics
understanding. I will have more to say about diSessa’s theory in later chapters, particularly in
Chapter 5.
Problem solving in physics
Contemporaneous with this research into conceptual understanding and naive physics, a
second and essentially disjoint set of studies were examining how students—and, to a lesser
extent, expert physicists—solve textbook physics problems (Bhaskar & Simon, 1977; Chi,
35
Feltovich, & Glaser, 1981; Larkin, McDermott, Simon, & Simon, 1980). In its earliest
incarnations, these studies of physics problem solving were strongly influenced by prior research
into human problem solving, conceived more generally. For example, some early research in
artificial intelligence and cognitive science examined how people approached the solution of the
famous “Tower of Hanoi” puzzle. (Simon, 1975, is a typical example.) The Tower of Hanoi
puzzle involves a set of rings that can be moved among three pegs (Figure Chapter 1. -7). The
object of the puzzle is to move all of the rings from the left hand peg to the far right hand peg.
The catch is that at no time can a larger ring be on top of a smaller ring. Furthermore, only one
ring may be moved at a time.
Figure Chapter 1. -7. The Tower of Hanoi puzzle
An essential feature of this puzzle is that, at any time, there are no more than a few possible
moves. You can only move the topmost ring on any peg to one of the other pegs. And many such
moves will be prohibited because of the rule that larger rings cannot be on top of smaller rings.
Furthermore, the current state and desired end-state of the puzzle are clearly and simply defined.
In this early cognitive science literature, the solution of the Tower of Hanoi puzzle was
described as a search through the space of the possible states of the puzzle. From this viewpoint,
the most interesting question to be answered was exactly how a person goes about searching this
space; in other words, we would want to know how, at any time during the solution of the puzzle,
a person chooses among the few possible available moves. What Simon (1975) found was that
problem-solvers could be well described by a relatively simple strategy, what he called “meansends analysis.” In means-ends analysis, a problem solver chooses the move that appears to most
decrease the perceived difference between the current state and the goal state.
Initially, this research was extended to physics problem solving in a quite straightforward
manner (Bhaskar & Simon, 1977; Larkin et al., 1980). The notion was that moves in the space of
equations written on a sheet of paper could be treated much like moves in the solution of a puzzle.
More specifically, physics problem solving was seen to work like this: The student or expert reads
the problem and takes note of the quantities that are given in the problem and the quantities that
are needed. Then the problem solver writes, from memory, equations that contain these various
quantities as variables. Once the equations are written, the search can begin; the equations are
36
manipulated until a route from the given quantities to what is given is obtained. The problem
shown in Figure Chapter 1. -1 provides a simple example. In that problem, the student is given the
initial velocity, vo , the acceleration, a, and asked for the time. Once the equation v = vo + at is
written, it can simply be solved for the time, t.
Of course, in more complicated problems, some type of strategy is needed to find your way
through the space of equations from the givens to the solution, just as means-ends analysis was
needed in the Tower of Hanoi problem. Larkin and colleagues (1980) found the interesting result
that novices work by “backward chaining:” They start with the quantity that they want to know,
then write down an equation in which that quantity appears. Each of the unknowns in this new
equation then becomes a new quantity to be determined, and the process iterates. In contrast,
experts were found to “forward chain;” that is, work forward from the quantities given by writing
equations in which these quantities appear, and then using these equations to find new quantities.
ÒUnderstandingÓ and problem solving
I am now approaching the point where I will say how the current inquiry fits in with all of this
prior research. First, notice that there are some obvious limits in the above problem solving
research. As I have presented it, the physics problem solving research does not posit much of a role
for anything like “understanding” during problem solving and the use of equations. The story is
simply that you see the quantities that appear in the problem, you dredge equations out of
memory and write them down, then you manipulate the equations according to simple rules to
find an answer. So where is “understanding?” Larkin and colleagues (1980) themselves comment:
The major limitation we see in the current models is their use of an exceedingly primitive problem
representation. In fact, after the initial problem representation, these models work only with algebraic
quantities and principles described by algebraic equations. Thus, in a sense, they have no knowledge of
physics, but only the algebraic representation. This may actually not be too bad for capturing the
performance of novice solvers, who have little knowledge of physics, but much more of algebra.
However, it is certainly inadequate for capturing the work of more competent solvers.
But “understanding” may be tricky to define and to recognize. Perhaps, knowing equations
and knowing how to execute the above problem solving process is exactly what constitutes
understanding. Even if we adopt this improbable—and, I think, unpalatable—stance, it is still
possible to find some quite obvious and definite lacks in the above view of equation use. Most
obvious of these lacks is the fact that, if equations are only written from memory and are not
“understood” in any deeper sense, then how is it possible that people ever compose completely
new equations, corresponding to newly discovered principles and never-before-modeled systems?
37
There are some obvious ways out of this bind. One could maintain, quite reasonably, that the
above model of problem solving is only designed for a certain regime of problem solving involving
well-known principles and systems. But, if we accept this view, then we are simply left with the
new problem of describing the unaccounted for regimes of equation use and problem solving.
Furthermore, one might maintain that the composition of new equations is something that is only
done rarely and by scientific geniuses, like Einstein and Feynman. The rest of us can then make
relatively mechanical use of the equations that they develop for us. I will attempt to show, in
succeeding chapters, that this is simply not how things are.
The “naive physics” researchers may seem to have an advantage here, since they discuss
behavior that, at least intuitively, appears closer to “understanding.” But these researchers have yet
to tell us precisely how intuitive physics and conceptual knowledge is employed during the use of
equations. In what way do physicists make use of the sort of qualitative understanding that these
researchers are concerned with during problem solving? Is there any relation or are there simply
two sets of things to learn to become a physicist?
This is where I come in. In a sense, this inquiry will combine these two areas of research that,
to this point, have remained relatively disjoint. I will be examining phenomena akin to
misconceptions, naive theories, and p-prims, but in the context of problem solving and equation
use. Strikingly, we will see that there is actually a very strong connection between intuitive physics
knowledge and equation use; equations are understood and generated from knowledge that is
very closely related to intuitive physics knowledge. Describing this tight link between equations
and understanding is the central focus of this work.
Before proceeding, I want to note that the above account of the problem solving literature is a
little unfair. The same researchers have subsequently elaborated their work to account for some of
the difficulties that I have mentioned. A notable example is Larkin (1983), which I will discuss
extensively in Chapter 3. Also, the substantial success of the problem solving literature deserves
some emphasis. These researchers were very successful in building models that accurately
described subjects’ behavior in the regimes of problem solving that they studied. Thus, it is likely
that future research can profitably build on this work.
On the teaching of Òconceptual physicsÓ
I want to close this section with a note concerning how the results of the above research on
physics learning, particular the disastrous results reported by the misconceptions research, have
38
been interpreted as a prescription for improving physics instruction. The fact that intuitive physics
exists has led to the notion that it should be directly addressed in some way. Take the example of
the spiral tube misconception and students’ failure to understand the implications of Newton’s
First Law. Why not take the time to emphasize the (qualitative) principle that objects continue to
move in a straight line with a constant speed in the absence of intervening forces?
This prescription—emphasize the qualitative principles at the expense of problem solving—has
been taken up in a variety of courses that I will refer to as “Conceptual Physics” courses.3 These
courses operate under the principle that they should emphasize the “conceptual” content of
physics. In practice, this means that there is little problem solving, and few equations. Though, of
course, this varies greatly from course to course.
Given the viewpoint I am beginning to lay out here, it is clear that we should at least take the
time to reconsider this prescription for instruction. I have proposed replacing algebraic notation
with a programming language. Conceptual Physics courses make a similarly strong move, they
replace equations with plain language statements. If knowledge is inextricably representational,
then what are the consequences if the equations are removed altogether? What would Feynman
say?
I believe that the consequences are not necessarily bad; they may very well not be worse than
the consequences of my proposal to use programming languages. But I do want to make the point
that we need to understand what we’re doing if we choose to teach physics in the manner of
Conceptual Physics. I challenge the assumption that—in physics or any domain—we can separate
the “conceptual” from the symbolic elements of a practice for the purposes of instruction. In
removing equations from the mix, we change the constitution of understanding. This does not
imply that we cannot teach physics without equations. However, it does imply that equation-free
courses will result in an understanding of physics that is fundamentally different than physics-asunderstood-by-physicists.
Again, I must make clear that I do not mean to critique Conceptual Physics courses, rather I
hope to better understand the consequences of omitting equations. In fact, in the vein of these
more “conceptual” techniques, there is already a history of striking successes and exciting
proposals. For example, White and Horwitz designed and implemented a middle school
curriculum based around computer models. The students in this curriculum performed better on
3
See, for example, the textbook ÒConceptual PhysicsÓ (Hewitt, 1971). This textbook has been reprinted
in a number of more recent additions.
39
many tasks than students in a traditional high school course (White, 1993b). Clement, through his
technique of “bridging analogies,” has had success in training expert intuition (Clement, 1987).
And I suspect that even many working physicists could learn a great deal from Haertel’s
impressive reworking of the entire topic of electricity and magnetism in qualitative terms (Haertel,
1987). All of this published research is in addition to the many instructors at universities and high
schools that are enjoying success in designing and teaching Conceptual Physics courses. My work
here will help us to better understand the character of conceptual physics, so that we can
understand and better judge the results of such an approach, in clear contrast to alternatives.
Incidentally, the instructional move embodied in Conceptual Physics courses appears to be
common in much of the recent innovation in mathematics and science instruction. “Teaching for
understanding” seems to frequently involve a scaling back in the use of formalism. I do not intend
necessarily to question whether this is a productive instructional maneuver. However, if the
concerns expressed here are valid, then this sort of “teaching for understanding” may lead to new
sorts of understanding of mathematics and science.
Chapter Overview
Part 1. Algebra-physics and the Theory of Forms and Devices
The remainder of this dissertation is divided into two major parts. In Part 1 I will describe the
Theory of Forms and Devices, drawing on examples from the algebra-physics portion of the data
corpus. Thus, the first part of the dissertation will perform the dual roles of describing the theory
and applying it to describe algebra-physics.
Part 1 is divided into six chapters, Chapters 2 through 7. In Chapter 2 I take some steps
preparatory to the introduction of the major theoretical constructs in the following chapters. This
chapter will include some accounts of students using equations, accounts which are designed
simply to show that students can use equations in a meaningful manner. At the end of this chapter
I will also introduce the notion of “registrations.”
In Chapter 3 I will embark on a detailed discussion of symbolic forms. I will explain the
nature of symbolic forms in more detail, and I will individually discuss each of the specific
symbolic forms needed to account for the observations in the algebra data corpus. This will
include numerous examples from the data. In addition, I will argue against some competing
viewpoints, including what I call the “principle-based schemata” view.
40
Chapter 4 introduces and describes representational devices. As with symbolic forms, I will
discuss each of the individual devices needed to account for the observations in the data corpus
and I will draw examples from the data to illustrate my points. Along the way, I will have reason
to discuss what it means to “interpret” a physics equation. Importantly, we will see that the
“meaning” of a physics expression, as it is embodied in students’ interpretive utterances, is highly
dependent on the concerns of the moment. There is no single, universal manner in which meaning
is attached to equations.
In Chapter 5, I will speculate on the development of the knowledge system described in the
preceding chapters. My basic tactic here will be to draw on existing accounts of students’
knowledge as it is found prior to instruction, namely the research on intuitive physics knowledge
that I described above. Most prominently, I will draw on diSessa’s theory of the sense-ofmechanism. By comparing diSessa’s account to my own model, I will be able to comment on the
relationship between new and old knowledge, and I will construct hypothetical developmental
paths to describe how symbol use may contribute to the development of physics understanding.
One important result of this discussion will be the observation that expert “physical intuition” is
sensitively dependent on the details of the symbolic practices of physics.
In Chapter 6, I will present the results of applying the model systematically to describe the
whole of the algebra data corpus. In that chapter, I will describe the algebra corpus in some detail,
as well as the iterative process through which the data corpus was analyzed. In addition, the
chapter will contain an attempt to characterize aspects of the algebra corpus in a quantitative
manner. One reason for this quantitative characterization is to provide the reader with a means of
drawing conclusions concerning the prevalence of the phenomena that I describe, without having
to read the entirety of the data corpus. For example, through this quantitative characterization, we
will arrive at a measure of how often students construct novel equations and interpret equations.
Both of these phenomena turn out to be quite common. I will also present results concerning the
frequency of individual forms and representational devices.
The reader should be warned that, although the analysis process I will describe in Chapter 6 is
meticulous and detailed, it does not provide an exact and rigorous method of extracting my
model from the data. Instead, it is a highly heuristic process, and the recognition of specific
knowledge elements in the data should be understood as rooted in my intuition and theoretical
predispositions, as well as in the data itself. For this reason, much of the rhetorical burden of
arguing for my view must be carried by Chapters 2 through 5.
41
In Chapter 7—the last chapter of Part 1—I will reflect on the theory, draw out some
implications of the theory, and I will attempt to generalize the observations of the preceding
chapters outside the immediate concerns of symbol use in physics. Included in this chapter will be
a discussion of the implications of the preceding results for the relation between physics
knowledge and external representations. In addition, I will contrast my own theoretical position
with a variety of other theoretical accounts.
Part 2. Programming-physics, its analysis, and a comparison to algebra-physics
In Part 2, the discussion turns to programming-physics. This is the first time that
programming will appear following this introductory chapter. The primary concerns of this part of
the dissertation are to apply the Theory of Forms and Devices to describe programming-physics,
and to use the theory to compare programming-physics and algebra-physics.
Chapter 8 introduces the reader to the particular practice of programming-physics that was
the subject of this study. I will describe the programming language used and give examples to
illustrate the types of programs written by students.
In Chapter 9, I apply the Theory of Forms and Devices to programming-physics. This will
include a discussion of individual forms and devices with examples drawn from the programming
data corpus. Particular attention will be paid to the forms and devices that do not also appear in
algebra-physics.
Chapter 10 presents the systematic analysis of the programming data corpus. The content of
this chapter will be parallel to that of Chapter 6, in which I described the systematic analysis of the
algebra corpus.
In Chapter 11, I will use the analyses of algebra-physics and programming-physics in terms of
forms and devices to compare these two practices. To briefly preview the results, I found that the
distribution of forms in programming-physics was substantially similar to that in algebra-physics;
however, an important new class of forms appeared that was completely absent in the algebra
corpus. These new forms relate to schematizations of processes. With regard to representational
devices, there were some interesting differences in the frequency with which the various classes of
devices were employed.
Finally, in Chapter 12, I will summarize the thesis and trace the instructional implications of
this work. This will include a discussion of the implications of this work for traditional physics
42
instruction. I will argue that, given an improved understanding of how students use and interpret
equations, we may be able to recognize and remediate some types of student difficulties.
Finally, not least among the educational implications of this work is the fact that it considers
the possibility of a substantially altered practice of instruction, built around student programming.
In the last chapter, I will finally calibrate the strengths and weaknesses of this alternative approach,
to the extent that this study allows.
43
Chapter 2. Prelude to Theory
Rij − 12 gij R − Λgij = 8πGT ij
In the years around 1916, Albert Einstein presented the world with a new theory of gravitation.
And a truly new theory it was. In the existing Newtonian theory, gravity was “explained” by the
simple statement that there is an attractive force between every two masses. Pick any two masses,
m1 and m2, separated by a distance r. Then each of those masses feels a force, directed toward the
other mass, of size
F=
Gm 1 m 2
r2
In Einstein’s theory, the nature of gravity was wholly reconceptualized. Rather than explaining
gravity by an appeal to some unexplained “action at a distance,” Einstein reduced gravity to the
tendency of matter to distort space. An object, like the sun, reshapes the space around it. Then
when other objects—like the Earth—travel through this distorted space, their motion is directed
by the shape of space, just like a marble moving around the inside of a curved metal bowl. At the
heart of this new theory is Einstein’s “Field Equation,” written at the head of this chapter, which
relates the curvature of space to the distribution of matter. What we need to note about this
equation is that, somehow, a portion of Einstein’s reconceptualization of gravity is embodied in
this expression. Einstein began with the notion that matter distorts geometry, mixed in the old
Newtonian theory and some essential principles, and produced a new equation, one that did not
exist before.4
As we saw in Chapter 1, the existing literature concerning physics problem solving is not
capable of accounting for Einsteinian behavior of this sort. In that literature, problem solving is
dependent on a database of known equations, so it cannot explain the development of wholly new
4
Physicists might complain that textbooks do, in fact, contain ÒderivationsÓ of EinsteinÕs field
equation. However, a close examination of these ÒderivationsÓ reveals that they do not begin with
accepted physical laws and then perform logical manipulations to derive consequences of these laws.
Rather, the Field Equation is derived in part from some assumptions concerning what a geometric
theory of gravity must look like, such as what quantities the curvature of space may reasonably depend
on.
44
equations, or modifications based on conceptual considerations. However, it may not be fair or
appropriate to think of this as a deficiency in that literature or even to think of it as a task for
education research. The focus of the problem solving literature (and of this study) is on physics
learning and instruction. Our job here is to describe what students do and what we can reasonably
expect them to learn in an introductory course. And it’s possible that the construction of novel
equations is solely the province of physical geniuses, like Einstein and Feynman, or at least expert
physicists. Even if such invention does not require special mental prowess, it may nonetheless
depend on possessing some esoteric skills. Thus, the rest of us, including introductory physics
students, should perhaps only expect to use the equations that the experts write. If this is the case,
then accounting for this behavior may be less important for educationally directed research, both
because it is not really relevant to learning, and simply because it is relatively rare.
In the next sections of this chapter, I will take the first step toward showing that this view is
untenable. Equations have a perceived meaningful structure for all initiates in physics, even
introductory physics students. We will see that students can and do construct novel equations,
and they use equations in a meaningful manner, flexibly constructing their own interpretations. In
this chapter, I will begin by presenting some examples designed only to show that students can, at
least sometimes, construct and interpret equations. These examples will also provide a first taste of
what these behaviors look like. Then, in the latter part of this chapter, I will take some of the first
steps toward a theoretical account of this behavior. In subsequent chapters I will complete the
theoretical account and I will show through systematic quantitative analysis that these phenomena
are actually quite common.
Young Einsteins
Of all the tasks that I asked students to do, students found a task that I call “The Shoved
Block” problem the most exasperating. They found it exasperating, not because it is difficult, but
because it is a seemingly easy problem that produces counter-intuitive results. In the Shoved Block
problem, a block that rests on a table is given a hard shove. The block slides across the table and
eventually comes to rest because friction between the block and the table slows the block down.
The problem then asks this question: Suppose that the same experiment is done with a heavier
block and a lighter block. Assuming that both blocks are started with the same initial velocity,
which block travels farther?
45
v = vo
v = vo
Figure Chapter 2. -1. A heavier and a lighter block slide to a halt on a table. If they are shoved so as to have
the same initial speed, they travel exactly the same distance.
The counter-intuitive answer to this question is that the heavier and lighter block travel
precisely the same distance (refer to Figure Chapter 2. -1). In a later chapter I will describe how
students used equations to obtain this result. However, for the present purposes, I want to describe
how the dissatisfaction of a pair of students with this counter-intuitive result led them to invent
their “own brand of physics.”
Students actually express two competing intuitions concerning the Shoved Block. The most
common intuition is that the heavier block should travel less far. The reason given is that this is
true because the heavier block presses down harder on the table, and thus experiences a stronger
frictional force slowing it down. The alternative intuition is that the heavier block should travel
farther, since it has more momentum and is thus “harder to stop.” In fact, it is because these two
effects precisely cancel that the heavier and lighter block travel the same distance.
The pair of students I am going to discuss here adopted the latter intuition, that heavier blocks
should travel farther. And they clung to this intuition, even after they had used equations to derive
the “correct” result. This was especially true of one student in the pair, Karl5, who argued that it
was simply “common sense” that the heavier block should travel farther.
Karl
Yeah, that's true. But I still say that the heavier object will take the longer distance to stop
than a lighter object, just as a matter of common sense.
His partner, Mike, was somewhat less worried by the counter-intuitive result, but he was
willing to stop and discuss what might be changed in their solution. Eventually, Karl came to the
conclusion that “in real life” the coefficient of friction probably varies with weight in such a way
that heavier blocks travel farther.
Karl
I think that the only thing that it could be is that the coefficient of friction is not constant.
And the coefficient of friction actually varies with the weight.
The coefficient of friction is a parameter that determines the size of the frictional force. Students
are taught in introductory physics classes that this parameter can be treated as a constant that
46
depends only on the nature of the materials involved, such as whether the objects are made out of
wood or steel.
Given the counter-intuitive result, Mike and Karl were willing to reconsider the assumption
that the coefficient of friction is a constant. After some very minimal cajoling, (I said only “Do
you want to try putting in maybe some hypothetical weight dependence to that”), they set out to
compose their own expression for the coefficient of friction. Karl began by expressing their core
assumption:
Karl
I guess what we're saying is that the larger the weight, the less the coefficient of friction
would be.
To obtain their desired result—that heavier masses travel farther—they needed to have the
coefficient of friction decrease with increasing weight. In other words, friction must work less on a
heavier mass. Over approximately the next nine minutes, they laid out some additional properties
that they wanted their new expression to have. Here’s an excerpt (refer to Appendix A for a key to
the notations used in transcripts):
Karl
Well yeah maybe you could consider the frictional force as having two components. One
that goes to zero and the other one that's constant. So that one component would be
dependent on the weight. And the other component would be independent of the weight.
Mike
So, do you mean the sliding friction would be dependent on the weight?
Karl
Well I'm talking about the sliding friction would have two components. One component
would be fixed based on whatever it's made out of. The other component would be a
function of the normal force. The larger the normal force, the smaller that component. So
that it would approach a - it would approach a finite limit. It would approach a limit that
would never be zero, but the heavier the object, the less the coefficient of friction at the
same time. (…)
Mike
I don't remember reading that at all. [laughs]
Karl
See, I'm just inventing my own brand of physics here. But, if I had to come up with a way if I had to come up with a way that would get this equation to match with what I think is
experience, then I would have to - that's what I would have to say that the=
Mike
Actually, it wouldn't be hard to=
Karl
=the coefficient of friction has two components. One that's a constant and one that varies
inversely as the weight.
In this passage, Karl outlines what he wants from the new expression. He decided that the
coefficient of friction should have two components. One of these components is independent of
weight, the other decreases with increasing weight. After some fits and starts, Mark and Karl,
settled on this expression for the coefficient of friction:
µ
µ = µ 1 + C m2
5
All student names are fictitious.
47
Here µ1, µ2, and C are constants and ‘m’ is the mass. This expression does have some problems,
most notable of which is the fact that the coefficient of friction tends to infinity as the mass goes
to zero. But this expression does capture much of what Mike and Karl wanted to include, it
decreases with increasing mass, and it approaches a constant as the mass becomes infinitely large.
The most important thing to note about this expression is that you will not find it in any
textbook. Mike and Karl did not learn it anywhere, and they didn’t simply derive it by
manipulating equations that they already knew; instead, they built this expression from an
understanding of what they wanted it to do. As Karl says, he was “just inventing his own brand of
physics.” Their expression is not as fancy and subtle as Einstein’s Field equation, but it is clearly a
non-trivial, novel construction. It seems that the construction of novel equations is not only the
province of experts like Einstein.
The question for us to answer in the upcoming chapters is: How did they do this? How did
they know to write a ‘+’ instead of a ‘×’ between the two terms? How did they know to put the m
in the denominator? These are precisely the behaviors we need to be able to explain; we want to
know what knowledge gives students the ability to connect conceptual content and equations.
After a few more preliminaries, that is what we will do.
Sensible Folk
Episodes like the last one were a little uncommon in my data corpus, though not really rare. In
fact, like Mike and Karl, all of the pairs in my study demonstrated an ability to compose new
expressions at some point during their work, and without obvious difficulty. Nonetheless, these
episodes were uncommon because most of the physics problems that students are asked to solve
do not require them to invent novel equations. In this next example, I am going to describe an
episode that should appear almost painfully mundane to those who are familiar with introductory
physics. The important fact about this episode is that it includes an instance of a student
interpreting an equation. Such episodes are not uncommon because, although students to not need
to invent totally new expressions very often, they do frequently need to interpret equations that
they derive.
In this episode, Jack and Jim were asked to solve the following problem, in which a mass hangs
at the end of a spring:
A mass hangs from a spring attached to the ceiling. How does the equilibrium position of the mass
depend upon the spring constant, k, and the mass, m?
48
This problem was extremely easy for them and they spent only about 2 1/2 minutes working on
it. Their board work is reproduced in Figure Chapter 2. -2.
k
x
F=-kx
m
F=ma
kx=mg
x=
mg
k
Figure Chapter 2. -2. Jack and Jim's solution to the mass on a spring problem.6
Jack and Jim began by explaining that there is a force from gravity acting downward, a force
from the spring acting upward, and that these forces must be equal for the mass to hang
motionless:
Jim
Mmm. Well, there's the gravitational force acting down. [w. Arrow down; F=ma next to
arrow.] And then there is
Jack
a force due to the spring // holding it up.
// a force due to the spring which I believe is,, [w. F=-kx] Is equal
Jim
to that.
Jack and Jim know that the force of gravity on an object is equal to mg where m is the mass of the
object and g is the gravitational acceleration, a constant value. They also know that springs apply a
force proportional to their displacement. If a spring is stretched an amount x from its rest
length—the length it likes to be—then it applies a force kx, where k is a constant known as the
“spring constant.”
Jack and Jim went on to equate these two forces, writing kx=mg. Then they solved for the
displacement x to obtain their final expression:
x=
mg
k
Then Jim immediately explained that this is a completely sensible result:
Jim
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k in x=mg/k] And that if you have a
6
All diagrams are my own reproductions of student work. In creating these reproductions I attempted to
retain the flavor of student drawing while rendering important elementsÑespecially symbolic
expressionsÑclearly.
49
stiffer spring, then you're position x is gonna decrease. [g. Uses fingers to indicate the
gap between the mass and ceiling.] That why it's in the denominator. So, the answer
makes sense.
Jim says that this expression makes sense for two reasons. First, he tells us that the equation says
that, if you increase the mass (with k held constant) then x increases. Second he considers the case
where the spring is made stiffer; i.e., k is increased. In this case, he says, the equation implies that x
will decrease. Presumably, these observations make sense to Jim because of knowledge that he has
concerning the behavior of springs.
I consider this statement by Jim to be an “interpretation.” Jim is not merely providing us with
a literal reading of the symbols on the page; he does not simply say “X equals M G over K.”
Rather, he tells us what the equation “says.” For example, one thing that the equation “says” is
that if you increase k then x decreases.
Even this very simply interpretation has a number of interesting characteristics that are worth
mentioning. In saying that as k increases x decreases, Jim is imagining a very particular process: k is
increasing rather than decreasing, and m and g are parameters that are implicitly held fixed.
Describing and accounting for these particulars is part of the job of this study. I would like to
know: What do interpretations look like, in general? Do they always involve holding some
parameters fixed and varying others? Furthermore, how does Jim know what the equation says?
How does he know, by looking at the equation, that if k increases then x will decrease?
Interpretations of this sort are a simple and important phenomenon that are not included in
existing accounts of equation use in physics. Interpretation has no place in a model in which
equations are just written from memory and then manipulated to produce a numerical result. This
might be acceptable if this behavior was rare or unimportant, but that is not the case. To those
familiar with introductory physics, Jim’s interpretation will appear so obvious and common that it
is almost unnoticeable. Later on, the systematic analysis will show just how frequently these
interpretations appear in my data corpus. In fact, almost every problem solving episode involves
some interpretations. And students interpret, in some manner, a relatively high percentage of the
equations that they write. So, in addition to any prima facie importance of these phenomena,
there is the simple fact that interpretations are extremely common in students’ problem solving,
and therefore should be taken into account in any model.
50
A Feel for the Numbers
The above examples begin to illustrate what it means for students to use equations
meaningfully. Mike and Karl composed an expression from an idea of what they wanted from the
expression. And Jim told us what an equations “says.” In both of these examples, the equations
were, in a sense, directly meaningful for the students; the meaning was somehow seen in the
structure of the equations itself.
The ability to check the sensibility of results is very important. “Is my work correct?” and
“Does this equation make sense?” are extremely useful questions to be able to ask and answer
when working with equations. The point I want to make with this next example is that equation
users have many methods available for determining if what they have written is correct; they do
not always need to see meaning directly in equations.
These alternative methods of checking the sensibility of expressions will not play as important
a role in this inquiry. I will not attempt to model the knowledge involved in these behaviors.
However, I want to do one brief example here in order to help clarify what I intend to exclude. In
this episode, Jon and Ella were working on a very standard problem:
Suppose a pitcher throws a baseball straight up at 100 mph. Ignoring air resistance, how high does it
go?
Jon and Ella’s path to the solution of this problem was very short. They began by writing, from
memory, an equation:
v 2f = vo2 + 2ax
This equation applies to cases in which an object moves with constant acceleration, either speeding
up or slowing down. It happens that gravity has (approximately) the effect of causing objects to
decelerate at a constant rate as they travel up into the air, so this equation is appropriate for the
task. Here, vf and vo are the final and initial velocity of the object, a is the acceleration, and x is the
distance traveled.
After the ball is thrown, it steadily decelerates at the fixed rate determined by gravity until, at
the peak, it stops momentarily. Thus, the ball’s maximum height can be found by substituting a
value of zero for vf and then solving for x. This is precisely what Jon and Ella did, and they wrote
51
vo2
=x
2a
When they substituted numerical values for vo and a, Jon and Ella obtained a result of 336ft for x.
Then they began to wonder if this is a reasonable result:
Ella
Yeah, okay. I guess that’s pretty reasonable.
Jon
Well that’s like a - a football field. [g. =336ft.]
Ella
Yeah. How far,,
Jon
A hundred miles an hour is,, (1.0) If you were in a car going a hundred miles an hour,
Ella
[laughs] straight up into the air.
Jon
straight up into the air, would you go up a football field? (0.5) I have no idea. Well it doesn’t
sound - it’s on the right order of magnitude maybe.
Ella
Yeah. Okay.
(5.0)
Jon
I mean, it’s certainly not gonna go (0.5) thirty six feet. That doesn’t seem far enough.
Cause a hundred miles an hour is pretty fast.
Ella
Hm-mm.
Jon
It’s not gonna go three thousand feet.
So, Jon and Ella concluded that 336 feet was a reasonable answer, or at least the right “order
of magnitude.” This method of checking the sensibility of a result is quite different than Jim’s
interpretation of an equation. Here, Jon and Ella did not look at the nature of the final symbolic
expression, rather they examined the numerical result that this expression produces. Once they
obtained the numerical result, they were able to draw on a wide variety of knowledge to get a feel
for whether this result was correct. The ball is thrown at roughly the speed that a car travels and it
goes about as high as a football field, does this make sense? Well, they argued, it is a little hard to
tell but “it’s on the right order of magnitude.”
Students have a selection of alternative routes available for checking the sensibility of their
work. As in the first two episodes I presented, they can see meaning “directly” in equations.
However, they also have some less direct methods available for checking whether their work
makes sense. For example, like Jon and Ella, they can consider the reasonableness of numerical
results. Or they can look at their expressions and see if they have the “right units.” As I
mentioned, these alternative routes will be less important in the work that follows and will not be a
central focus of my modeling efforts.
52
Steps Toward a Model
The above episodes, especially the first two, are intended to provide a first introduction to the
type of phenomena that I want to understand, as well as to show that they exist. In the first
example, Mike and Karl composed their own novel expression, starting from an understanding of
what they wanted from that expression. The observation that students have this capability has
implications beyond contexts where students have to invent their own equations. If students can
build equations as meaningful structures, why can’t they see existing equations, composed by
others, in terms of the same meaningful structures? The answer is that they can, and this is
precisely what was illustrated in the second episode. Jim told us what an equation said.
Furthermore, the second example provided a glimpse into what it means to “interpret” an
equation.
The task now is to build a model of symbol use in physics, in the spirit of cognitive science,
that incorporates these important phenomena. Before proceeding, it is worth a moment to
comment on what is involved in “the spirit of cognitive science.” One way that cognitive science
proceeds is by making models that are designed to make predictions and provide accounts of
human activity—what people say, do, and think. Furthermore, these models are of a particular
sort, they generally posit the existence of internal mental structures, including entities called
“knowledge.”
In spite of the fact that I will be working within this tradition, I will not develop a model that
is detailed enough that it can be “run” to generate predictions. I will posit the existence of
knowledge elements that are to be understood as elements of such a model. But to actually “run”
the model, in any sense, we would have to fill in many missing details with common sense and
intuition. This is required because the creation of anything approaching a complete model would
be very difficult given the present early stage of this research.
Note that this boils down to self-consciously applying a different tactic than that taken by the
existing physics problem solving literature. In that literature, certain phenomena are disregarded
and simplified situations are studied so that a runnable model is possible. Such an approach has
great merits, but simply will not do for my needs. We want to press on what “meaningful”
symbol use is, which precludes our ignoring phenomena like interpretation. This does not mean
that I am not throwing things out for the sake of simplicity. But preserving the phenomena that I
am interested in, even in a simplified form, is enough to make the creation of a runnable model
very difficult. As we will see, the interpretation of equations is a complex and idiosyncratic
53
business, sensitively dependent on details of context, and hence is not amenable to description by
any simple models.
Let me pin down a little further exactly what phenomena I propose to model and where my
attention will be focused. First, my model of symbol use in physics should primarily be
understood as a model that applies in the very moments of symbol use, when an individual is
actually working with an expression. The model will address questions such as: How does a
student know to write “+” instead of “×”? And: What kinds of things does a student say when
they have an equation in front of them? Interpersonal and developmental (learning-related)
phenomena will be treated as secondary, although I will do some tentative extending of the model
in each of these directions.
Second, my attention will be focused on only one particular slice of the knowledge associated
with symbol use in physics; prototypically, this is knowledge that is particularly evident when
novel expressions are constructed and expressions are interpreted. I will not worry about, for
example, the knowledge needed to correctly perform syntactic manipulations of expressions, nor
will I worry about how students know which equations to pull out of memory for a given task.7
This focus is justified because of the primary concern of this work, which is to understand the
relation between understanding and symbol use. The presumption is that, to understand that
relation, our effort is best focused at the places where symbols and understanding, in a sense,
come together.
First step: Registrations
Now we are finally ready to take some first steps in the direction of a model of symbol use in
physics. Let us begin by imagining a specific situation, a situation that is prototypical of our
concerns. A student is looking at a sheet of paper that has, written on it, a single equation. The
student has some particular interest in this equation, perhaps wondering if it’s written correctly,
perhaps wondering what its implications are. To take a first step toward constructing our model,
we can begin with a fundamental question: What does the student “see” when she looks at this
equation? On what features of the page of expressions does her behavior depend?
7
Chi et al. (1981) present evidence that students and experts can directly recognize problems as being of
a specific type and then equations are associated with particular problem types. This is one model of
how students draw on known equations from memory.
54
This may seem like a simple or silly question, but it is not altogether trivial. To illustrate, think
for a moment of the experience you have when looking at a word processing document on a
computer display. The text that you see is actually, at its lowest level, made up of a number of
small dots, pixels. Does this mean that you are seeing pixels when you look at the document on a
computer display? In a sense, yes. But, in an important sense, you also “see” letters. Notice that
there is no effort involved in going from the pixels to the letters, you just see the letters directly.
In fact, if you are reading a printed version of this document, the letters are also probably formed
from dots, but you are certainly not conscious of those dots as you read. And there are other
things that you “see” when you look at a text document. You can see words, groups of words,
paragraphs, and perhaps arbitrary groupings of text. You can even see the entire block of text on a
page as a unit.
Similar points have been made in a variety of literature with diverse concerns. It is a wellknown result that an account of perception which maintains that we simply perceive a neutral array
of colors is inadequate. Rather we see a world of objects and edges, a partly processed world of
things. I am not going to attempt at all a survey of the literature that has made related points.
However, I will borrow a term from Roschelle (1991). Roschelle uses the term “registration” to
describe the way people “carve up their sensory experience into parts, give labels to parts, and
assign those labeled parts significance.” This provides us with a way to rephrase the question we are
currently concerned with: How does our hypothetical student “carve up” her experience of the
equation written on the sheet of paper? Here is a preliminary list of some of the more important
registrations with which I will be concerned:
1. Individual symbols. If our hypothetical student is looking at a sheet of paper where the equation
v = vo + at is written, she may see the letter ‘v’ as an individual symbol. This is in contrast to
seeing some dots, or two straight lines that meet at a point. The student could also see the
letter ‘t’ or a ‘+’.
2. Numbers. Students can also see numbers. Jon and Ella saw the number “336” that they had
written on the blackboard. Note that this number can also be seen as comprised of three
individual symbols.
3. Whole equations. Our hypothetical student can see the equation v = vo + at as a single unit.
55
4. Groups of symbols. Arbitrary groups of symbols can also be seen as units. For instance, if I write
an expression for the total momentum of two objects as, P = m1v1 + m2v2, then m1v1 can be
seen as the momentum of one of the objects. Terms in expressions and symbols grouped in
parentheses are frequently seen as units, but arbitrary and complicated groupings are not
uncommon.
5. Symbol Patterns. Finally, part or all of an expression may be seen in terms of a number of
“symbol patterns”—patterns in the arrangement of symbols. These patterns will be discussed
extensively in the next chapter, along with many examples, so I will just give a brief
explanation here. Roughly, symbol patterns can be understood as being templates in terms of
which expressions are understood and written. In general, these are relatively simple templates.
For example, two symbol patterns that students see in expressions are:
=
+
+
…
The symbol pattern on the left is a template for an equation in which two expressions are set
equal. These two expressions can be of any sort—the
can be filled in with any expression.
Similarly, the symbol pattern on the right is a template for an expression in which a string of
terms are separated by plus signs. Note that symbol patterns differ from what I have called
groups of symbols because, in the case of groups, the arrangement of symbols within the group
is unimportant; the group is simply treated as a whole. For patterns, the arrangement of
symbols constitutes a recognizable configuration. Symbol patterns will play a key role in my
explanation of how students interpret expressions and compose novel expressions.
So, with this discussion of registrations, we have taken a first step into the physicist’s world of
symbols. What does a physicist or physics student see when looking at a sheet of equations? They
do not see an organic array of curving lines and marks; they see symbols, equations, terms, and
groups of various sorts. Furthermore, we will see that, for a physicist, a sheet of physics equations
is even more structured and meaningful than this first account implies. In the next chapter, we
dive hard into this world of structure and meaning.
56
Chapter 3. A Vocabulary of Forms
Rabbit or duck?
Art and Illusion, the well-known book by E. H. Gombrich concerning art and pictorial
representation, begins with a discussion of what Gombrich calls “the riddle of style” (Gombrich,
1960). For Gombrich, the riddle is this: “Why is it that different ages and different nations have
represented the visible world in such different ways? Will the paintings we accept as true to life
look as unconvincing to future generations as Egyptian paintings look to us?” Gombrich rejects the
view that the history of art can simply be understood as a development toward more accurate
representations of reality. However, Gombrich balances this rejection with an acceptance of a
certain kind of historical development. He believes that, over time, artists have developed an
increasingly varied and flexible “vocabulary of forms.” “But we have seen that in all styles the artist
has to rely on a vocabulary of forms and that it is the knowledge of this vocabulary rather than a
knowledge of things that distinguishes the skilled from the unskilled artist.” Every artist and every
person that looks at art learns the current vocabulary by looking at existing paintings. Then, new
paintings are created from this vocabulary and other paintings are seen in terms of the vocabulary.
The point I want to extract here is that the forms that we see when looking at a painting are
not somehow inherent in the painting, or even in the painting and the “raw” perceptual capabilities
of humans. People learn—and we, as individuals, learn—to see the forms, and then we bring this
capability with us when we look at new drawings and paintings.
This point is especially evident in some more marginal cases. Consider the simple drawing at
the top of this page, which I have taken from Gombrich. Is it a smiling duck? Is it a cute, but wary
rabbit? Whatever it is, this drawing consists of a very simple arrangement of lines. But when we
look at it we see more than those lines; to us, the image is much more than a simple sum of its
parts. If it was a simple sum of its parts, how could it possibly be both a rabbit and a duck?
Because we have learned to see certain forms, we see a meaningful image. In a sense, we bring the
rabbit and duck with us.
57
I can now add a basic point to the discussion of registrations I presented at the end of the last
chapter. The registrations available in any page of expressions is not only a function of the marks
on the paper, it also depends on the (learned) capabilities of the person looking at the display. To
see letters, for example, you have to know the alphabet. This will be my focus in this chapter: I will
describe the knowledge that people acquire that is associated with certain registrations in symbolic
expressions.
What is a Symbolic Form?
I am now ready to define the first major theoretical construct in my model of symbol use in
physics, what I will refer to as “symbolic forms” or just as “forms,” for short. A symbolic form is a
particular type of knowledge element that plays a central role in the composition and
interpretation of physics expressions. Each form has two components:
1. Symbol Pattern. Each form is associated with a specific symbol pattern. Recall that symbol
patterns are one of the types of registrations I described in the previous chapter. A symbol
pattern is akin to a template for an expression and symbolic expressions are composed out of
and seen through these templates. To be more precise, forms are knowledge elements that
include the ability to see a particular symbolic pattern.
2. Schema. Each symbolic form also includes a conceptual schema. The particular schemata
associated with forms turn out to be relatively simple structures, involving only a few entities
and a small number of simple relations among these entities. These schemata are similar to
diSessa’s (1993) “p-prims” and Johnson’s (1987) “image schemata,” which are also presumed
to have relatively simple structures. For example, as I discussed earlier, diSessa describes a
particular p-prim known as “Ohm’s p-prim” in which an agent works against some resistance
to produce a result. This p-prim can be understood, in part, as a simple schematization
involving just three entities, in terms of which physical situations are understood. Although I
will discuss both p-prims and image schemata in later chapters, the view that I will describe
shares many more of the commitments of diSessa’s theory.
One other point about forms is worthy of note at this very early stage. The schema component
of forms can allow specific inferences. I will understand this as working as follows: Generic
58
capacities (that I will not describe) act on the schemata to produce inferences. I will say more
about this in a moment.
To illustrate forms, I want to return to the very first example that I presented from my data
corpus. In Chapter 2, I described an episode in which Mike and Karl constructed their own
expression for the coefficient of friction:
µ
µ = µ 1 + C m2
When presenting this episode, I commented that this was an equation that you would not find in
any textbook; Mike and Karl had composed this equation on their own. Now we are ready to fill
in the first pieces of the puzzle in accounting for how they accomplished this feat.
Recall that the core of Mike and Karl’s specification for this equation was the notion that the
coefficient of friction should consist of two parts, one that is constant and one that varies inversely
with the weight:
Karl
=the coefficient of friction has two components. One that's a constant and one that varies
inversely as the weight.
I want to argue that two forms are evident in this specification, the first of which I call PARTS -OF -AWHOLE.
Like all forms, PARTS -OF -A-WHOLE consists of a symbol pattern and a schema. The symbol
pattern for PARTS -OF -A-WHOLE is two or more terms separated by plus (+) signs. And the schema
has entities of two types: There is a whole which is composed of two or more parts. This schema
can be seen behind Karl’s above statement; he says that the coefficient of friction consists of “two
components.” In other words, the coefficient of friction is a whole that has two parts.
Furthermore, the equation has the structure specified in the symbol pattern; it has two terms
separated by a plus sign.
To describe the various symbol patterns that we will encounter, I will employ a shorthand
notation. For PARTS -OF -A-WHOLE I write the symbol pattern as:
PARTS -OF -A-WHOLE
[
+
+…]
Here the ‘ ’ character refers to a term or group of symbols, typically a single symbol or a product
of two or more factors. The brackets around the whole pattern indicate that the entity
corresponding to the whole pattern is an element of the schema. The shorthand notation for
symbol patterns is briefly summarized in Appendix D.
59
The second form in Karl’s statement is what I call PROPORTIONALITY MINUS or just PROP-. The
schema for PROP- involves two entities: One quantity is seen as inversely proportional to a second
quantity. In Karl’s specification for this expression, it is the second component of the coefficient of
friction that is inversely proportional to the weight. As he says, there’s one component that’s
constant and one that “varies inversely as the weight.”
The symbol pattern for PROP- is relatively simple. When a student sees PROP- in an expression,
all they are attending to is the fact that some specific symbol appears in the denominator of an
expression. I write this as:
 … 
…x… 
PROP-
Here, x will usually correspond to a single symbol. In Mike and Karl’s expression, the m is the
symbol of interest and, in line with the PROP- form, it appears in the denominator of the second
term.
I want to digress for a moment here to make a couple of points about the forms we have seen
thus far. First, it may seem as though the list I am building will simply end up with a form
corresponding to each syntactic feature of algebraic notation. But this is not the case. We will see
that there are limits to what features people are sensitive to, and that identical patterns can
correspond to more than one form. Note, for example, that
PROP-
constitutes a very specific
hypothesis concerning what features are consequential for students during equation construction
and interpretation. All that is relevant for PROP- is that the symbol in question appears in the
denominator of the expression; it does not matter whether the quantity is raised to a power or
whether there are other factors surrounding it. With regard to this form, such details are not
consequential; both 1/m and 1/m2 can be seen as PROP-.
Second, I want to say a little about the inferences permitted by these two forms, just to give
the flavor of what I have in mind. Here is a sampling of inferences permitted by
PARTS -OF -A-
WHOLE:
•
If one part changes and the other parts are constant, the whole will change.
•
If one part increases and the other parts are constant, the whole will increase.
•
If one part decreases and the other parts are constant, the whole will decrease.
•
If the whole is necessarily constant and one part increases, then another part must decrease.
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And here is a sampling of inferences permitted by
PROP-:
•
If the denominator varies, the whole varies.
•
If the denominator increases, the whole decreases.
•
If the denominator decreases, the whole increases.
Some of these inferences may be playing an implicit role in Mike and Karl’s construction of their
equation for the coefficient of friction. Recall that Mike and Karl began with the requirement that
µ must decrease with increasing weight:
Karl
I guess what we're saying is that the larger the weight, the less the coefficient of friction
would be.
Inferences of the above sort could confirm that this requirement is met. If the weight is increased,
PROP-
tells us that the second term in Mike and Karl’s expression for µ must decrease. Then PARTS -
OF -A-WHOLE
permits the inference that if the second part is decreased the whole also decreases,
since the first term is constant. Thus, the whole coefficient of friction decreases if the weight is
increased.
Two additional forms also played a role in Mike and Karl’s building of their novel equation
and I will just comment on them briefly here. The first of these forms has to do with the ‘C’ in
front of the second term. This factor was inserted by Mike and Karl almost as an afterthought, at
the very end of the episode. I take this to be a case in which the “COEFFICIENT” form is in play:
COEFFICIENT
[x
]
In COEFFICIENT, a single symbol or number, usually written on the left, multiplies a group of
factors to produce a result. A number of features are typical of the coefficient part of the schema,
and I will explain these features later in this chapter. For now, I just want to comment that the
coefficient is frequently seen as controlling the size of an effect. This, roughly, was Mike and
Karl’s reason for inserting the coefficient. As Karl wrote, he commented:
Karl
You find out that uh you know the effect is small, or you find out the effect is large or
whatever so you might have to screw around with it.
So, for Mike and Karl, the factor of C was a way for them to tune their expression.
Finally, there is one important part of Mike and Karl’s expression that I have yet to comment
on; I have not said anything about the equal sign and the µ that appears on the left side of the
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equation. This constitutes an application of one of the simplest and most ubiquitous forms, the
IDENTITY
form.
x = […]
IDENTITY
In the IDENTITY form, a single symbol—usually written on the left—is separated from an arbitrary
expression by an equal sign. I will have more to say about IDENTITY later in this chapter; at this
point, however, I just want to note that identity transparently allows the inference that whatever is
true of the entity corresponding to the whole right hand side is also true of the entity on the left.
In the case of Mike and Karl’s equation for the coefficient of friction, since the whole expression
on the right decreases with increasing mass, µ can also be presumed to decrease with increasing
mass.
In the previous chapter I asked: How did these students know to write a ‘+’ instead of a ‘×’
between the two terms? And: How did they know to put the m in the denominator? Now I have
given the answers to these questions; or, at least, I have given a particular variety of explanation. I
have said that, to compose a new equation, these students had to essentially develop a specification
for that equation in terms of symbolic forms. Once this was done, the symbol patterns that are
part of these forms specified, at a certain level of detail, how the expression was to be written.
In addition to being knowledge for the construction of the expression, the forms are also the
way that the equation is seen and understood by Mike and Karl. Just as we can see a rabbit or a
duck in the drawing at the head of this chapter, Mike and Karl see
IDENTITY
PARTS -OF -A-WHOLE
and
when they look at their expression for µ. For Mike and Karl, their equation is not just a
collection of marks, or even a collection of unrelated symbols, it has a meaningful structure.
Forms and a More Traditional Problem
As I have mentioned, the episode discussed in the previous section is somewhat atypical. When
solving problems in an introductory physics course, it is rare that students must build elaborate
expressions from scratch. In this section, I want to turn to a more typical episode, involving more
familiar expressions. In addition, this episode is interesting because I believe it may provoke
skeptical readers to argue that knowledge other than forms is responsible for the students’ ability
to write the expressions that appear. Thus, I will need to do some work to argue that forms are,
nonetheless, a central part of the knowledge involved here.
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In this example, I am again going to look at the work of Mike and Karl. The reason for
sticking with these students is that I want there to be no doubt that the students we see doing the
more typical work described here have the abilities demonstrated in the above section. In this
episode, Mike and Karl were working together at a blackboard to answer the following question:
For this problem, imagine that two objects are dropped from a great height. These two objects are
identical in size and shape, but one object has twice the mass of the other object. Because of air
resistance, both objects eventually reach terminal velocity.
(a) Compare the terminal velocities of the two objects. Are their terminal velocities the same? Is the
terminal velocity of one object twice as large as the terminal velocity of the other? (Hint: Keep in mind
that a steel ball falls more quickly than an identically shaped paper ball in the presence of air
resistance.)
This problem asks about a situation in which a ball has been dropped under the influences of
gravity and air resistance. Gravity is pulling the ball down and, as the ball rushes through the air,
the resistance of the air tends to oppose the downward fall. When the ball is first dropped, it
initially accelerates for a while—it goes faster and faster. But eventually, because of the opposition
of air resistance, the speed levels off at a constant value. This maximum speed that the ball reaches
is called the terminal velocity. The question that this problems asks is: How does the terminal
velocity differ for two objects, one of which is twice as large as the other?
(b)
(a)
(c)
Fair
Fair
Fgravity
(d)
Fgravity
Fgravity
Fair
Fgravity
Figure Chapter 3. -1. (a) An object is dropped. (b) & (c) As it speeds up, the opposing force of air resistance
increases. (d) At terminal velocity, the forces of air resistance and gravity are equal.
One way to understand what’s going on here is that there are two forces acting on the ball, a
force down from gravity and a force from air resistance acting upward. Initially the ball is not
moving and the force from air resistance is zero. But gradually the ball speeds up and the force
from air resistance grows until, at terminal velocity, it’s exactly equal to the force from gravity
(refer to Figure Chapter 3. -1).
Now let’s look at what Mike and Karl had to say about this problem. They began by
commenting that the hint given in the problem “basically answers the question,” since it gives
away the fact that the terminal velocity of a heavier object is higher. The discussion below
63
followed, in which, Mike and Karl agreed that what they called the “upward” and “downward
accelerations” of a dropped object must be equal when it reaches terminal velocity:
Karl
So, we have to figure out (0.5) what, how do you figure out what the // terminal velocity is.
Mike
// terminal velocity is.
Well, okay, so, you have a terminal velocity, You have terminal velocity, that means it
falls at a steady rate. // Right, which means the force opposing it
// Right, which means
Karl
Karl
It means it has an upward acceleration that's equal to the downward acceleration.
Mike
Ri:::ght.
Following this discussion, Mike drew the diagram shown in Figure Chapter 3. -2, talking as he
drew.
f(v)=
Fa
v m
Fg
Figure Chapter 3. -2. Mike and Karl’s air resistance diagram.
Mike
Air resistance. You have a ball going down with a certain velocity. A for:::ce,, You reach
terminal velocity when air resistance, the force from the air resistance is the same as the
force from gravity. [g. points to each of these arrows as he writes, with spread thumb and
forefinger indicating size] But, we also know that force of air resistance is a function of
velocity. [w. f(v)=] Right?
In this last comment, Mike reiterates the notion that the influences of air and gravity must be
equal at terminal velocity, but here he talks in terms of opposing forces, rather than opposing
accelerations. Notice that Mike’s drawing is in agreement with the account I presented in Figure
Chapter 3. -1.
Finally, in the next passage, Mike and Karl compose an expression for air resistance:
a(t ) = −g +
f (v)
m
Mike
So, at least we can agree on this and we can start our problem from this scenario. [g.
Indicates the diagram with a sweeping gesture] Right? Okay? So, at any time,, At any
time, the acceleration due to gravity is G, and the acceleration due to the resistance force
is F of V over M. [w. g + f(v)/m] This is mass M. [w. m next to the mass on the diagram]
Karl
Ah:::.
Mike
Right.
Karl
Yeah.
Mike
Okay, // now they're opposing so it's a minus.
Karl
// (So) as your mass goes up,, [g. m in equation]
64
Mike
So, this is negative G. [g. down, makes a negative sign in front of the g; w. arrow up
labeled "+" near the top of the diagram] Positive direction. You have negative G plus F of V
over M [gestures down and then up]. That's you're acceleration at any time. Right?
Karl
Well, wait. You want them to have,,
Mike
This is the acceleration at any time T. [w. a(t) next to the expression]
In this last passage, Mike and Karl’s conceptualization of the situation in terms of two competing
influences takes shape as an equation. The construction involves a few forms:
OPPOSITION,
COMPETING TERMS,
and IDENTITY.
COMPETING TERMS
±
OPPOSITION
–
x = […]
IDENTITY
We encountered the
IDENTITY
±…
form in the previous episode. COMPETING TERMS and OPPOSITION
are new, so I will explain them briefly:
COMPETING TERM s. Each of the competing influences corresponds to a term in the equation. As
I will discuss later, this is one of the most common ways that physicists see equations, as terms
competing to have their way.
OPPOSITIO n. The two competing influences are competing in a specific manner, they are
opposing. Opposition is frequently associated with the use of a negative sign. “Now they’re
opposing so it’s minus.”
A Fundamental Contrast: Forms versus Principle-Based Schemata
The basic presumption underlying my account of Mike and Karl’s above work is that they
conceptualized the situation as involving competing influences and these competing influences get
associated with terms in equations. But notice that there are aspects of the equation that this does
not explain. In particular, I have not said anything about how they knew to write f(v)/m for the
term corresponding to air resistance. Likely this has to do with the equation F=ma, an equation
that Mike and Karl certainly remember. They stated that the force of air resistance depends on
velocity, which led them to write this force as f(v). Then, since they were talking about opposing
accelerations, they used F=ma to write the acceleration as a=F/m.
So, at the least, forms cannot be the whole story in this construction episode—and they will
not be the whole story in most construction episodes. More formal considerations and
remembered equations must also play a role. In my rendition of the above episode, a remembered
65
equation was used to fill in one of the slots in a symbol pattern. However, the above episode is
open to some interpretations that are significantly different than the one I provided. Most
notably, one might argue that Mike and Karl have memorized the equation:
∑F=ma
Here, the ‘∑’ symbol means ‘summation.’ This equation says that the sum of all the forces on an
object is equal to the mass of the object multiplied by its acceleration. Thus, it may be possible to
see Mike and Karl’s work as solely a straightforward application of this equation; they write the
sum of the forces and then divide by the mass to get the acceleration.
This alternative viewpoint is an important enough (and strong enough) alternative that I want
to devote significant effort to discussing it and contrasting it with the view I am developing in this
thesis. In addition, working through this alternative will help to clarify what is particular about my
own viewpoint. But, I will not try to argue that this alternative view is incorrect, or that it is even
strictly competitive with the one I am developing in this thesis. In fact, I believe that a more
complete description of the knowledge involved in expert symbol use would include knowledge
of multiple sorts, including knowledge more closely tied to formally recognized principles, as well
as symbolic forms. However, I will try to argue that some student behaviors are better explained
by symbolic forms and that symbolic forms—and the related behavior—persist into expertise,
complementing more formal aspects of knowledge.
A version of this alternative view was worked out by Larkin (1983). Larkin begins by
distinguishing what she calls the “naive” and “physical” representations. (Here, when Larkin says
“representation” she is talking about internal mental representations, not external representations.)
In Larkin’s scheme, everyone—novice AND expert—constructs a naive representation as a first
step toward analyzing a physical situation. A naive representation involves only “familiar” entities
and objects; it essentially uses our everyday vocabulary for describing the world. In contrast, the
physical representation, which is only constructed by physicists, “contains fictitious entities such as
forces and energies.” Once the physical representation is constructed, the writing of an equation is
apparently relatively easy. Larkin tells us that: “Qualitative relations between these entities can be
‘seen’ directly in this physical representation, and these qualitative relations then guide the
application of quantitative principles.”
To this point, Larkin’s view may be said to share many broad similarities with my own. There
is a qualitative mental representation that guides the construction of an equation. But Larkin soon
66
takes a step that strongly differentiates her view. She says: “The physical representation of a
problem is closely tied to the instantiation of the quantitative physics principles describing the
problem.” In contrast, in my own view, forms are not closely tied to specific physical principles,
such as Newton’s Laws.
Let’s take a moment to see how Larkin’s model works. According to Larkin, experts possess a
number of schemata that are directly associated with physical principles. These schemata contain
what she calls “construction rules” and “extension rules.” The construction rules act on the naive
representation to produce the physical representation. And the extension rules act on an existing
physical representation to add to that representation.
Larkin provides two examples of these schemata, the “Forces Schema” and the “Work-Energy
Schema.” These schemata are quite closely related to physical principles, as these principles would
be presented in a physics textbook. For example, Larkin says that the Forces Schema “corresponds
to the physical principle that the total force on a system (along a particular direction) is equal to
the system’s mass times its acceleration (along that direction).” This is essentially Newton’s
Second Law, F=ma. The construction rules in this schema correspond to “force laws,” which are
the laws that allow a physicist to compute the forces on an object given the arrangement of objects
in a physical system. So the idea is this: A physicist looks at a physical system and sees the
arrangement of objects involved. The Forces Schema then provides force laws that allows the
physicist to determine the forces (a particular kind of fictitious entity) acting on the objects. Then
the physicist plugs these forces into the equation ∑F=ma and solves for whatever is desired. Note
that this equation, ∑F=ma, is a template of a sort, and it allows the construction of equations that
are not literally identical to those seen before. But, it is has particular limits on its range of
applicability: the entities in the sum must be forces and they must be set equal to ma.
Of course, as I have already suggested, it may be possible to apply Larkin’s account to describe
Mike and Karl’s work on the air resistance task. This would go as follows: Using the construction
rules provided by the activated Forces Schema, Mike and Karl write the force on each object; -mg
is the force of gravity, which is negative because it acts in the downward direction, and the force
of air resistance is simply taken to be f(v). These forces are then summed and set equal to ma:
ma = −mg + f (v )
(Presumable, Mike and Karl do this step mentally, since it doesn’t appear as one of the equations
they write.) Finally, this equation is divided by the mass, m, to produce an equation for a.
67
I must emphasize that one crucial difference between Larkin’s view and my own is the fact
that her schemata are closely tied to physical principles. (This is in addition to some more global
differences in orientation.) For her, it is crucial that the influences in competition are forces, since
Newton’s laws only talk about forces. And the above equation is only written because the schema
tells us to—it’s a simple matter of “plugging-in.” In contrast, in my view, the situation is
generically schematized as competing influences, a notion that is supposed to cut across physical
laws. Many types of entities, other than forces, may compete.
For this reason, I will refer to Larkin’s account as the “principle-based schemata” view. The
modifier “principle-based” is important here since, as we have seen, a type of schema also appears
in my own model. However, in this section, I will at times simply refer to Larkin’s view as “the
schema view.”
Now that I have presented the principle-based schemata view and contrasted it with my own
view (the “forms” view), I want to take steps toward arguing for my own account of some
episodes. In the remainder of this section I will discuss a few episodes that are specifically selected
to contrast these two views. In addition, in later sections and later chapters I will argue for the
usefulness of my viewpoint on other grounds.
Step 1: Students do, at least sometimes, see symbolic forms
The first step I want to take is to point out that I have already presented enough evidence to
show that the schema view cannot be the whole story. In particular, there is no way for Larkin’s
schema view to explain Mike and Karl’s ability to construct their own equation for the coefficient
of friction.
µ
µ = µ 1 + C m2
This is because there are simply no principles or existing equations to guide this construction.
Furthermore, once we have accepted that Mike and Karl have abilities that transcend what is
allowed by the principle-based schemata model, we have to live with this observation in all
accounts that we give of all types of problem solving. Both of the above two episodes involved the
same pair of students, Mike and Karl. And, just because they were working on a more familiar
task, there is no reason to believe that their form-related capabilities will simply go away or not be
employed. Suppose, for a moment, that something like Larkin’s Forces Schema is responsible for
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Mike and Karl’s work on the air resistance task. Even if this is the case, because Mike and Karl have
the ability to see meaningful patterns—forms—they may, at any time, be seeing these patterns in
equations that they write.
I want to illustrate this point with an analogy. Suppose that we decided to teach some
mathematicians a little physics, so we teach them to write equations and solve problems.
Mathematicians already know a lot about equations. They know a linear equation when they see
one, and they can differentiate a linear equation from a quadratic equation. Thus, even if they
learn to write the expression v = vo + at from memory, they may still always see this equation as a
linear relation between the velocity and the time. The equation has a structure that a
mathematician can recognize.
I am trying to make an analogous point about forms. The coefficient of friction episode tells
us that Mike and Karl can sometimes compose expressions from symbolic forms. They may thus
see these forms in any equation that they write. If this is true, then the principle-based schemata
view cannot be “the whole story.”
Step 2: Deviations from the ÒschemaÓ model in the air resistance example
Mike and Karl’s work on the air resistance task warrants a closer look. Recall that they
bounced between describing the situation in terms of competing accelerations and in terms of
competing forces. This must, to some degree, be because Mike and Karl are aware that force and
acceleration are formally interchangeable via F=ma. However, I also take this as evidence that a
view of the situation in terms of opposing influences is the core driving element of these students’
work; thus, the specific nature of these influences is not of utmost importance to them at all stages
of the solution attempt. This observation really cuts to the heart of the difference between the
forms model and the principle-based schemata model. I am arguing that Mike and Karl’s work is
driven—at least, in part—by an understanding tied to a notion of opposing influences that cuts
across principles, not by a schema tied solely to forces and Newton’s laws.
For those who are less familiar with traditional physics practice, it is worth noting that Karl’s
use of the terms “upward” and “downward acceleration” would probably be considered
inappropriate by many physicists. To a physicist, the term “acceleration” is usually reserved for
describing actual changes in the speed of objects, not for virtual changes that may be canceled out
by other virtual changes. Thus, Mike and Karl’s work departs somewhat from a strict and careful
69
application of traditional methods that follow a universal schema associated with physical
principles.
Finally, notice that, as they work, Mike and Karl have interpretations readily available. For
example, Mike says that one acceleration is “opposing” the other. The fact that such interpretive
comments are ready-to-hand suggests that these notions are playing a role in these students’ work.
Step 3: Students do not rigorously deduce from principles
As the final step in my argument against the schema view, I want to present a battery of
evidence designed to further refute the notion that students (and experts) work by slavishly
following standard techniques and rigorously deducing from principles, even when these
techniques are available and appropriate. The purpose of this data is not only to argue against
certain aspects of the principle-based schemata view, but also to provide counter-evidence for a
broad range of opposing views and to give a feel for the character of student work. I will begin
with one more example concerning Mike and Karl; here, I look at their work on the “spring” task,
in which a mass hangs motionless at the end of a spring. In Chapter 2, I briefly examined Jack and
Jim’s work on this task.
The incident I want to recount actually took place during Mike and Karl’s final session, which
was in part used to wrap-up some issues and debrief the subjects. During this session, Mike and
Karl had returned to the spring task, and they had written the equation:
Fs = Fg
This equation says that, when the mass hangs motionless, the force upward due to the spring must
equal the force of gravity pulling down. Since this was a wrap-up session, I probed Mike and Karl
to see how, and in what way, they related this equation to known principles. I began by asking if
they could somehow include the equation F=ma in their solution. What I had in mind was that
they would say that, because they know that the sum of the forces is equal to ma, and the mass
isn’t accelerating (a = 0), the sum of the forces must be zero: Fs + Fg = 0 . Furthermore, if we are
willing to adjust some sign conventions, then this expression can be rewritten as Fs = Fg .
Immediately following my initial probe, Mike and Karl complained that it simply did not
make sense to employ F=ma in the solution of this problem:
Mike
It's not very natural.
Karl
We're looking at a static system.
70
So, I tried another approach. I pressed them as to how they knew that the forces of the spring and
gravity must be equal.
Mike
How do I know F S equals F G? Because it's in equilibrium.
Karl
It's defined that way. [They both laugh]
Mike
Because it's in equilibrium, right, so the forces they have to, um, add up to zero.
Bruce
Why do the forces have to add up to zero?
Mike
To be in equilibrium. [laughs] We're going in circles aren't we. If it's in equilibrium there's no
acceleration, is that what you're looking for?
In the above passage, Mike and Karl first seem to assert that it is essentially obvious that these
forces must be equal. But then, at the end of the passage, it seems like Mike has latched onto what
I had in mind, he says: “If it's in equilibrium there's no acceleration, is that what you're looking
for?” Strikingly, Mike’s further work made it clear that he was still not on my wavelength. He
went on to assert that the acceleration due to each force must be equal and wrote
as = ag
Apparently, he had in mind using F=ma simply to find an acceleration associated with each of the
forces, not to establish the dynamical truth of their equality.
Mike
There’s acceleration but they cancel, right? I mean there is - when there’s force on it and
there’s no - the result - there’s no resulting acceleration. There’s no net acceleration,
maybe that’s a better way to put it. (3.0) Cause one is accelerating one way [g. up] and the
other force is accelerating downward.
So, it seems that I had still not succeeded in getting Mike and Karl to state the argument that
I had in mind. Nonetheless, to this point in the episode, it is still possible that Mike and Karl were
well aware of how their equation followed form F=ma, but that they simply did not know what I
was looking for. Perhaps, for example, the explanation from F=ma is simply too simple to be
worth mentioning. To explore this possibility, I finally just told them what I had in mind, giving
the argument I presented above. Mike and Karl’s reaction to this made it clear that this was an
entirely novel idea to them. Karl was especially astounded and pleased by my little explanation of
how Fs = Fg followed from F=ma and a = 0. In fact, as the session continued and we moved on
to other issues, he occasionally returned to this issue to comment on how much he liked my
explanation:
Karl
I like - I like - this made sense to me because this took away the “well, this is true just
because.”
Bruce
Right.
Karl
And we get a lot of “just becauses.”
Mike
Uh-huh.
71
Karl
This explained why.
What I am hoping that this episode makes very clear is that fundamental principles like F=ma
are not always lurking secretly behind student’s work, simply unsaid. It is true that this episode
cannot quite confirm the kind of generality I have asserted for forms; for instance, it is possible
that Karl has something like a Forces Schema that is specialized for static situations. But this
episode can at least eliminate the most economical version of the principle-based schemata model,
in which all work flows from schemata associated with the most basic fundamental principles.
Of course, even if we accept that Mike and Karl’s work is not driven by principle-based
schemata, one could point out that these students are simply not experts and maintain that, in
expertise, behavior tends toward the description given by the principle-based-schemata view. It is
very likely that experts and more advanced students would be able to quickly produce the
argument that the equality of the two forces in the spring task follows from F=ma. Nonetheless,
we must keep in mind that these students are at least at an intermediate level of expertise. They
have completed two full semesters of physics and they can very reliably solve problems like the
spring task. The fact that this sort of behavior continues into a stage in which there is practiced
and reliable performance suggests that the associated knowledge may linger into expertise.
Adding to the plausibility of this statement is the observation that symbolic forms constitute a
useful type of knowledge. This knowledge is necessary, for example, in the construction of
completely novel expressions, such as Mike and Karl’s expression for µ. Because there is an
important niche for this knowledge in expertise, it is thus more plausible that it will continue to
play a role as students progress toward more expert behavior.
For the sake of completeness, I should mention that I would explain Mike and Karl’s
construction of the relation Fs = Fg by appeal to what I call the BALANCING form.
BALANCING
=
In BALANCING , two competing influences are seen as precisely equal and opposite. Note, once again,
the central point here: The BALANCING form is more general than any assertions about balanced
forces. Any types of influences can be balanced and, once they are seen as balanced, a relation in
line with the BALANCING symbol pattern can be written.
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Some additional examples
Mike and Karl were quite typical of my subjects; as a whole, the students almost never worked
by a strict following of traditional techniques that follow from principles. Furthermore, when
students did appear to work from principles—or, at least, maintained that they were working
from principles—these applications of principles often seemed to play the role of an after-the-fact
rationalization for work that was already done. And even these rationalizations often involved
some hand-waving. Let’s look briefly at a few examples.
First, I return to Jack and Jim’s work on the spring task. Recall that they began by writing the
force due to the spring as F=-kx and the force due to gravity as F=ma. Then, in a passage that I
did not cite earlier, they proceeded to equate these two forces, writing kx=mg:
Jack
So, okay, so then these two have to be in equilibrium those are the only forces acting on
it. So then somehow I guess, um, (3.0) That negative sign is somewhat arbitrary [g. F=-kx]
depending on how you assign it. Of course, in this case, acceleration is gravity [g. a in
F=ma] Which would be negative so we really don't have to worry about that [g.~ F=-kx] So I
guess we end up with K X is M G. [w. kx=mg]
The delicate issue that Jack is trying to deal with is, if these forces are going to be equated, then
what happens to the negative sign in F=-kx? Jack knows that he should somehow end up with
kx=mg, but he makes a pretense of being careful; he says that the acceleration due to gravity is also
negative so the negative signs cancel. This is not quite up to the standards of a rigorous argument;
such an argument would require that Jack carefully associate the signs of terms and directions on
the diagram. Jack does not need to do this however, because he knows the answer before he starts.
This sort of hand-waving concerning signs is extremely common in my data corpus. If anything, it
is the norm.
Finally, I want to talk about some student work on a task I have yet to mention, what I call the
“Stranded Skater” problem.
Peggy Fleming (a one-time famous figure skater) is stuck on a patch of frictionless ice. Cleverly, she
takes off one of her ice skates and throws it as hard as she can.
(a) Roughly, how far does she travel?
(b) Roughly, how fast does she travel?
What happens in this situation is that, when Peggy throws the skate, she moves off in the direction
opposite to the direction in which the skate was thrown. I am going to talk about the work of two
pairs on this problem, Jack and Jim and Alan and Bob. Jack and Jim began by announcing that
“it's basically a conservation of momentum problem.” Then they immediately went on to write
the expression
73
m pf v1 = ms v2
where m pf is the mass of Peggy Fleming and ms is the mass of the skate. Here, Jack and Jim are
implicitly using the relation P=mv, where P stands for the momentum of an object. The question
is, how did Jack and Jim know to simply equate the momenta of the skater and the skate? I
believe that they see this problem in terms of BALANCING . After the skate is thrown, the movement
of the skater and skate must somehow balance out.
Figure Chapter 3. -3. Alan and Bob’s diagram for the Stranded Skater problem.
After they had written this relation, Jack moved to rationalize it in terms of the principle of
conservation of momentum. He wrote the following expression and said:
m pf v1 + ms v2 = 0
Jack
Actually, it should probably be … [w. MpfV+MsV=0] This case we'll assume that this
velocity would be a negative velocity [g. above v of skate] so that we can go to that. [g.
first equation] Cause one's gonna go one way, one's gonna go the other.
So, in this episode, Jack began by writing a relation that he believed to be true and then moved to,
after the fact, rationalize that relation in terms of known principles. The principle applied here is
the conservation of momentum, which says that the initial and final momenta of a system must be
equal in the absence of any externally applied forces. In this case the initial momentum is zero,
since everything in the system is at rest before the throwing of the skate. Thus, an expression for
the total final momentum can be correctly equated to zero. This is what Jack does in the above
equation.
I want to note that it was common for students to see the Stranded Skater problem in terms
of BALANCING as well as in terms of a related form, what I call CANCELING (0 =
-
Ê
). Alan and
Bob began their work on this problem by writing the expression:
0 = ms vs − m pv p
Alan
And we expect her to move to the left to compensate (her) mass of Peggy times the
velocity of Peggy. [w.mod 0=MsVs–MpVp]
74
Here, Alan argues that Peggy must move in the direction opposite that of the skate in order to
“compensate” for the motion of the skate. The important point here is that the use of a negative
sign between these terms is not derived from a principle or from mechanically following
techniques. Rather, it is introduced based on an understanding of what is going to happen and
what the expression should look like. In my view, Alan sees “competing influences” that cancel
and he makes sure that his expression is in line with this symbolic form.
Final comments on principle-based schemata
In this section I have contrasted my model with a model that is truly different, what I have
called “the principle-based schemata” view. This contrast is extremely important because it gets to
the heart of some of the essential features of my model. Allow me to elaborate briefly. There is a
conventionalized version of physics knowledge that is presented in physics textbooks. But it is an
error simply to presume that, when people learn physics, the knowledge they acquire somehow has
the same structure as the conventionalized knowledge presented in a textbook.8 Physics
knowledge, as realized in individuals, may mirror textbook physics, but it may not; the question
must be settled empirically.
I have attempted to argue that at least one slice of physics knowledge does not have the same
structure as textbook physics. In particular, the knowledge constituted by forms is not organized
like a textbook; it is not tightly related to basic physical principles. Instead, notions like BALANCING
are central. And these notions cut across physical principles—you can have balanced forces OR
balanced momenta.
However, principle-based knowledge must have been playing some role in the episodes I
presented above. For example, even if we accept that some students understand the Stranded
Skater problem in terms of BALANCING , we still have to explain how they knew to balance momenta,
in particular. I believe that this depends, in part, on students ability to recognize problem types
and to know what principles apply to those problem types.9
This observation implies that the knowledge employed in symbol use, in its full and finished
expert form, must certainly have components that are tied to physics principles. In fact, I believe
that there is likely to be knowledge that is somewhat like principle-based schemata. This returns
me again to the point that a full model of symbol use in physics will be quite complicated, more
8
9
Andrea A. diSessa has made this point in a number of articles. See, for example, diSessa, 1993.
See, for example, Chi et al., 1981.
75
complicated than I can deal with here. So, I do not mean to discard the principle-bases schemata
view, but I do hope that I have shown that there are cracks in this model—some pretty big cracks,
in fact—that make room for other important classes of knowledge, and that greatly broaden the
range of student activity that we can expect and explain.
To sum up this section, we have seen that, first, there are episodes that the principle-based
schemata model simply cannot handle. These episodes constitute evidence for the kind of
knowledge that I am calling “symbolic forms.” Furthermore, I have argued that, because this
knowledge exists, it is plausible to presume that it gets employed during more traditional types of
symbol use. In addition, I presented a number of examples to illustrate that students do not
always work by rigorous deduction from principles; instead, they wave their hands and rationalize
to get the answer that they already know they want. This constitutes evidence that forms play a
role in what, at first glance we might take to be the strict following of principle-based tradition.
Forms: A Primer
In this section I am going to discuss all the forms that are necessary to account for the behavior
of students in my data corpus. Because this knowledge system is not organized according to the
principles of physics—and because it apparently has no other simple structure—there are no
shortcuts for describing the breadth of forms, I have to describe each type of form and give
examples. However, for the reader that forges ahead, this section is not without its rewards. There
are many interesting details associated with particular forms, as well as important and subtle
points to be made along the way. In addition, the broad scope provided by this overview is
important. The hope is that it paints a picture of a physicist’s phenomenal world of symbols—how
physicists and physics students experience a page of expressions.
Keep in mind that this list is intended only to account for the particular data in my algebra
corpus. If I had given students a different set of tasks, looked at a different population, and
certainly if I was concerned with a different domain, the symbolic forms in this primer would be
different. Nonetheless, I believe that this list captures much of the breadth as well as the most
common and central forms of introductory algebra-physics. Adding tasks would likely discover
more, but my belief is that these additions would primarily allow us to fill in some less common
forms around the fringes of the form system. I hope to support this viewpoint when I present the
systematic analysis in Chapter 6.
76
In saying that this list of forms “accounts” for my data corpus, I do not mean to imply that
the relation between this list and my data is straightforward. In Chapter 6, I will describe the
procedure whereby I make contact with my data. However, I must warn the reader that, even
there, I will not be presenting an exact procedure for extracting this list of forms from my
transcripts. Instead, the procedure I will describe is highly heuristic, and the resulting categories
depend heavily on theoretical arguments, as well as the data corpus itself.
Although there is no very simple structure to the forms knowledge system, it is possible to
organize forms according to what I call “clusters.” 10 Within a given cluster, the various schemata
tend to have entities of the same or similar ontological type. For example, forms in the
Competing Terms Cluster are primarily concerned with influences. In addition, the forms in a
cluster tend to parse equations at the same level of detail. For example, some forms tend to deal
with equations at the level of terms, while others tend to involve individual symbols. A complete
list of the forms to be discussed here, organized by cluster, is provided in Figure Chapter 3. -4. In
addition, a list of these forms with brief descriptions is given in Appendix D.
10
In using this term I am following diSessa, 1993. I will make close contact with diSessa and his use of
this term in Chapter 5.
77
Competing Terms Cluster
Terms are Amounts Cluster
PARTS -OF -A-WHOLE
[
+
–
BASE ± CHANGE
[
± ∆]
BALANCING
=
WHOLE - PART
[
-
CANCELING
0=
COMPETING TERMS
±
OPPOSITION
± …
SAME AMOUNT
–
Dependence Cluster
DEPENDENCE […x…]
NO DEPENDENCE
SOLE DEPENDENCE
…]
]
=
Coefficient Cluster
COEFFICIENT
[x
]
SCALING
[n
]
[…]
[…x…]
Other
Multiplication Cluster
INTENSIVE •EXTENSIVE x × y
EXTENSIVE •EXTENSIVE
+
IDENTITY
DYING AWAY
x=…
[e− x…]
x×y
Proportionality Cluster
PROP+
…x… 
 … 
PROP-
 … 
…x… 
x
y
 
CANCELING(B) …x… 
…x… 
RATIO
Figure Chapter 3. -4. Symbolic forms by cluster.
The Competing Terms Cluster
To a physicist, a sheet of equations is, perhaps first and foremost, an arrangement of terms
that conflict and support, that oppose and balance. The Competing Terms Cluster contains the
forms related to seeing equations in this manner, as terms associated with influences in
competition. Frequently—though not always—these influences are forces, in the technical sense.
The forms in the Competing Terms Cluster are listed in Table Chapter 3. -1.
COMPETING TERMS
±
OPPOSITION
–
BALANCING
=
CANCELING
0=
± …
–
Table Chapter 3. -1. Forms in the Competing Terms Cluster.
78
We have already encountered a number of forms from this cluster. During Mike and Karl’s
work on the air resistance task, they composed the equation:
a(t ) = −g +
f (v)
m
I argued that their construction of this equation involved the COMPETING TERMS and OPPOSITION
forms. Mike and Karl schematized the situation as involving two opposing influences, sometimes
understood as forces and sometimes as accelerations. Recall that it was the
OPPOSITION
form that
was associated with Mike’s insertion of a minus sign: “now they're opposing so it's a minus.”
Many of the interesting details here, in fact, have to do with how students select signs (+ or –)
for the terms in their Competing Terms expressions. In cases in which these forms are in play,
students frequently draw diagrams and associate the signs of influences with directions in the
diagram. For example, a student may decide that influences acting to the right are positive, and
influences acting to the left are negative.
We also encountered some instances of BALANCING in previous episodes. I maintained that
BALANCING
was responsible for the equating of the forces in Mike and Karl’s work on the spring
task; the force up due to the spring and the force down due to gravity were seen as “in balance.” In
addition, I argued that Jack and Jim were BALANCING the momenta of the skater and the skate in
the Stranded Skater problem when they wrote:
m pf v1 = ms v2
The last form in this cluster, CANCELING, was encountered briefly in Alan and Bob’s work on
the skater problem:
0 = ms vs − m pv p
Alan
And we expect her to move to the left to compensate (her) mass of Peggy times the
velocity of Peggy. [w.mod 0=MsVs–MpVp]
The CANCELING and BALANCING forms are quite closely related. Both of these forms involve equal and
opposite influences. However, they do differ in some important respects. First, with respect to their
symbol patterns, CANCELING is usually written with terms separated by a minus sign and set equal to
zero, rather than as two terms separated by an equal sign. In addition, the associated schemata differ
in a subtle manner. The schema associated with the BALANCING form involves only two entities, the
two influences that are in balance. In contrast, CANCELING can involve three entities; two entities are
79
combined to produce a net result that happens to be zero. Furthermore, the relations between the
entities in CANCELING frequently include a weak type of ordering of the two competing influences.
Sometimes, for example, one of the influences is seen as a response to the other influence. This type
of relationship is evident in Alan’s statement just above. It is as if Peggy’s motion is a response to the
throwing of the skate; she throws the skate and then moves “to compensate.”
BALANCING is an extremely common and important form, and thus merits a little further
discussion here. BALANCING is important because it provides the central dynamical constraint of
many problem solving episodes. For example, in Mike and Karl’s work on the spring problem
described above, it was BALANCING that was really answering the “why” question behind all their
work. Why do we write Fs = Fg ? Because the forces must balance.
The next couple of examples involve student work on a task I call the “Buoyant Cube”
problem. In this problem, as shown in Figure Chapter 3. -5, an ice cube floats at the surface of a
glass of water, and the students need to find how much of the ice cube is below the surface.
An ice cube, with edges of length L, is placed in a large container of water.
(a) How far below the surface does the cube sink? (ρice = .92 g/cm3; ρwater = 1 g/cm3)
L
L
Figure Chapter 3. -5. The Buoyant Cube Problem.
In brief, this problem is typically solved as follows (refer to Figure Chapter 3. -6). Two forces
act on the floating cube, the force downward from gravity, Fg , and a buoyant force upward from
the water, Fb . In order for the cube to be motionless, these two forces must be equal.
Furthermore, the force of gravity is equal to the weight of the cube, which is the product of the
density of ice, the volume of the cube, and the acceleration due to gravity: ρi gVcube . And the
buoyant force is equal to the weight of the water that is displaced by the cube: ρw gV water . If the
amount that the cube sinks below the surface is given the label s, then the volume of water
displaced is equal to L2s. With these relations in place, it is possible to solve for the value of s as
shown in Figure Chapter 3. -6.
80
Fb = ρ wgV water
Fb = Fg
ρ wgV water = ρ i gVcube
( )
( )
ρ wg L2s = ρ i g L3
ρ
s= i L
ρw
s =.92L
S
Fg = ρ igV cube
Figure Chapter 3. -6. A solution to the Buoyant Cube problem.
For almost all students, this problem strongly suggests BALANCING . Frequently they even oblige
us by saying the word “balance:”
Jack
Um, so we know the mass going - the force down is M G and that has to be balanced by the
force of the wa=
Jim
=the displaced water.
What is truly striking, is that students seem to really possess the sense for
BALANCING
at quite a
fundamental level. In Mark and Roger’s work on this task, Roger carried most of the load, with
Mark expressing confusion along the way. After Roger produced a solution, Mark tried to
rationalize Roger’s work for himself. In particular, Mark was a little mystified by Roger’s first
expression, in which he had equated the weight of the ice cube and the weight of the displaced
water.
W w = Wi
Mark ultimately satisfied himself that this equation makes sense with this comment:
Mark
We know one is gonna float. And then just sort of, I guess, I don't know from like all our
physics things with proportionalities and everything, that we'll have to have something
equal, like the buoyancy force should be equal to the weight of the block of ice. [g. pulling
down block]
Here, Mark does not appeal to forces or anything like the notion that the net force must be zero.
Rather he simply states that, if the block is going to float, then something has got to balance: “we’ll
have to have something equal.”
Instead of being an appeal to principle, Mark’s argument is an appeal to experience.
Furthermore, the experiences he appeals to are of a special sort: “…from like all our physics things
with proportionalities and everything, that we'll have to have something equal… .” Mark, like all
other physics students, has a wealth of experience with equations, writing expressions, setting
81
things equal, and manipulating to find results. From this symbolic body of experience, Mark has
developed the sense that setting things equal is a sensible, intrinsically meaningful action, and that
it is typical physics. I will return to this point in later chapters.
The final point I want to make about BALANCING is that its prevalence may actually extend to
overuse; students sometimes employ BALANCING in a manner that could be called inappropriate. To
illustrate, I will return to the Shoved Block problem, which I discussed briefly in Chapter 2. Recall
that this problem involves two objects that are shoved in such a way that they both start moving
with the same initial speed. The question is, given that one block is twice as heavy as the other,
which block travels farther?
As I mentioned earlier, the correct answer is that both blocks travel the same distance. The
derivation of this fact can be quite short. (Refer to Figure Chapter 3. -7). After the shoved block is
released the only force acting on it is the force of friction, which is F f = µmg . Since this is the
only force, it alone is responsible for any acceleration that the block undergoes as it slides along
the table. For this reason, this force can simply be substituted for F in the equation F=ma to give
the equation µmg = ma . Then the m that appears on each side of the equation can be canceled to
produce the final expression µg=a.
Direction of motion
F f = µN = µmg
Friction
∑ F = ma
F f = ma
µmg = ma
µg = a
Figure Chapter 3. -7. A solution to the Shoved Block problem.
The important thing to note about the final result is that this expression for acceleration does not
involve the mass. This means that it simply does not matter what the mass is, the shoved block
always travels the same distance.
We will have reason to return to this problem a number of times, but, for now, I just want to
consider a single interesting episode. Alan and Bob had produced a solution very similar to the
one presented in Figure Chapter 3. -7. In brief, their work looked just right. However, after they
were done, Alan pointed to the equation µmg=ma and added this little addendum to their
explanation:
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
82
I believe that this interpretation is clearly inappropriate. When Alan looked at the equation
µmg=ma, he saw two influences in balance, the “force frictional” and the “force it received.” More
specifically, he sees this equation as specifying a condition for an end state, in which the block
comes to rest because these influences balance. However, as I have mentioned, once the block is
released, only one force acts on it, the force due to friction. In other words, only one side of this
equation corresponds to an influence! Thus, Alan is seeing BALANCING in an inappropriate context.
The Proportionality Cluster
When a physicist or physics student looks at an equation, the line that divides the top from the
bottom of a ratio—the numerator from the denominator—is a major landmark. Forms in the
Proportionality Cluster involve the seeing of individual symbols as either above or below this
important landmark. Earlier in this chapter I argued that one of these forms,
PROP-,
was involved
in Mike and Karl’s construction of their novel expression for the coefficient of friction:
µ
µ = µ1 + C 2
m
Karl
The coefficient of friction has two components. One that's a constant and one that varies
inversely as the weight.
The idea was that Karl’s statement specified that the second component of their expression should
be inversely proportional to the mass. Thus, the PROP- form was invoked and the ‘m’ symbol was
written in the denominator of the second term. Table Chapter 3. -2 contains a list of all the forms
in the Proportionality Cluster.
PROP+
PROP-
…x… 
 … 
 … 
…x… 
x
y
 
CANCELING(B) …x… 
…x… 
RATIO
Table Chapter 3. -2. Forms in the Proportionality Cluster
We also encountered some other clear examples of
PROP+
and PROP- in Chapter 2, though
they were not noted at the time. I have in mind the interpretation that Jim gave of the final result
of the spring problem:
83
x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then your position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k] And that if you have a stiffer spring,
then your position x is gonna decrease. [g. uses fingers to indicate the gap between the
mass and ceiling] That why it's in the denominator. So, the answer makes sense.
In the first part of this interpretation, Jim sees the expression in terms of the PROP+ form, x is seen
as directly proportional to the mass: “as you have a more massive block hanging from the spring,
then you're position x is gonna increase.” And, in the second part of the interpretation, the
expression is seen through the lens of PROP-: “if you have a stiffer spring, then you're position x is
gonna decrease.”
There were many instances in which students interpreted final expressions of this sort by
announcing proportionality relations. For example, when solving the air resistance problem,
students typically derived an expression very similar to
v=
mg
k
for the terminal velocity. In this relation, k is just a constant. To cite an instance, after Alan and
Bob derived this equation, Alan commented:
Alan
So now, since the mass is in the numerator, we can just say velocity terminal is
proportional to mass [w. Vt∝m] and that would explain why the steel ball reaches a faster
terminal velocity than the paper ball.
And, concerning the same relation, Mark said simply:
Mark
So the terminal velocity of the heavier object would be greater.
Now I will move on to a form we have not before encountered, the RATIO form. In the RATIO
form, the symbol pattern involves an individual symbol in both the numerator and denominator
of an expression. The key to the RATIO form is that the ratio is seen as a comparison—it’s a
comparison of the quantities that appear on the top and the bottom of the ratio. In most cases, the
quantities involved are of the same type; for example, they may be both masses or both velocities.
To see an example, I return to Alan and Bob’s work on the Stranded Skater problem. Recall
that they began their work on this problem by writing this expression.
0 = ms vs − m pv p
84
With this expression in hand, they proceeded to solve for v p , the velocity of Peggy:
vp =
ms
vs
mp
Then Alan commented:
Alan
Well the mass of the skate [g. Ms] is approximately small - it’s smaller. It’s probably much
smaller than mass of Peggy. [w. Ms<< Mp] …So your velocity is gonna be really small. [w.
Vp<<Vs]
The cueing to activity of the RATIO form almost always leads to a certain question being asked:
Which is bigger, the numerator or the denominator? This question can be seen at the heart of
Alan’s comment. He notes that, not only is the mass of the skate smaller than Peggy’s mass, it is
probably much smaller. Alan’s concern with the relative size of numerator and denominator
suggests that he is seeing the RATIO form in this expression; in particular the ratio of the masses,
ms m p , is seen in terms of this form.
The RATIO form frequently appeared in a very similar context in the Buoyant Cube problem.
Student work on this problem usually produced an expression similar to the solution presented in
Figure Chapter 3. -6.
s=
ρi
L
ρw
This expression tended to encourage interpretations based on a comparison of the densities of ice
and water. The following passages are typical:
Jack
This does seem to make sense cause if ice had the same volume as water [g. ρi then ρw],
er, same density as water, then L would equal S which means the cube would be right at
the surface.
_______________
Mike
Does it seem sensible? Well, let's see. Um. If ice is lighter than water, [g. ρ(I) then ρ(w) in
last] then ice will float.
Karl
It makes sense because this is gonna be less // than one. [g. ρ(I)/ρ(w) in last]
// Less than one.
Mike
Karl
And we know that it's gonna float.
Mike
Right. So, yeah, that makes sense.
Karl
That makes sense.
The final form in the Proportionality Cluster is the CANCELING(B) form. This form bears the
name CANCELING(B) to distinguish it from the CANCELING form in the Competing Terms Cluster.
85
The reason for retaining the term “canceling” in both names is that students very commonly use
the word “cancel” in association with both of these forms.
Since, as we shall see in Chapter 6, I observed this form only rarely, I will present just a single
brief incident for illustration. This incident took place during an episode in which Mark and
Roger were discussing why all dropped objects fall at the same rate (neglecting air resistance). To
answer, Mark wrote the following expressions and commented:
a=
Mark
F
; F = mg
m
You know, however you have a force [g. F in a=F/m] that varies at the same proportion
directly - directly with the mass, don't you? The heavier - the heavier an object is, the more
force it will exert, right? Upon, like say I try to lift it. … So, if it's double the mass it's gonna
have double the force downward.…Two M G. Right? But the mass is also two M , so those
two M's cancel and you get - you're left with A equals G in all cases, aren't you.
The idea here is that, in a sense, Mark has mentally combined these two expressions to obtain
a=mg/m. This is Mark’s expression for the acceleration of a dropped object. With this expression
in mind, Mark imagines what would happen to the acceleration if the mass is doubled. The answer
is that nothing would happen, any change in the numerator is canceled by a corresponding change
in the denominator. This canceling of parallel changes in the numerator and denominator of a
ratio is the core of the CANCELING(B) form. Here, it constitutes the basis of Mark’s explanation
concerning why all objects fall at the same rate.
Forms and the limits of qualitative reasoning
This discussion of the Proportionality Cluster provides an appropriate context to consider an
important issue, what I will call the “qualitative limits” of forms. For some first examples, I want
to look to the air resistance problem. As I have discussed, two forces act on the dropped object in
the air resistance task, gravity and a force from air resistance. All students know an expression—a
force law—that specifies the force of gravity, Fg = mg. However most students do not know an
expression for the force due to air resistance. While some may have seen such an expression once
or twice, it is rare for students to have committed an equation to memory. For this reason, the
students have no option but to construct their own expression for the force of air resistance.
The “right” answer—the answer you will find in an introductory textbook—is
Fair = kv 2
86
where v is the current velocity of the object and k is a constant.11 In general, the students in my
study just stated that the force of air resistance should be proportional to velocity, and wrote
either the above expression or Fair = kv . Here are some typical examples:
Fair = kv
Bob
Okay, and it gets f-, it gets greater as the velocity increases because it's hitting more
atoms (0.5) of air.
_______________
R = µv
Mark
So this has to depend on velocity. [g. R] That's all I'm saying. Your resistance - the
resistor force depends on the velocity of the object. The higher the velocity the bigger the
resistance.
The important issue here is that the PROP+ form doesn’t specify whether the force of air resistance
should be kv or kv 2 . Some students ran headlong into this difficulty, realizing that understanding
that the force of air resistance must be proportional to the velocity was not sufficient to determine
what relation they should write. Jack went so far as to list the possibilities. In the following, the ‘∝‘
symbol means “is proportional to.”
Fu ∝ v
∝ v2
∝ v3
Jack
Somehow they’re related to the velocity but we’re not sure what that relationship is. All we
can say is that F U is either proportional to V [w. F∝v] or to V squared or to V cubed. [w.
∝v2; ∝v3]
This type of ambiguity seems to be a fundamental property of forms; there are limits to the
specificity with which forms describe equations. And this is true both when reading and writing
equations. When we see an equation in terms of PROP+ and say “it’s proportional,” it is simply not
consequential whether the proportionality relation is linear or quadratic.
The students in my study came up against these limits in a number of cases, and not only in
the context of the air resistance task. For example, when working on the spring task, Mike and
Karl could not remember whether the force due to a spring is F = kx or F = 1 2 kx 2 . Although
they knew that the expression needed to be consistent with PROP+, this was not sufficient to
distinguish between these two equations:
11
A more complete expression could include other parameters, such as the cross-sectional area of the
dropped object. However, for the present task it is sufficient to presume that these other parameters
are part of the constant, k, and that this constant is the same for both objects.
87
F = kx ; F = 1 2 kx 2
Karl
Okay, now, qualitatively, both K X and half K X squared do come out to be the same
answer because as (…) looking at it qualitatively, both half - both half K X squared and K
X, um, you know, increase as X increases.
Furthermore, this was no fleeting difficulty. Mike and Karl struggled with this question for about
25 minutes, but to no avail. Up to the level of their specification, these equations were
indistinguishable. As Karl says, both “increase as X increases.”
In Karl’s above statement, he expresses the intuition that two expressions are “qualitatively”
equivalent. Of course, what it means for two expressions to be qualitatively the same is far from
clear. Interestingly, however, there is a relevant body of literature that attempts to make this
intuition precise. This literature describes what the researchers involved call “qualitative reasoning
about physical systems” (deKleer & Brown, 1984; Forbus, 1984).
The qualitative reasoning research is concerned with describing a certain slice of physics
knowledge and reasoning. This work falls into the “formalization” category I described in Chapter
1; these researchers have attempted to create a formalization of a particular slice of physics
knowledge. The interesting observation, from the point-of-view of the current discussion, is that
the knowledge system formalized by the qualitative reasoning researchers apparently possesses
similar limits to those described here. For this reason, it merits a brief discussion.
Paradoxically, the focus of qualitative reasoning is on what physicists would call physical
quantities. The literature that discusses qualitative reasoning makes specific hypotheses about how
people reason about physical quantities to describe and predict behavior. One hypothesis concerns
the nature of the information about a quantity that people encode. The range of possible values
that a quantity can attain is seen as divided into a set of discrete regions, with the boundaries at
“physically interesting” places. Then, the hypothesis is that, for qualitative reasoning, only the
following information about a quantity is relevant: (1) the region the quantity is in, and (2) its
direction of change.
In addition, later literature in this vein made hypotheses about how people propagate
knowledge about changes in one quantity to changes in another. Of particular interest for us are
proportionality relations, which Forbus (1984) calls “qualitative proportionalities.” For example, if
we know that two quantities are qualitatively proportional, and we know the direction of change
of one quantity is such that it is increasing, then we know that the second quantity is also
increasing. Notice that, just as in the case of forms, it is not appropriate to ask whether the relation
is linear, second order, or otherwise. So the knowledge system formalized by the qualitative
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reasoning literature appears to possess the same limits as forms. As with
PROP+,
a qualitative
proportionality relation between force and velocity says simply that force increases with increasing
velocity (or decreases with decreasing velocity), and that is the limit of the specification.
These “qualitative limits” constitute an essential part of the hypothesis associated with the
statement that people construct equations from and see equations in terms of symbolic forms.
Relations and symbolic expressions are only schematized at a certain level, and only some details
are relevant. Really, this should not be a surprising result. If every distinction was relevant—if we
didn’t carve up our experience and register it in some way—then our worlds would be an
overwhelming mass of details. Of course, with learning, we can learn to register more detail
productively; we can learn a larger vocabulary of forms. It is precisely such a developmental
process that Gombrich hypothesized to explain the evolution of art and individual artists. The
skilled physicist—as well as the skilled artist—learns an increasing vocabulary of forms in terms of
which to see the world, as well as for seeing the world as represented on paper.
The Terms are Amounts Cluster
Forms in the Proportionality Cluster, for the most part, address expressions at the level of
individual symbols, with special attention to whether these individual symbols are located above
or below the division line in a ratio. In this cluster, I am going to again be describing forms that,
like those in the Competing Terms Cluster, address expressions at the level of terms. However,
these forms differ in an important way from forms like competing terms and opposition. In
opposition and other forms in the Competing Terms Cluster, terms are seen as corresponding to
influences. In a very general sense, these influences are directed and they work to somehow move
an object toward their desired ends.
In contrast, in the Terms are Amounts Cluster, terms are not treated as influences, they are
treated more like a quantity of a generic substance. Rather than describing a battle between
competing terms, these expressions concern the collecting of this substance, putting some in and
maybe taking some away. Thus, while signs in the Competing Terms Cluster are commonly
associated with directions in physical space, signs in this cluster generally signal adding on or
taking away from the total amount of stuff. The forms in the Terms are Amounts Cluster are
listed in REF _Ref336930235 \* MERGEFORMAT Table Chapter 3. -3 .
89
parts-of-a-whole [ + + …]
base ± change [ ± _]
whole - part [ - ]
SAME AMOUNT
=
Table Chapter 3. -3. Forms in the “Terms are Amounts” Cluster.
The first form in this cluster, PARTS -OF -A-WHOLE, played a role in Mike and Karl’s construction
of their novel expression for the coefficient of friction. Their friction had two “components” that
together comprised the whole coefficient of friction. However, that episode does not really
provide the clearest and most illustrative example of that form; in particular, it is a little hard to
see the coefficient of friction as a “quantity of substance.” Indeed, this is probably something of a
marginal instance. For that reason, I want to present an episode in which a more typical instance of
the PARTS -OF -A-WHOLE form appears. This episode involves students engaged in solving a problem
we have not yet encountered, “Running in the Rain.”
Suppose that you need to cross the street during a steady downpour and you don’t have an umbrella. Is
it better to walk or run across the street? Make a simple computation, assuming that you’re shaped like
a tall rectangular crate. Also, you can assume that the rain is falling straight down. Would it affect your
result if the rain was falling at an angle?12
Students generally thought that this question was a little silly, but they were game to give it a
try. A diagram from Alan and Bob’s work on this task is shown in Figure Chapter 3. -8. Early on
in their work, Alan and Bob had a little argument concerning whether rain would hit only on the
top of the crate, or on the top and front side as you (the crate) moved across the street. Alan
started the discussion by taking the stance that rain would only strike the top of the crate:
Alan
It’s falling straight down. If the rain is falling straight down we're only concerned with how
much water is actually hitting this portion right here. [w. draws in top and darkens]
Figure Chapter 3. -8. Alan and Bob's diagram for the “Running in the Rain” problem.
12
This problem is taken from The Flying Circus of Physics (Walker, 1975).
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But Bob disagreed, and the discussion below ensued:
Bob
That's if you're standing still!
Alan
If we're running, and this box is move=
Bob
=But see there's rain=
Alan
You're still only looking at the rain that hits the top of the box. [g. top of box]
Bob
Your picture - your picture's not accurate. Okay. There's rain everywhere. [w. draws rain
in front of the box]
Alan
Oh, so you're talking about the rain that=
Bob
=Okay, you run this way [w. arrow indicating direction of motion] - well, yeah, it's sort of
bizarre. Okay, it is bizarre.
Alan
So, you're saying it's gonna hit the raindrops on the side, so we need to account for this,
[w. darkens the front area]=
At the end of the last passage, Alan agrees that rain would hit the front of the crate. With
consensus on this fact, Alan and Bob went on to construct an expression for the total rain that
strikes the person which consisted of two parts, one part corresponding to what strikes the top of
the crate and one part corresponding to what strikes the front.
total rain =
Bob
# raindrops
+C
s
There's two - two sources of rain. Okay, rain that you walk into and rain that drops on you.
[g. rain hitting his head] Okay. Walk or run into. Okay, so, the number of rain per second
[g. #raindrops/s] is how much rain hits your head. … if you're a crate and you - you move,
move into some rain. Since you move from here to here, [g. two points on the board],
there's - since there's rain coming down at a constant rate [g. falling rain], there's always
rain in front of you. Okay, so there's raindrops in front of you. So if you walk forward you
hit it, hit that rain. Okay. And-and- I mean, you're gonna hit some rain when you walk
forward.
Alan and Bob’s expression follows from
PARTS -OF -A-WHOLE;
the total amount of rain is seen as
consisting of two parts. The first term corresponds to the rain that strikes the top of the crate, and
the second term corresponds to the rain that strikes the front of the crate. In this case, thinking of
the total rain as a total amount of “stuff” is relatively natural. You can almost imagine a container
filling up with all this rain, flowing in through the two possible surfaces.
One interesting aspect of the above expression deserves a brief comment. Notice that this
expression is not quite a syntactically correct equation. In the place of the first term there are just
some units, the number of raindrops per second. Furthermore, these aren’t even the correct units
for this term; ultimately, Alan and Bob built an expression in which the terms all had units of
volume. Given the “forms” view, such partially syntactic expressions are to be expected. Since
forms like
PARTS -OF -A-WHOLE
only describe equations at a certain level of description, it makes
sense that expressions might appear in a schematic form during early stages of their construction. I
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believe that this is precisely what we are seeing in Alan and Bob’s expression. The units written in
the position of the first term can be taken as functioning as a placeholder for a more complete
expression to be written later. In fact, this is exactly how this expression functioned in Alan and
Bob’s work. Ultimately both the first and second term were elaborated in terms of some useful
parameters.
The next two forms in this cluster, BASE +CHANGE and BASE -CHANGE , are flip-sides of the same
coin. Both of these forms typically correspond to expressions involving two terms. The first term
is a “base” quantity, and the second term is an amount that is taken away or added to this base.
The BASE +CHANGE form may appear indistinguishable form
PARTS -OF -A-WHOLE,
but this is not the
case. In the case of PARTS -OF -A-WHOLE, the two “parts” are semantically indistinguishable; within
the schema, these two entities are of precisely the same sort. In contrast, in BASE +CHANGE there is a
semantic asymmetry between the entities corresponding to each of the terms; in particular, there
will be some sense in which one of these entities is a base to which something is added.
In fact, the key feature to look for in identifying BASE +CHANGE or BASE -CHANGE is the notion
that some quantity is made greater or made less. The typical case is one in which a student already
has an expression of some sort and they know, for example, that they want a new expression that
is larger than the first. So what do they do? They add something onto the first expression.
Let’s look at an example. Part (b) of the air resistance task asked students the following
question:
(b) Suppose that there was a wind blowing straight up when the objects were dropped, how would
your answer differ? What if the wind was blowing straight down?
Students approached this problem in a number of ways. Sometimes they performed a reference
frame analysis, and sometimes they simply treated the wind as an additional force. But I do not
want to get into these particular details at this point. I want, instead, to jump to the final result.
First, recall that the final expression for the first part of the air resistance task was:
vt =
mg
k
This equation expresses the terminal velocity as a function of the mass, the acceleration due to
gravity, and a constant, k. Most students eventually derived an equivalent expression, although
they used a variety of symbols to stand for the constant.
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In Mark and Roger’s solution to Part (b), they treated the wind as an additional force. When
they were done, they had two new expressions for the terminal velocity.
vt =
mg − Fw
mg + Fw
; vt =
µ
µ
Mark and Roger knew that one of these expressions corresponded to the case where the wind is
blowing up and the other corresponded to the case where the wind is blowing down, but the
question was, which equation goes with which case? In the next passage Roger worked out the
answer to this question:
Roger
Okay, well for the up case - yeah, terminal velocity will be (0.5) // less [g. -Fw in Vt=(mgFw)/µ] by a factor F W over mu. [w. Fw/µ] Right?
// less
Mark
Mark
Yeah. // Yeah, by - By a constant. By a constant factor.
// And for the down case - okay, because we're assuming a constant wind right,
Roger
force?
Bruce
Uh-huh.
Mark
By a constant [g. Fw in (mg-Fw)/µ] - it'll be less by that much. [w.O Fw/µ, adds a minus sign]
And minus. And then you have to-
Roger
This would be greater by F W over mu. [w. +Fw/µ in lower area]
Roger decided that the expression with the minus sign goes with the case in which the wind is
blowing up. His reasoning was that, if the wind is blowing up, then the terminal velocity should be
less. And, following
BASE -CHANGE ,
in order to make a quantity less, you have to subtract
something.
There is a minor subtlety here. The force of wind, Fw , could potentially be treated as an
influence and we could see the expression mg − Fw in terms of COMPETING TERMS. In that case, the
sign of Fw would be chosen based on some association of signs with directions. But this is not
what Roger does in the above passage. He does not argue that there needs to be a negative sign in
front of the Fw because of the direction that the wind points. Rather, he argues that there needs
to be a negative sign because the terminal velocity must be less than it was in the base case—the
case in which there was not any wind. This is evidence for BASE -CHANGE .
I want to mention one other place that BASE +CHANGE appears, mainly because it involves an
equation that is learned very early and used frequently: v = vo + at . At a number of points during
the problem solving sessions, I pressed students about this equation, asking them such things as
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how they knew it was true, and whether they knew it “by heart.” During the final session, I also
each asked pair whether they thought that this equation was “obvious.” Mike and Karl responded:
Mike
Well yeah it is obvious because, well velocity will equal to V naught if it's not being
disturbed. But if it's being acc- If there's a f- acceleration action on it, then uh- and that's
constant, you know then velocity will be decreasing as time goes on. Or increasing,
whatever it works, I mean whichever it does. So, it's like whatever it is and then plus a
correction on the acceleration. So yeah it makes sense. It's obvious, yes it is.
…
Karl
What's obvious to me is that you have the final velocity is obviously going to be equal to
the initial velocity plus however much, however faster it gets. That's what's obvious to me.
What's not necessarily positively obvious is that the amount, the amount that it gets
faster is the acceleration times the time.
Mike begins by answering that, yes, v = vo + at is obvious. And it is obvious for Mike because it is
consistent with the BASE +CHANGE form, there is a base, vo , plus some “correction.” Karl agrees that
the equation is obvious, but only in a limited sense. In particular, he’s not sure that it’s necessarily
obvious that the second term must be precisely “at”. He agrees that there needs to be an initial
velocity plus some change that depends on the acceleration, but he does not know exactly what
that change should look like.
Here, again, we have run into the limits of forms. Forms specify equations to be written, but
only up to a certain level of detail. The interpretation of v = vo + at given by Mike and Karl was
the one provided by almost all students; they typically saw this equation in terms of
BASE +CHANGE
and at no finer level of detail. In fact, it was not uncommon for students to write the second term
incorrectly. For example, rather than writing v = vo + at , Mark and Roger wrote:
v = vo + 12 at 2
Mark
Cause we have initial velocity [w. circles vo term] plus if you have an acceleration over a
certain time. [w. circles 1/2at2] Yeah, I think that's right.
Although Mark’s expression is consistent with BASE +CHANGE —it has an initial velocity plus a
change involving the acceleration—this expression does not correctly give the dependence of
velocity on time for cases of constant acceleration. In fact, the units of the second term are not
even correct.
Again, we have students seeing an equation in terms of a meaningful structure, but it is a
structure that does not recognize or determine all details of the equation. The point here is really
not any different than points I made just above. Nonetheless, some physicists and physics teachers
may be a little surprised by these episodes since, to the physics expert, the equation v = vo + at
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seems very straightforward, at every level of detail. I will have more to say about this in Chapter
12.
The final two forms in the Terms are Amounts Cluster, WHOLE - PART and SAME AMOUNT ,
appear extremely rarely in my data corpus, so I will only mention them briefly here. In
PART,
WHOLE
-
rather than adding to or subtracting from some base quantity, a piece of a whole is
removed. Typically, this piece is an identifiable component of the original whole. SAME AMOUNT is
the counterpart of the BALANCING form. Rather than involving equal and opposite influences, it
states that two collections contain the same amount of stuff. A statement that energy is conserved,
for example, may be seen in terms of this form.
The ÒDependenceÓ Cluster
The forms in the Dependence Cluster are, in a sense, extremely simple. In fact, their symbol
patterns may hardly even merit being called “patterns.” Nonetheless, I will argue in Chapter 6 that
these forms are common and important. Table Chapter 3. -4 lists the forms in this cluster.
DEPENDENCE
NO DEPENDENCE
SOLE DEPENDENCE
[…x…]
[…]
[…x…]
Table Chapter 3. -4. Forms in the Dependence Cluster.
The forms in this cluster have to do with the simple fact of whether a specific individual
symbol appears in an expression or does not appear in an expression. Most basic of all these forms
is NO DEPENDENCE , whose symbol pattern involves the absence, rather than the presence of
symbols. When we see an expression in terms of NO DEPENDENCE , we are noting that a particular
symbol does not appear in the expression. For example, in the construction of their expression for
µ, Mike and Karl wanted to have one component that does not depend on the weight (mass). This
is an application of NO DEPENDENCe:
µ
µ = µ1 + C 2
m
Karl
Well yeah maybe you could consider the frictional force as having two components. One
that goes to zero and the other one that's constant. So that one component would be
dependent on the weight. And the other component would be independent of the weight.
A prototypical use of NO DEPENDENCE can be found in students’ work on the Shoved Block
problem. Recall that the “correct” answer to this question is that the heavier and lighter block
travel precisely the same distance, if started with the same initial speed. In order to arrive at this
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conclusion, the students needed to derive an equation in which the mass doesn’t appear. Alan and
Bob derived such an expression and then commented:
a = gµ
Alan
Bob
Right, so, actually, they should both take the same.=
=Wait a minute. Oh, they both take the same! [Surprised tone]
…
Bob
So, no matter what the mass is, you're gonna get the same, the same acceleration.
Here, Alan and Bob are seeing their equation in terms of NO DEPENDENCE . They state that, because
no mass appears in this expression, the acceleration doesn’t depend on the mass, so both objects
travel the same distance. Note that this really does constitute a specific way of “seeing” this
expression. To see this expression in terms of NO DEPENDENCE , Alan and Bob have to do the trick
that is always associated with seeing equations in terms of forms; they have to blur certain features
into the background, and highlight other features for themselves. In this case, the absence of the
symbol ‘m’ is the key registered feature. Furthermore, notice the character of Alan and Bob’s
reaction in the above passage. NO DEPENDENCE is not something subtle; it is striking and obvious to
them, and elicits a strong reaction.
The flip-side of NO DEPENDENCE and the next step up in simplicity is DEPENDENCE. When
seeing an expression in terms of DEPENDENCE, we simply note the appearance of a particular
individual symbol. In the construction of expressions, this form is frequently used in tandem with
other forms. For example, in the early specifications of their expression for µ, Mike and Karl
stated that one component should be dependent on the weight. (Refer to the last Karl passage
above.) This was later elaborated to the statement that this component should be inversely
proportional to the weight. Similarly,
DEPENDENCE
usually played a role in students’ steps toward
the construction of an expression for the force of air resistance:
R = µv
Mark
So this has to depend on velocity. [g. R] That's all I'm saying. Your resistance - the
resistor force depends on the velocity of the object. The higher the velocity the bigger the
resistance.
DEPENDENCE and NO DEPENDENCE are the basic and most common forms in this cluster. The
one additional form is, in a sense, a slight variation on this theme. In SOLE DEPENDENCE , one
quantity is seen as depending only on a single other quantity. Consider this passage from Alan and
Bob’s work on Running in the Rain:
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total rain(t) = #rain
t + ρrain hwx
s ( )
Alan
You're just gonna- (well) based upon that equation right there you're just gonna conclude
that you would feel more - you would have more raindrops hitting you if you walk because
it's gonna take you a longer time since the last term is just a constant. And you're just
saying it just depends on the time.
Bob
Right.
Alan
It's only dependent upon time.
In the above passage, Alan argues that the time needed to cross the street, t, is larger if you walk
than if you run. Thus the “total rain” is larger if you walk. (Based on an inference from PROP+.)
Furthermore, Alan goes on to emphasize that time is the only quantity that the total rain depends
on. Across the two cases of interest—walking and running—the time is the only varying quantity
that appears in the expression. This is an application of SOLE DEPENDENCE .
The Coefficient Cluster
In the COEFFICIENT form, a product of factors is seen as broken into two parts. One part is the
coefficient itself, which often involves only a single symbol and almost always is written on the
left. This is indicated in the symbol pattern shown in Table Chapter 3. -5.
COEFFICIENT
[x
]
SCALING
[n
]
Table Chapter 3. -5. Forms in the Coefficient Cluster
The COEFFICIENT form is distinguished from other forms because of some properties that are
typically possessed by the coefficient. Students will frequently comment, for example, that a
coefficient is “just a factor” or “just a number.” It is, in fact, the case that important dynamical
variables rarely appear in the position of the coefficient. For example, you will almost never see a
student treat a force as a coefficient. And you will not see velocities or positions that vary over the
course of a motion treated as coefficients. Instead, coefficients most frequently involve
parameters, such as the coefficient of friction, that, in a sense, define the circumstances under
which a motion is occurring.
The proportionality constant introduced by students in the air resistance problem is a typical
example of a coefficient.
F = kv 2
Jack
Right, all I did was introduce a constant of proportionality. [g. k] We have no idea what it
is.
97
Here, Jack sees this expression in terms of the
COEFFICIENT
form; the constant k is seen as
multiplying the rest of the expression. Furthermore, Jack’s comment that “We have no idea what
it is” is characteristic of the type of things that people say about symbols that are being treated as
coefficients. Jack makes this comment in an utterly flippant manner. The suggestion is that,
although they have no idea what this coefficient is, it doesn’t matter that they don’t know. For the
most part, coefficients only matter at some boring level of detail. They can tune the size of an
effect or influence, and in that way have a fine quantitative effect on a motion, but they are not
generally seen as influencing the overall character of a motion or the result in question.
These details constitute the stance that goes with the COEFFICIENT form. Notice that such a
stance constitutes a very specific orientation to an expression or portion of an expression. A
product of factors, such as abc, need not have any particular structure to it. We need not,
necessarily, see the ‘a’ as any different than the ‘b’ or ‘c.’ But, when a physicist does see
COEFFICIENT
here—when they do see the ‘a’ as a coefficient that multiplies ‘bc’— the implications
are significant: All of the above stance comes along in the bargain. This is part of what forms do.
They do not only blur features; in carving up an expression, they also highlight features of the
symbolic landscape and give meaning to those features. When a physicist sees the expression abc in
terms of the
COEFFICIENT
form, they are seeing a specific, meaningful structure.
The second and only other form in this cluster is scaling. Scaling is very similar to the
COEFFICIENT
form, with the single important difference being that, in scaling, the coefficient is
unitless. Suppose, for example, that we have an expression of the form
T2 = nT1
where n is a (unitless) number multiplied by a collection of factors T1 to produce the result T2 .
Because n is unitless, T2 is the same type of entity as T1, and we can think of the coefficient n as
an operator that acts to “scale up” or “scale down” T1. In addition, inferences associated with the
scaling form tended to be very salient in my data. Here are a few common examples:
•
If n > 1 then T2 > T1
•
If n < 1 then T2 < T1
•
If n = 1 then T2 = T1
•
If n >> 1 then T2 >> 1
98
The scaling form is nicely illustrated in some examples involving the Buoyant Cube problem.
Recall that the point of this problem is to find how much of an ice cube is below the surface of the
water when an ice cube floats at rest. Alan and Bob obtained this simple expression as a final result
and commented:
x =.92L
Bob
Um, okay. [w. x=.92L] Yeah. Point nine two L. (2.0) Does that make sense? Yeah.
Alan
Yeah, because like when you see an iceberg they say that most of the iceberg is below
the water, so that's why ships have to be really careful.
In this example, the result x is the same type of entity as L; both are lengths. The factor .92 scales
L to produce a length that is “most” of L, a result that Alan and Bob believe makes sense.
The Multiplication Cluster
The Multiplication Cluster is the final cluster to be discussed in this primer concerning forms.
To this point I have discussed five other clusters. Two of these clusters, Competing Terms and
Terms are Amounts, deal with expressions at the level of terms. Another two clusters,
Proportionality and Dependence, were concerned more with individual symbols. Forms in the
Dependence Cluster were sensitive to the simple presence or absence of these individual symbols.
And forms in the Proportionality Cluster were concerned with whether particular symbols
appeared above or below the division sign in a ratio.
The observation I am working toward here is that only the fifth cluster—the Coefficient
Cluster—was concerned with structure in the arrangement of a product of factors. Consider, once
again, an expression of the form abc. In terms of what forms can a student see this expression? The
Terms as Amounts forms and Competing Terms forms could allow the whole of the expression to
be seen as an influence or as some “stuff.” In addition, we could see this expression in terms of
DEPENDENCE
in three ways, as dependent on a, b, or c. Similarly we could see prop+ in three
separate ways, as focusing on any of the factors involved.
However, only
COEFFICIENT
breaks down this string of factors into a meaningful structure; it
splits off the first factor, a, and treats it differently. And this is a very specific structure that will
only be relevant in a limited number of cases. It is certainly not true that every time we multiply
two things, one can profitably be thought of as a coefficient. So we are left wondering: Isn’t there
some other way to see meaningful structure in a string of factors? Aren’t there some forms missing
here?
99
Of course, it may simply be the case that the vocabulary of forms just happens to be a little
sparse in this territory, at least for students working on the tasks in my study. In fact, to a certain
extent, I believe this is the case. However, there appear to be a couple of forms that cover some of
this territory. I will discuss these forms presently, but I do this with the qualification that these
forms were relatively rare in my data and their identification is somewhat more tentative then
others I have described.
INTENSIVE •EXTENSIVE
EXTENSIVE •EXTENSIVE
x×y
x×y
Table Chapter 3. -6. Forms in the Multiplication Cluster.
As shown in Table Chapter 3. -6, the two forms in the Multiplication Cluster are called
INTENSIVE •EXTENSIVE
and EXTENSIVE •EXTENSIVE . The use of the terms “intensive” and “extensive”
here is taken from some other researchers, who used them in similar contexts (Greeno, 1987; Hall,
Kibler, Wenger, & Truxaw, 1989). I will discuss this research at the end of this chapter.
Intensive and extensive quantities are distinguished in roughly the following away. An
intensive quantity specifies an amount of something per unit of something else. In contrast, an
extensive quantity is a number of units. So if, for example, we are solving an arithmetic word
problem that gives us a number of apples per basket, a number of baskets, and asks us how many
total apples there are, then the number of apples per basket is intensive, and the number of baskets
is extensive.
As the names imply, the INTENSIVE •EXTENSIVE form applies when an expression, such as xy, is
seen as a product of an intensive and an extensive quantity, and the EXTENSIVE •EXTENSIVE form
applies when such an expression is seen as a product of extensives. In my data, intensive quantities
were almost always densities or rates. So, the
INTENSIVE •EXTENSIVE
form usually involved the
product of a rate and a time or a density and a volume. The fact that this form applied in such a
narrow range was part of the reason that it was difficult to establish its general validity.
There are a few instances, however, in which the INTENSIVE •EXTENSIVE form was applied in a
more flexible manner, and seemed to slightly transcend this narrow range. A number of these
instances appeared while students worked on the Running in the Rain problem. For example,
while working on this problem, Jon and Ella defined a quantity that was essentially a flux rate for
the rain.
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 # drops 
 A•s 
Jon
Do we have a rate for the raindrops? We need the number of drops per-…per area and per
second. [w. #drops/A•s]
So, Jon and Ella’s rate is the number of drops that flow through an area per unit time. After
defining this quantity, they went on to multiply it by the area of the top of the crate and the time
to cross the street, in order to get an expression for the total amount of rain that flows through the
top of the crate.
 # drops  A
(
)(t )
 A • s  head walk
Jon
And then we multiply that by the area of our head. [w.mod (#drops/A•s)(Ahead)] Oh, maybe
we should just say - Times the time we're in the street (for) walking. [w.mod (#drops/A•s)
(A head)(twalk)]
A number of the features of this little episode are typical of cases in which intensive•extensive is
applied. First, the use of the term “rate” is an important indicator. Second, explicit discussion of
the units involved seems to be common. Often the units of the factors are written out and used as
a guide in composing the expression. This is precisely what Jon and Ella did in this episode.
I will not give any additional examples of the intensive•extensive form here, but I do want to
foreshadow an observation that I will make in Chapter 12. Recall that the ‘at’ in the equation
v = vo + at , while somehow obvious for experts, was not obvious for the students in my study. I
believe that it is partly through the INTENSIVE •EXTENSIVE form that this term becomes obvious for
experts; in particular, they see it as a rate times a time. Furthermore, the fact that students do not
see intensive•extensive in this equation may be symptomatic of the overall low appearance of forms
in this cluster.
The
IDENTITY
form
To wrap up this primer, I want to discuss a pair of forms that do not really fit into any of the
above clusters. To begin, note that few of the forms discussed so far involve the use of an equal
sign. The only exceptions are BALANCING and SAME AMOUNT . Now I want to mention another form
involving an equal sign that covers a very large percentage of the expressions that are composed
and interpreted: the IDENTITY form. In IDENTITY, a single symbol, usually written on the left, is
separated from an arbitrary expression by an equal sign. I mentioned this form earlier in my
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account of Mike and Karl’s construction of their novel expression for the coefficient of friction. It
was IDENTITY that allowed them to write the “µ=” on the left side of the expression.
IDENTITY
x = […]
IDENTITY is so common that it is nearly invisible and rarely commented upon directly. As I
mentioned,
IDENTITY
allows the results of inferences to transparently travel across the equal sign to
the symbol on the left. Whatever is true of the expression on the right is true of the quantity
identified with that expression. Furthermore,
IDENTITY
is not only involved in the construction of
expressions, expressions are frequently manipulated so as to be in alignment with IDENTITY before
they are interpreted. This was the case with the final expression from the spring task that Jim
interpreted:
x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k] And that if you have a stiffer spring,
then you're position x is gonna decrease. [g. Uses fingers to indicate the gap between the
mass and ceiling] That why it's in the denominator. So, the answer makes sense.
In this passage, Jim sees the right side of this expression in terms of PROP+ and PROP-. This means
that if x increases, the whole right side increases. IDENTITY tells Jim that what is true of the whole
right hand side, is true of x: If the whole right hand side increases, then x increases.
In truth, I have not been careful with my language in such situations. Throughout my
examples, whenever an equation was in line with
IDENTITY,
I spoke interchangeably of the whole
right hand side and the quantity identified with that expression. In what follows, I will continue to
be loose in this way. Only when precision is called for or when certain issues arise will I bother to
note the role being played by the IDENTITY form.
Meanings of the equal sign
While I am on the subject of the equal sign, I want to briefly mention some related literature
from mathematics education. This literature is concerned with how younger, mostly grade-school
students understand the “meaning” of an equal sign. Central to this literature is the observation
that novice algebra students tend to see an equal sign as a “do something signal” (Kieran, 1992;
Herscovics & Kieran, 1980). The equal sign is seen as simply separating the answer from the
computation to be performed, as in 5+4=9.
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These researchers take the stance that there is nothing particularly wrong with this way of
understanding an equal sign. However they believe that it would be profitable if students’
understanding of the meaning of equal signs and equations were enlarged in a particular manner.
They have in mind an understanding in which expressions involving an equal sign are taken as
symmetric and transitive relations, as in expressions like 5+4=6+3. Such an understanding is seen
to be critical for more advanced mathematics in which equations are treated as objects in their
own right.
There is a certain temptation to attempt to match my forms,
IDENTITY
and BALANCING , with
the “do something” and “symmetric relation” versions of an equal sign. But note that such a
correspondence is not easily made. The equal sign in the
IDENTITY
form doesn’t necessarily specify
an action to be performed; rather, it states that one quantity can be identified with an expression.
In fact, the shift from writing the single symbol on the right to writing it on the left may be
significant. In the case of
IDENTITY,
the single symbol is not the computed result of a
computation; instead, it announces the nature of the expression to follow.
But I do not want to push this comparison too far. Keep in mind that my subjects are
advanced college students and the “equal sign” literature deals with novice algebra students;
therefore, we should expect significant differences. However, I believe that these mathematics
education researchers can draw some important morals from the presentation in this chapter. If
the “forms” view is correct then, as students progress to expertise, we should not expect them to
move toward possessing only one or two stances toward the meaning of an equal sign, identified
with distinctions like “process” versus “object.” Instead, we should expect something more like an
idiosyncratic vocabulary of stances, that depends on the domain and the tasks undertaken in that
domain. I will touch on related issues in the chapters that follow.
D YING A WAY
Lastly, as the final entry in this primer, I want to note one form that is interesting because it
suggests how the vocabulary of forms may be extended in expert directions. The form is called
DYING AWAY
and the symbol pattern involves exponentiation to a negative power.
DYING AWAY
[e− x…]
This form appeared just a few times in my data corpus. When working on Running in the
Rain, Jack and Jim hypothesized that the probability of getting struck by a raindrop would die
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away exponentially with increasing velocity. Figure Chapter 3. -9 shows their graph and the
expression that they wrote. Their comments follow.
P
Ae−v
0
v
Figure Chapter 3. -9. Jack and Jim's graph and expression.
Jim
The faster you go, the less you're gonna get wet.
…
Jack
So, you could say the probability if you're traveling at infinite velocity is very extremely
small.
…
Jim
It’s probably- it’s probably if you fit a mathematical model to it, it's probably like an
exponential decaying curve, where you go up to infinity as you near zero velocity. And the
probability of getting hit by a raindrop the faster you go goes down and down but it never
reaches zero.
A number of features of this episode seem to characterize all of the few episodes in which
AWAY
DYING
appeared. First, talk about the limiting case, when the exponent goes to infinity, always
accompanied this form. Second, the use of this form was always accompanied by the drawing of a
graph, similar to Jack and Jim’s. The relationship between physics equations and graphs is an
interesting subject to which, unfortunately, I will not be able to devote significant time.
An interesting point to note about DYING AWAY is that, in a sense, it transcends the limits of
qualitative reasoning. Note that, according to the limits laid out by the qualitative reasoning
researchers, DYING AWAY and PROP- are equivalent—both say that one quantity decreases as another
quantity increases. But the DYING AWAY form is more specific than this brief description; it is
somehow tied to the particular shape of the curve associated with an exponential function, and the
notion of a gradual approach to a limiting value seems to be essential. Perhaps the use of graphs
and the vocabulary of limits serve as crutches that allow people to extend beyond the limits of the
qualitative reasoning system. For example, while certain distinctions may lie beyond the abilities
of qualitative reasoning, it may be easy for us to distinguish between the corresponding graphs. In
any case, I believe that we are seeing the beginnings of the extension of the vocabulary of forms to
include more expert features.
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ÒPatternsÓ in Arithmetic Word Problems
Before wrapping up this discussion of forms, I want to mention a collection of research that
bears some clear similarities—both superficial and deep—to my symbolic forms-based model of
symbol use. This research comes out of mathematics education, and deals with observations of
how some much younger students solve arithmetic world problems such as the following: John has
five apples and Mary gives him three more, how many does he have?
The research in question is a collection of papers that identify what Greeno (1987) has called
“patterns” in arithmetic word problems. (See, for example, Carpenter & Moser, 1983; Riley,
Greeno, & Heller, 1983; Vergnaud, 1982). There is some variation in the patterns identified by
these researchers, but a single example will suffice here. Riley and colleagues list four categories of
arithmetic word problems, what they call “change,” “equalization,” “combine,” and “compare”
problems. In change and equalization problems, addition and subtraction are described as actions
that lead to increases or decreases in some quantity. More specifically, in change, an amount is
simply added onto another quantity, as in the case of John and Mary’s apples. In equalization, an
amount is added to a base quantity so that it becomes equal to some third quantity: How many
apples must Mary give to John so that he has the same number as Fred?
In contrast, combine and compare problems “involve static relations between quantities.” In
combine problems, there are two distinct quantities that remain unchanged: How many apples do
John and Mary have between them? And in compare problems the difference between two
unchanging quantities is determined.
In addition to additive patterns of this ilk, some researchers also identified a number of
“multiplicative patterns” for problems that involve multiplication and division. Here is one place in
the literature that the “intensive” and “extensive” quantities that I mentioned earlier appear. In
some researcher’s schemes, for example, there are multiplicative patterns that differ according to
whether they involve the products of intensives, extensives, or combinations of intensives and
extensives (Greeno, 1987).
One question we can ask of this research is: What kind of thing are these patterns? It seems
that, in the first iteration, researchers considered the patterns to be categories of problems. Riley et
al., for example, call them “problem types.” Given this orientation toward the status of patterns,
the Patterns research constitutes a particular kind of empirical research strategy. You take the
problems that students have to solve and divide them up into different categories. Then you see
how students perform on the different categories. If there are significant differences across
105
categories then you’ve found something interesting. And researchers such as Riley and colleagues
did find some strong differences in how students performed across category types. For example,
certain types of compare problems were found to be particularly difficult for younger students.
But, as the Patterns researchers were quick to point out, the fact that students perform
differently on the different problem types tells us something about how students are
understanding and solving these problems. This led to some hypotheses in which the patterns
played a role in cognitive models of problems solving. For example, in Riley et al.’s model,
students learn to see arithmetic problems in terms of the patterns, which are associated with
problem schematizations. Then problem solving operators are tied to these schemata.
Clearly, this whole story bears some strong similarities to my own. “Patterns” seem to live at a
similar level of abstraction to my “forms.” And even some specific forms seem similar to patterns.
For example,
PARTS -OF -A-WHOLE
and BASE +CHANGE are similar to Riley’s combine and change
categories.
There are some noteworthy differences, however, and I will begin with the obvious ones. First,
the concerns of the Patterns literature are, in a number of respects, quite different than my own.
For example, the Patterns research looks at much younger students solving arithmetic problems,
not at young adults working on physics problems. There are also some more subtle differences in
concerns. Most notably, the Patterns researchers are concerned with describing problem solving,
how students get from the statement of a problem to a solution. They want to build a model that
has the same problem solving capabilities and goes through the same steps as students. In contrast,
I am interested in describing how students understand expressions and in the role that symbol use
plays in understanding. Thus, I am interested in building a model that can describe, for example,
the interpretations of equations that students give. By way of illustration note that, in the Patterns
literature, patterns are things that are seen in problems, not necessarily in equations. This is because
of the problem solving orientation; in problem solving, you go from problem specification, to
equation, to result.
Now I will move on to the most important and, perhaps, most subtle difference. It is
possible—though not necessary—to read the Patterns literature as implying that all of the possible
word problems that students will ever encounter can be described by a relatively small number of
patterns. This is implied by the fact that these researchers use a relatively simple vocabulary
involving “sets” and a few kinds of actions and relations between sets to describe the patterns. For
example, Carpenter and Moser (1983) say that arithmetic word problems can be characterized
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according to three dimensions: (1) whether there is “an active or static relationship between sets of
objects,” (2) whether the problem “involves a set-inclusion or set-subset relationship,” and (3) for
the case of problems that involve action, whether the action results “in an increase or decrease in
the initial given quantity.”
This sort of talk suggests a relatively easy to specify space of possibilities. In contrast, the
forms vocabulary that I have observed seems much more idiosyncratic and particular to physics.
Note, for example, that the “set” vocabulary doesn’t even apply to the influences that appear in
the Competing Terms forms. Furthermore, the semantics of influences that appear in these forms
may be somewhat particular to physics.
Taken as a critique of the Patterns literature, what I have just said is a little unfair. The
Patterns researchers are quite self conscious about looking at only a very limited class of problems,
as they are performed by a particular range of young students. It is certainly possible that the
patterns they have uncovered are sufficient to account for their data, and they do not explicitly
argue for the generality of their patterns or their vocabulary beyond this limited range. Overall, I
mean to portray a positive attitude toward this research. And, in fact, it has served as a useful
resource in my own endeavor: I used the patterns discovered by these researchers to aid in my
recognition of forms. But there is a possible trap to be recognized here if we do try to generalize
from the Patterns literature. We must be careful not to assume that the vocabulary and
distinctions that are useful for mathematicians and physicists, such as “sets,” will be the right
distinctions for describing student or even expert knowledge.
Conclusion: How Do I Know That Forms Are Real?
How do I know that forms are real? In this chapter, I have described symbolic forms and
listed the various kinds of forms that appeared in my data corpus. I have presented examples to
illustrate particular aspects of the hypotheses associated with positing the existence of forms, and I
have contrasted my view with alternatives and competitors. Still, I have not presented all of my
evidence that forms do exist and are important. Most importantly, I have not described the
details of how I make contact with my data corpus. This connection to the data will not be
described until Chapter 6, when I will explain how, through repeated iterations of systematic
analysis of the entire corpus, I arrived at the particular list of forms described here. In addition, I
will present specific results concerning the pervasiveness of the various types of events described
here.
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However, I believe that the content of this chapter contains many elements that can contribute
to an argument for my viewpoint. First, I have argued against some important competitors; most
notably, what I referred to as the principle-based schemata model and other models that propose
that physics knowledge is closely tied to physical principles. Some of the arguments against these
competitors were relatively straightforward. As we saw, these competitors are not constructed with
an intent to deal with some of the phenomena that I am treating as central, namely the
interpretation of expressions and the construction of novel expressions.
In addition, I believe that the large number of examples presented in this section can add
significantly to the plausibility of the view that I am describing. We saw, of course, that students
can and do construct and interpret equations, and I tried to give a feel for what these behaviors
look like. Furthermore, many of the examples I presented were quite suggestive of the types of
phenomena I hypothesize to be associated with forms as knowledge elements. Here is a list of
some of these suggestive examples:
•
Mike and Karl, two students that demonstrated great prowess in their ability to construct
novel expressions, did not know that the equality of forces in the spring problem followed
form F=ma.
•
We saw that students do not always work by carefully and straightforwardly applying
principles. Instead, principles are often used as a way of rationalizing expressions after they are
already composed.
•
To explain why, in the Buoyant Cube problem, the weight of water displaced had to equal
the weight of the cube, Mark argued simply from the observation that, in his experience with
problems of this sort, you “have to have something equal.”
•
We saw that students are not always capable of choosing between certain kinds of related
expressions. For Mike and Karl, F = kx and F = 1 2 kx 2 are “qualitatively” equivalent.
•
Similarly, we saw that the equation v = vo + at was “obvious” for students, but only in a
certain way. How can students, in the same breath, say that v = vo + at is obvious, but that
they can’t be sure of the ‘at’ term? If we say that students are understanding this equation in
terms of the BASE +CHANGE form, then this phenomenon makes sense.
•
While working on Running in the Rain, Alan and Bob composed an expression with
“placeholders.”
108
The point is that these surprising phenomena are well accounted for by the forms hypothesis. It is
part of this hypothesis that students have knowledge for constructing equations that cuts across
physical principles. And it is an essential property of forms that certain features of equations
become consequential at some times, and not at others.
Most of all, I hope that the presentation in this chapter is plausible to some readers because it
maps a territory that they recognize, a world that they have lived in. The idea of this chapter has
been to do part of the job of describing the meaningful structure that a physicist and physics
student see when looking at a sheet of equations. They do not see a meaningless mass of details or
only a collection of individually recognizable symbols and equations. Like Gombrich’s painters,
they bring to a sheet of expressions a known “vocabulary of forms.”
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Chapter 4. Representational Devices and the
Interpretation of Symbolic Expressions
x=
mg
k
“As you have a more massive block hanging from the spring, then you're position x is gonna
increase, which is what this is showing.” In this statement, Jim tells us what this equation “shows.”
But how did he conclude that this is the content of this equation? I have already argued that
symbolic forms are one piece of the answer to this question. In this statement, Jim announces a
proportionality relation between position and mass, which suggests that he is seeing the
PROP+
form. And the PROP+ form allows Jim to make various inferences, including the inference that, in
this case, increasing the mass will increase the position.
However, important elements of the story are still missing. The PROP+ relation between
position and mass is only one of the possible ways that a form can be seen in this equation. It
would be helpful if we could say something about what factors lead to Jim seeing this particular
form. Furthermore, Jim expresses this form in a very specific manner. He does not just say that x
“is proportional to” the mass, instead he talks through the specific case in which the block becomes
“more massive.” What does a “more massive” block have to do with this expression? Why does he
choose to talk about a more massive block rather than a less massive one?
Clearly a number of important aspects of Jim’s interpretation remain to be accounted for by
my model. One thing that might help here is a better understanding of what it means to
“interpret” an equation. If we understood, in general, what it means to interpret an equation, then
we might be in a position to explain the choices that Jim makes in his statement. The hope is that
we could find some general procedure for generating interpretations of equations, and then we
could view Jim’s interpretation as an application of this general procedure.
Let’s take a moment to work on what an interpretation of an equation could be. Intuitively, we
might say that to interpret something is to state its “meaning.” If this is right, then the
construction of a “theory of meaning” would be a significant first step toward understanding the
nature of interpretation. This might not seem like a very helpful observation. Why should it be
any easier to invent a theory of meaning than a theory of interpretation? However, it happens that
some potentially useful theories of meaning already exist. In particular, philosophers and linguists
110
have worked out theories of meaning for natural language sentences, so we could begin with these
existing theories and try to adapt them. Although I will ultimately adopt an alternative approach,
I want to take a moment to explore this possibility.
One type of theory of meaning involves what is called “model-theoretic semantics.” Modeltheoretic semantics associates the meaning of a sentence with its “truth conditions”—the situations
in the world, understood in terms of a certain class of models, in which the sentence is true. Of
course, employing such a definition requires that we have a way of determining, from a sentence,
what its truth conditions are. In model-theoretic semantics this works as follows: Individual words
and predicates get their meaning by referring to things in the world. Then the truth conditions of
a sentence are a function of the arrangement of words in the sentence and the referents of these
individual words.
To take this discussion a little closer to home, some mathematicians and philosophers have
tried to provide foundations for mathematics that employ a version of model-theoretic semantics
called set-theoretic semantics. They begin with a collection of axioms that are expressed in terms
of what are essentially meaningless symbols. Proofs are then performed by using the rules of logic
to deduce new statements from these axioms. All of these statements are still in terms of
meaningless symbols. The idea is that set-theoretic semantics specifies how the statements can be
given meaning, of a sort. The symbols in the axiomatic system are put in correspondence with a
model that consists of sets and various kinds of relations among sets. (See Lakoff, 1987, for a
relevant discussion and critique of set-theoretic and model-theoretic semantics.)
For either of these cases—for model-theoretic semantics or the special case of set-theoretic
semantics in mathematics—“interpretation” can be understood to be the announcing of the truth
conditions of a symbolic statement. So, to understand what an interpretation of an equation is, we
could embark on working through the details of this program for the case of physics equations.
This would mean that we would have to define a class of models—like the set-based models of
set-theoretic semantics—and then rules by which physics equations are put in correspondences
with circumstances in these models.
This could work as follows: We begin by defining a method for building models of physical
circumstances. These models could be something like Larkin’s “physical representation;” they will
include point-masses, forces, velocities and other entities from the physicist’s palette. Then
individual symbols, such as the letter ‘F’, can be put in correspondence with entities in these
models. In this case, ‘F’ might correspond to a force in a particular circumstance. Finally, the
111
arrangement of symbols in the equation somehow specifies a relation among these entities that
must hold. The set of circumstances in which this relation holds stated in the modeling
language—its truth conditions—constitutes the “meaning” of the equation.
Can such a project, in principle, be made to work? I believe that the project to invent a theory
of equation meaning is a useful and interesting project, and it is for that reason that I have
followed it this far. However, there is one important difficulty with this approach from the point
of view of the current work. Even if we can construct an internally consistent theory of equation
meaning, this theory may not be useful for describing students’ interpretive utterances or the
knowledge that generates those utterances.
Thus, although I will make some contact with theories of equation meaning at the end of this
chapter, this is not the approach I will adopt here. Rather, I propose to adopt an altogether
different approach, an empirical investigation of what, in practice, constitutes an interpretation of
an equation. In many ways this is an easier—if more tedious—project than the philosophical, a
priori approach. To execute this empirical project, I can begin by looking through my data corpus
for any utterance that sounds remotely as though it could be called an interpretation of an
expression. Along with Jim’s above interpretation, I have already presented many examples of
utterances that fall into this category:
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
________________
s=
Jack
ρi
L
ρw
This does seem to make sense cause if ice had the same volume as water [g. ρi then ρw],
er, same density as water, then L would equal S which means the cube would be right at
the surface.
________________
F = kx ; F = 1 2 kx 2
Karl
Okay, now, qualitatively, both K X and half K X squared do come out to be the same
answer because as (…) looking at it qualitatively, both half - both half K X squared and K
X, um, you know, increase as X increases.
________________
total rain(t) = #rain
t + ρrain hwx
s ( )
Alan
You're just gonna- (well) based upon that equation right there you're just gonna conclude
that you would feel more - you would have more raindrops hitting you if you walk because
112
it's gonna take you a longer time since the last term is just a constant. And you're just
saying it just depends on the time.
In a sense, such a list of examples constitutes a preliminary answer to my question; if you want
to know what an interpretation of an equation is, then look at these examples. Of course, an
answer of this sort would be far from satisfying, and I intend to do better. One obvious next step
would be to look for regularities that span these instances. For example, if every interpretive
statement seemed to involve varying some quantity and holding other quantities fixed, as in Jim’s
interpretation, that would be an important observation. Failing the existence of such universal
regularities, a further step would be to group interpretive utterances into categories that do seem
to share features. Then we could characterize the within-category regularities.
This last approach, the grouping of observed interpretive utterances into categories, is roughly
the tactic that I will employ. However, since this is a cognitive science-style inquiry, I will not talk
about these categories as simply constituting observed regularities in student discourse. Rather, in
the manner of cognitive science, I will explain these regularities in terms of mental structures,
specific abilities possessed by individuals. In particular, I will associate each of these categories of
utterances with an interpretive strategy that can be known by an individual. I call these interpretive
strategies “representational devices.” Here, the use of the term “device” is supposed to call to
mind “literary devices.” We will see why in a moment.
Representational devices are the second major theoretical construct in my model of symbol
use in physics, and a discussion of devices is the main concern of this chapter. Devices are to be
understood as a type of knowledge, and, as knowledge, devices constitute an important part of
what physics students must learn on their way to expertise. In the last chapter, we explored the
structure—a meaningful structure—that a physicist sees in a page of expressions. The point of the
present chapter is that, to enter into this meaningful world of symbol use, it is not sufficient to
know symbolic forms. In addition, initiates in physics must possess the complex repertoire of
interpretive strategies that I call “representational devices.” With forms, this repertoire of devices
defines the character of the meaningful symbolic world of physics.
Two very different examples
I will start by looking a little more closely at two examples that we have already encountered.
First, here is Jim’s interpretation of the spring problem result, reproduced once again in full.
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x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k] And that if you have a stiffer spring,
then you're position x is gonna decrease. [g. Uses fingers to indicate the gap between the
mass and ceiling] That why it's in the denominator. So, the answer makes sense.
Jim’s statement involves two separate interpretations. In the first of these interpretations, Jim
describes what the equation says will happen to the position if the object becomes more massive.
In the second interpretation, he says what will happen if the spring is made stiffer; that is, k is
increased. It does turn out that a certain percentage of students’ interpretive utterances do involve
an imaginary process of this sort, in which some quantities are held fixed and one quantity is
varied. I call the device involved in such interpretations CHANGING PARAMETERS. Both of the
interpretations in Jim’s statement involve the use of this representational device.
Now we move on to a second instance that we have already encountered. Recall that at the
end of Alan and Bob’s work on the Shoved Block problem, Alan made what I took to be an
inappropriate statement concerning an equation. In particular, he incorrectly maintained that a key
equation stated a condition that must be satisfied for the motion to terminate:
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
At this point, it is not especially relevant for the discussion at hand that this interpretation is
inappropriate. What is important is that this interpretation is very different than Jim’s
interpretations of the spring equation. Note that it does not seem correct to describe Alan’s
interpretation in terms of quantities that are varied and quantities that are held fixed. Rather, Alan
projects this equation into a particular moment in the motion. I refer to the device that Alan
employs here as the SPECIFIC MOMENT device.
Jim’s and Alan’s interpretations are typical of two of the major classes of representational
devices, what I call “Narrative” and “Static” devices. I will not explain these major types further at
the moment; for now, I simply want to note that there are interpretations that seem to be of
significantly different types. Thus, we can already see that my project of putting interpretive
utterances into categories is not going to end up with all utterances in the same category.
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On the form-device relation
I opened this chapter by wondering what factors determine which forms are seen in a given
equation, by a given individual at a particular time. I want to now discuss the role that
representational devices might play in determining the forms engaged. In this vein, I begin with a
hypothetical story concerning Jim’s interpretation of the spring equation, a story that I will
ultimately question:
1. Jim starts with the CHANGING PARAMETERS device and looks to see the effects of increasing the
mass.
2. This draws his attention to the ‘m’ variable, highlighting its location in the equation and
causing other features of the equation to be less salient. This leads to the
PROP+
form being
engaged.
3. Then PROP+ (with IDENTITY) allows the inference that, as mass increases, position increases.
The key feature of this story is that the choice of representational device comes first. The device is
chosen and then this choice can be said to be determining the symbolic form seen by Jim.
Similarly, in Alan’s interpretation, the SPECIFIC MOMENT device may lead Alan to see BALANCING
in the equation. Perhaps the choice to see the equation as describing a particular moment in the
motion causes Alan to “look for” BALANCING in the equation. However, there is another very
plausible alternative to this story: The form could come first and determine the device, rather than
the reverse. Perhaps, for Alan, the equation in question strongly cues a BALANCING interpretation.
This is especially plausible since, as I have said, students are predisposed to overuse BALANCING .
So which of these stories is correct? Do forms determine devices or devices determine forms?
My answer to this question is that neither story is always correct; I believe that there are multiple
ways that the process of interpretation is structured, including the two possibilities described
above. However, I will not undertake a description of these processes in this thesis. Instead, for the
sake of simplicity, I propose to adopt a more heuristic stance in which forms and devices are seen
as elements in a system of constraints that, working as a whole, adjusts itself until the system is
satisfied and an interpretation is produced. In adopting this model, I am intentionally glossing
some of the details of the interaction between forms and devices.
As shown in the simple schematic in Figure Chapter 4. -1, in addition to forms and devices,
the system includes the equation itself and what I have simply called the “rest of the context.” The
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point of including the “rest of the context” is that the form and device engaged must depend on
considerations of the moment, such as what the question asks. This issue will be a major focus of
the final sections of this chapter.
symbolic expression ⇔ form ⇔ device ⇔ rest of the context
Figure Chapter 4. -1. The form-device relation.
Let me take a moment to clarify the roles that these two types of knowledge play in this story.
As shown in my schematic, forms are most closely connected with equations; they involve directly
seeing patterns in equations. Furthermore, everything to the right of forms in my schematic plays
a role in constraining what forms are seen in the equation. Now, I could have chosen to just wrap
everything to the right of forms in a black box and say that I am not going to explain it further in
this study. But I have attempted to do just a little bit better. My model includes an additional
layer between forms and the rest of the context, what I have called representational devices.
Representational devices are interpretive stances and strategies that more broadly characterize an
orientation to an equation, and thus influence the forms seen in a symbolic expression.
Representational devices and the construction of equations
I want to clarify one final point before diving in and systematically discussing each of the
types of representational devices. The schematic shown in Figure Chapter 4. -1 is designed to
apply to the construction of new equations, as well as to the generation of interpretive utterances
concerning equations that are already written. Notice that the arrow that connects forms to the
equation points in two directions. The idea is that, just like the other elements of the system, the
equation can change as the system adjusts itself.
All of this implies that the construction of equations involves representational devices. To
illustrate, on the way to composing the novel equation,
µ
µ = µ 1 + C m2
Karl commented that:
Karl
I guess what we're saying is that the larger the weight, the less the coefficient of friction
would be.
The thing to notice here is that a CHANGING PARAMETERS device is built into Karl’s specification for
this equation. In an important sense, equations are constructed with interpretations in mind.
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The Variety of Devices
In this section, the project of categorizing interpretive utterances and listing representational
devices moves into full swing. As shown in Figure Chapter 4. -2, this discussion divides the
representational devices observed into three “classes:” Narrative, Static, and Special Case. I will now
discuss each of these three classes in turn.
Narrative
Static
CHANGING PARAMETERS
SPECIFIC MOMENT
PHYSICAL CHANGE
GENERIC MOMENT
CHANGING SITUATION
STEADY STATE
Special Case
STATIC FORCES
RESTRICTED VALUE
CONSERVATION
SPECIFIC VALUE
ACCOUNTING
LIMITING CASE
RELATIVE VALUES
Figure Chapter 4. -2. Representational devices by class.
The Narrative Class
Representational devices in the Narrative Class embed an equation in an imaginary process in
which some type of changes occur. As a whole, I call these devices “narrative” because they often
lead to interpretive utterances that are like a very short story or short argument. It is because of
these devices—and the fact that they constitute the most common class—that devices deserve
their name. The use of the term “device” is designed to suggest that representational devices are
like the literary devices for the very short stories in narrative interpretations. The devices in this
class are listed in Table Chapter 4. -1.
CHANGING PARAMETERS
PHYSICAL CHANGE
CHANGING SITUATION
Table Chapter 4. -1. Devices in the Narrative Class.
The analysis I will present in Chapter 6 will suggest that the CHANGING PARAMETERS device was
the most common of all representational devices in my data corpus. In CHANGING PARAMETERS,
some parameter is picked to vary and other parameters are held fixed. As I already mentioned,
Jim’s interpretation of x=mg/k involved the use of this device.
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CHANGING PARAMETERS is frequently applied as a way to get at the properties of more elaborate
expressions. For example, during their work on the Running in the Rain problem, Jon and Ella
arrived at the following expression for the total rain that strikes a person as they run across a street.
In this expression, vR is the running velocity, how fast the person runs across the street.


h
w
 # drops  
+
=
wx
vd2 v R 
 A⋅s  
 v d 1+ 2

vR




Extracting meaning from a complicated expression of this sort can be quite difficult. By way of
interpretation, Ella tried to determine what this expression says about how the total rain will
change if the speed increases, that is, if the person runs instead of walking across the street.
Ella
Okay, V R is bigger. Then (10.0),, This is gonna get smaller, [w. arrow pointing to second
term] but this is gonna get bigger. [w. arrow pointing to first term] I think.
Jon
Yeah, I think - that gets smaller [g. second term] cause V walking is smaller than V
running.
Ella’s interpretation is based on the CHANGING PARAMETERS device; she imagines that v R increases
and that all of the other parameters are held fixed. The result, in this case, is somewhat
inconclusive. She infers that the second term will get bigger but that the first term will get smaller.
I will have more to say about these inconclusive outcomes later in the chapter.
The CHANGING PARAMETERS device is not only used as a way of extracting information from
final expressions; it is also used to check the sensibility of equations that constitute intermediate
steps in a solution. For an example, I return to Mike and Karl’s expression for the acceleration of a
dropped ball under the influence of gravity and air resistance:
a(t) = −g +
Karl
f (v)
m
Okay, this sign is negative [g. -g] this number [g. f(v)/m] as the mass- as you increase the
mass,, Well, one problem- from one ball to the other the mass is doubling. Right, so you
double this mass [g. m] so this number [g. f(v)/m] is getting smaller.
Here, Karl’s interpretation involves varying the mass and holding the other parameters fixed.
More specifically, Karl imagines that the mass is doubled. Such a choice is not very surprising since
the task under consideration—the Air Resistance problem—asks the students to compare the
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terminal velocities of two masses, one of which is twice as heavy as the other. This occasional
dependence of device choice on the task will be a major focus of the final section of this chapter.
I have yet to comment on one very important feature of CHANGING PARAMETERS. In the
CHANGING PARAMETERS
device, the changes in question are not changes associated with a motion
under consideration. For example, in Karl’s interpretation above, the process he imagines involves
the mass doubling. This is not a process that takes place through the temporal sequence of the
motion, as the ball drops. The same is true of Ella’s interpretation. She is not telling the story of a
single individual running across the street and what happens along the way; rather, she is talking
about two separate instances in which someone runs across the street, but with different speeds.
There are, in fact, interpretations in which an equation is placed in the process of the motion
itself, and which concern changes in quantities that do occur through the temporal sequence of a
motion. I call the device associated with this separate category of interpretation the
CHANGE
PHYSICAL
device. Bob’s interpretation of this expression for the force of air resistance is an example:
Fair = kv
Bob
Okay, and it gets f-, it gets greater as the velocity increases because it's hitting more
atoms (0.5) of air.
The key thing to note about this little example is Bob’s statement that this happens “as the
velocity increases.” Just as in a CHANGING PARAMETERS interpretation, this interpretation involves a
quantity that is increasing, but in this case it is increasing through the time of the motion itself.
Later on in the same task, Alan provides us with another nice example involving PHYSICAL CHANGE :
∑F = mg - kv
Alan
And so I guess we were just assuming that this was [g. kv] de-accelerating - that we would
have gravity accelerating the ball and then as, as we got a certain velocity, [g. kv] this
would contribute to making accelerating closer - de-accelerate it and so we would just
have a function of G minus K V. And then we assumed that since we know that it reaches
terminal velocity, its acceleration's gonna be zero, we just assumed G minus K V would
equal zero.
Roughly, the story in Alan’s interpretation is that gravity starts the ball accelerating. Then, as the
ball speeds up, the kv term grows and grows until it eventually cancels mg and makes the
acceleration zero. Thus, the equation is placed in the story of the motion and the interpretation
tells us what happens to the equation as the ball falls.
So, interpretations based on PHYSICAL CHANGE involve changes in parameters that actually vary
during the motion itself. In contrast, in CHANGING PARAMETERS, an equation is placed in a story in
which one of the parameters that affects the motion, viewed as a whole, varies. These are
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parameters that, like the mass, do not vary during the time of the motion itself. Note that the
changes induced by varying these parameters are typically changes in degree only, the motion is
still motion of the same sort and the same influences and entities are involved.
In the final device in this class, CHANGING SITUATIONS , an equation is given meaning by
embedding it in a comparison between the current situation and a situation that is fundamentally
different; that is, it differs by more than simply the values of the parameters that define the
motion. Recall, for example, that part (b) of the Air Resistance task asked students to consider
how the terminal velocity of a dropped object would change if there was a wind blowing straight
up or straight down. The addition of the wind is precisely the type of change I have in mind here
and, in fact, this problem did tend to engender the use of the changing situations device. In solving
this problem, students typically obtained an expression for the terminal velocity which was the
same as the expression obtained without wind, but had an additional term. Close to every pair
gave a CHANGING SITUATIONS interpretation of their version of this expression.
vt =
Karl
mg
− vw
k
Which is what I was saying [g. final expression] that the terminal velocity will be the velocthe terminal velocity in still air plus the velocity of the wind.
________________
vt =
Bob
m
g − vo
k
If the wind is blowing up, the velocity - terminal velocity is going to be less than it was in
this case. [g. vt=m/k g] I mean, we have the same mass. This is M K G. [w. vt=m/k g; draws
a box around it] This is no wind. [w. "no wind"] No wind. And this is with wind. [w. "w/ wind"
and boxes wind result] Okay, so if the wind is blowing up, the terminal velocity is gonna be
less.
________________
vt =
Roger
(mg − Fw )
µ
Okay, well for the up case - yeah, terminal velocity will be (0.5) less [g. -Fw] by a factor F W
over mu.
In each of these examples, the student’s interpretation of their expression is framed by a
comparison to the case in which there is no wind. The adding or taking away of the wind is a
change in the situation of the sort that is associated with the use of the CHANGING SITUATION device.
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The Static Class
As we have just seen, Narrative devices embed an equation in a very short story that involves a
hypothetical process of change. In contrast, devices in the Static Class project the equation into a
static situation. Rather than playing a role in a story, the equation is treated more like a snapshot
of a moment. Furthermore, Static devices are often used in close concert with diagrams, which
can also be used to depict instants in a motion. The devices in the Static Class are listed in Table
Chapter 4. -2.
SPECIFIC MOMENT
GENERIC MOMENT
STEADY STATE
STATIC FORCES
CONSERVATION
ACCOUNTING
Table Chapter 4. -2. Devices in the Static Class.
Alan’s interpretation of the equation µmg=ma is an example of an interpretation involving the
SPECIFIC MOMENT
device.
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
In specific moment interpretations, an equation is treated as describing one particular moment in a
motion. As I have mentioned, in Alan’s interpretation above he associates this equation with the
particular instant in which the shoved block comes to rest.
SPECIFIC MOMENT is often employed (correctly) to describe a moment when two forces are in
balance. This was frequently the case in students’ work on the Air Resistance task, where this
device was used to treat an equation as describing the moment when the force of air resistance has
just become equal to the force of gravity. In the following excerpts, note the appearance of some
key suggestive phrases, such as “at some time” and “that’s when.”
R = mg
Roger
It means that the velocity won't increase anymore.
Mark
Okay.
Roger
That's after a certain point.
Mark
Oh, that's when the air resistance equals the=
Roger
=Yeah!
Mark
That's when the forces are equal then, right?
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Roger
Yeah, because then you wouldn't have acceleration anymore. [g. g in mg]
Mark
Right. (…) When the force is zero the acceleration's zero.
Roger
Okay. [w. R=] I guess. Okay. After a certain time
Mark
R equals G.
Roger
At T, some T.
________________
Cv 2 = mg
Ella
The terminal velocity is just when the - I guess the kind of frictional force from the air
resistance equals the gravitational force?
In both of these excerpts, the equation is associated with the specific moment when the two forces
are equal and the terminal velocity has been reached.
I have stated a number of times that Alan’s interpretation of the equation µmg=ma is
inappropriate since it implies the existence of a second force that does not exist in the standard
account. Now I would like to deal with the question of what would constitute a more appropriate
interpretation of this expression. An alternative stance that can be taken toward this expression is
that it is not just true at the end of the motion, it is true at any time along the entirety of the
motion of the shoved block. More specifically, the equation can be thought of as providing a
means of computing the acceleration at any time along the motion. When we see an equation in
this manner, as true at any time, then we are employing the “GENERIC MOMENT ” device.
A very typical example of this device can be found in Mike and Karl’s work on the Air
Resistance task, when they composed an equation for the acceleration of a dropped ball. In the
following passage, note their use of the phrase “at any time.”
a(t ) = −g +
Mike
f (v)
m
So, at least we can agree on this and we can start our problem from this scenario. [g.
Indicates the diagram with a sweeping gesture] Right? Okay? So, at any time,, At any
time, the acceleration due to gravity is G, and the acceleration due to the resistance
force is F of V over M. [w. g + f(v)/m]
…
Mike
You have negative G plus F of V over M [gestures down and then up]. That's you're
acceleration at any time. Right?
Karl
Well, wait. You want them to have,,
Mike
This is the acceleration at any time T. [w. a(t)=]
In this episode, Mike and Karl composed an expression for the acceleration that is valid at any
time as the ball drops. It is true at the moment the ball is released, as the ball speeds up, and even
after it has reached terminal velocity.
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At first blush, it may seem a little strange that it is possible to describe a “generic” instant in
such a varying sweep of events. Indeed, the ability to provide such descriptions—to find the level
of generality from among the particulars—is one of the great powers of physics, and the ability to
create and interpret such descriptions is one of the skills that physics students must learn.
In fact, there are a number of supports built into the representational practice of introductory
physics that aid students in constructing equations that are to be interpreted in the manner of the
GENERIC MOMENT
device. The ubiquitous “free-body diagram” can be seen as an example of a
support of this kind. In learning to use free-body diagrams, students are taught to isolate an object
and to draw all of the forces that act on that object. Mike and Karl’s diagram for the air resistance
task, which I presented in Chapter 3, is an example (refer to Figure Chapter 4. -3). In acting out
the set of procedures associated with the use of a free-body diagram, students are led to produce
equations that are true at any time during the motion, and which are amenable to GENERIC
MOMENT interpretations.
f(v)=
Fa
v m
Fg
Figure Chapter 4. -3. Mike and Karl’s air resistance diagram.
Readers familiar with the practice of introductory physics will be well aware that the use of
free-body diagrams to construct equations of this sort is extremely common. I will present just
one additional example here. When working on the spring task, Alan and Bob set the sum of the
forces equal to ma, saying that this described the spring at each point along its oscillation:
ma = -ky + mg
Alan
What's gonna happen is that, at most points, except for I believe this point right here,
[g. rest position of spring] you're gonna have a net force that's gonna cause the spring to
accelerate. [g. ma] Right? So, I'm gonna set this up as this equation [w.mod ma=-ky+mg]
The above diagram-oriented work is typical of most cases involving the
GENERIC MOMENT
device. There are some cases, however, in which a student simply emphasizes that a given relation
is true at any time along the motion and can be used to compute results that are true for any
particular time. For example, Mark and Roger manipulated the relation v = at to obtain an
expression for t and then commented:
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t=
v
a
Mark
Oh, it gives us (st…) say we want to find the velocity, [g. v] we can if we know the time. [g.
t] Or, vice-versa, if we know the velocity we can find how far into the free fall it =
Roger
= I mean this is a relationship between p- T and V. [g. t, v]
As Mark and Roger emphasize, this relation is true at any time. You can pick a time that you want
to know about and get a velocity, or you can substitute a value for the velocity and compute the
associated time.
The next device in this cluster, STEADY STATE , is a companion to the above two. There is a
special case to be considered here, the case when there is no distinction between
GENERIC MOMENT
and SPECIFIC MOMENT . This will be true when no parameters of the system vary with time, as in the
case of the spring problem. Consider the following two examples:
kx = mg
Jack
So, okay, so then these two have to be in equilibrium those are the only forces acting
on it. … So I guess we end up with K X is M G. [w. kx=mg]
________________
1 kx 2
2
Mike
= mg
Now, when it’s hanging, you have the force from the spring pulling it up and the weight
from the mass pulling it down. And those two forces would have to equate [g. the force
diagram], and you’ve got that right here. [g. 1/2kx2=mg]
These examples can be seen to share aspects of both GENERIC MOMENT and SPECIFIC MOMENT . The
expressions are true at any time, but such a fact is trivially true since nothing about the system
varies with time. And, in fact, these examples can be seen to have features I have associated with
SPECIFIC MOMENT .
For example, there are balanced forces involved. And note Mike’s use of the
word “when” in the phrase “when it’s hanging.” Thus, I consider both of these examples to be
instances of a third device, the STEADY STATE device.
The STATIC FORCES device is designed to account for another kind of ambiguity. Above, I
argued that we could think of free-body diagrams as a representational support for the
construction of GENERIC MOMENT expressions. Students also learn to draw diagrams of this sort that
depict a particular moment in a motion, and that depict steady state situations. The problem is
that, in many cases, although it may be clear that students are projecting an equation into some
static situation involving forces, it may not be clear which of the three static situation the student
has in mind. Furthermore, there is no reason to believe that the student “intends” any of the three.
Consider, for illustration, the following example:
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∑F = mg - kv
Bob
There's two forces. Okay. M G. And then there's a force upward which is dependent on
velocity, K V. [w. a little diagram showing up and down forces] Alright, so the sum of the
forces is equal to that. [g. ∑F equation]
I believe that there is a relatively well-developed nexus of equation construction activity here,
which takes on a kind of life of its own. Students spend a lot of time drawing diagrams with
forces, then building equations from these forces. In the STATIC FORCES device, it is as if an
equation is projected into one of these diagrams, rather than into a motion or physical situation.
Because these diagrams may describe the sort of situations associated with SPECIFIC MOMENT ,
GENERIC MOMENT , OR STEADY STATE, STATIC FORCES
interpretations cannot be treated as belonging
to one of these other categories.
Now on to the fifth device in this class, what I call the “CONSERVATION” device. The
CONSERVATION
device is named after its most common situation of use, problems that involve the
application of conservation principles. It is similar in some respects to SPECIFIC MOMENT , except
that each side of the equation is associated with a different moment in the motion. As an example,
I want to present some student work on a problem that I have not discussed much to this point,
the problem in which a ball is thrown straight up and students are asked to find the maximum
height reached. In Chapter 2, I briefly described Jon and Ella’s work on this task, with emphasis
on their discussion of whether a numerical result was reasonable. In their solution, Jon and Ella
employed the memorized equation vf2=vo 2+2ax.
In contrast, Alan and Bob used the principle of “conservation of energy” to solve this problem.
This principle tells Alan and Bob that the total energy of the ball is the same at any point along its
motion. They also know some other information that helps them here. They know that the total
energy at any moment is the sum of the “kinetic energy” and the “potential energy.” Furthermore
they know expressions for the kinetic and potential energy:
kinetic energy = 12 mv 2 ; potential energy = mgh
These expressions give the kinetic and potential energy in terms of the mass, m, the height, h, the
speed of the ball, v, and the acceleration due to gravity, g. With these relations and the
conservation of energy in mind, Alan and Bob assembled the following expression:
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mgh = 12 mv 2
Alan
Because at the top - we take the initial energy which is only kinetic energy. Potential
energy at the height of zero on the planet we arbitrarily set that to zero. So we only have
kinetic energy which is one half M V squared. [g. 1/2mv2 ] At the top we're not - the
velocity's going to be zero. [g. top of graph] It's going to be standing still for a split
moment. So we only have potential energy that's only gonna be M G H.
In this expression, Alan equates the potential energy at the maximum height with kinetic energy at
the moment the ball is released. This makes sense because, as Alan argues, the total energy at the
top consists only of potential energy and the initial energy is purely kinetic. The key point to note
here is that the left side of the equation is associated with a particular moment in the toss, the
instant when the ball reaches its apex, and the right side is associated with the instant that the ball
is released. Each side is thus associated with a different moment in the motion (refer to Figure
=
0
Chapter 4. -4). This is the CONSERVATION device.
Etop = 12 mv2 + mgh = mgh
=
0
mgh = 12 mv 2
Einitial = 12 mv2 + mgh = 12 mv 2
Figure Chapter 4. -4. Solving the tossed ball problem using conservation of energy. The instant of release and
the instant when the apex is reached are each associated with a side of the equation.
The last device in the Static Class, ACCOUNTING, is of a somewhat different sort than the other
devices in this class. In applying this device, a student imagines that an equation provides an
accounting of all of some entity or substance. Although I did not find that this device was
especially common in my data corpus, it is associated with what might be the most stereotypical
stance that can be taken toward an equation: We imagine that the job of an equation is to tell us
“how much” or “how many.”
The ACCOUNTING device belongs in this class because interpretations based on ACCOUNTING can
be thought of as describing a static situation. In fact, like the other elements of this class it tends
to be closely tied to depictions in diagrams. To elaborate, when students use free-body diagrams
to construct GENERIC MOMENT or SPECIFIC MOMENT expressions, they walk around the diagram,
126
using it to systematically enumerate each of the forces that are acting. When expressions are being
constructed using
ACCOUNTING,
the diagram plays a similar role; it can help to systematically
account for all of the entity or substance that the equation is supposed to be computing.
The context provided by the Running in the Rain problem consistently engendered use of this
device. Alan and Bob’s work on this task, which we have already encountered, was typical. In this
example, note the systematic listing of “sources” and Bob’s use of the phrase “how much.”
total rain =
Bob
# raindrops
+C
s
There's two - two sources of rain. Okay, rain that you walk into and rain that drops on
you. [g. rain hitting his head] Okay. Walk or run into. Okay, so, the number of rain per
second [g. #raindrops/s] is how much rain hits your head. … if you're a crate and you you move, move into some rain.
Here, Bob’s stance is that this equation is doing the “accounting” job, it tells us how much rain
there is. Jon and Ella made a similar accounting of the rain that hits your head and face as you run
through the rain:
# drops   xwh w 2 x 

+
# drops total =

 A ⋅ s   vd
vr 

Ella
And uh this term [g. first term]. This is from the drops that hit you on your face. And this
term [g. second term] is from the drops that hit you on your head.
The Special Case Class
For interpretations based on all of the above devices, the statements made are presumed to be
true for any values of the quantities that appear in the expression. For example, when Jim says that
the expression x=mg/k implies that when m increases, x increases, this observation is assumed to be
true independent of the values of the quantities involved. In contrast, in Special Case devices,
conclusions are drawn for cases in which the values of quantities that appear in an expression are
somehow restricted. The devices in this class are listed in Table Chapter 4. -3.
RESTRICTED VALUE
SPECIFIC VALUE
LIMITING CASE
RELATIVE VALUES
Table Chapter 4. -3. Devices in the Special Case Class.
To illustrate, consider the following example from Alan and Bob’s work on the Mass on a
Spring problem. In this example, Alan checks the sensibility of the expression F = -ky that he has
127
written for the force applied by the spring. In his interpretation, Alan refers to a diagram he has
drawn, which I have reproduced in Figure Chapter 4. -5. The dotted line in the diagram marks
the location where y is equal to zero, the rest position of the spring in the absence of any applied
forces. Locations above this line are positive and locations below are negative.
y=0
m
Figure Chapter 4. -5. A diagram from Alan & Bob's work on the spring problem.
To check the sensibility of his equation for the force, Alan considers cases where the position
of the spring, y, is positive, negative and zero:
F = -ky
Alan
If you look at this point [g. point where dotted line intersects spring] there'd be no force
due to Hooke's law, and that's where it'd remain at rest. If you're looking above this axis
[g. dotted line then above dotted line] you're looking at a negative force so that's gonna
push it down. So, if you get closer to the ceiling, the spring force is gonna push you down.
If you're below this axis um you're going down but now you're gonna have a force because you're gonna have two negatives [g. -ky term] you're gonna have a force that's
sending you back up towards the ceiling. So it's just a question of signs.
In this interpretation, Alan concludes that his expression for the spring force makes sense because
his sign analysis yields forces that are in the correct directions. For example, for positions below
the y=0 line, y is negative. Substituting a negative value for y yields a positive value for the force,
which implies that the force is acting upward. This makes sense since a spring that is pulled away
from its rest position exerts a force back toward that rest position.
In the above example, Alan is applying what I refer to as the
RESTRICTED VALUE
device, he
checks whether the equation makes sense by determining its behavior when the range of some
quantities are restricted to a limited region. In this case, Alan considers the cases when the value of
y is restricted to being negative (less than zero), and then when it is restricted to being positive
(greater than zero).
In addition to considering the cases where y is positive and negative, Alan also considers the
special case when y is precisely zero, noting that the spring force is zero when y=0. This is an
instance of the SPECIFIC VALUE device. In SPECIFIC VALUE, rather than simply restricting a quantity
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to a range of values, students consider the case when a quantity is restricted to a precise value.
Presuming a quantity to be zero is particularly common, but any value can potentially be used. In
this next example, Mark and Roger are trying to determine if the expression v = vo + at makes
sense, and they consider a few specific values. In addition to this expression, they have also written
a statement on the board specifying that the value of the acceleration is a=32 ft/s2.
v = vo + at ; a = 32 ft 2
s
Mark
Say it starts at zero, right? And it has an acceleration of this much. [g. 32ft/s2] How far
has it gone- what's the velocity after three seconds? Acceleration [g. 'a' in v=v o+at; g.
32ft/s2] times three, right? Cause every second it's increasing by thirty two feet per
second. This is you can say is equal to thirty-two feet per second, per second, right?
Roger
Uh-huh.
Mark
So per second it increases by this much [g. 32ft/s2] in velocity. Right? So, if it starts at
zero, after one second it's going this fast, after two it's going sixty-four, after three it's
going ninety six. See what I'm saying?
In his above statements, Mark presumes throughout that the value of the initial velocity, vo , is zero,
and then he considers several specific values for the time. Initially the velocity is zero, then after
one second, it’s 32, after two seconds, it’s 64, and after three seconds, it’s 96.
Notice that, in the above example, the values that Mark considers each correspond to a
particular moment in the motion. This is frequently the case when the SPECIFIC VALUE device is
applied; a student will elect to restrict a quantity to a specific value because that value corresponds
to some interesting time during the motion. Does this mean that these cases could also be
accounted for by the SPECIFIC MOMENT device? The answer is no, and to explain why I will refer
back to one of the examples I previously presented of SPECIFIC MOMENT . Here, Ella is looking at an
expression from the Air Resistance task, in which the forces of gravity and Air Resistance are
equated.
Cv 2 = mg
Ella
The terminal velocity is just when the - I guess the kind of frictional force from the air
resistance equals the gravitational force?
The point is that, in this interpretation, Ella does not restrict the expression to a specific moment
in the motion; rather, this equation is only true at a specific moment. When we apply SPECIFIC
MOMENT,
we are saying that an equation is only valid for describing a particular moment in a
motion. In contrast, when employing a Special Case device, we consider the behavior of an
expression within a narrower range than its presumed regime of validity.
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The LIMITING CASE device is a version of SPECIFIC VALUE in which a quantity takes on an extreme
or limiting value. Recall that, for the Running in the Rain task, Jack and Jim composed an
exponential expression for the “probability” of being struck by a raindrop.
Ae −v
Jack
So, you could say the probability if you're traveling at infinite velocity is very extremely
small.
Here, Jack states that if the velocity, v, becomes infinite then the probability of being struck by a
raindrop is very small. This is an example of LIMITING CASE .
In Jon and Ella’s work on the Running in the Rain task we find an instance from the other
extreme. In this example, Jon and Ella are attempting to interpret another complicated expression
that they generated while working on this task. Recall that v R is the speed with which the person
crosses the street. Here they get some work out of considering the case where vR is zero:
vRh
# drops  
w
# drops running = 
wx 
+ 
 A ⋅ s   v v2 + v2 vR 

 d R d
Ella
Jon
Yeah. // And in an extreme case where you’re just standing still then V R [g. vr in bottom of
first term]
// You just put in-
Jon
goes to zero.
Ella
would equal zero.
Jon
And you get=
Ella
=Then this (…) go to zero then you’d be in trouble. [g. last term]
Jon
Yeah, that would be a problem.
Ella
Oh that’s because you’d be standing there forever. (3.0) So I guess that makes sense.
Jon
Oh, yeah. So it go to an infinite.
In this case, imagining that vR is zero led Jon and Ella to a striking conclusion: Although the first
term is simply zero, the second term in the expression is infinite, which means that the person
crossing the street becomes infinitely wet. This is very wet indeed! However, after a moment’s
worry, Jon and Ella decided that this result is precisely right. Traveling at zero speed means that
the street-crosser is “standing there forever,” in which case, it is not surprising that they would get
infinitely wet. Thus, the result of this interpretation is consistent with the conclusion that this
complicated expression is correct.
It is interesting note that focusing on one quantity in the expression and imagining that this
quantity is zero can highlight certain aspects of the structure of the equation and makes many
details irrelevant. For example, the radical in the first term is simply wiped out of consideration.
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Of course, this interpretive strategy has the drawback that it does not allow Jon and Ella to draw
detailed conclusions concerning the validity of this portion of the expression.
When we examine the special case in which a quantity is zero or infinite, as in the above
example, this typically has the effect of highlighting broad aspects of the structure of an
expression. Consider what happens when we substitute zero for a quantity. If the quantity appears
in the numerator of a term, then that term is simply wiped out; the whole term becomes zero.
There is no need to know any of the other quantities that appear in the term, as long as the
denominator is non-zero. If the quantity in question happens to be in the denominator, then the
term blows up to infinity. Again, the values of other quantities in the term are irrelevant. Note that
this is just the kind of sensitivity to the structure of the equation that is associated with forms; a
key feature here is whether a symbol is in the numerator or denominator, and structure at the level
of terms is highlighted.
The final device in this class is the RELATIVE VALUES device. In all of the other devices in this
class, the value of a single quantity that appears in an expression is somehow restricted. In RELATIVE
VALUES,
two quantities are picked for attention, and the values of these two quantities are
restricted relative to each other. For example, we might specify that one quantity is larger than the
other, that one quantity is much larger than the other, or that the two quantities are precisely
equal.
A variety of examples can be found in the Buoyant Cube problem. As we have seen, each pair
of students derived a relation for the amount of the ice cube that lies below the surface of the
water, and this relation involved the ratio of the densities of ice and water multiplied by the length
of a side of the cube. To interpret this expression, students varied the relative values of the two
densities involved. Sometimes, as was stated in the problem, the density of ice was presumed to be
less than that of water. But, in some cases, students imagined that the densities were equal or even
that the density of ice was greater than the density of water.
D=
ρi
L
ρw
Mike
Does it seem sensible? Well, let's see. Um. If ice is lighter than water, [g. ρi then ρw in last]
then ice will float.
Karl
It makes sense because this is gonna be less // than one. [g. ρi/ρw in last]
// Less than one.
Mike
Karl
And we know that it's gonna float.
Mike
Right. So, yeah, that makes sense.
Karl
That makes sense.
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________________
ρL
S= i
ρw
Jack
This does seem to make sense cause if ice had the same volume as water [g. ρi then ρw],
er, same density as water, then L would equal S which means the cube would be right at
the surface. [g. diagram]
________________
ρL
S= i
ρw
Jim
Well, if - if you - okay. Let's say rho of ice [g. ρi in last] wasn't point nine two, it was
something greater than water,
Bruce
Uh-huh.
Jim
So, it's something greater than one [g. ρi in last]. That means this would go up [g. L] and S
would go up [g. S in last] so which means it's sinking, which would make sense, cause if
the block was heavier then it would sink more.
The examples above provide clear instances of this last device. In the first passage, Mike and Karl
presume that, as stated in the problem, the density of ice is less than the density of water. In the
second example, Jack considers the case where the densities are equal, which implies that the
amount of the cube below the surface is equal to the length of a side of the cube. Finally, in the
last example, Jim considers the case where the density of ice is greater than the density of water,
which appears to suggest that the amount of the cube below the surface is greater than L. To Jim,
this implies that the cube is “sinking.” As we will see in a moment, there are some problems with
this last interpretation.
It is worth taking a moment to remind the reader of how I explained these examples in terms
of symbolic forms. Two forms are involved in the interpretations of these equations. First, the
ratio of the densities of ice and water is seen in terms of the RATIO form. In addition, this ratio,
treated as a unit, is seen as SCALING the length L.
With these forms in mind, we can now get a little of a feel for how devices and forms work
together in these interpretations. Recall that the RATIO form is associated with comparisons of the
quantities that appear in the numerator and denominator of the ratio. That’s what going on here,
the students are basing their interpretations on comparisons of the densities. Furthermore, the
forms are allowing inferences to be drawn. The RATIO form permits the conclusion, for example,
that if ρi is greater than ρw , then the ratio is greater than one. And SCALING allows inferences
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concerning whether the amount of the cube below the surface is more or less than the length of an
entire side.
Inventive combinations of devices
To this point, I have been proceeding under the conceit that I can easily place interpretive
utterances into distinct categories. Although the majority do fall neatly into the categories I have
outlined, some interpretive utterances seem to involve aspects of more than one category. This is
not a problem as long as we think of representational devices as a repertoire of interpretive
strategies that can be combined in an inventive manner.
Note that this is one place that talking about representational devices as knowledge pays off.
Even though some interpretive utterances may not line up uniquely with the categories I defined,
we can still think of these utterances as generated by the set of devices that I listed. Thus, the
hypothesis is that representational devices constitute elements of knowledge that can be learned as
separate elements, and then employed, separately and together, to generate interpretive stances.
I want to present just one example of an utterance produced by an inventive combination of
devices. This example pertains to Mike and Karl’s work on the Air Resistance task. I have had a
number of occasions to discuss their equation for the acceleration:
a(t ) = −g +
f (v)
m
Like all of the subject pairs, Mike and Karl decided that the force of air resistance should be
proportional to velocity, and they wrote the relation f(v) = kv. Substituting this relation in the
above expression for the acceleration yielded
a(t ) = −g +
kv
m
Now I want to jump to Mike and Karl’s work on part (b) of the Air Resistance task, in which
the students were asked to consider the effects of a wind that blows straight up. To account for
this wind, Mike and Karl modified their expression so that the velocity of the wind, vw , was added
to the velocity of the ball:
a(t ) = −g +
k
(v + vw )
m
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Finally, with this expression in hand, Karl proceeded to interpret it as follows:
Karl
So, if the wind is blowing up, it causes the terminal velocity, this acceleration [g.
k/m(v+vw)] to reach this acceleration [g. -g] faster.
Karl’s interpretation involves two imaginary processes through time, one process in which there is
wind, and one in which there is not. In both of these processes, the second term grows in time
until it cancels the first term. The point of Karl’s interpretation is that this happens more quickly
in the case in which there is wind.
I understand this interpretation as involving two representational devices: (1) There is a
comparison across two fundamentally different situations (wind and no wind) and (2) there is a
process through the time of the motion. Thus, this interpretation is based on an inventive
combination of two representational devices, CHANGING SITUATION and PHYSICAL CHANGE .
Inappropriate and Inconclusive Interpretations
In the previous section, we sifted through students’ repertoire of interpretive strategies. The
extent and flexibility of this repertoire is, in itself, striking. It seems that, when it comes to
interpreting equations and using them meaningfully, third semester physics students already
possess a great deal of expertise. However, although I tended to emphasize the positive in this
chapter, there are limits to students’ abilities. As we have already seen in some examples, students
do not always interpret equations flawlessly; they sometimes give inappropriate interpretations and
sometimes get stuck with inconclusive results. These inappropriate and inclusive interpretations are
the subject of this section.
To begin, as I have already mentioned, Alan’s BALANCING /SPECIFIC MOMENT interpretation of
the equation µmg=ma can be confidently described as “incorrect.” It presumes the existence of a
second influence and inappropriately associates the equation with a single moment in the motion,
the moment when the block comes to rest.
Students also produced a number of inappropriate interpretations of their final expression
from the Buoyant Cube problem. We encountered one of these interpretations during my recent
discussion of Special Case devices.
ρL
S= i
ρw
Jim
Well, if - if you - okay. Let's say rho of ice [g. ρi in last] wasn't point nine two [g. .92 on
block], it was something greater than water.
Bruce
Uh-huh.
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Jim
So, it's something greater than one [g. ρi in last]. That means this would go up [g. L] and S
would go up [g. S in last] so which means it's sinking, which would make sense, cause if
the block was heavier then it would sink more.
In this passage, Jim imagines the case in which ρi is greater than ρw , which seems to imply that S
is greater than L. But there is a big problem with this interpretation. In order to derive the above
expression, Jack and Jim equated the upward and downward forces on the block and, in so doing,
they implicitly assumed that the block does float. However, if ρi were greater than ρw , then the
block would sink and the assumption that the forces are equal does not hold. This means that the
above equation cannot correctly be used to describe the case in which ρi is greater than ρw . Thus,
Jim’s interpretation is inappropriate.
It is interesting that such an innocuous seeming expression can be subject to serious errors in
interpretation. We might be tempted to think that, once this expression is derived, then all of the
hard work associated with solving the problem is complete. But the interpretation of an expression
can pose a serious challenge. In fact Alan and Bob made precisely the same error in their work on
the Buoyant Cube problem, though Bob immediately caught the mistake:
ρ L
x = ice
ρh2 o
Bob
So, okay, so if the density [g. ρice] is greater than water [g. ρh20]. Um, X is gonna be greater
than L. Um, okay, makes sense.
Alan
The density is // greater than (one).
Bob
// This - we assumed, we assumed it floated actually. [w. "floated"] We have
to assume it floated. If we didn't assume it floated, then this wouldn't give us the right
answer.
Now I want to consider another way in which limits in a student’s abilities can manifest
themselves during attempts to produce an interpretation. In some instances, students struggle to
produce any useful interpretation of an expression. The interesting thing about these instances is
not that students say things that are wrong, but that they have trouble saying anything cogent
about an equation or drawing a useful conclusion from an equation. An interesting set of related
cases involve what I call “inconclusive” interpretations. In these interpretations, a student sets out
to construct a certain type of interpretation of an expression, but, in some manner, the
interpretation fails to produce useful results.
We have already encountered some examples of inconclusive interpretations. In attempting to
interpret one of their complicated expression for the Running in the Rain task, Jon and Ella
explored a tactic that failed to produce useful results:
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

h
w
 # drops  
+
=
wx
vd2 v R 
 A⋅s  
1+
v
 d

v 2R




Ella
Okay, V R is bigger. Then (10.0),, This is gonna get smaller, [w. arrow pointing to second
term] but this is gonna get bigger. [w. arrow pointing to first term] I think.
Jon
Yeah, I think - that gets smaller [g. second term] cause V walking is smaller than V
running.
Jon and Ella have run into a problem here. They see that if vR gets bigger, then the second term in
their expression gets smaller. However, under these circumstances the first term gets larger. So,
given these conflicting observations what can Jon and Ella conclude about the overall expression?
The result is simply inconclusive; this method of interpreting does not tell them whether the total
rain becomes greater or less. Jon and Ella understood this difficulty clearly:
Jon
And, um, we’re trying to get it in a form that we can look - we can say, well V walking is less
than V running and then compare the equations. [g. expression] But right now since this
thing is getting bigger [g. second term] - I mean this thing is bigger when you are walking
and this thing [g. first term] is smaller when you’re walking,
Bruce
Hm-mm.
Jon
They sort of - we don’t know which one is bigger than the other.
Bruce
Hm-mm.
Jon
Or has more significance than the other.
So, sometimes students reach an impasse when trying to construct a useful interpretation of an
expression. In this case, I am not sure that an expert could make a CHANGING PARAMETERS
interpretation of this expression do much more work for them. Because Jon and Ella made some
errors during earlier steps in their work, they produced a somewhat less tractable and more
difficult to interpret expression than other students in the study.
Before wrapping up this section, I want to present one additional example of an inconclusive
interpretation. This example involves the Shoved Block problem, in which a heavier and lighter
block are shoved so that they have the same initial speed. The question that students must answer
is: Which block goes farther? As I have discussed, my subjects displayed two different intuitions
concerning this problem. The first intuition is that the heavier block should go less far because it
presses down harder on the table and thus is subject to a higher frictional force. The other
intuition produces the opposite conclusion; the heavier block should go farther because heavier
things have more momentum and are thus “harder to stop.” In their work on this problem, Alan
and Bob quickly mentioned both of these intuitions. Alan began with:
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Alan
And then if the block was heavier, it'd just mean that our friction was greater, and it would
slow down quicker.
Then, shortly later, Alan stated the second intuition:
Alan
That seems kind of- Something seems strange about that because if you had like a block
coming at you, a big heavy block, and a small lighter block. Two different instances. And
you have this big heavier block coming toward you and you want it to slow down, it
certainly takes much more force for you to slow down the bigger heavier block than the the lighter one.
Bob
Oh wait, that's true!
Alan
Because, even though they're both working on the same frictional surfaces, just logically I
know the big heavier block takes-, I mean football is all about that, that's why you have
these big three hundred pound players.
So, which intuition is correct? Answering this question is a job for equations. Bob first attempted
to use only the equation F=ma to determine which intuition is right:
F = ma
Bob
I mean, F is obviously going up as you get um - The frictional force is going up as you get
heavier [w. an arrow going up under the F], and the mass is going up [w. an arrow under the
m]. So, I mean it depends if.
As indicated by the arrows that Bob drew, both the force and the mass increase as you go between
the light and heavy-block circumstances. Essentially, Bob is running a CHANGING PARAMETERS
device that involves simultaneously changing the force and the mass. His hope, we can presume, is
that he will find out how the acceleration changes as the force and mass are increased. However,
the result of this interpretation is inconclusive. The fact that the force is increasing implies that the
acceleration should also increase, but the increasing mass implies that the acceleration should
decrease. As we know, it turns out that these effects precisely cancel, but Bob’s above method is
not sufficient to determine this fact. Following this inconclusive interpretation, Alan and Bob went
on to do a more complete solution and they did determine that these two effects precisely cancel,
and therefore that both blocks travel the same distance.
The upshot of this section is that interpreting equations can be difficult. Sometimes students
produce incorrect interpretations, and sometimes they have trouble producing any useful
interpretation at all. These observations are important for several reasons.
First, the difficulties observed here point to issues that must be dealt with by physics
instruction. Our instruction should lead to students having the ability to extract correct and useful
interpretations from the expressions that they write. Such a goal is a worthy end in its own right,
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but it is important to note that interpretations also play a role in the problem solving process. To
take a simple example, if a student decides that an expression does not “make sense,” then it is
likely that they will try to fix the solution that led to the expression. For such reasons, the ability
to correctly interpret expressions has implications for successful problem solving and therefore
must be addressed in instruction.
Furthermore, the observations in this section help to support the notion that it is important to
account for interpretive utterances in our models of symbol use. If interpretation were always
smooth and unproblematic, we could perhaps wrap it in a black box and take the ability to
generate interpretations for granted, just as I am taking for granted students’ ability to plug
numbers into an equation and their ability to perform syntactic manipulations. However, the
above examples make it clear that the act of interpreting an equation is not always unproblematic.
Finally, I believe that the examples in this section can generally help to support the fact that
interpretive phenomena are real—there really is a class of utterances worth calling “interpretations”
and there really is knowledge devoted to interpreting expressions. These examples help because,
like many other capabilities, the ability to interpret expressions reveals its existence and nature in
cases of breakdown. Alan’s interpretation of the equation µmg=ma sticks out like a sore thumb in
the protocol of the session in which it appears, and Jon and Ella’s frustration when they could not
generate a useful interpretation of their complicated expression was dramatic. Thus, these
inappropriate and inconclusive interpretations stand out as real phenomena to be accounted for.
And once we acknowledge that there are inappropriate interpretations, then we have already
implicitly acknowledged that there is such a thing as an interpretative utterance, of any sort.
Interpretation as ÒEmbeddingÓ
I have, at times, described the project in this chapter as taking interpretive utterances, putting
them in categories, and then characterizing the common features of utterances within each of the
categories. Now I want to take a moment to think about whether we can come up with any
characterizations that span the categories. Are there any overarching statements we can make
about all of the interpretive utterances that we have seen? Are there any general ways to
characterize interpretations of equations?
A possibility that I mentioned was that interpretation could be equivalent to the stating of
truth conditions. I will have more to say about this possibility and theories of equation meaning in
a moment. Another possibility is that interpreting an equation may boil down to making
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correspondences between the equation and the motion that the equation is supposed to describe.
At least at first blush, this works well for some interpretations we have seen, such as SPECIFIC
MOMENT
interpretations, which project an equation into a single instant in a motion. However, it is
a little less clear how this correspondence characterization will work for some other varieties of
interpretations, such as those involving the CHANGING PARAMETERS device. CHANGING PARAMETERS
interpretations incorporate stories that span multiple versions of the same motion, and CHANGING
SITUATION
interpretations interpret an equation by considering motions other than the one that is
specifically the subject of study. Thus, it does not seem like the notion of correspondence between
equation and the physical world does a good job of characterizing the breadth of interpretations
that we have encountered, at least in its simplest form.
But my goal here is not to list a variety of possibilities and argue among the alternatives.
Thoroughly arguing for a choice among such alternatives would be an extensive undertaking.
Rather, I want to present what I think is a passable, rough characterization, what I call the
“interpretation as embedding” view. This characterization is not designed to serve as anything like
a definition or even as a rigorous description. Instead it is designed to function as a mnemonic
that captures some features of what I have found, particularly features that some may find
surprising and that distinguish my viewpoint from competitors. In this section and the next two, I
will work through some of the aspects of this “embedding” view of interpretation, and, in the
process, I will attempt to uncover some of the general conclusions concerning the nature of the
interpretation of equations that can be drawn from the observations in this chapter. Although I will
be building on the examples and observations presented earlier, my comments in these sections
will be, in general, much more speculative than those in the rest of this chapter, and I will be going
somewhat beyond the core of my theoretical position.
The first thing I will do is make clear what it means to interpret by embedding. To do this,
consider the equation that Jim interpreted:
x=
mg
k
Now I will present my own interpretation of this equation, which is designed specifically to help
illustrate the nature of interpretation by embedding:
Bruce
Suppose you’ve written this equation on a sheet of paper. Then you plug in some value for
the mass and get out a result. Then you can plug in another, larger value for the mass and
get out another result. If you do this, then the answer you get for the larger mass will be
bigger.
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In this passage, I have interpreted this equation by placing it within a little story. We can, if we
like, construe this story as being about the equation itself, but we need not. The equation is only
one element in a story that includes other elements, playing other roles. The story includes, for
example, a person using the equation, writing it on a sheet of paper and plugging in values. And
there are narrative elements that are not necessitated by the equation, such as the choice to first
substitute one value, then substitute a larger value.
This is the essence of the “embedding” characterization of interpretations: The equation is set
within a context or framework that is larger than any possible listing of the entailments of the
equation. The framework is not fully dictated by the equation, and not every aspect of the
framework corresponds to some feature of the equation. Thus, in this view, it is simply not
appropriate to ask how features of the equation correspond to aspects of some world. If we are
looking to ask a question in this vein, we should instead ask how the equation articulates with or
fits with the other elements of the story; in other words, we should ask how the equation is
integrated into the larger framework of the interpretation.
Ambiguity and the Abstractness of Devices
There are some additional, somewhat more subtle points to be made about the nature of
embedding interpretations, as they appear in the student utterances in my data corpus. Consider
the following CHANGING PARAMETERS interpretation, in which Karl interprets an equation for the
acceleration from the Air Resistance task:
a(t) = −g +
Karl
f (v)
m
Okay, this sign is negative [g. -g] this number [g. f(v)/m] as the mass- as you increase the
mass,, Well, one problem- from one ball to the other the mass is doubling. Right, so you
double this mass [g. m] so this number [g. f(v)/m] is getting smaller.
Contrast Karl’s interpretation with the contrived example I presented in the previous section. In
the contrived example, certain aspects of the story I told were very clear. For example, it was quite
clear where the story took place and who the actors were; the story involved a person working with
an equation, writing things on a sheet of paper. However, in Karl’s above interpretation, it is
somewhat less clear where the story takes place. As in my contrived interpretation, this one
involves a changing mass. But we can imagine this change as occurring in either of two possible
settings. First, just as in my contrived example, we could treat Karl’s interpretation as taking place
140
in the world of a problem solver, in which someone is using the equation. In this setting, there is a
hypothetical person plugging-in different values for the mass and obtaining various results.
But we can also imagine Karl’s interpretation as taking place in an alternative setting, the world
of objects and motions. If we adopt this stance, then Karl’s story is about a sequence of physical
experiments in which someone takes objects of various mass to a great height and then drops
them.
So which of these settings does Karl have in mind in his above interpretation? What does Karl
mean when he says “as you increase the mass?” Does this increase take place in the physical world
or in the here-and-now of the problem solving world?
Of course, there is the possibility that Karl’s interpretation just happens to be more ambiguous
than the norm. To see that this is not the case, the reader is encouraged to flip back through this
chapter. Such a perusal will reveal that the majority of interpretations are ambiguous in the same
manner. However, I will present one additional very brief example here. This is a typical
interpretation of the final expression from part (b) of the Air Resistance task, in which the effect of
wind is considered.
vt =
Bob
m
g − vo
k
If the wind is blowing up, the velocity - terminal velocity is going to be less than it was in
this case. [g. vt=m/kg] I mean, we have the same mass. This is M K G. [w. vt=m/kg; draws
a box around it] This is no wind. [w. "no wind"] No wind. And this is with wind. [w. "w/ wind"
and boxes wind result] Okay, so if the wind is blowing up, the terminal velocity is gonna be
less.
As is typical of interpretations of this expression, Bob compares this result to the case in which
there is no wind. The question here is: what kind of things are these two “cases”? Are they two
real-world experiments? Or is each “case” a problem to be solved in a physics class?
I believe that this ambiguity is built into the nature of these interpretive utterances. Although
some interpretations may sound more or less like they take place in one of these two possible
“worlds,” in most cases it is hard to decide which world students have in mind. Furthermore, I do
not believe that this is just a matter of limitations in what we, as analysts can reliably say about
protocol data. Rather, I believe that there is a fundamental sense in which these utterances are
truly ambiguous. According to this view, it is incorrect to ask whether an interpretive utterance is
about the physical world or about the here-and-now world of symbol use. This means, for
example, that when a person says the word “velocity,” they mean neither the velocity symbol nor a
velocity in the world, they somehow mean both at once.
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I want to propose a way of thinking about this observed ambiguity that I hope takes away a
little of the mystery and strangeness. The proposal is that we can think of the interpretation of
physics equations as involving embedding in worlds or settings that are abstractions of “real”
settings or real processes. The idea is that these specially adapted hypothetical settings involve less
information and are less thoroughly defined than real settings in which humans live and act. And
because they are more abstract, they can simultaneously be consistent with more than one of these
real settings for human activity.
To illustrate, I want to point out some of the other ways that student interpretations tend to
be ambiguous. In CHANGING PARAMETERS interpretations, we describe a process in which some
parameter increases or decreases. There are a couple of possibilities concerning how this change
might occur. First, the change could be a continuous change, in which the parameter is supposed
to pass through every possible value. Second, the parameter could change discontinuously, from
one value to another. Again, student utterances were generally ambiguous in this regard.
Furthermore, it is often not possible to determine whether the supposed change even happens
through time. Do you do one experiment and then the other? Or are the two experiments simply
two cases to be considered in parallel, with no assumed time-ordering?
Again, the idea is that the worlds and processes described in interpretations can be considered
to be abstract and there are thus limits to what we can say about these interpretations. We can say
that they embed the equation in a hypothetical process of change; i.e., they employ the
PARAMETERS
CHANGING
device. But we cannot say that this process takes place in one world or another, and
we cannot necessarily claim that the process is continuous or discrete. The point is that if we try to
understand Karl’s interpretation as taking place in a real setting—as describing some specific
experience that we could really have, such as dropping a ball or writing an equation—then we are
filling in details that are not part of or even implicit in Karl’s statement.
I believe that it is plausible that this very particular kind of abstractness is also characteristic of
the interpretations of physicists—not just the moderately expert students in this study. If this is
true, it points, once again, to the fact that there is a lot for students to learn here; there is much
that is special about interpreting physics equations. This is not really a surprising observation.
Physicists have a standard way of talking with and about equations, and it makes sense that there
will be some particularities to this discourse. In learning to interpret equations, students are
learning to make utterances about expressions that are like those that a physicist would make, and
it turns out that these utterances involve the sort of abstract constructions I have described here.
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Postscript: A function for ambiguity
In this section, I have argued that interpretive utterances tend to be ambiguous in certain
specific respects. And my stance has been purely descriptive; I have simply argued that “this is how
physics discourse is.” The question arises of whether we can say why it is that interpretive
utterances are ambiguous in this way. Does it serve any obvious function?
I have one guess at an answer to this question. Above, I distinguished two possible settings in
which an equation could be embedded. The first is the world of problem solving, in which a
person is writing and manipulating equations. The second is the physical world that the equation is
presumed to describe. It is clear how an equation can be embedded in a story that takes place in
the first kind of world—equations are one of the things that appear in a problem solving setting.
But, what could it possibly mean to embed an equation in a story about the physical world?
Stories about Newton’s physical world include objects and motions, not equations.
In discarding correspondence as a fundamental means of connecting the symbolic and the
physical, we are left with a problem. Without correspondence, how can we make an equation have
anything at all to do with the physical world? This is where the above type of ambiguity can
actually help. The abstractness built into interpretations allows a blurring—a sort of welding
together—of the physical world and the here-and-now of the problem solving world.
Interpretations thus embed equations in a story that is simultaneously about the physical world
and the problem solving world. This permits a connection, of sorts, between equations and the
physical world.
On the ÒMeaningÓ of an Equation
I began this chapter by wondering if we could better understand interpretations of equations
by first constructing a “theory of meaning” for equations. The route that I ultimately took was a
different one, however. Rather than beginning with an a priori theory of meaning, I instead took
an empirical approach to discovering what an interpretation of an equation is, in practice. Now this
project can come full circle. Assuming that what we have seen in this chapter is broadly
characteristic of interpretation in physics, what can we say about the meaning of equations?
Let’s begin by looking closely at a brief example. In this example, Bob gives an interpretation
that involves the NO DEPENDENCE form and the CHANGING PARAMETERS DEVICE. The equation in
question is from the Shoved Block task.
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a = gµ
Bob
So, no matter what the mass is, you're gonna get the same, the same acceleration.
According to Bob, this equation says that the acceleration of the block is independent of the mass.
That is Bob’s statement of a meaning for this equation. Now, notice that this interpretation has a
strange feature: The meaning that Bob gives for this equation involves a quantity that does not
even appear in the expression! For this reason, this interpretation is not consistent with simple
versions of a theory in which meaning is something that can be just “read out” of an equation; the
meaning cannot be intrinsic to the arrangement of symbols in the expression. In particular, there is
no way that the arrangement of symbols can explain why Bob chooses to mention that this
expression does not depend on the mass, rather than mentioning all of the other many parameters
that do not appear in this expression.
In this next example, the plot thickens further. Here, once again, is Karl’s interpretation of an
equation from the “wind” problem that combined the PHYSICAL CHANGE and CHANGING PARAMETERS
devices:
a(t ) = −g +
Karl
k
(v + vw )
m
So, if the wind is blowing up, it causes the terminal velocity, this acceleration [g. second
term] to reach this acceleration [g. first term] faster.
The noteworthy feature of this interpretation for the present discussion is that the meaning that
Karl assigns involves a comparison to a fundamentally different physical situation. In fact, as we
saw, many interpretations from the wind task involved the CHANGING SITUATION device and
comparisons of this sort. For example, students frequently interpreted the final result for the
terminal velocity by comparing it to the terminal velocity for the case in which there is no wind:
vt =
Bob
m
g − vo
k
If the wind is blowing up, the velocity - terminal velocity is going to be less than it was in
this case. [g. vt=m/k g] I mean, we have the same mass. This is M K G. [w. vt=m/k g; draws
a box around it] This is no wind. [w. "no wind"] No wind. And this is with wind. [w. "w/ wind"
and boxes wind result] Okay, so if the wind is blowing up, the terminal velocity is gonna be
less.
The point I want to make about these interpretations is that they are not solely about the physical
situation that the equation presumably describes. Instead, these interpretations are based around a
comparison to an altogether different physical situation. This observation forces us to the strange
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conclusion that the meaning of the above equations, as embodied in students’ interpretations, has
something to do with a physical circumstance that, in a sense, is not “mentioned” in the equation.
This point is central to the embedding view of interpretation. The building of “meaning” for
an equation involves dragging in more than what is straightforwardly entailed by the symbols that
appear in the expression. The existence of CHANGING SITUATION interpretations, which involve
comparisons to altogether different physical circumstances, helps to make this point evident.
Because Bob’s interpretation involves a “less than” comparison, the meaning is intrinsically
relational. Furthermore, it is not clear that there is anything about the equation itself that instructs
the equation-user what physical circumstance to choose for comparison.
These observations lead to another set of questions. If anything can be used as part of the
meaning of an equation, what determines what extraneous material is dragged in? And, more
generally, if meaning is not a simple function of the arrangement of symbols in an equation, what
else determines the meaning of the expression?
symbolic expression ⇔ forms ⇔ devices ⇔ rest of the context
To investigate this question, we can return to my schematic of the form-device system. Recall
that in my heuristic model, the equation, forms, devices, and the rest of the context form a
system of constraints that adjusts itself to generate an interpretation. Thus, the forms seen by the
student and the choice of device are not only influenced by the equation, but also by the “rest of
the context.” The conclusion I draw from this is that the meaning of an equation as embodied in
interpretive utterances is context dependent; it depends upon the concerns of the moment. What
we choose to compare an equation to and what other entities and circumstances we drag in, will
depend on what’s going on in the here-and-now world of problem solving. With this conclusion in
mind, it is not surprising that students interpret the result of the wind task by comparing it to the
no-wind case since, only minutes before, they had solved the same problem for the case in which
there is no wind. Thus it makes sense that the meaning of the equation for students, in this context,
would involve a comparison between these two cases.
This driving of interpretation by concerns of the moment can take an even more trivial form.
In some cases, the choice of representational device can be seen to be driven directly by the
problem statement. Consider, for example, Bob’s above interpretation of the equation a=gµ in
which he notes that this expression does not depend on the mass. Bob has chosen to be concerned
with mass dependence for the simple reason that this is what the question asks; it asks whether a
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heavier or a lighter block will travel farther. Similarly, interpretations of expressions written for the
Air Resistance task frequently involved
CHANGING PARAMETER
v=
Alan
devices in which the mass varied:
mg
k
So now, since the mass is in the numerator, we can just say velocity terminal is
proportional to mass [w. vt∝m] and that would explain why the steel ball reaches a faster
terminal velocity than the paper ball.
________________
a(t) = −g +
Karl
f (v)
m
Okay, this sign is negative [g. -g] this number [g. f(v)/m] as the mass- as you increase the
mass,, Well, one problem- from one ball to the other the mass is doubling. Right, so you
double this mass [g. m] so this number [g. f(v)/m] is getting smaller.
Again, it is not surprising that these students chose to interpret these equations in this manner
since the problem statement specifically asks how the terminal velocity would differ in the case
when the mass is twice as large.
The upshot of these last few paragraphs is that the meaning of an equation will be dependent
on the concerns of the moment. I want to emphasize, however, that I have tried to do more than
simply announce that equation meaning is “context dependent.” I believe that one of the main
contributions of this work is that I have provided a partial explanation of how concerns of the
moment interact with equation use. What I have argued is that people know a set of interpretive
stances and strategies that I call representational devices, and that these function as a layer between
forms and the “rest of the context.” The concerns of the moment thus feed through
representational devices to determine how equations are interpreted.
So, the bottom line is that equation meaning, as embodied in student utterances, is not a
simple function of the symbols in an equation. And there is no single, universal procedure for
reading out the meaning of an equation. Instead, meaning is built around an equation by the
symbol user via representational devices, and it depends in a complex manner on the concerns of
the moment.
In conclusion, we have once again taken a foray into the physicist’s world of symbols. In the
last chapter, we explored the structure—a meaningful structure—that a physicist sees in a page of
expressions. Now, in this chapter, I have argued that, to enter into this meaningful world of
symbol use, it is not sufficient to know symbolic forms. In addition, initiates in physics must
possess a complex repertoire of interpretive strategies that I have called “representational devices.”
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With forms, this repertoire of devices defines the character of the meaningful symbolic world of
physics.
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Chapter 5. Symbol Use and the Refinement of
Physical Intuition
The model described in the preceding chapters is designed to do a very limited job. To pin down
what this job is, it is helpful to think about time scale. So far in this project, I have only been
working to describe phenomena, such as the interpretation of an equation, with a time scale on the
order of a few seconds. But there are many other phenomena, with a range of associated time
scales, that could reasonably be taken to be part of this project. Here are some time scales that we
might want to consider:
1. The evolutionary time scale (~10 million years). As brains evolved so did cognition. This is, in
part, the sort of story I attributed to Merlin Donald in the first chapter.
1. The historical time scale (~10,000 years). Some authors have wondered whether the nature of
human thought has changed over the course of recorded history. This time scale is the purview
of researchers like David Olson and Jack Goody, both of whom have worried about the effects
of the development of writing on thought.
1. The sociocultural time scale (~100 years). We can imagine that there are ebbs and flows in the
thinking of particular cultures and of the practitioners within a domain.
1. The ontogenic time scale (~10 years). The ontogenic time scale is Piaget’s realm. Piaget was
interested in characterizing broad changes in the nature of thought, as individuals progress
from birth to adulthood. While Donald’s stages are phylogenic—they pertain to the
evolutionary development of the human species—Piaget’s stages describe the development of
individuals.
1. The learning time scale (~1 year). Even when no broad changes in the nature of thought are
implicated, minds can still change and we can call these changes “learning.” For example,
during their first one or two years in college, students can learn some physics.
1. The problem solving time scale (~10 minutes). This is the time scale of the problem solving
research I discussed in Chapter 1. Problem solving research is primarily concerned with
describing behavior of individuals that spans from a few minutes to a couple of hours, as they
work out the solution to problems.
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1. The thought time scale (~1 second). The thought time scale is associated with phenomena like
the generation of an interpretation of an equation: A student looks at an equation, comes to
see it in a certain way, and then says something.
For the most part, I have thus far been dealing with phenomena in time scale (7), with a little
thrown in about (6). The “model of symbol use” I have presented is intended to describe short
time scale phenomena; the form-device system adjust itself in a few seconds. Of course, as I have
discussed, there are implications of my model for problem solving. Because students can see
equations as meaningful, they can construct novel equations and they are likely to take certain
subsequent actions. Nonetheless, my model does not really describe events on the order of
minutes. For example, I have not been directly concerned with the order and progression of
expressions that are written on the way to a solution.
In this chapter I want to extend my scope of interest, not to include phenomena in time scale
(6), but up yet another order of magnitude to the learning-related phenomena in time scale (5).
My goal here is to provide some account of the learning that takes place over the period of the
first few years of physics instruction. Since my data corpus does not include a longitudinal study,
this extension of my model will necessarily be somewhat more speculative than the core of the
model, which was presented in the previous two chapters.
There are, however, some reasons to believe that I can do pretty well with these speculations.
First, the students in my study are neither complete novices nor true experts, they are at an
intermediate level of expertise. For this reason, the “snapshot” that I have in my data corpus
should have useful properties from the point of view of speculating about the broad sweep of
development. Second, although the data corpus that is at the center of this work does not include
longitudinal observations, there is other research that I can draw on. In fact, the educational
research literature is replete with characterizations of the knowledge possessed by younger
students and of characterizations of “intuitive physics” knowledge, the knowledge that students
bring to physics instruction. Therefore, in essence, I have snapshots of some knowledge systems as
they exist prior to physics instruction, and some knowledge that exists during an intermediate
stage in physics instruction. Given these snapshots, I can speculate about the processes that connect
these knowledge systems.
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Cognitive Resources: Prior Knowledge and Physics Learning
How do we interpolate between snapshots of knowledge systems? To deal with this question,
we need to develop a general means of thinking about how old resources come to be used in new
contexts. There is a set of related questions to be addressed here:
•
How must existing cognitive resources be adapted for use in new contexts?
•
Under what circumstances should we say that new resources have developed out of
existing resources, rather than old resources adapted?
•
When new resources develop, in what ways do these resources relate to the previously
existing resources?
All of these questions relate to the flexibility and generality of cognitive resources. Certainly, we
expect that cognitive resources will change and adapt. We must also expect that, as individuals
learn sufficiently new behaviors, new cognitive resources will develop out of the existing resources.
These issues arise in our study of the learning of symbol use in physics. We must ask: In what ways
does symbol use in physics make use of existing resources and in what ways does it require the
development of new cognitive resources?
In this paper, I will do little to answer these very general questions concerning the specificity of
resources and the manner in which new resources develop from existing resources. My main
contribution will be to point roughly to relevant prior resources and to indicate where I believe we
should say that new resources have been developed. To make this task simpler, I will introduce a
little terminology. I will use the term “cognitive resource” to describe relatively functionally
specific resources, and I will also use the term “cognitive resource cluster” to describe collections of
such resources, grouped according to use in narrowly defined activities.
Cognitive resources and the learning of physics
The above brief discussion of cognitive resources lays out the issues to be considered here very
generally. People have cognitive resources and, when they learn new things, this means that they
adapt these resources and develop new resources. Now I want to start to fill in the particulars of a
story of this sort for physics learning.
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What are the relevant cognitive resources that exist prior to physics instruction? We can narrow
the scope here a little by saying that we are interested in resources that contribute to symbol use in
physics, as I have been describing it. Much prior knowledge is implicated in the development of
even this narrow slice of physics knowledge, so I will limit my discussion to describing a few of the
particularly interesting relations between existing and new resources.
First, students come to physics instruction with a cluster of resources that pertain to the
solving of mathematics word problems. I call these cognitive resources the “algebra problem
solving” cluster (refer to Figure Chapter 5. -1.) This knowledge is certainly relevant to learning to
solve the types of textbook problems that students encounter in their physics courses. The algebra
problem solving cluster is almost exclusively associated with schoolish activities and it is employed
in solving the traditional algebra problems that one would find in a school textbook. It includes
resources relevant to performing the small variety of tasks found there, such as solving given
equations, solving simple word problems, or deriving equations by manipulating given equations.
Of course, there is a wide variety of literature that attempts to describe the knowledge associated
with this activity. Of particular relevance for us is the “patterns” research, described in Chapter 3.
Recall that this research argued that students apprehend arithmetic word problems in terms of a
number of simple patterns, such as “part-part-whole” and “equalization.”
Quantity
e.g. qualitative
proportionalities
Algebra Problem
Solving
Intuitive Physics
e.g. p-prims
e.g. "patterns"
Algebra-Physics
e.g. forms, devices
Figure Chapter 5. -1. A resource cluster story for physics learning.
A second relevant cognitive resource cluster I refer to as the “intuitive physics” cluster. The
intuitive physics cluster is associated with the day-to-day activity that occurs throughout a
person’s life, interacting with the physical world. In particular, I have in mind the activities of
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describing and explaining physical phenomena. Our capabilities in this regard are, of course, quite
extensive and varied. We can describe the geometrical arrangement of objects in space as well as
make predictions concerning the behavior of a physical system. In what follows, I will be
particularly interested in a particular resource within this cluster, diSessa’s “sense-of-mechanism,”
which I discussed in Chapter 1 (diSessa, 1993).
In large part, learning expert physics means learning to quantify the physical world. For this
reason I have included a cluster of resources relating to quantity in Figure Chapter 5. -1. As with
the other clusters of resources, I presume that students enter physics with many resources relating
to dealing with quantities in the world. Children learn to count objects and variously use numbers
across a variety of circumstances. In addition, people develop the ability to deal more generally
with amount, even when the entities involved are not simply denumerable, as when we are talking
about a quantity of liquid. Among the resources that we could consider, I have already discussed
the abilities that people possess in the area of “qualitative reasoning about physical systems.”
This third cluster of resources is less functionally specific than the above two; in fact, it
certainly plays some role in algebra problem solving and intuitive physics activities. In general, we
can expect there to be interesting overlaps and inter-relations among resource clusters. For the sake
of keeping this story simple, I am not going to worry about these inter-relations. My purpose here
is only to give a very simple account that highlights what I believe are some of the more important
resources involved.
The point I have been working toward here is that learning symbol use in physics involves the
formation of a new cluster of resources that I refer to as “algebra-physics.” This cluster involves its
own characteristic resources, including the form-device system I described in the previous two
chapters. Furthermore, I intend for the diagram in Figure Chapter 5. -1 to depict two different
kinds of relations between algebra-physics and the previously existing resource clusters. The first
of these relations is genetic: Algebra-physics resources develop, in part, out of the existing
resources. Second, once the new resources have developed, connections may remain between the
new resources and the previously existing resources.
My purpose here is not to elucidate the mechanisms involved in these two relations. For the
present, I do not intend this list of relations to be much more than an accounting of some of the (a
priori) possible relations between old and new resource clusters. The statement here is really a very
simple one: New resources can develop out of existing resources and, once developed, they may
maintain some types of relations to these prior resources.
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Furthermore, I have no intention of even working through all of the possible relations shown in
Figure Chapter 5. -1. Instead, I will focus primarily on one particular relation between old and
new resources: the relation between diSessa’s sense-of-mechanism and the form-device system.
There are a few reasons for focusing on this particular relation. First, we shall see that the elements
in the sense-of-mechanism—p-prims—are closely related to symbolic forms (in the sense of both
types of relations that are supposed to be depicted in Figure Chapter 5. -1). Second, this focus will
help to show how symbol use can have effects that permeate right down to what, I will argue,
must be considered the roots of physics understanding. And finally, a number of important issues
relating to learning will be clarified by working through this particular relation.
The Developing Sense-of-Mechanism
In this section I begin my focus on the sense-of-mechanism and its role in the development of
the form-device system. I briefly discussed diSessa’s account of this cognitive resource in Chapter
1 and I will provide a somewhat extended discussion here. All of this presentation is derived from
diSessa’s Toward an Epistemology of Physics (1993) and the interested reader should look there for
the authoritative account.
The sense-of-mechanism is a portion of our intuitive physics knowledge—the knowledge that
we gain through our day-to-day experiences in the physical world. One function of this
knowledge is to contribute to our ability to interact in the physical world; it plays a role in the
pushing, pulling, throwing, and pouring that we do in order to live in the world. But the sense-ofmechanism also has some functions that apply more obviously and directly to physics learning: It
allows us to judge the plausibility of possible physical events, make predictions, and it plays a
crucial role in our construction of explanations of physical events.
The sense-of-mechanism consists of knowledge elements that diSessa calls “phenomenological
primitives” or just “p-prims” for short. They are called “primitives” because, as I have already
explained, p-prims constitute the primitive level of our explanations of physical phenomena. In
Chapter 1 I described an example situation in which someone places their hand over the nozzle of
a vacuum cleaner and the pitch increases. A typical explanation of this phenomenon, diSessa tells
us, is that the vacuum cleaner must work harder to overcome the resistance that is applied by the
hand. This boils down to an appeal to what diSessa calls OHM’S P -PRIM as the basis for the
explanation. In OHM’S P -PRIM , the situation is schematized as having an agent which works against
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some resistance to produce a result. The idea is that OHM’S P -PRIM provides the primitive basis for
this typical explanation; the explanation goes precisely this deep and no deeper.
The “phenomenological” part of “p-prim” also merits some comment. P-prims are described
as phenomenological because they develop out of our experience in the physical world. We have
many experiences in the physical world, pushing and lifting objects, and p-prims are abstractions
of this experience. Furthermore, once they are developed, we come to see p-prims in the world. In
sum, p-prims are basic schematizations of the physical world that we learn to see through repeated
experience in the world.
Force and Agency
OHM’S P -PRIM
Constraint Phenomena
SPONTANEOUS RESISTANCE
SUPPORTING
FORCE AS MOVER
GUIDING
BLOCKING
DYING AWAY
Balance and Equilibrium
DYNAMIC BALANCE
ABSTRACT BALANCE
Figure Chapter 5. -2. A sampling of p-prims.
The variety of p-prims
In order to give a feel for the variety and scope of phenomena covered by the sense-ofmechanism, I want to discuss a selection of p-prims (refer to Figure Chapter 5. -2). I begin with
some p-prims from what diSessa calls the “Force and Agency” Cluster. OHM’S P -PRIM is an
example of a p-prim in this cluster. Recall that in OHM’S P -PRIM a situation is schematized as
involving some agent that works against a resistance to produce a result. A related p-prim in this
cluster is SPONTANEOUS RESISTANCE . The resistance in SPONTANEOUS RESISTANCE is different than that
in OHM’S P -PRIM because it is intrinsic to the patient of some imposed effort. For example, the
difficulty that we have in pushing a fairly heavy object can be attributed to SPONTANEOUS
RESISTANC e.
Compare this to the resistance that is imposed by a hand in the vacuum cleaner
situation.
I want to mention two other p-prims in the Force and Agency Cluster. The first,
MOVER,
FORCE AS
has clear relevance to physics learning. In FORCE AS MOVER , a push given to some object is
seen as causing a movement of the object in the same direction as the push. In some
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circumstances, the predictions made by FORCE AS MOVER agree with Newtonian physics, but, in
other circumstances,
FORCE AS MOVER
contradicts the predictions of Newtonian physics. According
to Newton’s laws, an object only moves in the direction of an applied force if the object is initially
at rest, or if the push happens to be in the direction that the object is already moving. Otherwise,
the object will be deflected by the applied force.
The final p-prim I want to mention in the Force and Agency Cluster is DYING AWAY . Like
FORCE AS MOVER ,
this p-prim is associated with the drawing of non-Newtonian conclusions about
the world. The idea behind DYING AWAY is that all motion must, in due time, die away to nothing.
In contrast Newton’s laws predict that, in the absence of any applied forces, objects in motion
continue to move indefinitely.
A second cluster of p-prims pertains to constraint phenomena. These p-prims explain
phenomena by appeal to the constraints imposed by the geometric arrangement of physical
objects. For example, if a rolling ball runs into a wall, the ball stops. We might explain this by
saying that the wall blocked the ball’s motion, or the wall simply “got in the way.” This is an
application of the BLOCKING p-prim. Compare this explanation with one that states that the ball
stopped because the wall applied a force to it.
Another p-prim in this cluster, SUPPORTING , is a special case of BLOCKING in which the motion
opposed, or the motion that would have happened, is due to gravity. Why doesn’t a book placed
on a table fall? Because the table supports it.
The last p-prim from this cluster that I will mention is GUIDING . As an example, imagine a
metal ball rolling in a groove made in a wood surface. It is not surprising to us that the ball follows
the path of the groove. Note, again, that we explain this without appeal to forces; the groove
simply guides the ball because of its geometric nature.
Finally, I want to mention two p-prims from the “Balancing and Equilibrium” Cluster. The
first of these p-prims is DYNAMIC BALANCE . A situation involving two equal and opposite forces
would likely be explained by appeal to this p-prim. diSessa contrasts this p-prim with a second
that he calls ABSTRACT BALANCE. In ABSTRACT BALANCE, the balancing of the quantities involved is
required either by the definition of these quantities (as in one kilogram is 1000 grams), or because
of universal principles (such as the conservation of energy). As diSessa says: “abstractly balancing
things should or must balance; dynamic balancing is balancing by accident or conspiracy.” I will
discuss the relation between these p-prims and balance-related symbolic forms below.
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A mechanism for p-prim activation
The above discussion is designed only to provide a feel for the scope of the sense-ofmechanism. diSessa’s list is somewhat longer and he suggests where many p-prims exist beyond
those that he names. My listing of p-prims will end here, however, and I will instead move on to
another piece of diSessa’s account of the sense of mechanism. To this point, I have not said very
much about the mechanism that determines which p-prims gets used at which time. So far, the
only mechanism that we have is “recognition,” p-prims are just recognized in circumstances.
But diSessa extends his account beyond the simple statement that p-prims are recognized. The
key question is when and how a p-prim is “cued to an active state.” diSessa argues that the
activation of a p-prim depends on other aspects of the current “mental context,” which includes
what we perceive in the world, what other p-prims are active, and any additional active
knowledge, including “conscious ideas.” Furthermore, diSessa defines two terms that are designed
to provide characterizations of how likely a given p-prim is to be activated. The first of these
terms, “cuing priority,” describes the likelihood that a given p-prim will be activated given some
perceived configuration of objects and events in the world. In effect, cuing priority tells us how
much activation must be contributed by other elements of the mental context in order to cue a
given p-prim in a particular physical context. In cases where the p-prim clearly applies—when it
has a high cuing priority—little additional activation is needed from the rest of the mental
context. Note that the cuing priority of a p-prim is context dependent; it depends on the nature
of the physical circumstance that is currently of concern. However, I will sometimes use the term
“cuing priority” more roughly as an overall measure of the frequency with which a given p-prim is
activated.
The second term concerning p-prim activation that diSessa defines is “reliability priority.”
Reliability priority provides a measure of how likely a p-prim is to stay activated once it is
activated. The point is that, once a p-prim is activated, this activation contributes to a subsequent
chain of mental events that may or may not involve the p-prim continuing to be activated. Taken
together, cuing priority and reliability priority constitute what diSessa calls “structured priorities.”
Although diSessa talks in the language of structured priorities throughout most of Toward an
Epistemology of Physics, he also provides a model of the sense-of-mechanism as a connectionist
network. P-prims are nodes in the network and there are weighted connections between these
nodes. Given this model, cuing and reliability priority can be reduced to behavior of the network
due to the values of these various weightings. Thus, in principle, we could discard the language of
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structured priorities in favor of descriptions given solely in terms of this connectionist network.
diSessa argues, however, that is worth retaining “cuing priority” and “reliability priority” as
technical terms because these terms provide qualitative characterizations of the connectionist
network that are more easily put in correspondence with human behavior. For example, given the
activation of a high reliability p-prim, a person is more likely to stick to their characterization of a
situation in terms of this p-prim, and to assert a high level of confidence for this characterization.
The development of the sense-of-mechanism
With this account of the sense-of-mechanism in hand, we can begin to talk about how the
sense-of-mechanism develops during the learning of physics. I am going to begin here by
presenting what diSessa says about how this cognitive resource develops, and then I will add some
of my own details about the role that symbol use plays in this development. diSessa describes
three types of changes that the sense-of-mechanism undergoes during physics instructions:
1) Weightings change and the sense-of-mechanism is restructured. One type of development that
occurs is changes in the weights in the connectionist network, which can alternatively be thought
of as changes in the priorities of individual elements. We can imagine that these are incremental
adjustments to the weighting values that occur through repeated experiences in physics
instruction. However, although individual changes may be small, diSessa believes that these
incremental adjustments ultimately lead to a change in the overall character of the p-prim system.
Before any physics instruction the sense-of-mechanism is relatively flat, it has only very local
organization with individual p-prims having connections to only a few others. Certainly some pprims have higher priorities than others, but there are no central p-prims with extremely high
priority.
With the development of expertise, this situation changes. diSessa hypothesizes that the
priority of some elements is greatly increased and the priority of others greatly decreased. The
result is a system with central, high priority elements. Thus, there is a change in the character of
the sense-of-mechanism; it undergoes a transition from having little structure to having more
overall organization.
2) New p-prims develop. As students learn to attend to different pieces of the world and their
experience, new elements may be added to the sense-of-mechanism.
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3) P-prims take on new functions. Finally, the new activities associated with classroom physics and
the new types of knowledge that are acquired provide opportunities for p-prims to perform new
functions. For example, p-prims may come to serve as “heuristic cues” for more formal
knowledge; when a p-prim is cued it can lead more or less directly to the invocation of some
formal knowledge or procedure. In addition, p-prims may play a role in “knowing a physical law.”
For example, we may in part understand Newton’s second law and F=ma through SPONTANEOUS
RESISTANCE,
the tendency of objects to continue moving in the direction that they are already
moving.
The result of all of these changes is a physical intuition that is refined and somewhat adapted
for use in expert physics. diSessa also provides a wealth of details concerning the specific
refinements that he expects to occur, of which I will only mention a few. Some parts of the senseof-mechanism can have little use in expert physics and thus should be substantially suppressed. For
example, in expert physics, constraint phenomena are no longer explained by a simple appeal to
the geometry of objects. Instead, these phenomena must be explained in terms of forces applied
by obstructing objects. Thus, the priority of p-prims, such as BLOCKING, that were previously
associated with constraint phenomena should be greatly decreased, and the cuing priorities of pprims in the Force and Agency Cluster should be increased for these circumstances.
The changes associated with constraint phenomena are typical of a more widespread trend
predicted by diSessa. He conjectures that, in general, the role of agency will be greatly expanded
in the sense-of-mechanism. As students learn to see more and more circumstances in terms of
forces, the range of application and priority of p-prims in the Force and Agency Cluster will be
increased.
Symbol use and changes in the sense-of-mechanism
The above sub-section described the types of changes that we can expect the sense-ofmechanism to undergo during physics instruction. But how does physics instruction cause these
changes? What, precisely, are the experiences that foster these developments? Since physics
students spend such a significant amount of time manipulating equations and solving problems, it
would be comforting to know that some tuning of the sense-of-mechanism occurs during the
symbol use that is typical of these activities. In this sub-section, I will show that this is plausible
and I will give a little of a feel for how I believe it works.
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Let’s start by considering a hypothetical and clear type of circumstance. Imagine that a
student is working to understand a physical situation and two competing and contradictory pprims are cued to activation. From the point of view of the sense-of-mechanism, this cuing of
conflicting p-prims is a problem. Since, in its naive state, the sense-of-mechanism is rather flat and
only weakly organized, no p-prim has a much higher priority than any other p-prim. Thus, the pprim system is not very good at resolving this type of conflict.
Equations and symbol use can provide a way out for this stymied student. It is possible that,
by manipulating equations, the student can find a solution to the problem and thus resolve the
conflict. If this happens, then it is likely that the priority of the “winning” p-prim will be
incrementally increased, and the priority of the losing p-prim will be incrementally decreased.
Through such experiences the sense-of-mechanism can be nudged toward alignment with expert
intuition.
As we have seen, students frequently stated two conflicting intuitions when working on the
Shoved Block task. These two intuitions were: (1) The heavier block experiences a greater
frictional force, thus it slows downs more quickly and travels less far. (2) Heavier things are
“harder to stop” so the heavier block travels farther. In Chapter 4 we saw that Alan stated both of
these intuitions quite clearly:
Alan
And then if the block was heavier, it'd just mean that our friction was greater, and it would
slow down quicker.
________________
Alan
That seems kind of- Something seems strange about that because if you had like a block
coming at you, a big heavy block, and a small lighter block. Two different instances. And
you have this big heavier block coming toward you and you want it to slow down, it
certainly takes much more force for you to slow down the bigger heavier block than the the lighter one.
Bob
Oh wait, that's true!
Alan
Because, even though they're both working on the same frictional surfaces, just logically I
know the big heavier block takes-, I mean football is all about that, that's why you have
these big three hundred pound players.
After Alan stated these conflicting intuitions, Alan and Bob went on to use equations to find the
solution to this problem. As we have seen, they derived the following equation and noted that it
did not depend on the mass:
a = gµ
Alan
Right, so, actually, they should both take the same.=
Bob
=Wait a minute. Oh, they both take the same! [Surprised tone]
…
Bob
So, no matter what the mass is, you're gonna get the same, the same acceleration.
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So, in this little episode, Alan and Bob have resolved this conflict between two intuitions in a way
that is a little surprising to them. It turns out that neither intuition is exactly right, both blocks
travel exactly the same distance.
Now I want to talk about this episode in terms of p-prims. The first intuition can be
considered to be a somewhat refined application of
FORCE AS MOVER ;
the presence of the frictional
force causes the block to slow down. The second intuition involves the SPONTANEOUS RESISTANCE pprim; the block has an intrinsic resistance to changing its speed by slowing down. In addition,
both intuitions probably involve the activation of OHM’S P -PRIM since there is an effort working
against a resistance to produce a result. But this last p-prim is applied somewhat differently in each
instance. In the case of the first intuition, the change in intrinsic resistance is not a salient aspect of
the difference between the heavy block and light block situations.
It is worth noting that both of these intuitions already involve somewhat refined applications
of the sense-of-mechanism. The use of FORCE AS MOVER to account for changes in speed rather than
as an explanation for motion in some direction constitutes a refined use of this p-prim. More
dramatically, it is likely that complete novices would explain this motion by an appeal to entirely
different p-prims such as DYING AWAY . If we apply DYING AWAY to explain the shoved block then we
state that the motion dies away simply because that is what motions do. In contrast, Alan and Bob
are capable of attributing the slowing down of the block to a particular agent, a force applied by
the table. This suggests significant progress in the direction of expertise.
So can we say anything about what effects this brief experience may have on Alan and Bob’s
senses-of-mechanism? First notice that this episode is not exactly the same as the hypothetical
circumstance described above, in which the conflict between two p-prims was resolved in favor of
one of them. In this case, symbol use does not choose between two competing p-prims, instead it
suggests that both of these intuitions have some validity.
One further observation from my data can help here. It turns out that the FORCE AS MOVER
intuition was somewhat more common and was always produced before the SPONTANEOUS
RESISTANCE
intuition. Furthermore, only one pair (Mike and Karl) expressed a preference for the
second intuition, and even that pair generated the FORCE AS MOVER intuition first. Of the other
pairs, only Alan and Bob generated the SPONTANEOUS RESISTANCE intuition without some explicit
prompting on my part, though all pairs were quick to acknowledge the plausibility of this second
intuition.
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I therefore speculate that this episode may have the effect of incrementally increasing the
priority of the SPONTANEOUS RESISTANCE p-prim. In particular, I hypothesize that this experience
may increase the cuing priority of this p-prim for cases in which the mass of an object can be
treated as an intrinsic resistance. This is a rather satisfying outcome since the resistance of masses
to changes in motion is one of the fundamental aspects of Newton’s laws. In the next passage, Bob
sums up the results of this problem. As he does, note that he particularly emphasizes the validity
of the SPONTANEOUS RESISTANCE intuition:
Bob
So, well, when there isn't like equations down there sort of, I have a tendency just to sort
of say: oh yeah, it just=
Alan
=Right.=
Bob
=it would slow down quicker. But, so if - I mean, you do - That's what physics is, is you
look at the equations and figure it out. But, um, okay=
Bruce
=So, what do you think of that result that they slow - that they take the same amount of
time to stop?
Bob
Um=
Bruce
=Is it surprising? Is it not so surprising in retrospect?
Bob
In retrospect it isn't too surprising. But, um, I mean, I guess, I thought (2.0) Why did I
think that the heavier block would slow down quicker? I (st…) just wasn't, I mean. You're
pushing on it - pushing against with a greater force so (…) it seems like it would slow down
quicker. And that was my first thought. Um, but since it is bigger you obviously need a
greater force to stop it. So, I wasn't - I just wasn't thinking about that. I was just thinking:
Oh well, you got two blocks. One's experiencing a greater force and one, one - so this
one's gonna slow down quicker. But, I mean they're not equal. This one's bigger than this
one. Um, so, it isn't too surprising. I mean, it makes sense that, that you need to push, I
mean, you got this big boulder coming at you, you have to push harder than if a pebble's
rolling at you. Um,, To stop it.
In the above passage, Bob starts off by telling us what his “first thought” was, that the heavier
object should slow down faster. But then he goes on to tell us that there is a second effect that
must be considered and that, in retrospect, “isn’t too surprising.” He explains this second intuition
with an example: “it makes sense that, that you need to push, I mean, you got this big boulder
coming at you, you have to push harder than if a pebble's rolling at you.”
The point here is that Bob ends his summary by essentially emphasizing the importance of the
intrinsic resistance of masses. Although the SPONTANEOUS RESISTANCE p-prim did not “win” in this
problem—the outcome was a draw—from Bob’s point of view the moral is that the effects of
SPONTANEOUS RESISTANCE
must not be overlooked. Given his experience in this episode, it is
plausible that, in future episodes, Bob will be slightly more likely to see masses as having intrinsic
resistance. The priority of the SPONTANEOUS RESISTANCE p-prim has been incrementally increased.
Of course, it would be possible to give many competing analyses of this episode. The above pprim account is very speculative, and I have not tried to argue against competitors. This is
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appropriate since my purpose here is only to illustrate, with an example, a possible mechanism
through which physical intuition could be altered through symbol use.
As a slight aside, I want to also mention that the use of equations can do more than adjudicate
between two p-prims; in addition, the use of equations will also tend to cue p-prims in places
where they might otherwise not be cued or shift how they are applied. This added cuing of certain
p-prims should lead to the increased priority of these p-prims. As an example, recall that during
their solution to the Shoved Block problem, Alan saw one of the written equations as expressing
BALANCING :
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
As I have pointed out, the use of BALANCING here is not really appropriate. Nonetheless, this does
illustrate how equations can lead to the activation of p-prims that might otherwise have not been
employed.
I believe that there are two important morals to be derived from the story of Alan and Bob’s
work on the Shoved Block problem. The first of these morals I have already suggested. Alan and
Bob’s experience here can lead to changes in the weighting of connections in the sense-ofmechanism; symbol use can have the effect of increasing the priority of certain p-prims and
suppressing others.
The second moral is a little more subtle. My telling of the above story essentially presumes
that Alan and Bob do not have a fully developed (expert) intuition at the time of the episode. In
particular, I suggested that their sense of SPONTANEOUS RESISTANCE may be a little weaker than that
of an expert physicist. However, even with this relatively unrefined physical intuition, Alan and
Bob were able to successfully complete this problem and produce the correct answer! So, from the
point of view of solving problems, it does not seem to matter very much that their physical
intuition was not fully developed.
This observation leads to the following question: How much work should the sense-ofmechanism and other aspects of physical intuition be able to do? It seems reasonable that
physicists will have an increased priority for SPONTANEOUS RESISTANCE . But is it necessary for the
sense-of-mechanism to become so finely tuned that it can not only activate both SPONTANEOUS
RESISTANCE
and FORCE AS MOVER , but that it can actually produce the result that these effects
precisely cancel?
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Clearly there are limits to what we need from our physical intuition. Furthermore—and this is
the big point I have been working toward—these limits are in part determined by what symbol
use is capable of doing. Since, Alan and Bob can always rederive this result, it does not need to be
wired into their sense-of-mechanism.
This is akin to the external representation as “crutch” story that I told in the first chapter.
Symbol use can act as a support that allows certain aspects of the sense-of-mechanism to atrophy,
and that requires other aspects to be strengthened. More generally, the moral here is that the
character of physical intuition will depend on specific characteristics of the practice of symbol use
employed. This is true not only because weightings are adjusted through symbol use, but because
the physical intuition of physicists is adapted specifically for the purpose of complementing and
functioning in the activity of symbol use. Thus, the cluster of resources that corresponds to
physical intuition is not necessarily refined so that it approaches Newton’s laws as a limit. Instead,
there is the possibility that this refinement is idiosyncratic, with the idiosyncrasies dependent on
the specific capabilities and requirements of symbol use. I will return to this point in Chapter 7.
The Development of New Resources and the Relationship
Between Forms and P-prims
To this point, all of this discussion of the development of intuition has been in terms of
diSessa’s model of the sense-of-mechanism. This means that I have really only been talking about
changes inside the “intuitive physics” oval in Figure Chapter 5. -1. The story has been that, with
the development of expertise, the sense-of-mechanism gets refined and adapted for new roles. In
this section I want to talk about the new cluster of resources that develops for symbol use in
physics, what I have called “algebra-physics.”
I want to start by pointing out that the notion that the sense-of-mechanism can be refined
during symbol use is, from a certain point of view, slightly bizarre. Prior to physics instruction, the
sense-of-mechanism was built and tuned through experiences in the physical world. Thus, the
advent of physics instruction means not only that there is a whole new set of pressures acting on
the sense-of-mechanism, these pressures are of a new kind. Furthermore, it appears that p-prims
are going to be called upon to perform new types of functions.
These observations lead to some questions: How far can the sense-of-mechanism be stretched
before it should not even be called the sense-of-mechanism anymore? If p-prims are taking on
radically new functions and being cued in radically new circumstances, should they still be called
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p-prims? Notice that, throughout the above discussion, I always talked as if the individual p-prim
elements themselves were unchanged. What changed were the weightings of connections between
these elements and the jobs these elements were to perform. But isn’t it also possible that the
character of individual elements could change?
In building models of the development that occurs during physics learning, there are thus
several possibilities that we could pursue with respect to modeling the development of the senseof-mechanism. Consider the following simple palette of modeling possibilities:
1. The sense-of-mechanism is only tuned and adapted. In this model, individual elements remain
essentially unchanged. Weightings change and the system is adapted for new uses.
1. The sense-of-mechanism mutates into something new. It is possible that the sense-of-mechanism
could be transformed into something that is different enough that it is not even appropriate to
call it the sense-of-mechanism anymore. Even though the new system may develop out of this
old one, the individual elements may be substantially different than p-prims. This model
presumes that the sense-of-mechanism, as we knew it prior to physics instruction, no longer
exists.
1. The sense-of-mechanism continues to exist, but new related resources develop. In this model, a
new cognitive resource develops from prior resources and it maintains links to these prior
resources. Since the prior resources continue to exist, they may be tuned and adapted in the
manner of model (1).
I believe that I have already made it clear that I plan to adopt model (3): Algebra-physics
requires the development of new cognitive resources. More particularly, I plan to argue that forms,
in part, develop from p-prims and they retain close ties to p-prims. My reasons for choosing this
particular model (and the reasons that I believe it is the most accurate model) are rooted in the
argument I began in Chapter 3 and which I will complete in the next two chapters. In particular, I
have been arguing that physics learning involves the development of knowledge elements that I
call symbolic forms and that these knowledge elements have certain specific properties. Some of
these properties, such as the inclusion in forms of the ability to see symbol patterns, distinguish
forms from p-prims. Furthermore, as we will see in a moment, although some forms bear a close
relation to specific p-prims, the majority of individual forms do not line up with individual p-
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prims. Thus, the view I have been arguing for asserts the existence of a new cognitive resource that
bears some relation to the sense-of-mechanism, but nonetheless is substantially different.
Intuitive Physics
(p-prims)
Algebra-Physics
(forms)
Figure Chapter 5. -3. Forms develop in part out of p-prims but the sense-of-mechanism continues to exist.
In addition, there are many reasons to believe that the sense-of-mechanism continues to exist
in something close to its original form. Even expert physicists must continue to participate in the
same old activities in the physical and social worlds that require the sense-of-mechanism. They
still have to push and pull objects. And they still have to participate in the usual informal discourse
about the behavior of the physical world. For this reason, I believe that it makes sense to talk as if
the sense-of-mechanism continues to exist in a form very close to that described by diSessa.
Forms are abstracted, in part, from the symbolic world of experience
So my view is that a new cognitive resource develops that includes symbolic forms as elements.
Forms are descended from and maintain fairly direct connections to p-prims, as well as to other
previously existing resources. Furthermore, I want to emphasize that I think of forms as being very
similar knowledge elements to p-prims. I take them to be nearly primitive knowledge structures
that involve simple schematizations of situations.
But now I want to point to some of the differences between forms and p-prims, differences
that should clarify why I choose to treat symbolic forms as new knowledge. Forms differ from pprims because they are not abstracted only from experiences in the physical world, they are
abstracted from the worlds of physical AND symbolic experience, and can be cued in both.
What does it mean that forms are “abstracted out of symbolic experience?” I will not provide
much detail concerning this process beyond the statement that I intend it to be much like the
abstracting of p-prims out of physical experience. For illustration, first consider a p-prim example:
People have repeated experiences in the physical world pushing on objects that are at rest to get
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them moving. And when you push on something at rest, it moves in the direction that you push it.
These repeated experiences can lead to a simple schematization of these similar events of the sort
embodied in the FORCE AS MOVER p-prim, which says that pushed objects move in the direction of a
push. Similarly, students have experience in the symbolic world adding numbers onto other
numbers. And when they add a number onto another number, they get a result that is bigger than
the original number. This can lead to the abstraction of the BASE +CHANGE form.
Primarily, I intend for my discussion in Chapter 3 to provide support for this view, but I will
remind the reader of a brief example here. Recall that when working on the Buoyant Cube
problem, Mark asserted that the weight of the ice and the weight of the displaced water (the
buoyant force) had to be equal for the following reason:
W w = Wi
Mark
We know one is gonna float. And then just sort of, I guess, I don't know from like all our
physics things with proportionalities and everything, that we'll have to have something
equal, like the buoyancy force should be equal to the weight of the block of ice. [g. pulling
down block]
As I mentioned earlier, Mark is arguing simply from an appeal to his experiences of symbol use, he
says “we have to have something equal.” When students enter physics instruction, DYNAMIC
BALANCING
is a primitive element of the way they understand and explain physical phenomena. For
situations in which there is no motion because opposing forces are in balance, no further
explanation is necessary. In this example, however, Mark appeals to a new sort of intuition,
intuition that is rooted in symbolic experience.
Similar stories can be told about many symbolic forms. Let’s think about the
form for a moment. Certainly
DEPENDENCE
DEPENDENCE
has some roots in physical experience. We have
experiences in which the outcomes of events depend on aspects of the physical world. But we also
have relevant experiences in the symbolic world. For example, we can plug numbers into
expressions and get out numerical results. And these results depend on the values you plug in; if
you plug in a different number, you get a different result.
Hypotheses concerning relations between individual forms and p-prims
I want to take a moment to hypothesize about the relations between specific forms and pprims. To begin, it should be clear to the reader that I intend to suggest a close relationship
between forms and p-prims that have the same or similar names. For example, the
form is supposed to be closely related to the
DYING AWAY
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DYING AWAY
p-prim. Similarly, the BALANCING form
bears a close relation to the DYNAMIC BALANCING p-prim. The hypothesis is that, in such cases, the
individual form is a direct descendant of the p-prim and that activating the form will have a
strong likelihood of activating the p-prim.
I also hypothesize that there are connections of varying strengths between forms and p-prims
in cases where there is not a simple correspondence between individual elements. In addition,
there are connections between individual forms and other previously existing cognitive resources.
Here, proceeding by cluster of forms, I will speculate about some of the stronger connections.
The Competing Terms Cluster. Across the board, forms in the Competing Terms Cluster will be
closely tied to Force and Agency p-prims. A key observation here is that these forms involve what
diSessa calls the same “attributes” or “base vocabulary” as Force and Agency p-prims. What this
means is that these particular forms and p-prims tend to break down a physical situation into
similar components—both involve agents, usually forces, working for results. Once a form or pprim with a given base vocabulary is activated, the likelihood of elements with the same base
vocabulary being activated is increased. In addition, there will be close connections between some
forms in this cluster and Balance and Equilibrium p-prims, such as between the BALANCING form
and the DYNAMIC BALANCING p-prim.
The Proportionality Cluster. The genetic roots and connections of Proportionality forms are
complex and multifarious. First, these forms will be tied to Force and Agency p-prims. In
particular, I hypothesize that PROP+ and PROP- are strongly connected to physical notions of effort
and resistance. To illustrate, consider the equation F=ma rewritten as a=F/m. The right side of this
equation can be seen in terms of the PROP- form—the acceleration is inversely proportional to the
mass. The hypothesis is that the use of PROP- here will tend to activate the SPONTANEOUS
RESISTANCE
p-prim with the mass seen as an intrinsic resistance that resists acceleration. In addition,
the right hand side may be seen in terms of
PROP-
PROP+.
Taken together, the activation of PROP+ and
may tend to cue OHM’S P -PRIM with, in this case, force taken as the effort, mass as the
resistance, and acceleration as the result.
In fact, the very same story may be told about the equation associated with Ohm’s
Law—V=IR—the law for which OHM’S P -PRIM is named. If we rewrite this equation as I=V/R it
says that the current, I, is proportional to the voltage, V, and inversely proportional to the
electrical resistance, R. In this case, the voltage can be thought of as the “effort” and, of course, the
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electrical resistance as the “resistance.” Again, the point is that the
PROP+
and PROP- forms will be
connected to p-prims involving effort and resistance.
It is important to note that Proportionality forms also have their roots in cognitive resources
other than the sense-of-mechanism. For example, I noted that some researchers have made
“qualitative proportionality” relations a central element in their accounts of how people reason
about physical quantities. Proportionality forms will remain connected to these quantity-related
resources.
The Dependence Cluster. The connections of Dependence forms are also complex and diverse. As
I will argue in Chapter 11, we can understand DEPENDENCE as subsuming a wide range of
asymmetric relations that can exist between entities of a variety of types. For example, one event
can cause another event, or one quantity can determine the size of another quantity. And such
relations can be found in many p-prims. In OHM’S P -PRIM , the size of the result depends on the
both the amount of effort and the amount of resistance; in
FORCE AS MOVER
the amount of motion
depends on the size of the force; and in GUIDING the direction of motion depends on the shape of
the constraining geometry. To a varying extent, Dependence forms should be tied to all of these
p-prims.
This wide applicability of the DEPENDENCE form can be understood if we think of
DEPENDENCE
as similar to what diSessa calls a meta-p-prim—a p-prim that embodies a higher level abstraction
that is common to many p-prims. Furthermore, it is interesting to note that, because dependence
lives at a higher level of abstraction, much of the nuance associated with the diversity of p-prims is
lost. A similar loss of nuance can be seen in the PROP+ and PROP- forms—proportionality is a more
generic relation than that which holds, for example, between an effort and result.
I believe that this washing out of nuance is a fundamental feature of the move from intuitive
physics to algebra-physics. One of the hallmarks of expert physics practice is its ability to quantify
the entirety of the physical world; everything is described in terms of numbers and relations
between numbers, and all equations look the same whether the quantities that appear are forces or
velocities. I call this tension between the homogenizing influence of algebra and the nuance
inherent in intuitive physics the “fundamental tension” of physics instruction. The set of forms, as
I have listed them, constitute the end product of this tension; much of the nuance inherent in
intuitive physics is lost, but more distinctions are preserved than those inherent in the syntax of
symbolic expressions.
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The Terms are Amounts, Multiplication, and Coefficient clusters. Forms in the remaining three
clusters primarily have their origins in cognitive resources other than the sense-of-mechanism. The
PARTS -OF -A-WHOLE
form is an example. As I have mentioned, similar knowledge structures were
posited to account for arithmetic problem solving behavior by the “patterns” researchers. Thus,
this form will have primarily developed from and have connections to cognitive resources in what I
have called the “algebra problem solving” and “quantity” resource clusters. There are some
exceptions among these forms, however, and a noteworthy example is the
SAME AMOUNT
form. In
addition to being strongly connected to algebra problem solving resources, I believe that SAME
AMOUNT
is tied to the ABSTRACT BALANCING p-prim. An example situation that might activate this
form and p-prim is when the principle of Conservation of Energy is applied. If we write
Efinal=Einitial, then this relation between the initial and final energies can be seen in terms of SAME
AMOUNT
and understood as required by ABSTRACT BALANCING.
This concludes my discussion concerning the connections between individual symbolic forms
and previously existing resources. Before moving on to the final section of this chapter, however, I
want to observe that p-prims in the Constraint Cluster were not mentioned at all in this discussion
of connections between individual forms and p-prims. This is for the simple reason that, as
hypothesized by diSessa, Constraint p-prims play little role in expert physics. As I have
mentioned, with expertise the physical circumstances associated with these p-prims must come to
be understood in terms of the action of forces.
Finale: Symbol Use and the Development of Expert Intuition
Let’s summarize where we have been in this chapter. First, we began with “snapshots” of
cognitive resources and I focused on the relation between two of these resources. One, the senseof-mechanism, exists prior to any physics instruction. The second resource—the form-device
system—is a new resource that develops during physics instruction. My goal was somehow to
interpolate between these two resources to describe some features of the development of physics
knowledge. In this regard, I made a couple of points. First, I described the sense-of-mechanism
and how it might develop because of symbol use. I stated that the experiences of symbol use
would lead to certain elements being reinforced and I argued that the sense-of-mechanism would
have to develop so as to be adapted for the specific roles required of it by symbol use. Second, I
proposed some specific relations between the sense-of-mechanism and the new resources. I argued
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that the form-device system develops in part from the sense-of-mechanism. In addition, I
maintained that the sense-of-mechanism continues to exist in a refined form and is connected to
these new resources. In particular, I argued for a strong relation between p-prims and forms. Now,
with this broad picture in hand, I can clarify some points I made earlier and I can make some final
observations.
In the early part of this chapter I talked about some of the details concerning how I expect the
sense-of-mechanism to be refined during symbol use. I argued that symbol use can have the effect
of reinforcing some p-prims and suppressing others. Now I am in a position to elaborate just a
little on how this works. First, we know that forms can be cued directly by equations, they are
essentially “seen” in expressions. Furthermore, I have also asserted that the activation of forms can
lead to the activation of related p-prims. This equation–form–p-prim chain is thus a
particular—and rather direct—mechanism by which p-prims can be activated during symbol use.
As an example, take the episode in which Alan sees the equation mgµ=ma in terms of the
BALANCING
form. Once this form is cued, this can lead directly to the activation of the DYNAMIC
BALANCING
p-prim. The upshot of such an episode is that both the form and p-prim are reinforced.
The broad picture can also help me to clarify my stance on another issue: What is expert
“physical intuition?” It will be my stance that both the refined sense-of-mechanism and the newly
developed form-device system constitute components of an expert’s physical intuition. I realize
that it may seem a little irregular to think of forms as part of physical intuition since they are
derived in part from experience with symbol use. Nonetheless I believe that it is appropriate to
think of forms as part of the very roots of expert intuition. I will argue for this stance in Chapter 7.
Finally, with the model laid out in this chapter, I am now in a position to state one of the
fundamental arguments that I will make in this document. The idea is that the forms available in a
symbolic practice and the frequency of use of those forms have a strong influence on the nature of
expert intuition. If a form gets used frequently, then that form and related p-prims both get
reinforced. This is one way to think about how the details of a symbolic practice, such as algebraphysics, can determine the nature of expert physical intuition. Algebra-physics involves a
“vocabulary of forms,” and the nature of these forms and their frequency of use is indicative of the
character of expert intuition that is associated with algebra-physics.
Furthermore, this argument will provide one of my fundamental points of leverage in
comparing algebra-physics and programming-physics. What I propose is to actually count the
frequency of individual forms and devices as they appear in my programming-physics and
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algebra-physics data. When this is done, I will essentially have characterizations of these two
practices that I can contrast. In the next chapter, I will show how I went about obtaining these
counts for algebra-physics, and I will present the results.
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Chapter 6. Systematic Analysis of the Algebra
Data Corpus
To this point, I have devoted myself to describing my model of symbol use and arguing more
broadly for my theoretical orientation. I have presented my two main theoretical
constructs—symbolic forms and representational devices—and I have talked about how these new
resources develop out of existing prior resources during physics learning. Although the main goal
of the preceding chapters was to lay out the theoretical view, my presentation was not totally bare
of support for that view. I argued against competing viewpoints where appropriate. And I
employed examples from my data corpus to illustrate my points and to add to the plausibility of
my position.
In this chapter, I reveal more of the details of the methodology used to make contact with my
data corpus. This will include a more comprehensive portrayal of the participating subjects as well
as a description of the methods used for collecting and analyzing the data corpus. In brief, my
methods involved first videotaping students solving a range of physics problems and transcribing
the resulting videotapes. A sizable subset of the resulting transcripts were then systematically
coded. This required going through the corpus equation-by-equation and utterance-by-utterance
looking for forms and devices.
There are two major reasons that the effort necessary to produce this systematic analysis is
warranted. The first reason is that it is through the systematic analysis that the lists of forms and
devices, as I presented them in Chapters 3 and 4, were developed. Through an iterative process of
coding and recoding, these lists were refined until they were just sufficient to account for all
required aspects of the data. Thus, in recounting for the reader how the systematic analysis was
performed, I am essentially explaining how I arrived at the lists of forms and devices.
Second, the results of the systematic analysis paint a particular kind of picture of the data
corpus. These results can provide the reader with a sense for broad features of the data, such as
how regularly students constructed and interpreted equations, and how frequently each type of
symbolic form and representational device appeared. In addition—though more
speculatively—these frequencies can be taken to provide a picture of algebra-physics knowledge.
One of the goals of this work is to understand the character of expert intuition and, as I began to
argue in the previous chapter, the form-device system is one important component of expert
intuition. The underlying presumption here is that the frequency with which specific forms and
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devices are used reflects the priority of these elements—in diSessa’s technical sense—within expert
intuition.
There are, however, some limits to the analysis I will present here. First, the analysis process I
describe in this chapter does not provide an exact means of extracting categories from a data
corpus, and I did not develop a specification so that my results could be corroborated by another
coder. Although I believe that my lists of forms and devices are supported by my data and this
analysis, the production of these categories was still strongly driven by theoretical concerns and
intuition; as I will describe here, it was a highly heuristic process. Even so, I hope that the reader
will understand this analysis as a useful first step toward future, more exact and replicable
methods of employing analyses of this sort to identify knowledge elements.
There are also limits in this analysis, not only due to the techniques I employed, but also due
to the nature of the data corpus itself. Most notably, the relatively small number of subjects and
limited task selection limit the extent to which we can expect to generalize from this corpus to
make claims about the nature of algebra physics. Even with these limits, I believe that the results of
this analysis provide a useful window into algebra-physics and, when similar data is obtained for
programming-physics, the results will allow a revealing comparison of these two practices.
In the following sections of this chapter I will describe how and what data was collected, how
that data was analyzed, and the results of the analysis.
About the Algebra Data Corpus
Subjects
All of the participants in the study were students at U.C. Berkeley and were currently enrolled
in “Physics 7C.” Physics 7C is a third semester introductory course in a series entitled “Physics for
Scientists and Engineers.” The title tells us at whom this course is targeted; it is for students who
intend to major in a “hard” science or engineering discipline. In contrast, students that intend to
go to medical school or who are headed toward a “softer” form of engineering (e.g., architecture)
generally enroll in the “Physics 8” series.
Two preceding courses in the Physics 7 series, Physics 7A and 7B, are prerequisites for
participation in 7C. Physics 7A is primarily an introduction to classical mechanics and 7B covers
topics in electricity and magnetism, as well as some thermodynamics. The third semester, Physics
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7C, covers optics and electromagnetic waves, in addition to some modern physics topics,
including special relativity and quantum physics.
It is thus the case that the subjects in this study were fairly advanced physics students at a
prestigious university. Nonetheless, these students were certainly not experts in physics; rather,
they were at an intermediate level of expertise, somewhere between true novices and practicing
physicists. For several reasons, this intermediate level of expertise was necessary for this study.
First, since I wanted to make generalizations about the nature of algebra-physics practice, I needed
students that were already somewhat on their way to learning this practice. However, I also
wanted to be able to speculate about the development of algebra-physics knowledge. For this
reason, I did not want to look at true experts; I needed students that were still at an early enough
stage of their studies that I could expect to observe the types of difficulties encountered by
students in introductory courses.
Finally, there was an additional pressure to not have subjects closer to the novice end of the
spectrum. It was of great practical benefit to have students whose behavior on the given tasks was
relatively stable over the period of the semester. Since the tasks were primarily drawn from
mechanics topics, the new material learned in Physics 7C should not have contributed too
significantly to the ability of students to accomplish the tasks. This meant that each pair could
spread their sessions over the semester in a manner convenient with their schedules.
To recruit participants, I went to a Physics 7C lecture early in the semester and asked for
volunteers. Students were told that they would be paid $5 an hour for their work, and interested
students filled out sheets with their names and telephone numbers. They were permitted to
volunteer in pairs or singly. Later, I shuffled the stack of sheets and telephoned students at
random. I scheduled 7 pairs hoping that at least 5 would complete the study. In fact, precisely 5
of the 7 pairs completed all of the tasks.
In total, 21 students volunteered to participate, 17 men and 4 women. (This high percentage
of male volunteers is not surprising given that the class was predominantly male.) Of the 14
students that began the study, 11 were men and 3 women, with 4 pairings of two men, and 3
pairings of a man and a woman. The specific pairings chosen were almost entirely dictated by the
students’ scheduling constraints. Unfortunately, the two pairs that dropped out of the study were
both male/female pairings. One of these pairs dropped because the male had to leave school for
personal reasons. The other pair asked to drop out, pleading that they were simply too busy. In
fact, both members of this latter pair seemed very uncomfortable during the two sessions that
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they attended, and I was contemplating asking them if they wanted to cease participation in the
study.
It may seem as though five pairs is a relatively small number for this study, but even this
number pushed the boundaries of what was practical. As will become clear shortly, the methods
employed required the transcription and analysis of a large number of hours of videotape, for both
the algebra and programming portions of the study. Furthermore, the small number of subjects
does not pose a significant difficulty for most of the claims that I will make. At no point will I try
to make claims concerning differences and similarities between individual students or pairs of
students. Instead, all of the (algebra-physics) data is amalgamated into a single set of numbers
before drawing conclusions.
The Tasks
The participants in the study were asked to solve a range of problems and to answer a variety
of questions. These tasks, as presented to the students, are reproduced in full in Appendix B. As
shown in Table Chapter 6. -1, the tasks were divided into four types, administered in the order
shown.
Open-Ended
Main Body
“Quick” Tasks
Cross Training
4 questions
11 problems
7 questions
4 problems
1/2 - 1 hour
3 - 3 1/2 hours
1/2 hour
1/2 - 1 hour
Table Chapter 6. -1. The four main types of tasks in order of execution.
Students began their work in the first session with four “open-ended” questions. These
questions did not ask students to solve specific problems or derive particular equations; instead,
the purpose of these tasks was to get students talking with only very limited prompts. The
questions were introduced with the following written instructions:
These first few tasks are slightly different than the others I’m going to ask you to do later.
For these, you should just explain as much as possible about what happens in the
situations described.
The “Shoved Block” task, which was described in earlier chapters, was amongst the situations
asked about in these open-ended questions. Another of the questions asked students simply to talk
about what happens when a ball is tossed into the air:
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(1) A ball is tossed straight up in the air. What happens? Make sure to say something
about forces.
Following these open-ended questions, students moved on to the main body of tasks, which
consisted of 11 problems to be solved. The solving of these tasks occupied the vast majority of
student time in the study and work on these tasks was always spread over several sessions. There
were a range of problems here. Some of these problems, such as the Mass on a Spring problem,
were completely typical of problems that students solve during the first semester of introductory
physics. Others were somewhat more unusual. For example, one of these problems asked students
to answer questions about an alternative universe—“The Geeb Universe”—in which the laws of
physics differ from our own.
After the main body of problems students were given two additional types of tasks. The first,
which I simply call the “Quick” tasks, was a series of seven questions that students were specifically
told they should expect to answer quickly. In general, this portion of the sessions provided an
opportunity for me to ask students some specific questions and probe for particular information
without disrupting the normal problem solving activity I wanted to observe during the main body
of problems. For example, one of these questions asked students whether the equation v = vo + at
is “obvious.”
(3) Some students say that the equation v=vo+at is “obvious.” What do you think?
The final type of tasks constituted what I refer to as “Cross Training.” During this portion of
the study, I taught students a modified practice of algebra-physics that was specifically designed
to incorporate some of the features of programming representations. For example, in this
modified practice, an equal sign was sometimes replaced with an arrow that points from one side
of the equation to the other. The students were first given some written instructions that explained
how to solve problems using this modified representational system, then they were asked to solve
four problems using the modified system. The results of the Cross Training will not be discussed
in this document due to limits of space. Interested readers can examine the instructions and tasks
given to students, which are provided along with the other tasks in Appendix B.
For the purposes of the systematic analysis a subset of the above tasks was selected. This was
necessary because it would have been prohibitively time-consuming to analyze the entirety of the
data corpus in detail. In narrowing the tasks for analysis, I included tasks that, for the most part,
are typical of the algebra-physics practice in introductory courses. This is sensible because one of
the purposes of the systematic analysis is to paint a picture of this practice. Use of this criterion
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eliminated the “Quick” and “Cross Training” tasks from the systematic analysis, as well as some
of the more unusual problems, such as the “Geeb Universe” task. The most unusual problem
included in the systematic analysis was “Running in the Rain.”
A second, minor criterion was also employed to narrow the scope. I wanted to have coverage
from the point of view of the “physical principles” involved in the problems, without too much
redundancy. For example, the main body of tasks included two problems that are traditionally
solved using the principle of Conservation of Momentum. Only one of these two problems was
included in the systematic analysis.
1. Shoved Block
Now the person gives the block a shove so that it slides across the table and then
comes to rest. Talk about the forces and what’s happening. How does the situation
differ if the block is heavier?
2. Vertical Pitch
(a) Suppose a pitcher throws a baseball straight up at 100 mph. Ignoring air
resistance, how high does it go? (b) How long does it take to reach that height?
3. Air Resistance
For this problem, imagine that two objects are dropped from a great height. These
two objects are identical in size and shape, but one object has twice the mass of the
other object. Because of air resistance, both objects eventually reach terminal
velocity.
(a) Compare the terminal velocities of the two objects. Are their terminal velocities the
same? Is the terminal velocity of one object twice as large as the terminal velocity of
the other? (Hint: Keep in mind that a steel ball falls more quickly than an identically
shaped paper ball in the presence of air resistance.)
4. Air Resistance
with Wind
(b) Suppose that there was a wind blowing straight up when the objects were
dropped, how would your answer differ? What if the wind was blowing straight down?
5. Mass on a
Spring
A mass hangs from a spring attached to the ceiling. How does the equilibrium
position of the mass depend upon the spring constant, k, and the mass, m?
6. Stranded Skater
Peggy Fleming (a one-time famous figure skater) is stuck on a patch of frictionless
ice. Cleverly, she takes off one of her ice skates and throws it as hard as she can. (a)
Roughly, how far does she travel? (b) Roughly, how fast does she travel?
7. Buoyant Cube
An ice cube, with edges of length L, is placed in a large container of water. How far
below the surface does the cube sink? (ρice = .92 g/cm3; ρwater = 1 g/cm3)
8. Running in the
Rain
Suppose that you need to cross the street during a steady downpour and you don’t
have an umbrella. Is it better to walk or run across the street? Make a simple
computation, assuming that you’re shaped like a tall rectangular crate. Also, you can
assume that the rain is falling straight down. Would it affect your result if the rain was
falling at an angle?
Table Chapter 6. -2. The eight tasks selected for systematic analysis.
Table Chapter 6. -2 lists the seven tasks selected for systematic analysis, along with the
complete text of each of these tasks. However, notice that part (b) of the Air Resistance problem
has been listed as a separate entry in this table. In what follows, I will consider part (b) of the Air
177
Resistance problem to be a separate task, since students frequently spent a significant amount of
time on this portion of the problem and their work on the two portions was easily separable. Thus,
throughout the rest of this discussion, I will speak in terms of the eight tasks selected for
systematic analysis.
Finally, I want to note that, since the forms and devices we observe are likely to depend
critically on the specific tasks selected, it is important to have a selection that is appropriately
representative. Although I did, as described above, employ some criteria, the ultimate selection of
tasks was still somewhat restricted and arbitrary. This is one of the most important limits on the
conclusions that can be drawn from this analysis and I will discuss this issue further in later
sections of this chapter, and in more detail in Chapter 10.
Details of the experimental sessions
All of the experimental sessions were performed in a laboratory setting. Each pair worked
standing at a whiteboard, and a single video camera was positioned so as to record the movements
of the two students as well as everything they wrote on the board. I operated the video camera
which, at most times, remained fixed in a single location and orientation. Occasionally I
repositioned the camera to account for student movements.
At all times, I did my best to maintain an open and relaxed atmosphere. Students were
encouraged to comment on whatever came to mind, and to air any concerns they might have
about their solutions. In fact, part of the purpose of the open-ended tasks (the first tasks given to
the students) was to encourage a slightly more open attitude to the tasks that followed. In
addition, I believe that having a partner present helped to make students more comfortable. The
hope was that this “open and relaxed” atmosphere would be somewhat characteristic of the group
and pair problem solving that students do outside of their courses—more characteristic than the
“performance” atmosphere that laboratory sessions might tend to engender.
Furthermore, having students work in pairs provided a natural way to encourage them to
explain what they were doing and why. When, in education research, we attempt to study
students while they work independently to solve problems, the students must frequently be
prompted to reflect aloud on their actions. In contrast, when a student works with a partner, the
partner must be consulted on each step of the solution process. Furthermore, using pairs is
certainly a more ecologically valid means of getting students to explain their thought processes.
Presumably, students do not typically reflect aloud as they solve physics problems alone in their
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dormitory rooms. But discussing work with a partner while solving a problem in tandem is a
natural part of algebra-physics discourse.
Each task proceeded as follows. First, I handed each student a sheet of paper that had the
problem or question written on it. The students could ask for clarification after reading the
question or at any time while they worked on the problem. Once the students had read the
problem and asked any immediate questions, they proceeded to solve the problem and, when
they were done, I asked them a few questions about their solution.
Ideally, I always hoped that students would be able to proceed from start to end of the
solution without interruption. However, I did interrupt students occasionally for various reasons. I
sometimes prompted students for clarification. In addition, when students said that they were
“stuck” or they had spent a significant amount of time exploring what I knew to be an unviable
avenue, I interrupted them and gave them some hints to start them on a more useful track.
It is important to understand that these interruptions and hints pose only a very minor
difficulty for this study. It is true that these hints, by their nature, disrupt the natural flow of
problem solving. However, this study is not centrally concerned with the “flow of problem
solving.” In this systematic analysis, I will not attempt to describe the process by which a solution
is reached. At no time, for example, will I attempt to characterize the order in which equations are
deployed. Instead, I am interested in some specific kinds of events, namely the construction and
interpretation of equations.
What is potentially more problematic is the fact that my prompts to explain solutions, even
though primarily at the end of episodes and generally minimal, could tend to alter the frequency
of these events. This is true, first, because such prompts can directly provoke interpretation events.
In addition—and perhaps more profoundly—these prompts could tend to suggest to students
that I am particularly interested in behavior of this sort.
The reason for including these prompts is that, no matter what I did, the experimental sessions
were not going to be precisely the same as a session with a few undergraduates working together
on a problem set. Furthermore, the bigger danger—I worried—was that students would tend to
be particularly hesitant to voice their thoughts in a laboratory setting, in front of a video camera.
For this reason, as I said, I felt it was appropriate to cultivate a relaxed atmosphere, one open to
reflection and comment. Thus, I may have, in some sense, over-stimulated the production of
interpretations, and this must be kept in mind. However, importantly, this worry does not pose
difficulties for the assertion that students possess the knowledge necessary to interpret expressions.
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Sessions were typically between 1 and 1.5 hours in length, with pairs participating in between 4
and 6 sessions (refer to Table Chapter 6. -3). The total time spent by all students working on tasks
was approximately 27 hours. When these 27 hours are narrowed to include only student work on
the tasks selected for systematic analysis, the result is 11 hours of videotape.
# of Sessions
Total Hours
Analyzed Hours
Mike&Ken
Jack&Jim
Mark&Roger
Jon&Ella
Alan&Bob
4
5.0
2.0
5
4.5
1.5
5
6.0
2.0
6
6.0
3.0
6
5.5
2.5
Totals
26
27.0
11.5
Table Chapter 6. -3. The time that students spent in the experimental sessions.
Analysis Techniques
The above sections described the nature of the algebra data corpus: the participants involved,
the tasks they performed, and the experimental setting. Now I want to describe the method used
to analyze the resulting videotapes and I will begin with a brief overview of the technique
employed. When I began the analysis, I had not yet narrowed my focus to only 8 tasks, but I did
quickly decide to exclude the Quick Tasks, Cross Training, and one task from the main body
which pertained to electrical circuits. I then made a first pass at transcribing student work on all of
the remaining tasks—a total of about 20 hours of videotape.
Next, one-by-one, I selected individual episodes for coding. Here an “episode” is one pair’s
work on a single task. After an episode was selected, I viewed the corresponding videotape again
and polished the transcript. Then the episode was coded and the coding technique refined so that
the episode was comfortably accommodated. In part, refining the coding technique meant
adjusting the lists of forms and devices so as to appropriately cover the data. Then another episode
was selected and the process iterated. The idea of this initial coding of a few episodes was to refine
the coding techniques to a passable level.
After coding 7 individual episodes in this manner, I felt that the coding scheme was stable
enough to apply to a large portion of the data corpus. At this point I narrowed my focus to 8
tasks, as described above, corresponding to 40 problem solving episodes. All of the corresponding
tapes were watched again, and the transcripts polished. Then these 40 episodes were coded, the
coding techniques refined once more, and then all 40 episodes were coded one final time.
The following sub-sections fill in the details of this analysis technique.
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Transcription
The first step in analyzing any episode was to transcribe that episode into a written form. For
several reasons, transcription is extremely helpful in performing almost any type of fine-grained
analyses of videotape-based data. First, executing this type of analysis requires reviewing episodes
repeatedly and it is typically faster to read a transcript than to watch a tape, particularly once the
analyst has some familiarity with the videotape. In addition, analyses of the sort I will describe
here require the comparison of key moments within and across episodes. This sort of comparison
is very difficult to do without transcripts. In fact, even if I wanted to review the original videotapes
at every stage of the analysis, I would likely need transcripts to serve as an index to the tapes.
As anyone who has transcribed spoken utterances is aware, verbal communication is somewhat
different than written communication. For this reason, it is necessary to develop some special
techniques to record what people say. To a certain extent, this means inventing an extended set of
punctuation for use in the transcripts. The special notations that I used are described in Appendix
A. In addition, it was also necessary for me to keep track of certain types of non-linguistic
acts—what people do other than what they say. As described in Appendix A, these non-linguistic
acts were also recorded in the transcripts, set off in brackets. I will give an example in a moment.
For the purposes of this study it was extremely important to have an accurate record of
everything that students wrote on the whiteboard. Creating this record posed some difficulties.
One option would have been to simply describe what students wrote and drew on the board
within the written transcript. However, it is often very difficult and time-consuming to describe
diagrams and equations in textual form. Thus I chose to supplement the written transcripts with
drawn images of the board: Any time that an equation was written, it was added to the drawn
image and a more modest notation was made in the written transcript.
The images of students’ board work were created using standard computer-based illustration
tools, such as Adobe Illustrator. More recent versions of these tools have one feature that is
particularly useful in this regard: They allow images to be built up in layers. This was extremely
useful because by building up board images in layers I was able to keep track of successive
additions and modifications to what was written.
I will illustrate how this worked with an example. In some cases, students proceeded in a very
neat and orderly manner, writing one equation under another. In such cases, it was sufficient for
me to simply reproduce the final state of the whiteboard, as it appeared at the end of the problem.
Mike and Karl’s work on the Air Resistance task, shown in Figure Chapter 6. -1, proceeded in
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such an orderly manner. They worked from the top to the bottom of what is shown, without
modifying or erasing along the way. Of course, this reproduction of their board work does not
contain all information concerning the order in which things were drawn and written, but
combined with the written transcripts it is possible to reconstruct this ordering information.
+
(kv)
f(v)= Fa
m
v
Fg
a(t)= g +
f(v)
m
kv
=0
m
−Fg + kv
= −g +
a(v)=
v=
m
m
g
k
Figure Chapter 6. -1. Whiteboard image from Mike and Karl's work on the air resistance task.
+
+
(kv)
f(v)= Fa
+
(kv)
f(v)= Fa
(kv)
f(v)= Fa
m
Fw
m
m
Fw
Fw
v
Fg
a(t)= g +
v
v
Fg
f(v)
m
mph
kv +50
= −g +
=0
m
−Fg + kv
a(v)=
m
m
v=
g
k
Fg
f(v)
m
h
k +50mp
kv
(v
+
v
)
== −g
+
=0
w
−g + m = 0
m
−Fg + kv
a(v)=
m − V
m
v=
g
w
k
a(t)= g +
f(v)
m
h
k +50mp
kv
(v
+
v
)
== −g
+
=0
w
−g + m = 0
m
−Fg + kv
a(v)=
m −+ V
m
v=
g
w
k
a(t)= g +
Figure Chapter 6. -2. Three successive board images showing Mike and Karl's modifications of their Air
Resistance work to account for wind. Changes made in each panel are marked with gray arrows.
When students erased and modified their work on the board, the situation was somewhat
more complicated. As students made modifications, these were added as layers over their existing
182
work. For example, after completing the Air Resistance task, Mike and Karl modified the solution
shown in Figure Chapter 6. -1 to account for the presence of a wind. Figure Chapter 6. -2 shows
three successive sets of modifications that they made to their work. The changes made in each
panel are indicated with gray arrows.
Figure Chapter 6. -3 contains the segment of transcript that corresponds to the modifications
made in the first panel of Figure Chapter 6. -2. Note that the first and last utterances in this
segment of transcript contain notations concerning changes to what is written on the board. These
written changes are set off in brackets and are preceded by a “w.” written just inside the bracket.
These notations tell us that, during the first utterance, Karl drew a circle around the kv/m term, as
shown in Figure Chapter 6. -2. Similarly, during the last utterance, Karl wrote “+50mph” above
this term. The point of this example is to illustrate how the combination of written transcript and
board image allow the progress of student work to be reconstructed. Note, for example, that it is
not necessary to say in the transcript precisely where the “+50mph” was written on the whiteboard.
Karl
Let me go back to, let me go back to this term. This term right here. Okay. [w. circles
kv/m term] This term right here is the force opposing
Mike
The acceleration
Karl
The acceleration due to gravity. And it's proportional to the velocity of the ball.
Mike
Right.
…
Karl
Well, what I'm saying is that the, this force [g. circled term], this force is relative- This
force is proportional to the velocity. But it's proportional to the velocity of the relative air
and ball system. It doesn't care about the really absolute, the absolute, uh, speed of the
ball relative to some outside observer.
Mike
It's just the relative between the two?
Karl
(This) only cares about the speed relative to between the ball and the air. That's what cThat's what counts this factor. [g. circled term] So, if you have a speed, if you have a
wind that's blowing up initially, then this speed [g. circled term] is the speed relative to
the ground plus say fifty K P H. [w. +50mph)]
Figure Chapter 6. -3. Portion of the transcript that corresponds to the modifications shown in the first panel of
Figure Chapter 6. -2.
Notice that the transcript given in Figure Chapter 6. -3 contains some additional nonlinguistic acts that are preceded by “g.” rather than “w.” These non-linguistic acts are gestures;
they indicate that the student pointed to something written on the board. Such notations are
essential to understanding many student utterances involving written expressions. For example,
when Karl says in this segment:
Karl
Well, what I'm saying is that the, this force [g. circled term], this force is relative- This
force is proportional to the velocity. …
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it would be difficult to understand what he means by “this force” if we did not know what he was
pointing to as he made this statement.
There is an important subtlety relating to the transcription of gestures. When a person points
to a given location on the whiteboard, how do we know what they are pointing to? For example, if
Karl points right at the vertex of the letter ‘v’ in the kv/m term, he could be intending to point at
this letter, the kv/m term, at the entire equation, or even at a region of the board. Given this
inherent ambiguity of gestures, one might decide that it is more appropriate to transcribe gestures
in a more neutral manner. For example, we could divide the whiteboard into a fine grid and then
just note where in the grid the student pointed. However, I do not believe that it is necessary to
use this extremely neutral method. In most cases, I found that—as in the example above—I could
use other aspects of the context to comfortably determine what students intended to indicate
with a pointing gesture.
Using context to disambiguate the nature of an utterance for transcription is not
unprecedented; we must do precisely the same thing in transcribing speech. For example, if a
student says: “I didn’t hear what you said,” we know to write the word “hear” and not “here,”
because that is what makes sense given the context of the sentence. Imagine the extreme case in
which we restrict ourselves to a purely phonetic rendering of vocalizations. Although this would
likely be useful for some purposes, it would be very difficult and time consuming, and researchers
do not consider it to be necessary for all types of analysis. I take the same stance with regard to
transcribing gestures; I use context—where I can—to determine what a student intended to point
to.
The base-level coding scheme
Once an episode is transcribed in the manner described above, the coding of that episode can
commence. First, note that the goal of the systematic analysis is to code every episode in terms of
symbolic forms and representational devices. But where should we look for forms and devices in a
transcript? Transcripts certainly do not come equipped with annotations that say “Look for
symbolic forms here” and “Look for representational devices here.”
For this reason, there needs to be an intervening step between the production of a transcript of
an episode and the coding of that episode in terms of forms and devices. In this intervening step
in the analysis, which I call the “base-level” coding, the events that must be explained in terms of
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forms and devices are identified. In addition, the base-level coding provides some basic
information concerning the character of the data corpus.
The base-level analysis is designed to identify the two types of events that must be explained
in terms of forms and devices:
(1) A symbolic expression is constructed by a student.
(1) A student interprets an expression.
Identifying events of type (1) requires an expression-centered analysis of the episode. In order to
find these events, I have to look at every expression that is written and decide if it was
constructed. Events of type (2) require an utterance-centered analysis. To identify these events, I
have to look at every utterance made by each student and decide if it is an interpretation of an
expression. In what follows I describe, in turn, the expression-centered and utterance-centered
components of the base-level analysis.
-mg
F
m
µ
mg
F=µN
f
Figure Chapter 6. -4. Board image from Jack & Jim's work on the Shoved Block task.
Expression-centered coding: Identifying constructed expressions
In the expression-centered coding, I look systematically at every expression that is written in
the episode under analysis. With a few exceptions, every one of these expressions is coded as
remembered, derived, constructed, or as a combination of two or more of these categories. Of
course, we need to have a way of deciding into which of these categories a given expressionwriting event falls, and I will tackle that issue in a moment. But there is an even more basic issue
that must be considered first; namely, I have to specify what counts as “an expression.” To begin,
isolated symbols written on the whiteboard do not count as expressions. For example, if a student
writes the letter ‘m’ on an object in a diagram to indicate that this object has a mass equal to m, I
do not treat this as an expression. However, any sequence of more than one symbol does count as
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an expression. Thus, there are 3 expressions in the board image shown in Figure Chapter 6. -4; mg,
-mg and Ff=µN are all expressions since they involve more than one symbol, but the isolated
symbols F, m, and µ do not count as expressions. Notice that expressions need not have equal
signs.
There are some more difficult cases to consider in deciding what constitutes an expression to
be coded. As in the example shown in Figure Chapter 6. -2, students sometimes modified
expressions that were already written. In such cases, it is necessary to keep track of both the
original expression and the expression produced by the modification, since forms and devices
could play a role in both events. When actually coding my data, I kept track of modifications
separately. Ultimately, however, I pooled all the data so that expressions produced by
modifications were treated the same as other expressions.
After the expressions in an episode were recognized, each one was coded as remembered,
derived, constructed, or as a combination of two or more of these categories.
Remembered expressions
A remembered expression is one that a student appears to have simply written out “from
memory.” For example, students will often begin work on a task by writing F=ma on the
whiteboard, usually with little comment. When this happens, we can be relatively certain that the
student is not rederiving the expression or reconstructing the expression.
A variety of types of evidence can be brought to bear in determining that an equation is
remembered. If a student begins a problem by immediately writing:
x = xo + vot + 12 at 2
I would code this as remembered. There are several reasons that this is justified: (1) This expression
is relatively complicated and it was written without explanation or much evident prior work. (2) I
am aware that this is an expression that is taught in introductory physics courses. (3) When this
expression is taught it is often written in exactly this manner, using these symbols (x for position,
xo for initial position, etc.) and with the terms in this order.
For further illustration, consider the following two equations and the comments that students
made as they wrote them:
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v 2f = vi2 + 2ad
Jim
So, I think I recall the equation for this being V F squared equals V I squared plus 2 A D?
[w. Vf2=Vi2+2ad]
________________
x = xo + vot + 12 at 2
Ella
So how does that equation go? It goes like X equals [w. x=]
Jon
X // naught.
// X naught [w. xo]
Ella
Jon
But we don’t need to know that.
Ella
plus V
Jon
T
Ella
naught T plus one half A T squared. [w. vot+1/2at2]
In each of the above cases, the students wrote an equation that is usually taught early on in an
introductory physics course, and they wrote it in a standard form. Furthermore, the equations are
apparently written with little effort. In fact, in the first excerpt, Jim actually tells us that he
“recalls” the equation that he is writing. And Ella starts out by saying “how does that equation
go?”, a clear sign that she is trying to remember an equation that she has used before. Each of
these expression-writing events can confidently be coded as remembered.
Derived expressions
A derived expression is one that is written by applying algebraic rules-of-manipulation to one
or more expressions that are already written on the whiteboard. In the prototypical case, a single
equation is manipulated to produce a new equation, as kx=mg can be manipulated to produce
x=mg/k. Of course new expressions may also be derived by combining multiple lines.
For the subjects in my study, the use of symbolic manipulation to produce new expressions
was a quick and fairly effortless process, and almost universally error-free. Thus, students often
said little during such derivations. For example, when solving the Vertical Toss problem, Mike
and Karl began by writing this expression for the distance
s = vot − 12 gt 2
After doing some work they found an expression for the time that it takes for the ball to reach its
maximum height:
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v
t= o
g
Then Mike said “plug it back in” and he proceeded to substitute this relation for the time into the
above expression for the distance. He did this with little comment, only mumbling a “V naught”
here and there. When he was done he had written three new lines:
v 
v 
s = vo  o  − 12 g o 
 g
 g
2
v2
v2
= o − 12 o
g
g
2
v
s = 12 o
g
Each of these lines would be coded as derived. The first line was produced by substituting the
relation for time into the expression for distance. Then the second line was obtained by
manipulating the first line, and the third by manipulating the second.
Constructed expressions
Except for some rare and marginal cases that I will discuss in a moment, all remaining
expressions fall into the category of constructed expressions. If a student did not remember an
expression and they did not derive it, then where did the expression come from? The answer is
that they must have just invented it. These invented expressions were coded as constructed.
In the preceding chapters, I described a number of episodes in which students constructed
expressions. Mike and Karl’s construction of an expression for the coefficient of friction that
depends on mass—described in Chapter 2—is a clear example.
µ
µ = µ 1 + C m2
The evidence that this expression was constructed is the complement of the evidence I would use
to argue that an expression was remembered. First, it was preceded by significant work and
accompanied by an explanation. Second, it is not recognizable as a relation that is typically taught;
you will not find this expression in any physics textbook.
Combinations of remembered, derived, and constructed
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Frequently, expressions do not fall neatly into one of the above three categories. This is not a
significant difficulty; in such cases, multiple codings are assigned. For example, suppose that when
solving the Mass on a Spring problem a student wrote
Fs = Fg
to indicate that the force of the spring must be equal to the force from gravity. Further suppose
that this is the very first thing written—no other expressions are yet on the whiteboard. Then if
the student writes
kx = mg
then this second expression would be coded as both derived and remembered. To understand this
coding, imagine that the students had written the following two expressions for the forces of
springs and gravity:
Fs = kx ; Fg = mg
Then they could have derived kx = mg by substituting these two relations into Fs = Fg . However,
since at the time that kx = mg was written these two expressions were not written on the board, I
treat the “kx” and “mg” as drawn from memory. The general rule being applied here is this:
Whenever an expression is produced using expressions that are currently written on the
whiteboard it receives a derived coding. And whenever an expression draws on known expressions
that are not currently written anywhere on the whiteboard it receives a remembered coding. In this
case, the expression kx = mg draws on both remembered expressions and expressions written on
the whiteboard.
Marginal cases
Even allowing for combinations of the above categories, some expression-writing events still do
not fall into any of these categories and must be treated separately. I start by noting that the
“remembered” category is intended to be quite narrowly defined. When a student writes
x = xo + vot + 12 at 2
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this is coded as remembered because the student is producing a literally remembered string of
symbols. You will find this expression written in textbooks using the same letters and written in
precisely this order. In coding expressions, I do allow some deviations from this strict rule. For
example, occasionally students will write the above expression with the terms on the right hand
side in a different order. But, for an expression to be coded as remembered, it should be evident
that the student is working from a literally remembered string.
Constructed expressions, on the other hand, are intended to lie at the other extreme, they are
supposed to constitute true inventions by the student. However, there are some cases that lie
between these two extremes and which I therefore treat separately. There are three cases to be
considered here, and I consider each in turn.
Remembered templates. Students seem to employ a set of templates that define a narrow class of
expressions. Unlike true “remembered” expressions, these templates do not literally determine the
symbols that are to be written—they do not specify that particular symbols to be written in a
specific order. Nonetheless, the expressions produced by these templates seem to fall short of the
other extreme, constructed expressions; although they have the template-like character associated
with symbol patterns, they are more specific to content than forms. An example of a remembered
template is the following expression template for the volume of a rectangular parallelepiped:
volume = length × width × height
When students employ this template, they do not always write literally the same expression; for
example, they might write V=lwh or V=xyz. Although these expressions are thus not literally
remembered strings, it would certainly be inappropriate to claim that a student invented such
expressions. Furthermore, like remembered expressions, expressions associated with remembered
templates are usually written quickly and with little comment. This is further evidence that they
should be treated more like remembered expressions than novel constructions. Table Chapter 6. 4 contains a list of all of the remembered templates that appeared in the algebra data corpus.
Remembered Templates
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volume = length × width × height
area = length × width
total = rate × time
total = density × volume
average =
sum of items
number of items
change = final − initial
Table Chapter 6. -4. The remembered templates used in the data corpus.
Value assignment statements. This class of expressions are always of the form variable=number. For
example, if a student writes vo =10 this would be coded as a value assignment statement. Such
expressions are not something that one could copy out of a textbook, they are specific to the
circumstance. Nonetheless, they are hardly worth calling “inventions.” Furthermore, these very
simple expressions are actually quite common—common enough that they would make a
significant contribution to any category that they were placed in. For this reason, I believe it is
worthwhile to separate out these expressions and treat them as a special case to be coded
independently.
Principle-based expressions. This last category is the most subtle and I believe the potentially most
controversial of these marginal cases. “Principle-based” expressions constitute another class of
expressions that are not literally remembered but also may not be worthy of being called
“inventions.” To explain this category, I have to return to the discussion of principle-based
schemata that appeared in Chapter 3. In that chapter I argued against the alternative view that
students always derive novel expressions by beginning with physical principles and then deriving
equations from those principles. In this alternative view, students possess schemata that are
associated with physical principles and these schemata provide templates for the construction of
expressions. For example, there is a template of the form
initial momentum = final momentum
that is associated with the principle of Conservation of Momentum.
I argued in Chapter 3 that, even when students claimed to be working from principles of this
sort, symbolic forms could be seen to be playing a role in their constructions. Furthermore, they
seemed to often use principles more for after-the-fact rationalization than as strict guides to
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construction. I stand by these claims. However, in a very small number of cases, students made a
show of working from principles and there was no evidence to the contrary. In these cases the
resulting expression was coded as principle-based.
In addition, throughout this work I have taken the stance that multiple types of knowledge
must be involved in any construction episode. Therefore, even in construction episodes where
forms and devices are clearly implicated, knowledge tied to principles may be playing some more
or less auxiliary role. To keep track of such occasions, where appropriate I coded expressions as
both constructed and principle-based.
k
x
F=-kx
m
F=ma
Figure Chapter 6. -5. Jack & Jim's diagram for the Mass on a Spring problem.
An example is provided by Jack and Jim’s work on the Mass on a Spring problem. They began
by drawing the diagram shown in Figure Chapter 6. -5 and commenting that there were two
forces, a force upward due to the spring and a force downward from gravity. Then Jack wrote an
expression in which these two forces were equated:
kx = mg
Jack
So, okay, so then these two have to be in equilibrium those are the only forces acting on
it. So then somehow I guess, um, (3.0) That negative sign is somewhat arbitrary [g. F=-kx]
depending on how you assign it. Of course, in this case, acceleration is gravity [g. a in
F=ma] Which would be negative so we really don't have to worry about that [g.~ F=-kx] So I
guess we end up with k x is m g. [w. kx=mg]
This expression is actually assigned three codings: derived, constructed, and principle-based. It is
derived because it makes use of expressions that are already written on the whiteboard, namely
“kx” from F=-kx and the “ma” from F=ma. Furthermore, I argued in Chapter 3 that the
construction of this expression could not be completely explained by the statement that Jack is
applying known principles; the noteworthy feature in this regard was Jack’s “hand-waving”
treatment of the signs involved. For this reason, the expression is coded as constructed.
Finally, I believe that knowledge that is more closely tied to principles plays some role in
constructions of this sort. Although I have argued that it is not justified to claim that the work
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here is being driven solely by a problem solving schema that strictly guides what is done, my guess
is that there is nonetheless knowledge active that is closely related to formal principles and
explicitly sanctioned procedures. For example, in this case it is likely relevant that Jack and Jim
have learned to draw force diagrams and to write equations involving the forces identified. For this
reason, I also code this expression-writing event as principle-based.
As an argument for including this additional information in the coding scheme, note that one
point of the systematic analysis is to paint a picture of the data corpus for readers. By including
information concerning how many expressions may plausibly have been constructed from
principles, I am allowing readers—especially skeptical readers—to draw some of their own
conclusions concerning the nature of algebra-physics.
Summary of expression-centered coding
In summary, the first part of the base-level analysis is the expression-centered coding. In this
coding, every expression in an episode is identified, then each of these expressions is coded as
remembered, derived, constructed, as one of three “marginal” cases, or as a combination of these
categories. (Refer to Table Chapter 6. -5.) The main function of this component of the base-level
analysis is to identify constructed expressions so that these can later be coded in terms of forms
and devices.
Expression-Centered
Coding Scheme
Remembered
Derived
Constructed
Marginal Cases:
Remembered Template
Value Assignment
Principle-Based
Combinations
Table Chapter 6. -5. Summary of categories for coding expressions in the base-level analysis.
Utterance-centered coding: Identifying interpretations of equations
The second component of the base-level analysis is the utterance-centered coding. The point
here is to identify places in an episode where a student “interprets” an equation. This is an
193
inherently tricky process since part of the purpose of analyzing the data corpus is to determine
what constitutes an interpretation of an equation.
My way out of this bind is to cast a wide net. In order to identify interpretive utterances, I
began by operationally defining an interpretation to be any statement that refers to or incorporates
part or all of an expression. For example, a student might point to an expression as they say
something, or a student might mention a portion of an expression in an utterance. Consider this
interpretation given by Jon & Ella, which I discussed in Chapter 4.


h
w

#
drops


+
=
wx
vd2 v R 
 A⋅s  
 v d 1+ 2

vR




Ella
Okay, V R is bigger. Then (10.0),, This is gonna get smaller, [w. arrow pointing to second
term] but this is gonna get bigger. [w. arrow pointing to first term] I think.
Jon
Yeah, I think - that gets smaller [g. second term] cause V walking is smaller than V
running.
Here, as Ella talks, she indicates portions of an expression by drawing arrows on the whiteboard.
Then, in Jon’s response, he also indicates a part of the expression, this time by simply pointing at
it. These statements would be coded as interpretations because an equation is being referred to by
the students.
This operational definition may appear to be very broad; however, I needed to narrow it only
slightly to have a useful rule for identifying interpretations. In particular, four types of utterances
that would otherwise fall under the broad rubric of this definition, were exempted:
(1) Literal reading of expressions. I excluded utterances in which a student simply reads the symbols
that appear in an expression. Bob’s statement below is an example:
2
v
h = 12
g
Bob
Um, so, H equals one half V squared over G. Right?
(2) Descriptions of problem solving actions. Students frequently recounted the steps that they took
to arrive at a solution. For example, a student might have stated that they had started with a
certain equation and then solved for a given variable in that equation. Utterances of this sort were
not considered to be interpretations. This next passage, in which Jim recounts his work on the
Vertical Toss problem, is typical:
194
Jim
But, what I did first of all … unit conversions which I've erased. And then I remember the
formula which I've also erased, V F squared equals V I squared plus two A D [w. rewrites
equation] So, using this I plugged in the converted values [g. 0=…] and I found the
distance, which for me was three hundred and thirty six feet. Using this I plugged it back
into the formula which is over here [g. x=…] and you do get a quadratic, depending on T.
Which I plugged into the ever reliable quadratic formula up here and it spat the value of
four point six eight seconds.
(3) Comments made during constructions. As students struggled to construct novel expressions, they
made many comments about the expression they were working on. These utterances are not
treated as interpretations. Instead, as described below, they were taken into account when coding
the construction effort itself—the expression-writing event—in terms of forms and devices. In
general, only utterances following the writing of a novel expression were considered to be
candidates for interpretive utterances; interpretive utterances were always interpretations of
existing, fully written expressions.
(4) Association of symbols with a physical entity. Sometimes a student will indicate a symbol or
group of symbols and associate it with a physical entity. Jack’s statement below is an example:
x=
Jack
mg
k
(An:::d) x equals m g over k. [w. x=mg/k] Where x is the distance from the ceiling.
Here, Jack tells us what ‘x’ is, it is the distance of the block from the ceiling. Similarly, a student
might point to ‘mv’ written in an expression and associate it with the momentum of some object
in the problem situation. Such statements were not treated as interpretations.
Since this last exemption may seem problematic I will elaborate on my rationale. In brief, the
reason for excluding these utterances is that, although it may be reasonable to describe them as
“interpretations,” these statements should not be explainable in terms of symbolic forms and
representational devices. Recall that, in my discussion of registrations in Chapter 2, I listed four
types of important registrations: (1) individual symbols, (2) numbers, (3) whole equations, (4)
groups of symbols such as individual terms, and (5) symbol patterns. Forms are associated only
with the last type of registration, symbol patterns. In contrast, utterances that associate a symbol
or portion of an expression with a physical entity involve registrations of type (1) or (4). These
types of registrations do not involve seeing the type of expression-spanning structure that is
associated with symbol patterns. For example, if I point to ‘mv’ in an expression and associate it
with the momentum of some object, then I am not recognizing structure in the expression, only a
component of the expression, whose substructure is not relevant. For this reason, utterances of this
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sort should not be explainable in terms of forms and devices, and, since the point of this portion
of the analysis is to identify events that should be explainable in these terms, these utterances
should be excluded.
So, the above operational definition plus the four exemptions were used to identify
interpretive utterances in episodes. Only one minor potential complication remains to be
mentioned. Before deciding whether an utterance is an interpretive utterance, the transcript must
essentially be segmented into utterances. Another way to say this is that we need to have a way of
knowing where one interpretive utterance ends and the next begins. The bottom line is that this
turns out to not be much of a difficulty since interpretive utterances tended to be fairly isolated
events. For example, a pair of students might solve an entire problem without interpreting any
expressions and then provide an interpretation of the final result. Or a student might interpret an
expression and then go on to begin deriving some other expression. Furthermore, since the
majority of interpretive utterances are isolated in this manner, it was possible to bootstrap out of
my developing understanding of the nature of interpretations to deal with the few difficult cases.
In other words, once I had developed the categories that define what an interpretation of an
equation is, I could use these categories, in difficult cases, to break a stream into separate
interpretive utterances.
A sample base-level coding
Before going on to describe the next stage of the analysis, I want to present a base-level coding
of an entire problem solving episode. The episode is Jack and Jim’s work on the Mass on a Spring
problem, and it is one of the shortest episodes included in the systematic analysis—it was only 2
1/2 minutes in duration. The complete board image from their work is shown in Figure Chapter
6. -6.
196
k
x
F=-kx
m
F=ma
kx=mg
x=
mg
k
Figure Chapter 6. -6. Reproduction of Jack & Jim's board work for the Mass on a Spring problem.
The entire transcript for this episode is given in Figure Chapter 6. -7 and is annotated with the
appropriate base-level codings. Perusing this transcript and the board image tells us that Jack and
Jim began by writing two equations from memory: F=-kx and F=ma. Then they wrote the
expression kx=mg, which is coded as a combination of derived, constructed, and principle-based for
reasons discussed above. The last expression that they wrote, x=mg/k, was derived by manipulating
kx=mg. Finally, the episode concludes with two interpretive utterances, both given during Jim’s
final turn in the conversation. Note that this is an instance in which a knowledge of standard
interpretation types is used to segment a series of contiguous statements into two separate
interpretive utterances. In this case, Jim gives two CHANGING PARAMETERS interpretations of the
equation x=mg/k. In the first, he imagines that the mass varies while all other parameters are held
fixed, and in the second the spring constant varies.
197
Jack
Jim
Jack
Jim
Jack
Jim
Jack
Jim
Jack
Jim
Jack
Jim
Bruce
So, let's see, so we've got the ceiling. [w. draws ceiling] With our
trusty chunk o' spring. [w. draws spring] And a mass. [w. mass w/
label.]
Spring constant K. [w. k]
Okay, um, and we're assuming that it's at equilibrium so it's at a
distance we'll call this X. [w. marks off height, x] From the ceiling
and that's pretty much constant. Um. (Do you) put in force
vectors, the various forces acting on it?
Mmm. Well, there's the gravitational force acting down. [w. arrow
down; w. F=ma] And then there is
a force due to the spring // holding it up.
// a force due to the spring which I
believe is,, [w. F=-kx] Is equal to that.
Right, cause the::: The spring potential is one half K X squared.
So, okay, so then these two have to be in equilibrium those are
the only forces acting on it. So then somehow I guess, um, (3.0)
That negative sign is somewhat arbitrary [g. F=-kx] depending on
how you assign it of course in this case acceleration is gravity [g. a
in F=ma] Which would be negative so we really don't have to worry
about that [g.~ F=-kx] So I guess we end up with K X is M G. [w.
kx=mg]
Hm-mm.
An:::d how,, How does the equilibrium position of the mass
depend on the spring constant. [Read from sheet.] So I guess
we're solving // for X. [w. underlines x in mg=kx]
// for X equals M G over K.
(An:::d) X equals M G over K. [w. x=mg/k] Where X is the distance
from the ceiling.
Okay, and this makes sort of sense because you figure that, as
you have a more massive block hanging from the spring, [g. block] then
you're position x is gonna increase, [g.~ x in diagram] which is what
this is showing. [g.~ m then k in x=mg/k.] And that if you have a
stiffer spring, then you're position x is gonna decrease. [g. Uses fingers
to indicate the gap between the mass and ceiling.] That why it's in the
denominator. So, the answer makes sense. [Looking at Bruce as
he says this last.]
Hmm-mm, hmm-mm. (6.0) Okay. Is that all it asks on that one?
F=ma.
Remembered expression.
F=-kx.
Remembered expression.
kx=mg.
Constructed+derived+
principle-based expression.
x=mg/k.
Derived expression
“as you have a more massive
block hanging from the
spring…”
Interpretive utterance.
“if you have a stiffer spring …”
Interpretive utterance.
Figure Chapter 6. -7. Complete transcript of Jack and Jim’s work on the Mass on a Spring task. Base-level
codings are given in the right hand column.
Coding symbolic forms
Following the base-level analysis of an episode, the next stage of the analysis is to code every
interpretive utterance and construction event in terms of forms and devices. In this section I will
discuss the coding of forms. The first thing to note about the coding of the algebra data corpus in
terms of forms is that this process requires doing two things simultaneously: (1) The list of forms
must be defined and (2) each individual event must be coded in terms of the forms that are on
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this list. In what follows, I will separately discuss each of these two components of the coding of
forms.
Defining the set of forms
As I described above, iterative coding and recoding of excerpts was an essential element of
defining the set of forms. I began with a preliminary list derived from a pilot study. Then I coded
7 selected episodes, refining the set of forms along the way. Finally, I coded the work of all pairs
on the 8 tasks selected for systematic analysis, refined the categories, and then coded all of these
episodes again.
I did not develop hard and fast rules for looking at a data corpus and specifying the symbolic
forms involved. Instead, following diSessa (1993), I employed a list of heuristic principles. diSessa
employed a similar list of principles to identify knowledge elements in the sense-of-mechanism,
and some of my principles are similar or identical to his.
1. Coverage of the data corpus. The set of symbolic forms must be able to account for every
constructed expression and interpretive utterance identified according to the procedure
described above. If it cannot, then new forms must be added or the set adjusted in some other
manner.
1. Parsimony. In tension with the requirements of the coverage principle is the need for parsimony
in theoretical work. It would not be scientifically useful, for example, to account for every
event in terms of a unique symbolic form. Instead, we want the minimum set that provides
adequate coverage.
1. Inclusion of a symbol pattern. Every symbolic form, by definition, involves a symbol pattern.
Thus, when positing the existence of a form it must be possible to specify a particular symbol
pattern that is associated with the form. This heuristic can help greatly in narrowing the
possible candidates for forms. For example, we might notice that students use intuitions like
“blocking” or “guiding” during problem solving. However, if we cannot associate these
intuitions with specific symbol patterns, then they cannot be considered legitimate candidates
for forms. Furthermore, by recognizing syntactic structures that exist in algebraic expressions,
we can find additional candidates for symbolic forms.
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1. Symbol patterns only preserve distinctions consequential for interpretive behavior. Of course, not
all of the candidate symbol patterns that we find by examining syntactic structure will be
associated with any forms. The heuristic here is that symbol patterns should only involve a
sensitivity to distinctions that are consequential for the behavior associated with construction
and interpretation events. For example, we saw that when the students in this study
interpreted expressions, it was not generally consequential whether symbols appeared to the
first, second, or a higher power. Thus, these syntactic distinctions are not preserved in symbol
patterns.
1. Impenetrability. Symbolic forms are the base-level “vocabulary” in terms of which people
specify and understand equations. This vocabulary should be “impenetrable” in the sense that
elements cannot be decomposed in terms of other knowledge.
1. Continuity with prior knowledge. In defining the set of forms, we must be sensitive to the fact
that, as discussed in Chapter 5, this knowledge develops out of existing cognitive resources. In
addition to constituting a very general constraint on the nature of any forms we might
propose, this principle also provides us with a means to locate candidate symbolic forms. For
example, we can look to diSessa’s list of p-prims for candidates and we can look at the
“patterns in arithmetic word problems” that, as discussed in Chapter 3, have been uncovered
by some researchers in mathematics education. In fact, both of these sources were extremely
helpful for uncovering candidate forms.
1. Approximation to expertise. Finally, we must keep in mind that we know something about the
behavior that physics instruction is ultimately trying to produce; students are supposedly
learning to use equations “in an expert manner.” Even if we do not know the precise nature of
expert physics knowledge, we do know, in many cases, what equations and solution methods
experts use. Thus, if we presume that the set of symbolic forms will ultimately contribute to
expert physics problem solving, we can make some guesses concerning the reasonableness of
specific forms. For example, since we are looking at somewhat advanced students, we should
see forms that are useful for constructing the types of expressions that experts write. The many
Competing Terms expressions involving forces are a good example here. And forms should be
associated with intuitions that we expect to play a useful role in expertise.
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The items on the above list differ considerably in how specific they are to the project at hand.
The first two principles are quite general and could potentially be applied to a wide range of
cognitive modeling projects. In contrast, principles (3), (4), and (5) are the most specific to this
project and they do the most work in narrowing the survey guided by these heuristics to symbolic
forms. Principle (6) is another very general heuristic. In attempting to identify knowledge we
should always be sensitive to the constructivist principle that this knowledge must develop from
existing knowledge. (See diSessa, 1993, Smith et al., 1993, and Roschelle, 1991 for more
discussion of this point.) However, in saying that we should look to p-prims and “patterns” for
continuity with forms, I have specialized this principle to the current project and I have made it
much easier to apply. Finally, though the last principle is not quite universal, it applies to a broad
range of circumstances that might be termed “instructional.” When people are learning to
participate in an existing practice, we can use a recognition of the established norms of that
practice to guide our search for the knowledge necessary to participate in the practice.
Recognizing forms in specific events
Once the list of forms is defined, individual events within episodes can then be coded in terms
of this list. Of course, as I have made clear, these tasks were not actually done sequentially; the list
of forms was defined through an iterative process of coding and recoding individual events.
Nevertheless, in order to simplify this discussion, it is productive to think of these two steps as
happening one after the other. First the list of forms is defined, then we write rules for recognizing
individual forms in a given event. Appendix D has a list of all the forms coded, a description of
each form, and a summary of the clues I used to recognize the form. Here, I will just briefly
discuss the types of evidence that were used to recognize forms in individual events.
1. Symbol patterns. We can begin by looking at the expression that is being written or
interpreted. What possible symbol patterns are there in this expression? This source of
evidence does not greatly restrict the forms that can be coded since a given expression may be
consistent with many symbol patterns and the associated forms. For example, almost any
expression is consistent with PROP+. However, this evidence can help in some cases. For
example, not every expression is consistent with PARTS -OF -A-WHOLE; the expression must have
more than one term.
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1. Verbal clues. Of course, what students say is the main source of evidence. In some cases, the use
of specific words is considered evidence for certain forms. For example, if a student uses the
phrases “depends on” or “is a function of,” these are clues that the dependence form should be
coded for an event (refer to Appendix D). In addition, it is also very useful to look at what
sort of physical entities are mentioned in a statement and what level of the expression these
entities correspond to. Consider the following two examples, both of which were coded as
involving the BALANCING form:
kx=mg
Jack
So, okay, so then these two have to be in equilibrium those are the only forces acting on
it.
________________
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
There are several clues in these statements that point to the BALANCING form. In both cases, the
student is talking about influences—forces, in particular. This is a clue that forms in the
Competing Terms cluster are involved. Furthermore, these influences are associated with sides
of the equation; one force goes with the left side of the equation and one force goes with the
right side. This is evidence that the student is seeing the expression in terms of the symbol
pattern associated with BALANCING . Contrast this with a statement in which a student mentions
entities corresponding to individual symbols in these expressions, such as the mass or the
spring constant.
3. Inferences. This is really a special case of verbal evidence. In Chapter 3 I stated that individual
symbolic forms permit specific inferences. For example, if b/x is seen in terms of the PROPform, and x increases, then we know that b/x decreases. If a student states an inference
associated with a form then this constitutes partial evidence for the form.
3. Graphical concomitants. Some forms and form clusters have “graphical concomitants”—items
that are a commonly drawn on the whiteboard in association with the form. Graphs are one
type of graphical concomitant. For example, as I have mentioned, the
DYING AWAY
form is
frequently seen in combination with a graph of an exponentially decaying function. In
addition, free-body diagrams and diagrams of other sorts are used in a variety of ways that are
characteristic of particular forms. Forms in the Competing Terms cluster are frequently
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accompanied by the use of a diagram in which the competing influences are labeled with
arrows and the signs of influences—positive or negative—are associated with directions in the
diagram.
3. Symbolic conventions. I have already suggested that there are some quite narrow conventions
associated with symbol patterns. For example, when an expression is written in accordance
with the COEFFICIENT form, it is almost always written with the coefficient on the left.
Similarly, expressions in-line with the IDENTITY symbol pattern almost always have the single
symbol on the left side of the equal sign: x=[…] . When expressions are in agreement with
these more narrow conventions, rather than just agreeing with the symbol pattern, more
broadly construed, this constitutes additional evidence for a form. Furthermore, conventions
can also extend beyond structural regularities. For example, when a symbol is being treated as
a coefficient, it is common to use a ‘C’ or a ‘K’ for that symbol. Again, agreement with such
conventions constitutes evidence that a form is operating.
3. Global as well as local evidence. Finally, the coding of an individual event need not be based
solely on evidence local to the event. For example, sometimes a student will repeat an
interpretive utterance later in an episode, in slightly different terms. Such a later interpretation
may guide us in coding the earlier episode. More generally, if a pair of students has been
speaking in terms of influences throughout an episode, this may skew the coding of forms
toward forms in the Competing Terms cluster. Furthermore, I believe this acceptance of
global evidence may be further extended to cross episode and even student boundaries. For
example, suppose that a student gives a very suggestive interpretation of an equation that,
nonetheless, poses problems when we attempt to code it in terms of forms. If other students
have, in a similar situation, given a clearer—but somehow similar—interpretation of the same
expression, we can use the clearer interpretation to support a particular coding of the
problematic instance.
On the Òsubstitution machineryÓ
In this section, I have thus far explained how I can take every constructed expression and
interpretive utterance and code it in terms of symbolic forms. However, just as in the base-level
analysis, there are some exceptions and marginal cases. Stated bluntly, it turns out that not quite
every relevant event can be coded in terms of forms. In some cases, even when a device is clearly
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involved, students appear to be drawing on resources other than forms to do the work that we
would normally expect from forms.
Here I will only attempt to briefly characterize these additional cognitive resources. When
students enter college physics instruction at UC Berkeley they have a number of well-developed
skills pertaining to the use of symbolic expression. Among these skills are the ability to manipulate
expressions to derive new expressions, and the ability to substitute numbers into expressions. If the
students in my study were given an expression and some numbers to substitute into that
expression, they could perform this substitution and obtain a result with extreme reliability. The
point here is that some interpretations given by students seem to draw on elements of this
“substitution machinery” rather than on symbolic forms. This is particularly true of many
interpretations involving Special Case devices.
As an example, consider Alan’s interpretation of Hooke’s law, which I presented earlier:
F = -ky
Alan
If you look at this point [g. point where dotted line intersects spring] there'd be no force
due to Hooke's law, and that's where it'd remain at rest. If you're looking above this axis
[g. dotted line then above dotted line] you're looking at a negative force so that's gonna
push it down. So, if you get closer to the ceiling, the spring force is gonna push you down.
If you're below this axis um you're going down but now you're gonna have a force because you're gonna have two negatives [g. -ky term] you're gonna have a force that's
sending you back up towards the ceiling. So it's just a question of signs.
In this statement, Alan draws on aspects of his substitution machinery. He knows that substituting
a value of zero for y will give a result of zero for the force. And he knows that substituting a
negative value will yield a positive value for the force. Interpretations of this sort are not given any
coding in terms of forms (unless some forms are also operating). Instead, they are simply assigned
to their own separate category.
This is not to say that these events are completely ignored in the analysis. These events do
involve representational devices, so they must be included if a full accounting of devices is to be
made. Furthermore, these interpretive utterances appear to perform very similar functions to the
interpretations that are central to this study. In the above interpretation, for example, Alan is
explaining why an expression “makes sense.” Checking and explaining the sensibility of
expressions is one of the main functions of interpretive acts. Thus, these events are counted as
interpretive acts and coded in terms of devices; they are simply not assigned form codings.
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Coding representational devices
When coding representational devices, their is a fundamental difficulty equivalent to the one
associated with coding forms. In order to code the representational devices in the algebra data
corpus, I needed to simultaneously define the set of devices and code individual construction and
interpretation events. I now discuss, in turn, each of these aspects of the coding of devices.
Defining the set of devices
As with forms, the key to finding the set of devices was iterative analysis of the data corpus. I
began with a preliminary set of devices derived from my pilot study and then refined this set
through an iterative analysis based on a set of heuristic principles:
1. Coverage of the data corpus. This is the same as the heuristic principle employed to identify
forms.
1. Parsimony. Same as for forms.
1. Close relation to the structure of interpretive utterances. Because they are more closely related to
visible aspects of behavior, devices are somewhat easier to recognize than forms. When
attempting to account for student behavior, forms function at a relatively deep explanatory
level. As analysts, we must therefore fill in several steps to get from forms to any visible aspects
of behavior. In contrast, devices relate much more directly to behavior; in particular, they are
closely related to the structure of interpretive utterances. As I said in Chapter 4, to find the set
of devices it is reasonable to think of taking all of the interpretive utterances in the data corpus
and putting them in categories. In fact, an important technique I employed for identifying
candidate devices was to look for turns-of-phrase that re-occur in interpretive utterances.
Examples of such phrases are: “At any time…”, “This is when…” and “As this goes up… .”
1. Restriction to reliably codable distinctions and allowance for ambiguity. When inventing categories
for devices there is a temptation to want to fill in “logical gaps” in these categories or to want
to break categories into multiple parts based on a priori possible distinctions. For example, it is
possible that the CHANGING PARAMETERS category should be broken into two distinct subcategories, one in which the change happens continuously, and one in which the change
happens as discrete hops between values. In practice, however, it is not possible to reliably place
individual interpretive utterances into one of these hypothetical categories or the other;
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student utterances seem to usually be ambiguous in this regard. Consider, for example, this
interpretation by Jim which we have examined many times:
x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k].
The question here is: Is Jim talking about a continuous increase in the mass or is he talking
about a discrete jump in the mass from one value to another? I believe that this utterances is
simply ambiguous in this regard. In defining the categories for devices, I only retained
distinctions that could be reliably made when coding events.
5. Continuity with prior knowledge. This principle is equivalent to the principle of the same name
for forms. However, for the case of devices, there is less to work with in the existing literature.
Although researchers have studied the discourse of mathematics classrooms, this literature has
not adopted a focus close enough to the concerns of this work to be directly useful for
identifying individual representational devices. An exception is the work of Nemirovsky (in
press) which I will discuss in Chapter 7.
5. Approximation to expertise. Again, the idea here is essentially the same as for forms; we can
presume that most representational devices will be useful for doing the types of tasks that
physicists must perform.
Recognizing devices in specific events
Because of the somewhat more behavioral character of devices, recognizing them in particular
events is somewhat easier than recognizing forms. A list of all representational devices coded and
clues for recognizing them are given in Appendix E. Here, I again list only the types of evidence
used.
1. Verbal clues. The most important clues for coding an event in terms of devices are found in
what the students say. In fact, the phrases used can be very strong clues for a specific device.
Examples of such phrases were mentioned just above—“At any time…”, “This is when…,”
“As this goes up… ”—and more can be found in Appendix E. In addition, we can attend to
the “plot line” of an interpretive utterance. For example, does the interpretive utterance focus
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on a particular quantity that appears in the expression? If so, is that quantity staying fixed or
varying?
1. Graphical concomitants. As with forms, other external representations drawn on the whiteboard
can provide clues concerning devices. For example, Static devices are generally seen in tandem
with a diagram of the physical situation. Because Static devices typically involve associating an
equation with a “moment” in a motion, it is relatively straightforward to draw a diagram to go
with the device. Although diagrams for Narrative devices are somewhat trickier, they are still
possible; a series of pictures may be drawn to depict the changes in a narrative. Figure Chapter
6. -8 shows a portion of the board image for Alan and Bob’s work on the Air Resistance task.
Note that they have drawn separate diagrams for the heavier and lighter mass. Such diagrams
can function as a graphical concomitant for the CHANGING PARAMETERS device.
m
∑F =
2m
mg − kv
Figure Chapter 6. -8. Some of Alan and Bob's work from the Air Resistance Task. A separate diagram is
drawn for the heavier and lighter mass.
3. Global as well as local evidence. As with forms, it is necessary to use global as well as local
evidence to code individual events in terms of devices. We can look at other events involving
the same students, or at other pairs in similar circumstances.
Coding of the sample episode
Before presenting the results of the systematic analysis, I want to briefly show how Jack and
Jim’s work on the Mass on a Spring task was coded in terms of forms and devices. This is the
episode for which I presented the base-level analysis in the previous section. Recall that Jack and
Jim began by drawing a diagram and then writing the equations F=ma and F=-kx. Since these two
expressions were coded as remembered in the base-level analysis, it is not incumbent upon me to
account for them in terms of forms and devices. Next, Jack wrote the expression kx=mg, which I
coded as constructed+derived+principle-based:
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Jack
Right, cause the::: The spring potential is one half K X squared. So,
okay, so then these two have to be in equilibrium, those are the only
forces acting on it. So then somehow I guess, um, (3.0) That negative
sign is somewhat arbitrary [g. F=-kx] depending on how you assign it of kx=mg.
course in this case acceleration is gravity [g. a in F=ma] Which would be Form: BALANCING
negative so we really don't have to worry about that [g.~ F=-kx] So I
Device: STEADY STATE
guess we end up with K X is M G. [w. kx=mg]
Because this expression is coded as, in part, constructed, I need to analyze the forms and
devices involved. First, the balancing form is implicated here. One important piece of evidence for
this coding is Jack’s use of the term “equilibrium” (refer to Appendix D). In addition, note that
Jack is talking about two influences that correspond to sides of the expression. He says “these two
have to be in equilibrium, those are the only forces acting on it.” This is strong evidence for
BALANCING .
The device involved here is the STEADY STATE device. When Jack says “these two have
to be in equilibrium,” he is describing a static, unchanging situation that continues through time.
Finally, after a single manipulation to produce the equation x=mg/k, Jim made two
interpretive utterances:
Jim
Okay, and this makes sort of sense because you figure that, as
you have a more massive block hanging from the spring, [g. block] then
you're position X is gonna increase, [g.~ x in diagram] which is what
this is showing. [g.~ m then k in x=mg/k.] And that if you have a
stiffer spring, then you're position X is gonna decrease. [g. Uses fingers
to indicate the gap between the mass and ceiling.] That why it's in the
denominator. So, the answer makes sense. [Looking at Bruce as
he says this last.]
“as you have a more massive
block …”
Form: PROP+
Device: CHANGING
PARAMETERS
“if you have a stiffer spring …”
Form: PROPDevice: CHANGING
PARAMETERS
Each of these interpretive utterances must be coded in terms of forms and devices. In both cases,
Jim picked out a single quantity and imagined that it varied; in the first, the mass varied, in the
second, the spring constant varied. Thus, both of these interpretations involve the changing
parameters device. There are also a number of clues here concerning the forms involved. The fact
that Jim attends to the equation at the level of individual symbols is an important clue. In
addition, a clear inference is evident in each of these utterances: Jim infers that the position will
increase if m increases, and the position will decrease if k increases. Thus, the first interpretive
utterance involves the PROP+ form; the position is seen as proportional to the mass. And the
second interpretive utterance involves the PROP- form, with the position seen as inversely
proportional to the spring constant.
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Analysis Summary
For future reference, the steps for analyzing a given episode (one pair’s work on a single task)
are summarized in Figure Chapter 6. -9. In the base-level phase of the analysis, symbolic
expressions and interpretive utterances are identified. Next, the identified expressions are further
coded as remembered, derived, constructed, or as belonging to one of the marginal cases. Finally,
in the last phase of the analysis of an episode, each constructed expression and interpretive
utterance is coded in terms of forms and devices. In this last phase, some events may not receive
any codings in terms of forms and instead are coded as involving the substitution machinery.
Episode Transcript
Symbolic Expressions
Interptive Utterances
Base-Level
Analysis
Remembered
Derived
Constructed
Forms
Marginal
Cases
Forms
Devices
Devices
Coding of
Forms and
Devices
Figure Chapter 6. -9. Overview of the analysis of an episode.
Summary of Results
Base-level results
I will begin the summary of results by presenting the outcomes of the base-level analysis.
Recall that the primary purpose of the base-level analysis was to identify events in the transcript
that must be coded in terms of forms and devices. Nonetheless, the base-level analysis does, in its
own right, reveal some important features of the data corpus.
The expression-centered component of the base-level analysis found a total of 547 expressions
in the episodes selected for systematic analysis. Of these expressions, 470 were written “from
scratch” and 77 were modifications of existing expressions (refer to Table Chapter 6. -6).
Expressions “from scratch”
Modifications
Total expressions
470
77
547
Table Chapter 6. -6. Expressions found in the base-level analysis.
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Table Chapter 6. -7 shows the result of coding these 547 expressions as remembered, derived,
constructed, or as belonging to one of the three marginal cases. The sum of all codings is somewhat
higher than 547 since individual events may be given multiple codings. The far right hand column
in Table Chapter 6. -7, labeled “With Other Codings,” is intended to provide some additional
information in this regard. The numbers in this column tell how many instances of a given coding
were made in combination with some other coding. For example, 33 of the expressions coded as
remembered were also assigned another coding.
Remembered
Derived
Constructed
Marginal Cases
Total
Codings
With Other
Codings
157
287
75
110
33
46
42
51
Table Chapter 6. -7. Base-level coding of expressions.
The highlighted row in Table Chapter 6. -7 is worthy of special attention since the primary
goal of the expression-centered coding is to identify constructed expressions. As shown in the table,
75 constructed expressions were identified, 42 of which were assigned some additional coding.
Table Chapter 6. -8 provides a breakdown by type of the 110 marginal cases. It is worth
noting that 41 of these expressions were the extremely simple type of expression that I called
“value assignments.” The reader should also note that 32 of the 36 expressions coded as principlebased also received some other coding. Thus, a very small number of expressions were coded as
purely principle-based.
Total
Codings
With Other
Codings
33
41
36
19
0
32
Remembered Template
Value Assignment
Principle-based
Table Chapter 6. -8. Breakdown of marginal cases by type.
The results of the second component of the base-level analysis—the utterance-centered
analysis—are fairly simple to report since this analysis only identifies interpretive utterances; it
does not further code them in any way. As stated in Table Chapter 6. -9, a total of 144
interpretive utterances were identified in the episodes included in the systematic analysis. One
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other piece of information I will provide here is that these 144 interpretive utterances were
associated with 93 expressions. The number of associated expressions is less than 144 since some
expressions were interpreted more than once by students.
144
93
Interpretive Utterances
Interpreted Expressions
Table Chapter 6. -9. Results of the utterance-centered analysis.
These 144 interpretive utterances, together with the 75 constructed expressions identified,
constitute the events that must be coded during the next stage of the analysis. In total there are
thus 219 individual events that must be coded in terms of forms and devices (refer to Table
Chapter 6. -10). It is these 219 events that I iteratively coded to arrive at the final sets of forms
and devices.
144
75
219
Interpretive Utterances
Constructed Expressions
Total Events
Table Chapter 6. -10. Events that must be coded in terms of forms and devices.
Before commenting further on these results, I want to present one additional table of values
from the base-level analysis. Table Chapter 6. -11 gives some of the results of this analysis by
subject pair. Note first that there was fairly substantial variation in the total number of expressions
written by the pairs. Mike and Karl wrote only 72 expressions, the smallest number in the study.
In contrast, Alan and Bob wrote 136, the most of any pair.
However, this variation across students in the total expressions written does not seem to drive
differences in the number of construction and interpretation events. Interestingly, there do not
appear to be strong correlations of the total number of expressions with interpretive utterances
(r=-.07), constructed expressions (r=.11), or the total interpretation and construction events
(r=.06).
Expressions
Interpretive Utterances
Constructed Expressions
Interpretations+Constructions
M&K
A&B
J&E
M&R
J&J
Totals
72
31
13
44
136
33
13
46
122
26
15
41
106
33
12
45
111
21
22
43
547
144
79
219
Table Chapter 6. -11. Results of the base-level analysis by subject pair.
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I want to speculate briefly on this lack of correlation to total expressions. For illustration, I will
compare Alan and Bob with Mike and Karl. It turns out that Alan and Bob wrote 18 value
assignment statements while Mike and Karl only wrote 3. These expressions are generally not a
very important aspect of student work, when students write a value assignment statement they are
usually just transcribing aspects of the problem statement onto the whiteboard. Similarly, there
was a big difference between these two pairs in the number of expressions coded as derived. For
Mike and Karl there were 30; for Alan and Bob there were 75. In part, this difference reflects a
particular stylistic variation between the two pairs: Alan and Bob tended to write out many steps
when manipulating expressions. Thus, my hypothesis is that the large differences in total number
of expressions are traceable to certain stylistic differences that are generally not closely related to
construction and interpretation events.
This is not to say that I do not expect there to be differences among students in how
frequently they interpret and construct expressions. In fact, Table Chapter 6. -11 already suggests
possible differences among my subjects. But, interestingly, even where there are differences in the
number of interpretations or constructions, there is some balancing out of these differences in the
last row of the table. In fact, the data does indicate a negative correlation between interpretive
utterances and constructed expressions (r=-.94). A possible reason for this is that statements made
during the construction of an expression might, to some extent, obviate the need for later
interpretation of the statement. (Recall that statements made prior to or during the writing of an
expression were used only in coding the expression; they were not treated as interpretive
utterances.) But, given the small number of subjects this observation is somewhat shaky and I will
not draw on it in further discussions. In any case, this particular constancy is actually quite
desirable for this study since it means that pairs contributed roughly equally to the forms and
devices coded in the next phase of the analysis.
Now I will sum up the results of the base-level analysis. I believe that the most important
conclusion to be drawn from the base-level analysis is that the construction and interpretation of
expressions are not rare events. Recall that, from the very first chapter of this dissertation, I have
argued that this work is important precisely because these phenomena have not been accounted for
by existing research. However, if it had turned out that these phenomena were simply rare—if, for
example, students almost never constructed expressions—then we might have decided that it was
reasonable to ignore these phenomena. At the least we would have had to accord them less
importance than they have been given by this research.
212
Since this point is extremely important, it is worth taking a moment to argue that the results
presented in this section can be interpreted as suggesting that construction and interpretation are
“not rare.” First, I believe that the sheer number of events is somewhat convincing. As we saw,
there were a total of 219 construction and interpretation events spread over 11.5 hours of student
problem solving work.
And these results can profitably be viewed from some other angles. First, let’s compare the
frequency of interpretations and constructions with the total number of expressions written. Of
the 547 expressions that students wrote, 75—14%—received a coding of constructed. This is not
at all a vanishingly small percentage. Furthermore, there were a total of 144 interpretive utterances
applied to 93 separate expressions. This means that 17% of all the expressions written were
interpreted in some manner. Again, this is a non-trivial fraction of all expressions.
Finally, I want to consider the prevalence of these events from one additional angle, by
episode. It turns out that of the 40 total episodes included in the systematic analysis, 34 included
construction events, 37 included interpretation events, and 39 included either a construction or
interpretation. Furthermore, in the one odd episode that involved no constructions or
interpretations, the students solved the task without writing any expressions. Thus, in every
episode that could potentially have involved interpretations or constructions, these events
appeared. Taken altogether, I believe it is reasonable to describe these results as indicating that the
construction and interpretation of expressions are not “rare” events.
Results from the coding of symbolic forms
The results of coding all of the 219 events identified by the base-level analysis in terms of
symbolic forms are given in Table Chapter 6. -12. Results are listed both by cluster and by
individual form. (Refer to Appendix D for a brief description of each form.) The values in the
column labeled “count” represent the raw number of events coded with the corresponding form or
cluster. The numbers in the “percent” column were obtained by dividing this raw count by the
total number of codings; that is, by dividing the associated count by the total counts of all forms.
These percentages thus provide a measure of the relative frequency with which the various forms
were coded.
213
Count
Percent
BALANCING
75
30
12
28
27%
11%
4%
10%
CANCELING
5
Proportionality
PROP+
PROP-
Competing Terms
COMPETING TERMS
OPPOSITION
RATIO
CANCELING(B)
Count
Percent
SOLE DEPENDENCE
63
40
16
7
23%
15%
6%
3%
2%
Coefficient
14
5%
77
52
28%
19%
COEFFICIENT
SCALING
5
9
2%
3%
14
9
2
5%
3%
1%
Multiplication
INTENSIVE •EXTENSIVE
EXTENSIVE •EXTENSIV
5
5
0
2%
2%
0%
2
1
1%
0%
Dependence
DEPENDENCE
NO DEPENDENCE
E
Terms are Amounts
PARTS -OF -A-WHOLE
BASE ± CHANGE
38
10
23
14%
4%
8%
WHOLE - PART
1
4
0%
1%
SAME AMOUNT
Other
DYING AWAY
NO EVIDENCE
Table Chapter 6. -12. Results for the systematic coding of symbolic forms.
As I discussed above, some events could not be coded in terms of forms. Instead, these events
seemed to employ aspects of what I referred to as the “substitution machinery.” Of the 219 total
events, 12 received no form codings and were coded as involving only the substitution machinery.
In Figure Chapter 6. -10, the breakdown of codings by cluster is displayed in a pie chart.
Looking at this figure, it is evident that no one cluster of forms is especially dominant. Rather,
three clusters stand out in this chart. The Competing Terms and Proportionality clusters had the
most codings, with the Dependence cluster running a close third. Together, these three clusters
account for 77% of all codings. Of the remaining three clusters, Terms are Amounts, was most
common, occupying a middle range in the data. Finally, the Coefficient and Multiplication
clusters were coded rarely, together accounting for only 7% of the total number of codings.
214
Multiplication
2% Other
Coefficient
1%
5%
Competing Terms
27%
Terms are Amounts
14%
Dependence
23%
Proportionality
28%
Figure Chapter 6. -10. Percentage of codings by cluster.
Referring to Table Chapter 6. -12 we can see that the two most common individual forms
were PROP+ and DEPENDENCE. These two forms alone account for about a third of all codings. It is
interesting to note that both PROP+ and DEPENDENCE involve a sensitivity to very basic aspects of
the structure of expressions. Recall that, when a student sees
DEPENDENCE,
they are only
recognizing that a particular symbol appears somewhere in an expression. PROP+ also does not
involve a sensitivity to very complex aspects of structure. Seeing
PROP+
means simply attending to
the fact that a particular symbol appears in the numerator of an expression. (In some cases, as in
expressions like F=-kx, there may not even be a denominator!) Thus, a very large percentage of the
work of seeing meaning in symbolic expressions can be traced to very basic aspects of the structure
of expressions.
The next two most common forms, COMPETING TERMS and BALANCING , together made up
about 20% of all codings. Thus, over 50% of all of the codings are accounted for by only 4 of the
20 forms in my list. This suggests the interesting conclusion that a relatively small number of
forms are accounting for much of the meaning associated with expressions.
The prevalence of the COMPETING TERMS and BALANCING forms can be better understood if we
examine the type of contexts in which these forms appeared. Recall that both of these forms are
from the Competing Terms cluster and that forms in this cluster involve treating an equation at
the level of terms, with terms corresponding to competing influences. The point I want to make is
that, in many instances, the influences involved were forces, in the technical sense of this term. In
fact, of the 28 cases in which BALANCING was coded, 24 involved balanced forces. The COMPETING
TERMS
form was deployed in a slightly more varied manner, with 11 of the 30 instances involving
forces. I believe that these observations help to put the prevalence of these two forms in
215
perspective, and can provide the reader with a sense for the types of circumstances in which these
forms were coded. Since forces can be used in a productive manner to solve many of the tasks that
I gave to students, it is not surprising that these two forms appeared frequently in the data corpus.
Some other aspects of Table Chapter 6. -12 require further explanation. The reader may have
noticed that several of the forms listed received a very small number of codings. For example,
DYING AWAY
was coded twice and WHOLE - PART was only coded once. If these were the only events
in which I had seen these forms, I would have been hesitant to conclude that they exist. However,
in all of these cases I saw additional instances of the forms in episodes not included in the
systematic analysis.
The very small number of codings in the Multiplication cluster, including zero codings for the
EXTENSIVE •EXTENSIVE
form, also requires some comment. In part this result is indicative of some
real sparseness in this territory—these forms simply do not appear very often. But such a result
must be at least somewhat troubling. Does this mean that students are nearly incapable of
constructing expressions involving the product of factors? Are they simply not very sensitive to this
type of structure in expressions? Or is it that the particular tasks I asked students to perform
simply did not require them to use forms relating to the product of factors?
Remembered Templates
count
volume = length × width × height
area = length × width
total = rate × time
total = density × volume
10
3
5
8
2
sum of items
average = number of items
change = final − initial
3
Table Chapter 6. -13. Remembered Templates with counts. The top four templates relate to multiplication
forms.
The answer seems to be, in part, that my analysis in terms of symbolic forms actually misses
some of the action relating to the product of factors. Recall that when expressions were coded in
the base-level analysis a number of expressions were placed in three “Marginal Cases.” One of
these Marginal Cases was what I called “Remembered Templates.” The full list of Remembered
Templates is reproduced in Table Chapter 6. -13 along with the number of times that each of
these templates was employed. The important fact to notice is that four of the six templates
shown in Table Chapter 6. -13 could be constructed or interpreted in terms of the two forms in
216
the Multiplication cluster. In particular, the rate × time and density × volume templates could be
usefully interpreted in terms of the
INTENSIVE •EXTENSIVE
form. And the templates for computing
volume and area could be usefully interpreted in terms of the
EXTENSIVE •EXTENSIVE
form. This, in
part, is where work relating to the product of factors is hiding in my analysis.
So why not code the associated expression-writing events in terms of the INTENSIVE •EXTENSIVE
and the EXTENSIVE •EXTENSIVE forms? The reason has to do with what these events actually look
like in the data corpus. In particular, students generally wrote the expressions associated with these
templates quickly and with little comment. For example, they did not try to explain the fact that
the area of a rectangle was the product of its length and width; instead, this seemed more like a
remembered relation that they were just using. Thus, I believe that to code these events as
constructed would be misleading. However, it is still important to note that these templates, while
more limited in their potential range of use than forms, are still somewhat flexible. This is
especially true of the rate × time and density × volume templates. The “rate” in rate × time, for
example, can be any type of rate, such as a velocity or an acceleration.
Now I will comment more generally on the meaning of the numerical results presented in this
sub-section. The results in Table Chapter 6. -12 state the frequency with which particular symbolic
forms appeared in the data corpus. Of course, there are many limits to how far we can trust these
numbers, and I will comment on these limits shortly. But, accepting these numbers for the
moment, what can we conclude about the nature of student knowledge and the character of
“expert intuition?” As I mentioned briefly in the introduction to this chapter, the move I want to
make is to presume that the frequency with which forms are employed by students reflects the
priority of the associated knowledge elements possessed by students. Here I am using the term
“priority” in the sense I ascribed to diSessa: It is a measure of the likelihood of a knowledge
element being activated. In diSessa’s picture of the sense-of-mechanism, intuitive physics
knowledge gets its character from the priorities of individual p-prims. Similarly, I am now
asserting that expert intuition, in part, takes its character from the relative priorities of individual
forms.
There are two reasons that I believe that coding frequencies can be presumed to reflect the
priority of elements. The first reason is that, by definition, higher priority elements are more likely
to be used. So, if elements appear more often, this is a sign that they have a higher priority. The
second reason—which is closely related to the first—is that the problem solving activity that is
contained in the data corpus is precisely the kind of activity that is supposed to determine the
217
priority of individual forms. If a particular form is called on repeatedly during problem solving
then the priority of that form should be increased. Thus, this is another reason that we can
presume that the forms that appear frequently are the high priority forms.
There are some problems with these arguments for interpreting frequencies as priorities,
however. In particular, recall that the cuing priority of a knowledge element reflects its likelihood
of being activated in a particular circumstance. According to the strict usage of this term, cuing
priority is context dependence; there is no overall cuing priority that spans contexts. Thus, the
“priorities” that I measured here must be understood to be a coarser measure of the likelihood of
elements being used than would be given by a true measurement of context-specific priorities. In
essence, I have a measure of the priority of elements averaged over the circumstances in the study.
Nonetheless, I believe that this “coarser” measure is still useful for getting a feel for the rough
character of expert intuition.
The frequencies in Table Chapter 6. -12 also provide some indirect evidence concerning other
aspects of expert intuition. In the last chapter I hypothesized that forms are tightly connected to
some specific types of previously existing cognitive resources, especially p-prims. When a
particular form is activated, it is more likely that related p-prims will be activated. And when pprims are activated, they are reinforced and their priorities are increased. Thus the frequency data
not only provides information about the priorities of forms, it also provides some indirect
evidence concerning the priorities of p-prims in expert intuition, at least for the medium-level
experts that participated in this study.
Given the viewpoint that the coding frequencies tell us about the character of intuition, I want
to reflect just a little more about the above results. In the discussion of the development of expert
intuition in the previous chapter, I explained that diSessa hypothesizes that agency plays an
expanded role in expert intuition; and, in particular, much of this agency is localized in the
concept of force. For example, when talking about a book resting on a table, a novice might say
that the table simply “supports” the book. In contrast, a physicists sees the table as an agent—the
table applies a force upward on the book that opposes the gravitational force.
The high frequency of forms in the Competing Terms cluster may be indicative of this
expansion in the role of agency. In these forms, terms are associated with influences that oppose
and support, and frequently they are associated with forces. Thus, the high use of forms in the
Competing Terms cluster could be viewed as part of the developmental trend toward seeing all
objects as potential agents that apply forces.
218
The relatively high frequency of one form in the Competing Terms
cluster—BALANCING —might be considered surprising. From the point of view of standardized
physics, situations in which influences are in balance are special cases—the influences just happen
to be equal. Compare the COMPETING TERMS form. COMPETING TERMS is, in a sense, more general; it
is used in situations where influences are seen as competing, but not necessarily in balance or
canceling. Referring to Table Chapter 6. -12 we see that BALANCING was coded roughly the same
number of times as COMPETING TERMS. Does this mean that expert intuition is skewed toward
seeing situations in terms of BALANCING ? It is possible that this result is simply an artifact of the
particular selection of tasks that I gave students. However, I believe that we are seeing something
real and important here. Recall, for example, Alan’s inappropriate BALANCING interpretation of the
expression mgµ=ma. As I have mentioned, I believe that this is indicative of a tendency to overuse
BALANCING .
The high frequency of Proportionality forms may also, in part, reflect the developmental
trend toward an increasing role for agency. As I argued in Chapter 5, the PROP+ and PROP- forms
may be related to the notions of effort and resistance. I also believe that the prevalence of these
forms is due to the fact that, in a fundamental sense, learning algebra-physics means learning to
quantify the physical world. As I have mentioned, proportionality relations may play an important
role in our abilities for reasoning about quantities; therefore, it is not surprising that these
associated symbolic forms appear frequently in algebra-physics.
M&K
A&B
J&E
M&R
J&J
Totals
Competing Terms
Proportionalit
Dependence
Terms are Amounts
Coefficient
Multiplication
16
14
9
11
2
0
19
14
13
15
3
0
15
9
15
2
2
2
16
18
18
5
2
1
9
22
8
5
5
2
75
77
63
38
14
5
Totals
52
64
45
60
51
272
Table Chapter 6. -14. Form cluster codings broken down by subject pair.
All of the data concerning symbolic forms that I have presented so far has been aggregated to
produce the total instances found in all episodes. Before concluding this section, I want to
decompose this data along two dimensions. Table Chapter 6. -14 breaks down the coding of
symbolic form clusters by subject pair. Looking at this table, we see that the distribution is not the
same for all subjects. This suggests several reasons for caution in interpreting the results presented
219
in this section. First, if the differences in Table Chapter 6. -14 are indicative of real differences
that exist among individuals, then it could mean that my results depend sensitively on the
particular subjects that participated in the study. In addition, these observations limit the extent to
which we should even attempt to look for anything like a universal distribution.
Of course, I have always assumed that there would be individual differences in students’ formdevice knowledge. The behavior of the form-device system is determined by many cuing and
reliability priorities, and there is no reason to believe that these priorities would be precisely the
same across subjects. I have even explicitly argued that these priorities would change during
learning; thus, since my subjects may be more or less far along in their learning of physics, there is
particular reason to expect that priorities should differ. Nonetheless, I was hoping that there
would be some generalities that could be extracted by an analysis of this sort.
There are a few across-pair uniformities in Table Chapter 6. -14. For example, for all subject
pairs, the Multiplication Cluster received less than 5% of all codings for that pair, and the
Proportionality Cluster always received at least 20% of all codings. Also, the top three clusters in
Table Chapter 6. -14— Competing Terms, Proportionality, and Dependence—always accounted
for at least 70% of the codings for a pair. I do not mean to play down the importance of
differences here, only note that there are some similarities across subject pairs.
When, the results are broken down by task, as in Table Chapter 6. -15, we see that the results
for each task were quite different. To begin, note the extreme variation in total codings across
tasks. This is not too surprising since students spent quite a different amount of time on each task.
But there is also great variation in the distribution of forms within tasks. For example, the Air
Resistance task had the most total codings, 73, but had no codings of forms in the Terms are
Amounts cluster. Similarly, the Running in the Rain task had no codings of Competing Terms
forms. Again, such results are not surprising since different tasks require different methods.
However, even though this non-uniformity across problems may not be surprising, it may still
have negative implications with regard to the universality of the frequencies derived from my data
corpus. In particular, as with the variation across pairs, it implies that when averaging over tasks to
get a measure of an “overall” priority, the result is sensitively dependent on the selection of tasks
that I gave to students. I will comment on this issue in the last section of this chapter, and I will
discuss it in slightly more detail in Chapter 10.
220
Shove
Competing Terms
Proportionalit
Dependence
Terms are Amounts
Coefficient
Multiplication
Totals
4
15
15
4
Toss
Skater
Buo anc
8
3
8
6
3
1
6
1
11
Spring
Air
Wind
9
8
5
3
25
22
22
21
4
3
14
Rain
Totals
19
17
5
1
5
75
77
63
38
14
5
47
272
4
38
12
14
21
25
73
42
Table Chapter 6. -15. Form cluster codings broken down by problem.
Results from the coding of representational devices
The result of coding all 219 events identified by the base-level analysis in terms of
representational devices is shown in Table Chapter 6. -16. In addition, the results collected by
class are displayed in a pie chart in Figure Chapter 6. -11. Referring to this pie chart, it is clear that
Narrative devices dominate. In fact, the three devices in this class are three of the four most
common individual devices, and the
CHANGING PARAMETERS
device far outstrips all other devices,
accounting for a third of all codings.
Narrative
CHANGING PARAMETERS
CHANGING SITUATIONS
PHYSICAL CHANGE
Special Case
SPECIFIC VALUE
LIMITING CASE
RESTRICTED VALUE
RELATIVE VALUES
Count
Percent
128
79
21
28
54%
33%
9%
12%
26
9
6
11%
4%
3%
3
8
1%
3%
Count
Percent
83
17
5
15
35%
7%
2%
6%
17
6
23
7%
3%
10%
2
2
1%
Static
SPECIFIC MOMENT
GENERIC MOMENT
STEADY STATE
STATIC FORCES
CONSERVATION
ACCOUNTING
Other
NO EVIDENCE
Table Chapter 6. -16. Results for the systematic coding of representational devices.
The drop off is precipitous as we move to the next two classes of devices. Though devices in
the Static class received a respectable 35% of the total codings, this is somewhat less than the 54%
of codings accounted for by Narrative devices. Finally, Special Case devices constitute a mere
11% of the total.
221
Special Case
11%
Other
1%
Narrative
53%
Static
35%
Figure Chapter 6. -11. Representational devices by class.
These results point to Narrative devices as a clear winner. Not only are these the most
prevalent devices by class, every one of the individual devices in this class seems to be extremely
common. And one device in particular,
CHANGING PARAMETERS,
far outstrips all others. This leads
to the question: Is this what we would expect?
From at least one perspective this result is somewhat surprising. To illustrate, I want to take a
moment to reflect on what students are taught in introductory physics classes. Representational
devices, per se, are not an explicit component of introductory physics courses. You certainly will
not find sections on the SPECIFIC MOMENT and CHANGING PARAMETERS devices in a physics textbook.
However, some representational devices are loosely associated with more or less explicit elements
of instruction. For example, instructors have been known to exhort students to consider “limiting”
or “special cases.” Although these instructors likely do not think of these heuristics as “strategies
for interpreting equations,” they may occasionally think of them as a way to check whether an
equation “makes sense.” Thus, some Special Case devices may actually be closely related to
relatively explicit—though peripheral—aspects of instruction.
Static devices are also related, though somewhat less directly, to some explicit and more
central aspects of instruction. I argued in Chapter 4 that we can think of free-body diagrams as a
means of constructing equations to be interpreted with some devices in this class. Students learn
free-body diagrams as an explicit technique for solving problems—you isolate an object, then
draw all of the forces acting on the object. And many introductory textbooks explicitly mention
free-body diagrams. Thus, although instructors are not teaching interpretive strategies, per se,
these particular interpretive strategies do appear to be frequently embedded in well-worn and
clearly recognized elements of problem solving practice.
In contrast, Narrative devices are neither expressly taught nor tied to any explicitly recognized
aspects of practice. Rather, Narrative devices seem to constitute an altogether tacit aspect of
physics practice that cuts across the gamut of problem solving activities. It is for this reason that
222
the results presented in this section can be considered surprising. It is slightly ironic that the one
class of interpretive heuristics that is most explicitly taught—the Special Case class—turns out to
be comparatively rare. Instead, the majority of interpretations of expressions involve Narrative
devices, which are neither recognized nor expressly taught.
As with forms, it is helpful to break down the results of the device analysis by subject pair and
task. Table Chapter 6. -17 gives the devices in each class as they were coded for each subject pair.
Again there are notable differences. Jack and Jim received more than three times as many
Narrative codings as Static codings, while, in contrast, Alan and Bob actually received slightly less
Narrative codings than Static codings. These differences lead to the same caveats as those
mentioned for the form-related results.
But, again, there are some weak across-pair uniformities. The Special Case Class was always
the least common, accounting for less than 20% of the codings for each pair, while the Narrative
Class always accounted for at least 40%. Furthermore, the Static Class fell in between for all pairs
except Alan and Bob.
M&K
Narrative
Static
Special Case
Totals
25
14
9
48
A&B
J&E
21
26
6
53
21
16
5
42
M&R
30
18
1
49
J&J
Totals
31
9
5
46
128
83
26
237
Table Chapter 6. -17. Device class codings broken down by subject pair.
Table Chapter 6. -18 breaks down the devices in each class by task. As we saw with the forms
data, there is great variation across task. The total codings for individual tasks extend over a great
range, from 9 to 60, and the distribution within problems appears somewhat different. There are
some weak suggestions of uniformity here, however. For 5 of the 8 tasks, the ranking of the classes
by frequency is the same as for the overall distribution. Furthermore, the remaining 3 tasks happen
to be the 3 tasks with the lowest total number of codings. Still, the variation here is reason for
caution in interpreting the aggregated results.
223
Shove
Narrative
Static
Special Case
Totals
22
4
3
29
Toss
2
11
1
14
Skater
6
3
9
Buoyancy
2
9
7
18
Spring
11
10
5
26
Air
Wind
40
19
1
60
25
14
4
43
Rain
Totals
26
10
2
38
128
83
26
237
Table Chapter 6. -18. Device class codings broken down by task.
Conclusions to be Drawn From the Systematic Analysis
What was the purpose of this analysis? I want to take a moment to reflect on the nature of the
conclusions that we can draw from the presentation in this chapter. To be sure, the results
presented in this chapter are not the output from any typical experimental design. For example,
there were not a “control” group and an “experimental” group that were compared. Although in a
later chapter I will compare algebra-physics and programming-physics data, the central results of
this chapter do not involve comparisons of any sort. So what do all of these numbers tell us?
The analysis described in this chapter, rather than producing a comparison, can be understood
as producing a description of algebra-physics knowledge. This description has several elements.
First, the lists of forms and devices themselves—even without any frequencies—constitute part of
the description of algebra-physics. Although I first presented these lists in Chapter 3 and 4, the
current chapter adds an account of how I made contact between these lists and my data corpus.
Providing this account is perhaps the most important purpose of this chapter. Along with the
examples presented in chapters 3 and 4, this chapter does the important work of clarifying the
relationship between my theoretical categories and the data I had available.
In filling in frequencies and providing other numerical results, the analysis presented in this
chapter also paints a more detailed picture than is provided by a simple listing of forms and
devices. There are two different senses in which these results provide a “picture.” First, the
numerical results obtained here perform the function of summarizing the data corpus for the
reader. By looking at these numbers, a reader can get a broad feel for the features of the corpus,
such as the frequency of interpretation and construction events, without having to watch every
videotape or read every transcript. This summarizing function is one of the usual purposes of
characterizing a data corpus with statistics.
In addition to providing an overview of the data corpus, these results can also be understood as
providing a picture of algebra-physics knowledge. Earlier in this chapter I argued that the frequency
224
results provide information concerning physical intuition, because they can be interpreted as an
indication of the priorities of knowledge elements. But any such interpretation of the results is
necessarily more speculative.
However there are reasons for caution in interpreting all of these results, particular for any
conclusions relating to the nature of individual algebra-physics knowledge. First, as I have
commented, the lists of forms and devices were not derived from the data using exact techniques.
Instead, I employed a heuristic process rooted in my intuitions and theoretical predispositions, as
well as the data. Furthermore, the content of these lists was not confirmed to any extent by a
second coder. Because of the heuristic nature of the process, training another coder would have
been an extensive activity in itself. Thus, I hope that the reader will understand this analysis as a
first step toward future, more exact and replicable methods of using analyses of this sort to
identify knowledge elements.
Second, even accepting the lists of forms and devices as I presented them, the coding of
individual episodes was not an entirely straightforward process; thus, the numbers that I associated
with the various categories could be disputed. The coding of episodes in terms of given categories
(forms and devices) would actually be more amenable to confirmation by additional coders than
the production of the categories themselves, and I hope to do this in the near future. Nonetheless,
there is still a significant effort required to train coders in this procedure and, due to constraints of
time, such a confirmation was not included here.
Third, even if the numbers presented in Table Chapter 6. -12 and Table Chapter 6. -16 are
taken as, in some sense, accurately describing my particular data corpus, there are still reasons for
caution. I commented, in my discussion of the experimental sessions, that the sessions themselves
may not be entirely typical of any type of naturally occuring activity, such as a group problem
solving session. Most notably, in occasionally prompting students to explain their work, I may
have tended to increase the frequency of interpretation events.
In addition, we saw that the frequency results varied across subjects pairs. This is a problem
because it suggests that my results could be sensitive to the particular subjects that participated,
and because it casts doubt on the possibility of generalizing across subjects. These worries were
slightly moderated by the observation of some weak uniformities across subjects.
The results also showed a strong variation across tasks. This result is far from surprising, but it
nonetheless poses difficulties for drawing conclusions. Of course, I did attempt to engage students
in a spread of tasks, so there is some hope that the variation averages to a useful measure of the
225
overall priority of elements. But I did not attempt to rigorously establish that the selection of tasks
is, in any sense, representative. In fact, it is not even clear what dimensions should be used to
determine the “representativeness” of tasks. Thus, we must accept the likelihood that changing the
selection of tasks would have altered the results obtained, and that my selection may not be
especially representative.
Nonetheless, I believe we can still draw some general—though limited—conclusions. To end
this chapter, I want to present these conclusions, and attempt to calibrate the confidence we can
have in each of them. First, I hope to have shown that construction and interpretation were not
rare events, and that it was not unusual for students at this level to have the ability to construct
and interpret equations. All pairs constructed and interpreted expressions approximately the same
number of times per task, and there was at least a single construction or interpretation event in
every problem solving episode except one. Given the variation observed across problems, the
precise frequency with which these events occur is somewhat more questionable. Furthermore, my
prompts may have tended to skew my results toward seeing more interpretations. But the fact
that all of the subject pairs demonstrated the ability to construct and interpret on numerous
occasions—whether prompted or not—means that we can be relatively certain that most students
have the requisite capabilities. And the universal existence of these capabilities strongly suggests
that this knowledge and behavior are important.
Assessing confidence in the lists of forms and devices is somewhat more difficult. The
production of these lists was the component of my analysis most dependent on heuristic
procedures and my own intuition. Thus, our confidence in these lists must depend significantly on
the plausibility that follows from the arguments in the preceding chapters.
But, if we accept the categories, what can we conclude from the particular frequency counts I
presented in Table Chapter 6. -12 and Table Chapter 6. -16? Even with the worries about crosssubject and cross-task variation, I believe it is appropriate to take a few tentative conclusions into
succeeding chapters. First, beginning with the device-related results, the aggregated counts
indicated quite large differences between classes: Static devices were coded almost three times as
much as Special Case devices, and Narrative almost five times as much as Special Case. Of course,
this distribution was not uniformly represented across subjects and tasks, but there were some
weak reflections of this distribution across these dimensions. Thus, for the chapters that follow, I
will tentatively presume that Narrative devices are generally most common in algebra-physics,
Special Case rarely used, and Static devices in the middle.
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We are in a similar position with regard to the frequency of forms. Again, the analysis found
some substantial differences across clusters. For example, there were 77 Proportionality codings
and only 5 codings in the Multiplication cluster, and the three most common clusters accounted
for at least 70% of the codings for every subject pair. In addition, there were substantial
differences in how many times the various individual forms were coded. The
DEPENDENCE
PROP+
and
forms were coded 52 and 50 times respectively, while some forms received only a
few codings. More generally, as I have commented, the four most common forms—PROP+,
DEPENDENCE, COMPETING TERMS,
and BALANCING —accounted for 50% of all codings.
Furthermore, I believe that these results are also somewhat plausible. The proportionality and
dependence forms can appropriately be used to interpret expressions in a wide range of
circumstances. And, because of its observed relation to forces, the high frequency of Competing
Terms forms is not very surprising.
Thus, even with the observed variations across tasks and subjects, I believe that it is appropriate
to take these results as indicating something about the character of algebra-physics. In what
follows, I will tentatively presume that the Competing Terms, Proportionality, and Dependence
clusters are somewhat more common than the other three clusters, and that the common forms
within these clusters— PROP+, DEPENDENCE, COMPETING TERMS, and BALANCING —are particularly
important in algebra-physics.
So, the result of this analysis is an account of how I arrived at the lists of forms and devices, as
well as a picture of the data corpus that can also be taken as a rough image of algebra-physics
knowledge. In later chapters, I will want to compare this image to a similarly constructed image of
programming-physics knowledge. At that time, we will see that the results of the corresponding
analyses are sufficient to uncover a number of revealing contrasts.
Finally, I want to conclude with one additional important conclusion that can be drawn from
the analysis presented in this chapter. I believe that the results of this analysis can be taken as
evidence for my theoretical viewpoint, understood more broadly. Included in this view is the notion
that students use expressions meaningfully; they can build symbolic expressions from an
understanding of what they want the expression to convey, and they can interpret expressions.
Also included in my viewpoint is the notion that the cognitive resources that are central to
meaningful symbol use have symbolic forms and representational devices as elements.
In a sense, the systematic analysis has put the totality of this viewpoint to test by attempting to
use it to describe the data corpus. In the process, we have seen that there are a few marginal cases
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which could not be accounted for in terms of forms and devices. For example, some of the action
involving the product of factors had to be accounted for by “Remembered Templates,” rather
than forms and devices. And some interpretive utterances that could be coded in terms of devices
had to be described as employing the “substitution machinery,” rather than forms. However, I do
not interpret the existence of these marginal cases as signs of the breakdown of the theory.
Instead, I take them as symptomatic of the complexity that the theoretical view presumes to exist.
The form-device system is just one cognitive resource and we should not expect this resource to
do everything. What is impressive, I believe, is that this analysis suggests that forms and devices
do quite a lot of work in students’ problem solving. We saw many instances of students
constructing and interpreting expression and the vast majority of these could, in part, be explained
by an appeal symbolic forms and representational devices.
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Chapter 7. Reflections on the Theory
For the physicist and physics student, a sheet of physics expressions is structured and meaningful.
This is a statement that I have made a number of times. But what, precisely, does this statement
mean? In what sense is a sheet of expressions more or less “meaningful?” Let’s back up for a
moment and work up to a more precise and comprehensible version of this statement.
We begin with a thought experiment. Imagine that we take a sheet of paper and fill it with
physics expressions, perhaps with the solution to a standard physics problem. Then we take this
sheet of paper and drop it on an island where the inhabitants have never seen any sort of writing
before. What will they make of this sheet of paper? If they have never seen paper before, they may
very well be fascinated by the sheet of paper as an object in itself. But what of the markings on the
paper? Whether the marks on the paper are interesting to the islanders or not, the details in the
arrangement of marks will not be consequential for them. Since they are not privy to the
conventions that we attach to these markings, the markings, in their details, are no more
significant for the islanders than the arrangement of grooves in the bark of a tree. Just as with these
grooves, fine differences in the markings are not important.
Now imagine that we send in a team to retrieve the sheet of paper from the island of nonliterate peoples. Once we have the sheet of paper, we then take it and show it to some people who
are literate in English, but who know no physics and little mathematics. What will these people see
when they look at the sheet of physics expressions? They will certainly recognize many individual
letters and perhaps some mathematical symbols like the equal sign and a plus sign. It is also likely
that they will recognize that these letters are arranged into short horizontal strings.
Next, we show the sheet of expressions to some students who are in the very early stages of
physics instruction. These students will likely recognize that the sheet is filled with equations.
Maybe they have learned the equation v=vo +at. If this equation is written on the sheet of paper
they might recognize it. They might then know, for example, that the v in the expression stands
for “velocity,” and they might know something about the conditions under which this equation is
used.
In the final step of this thought experiment we show the sheet of expressions to some people
who have been significantly initiated into the practice of physics. These initiates will recognize
many of the expressions that appear on the sheet of paper. If F=ma is written they will recognize it
and may associate it with Newton’s second law. If F=-kx is written they may know that this is
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Hooke’s law for the force applied by a spring. Most importantly, I have argued that not only will
they recognize individual symbols and whole expressions, they will recognize structures in these
expressions that I have called “symbol patterns,” and these symbol patterns are associated with a
particular kind of knowledge element that I have called “symbolic forms.”
The purpose of this thought experiment was to display a continuum of cases in which the page
of expressions is increasingly “meaningful” for the people involved. As we progressed through the
cases, the expressions were more and more meaningful in a particular sense: The individuals
involved were sensitive to more varieties of structure in the markings, with the perceived structures
associated with knowledge possessed by the individuals. This sensitivity to details of structure
combined with rich associations with knowledge is what it means for the page of expressions to be
“meaningful.”
For the physics initiate, a page of expressions is richly meaningful in this sense. To this point, I
have been engaged in showing how I believe this works. I have worked on describing what the
physics initiate sees when they look at an expression—what features they are sensitive to and how
what they recognize depends on concerns of the moment. Really, I have focused on just one
aspect of this story, the aspect associated with the seeing of symbol patterns in expressions. The
result was a map of this one aspect of the physicist’s meaningful world of symbolic experience.
Characterizing the knowledge associated with the seeing of this meaningful structure has been
the job of the preceding five chapters. The resulting model of this cognitive resource—the formdevice system—is the core theoretical result of this dissertation. In this final chapter of Part 1, I
want to reflect a little on the nature of this model and the larger theoretical view in which it is
embedded. This job will be divided into two parts. First, I will summarize the model and engage
in some quite speculative extrapolation, drawing out some of the model’s implications, and
attempting to generalize the model to other representational practices. A central goal of this first
portion of the chapter will be to extract implications from the model concerning the relationship
between external representations and physics knowledge. Then, in the second portion of this
chapter I will contrast the viewpoint I have adopted with four other research perspectives.
How Physics Knowledge is ÒIntrinsically RepresentationalÓ
In Chapter 2, I began my foray into the physicist’s world of symbol use with some examples
designed to show that students do sometimes construct and interpret equations. These examples
were important because they provided a first glimpse into the phenomena that are the focus of this
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project. In fact, a significant product of this research is the identification and description of these
phenomena.
Then, at the end of Chapter 2 and in subsequent chapters, I proceeded to describe how this
construction and interpretation works. I described my model of the cognitive resources involved
and how they develop:
Registrations. First, following Roschelle (1991), I introduced the term “registration,” to refer to
how people “carve up their sensory experience into parts.” I listed several kinds of registrations that
are relevant to symbol use in physics: individual symbols, numbers, whole equations, groups of
symbols, and symbol patterns.
A vocabulary of symbolic forms. Next I introduced the first of the two major theoretical constructs
in this work, what I have called “symbolic forms.” A symbolic form consists of two components:
(1) a symbol pattern and (2) a conceptual schema. Taken together the set of forms constitutes a
conceptual vocabulary in terms of which expressions are composed and understood.
Representational devices and the interpretation of symbolic expressions. In addition to symbolic
forms, I argued that physics initiates possess a complex repertoire of interpretive strategies that I
called “representational devices.” The project to identify representational devices involved, in part,
the collecting and categorizing of interpretive utterances.
The refinement of physical intuition. After describing the form-device system, I speculated on how
this knowledge develops during physics learning. I argued that the form-device resource develops
out of existing resources, including intuitive physics knowledge and knowledge relating to the
solving of algebra problems, and that these previously existing resources continue to exist and to
be refined.
The above summary, together with the specific lists of forms and devices that I provided in
Chapters 3 and 4, constitutes the core of the Theory of Forms and Devices. Now, in the
remainder of this section, I want to reflect on this model and extract some of its implications. As I
mentioned, this reflection will be speculative; I will be going somewhat beyond what can be
claimed with any confidence given the data and arguments I have thus far presented.
One key issue to which I want to pay particular attention is what I have called the “relation
between knowledge and external representations.” In Chapter 1 I discussed some researchers for
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whom this relation is variously deep and fundamental. Exemplars that I referred to often were
Jack Goody, who posited a central role for external representations in the historically developing
knowledge of society, and Richard Feynman, who argued that to truly appreciate the beauty of
nature it is necessary to have the experience of manipulating physics expressions.
The possibility of these deep relations led me to urge caution in a few places. I pointed out
that, given these possibilities, algebra-physics and programming-physics practice may involve
fundamentally different understandings of physics. I was also led to question some of the
assumptions underlying the teaching of “Conceptual Physics” courses, which teach physics largely
without the use of equations. In particular, in this thesis, I challenge the assumption, usually
implicit, that an expert’s understanding can be separated into two pieces, one conceptual, the
other symbolic-mathematical, so that the conceptual part can be taught separately. To anticipate
some of what follows, this assumption is called into question by the observation that the formdevice system does not fall neatly into either of these two categories; it is both conceptual and
symbolic-mathematical. To be clear, I have not meant to dispute whether Conceptual Physics is a
useful instructional method; rather, my intent has been to push for a better understanding of the
results of such instruction.
In the present section, I will explore what the model described in the preceding chapters can
tell us about the relation between external representations and physics knowledge. My approach
here will be to argue for the stance that physics knowledge is “intrinsically representational,” and
to attempt to describe the precise sense in which this is true. In making this claim, I mean to be
asserting that the physics knowledge of individuals is in some way inseparable from the external
representations that we use.
Thus, in what follows, I will use the model to argue that physics knowledge is intrinsically
representational, and to state some specifics of the deep relation between physics knowledge and
external representations. However, I will not be able to discuss this deep relation in full generality.
This is the case because I have only studied a small portion of the knowledge involved in physics
understanding. As I will argue below, a more complete discussion of the relation between
knowledge and external representations would require a more thorough analysis of the knowledge
involved.
Before jumping in, I want to remind the reader of a few details of my discussion of the
relation between knowledge and external representations in Chapter 1. In that chapter, I began by
distinguishing between “direct” and “indirect” effects of external representations on individual
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knowledge. Direct effects arise out of an individual’s actual interaction with an external
representation. In contrast, indirect effects arise because a person working with an external
representation may produce various products that can then be “fed back” into the knowledge of
individuals. This distinction was built into the simple framework that is reproduced in Figure
Chapter 7. -1.
Representational
Practice
Direct
Effects
Literate
Individual
External
Representation
"Products"
Indirect
Effects
Cultural
Mediation
Literate or
Non-Literate
Individual
Figure Chapter 7. -1. Framework separating direct and indirect effects of external representation on the
knowledge of individuals.
The model I described in the preceding chapters will primarily allow us to draw conclusions
concerning direct effects. In what follows, my discussion is organized around two views of direct
effects described in Chapter 1: (1) the point that knowledge must be adapted by and for symbol use
and (2) the notion that literates derive intuitions from a symbolic world of experience. I will now
reflect on my model from each of these two points of view.
Knowledge is adapted by and for symbol use
In Chapter 1 I mentioned that it is sometimes illuminating to treat a person and external
representation as a system whose cognitive abilities are more powerful than a person acting alone.
For example, the system consisting of a literate individual together with a pencil and a sheet of
paper may be said to have a “better memory” than the individual alone. An important feature of
this stance is it allows us to state that external representations significantly enhance the cognitive
abilities of individuals, without positing any substantial changes in the character of individual
thought processes in the absence of the external representation.
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However, I went on to argue that we should still expect certain types of “residual effects” on
the knowledge of individuals. Because external representations essentially alter the tasks that
individuals must perform, their knowledge and thought processes must be adapted in order for
individuals to participate successfully in a representational practice. In short, I argued that
although there may not be changes in the overall character of thought, the knowledge of
individuals must still be adapted by and for symbol use.
In my discussion in the previous chapters, I described some of the specifics of how the
knowledge of initiates is refined by symbol use in physics. Recall the hypothetical scenario I
presented in Chapter 5. I described a situation in which two competing p-prims were activated
and I explained how symbol use could resolve the conflict between these two p-prims. Then I
argued that the result of such an experience would be that the priority of the “winning” p-prim
was incrementally increased. In such a manner, incremental changes could be made to the senseof-mechanism and the sense-of-mechanism gradually refined. This is a “residual effect” on
individual knowledge due to direct interaction with an external representation. Similarly, I argued
that incremental changes could follow from p-prims being cued during symbol use.
In addition to being adapted by symbol use, it is also useful to think of physics knowledge as
being adapted for symbol use. In order to construct a novel expression, a student’s notions must be
cast in symbolic forms. This requires that existing resources, like the sense-of-mechanism, must be
refined so that this is possible.
An important point—which I also made in Chapter 5—is that the precise nature of the
refinements made in this way depends intimately on idiosyncrasies of the symbolic practice. To
see how important this effect is, consider an alternate viewpoint. We might hypothesize that, with
the development of expertise, physical intuition develops toward an ideal form that is determined
by the conceptual structure of Newtonian physics and, ultimately, by some sort of fundamental
properties of physical reality. In this view, there are still limits to what physical intuition can do;
we still will not expect that physical intuition, without the aid of symbol use, can make detailed
predictions in all circumstances. But, within the limits of its nature, physical intuition gets as close
as it can to this ideal form.
The view I am presenting here differs sharply with this alternative hypothesis. I have argued
that an expert’s physical intuition is specifically adapted for the purpose of complementing and
functioning in the activity of symbol use. The point here is that it makes sense to say that physics
knowledge is intrinsically representational, not because symbol use can directly cause changes in
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individual knowledge, but because the precise way that knowledge develops depends on the
details of the symbolic practice. Not only is there a “residual effect” of symbol use, this residual
effect bears the unique fingerprint of the symbolic practice that left it behind.
For example, consider the prevalence of the BALANCING form in my data corpus and recall that
this suggests that the BALANCING p-prim has a relatively high priority in expert intuition. As I
argued in Chapter 6, there is nothing particularly central about situations involving balanced
influences in the conceptual structure of Newtonian physics. Situations that involve such balanced
influences are special cases; the influences just happen to be equal. Thus, the prevalence of
BALANCING
intuitions in expert physics may have more to do with the nature of algebra-physics
practice than with the conceptual structure of Newtonian physics. In retrospect, this is not
surprising. The business of algebra-physics is the writing of equations, constraints that express
equalities. And if we want to write an equation, we have to find things in the world that we can set
equal. Thus, the intuition of the expert physicist must, to a certain extent, be geared particularly
toward searching out BALANCING in the world.
So what does all of this have to do with my model? Couldn’t we have made most of these
arguments without all of the work necessary to develop my model of the form-device system? It is
true that it is very nearly a trivial statement to say that the knowledge of physics initiates must, to
some extent, be adapted specially for symbol use. For example, physics students must learn
certain specific equations and they must learn specialized techniques for solving problems and
deriving new equations.
However, what my model adds—and what is really interesting—is that symbol use in physics
draws directly and significantly on knowledge at the level of physical intuition. We might have
found that people write and understand symbolic expressions only at the level of whole equations
associated with formal principles. Instead, I argued that people see meaningful structure at the
level described by symbolic forms which, in turn, are strongly tied to elements of the sense-ofmechanism. Thus, it is because of the fact that physicists see this type of meaningful structure that
the details of symbol use can have direct effects that percolate down to the very roots of physics
understanding.
A symbolic world of experience
Now we move on to the second way of thinking about how direct interaction with external
representations can lead to residual changes in individual knowledge. The idea is that, through
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participation in a representational practice, a new variety of intuition can be born, rooted in this
experience. This is essentially the notion that Feynman expresses in the quotes presented in
Chapter 1. Feynman apparently believes that the experience of manipulating mathematical
expressions to solve problems and derive results contributes in some significant way to a person’s
appreciation of “the beauties of nature.” One of the central goals of this dissertation has been to
put some theoretical meat on this idea.
The theoretical meat can be found in my discussion of symbolic forms and representational
devices. Let us take a minute to reflect on the nature of the form-device knowledge system.
Importantly, this resource has much the character of the intuitive knowledge systems that we have
discussed, such as the sense-of-mechanism. It has relatively small and simple elements that are
multiply connected to the world and other knowledge. And the form-device system is only
weakly structured; there are no central organizing elements that tie the whole system together and
give it a straightforward organization.
For these reasons, I believe it makes sense to consider the form-device system to be a
component of expert physical intuition. Thus, the suggestion here is to think of expert physical
intuition as consisting of refined versions of pre-existing resources, such as the sense-ofmechanism, together with the form-device knowledge system. If we accept this proposal, then
there is a component of expert physical intuition that has its roots solidly in symbolic experience.
This is what I mean when I say that there is a “symbolic basis of physical intuition.”
Throughout the preceding chapters, we have seen examples of forms and devices functioning
like other intuitive resources. For example, just like p-prims, symbolic forms constitute a primitive
level in terms of which physical phenomena are explained and understood. I want to present an
example to emphasize this point. In this episode, which I also discussed in Chapter 3, Mark and
Roger were attempting to explain why all objects fall at the same rate in the absence of air
resistance. Eventually, Mark wrote two expressions and explained as follows:
a=
Mark
F
; F = mg
m
You know, however you have a force [g. F in a=F/m] that varies at the same proportion
directly - directly with the mass, don't you? The heavier - the heavier an object is, the
more force it will exert, right? Upon, like say I try to lift it. … So, if it's double the mass it's
gonna have double the force downward.…Two M G. Right? But the mass is also two M , so
those two M's cancel and you get - you're left with A equals G in all cases, aren't you.
Above, Mark has “mentally” combined these two expressions to obtain a=mg/m, which is an
expression for the acceleration of the dropped ball. He then goes on to explain that any changes in
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the mass will have no effect on the acceleration, since any change in the numerator is canceled by a
corresponding change in the denominator.
So, how does Mark understand the fact that all objects fall at the same rate? What is his
explanation? I would argue that Mark’s explanation is based on the CANCELING(B) form. At the
heart of his explanation is the idea that there are two effects—one associated with the force and
one with the mass—that precisely cancel. Furthermore, there is no reason to presume that Mark’s
understanding of “canceling” is entirely separate from his experience with its mathematical
manifestation. This is precisely what it means to say that Mark’s explanation is based on the
CANCELING(B)
form. His explanation and understanding of the fact that all objects fall at the same
rate is rooted, at least in part, in his symbolic experiences of canceling. This is a very strong sense
in which physics knowledge is “intrinsically representational.”
The same point can be made from another angle. Suppose we were going to teach a
Conceptual Physics course and we believed that the knowledge of physicists could be separated
into “symbolic-mathematical” and “conceptual-physical” components. I have argued that
symbolic forms are part of the knowledge that expert physicists possess. This leads to the
question: Are forms “symbolic-mathematical” or “conceptual-physical?”
The answer is that symbolic forms do not seem to fit neatly into either of these categories;
they have their origins in both physical and symbolic experience. For example, consider the
dependence form. This is rooted in our physical experiences of causation and physical
dependence, but it also derives from our experience of symbolic dependence using expressions
and plugging various numbers into an expression to get a result. Thus, the physical and symbolic
are welded together in this single knowledge element.
In conclusion, this section has argued for a strong sense in which physics knowledge is
intrinsically representational. I maintained that, because it has the character of intuitive
knowledge, the form-device knowledge system should be considered a component of expert
physical intuition. And the knowledge elements of this system—basic elements of an experts
physical intuition—are themselves rooted in symbol use.
Summary: How would Conceptual Physics knowledge differ?
What do the comments in this section imply about the consequences of a move to Conceptual
Physics instruction? First, my arguments imply that we can expect a differently tuned sense-ofmechanism. In algebra-physics, the sense-of-mechanism and other pre-existing elements of
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physical intuition are specially tuned for the role that they play in symbol use. In contrast, a
practice of Conceptual Physics may tend to tune the resources more for producing a certain type
of qualitative explanations of phenomena.
Overall, we may guess that intuition will be tuned more for providing correct predictions and
explanations in the absence of equations. This, in fact, is part of the rationale behind Conceptual
Physics instruction. In Chapter 1, I discussed some of the striking failures of traditional
instruction. This included the observation that students do not seem to learn some of the basics of
Newton’s laws, such as the fact that objects continue to move in a straight line in the absence of
applied forces. Given the fact that Conceptual Physics focuses on issues of this sort, it is very
plausible that students will do better in this regard. And, in fact, as I discussed in Chapter 1, there
is already much evidence of success.
But do students come away from Conceptual Physics instruction with an “appreciation of the
beauties of nature?” Of course, this depends what we decide counts as such an appreciation. What
we can say is that students will at least come away with a somewhat different appreciation of
physics, a divergent understanding. I have argued that, for initiates in algebra-physics, there is a
component of physical intuition that is rooted in symbolic experience, and this component will
necessarily be less developed in Conceptual Physics instruction. Again, I do not mean to imply
that Conceptual Physics is any less a “real” or respectable version of physics, and I certainly do not
mean that Conceptual Physics cannot provide useful preparation for algebra-physics. I am simply
trying to say that, at an important and deep level, the understanding associated with conceptual
physics is different.
The Theory Generalized
In the above sections I reviewed some of the specifics of my model of symbol use in physics,
and I drew out some of the implication of this model for the relation between external
representations and physics understanding. Strictly, all of the claims made above are specific to
the particular representational practice that I studied, algebra-physics. Now I want to consider the
extent to which my theory and the associated claims can be generalized outside of this narrow
realm. Of course, a major task of this thesis—which I will undertake in Part 2—is the extension of
the theoretical framework to programming-physics. But, in this chapter, I want to first think more
broadly about the generality of the framework and its implications. I will begin with a brief
discussion of where and how the form-device model of symbol use can be generalized and
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extended. Then, as far as is possible, I will attempt to make a few broad claims about the relation
between knowledge and external representations.
How general is the form-device model?
Clearly, many aspects of my discussion of registrations, symbolic forms, and representational
devices are particular to the use of algebraic notation in physics. For example, since each individual
form is associated with a specific algebraic symbol pattern, the individual forms that I listed are
inescapably tied to algebraic notation. And since some devices have to do with projecting an
equation into a physical motion, these devices, at least narrowly understood, must be specific to
physics.
Nonetheless it is possible that my main theoretical constructs, understood more generally,
may have wider applicability. Take registrations, for example. Although specific registrations will
certainly not generalize across representations and practices, it seems like a reasonable project to
characterize, for any representational practice, the features and structures to which individuals are
sensitive. Thus my hypothesis is that this notion—which, recall, I borrowed from Roschelle
(1991)—has wide applicability.
However, the two novel theoretical constructs introduced in this research, symbolic forms and
representational devices, will likely be less generalizable to other representational practices. Note
that the hypothesis associated with the positing of these constructs is really rather strong.
According to the model that I presented, the seeing of meaning in expressions involves a very
particular sort of two-layer system: Representational devices set a stance, and expressions are seen
in terms of a number of specific symbolic forms, each of which is associated with a well-defined
symbol pattern. It seems unlikely to me that this very particular model will extend to all other
representational practices. Consider natural language, for example. It is quite implausible that we
would could capture much of the range of people’s abilities to construct meaningful statements in
terms of a few simple conceptual schemata and symbolic structures. The extensive efforts exerted
by linguists to construct grammars for natural language and to understand the nature of
utterances are sufficient testimony to this fact.
Registrations, forms, and devices in Ògraphing-physicsÓ
In some cases of “near transfer,” however, I believe that the form-device framework may turn
out to be quite helpful. Of course, one of the major presumptions of this work is that this model
can be extended to describe programming-physics, and I will undertake that project in the next
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part of this dissertation. In this section, I want to do a small amount of work to try out the model
on another closely related practice: graphing-physics. My purpose here is just to give a feel for how
the framework presented in the preceding chapters might be applied to other practices.
For the case of graphing-physics, the framework seems to work quite well at the level of
theoretical constructs (registrations, forms, and devices). In what follows, I briefly discuss how
each of the major theoretical constructs might be manifested in graphing-physics.
Registrations. As in algebra-physics, there are many possible types of structure to which people
might be sensitive. Some examples are:
•
The overall shape of a curve.
•
The average slope of a segment.
•
The location (relative to the two axes) of a particular location on a curve.
Figure Chapter 7. -2. A possible graphing-physics graph.
Symbolic forms. Associated with these registrations, there are likely a variety of symbolic forms in
graphing-physics. For example, students could learn to associate “DYING AWAY ” with the overall
shape of the graph shown in Figure Chapter 7. -2. Similarly, there could be a “ STEEP ” form that
includes the ability to register a high-valued slope in a segment and associated with a conceptual
notion that is something like rapid increase or rapid change (refer to the boxed region in Figure
Chapter 7. -2).
Representational devices. In the case of representational devices, I believe that the framework can be
extended quite directly from my analysis of algebra-physics. Not only do I believe that we can
find devices in graphing-physics, I believe that some of these devices will be very similar to devices
encountered in algebra. For example, in the manner of Narrative devices, a student could trace
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along a graph, telling the story of a motion as they go. Similarly, a particular location on a curve
might be associated with a specific moment in a motion. Such an interpretation would be similar
to the Static interpretations I described in Chapter 4.
It turns out that at least one researcher, Ricardo Nemirovsky, has argued for a theoretical
framework very close to the one presented in this section (Nemirovsky, in press). Nemirovsky
describes what he calls “mathematical narratives,” which he defines to be narratives that are
“articulated with mathematical symbols.” Strikingly, he gives examples that are very reminiscent
of the examples I have given of Narrative interpretations. Consider, for example, the following
graph and narrative, which Nemirovsky provides for illustration:
Figure Chapter 7. -3. Example graph from Nemirovsky (in press).
1.
2.
3.
First it rained more and more and it started to become steady (pointing to piece (a) of graph).
Then it rained steadily (marking piece (b) of graph).
Then it rained more and more (pointing to piece (c) of Graph).
Furthermore, in presenting this example, Nemirovsky argues that a major feature of such
narratives is the splitting of the continuous graph into phases or episodes. Such an observation is
closely related to many of the points I made in discussing symbolic forms. What Nemirovsky is
saying is that people recognize pieces of graphs, like the segments in the graph in Figure Chapter
7. -3. In fact, Nemirovsky goes on to hypothesize that people learn a “grammar of graphical
shapes” for graphs, a view that I take to be equivalent to saying that they learn symbolic forms
(Nemirovsky, 1992). As an example, he explains that the two “iconic shapes” shown in Figure
Chapter 7. -4 can express the distinction between “growing steadily” and “growing but slowing
down.” Thus, because of the great similarity of the view expressed by Nemirovsky, we can look to
these papers for arguments and evidence that the framework presented here can be applied to
graphing.
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Figure Chapter 7. -4. Possible elements of a “grammar of graphical shapes” (Nemirovsky, 1992).
Representations and understanding, in general
Finally, I want to conclude this section with one last piece of reflection and speculation before
turning to some contrasts. In Chapter 1, I framed this project, to a certain extent, as a special case.
I raised the issue generally of the relation between individual knowledge and external
representations, and I argued that equation use in physics was a good place to begin to study these
issues. Now that I have completed the first stage of this project, I would like to pause and ask
what more general morals we can draw. What can the work here tell us about the relation between
knowledge and external representations, especially for the more pervasive symbol systems in our
activity, such as natural language and writing?
First, notice that I did not even do the whole job for equation use in physics. I only looked at
one new cognitive resource that develops—the form-device system—and I talked a little about
how a few other existing resources, especially the sense-of-mechanism, would change through
symbol use. To tell a complete story, I would really need to do more analysis of this sort. And, in
principle, there could be as many effects as there are relevant knowledge systems.
Unfortunately, the same is true for any external representations. Really, the project here must
be repeated for each external representational practice that we are interested in. There’s no
shortcut from doing the knowledge analysis; we need to understand the relevant knowledge
systems and how they change, and then we can delineate the specific relations that hold. There is
no reason to believe that there will be simple generalizations, or that we can make assertions
without understanding the nuances of the knowledge systems involved. Furthermore, for
representations that, like natural language and writing, appear across much of our activity, the
knowledge systems involved will be extremely numerous.
This conclusion may seem disappointing. Rather than answering any questions about, for
example, the relationship between thought and literacy, I have simply proposed a quite extensive
program of research. Nonetheless, before we set out on this program in future work, there are a
few helpful morals we can draw from the analysis presented in Part 1 of this document. First, a
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point that I made just above is important: For pervasive external representations, there will be
effects on many types of knowledge. The interesting thing about this observation is that it seems
to be in agreement with the notion that the existence of pervasive representations, like natural
language, has extensive effects on the nature of human thought. Furthermore, if we accept the
program I propose here, we can have these extensive effects without positing any absolutely unique
influences for specific representations like language. There may be many and multifarious effects,
and these effects may permeate through and relate to much of our knowledge, but they can do so
without producing any simple, global effects on the character of thought.
This does not mean that the project to study the relationship between thought and natural
language is impossible. In fact, I believe that there are likely some typical types of interactions that
an analysis can uncover. But these specific types of interactions are what we should be looking for,
not for single, all-encompassing effects.
A second moral we can draw from Part 1 is that new knowledge that develops through symbol
use, knowledge that is closely tied to the use of an external representation, may be more
interesting than one might first think. I have tried to argue that representation-related knowledge
cannot be relegated to the realm of non-meaningful algorithms and procedures. This moral is
especially important to keep in mind for technical representations, such as those used in science
and mathematics.
The final moral really subsumes the last two. Throughout this work, I have pushed for a view
of symbol use as meaningful activity, much like other activity. As humans, we remake our world,
populating it with representations, and these representations become part of our experience. This
viewpoint smoothes things out a little; it suggests a stance in which external representations do not
play an absolutely unique role. Nonetheless, because it treats representations as a meaningful and
pervasive part of our experience, it still leaves the door open for deep effects on the nature of
individual knowledge.
Comparison 1: The Symbol-Referent View of Representation
In the above section, I reflected on the properties of the theory presented in the preceding
chapters. Now I will go still further to bring out some features of the theory by contrasting it with
a number of alternative viewpoints. Actually, I have already presented a few such contrasts. Most
notably, I took some time to contrast symbolic forms with what I called the “principle-based
schemata” view. In many ways, that viewpoint was very closely related to the one presented in this
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thesis. Like forms and devices, principle-based schemata are hypothesized, in part, to explain how
certain types of physics equations are written. Now, in this and upcoming sections, I will travel a
little farther afield and look at some viewpoints relating to the use of external representations that
have been developed for a more wide-ranging collection of purposes.
The first alternative view I will present is what I will call the “symbol-referent” view of
representations. Although this view is not always made explicit, I will argue that it is a common
view in educational literature. For this reason, this first contrast is perhaps the most important of
the ones I will present, and I will therefore devote the most time to it.
In what follows, I will first simply lay out the symbol-referent viewpoint and how it is applied,
then I will undertake to contrast it with the viewpoint adopted in this manuscript. In contrasting
the symbol-referent view with my own, the point will not be to critique this view. Instead, my
purpose will be to see what the symbol-referent view can and cannot do for us, and to understand
what sort of theory it is.
The basic symbol-referent view
To explain the symbol-referent view of representation, I will begin with an exposition by
Stephen Palmer (1977). In turning to Palmer, I am moving somewhat far from the concerns of
this work. Palmer is actually concerned with representations of a very different sort; his central
interest is in internal “mental” representations. Nonetheless, it is useful to begin with Palmer
because he lays out his position quite clearly. Furthermore, as we shall see, he has been cited by
researchers with concerns closer to those of this work.
Palmer begins by stating that a representation is “first and foremost, something that stands for
something else.” He then goes on to say that this description implies the existence of two
“worlds,” a “represented world” and a “representing world,” with correspondences between these
worlds. Thus, he argues, to fully specify a representational system, one must lay out the following
five aspects:
4. What the represented world is.
(5) What the representing world is.
(6) What aspects of the represented world are being modeled.
(7) What aspects of the representing world are doing the modeling.
(8) What are the correspondences between the two worlds.
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This, in a nutshell, is the symbol-referent view of representation. There are two separate
worlds—one relating to the symbols and the other to the referents of those symbols—and there
are correspondences between specific aspects of these worlds.
Figure Chapter 7. -5. A represented world and some representations.
(From Palmer, 1977. Adapted with permission.)
Palmer provides a very clear example to illustrate what he has in mind. In Figure Chapter 7. 5, the represented world—a collection of rectangles of various widths and heights—is shown in
(A). Then, in (B) through (G), Palmer gives various alternative representations of this represented
world. For example, in (B), the height of the rectangles is represented by the length of some lines.
Similarly, in (C) and (D), the width and area of the rectangles are represented by line-length.
Thus, in each of these three representations, the same representing dimension is doing the
representing, but different aspects of the represented world have been selected. In other words, for
each of these, the answers to question (4) are the same, but the answers to question (3) differ.
Like (B), representations (E), (F), (G), and (H) all select the height of the rectangles for
attention; however, they represent this dimension in different ways. Thus, for these
representations, we would give the same answers to question (3), but different answers to question
(4).
This example suggests that the symbol-referent view can be applied to distinguish among
representations of various types. In fact, it turns out that sorting representations is one of the main
uses of the symbol-referent view. Palmer does a little of this. For example, he uses the term
“informationally equivalent” to describe representations that represent the same aspects of the
represented world.
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Views like Palmer’s have appeared in multiple guises, and designed for many purposes. One
very notable version is presented by Nelson Goodman in his book Languages of Art (Goodman,
1976). Goodman’s purpose in this work is quite different than Palmer’s. Rather than being
interested in mental representations, he is interested in fleshing out some intuitive distinctions that
he believes exist among art forms. For example, a guiding question for Goodman is: “Why is
there only one original of a painting while a musical score can legitimately be played over and
over?”
Goodman’s view is elaborate and well worked out, but I will only give a taste of it here. Most
important to this project is Goodman’s discussion, in Chapter 4 of his book, of what he calls “The
Theory of Notation.” His purpose in this chapter is to work up to defining a class of
representational systems that he calls notations. Toward this end, he begins by defining a symbol
scheme as consisting of “characters, usually with modes of combining them to form others,” and
he goes on to explain that “characters are certain utterances or inscriptions or marks” (p. 131).
Then, later, he defines a symbol system as consisting of a “symbol scheme correlated with a field of
reference” (p. 143). Ultimately, Goodman goes on to describe requirements for the symbol
scheme and its relation to the field of reference that define when a representational system is
“notational.”
I am leaving out many interesting details here. Goodman actually develops his view quite
generally since he wants it to apply to a wide range of representational systems, including
paintings as well as musical and mathematical notations. For this reason, what he means by
“character” is quite flexible, and he is careful to include a great range of types of correspondence
between the symbol scheme and field of reference. Nonetheless, it is clear that there are some
basic similarities between the views of Goodman and Palmer. Both talk in terms of two worlds,
with correspondences between those worlds.
Now I will take a first step toward illustrating how the symbol-referent view is applied for
various purposes. One possible application of this view is to guide the design of representations, a
use to which Donald Norman (1991) has put the symbol-referent view. Among other things,
Norman is interested in judging the appropriateness of particular representations for specific tasks.
Toward this end, he discusses what he calls the “naturalness” of a representation. At issue, is how
difficult it is to describe the mapping between the represented and representing worlds:
The "naturalness" of a mapping is related to the directness of the mapping, where directness can be
measured by the complexity of the relationship between representation and value, measured by the
length of the description of that mapping.
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The basic idea here is to look at the match between the represented and representing worlds, but
Norman allows for some complexity in this story. He is careful to point out that it matters what
“terms” are used in the mapping description, and he emphasizes that the proper terms are
“psychological, perceptual primitives.” In addition, he also maintains that, after working with a
representation for a while, experts can derive new mapping terms, thus increasing the feel of
naturalness.
Some useful examples appear in Norman (1991), as well as in Zhang and Norman (1994). In
the latter paper, following Stevens (1946), the authors describe four types of “psychological scales”
that one might want to represent within a certain class of representations: nominal, ordinal, interval,
and ratio. In a nominal scale, there is no ordering among the instances, the instances are simply of
different types. Next up this ladder is an ordinal scale, in which instances can be ordered. In an
interval scale, the separation between ordered instances has meaning. Finally, instances in a ratio
scale are true magnitudes; they are values relative to a zero measurement point.
Figure Chapter 7. -6. Two representations of nominal data. In (a) the representing scale is not matched. In
(b), the representing scale is matched.
(From Zhang & Norman, 1994. Adapted with permission.)
Although Zhang and Norman are not completely strict in this regard, they generally argue for
a matching between the scale types of the representation and the represented information: “Thus,
in order for a representation to be efficient and accurate, the scale types of the represented and
representing dimensions should match.” This matching of scale types is illustrated in Figure
Chapter 7. -6, which is adapted from Zhang and Norman (1994). In this example, the data to be
represented is the type of each file in a computer directory. Thus, the represented scale here is
nominal; there is no sense in which the instances are ordered. Figure Chapter 7. -6(a) uses a
representing scale that is not matched to this data, a ratio scale. Figure Chapter 7. -6(b) uses a
scale that is matched; each file type is represented by a different shape.
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The symbol-referent view and mathematics education
Now I will discuss a second type of application of the symbol-referent view that is much closer
to the concerns of this project. The symbol-referent view has been used to help explain certain
types of difficulties that students have in using external representations and, following these
diagnoses, to aid in making prescriptions for instruction. This has especially been the case in
mathematics instruction. James Kaput’s Toward a theory of symbol use in mathematics is a notable
example, both because it is relevant, and because Kaput’s account is rich and extensive. I will now
discuss this work in some detail in order to illustrate how the symbol-referent view has been
applied in theorizing about mathematics learning and instruction (Kaput, 1987).
Kaput presents some definitions that are entirely along the lines of the symbol-referent view, as
we have already encountered it. He begins by explicitly recounting Palmer’s definition, just as I
cited it above. Then he lays out some further definitions that are very reminiscent of Goodman.
He defines a symbol scheme as “a concretely realizable collection of characters together with more
or less explicit rules for identifying and combining them,” and a symbol system as “a symbol
scheme S together with a field of reference F, and a systematic rule of correspondence c between
them … .” Clearly, Kaput is operating within the territory of the symbol-referent view.
Kaput goes on to specialize this view to mathematics in some interesting ways. He argues that
mathematical symbol systems are a special kind of representational system in which the
represented world is itself a symbol system. All of mathematics is then built up through a chain of
reference, which begins from “primary referents,” and then builds layers of symbol system on top
of symbol system, each one becoming the represented world for the succeeding layer.
This repeated consolidation of mathematics in layers is very interesting. In fact, there may be
some interesting connections to my own view here, since Kaput allows the consolidation of
previous experience with mathematical manipulations into meaningful structures. However, I will
not explore this line in what follows.
Instead, I want to work toward seeing how Kaput uses the symbol-referent view to make some
specific claims about mathematics learning. There are actually a number of interesting comments
that he makes, but here I will just focus on one important piece of his discussion. Kaput lists two
types of “cognitive activity” associated with symbol systems: (1) Reading information from a
symbol system or encoding information into a symbol system, and (2) the elaboration that occurs
once information has been encoded in a symbol system. Kaput further divides the latter category
into two types of activity, syntactic elaboration, in which the individual directly manipulates
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symbols (either in their head or on a sheet of paper), and semantic elaboration, in which the
individual reasons with the referents of the symbols. As an example, a person might add 23 and 9
by mentally visualizing symbols and performing two-column arithmetic (syntactic elaboration), or
by directly invoking knowledge of whole-number arithmetic (semantic elaboration).
Now, with these last distinctions laid out, Kaput is in a position to understand—or at least
provide a certain type of account of—some of the central difficulties experienced by students.
One of these difficulties is the observation that students seem to manipulate symbols without
understanding what they mean. For example, students may learn arithmetic algorithms, but not
know why they work, or how to modify and apply them. In Kaput’s terms, we can understand
students as stuck within the representing world of the symbol scheme. They work with symbols,
performing syntactic elaborations, but they do not read information out of this scheme to the
reference field, or perform the corresponding semantic elaborations.
Thus, for Kaput, symbol systems are a two-edged sword. One of the great benefits of symbol
systems is that they allow us to perform manipulations (syntactic elaborations) free of many of the
details of the reference field. But with this power comes a responsibility:
In each case we have alluded to, one sees an enormous increase in power resulting from the
development of a symbol system whose scheme’s syntax and correspondence faithfully preserve certain
relationships in its reference field. ... For individuals, however, the power and freedom of the
formalism brings an added responsibility—the ability to interpret the formal procedures back in the
reference field(s). Otherwise, all one has are meaningless symbols. (p. 180)
This quote exemplifies one of the most prominent ways that the symbol-referent view is used to
understand student difficulties in mathematics. The point is that working with mathematical
symbols is powerful, and it’s a power that we certainly want to give to our students. But, if we give
students this power, then they must—in Kaput’s words—live up to their responsibilities. They
need to know how to translate from the formalism to the reference field. If they do not, then all
that they have are “meaningless symbols.”
Though not always quite as explicitly, similar notions show up in many places in the
mathematics education literature, and we will see more examples in a moment. For now, I just
want to briefly mention one other related manifestation of a symbol-referent-like argument.
Mathematics educators are often concerned with the relations between symbols and
“embodiments,” which are more or less specific types of physical manifestations of mathematical
abstractions. For example, embodiments are frequently discussed in relation to instruction on
fractions, using such examples as a pie cut into slices or a collection of beads with several removed
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(e.g.,Post, Wachsmuth, Lesh, & Behr, 1985; Chaffe-Stengel & Noddings, 1982). In these cases,
the relation between the embodiment and the symbolic representation is often understood in
terms much like the symbol-referent view.
In addition, similar discussions appear in relation to the use of “manipulatives” in instruction,
which can be treated as a sort of embodiment. For example, Resnick and Omanson (1987)
undertook to give students a better understanding of arithmetic algorithms by carefully teaching
them correspondences between these algorithms and a particular manipulative, Dienes Blocks.
Interestingly, a careful empirical study of this instructional technique revealed largely negative
results.
How the symbol-referent view is converted to a cognitive theory
Now that I have described the symbol-referent view, I will contrast it with my own account of
symbol-use in physics. My purpose here will not be to argue that my view is correct and the
symbol-referent view is incorrect, though there will be places that I argue for caution in applying
the symbol-referent view. Instead, my position will be that these are theories of a different kind,
with different strengths and regimes of application.
To begin, recall that the Theory of Forms and Devices is a cognitive theory; in order to
describe and explain certain types of behavior, I posit the existence of particular mental entities. In
contrast, reflect for a moment on the nature of the symbol and referent worlds. Are they
knowledge systems? Are they cognitive entities of any sort? Although we could invent a theory in
which these worlds are cognitive entities (as we shall see in a moment), a pure version of the
symbol-referent view need not take these worlds to be cognitive in nature. Instead, in adopting the
symbol-referent view, we choose to study symbol systems, rather than cognitive systems, and these
symbol systems involve correspondences between a symbol scheme and a field of reference.
This observation does not invalidate the symbol-referent view. We have already seen that this
view may have some important uses, for example in distinguishing among types of
representations. The point is just to notice that symbol-referent theories are of a different sort than
the form-device theory. Nemirovsky (1994) makes a similar point in distinguish “symbol
systems” from “symbol use:”
With “symbol system” I refer to the analysis of mathematical representations in terms of rules. … On
the other hand with “symbol-use” I refer to the actual concrete use of mathematical symbols by
someone, for a purpose, and as part of a chain of meaningful events. Symbol-use may enact some rules
of the symbol system that is being used, but the central point is that it cannot be reduced to those rules.
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However, it turns out that there are cases where the symbol-referent view essentially gets
converted into a cognitive model. Sometimes this is implicit and piecemeal, but sometimes it is
quite explicit. For example, in Syntax and Semantics in Learning to Subtract, Lauren Resnick paints
a picture of a particular debate in mathematics education (Resnick, 1982):
Until quite recently, discussions of meaningfulness in arithmetic learning have been characterized by a
confrontation between those who advocate learning algorithms and those who argue for learning basic
concepts.
Her thesis in this particular paper is that it is not enough to learn “algorithms” or “basic concepts,”
they need to be hooked up correctly:
Proponents of conceptually oriented instruction have frequently argued that a major cause of difficulty
in learning arithmetic is a failure to relate rules of computational procedure to their underlying
mathematical concepts. ... The data presented here show that intuitions concerning the importance of
conceptual understanding are correct in a crucial respect: Difficulties in learning are often a result of
failure to understand the concepts on which procedures are based. But the data also show that even
when the basic concepts are quite well understood, they may remain unrelated to computational
procedure.
She goes on to display an algorithm for subtraction and says:
Because performing this algorithm requires no understanding of the base system, I claim that it
includes no semantics .... The evidence I present here suggests quite strongly that many children may
learn the syntactic constraints of written subtraction without connecting them to the semantic
information that underlies the algorithm.
The details of Resnick’s views – how, exactly, she believes that this “hooking up” occurs – are
actually quite relevant to this dissertation. However, for the present, I just want to look at how
Resnick paints the issues. The image she presents is of two separate categories of cognitive
resources, one associated with algorithms, the other with understanding of the “underlying
mathematical concepts.” This view appears very clearly in Resnick and Omanson’s (1987)
discussion of mapping instruction, which I mentioned above.
The idea was that the child came to mapping instruction with a mental representation of subtraction in
blocks that was attached to a rich knowledge base, including the principles of subtraction discussed in
this chapter. The child also entered instruction with a mental representation of written subtraction
that was purely syntactic in nature. According to the hypothesis, the detailed, step-by-step alternation
between the two routines would allow the child to attach to the written symbols the principles he or
she already knew with respect to the blocks. This would happen, according to our hypothesis, because
the child would come to think of the written symbols as representing the blocks—thus the emphasis in
our instruction on using the writing to record the results of blocks action.
Again, the idea is that there are two categories of mental resources, one associated with “written
subtraction” and “syntactic” manipulations of symbols, the other associated with a “rich
knowledge base.” Due to the negative outcome of their study, Resnick and Omanson came to
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somewhat recant the view expressed above. In fact, in the above passage they are describing a view
that they no longer held at the end of the study! Nonetheless, this passage and the ones above are
interesting for how they make use of the symbol-referent view. In these passages, I argue, the
symbol-referent view is embodied as a cognitive theory. In addition, this cognitive theory is of a
particular sort: The symbol world and referent worlds have been associated with distinct
categories of cognitive resources.
Furthermore, in associating the symbol and referent worlds with distinct categories of
cognitive systems, there seems to be an implicit presumption that each of the two categories of
cognitive resources has a somewhat different character. The cognitive resources associated with the
symbol world are described as “syntactic” and “procedural,” while those associated with the
referent world are “semantic” and “conceptual.” These types of descriptions tend to accord the
symbol-related resources a second-class status. The description of the symbol world constraints as
syntactic implies a certain kind of simplicity. We expect that, for some notations, the constraints
will be easy to specify. If the conventions can be thoroughly spelled-out, then work within the
symbol world becomes simple, almost trivial, and with this simplicity comes the idea that the
symbol world is impoverished. If symbol-related knowledge is only about the meaning-free
manipulation of symbols according to a simple set of rules, then any “understanding” must be
localized to the world of semantics, in referent-related knowledge.
This image of two categories of knowledge is a very powerful one, and it is strongly tied to the
notion, discussed earlier, that student problems are traceable to a failure to “translate” back to the
reference world. However, given the viewpoint I have presented in this work, there is some reason
to be skeptical about cognitive models of this form. Notice that, in the model that I presented,
there is no clear breakdown into categories of this sort. In particular, the form-device resource
does not belong clearly in either the symbolic of non-symbolic category. It is knowledge that is
clearly tied to symbols, but I have argued that it should be considered a component of our
deepest conceptual understanding. Again, this is not necessarily a problem with the symbolreferent view, only a problem with this particular type of manifestation of it as a cognitive theory,
though this discussion does suggest reason for caution.
The problem with this view is not the description of student behavior—at its worst—as
meaningless formal manipulation. It is possible that this particular family of cognitive models may
be useful for providing rough descriptions of earlier, less successful, mathematical behavior. The
problem is the assumption that symbols and conceptual knowledge remain separate even with the
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development of expertise. Notice, in some of the Resnick quotes above, the implication that
“concepts” exist to be learned separate from symbolic procedures: “…even when the basic
concepts are quite will understood, they may remain unrelated to computational procedure.”
In my view, as people learn to use computational arithmetic in its symbolic form, conceptual
understanding develops, with symbols as legitimate participants in meaning. Educationally, this
means that we cannot simply direct our instruction at forging links between an existing body of
conceptual knowledge and formal procedures. The integration and development that must occur
around forms is a non-trivial affair that depends, among other things, on the nature of the
computational manipulations employed.
To conclude this section, I want to emphasize that I do not mean to be overly critical of the
research I used to exemplify the symbol-referent view. I chose these researchers because they make
their assumptions clear and explicit, and because their work is rich and useful. Instead, my real
hope in trying to draw out the symbol-referent view has been to ultimately draw attention to these
assumptions where they have been left implicit. I believe that much recent educational innovation
in mathematics and science has been implicitly driven by the symbol-referent view and the
presumption that formal symbolic methods are somehow conceptually impoverished. These
assumptions, I believe, deserve to be considered more carefully.
Is the Theory of Forms and Devices a symbol-referent theory?
I want to continue my contrasting of the symbol-referent view and the Theory of Forms and
Devices. In this sub-section, however, I want to begin by turning the discussion around a little.
The question I want to ask here is: Can we see the Theory of Forms and Devices in terms of the
symbol-referent view? For example, we might decide that the collection of forms counts as a
reference field or represented world, with the external symbolic display playing the role of the
representing world.
Of course, whether this counts as a version of the symbol-referent view depends on how
loosely that view is defined. However, I believe that if the symbol-referent view is defined so as to
include this relation between forms and symbolic displays, then we have a version of the symbolreferent view that is so loose as to not be useful. Viewed at a very high level, form-device theory
involves mental structures that come into play depending, in part, on the particular symbols that a
person sees. But this structure is far from specific to the form-device theory; a structure in which
the activation of mental elements depends, in some way, on the elements of the environment
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present, is extremely generic. Therefore, if we decide that any type of association between mental
elements and symbolic displays counts as a symbol-referent relation, then the symbol-referent
view will not help particularly in distinguishing among cognitive models.
It is also helpful to look more closely at the particular type of association that holds between
forms and written symbolic expressions. This association is really somewhat looser than the sort of
alignments that are characteristic of stronger versions of the symbol-referent view. Going back to
my discussion of Palmer, recall his presumption that the representing and represented worlds are
divided into “aspects,” and then there are fairly strict correspondences between these aspects. In
contrast, associations between symbolic forms and equations do not have this structure. First,
there is no “world” that is neatly sliced up by a collection of forms. A student’s repertoire of forms
are a loose collection of related schemata, and this repertoire, taken as a set, does not decompose
any motion or class of physical situations. In addition, the alignment between forms and symbolic
expressions is not strict. Any expression can be seen in terms of forms in multiple and overlapping
ways. Thus, unlike Palmer’s example representations in Figure Chapter 7. -5, there is not any sense
in which there are clear aspects of the representing world that do the representing, and which
correspond to aspects of the represented world.
This is not to say that the relation between symbolic forms and equations is totally
unconstrained and not characterizable. I have described quite specific relations that exist between
forms and equations. The point is that these relations are not of the same clean sort as those
typically associated with the symbol-referent view.
Actually, these last comments suggest some reasons to worry that the symbol-referent view,
even in its non-cognitive form, will not be appropriate for some analytic tasks. In applying the
symbol-referent view to help understand instructional difficulties, the presumption is generally
made that this view will be useful, at some level, for describing what people do with symbols.
However, the observations and arguments that I have presented in this document may indicate
that this presumption does not hold in all cases. If the form-device theory is an accurate account
of symbol use, with its multiplicity and context dependence, then my observations may cast
doubt on our ability to give a useful symbol-referent account. We may not, in general, be able to
find clean refers-to relations that are either explicit or implicit in symbolic activity.
This observation has fairly wide-ranging implications. We are very accustomed to thinking of
representations as things that, as Palmer says, “stand for something else,” according to rules that
we could find if we worked hard enough. But we should realize this is an assumption that might
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not hold, or, at the least, may hold only in limited cases, such as the toy representation in Figure
Chapter 7. -5.
I believe that a more neutral stance is required, at least initially, with regard to how
representations function in human activity. We populate our experience with all kinds of artifacts
and, for most of these, we would not presume that we could find correspondences between
aspects of these artifacts and other entities. My proposal is to take a similar stance in studying the
use of external representations: We should attempt to characterize symbol use as we find it in the
world, perhaps positing knowledge along the way, without the assumption that we will find that
representations “refer.”
Comparison 2: Similar Views From Mathematics Education
I will now counterbalance the discussion in the preceding section with a comparison to some
mathematics education research that is somewhat closer to the viewpoint expressed in this work.
Actually, most of this research was already discussed in earlier chapters, where appropriate, so my
task here will be to just draw together those earlier discussions and summarize the comparison to
the present work.
Recall that, in Chapter 3, I discussed what have been called “patterns” in arithmetic word
problems (Greeno, 1987; Carpenter & Moser, 1983; Riley et al., 1983; Vergnaud, 1982). This
research asserts that students learn to recognize a relatively small number of schemata that guide
the solution of arithmetic word problems. These schemata are given names like “change”,
“equalization,” and “compare.” As I discussed, there are some relatively clear similarities between
this view and my own. Most notably, these schemata seem to live at a very similar level of
abstraction as forms, and there are even some specific forms that are closely related, such as
BASE +CHANGE .
However, there are also some clear differences. For example, the patterns research
is more interested in the role that the schemata play in driving a solution process than the role they
play as ways of understanding expressions.
Another set of closely related research, which I also discussed in Chapter 3, appeared under
the topic “meanings of an equal sign.” Researchers in mathematics education have contrasted
different stances that students adopt toward equations and the equal sign (Kieran, 1992;
Herscovics and Kieran, 1980). In some cases, students see an equal sign as a “do something”
signal. The contrasting stance is when an equation is treated as an object in its own right, with the
equal sign specifying a symmetric, transitive relation. Kieran (1992) argues that learning to adopt
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the object stance is a developmental achievement, with the process stance being more natural early
on in instruction, but both stances are seen as playing a useful role in expertise.
There is an important sense in which both of these collections of research may differ from the
view expressed in this project. For these other researchers, a relatively small number of stances or
schemata suffice to characterize the way students understand and write equations. Kieran
describes two main categories of stances toward an equation. And the patterns research
characterizes all of its schemata in terms of operations on and relations between sets. In fact, one
gets the feeling that we could write down all of the possible schemata, just by imagining the
possible operations that are possible on sets.
But this procedure would fail to generate many of the symbolic forms that I have identified.
Notice, for example, that the COMPETING TERMS form cannot be understood in terms of sets; it is
fundamentally about influences. The point here is that, in contrast to this mathematics education
research, I generally believe that stances and meanings will be multiple and varied.
These differences are partly explainable by the fact that there are some important differences
in the regime studied, as compared to the present project, not least of which is that most of this
mathematics-related research was done with much younger students, working on a limited class of
problems. Furthermore, these researchers do not suggest that their categories necessarily extend
beyond their regime of study. Nonetheless, I believe it is appropriate to sound a note of caution
here. We cannot presume that it will be easy to guess a repertoire of symbolic forms given any
individual viewpoint, such as that provided by operations on sets. Furthermore, as students learn
to use equations in new domains, we should expect new forms and devices to develop—neither
the vocabulary of sets, nor any other vocabulary, will suffice for all domains.
Note that this last comment relates to one of the central themes of this work. In saying that
new forms may develop for each domain, I am saying that there is important symbol-related
knowledge that is domain dependent. This is contrary to a view in which mathematics is a tool
that transcends domains of application.
Finally, I want to close this section with a discussion of the well-known “Students-Professors”
problem (Clement, 1982; Kieran, 1992). This is a seemingly very easy problem that surprised
researchers by being quite difficult for students:
Write an equation using the variables S and P to represent the following statement: “There are six
times as many students as professors at this university.” Use S for the number of students and P for the
number of professors.
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The correct answer to this question is the expression S=6P. Interestingly, Clement (1982) found
that only 43% of a population of university engineering students solved this problem correctly. In
addition, of those that got it wrong, 68% reversed the expression, writing 6S=P. Furthermore,
Clement found that this latter result was quite robust; students did not rapidly change their
responses, and the mistake appeared even when some hints were included in the question.
Clement explains these observations by saying that students apply one of three cognitive
schemes in solving the students-professors problems. In the first, which he calls “word order
match,” the equation is essentially written to follow the ordering of key words in the problem
statement. The second scheme, “static comparison,” involves a direct comparison between two
related groups, in this case, the 6 students and 1 professor. Evidence for this is that, when writing
their expressions, students made statements of the form: For each 6 students there’s a professor.
Clement describes these first two cognitive schemes as involving a “passive” association, and both
produce the reversed expression 6S=P.
Finally, in the last scheme, “operational equality,” the student imagines operating on the
number of professors to get the number of students. Clement characterizes this last scheme as
“active” rather than passive, and it is the only one of the three schemes that produces the correct
answer. Clement comments: “the key to understanding correct translations lies in the ability to
invent an operation … and to realize that it is precisely this action that is symbolized on the right
side of the equation… .”
Clement’s distinction between “passive” and “active” schemes can be discussed in terms of
Kieran’s (1992) distinction between structural and procedural stances toward equations, with the
interesting conclusion that a procedural stance (which is presumed by Kieran to be
developmentally less advanced) seems to be more successful here. Kieran (1992), in fact, makes a
similar comment.
But we can also understand Clement’s observations in terms of forms and devices. In the static
comparison scheme, there is likely some type of Static device in play, along with a BALANCING -like
form. In contrast, in the operational equality scheme, the student employs a device that narrates an
operation on the number of professors (i.e., employs a Narrative device), perhaps by imaging the
substitution of a value for P into the expression. Furthermore, this latter scheme would be
associated with some sort of forms involving an asymmetric relation between quantities, such as
DEPENDENCE
or the SCALING form.
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A little more care would be required to accurately account for Clement’s results in terms of
forms and devices. A more appropriate explanation might very well involve some forms and
devices other than the ones that appeared in my data corpus. For example, I believe that there are
some differences in nuance in the BALANCING -like form that appears in Clement’s static comparison
scheme. And the plugging-in-numbers device that appears in the active scheme does not precisely
match with any of the devices that I have previous mentioned, though it is clearly in the ballpark
of the Narrative devices I discussed.
Nonetheless, I believe that the form-device theory can nicely account for Clement’s
observations, just as it subsumes the other research discussed in this section. What we are seeing
here is that the form-device view can capture many of the insights associated with this crosssection of mathematics education research. We do not need entirely new theoretical work to
account for Clement’s schemes, the patterns research, or the contrast between structural and
procedural stances toward equations.
Comparison 3: Situated Cognition and the Ecological Perspective
Throughout this work, I have taken an explicitly cognitive approach, positing mental structures
as my key claims. But one collection of orientations that has appeared under such names as
“situated cognition” and “situativity theory,” questions the validity of this entire approach
(“Situated Action,” 1993). In this section, I will discuss this alternative view and how it relates to
the present work.
Let us take a moment to reflect on the program of cognitive science. As I have said, cognitive
science works by positing knowledge that is presumed to be possessed by individuals. Using such a
framework, we can think about, for example, how to describe the results of school instruction. In
school, a cognitive scientist might say, people learn certain kinds of knowledge and then carry that
knowledge out into the world, using it where appropriate. This essentially means positing invariant
structures in the minds of individuals—structures that are the same across multiple situations, such
as school and home.
Furthermore, cognitive science makes some presumptions about the nature of these invariant
structures. In particular, they are presumed to live at something like an “informational level.”
Cognitivists are not only interested in pointing to neurochemical regularities in the brain, they
want to be able to, perhaps, state that there are mental structures associated with such things as
“concepts” and “propositions.”
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For illustration, take a toy cognitive theory that purports to describe the knowledge that
people learn in mathematics instruction. We might hypothesize, for example, that students learn
arithmetic algorithms in their mathematics classes, and then, once these are learned, they carry
these algorithms around in their heads, using them in the world where appropriate. According to
this toy theory, this is what it means to “know math.”
But there is much evidence that this toy theory is far from good enough. Often cited in this
regard are the results of the “Adult Math Project,” a project conducted by Jean Lave and others
(Lave, Murtaugh, & de la Rocha, 1984). Among other things, this project set out to study the
mathematics that adults use while shopping in the supermarket. To this end, the researchers
followed people into a supermarket, watched what they did, and attempted to characterize their
mathematical activity. Looking at the observations and analyses produced by this project, it is
clear that it is far from sufficient to describe shoppers as simply taking algorithms learned in
school and applying them in supermarkets. Instead, in solving best-buy problems, for example, the
behavior of shoppers was wide-ranging and idiosyncratic, depending on the concerns of the
moment and what was available in the environment.
Results of this sort have been interpreted by some as a reason to reconsider some of the
fundamental assumptions of cognitive science. The program of describing knowledge seems
plausible if we believe that people learn procedures, like arithmetic algorithms, and then apply
those procedures in the world. But what if this is not the case, and the behavior of individuals is
highly idiosyncratic and context dependent? In that case, the whole project of looking for
knowledge, at least as cognitive science has traditionally done it, may not make sense, since it may
not be possible to identify structural invariants of an interesting sort.
This is where situated cognition comes in. The view of situated cognition is that cognitive
science has over-emphasized the determining effects of cognitive structure at the expense of the
influence of the material, social, and historical environments. For the case of supermarket
arithmetic, this means that you cannot expect to say much about this variety of math without
knowing about the details of the material and cultural context. This is frequently taken to suggest
that the very nature of what we mean by “knowledge” must be changed: Situated cognition
theorists often maintain that we should no longer think of knowledge as something localized in
the heads of individuals; instead, they say, we should think of “knowing” as existing only in the
interaction between person and environment.
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I believe that observations like those of the Adult Math Project are entirely sensible, and I take
the observed sensitive dependence on the situation as absolutely fundamental. In fact, my focus on
external representations is partly for this reason. However, I do not believe that these observations
require that we change the very nature of what we mean by knowledge. Instead, I believe that
these observations simply tell us something about the type of cognitive theories that we need to
build: We need to push down in grain-size, and the problem of describing knowledge is probably
going to be much harder than one might initially have thought, but I do not believe that this is a
reason to give up on knowledge, understood in the traditional sense.
Even with this major point of difference between my own view and that of situated cognition
researchers, it is nonetheless the case that their concerns are often close to my own. As I
mentioned, one of the concerns of situated cognition is to draw attention to the material
situatedness of cognition. Because of this emphasis, these researchers, like myself, tend toward a
particular concern with the role of external representations. For this reason, a more detailed
comparison is merited.
There is no uniform theory of situated cognition, only a collection of loosely related views,
and there is no way I can do justice to this variety. Instead, I will just look closely at two
researchers from a particular branch of situated cognition—the “ecological perspective”— that
happens to have concerns close to those of this work. Greeno and Engle (1995), who ascribe to this
perspective, describe their approach as follows:
we consider cognitive processes such as reasoning, understanding, and representing to be accomplished
by systems that include humans interacting with each other and with available material resources,
rather than as processes that only occur inside individual minds.
The idea is to look at a person working with symbols in some situation as a whole system of
interacting elements that act together, perhaps toward some end. The analysis then looks for
regularities in the interactions among elements, and tries to describe how the various constraints
within the system lead to some goals being reached. Thus, in contrast to cognitive science, the
structural invariants identified by the ecological perspective will, in general, span person and
environment.
In their particular study, Greeno and Engle looked at people using a physical device they call a
“winch machine.” This device consists of blocks that slide along numbered tracks, with the blocks
connected by strings to spools of various diameters. When the spools are turned, the blocks slide
across the track by an amount that depends on the number of turns and the diameter of the
spools. Greeno and Engle gave seventh and eighth grade students the task of making a table of a
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block’s position as a function of the number of turns of the spool, for spools of various sizes. Thus,
in this situation, the elements of the “ecology” include features of the winch machine, the written
tables, and the students.
Greeno and Engle performed several types of analyses on their transcripts of this activity. One
analysis was based around what they called “activity nests.” They decomposed transcripts of
activity into nested “functions,” saying which student activities contributed toward which
functions. In addition, they performed an analysis of “semiotic networks,” attempting to identify
networks of refers-to relations among elements of the ecology. Altogether, their image is of a
system, which includes students and external representations, interacting (in ways that they
describe) to achieve certain functions.
Learning, here, is a gradual attunement to constraints and affordances in the situation. This
means that a person develops a sensitivity to certain features of the environment, and can react to
these features in certain ways. This is how Greeno and Engel account for the types of change in
behavior associated with learning: If students are attuned differently, then they will interact
differently with the rest of the ecology, and the behavior of the system changes.
Though it may sound quite different, there are actually some similarities between Greeno’s
ecological view and the view presented here. Most notably, my argument that students become
sensitive to certain registrations, especially what I called symbol patterns, might be understood as
an attunement to an affordance. But there are important differences, even beyond the different
orientation to the nature of knowledge. In my discussion of the symbol-referent view, I already
questioned the possibility of finding refers-to relations in symbol-related activity. In addition,
Greeno and Engel’s function-based analysis departs from the analyses described in this document.
One of the main objectives of an ecological analysis is to explain how the interactions and
constraints in a system conspire to allow the system to reach some goal or perform some function.
In my analysis—as I have noted on a number of occasions—I did not attempt to explain how the
use of particular forms and devices contribute to the reaching of any particular goals, such as the
solving of a problem.
Before wrapping up this section, I want to look at one other piece of research, also done from
an ecological perspective, but which is closer to my own program in population and domain focus.
Rogers Hall looked at undergraduate computer science majors solving motion and work problems
involving linear rates (Hall et al., 1989; Hall, 1990). For example, one problem was as follows:
Two trains leave the station at the same time. They travel in opposite directions. One train travels 60
km/h and the other 100 km/h. In how many hours will they be 880 km apart?
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This problem can be solved using algebraic expressions, manipulated in the usual ways, and it is
likely that Hall’s college students have some experience with these techniques. However, Hall
found that even competent problem solvers engaged in techniques outside of these formally
recognized activities. Of particular interest was what Hall called “model-based reasoning,” in
which “a student ‘executes’ a model of the problem situation along the dimension defined by an
unknown quantity such as time, distance, or work.” For example, a student might step through
time, figuring out the location of each train at each moment of time. In addition to model-based
reasoning, Hall characterized a large number of tactics used by competent solvers to reach
solutions.
In summary, what Hall found was that these competent problem solvers do not just employ
the few formally recognized—and supposedly very general—techniques they learned in their
mathematics classrooms. Instead, they employ a wide variety of tactics and seem to construct
techniques as needed. Given these observations, Hall argued for a “constructive” rather than a
“recalled” view of problem solving. Notice that this view is very reminiscent of some of my
comments concerning equation use and problem solving in physics. I explicitly contrasted my own
account with what I called the “principle-based schemata” view of physics problem solving. In that
view, people possess schemata associated with physical principles, and these schemata dictate what
equations are written and how they are used to solve a problem. In contrast, the Theory of Forms
and Devices allows for the construction of novel expressions, and for interaction with equations at
a level other than that specified by formal principles.
Thus, there are some important similarities with Hall’s work. The differences come when, like
Greeno, Hall situates his observations within an ecological perspective (Hall, 1990). Like Greeno
and Engel, Hall is concerned with a discussion of how the ecology achieves certain goals. For
example, he discusses how the use of the various tactics he observed supports different types of
inferences. Nonetheless, the concerns of both Greeno and Hall are very close to those of this work.
They, like me, attend to the more informal and flexible aspects of activity with external
representations, thus escaping from the relatively rigid view of symbol use that sometimes prevails
in cognitive science.
Comparison 4: A Bodily Basis for Meaning?
Finally, I want to conclude this chapter with a comparison to a viewpoint that, strictly
speaking, is not particularly concerned with external representations. However, this is a worthwhile
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comparison to include here because it can serve to highlight certain important features of my
stance. In this section, I will address two related works: Women, Fire, and Dangerous Things, by
George Lakoff and Mark Johnson’s The Body in the Mind: The Bodily Basis of Meaning,
Imagination and Reason (Johnson, 1987; Lakoff, 1987).
Since Lakoff’s discussion, to a certain extent, subsumes Johnson’s, I will begin by following
Lakoff. Lakoff’s goals are quite broad and ambitious. In Women, Fire, and Dangerous Things he
sets out to define a new version of philosophical realism he calls “experiential realism” or
“experientialism.” Like many rejections of traditional realism, Lakoff begins by asserting that
there is not an objectively knowable world; our concepts do not exist separate from us to be
discovered. Then, what distinguishes experientialism is the notion that the concepts that we use
for understanding the world depend, in particular, on the bodily nature of our experience.
More specifically, the idea, for both Lakoff and Johnson, is that we have set of “preconceptual
experiences” that are “directly” meaningful. Then, our concepts get their meaning only indirectly,
by being tied to this preconceptual experience in certain specific ways. Lakoff explains:
“Experientialism claims that conceptual structure is meaningful because it is embodied, that is, it
arises from, and is tied to, our preconceptual bodily experiences” (p. 267).
Furthermore, the program seeks to identify types of “structure” that exist in our preconceptual
experience. Lakoff states that there are at least two kinds of structure. The first “basic-level
structure,” is closely related to what psychologists have called basic-level categories. The second
type of structure is called “kinesthetic image-schematic structure.” These “image schemata” are
primitive schemata that derive from the basic structure of our experience in the world. It is here
that Lakoff makes principle contact with Johnson’s book, which provides an extensive discussion
of these image schemata. One of Johnson’s principle examples is what he calls the “containment”
schema:
Our encounter with containment and boundedness is one of the most pervasive features of our bodily
experience. We are intimately aware of our bodies as three-dimensional containers into which we put
certain things (food, water, air) and out of which other things emerge (food and water wastes, air,
blood, etc.). From the beginning, we experience constant physical containment in our surroundings
(those things that envelop us). We move in and out of rooms, clothes, vehicles, and numerous kinds of
bounded spaces. We manipulate objects, placing them in containers (cups, boxes, cans, bag, etc.). In
each of these cases there are repeatable spatial and temporal organizations. In other words, there are
typical schemata for physical containment. (p. 21)
Other examples of image schemata given by Lakoff and Johnson are “balance,” the “path”
schema, “part-whole,” “up-down,” as well as some schemata relating to forces, including
“blockage,” and “compulsion.”
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Lakoff and Johnson’s program is not only concerned with identifying structure in
preconceptual experience. In addition, they set out to describe how “abstract conceptual structure”
arises from the preconceptual structure. Lakoff lists two ways that this occurs:
1. By metaphorical projection from the domain of the physical to abstract domains.
2. By the projection from basic-level categories to superordinate and subordinate categories.
The idea is that abstract conceptual structures get meaning by being tied to preconceptual
structure. In some cases, for example, this happens through metaphors to directly understood
domains.
One of the examples that Lakoff provides is the “more is up; less is down” metaphor. He gives
many example of how this metaphor is manifested in our language: “The crime rate keeps rising.”
“That stock has fallen again.” The point is that this is a case in which a particular abstract target
domain, “quantity,” is understood indirectly by being tied to the directly understood source
domain of “verticality.”
Interesting, like “patterns” in arithmetic word problems, image schemata seem to live at a level
of abstraction that is similar to that of forms. Varieties of the part-whole schema have appeared in
all three cases, and Johnson even has his own version of a balance schema. This is important, if only
because it means that Johnson’s list of image schemata can be used as a source of potential forms.
It is particularly illuminating to look at one relevant example given by Johnson. Johnson asserts
that “mathematical equality” is understood by metaphorical project from a version of the balance
schema:
Here the mundane TWIN -PAN BALANCE schema is projected onto the highly abstract realm of
mathematical computation and equation solution. The basis for this projection is a point-by-point
mapping of entities from the physical realm into the abstract mathematical realm … . (p. 90)
This sounds very similar to some of my own assertions, but there are important differences, even
beyond the observation that Johnson glosses what I have taken to be rich territory with a single
schema. The big difference is that the whole Lakoff-Johnson view is based on the presumption
that certain types of experience occupy a privileged status. There is the more concrete,
preconceptual variety of experience, and more abstract categories of experience. Furthermore, the
more abstract experience, like that associated with mathematics, is only understood indirectly, by
metaphorical project from structures in more basic experience.
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In contrast, in my view, there is not a stable background of preconceptual structures associated
with a special category of experience. Instead, there is a constant re-churning of our
understanding. One of the main points here has been to argue that through experiences with
symbols—and all other sorts of experience—the most primitive levels of our understanding
continue to evolve.
In fact, it is from the comparison to Lakoff and Johnson that this manuscript gets its title.
Recall that the sub-title of Johnson’s book is The Bodily Basis of Meaning, Imagination, and Reason,
a phrase that attempts to capture Johnson’s view that all meaning is ultimately traceable to bodily
experience. In contrast, the thesis of this work has been that experience with symbols is just as
basic as any other kind of experience, and that it has effects that percolate down to the most basic
levels of our understanding and intuition. Thus, my argument is that there is not only a bodily
basis of physical intuition, there is also a symbolic basis.
ReflectionÕs End
In this chapter, I wrapped up Part 1 of this thesis by reflecting on the nature of the theory
outlined in the preceding chapters. This actually was done in two parts. In the first half of this
chapter, I reflected on some properties of the theory itself. Then, in the second half, I worked by
contrasting my view to other research perspectives.
In reflecting on the nature of the Theory of Forms and Device, I discussed the sense in which
the theory implies that physics understanding is “intrinsically representational.” There were
essentially two answers. First, I argued that physics knowledge is refined specifically by and for
symbol use. Second, I argued that, through symbolic experience, a whole new resource develops,
the form-device system, and that this resource should be considered a component of physical
intuition.
In the first half of this chapter I also briefly discussed how the theory and its conclusions
might be generalized. I described how the Theory of Forms and Devices could be extended to
another representational practice, graphing-physics. In addition, I speculated concerning more
general implications for the relation between knowledge and external representations. My
conclusions in this regard were not definitive. I argued that there is simply no shortcut to avoid a
thorough analysis of the knowledge systems involved in any practice of symbol use, and that there
may potentially be as many relations between knowledge and external representations as there are
knowledge systems.
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Then, in the second half of this chapter, I contrasted my view to four other research
perspectives. Among the important conclusions was the simple observation that the theory
presented here is a cognitive theory, which distinguishes it from pure symbol-referent theories or
an ecological perspective.
In addition, I tried generally to contrast my view with perspectives that tend to treat symbolic
knowledge and experience as semantically impoverished. I do not believe that there are two
categories of cognitive resources—one syntactic and non-meaningful, the other non-syntactic and
meaningful—as sometimes implied by applications of the symbol-referent view. And I argued
that unmediated bodily experience is not the only kind of meaningful experience. These last
points, perhaps, are the central moral of Part 1 of this work.
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Chapter 8. An Introduction to Programming-Physics
From Ehrlich, 1973. Reprinted with permission.
A new technological innovation—the computer—is steadily becoming as much of a fixture in
schools as chalkboards and pencils. While some tout the computer as a potential catalyst for a
revolution in education, others believe that this technological marvel will turn out to be just
another educational flash in the pan. Certainly, the boundless enthusiasm that sometimes seems to
prevail may need some tempering as we proceed in our exploration of the potential of computers
in instruction, and it would doubtless be profitable for more researchers to adopt stances of
skepticism. As has been pointed out by Larry Cuban (1986, 1993) and others, (e.g., D. K. Cohen,
1988) broad change in our educational institutions does not come easily.
Although computers may not be the key ingredient in an incipient educational revolution,
there are some simple and compelling reasons to believe that some sort of computation or
information technology will ultimately become a permanent fixture in classrooms. Central among
these reasons is the simple fact that computers are useful for a wide range of tasks. The point is
that we do not need to argue that computers possess some very general properties, like
“interactivity” or “dynamism,” that explain why computers should be especially useful for
educational purposes. We simply need to note that, like all good tools, computers make some
tasks easier and new tasks possible. Thus, for the very same reasons that computers have
established a beachhead in homes and offices, we can expect to continue to find them in schools.
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Take the particular example of word processing software. Students should find this software
useful for many of the same reasons that business-people do.
I do not mean to imply that there are not unique instructional activities based around
computers—activities that would not be possible without computers. The existence of this new
tool with its specific capabilities does allow for some novel instructional applications. These
applications might even, in the long run, profoundly change how certain subjects are taught. The
second part of this dissertation, which begins with this chapter, is essentially an exploration of one
such novel application within the domain of physics. Here, we once again take up the discussion
begun in Chapter 1 and undertake to explore the possibility of improving physics instruction by
teaching students to program computers and to create certain kinds of physics-relevant programs.
A brief history of programming-physics
The proposal I will explore here, to teach physics through programming, is not new. To see its
roots, we must go back to a time prior to the use of computers in instruction. Even before the
wide appearance of computers in classrooms, some educators had realized the value of a related
technique, the use of numerical methods in physics instruction. For example, in his Lectures on
Physics, Richard Feynman used numerical methods as a key piece of his exposition on forces and
F=ma (Feynman, Leighton, and Sands, 1963).
As we progress, I hope to illustrate that the use of numerical methods is closely related to
programming-physics. In fact, many of the techniques and motivations underlying Feynman’s
numerical solutions also lie at the heart of programming-physics, with the important difference
that the availability of computers helps to make numerical methods more practical. For that
reason, it is worthwhile to begin this brief history with a little detail on Feynman’s exposition.
Feynman begins his discussion by asking this question: Suppose that we know the position and
velocity of an object at a time t, how can we find the position at a slightly later time t+ε? (The
symbol ε here stands for a small interval of time.) In answer, Feynman presents the following
expression:
x (t + ε ) = x (t ) + εv(t )
This equation says that, to find the position at the time t+ε, you add the quantity εv(t) to the
current position, where v(t) is the velocity at time t. This makes sense since, by definition, the
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velocity of an object is the amount that its position changes per unit time, so εv(t) is the amount
that the position changes in the time ε. In a similar manner, we can write an expression for the
velocity at the time t+ε:
v(t + ε ) = v(t ) + εa(t )
Here a(t) is the acceleration at the time t. Note that both of these relations are only
approximations because they neglect the fact that the velocity and acceleration may change over
the time interval ε.
Furthermore, if we know the force on an object as a function of its position and
velocity—F(x, v)—then we are in a position to employ a very powerful and general procedure for
predicting the motion of an object. The procedure works like this:
(1) Start with the position and velocity at some time t: x(t), v(t).
(1) Using the function F(x, v) and the equation F=ma, find the acceleration at the time t: a(t).
(1) Substitute x(t) and v(t) into the above expression for the position to obtain x(t+ε).
(1) Substitute v(t) and a(t) into the above expression for the velocity to obtain v(t+ε).
(1) Go back to step (1) using x(t+ε) and v(t+ε) and iterate.
Using this method we can trace the motion of an object forward into the indefinite future. We
start with the position and velocity at any time and then we step forward, one moment at a time,
along the motion. Of course, if each iteration only takes us a very small step forward in time, then
the calculations involved may be laborious.
The beauty of this method is its extreme generality and simplicity. As long as you know how
to find the force on an object, you can use a=F/m to find the acceleration, and then plug in and
iterate to find the motion of the object. The computations may be tedious, but the method will
work for every motion.
The alternative approach—what students are usually taught to do—is to essentially write and
solve differential equations to obtain analytic, closed-form solutions. The problem is that this
traditional approach quickly runs into mathematical difficulties. Not only is the mathematics
involved difficult, the majority of differential equations cannot be solved analytically at all. For
that reason, various tricks, approximations, and alternative techniques must be developed, each of
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which tends to be specific to only a small class of problems. Among this variety of techniques, the
underlying simplicity may be lost.
In his Lectures, Feynman went on to apply the numerical method in gory detail for two
motions, the oscillation of a mass on a spring, and the motion of a planet around the sun. Then,
following his last computation Feynman commented:
So, as we said, we began this chapter not knowing how to calculate even the motion of a mass on a
spring. Now, armed with the tremendous power of Newton’s laws, we can not only calculate such
simple motions but also, given only a machine to handle the arithmetic, even the tremendously
complex motions of the planets to as high a degree of precision as we wish! (p. 9-9 Volume 1)
As this quote suggests, the reason that Feynman spends time on these laborious computations is
that he is trying to convey to his students the fundamental power and generality of Newton’s
theory. One of the great virtues of numerical methods for instruction is that it makes this
generality and power fairly clear. As we shall see, this is one of the arguments that researchers
make in espousing the merits of programming-physics for instruction.
Nevertheless, in Feynman’s three-volume text, numerical methods are not to be seen outside
of this one exposition. There are certainly many reasons for this, not least of which is that
Feynman probably has some commitment to covering the material in its traditional form.
However, a major factor must also be the lack of a “machine to handle the arithmetic.” Although
computers were in use by researchers at the time of Feynman’s lectures, they were not available in
classrooms.
But computers were to appear on the scene shortly and, with them, the possibility of realizing
the benefits of numerical methods. In fact, it wasn’t long after the publication of Feynman’s
lectures that books appeared, some directed at physics teachers, that explained how to use the
teaching of programming in physics instruction (e.g., Bork, 1967; Ehrlich, 1973).The books were
written with the computers of the day in mind. They expected that students would be
programming in FORTRAN using punch cards, and they assumed that students would be seeing
output on a line-by-line printing device. This means that the user never looked at the sort of
computer display to which we have become accustomed; instead, all output was printed on a sheet
of paper that scrolled out of the top of a printer.
In fact, much cleverness and ingenuity was directed toward inventing methods for producing
interesting displays with these printing devices. The example shown in Figure Chapter 8. -1, which
is taken from Ehrlich (1973), shows the electric potential around two point charges. Notice that
the image is produced using only ‘+’ and ‘-’ symbols. (See also the image at the head of this
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chapter.) This image is quite effective, given the limits of the medium. But, no matter how clever
these early pioneers were, it was not possible for them to produce dynamic displays on a line
printer; they were restricted to static images.
Figure Chapter 8. -1. The electric potential around two point charges printed using only '+' and '-' symbols.
(From Ehrlich, 1973. Reprinted with permission.)
The publishing of Mindstorms by Seymour Papert is an important landmark in this brief
history (Papert, 1980). In Mindstorms Papert described his vision for a child-centered classroom.
At the heart of this vision was a new programming language, Logo, which was designed specially
for use by students. Though Mindstorms does contain some physics-related examples, Papert was
not especially concerned with programming-physics. Nonetheless this is an important landmark
for this history because of Papert’s well-known espousal of student programming, generally, as a
useful ingredient in instruction.
It is also worth noting that Logo was riding the crest of what was then the newest computer
technology. Notably, Logo was designed to be used with video displays rather than printers for
output. In this regard, a central innovation of the Logo language was the ability to program “turtle
graphics.” In Logo, students can give programming instructions to a small triangular object,
known as the “turtle,” which can be made to move around the video display. Thus, in contrast to
the early days of printed output, students could produce dynamic displays in which objects
actually appeared to move on a computer screen.
In subsequent years, as computers have become more powerful and user-friendly, the move in
physics education has been toward uses other than student programming. Rather, applications
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such as pre-made simulation environments, computer tutorials, and real-time data-collection tools
have come to dominate. (For a broad picture, see Redish and Risley, 1988.)
In some respects, this proliferation of alternative uses can be understood as a consequence of
the increased power and capabilities of computers. It is not surprising that the development of new
capabilities, such as the ability to produce dynamic output, would lead instructors to invent new
applications for computers in physics instruction. But I believe that this shift in application also has
to do with some changes in the prevailing attitude toward computers. As applications have
become increasingly powerful and elaborate, programming has become more and more the
province of expert programmers. In this age of large, powerful, and flashy applications, the notion
that students can easily program their own simulations may sound less plausible to the average
teacher.
But student programming in physics has not been abandoned through the 1980’s and 1990’s.
Materials are still being produced for teachers, though they are frequently targeted for auxiliary
courses in numerical techniques (e.g., Gould & Tobochnik, 1988). And similar approaches are
showing up in alternative forms, such as in applications of spreadsheets to physics instruction (e.g.,
Misner & Cooney, 1991).
A notable example of a currently active program is the M.U.P.P.E.T. project at the University
of Maryland, which is an ambitious attempt to integrate Pascal programming into the university
physics curriculum (Redish & Wilson, 1993; McDonald, Redish, & Wilson, 1988). Rather than
simply supplementing the standard curriculum with a few programming activities, Redish and
colleagues have set out to “rethink the curriculum entirely from the ground up.” One of their
main points is that limits in students’ mathematical abilities, as well as the inherent difficulty of
much of the mathematics involved in physics, has placed many constraints on the physics
curriculum: Topics must be ordered in a way that parallels the development of student
mathematical knowledge, and some topics and phenomena cannot be studied at all because the
mathematics involved is extremely difficult, even for expert physicists. By introducing numerical
methods implemented on a computer, many of these constraints are loosened. As Feynman
realized, the same, very general, numerical methods can be applied to study a very wide range of
physical phenomena. This allows the curriculum to be ordered in a more “natural” manner, and
the range of phenomena that can be studied is greatly broadened.
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The current project
I am now ready to discuss where the present research fits in this brief history. The work I am
going to describe in the second part of this dissertation is unique because of its tight focus on the
representational language itself. Earlier research has been very concerned with what student
programming allows. As Redish and colleagues have argued, if your students can program, then
you can re-order the topics in the curriculum, and you can study topics and phenomena that were
formerly inaccessible to students. From this point of view, the primary importance of the
programming language is that it provides a means of bringing the power of the computer into
play; if you want to use the computer to do some laborious computations for you, then you have
no choice but to give the computer instructions using a programming language. In this view,
algebra retains its status as the primary representational system of physics, and programming plays
an auxiliary role as the language we use to communicate with computers.
In contrast, I am interested in studying programming languages as representational systems in
their own right. I want to elevate programming languages to the status of bona fide
representational systems for expressing physical laws and relations, and I want to study the
properties of these systems. This means adopting the stance that a programming language can
function much like algebraic notation in physics learning. I believe that this move constitutes the
next logical step in this history and, when we take this step, we will be moving one notch further
toward a true programming-physics.
It is at this point that we can make contact with the broader concerns of this project. One of
the main goals of this dissertation is to argue for a tight relationship between knowledge and
external representations. In Chapter 1 I made the point that, if this tight relation exists, then
replacing algebraic notation with a programming language may have a fundamental effect on how
students understand physics. With that argument, the stage was set for the overall task of this
research project: a comparison of algebra-physics and programming-physics knowledge.
The first segment of this project, a description of a certain component of algebra-physics
knowledge, was completed in Part 1 of this dissertation. Now, in Part 2, I will move on to the
complementary segment, the production of a description of programming-physics knowledge
that can be compared to the results for algebra-physics. The remainder of this chapter lays out
some of the preliminaries for this second segment of the project. I will begin with an introduction
to the programming language used in my study, and then I will provide a description of the
particular practice of programming-physics that I taught to my subjects.
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An Introduction to Boxer Programming
All student programming in this study was done in a new and rather unique programming
environment known as “Boxer,” which employs its own programming language. The Boxer
programming environment and language are currently under development at U. C. Berkeley
(diSessa & Abelson, 1986 ; diSessa et al., 1991). In many ways, Boxer can be understood to be a
direct descendent of Logo, and it has been designed with the specific aim of correcting some of
the shortcomings of Logo while maintaining many of the same goals and strengths (diSessa,
1986). For example, as we shall see in a moment, Boxer involves a form of graphics programming
that is very similar to the turtle graphics found in Logo. But perhaps the most important feature
that Boxer shares with Logo is its overall philosophy: Unlike most programming languages, Boxer
is designed especially with education in mind and it is intended to be used by students of a wide
range of ages and abilities, from elementary school students through adults.
The primary reason that I chose Boxer for this study is that it is easy to use and learn. In my
pilot work I found that it was only necessary to give the students in this study a one-hour
introduction to Boxer programming before they could begin working on some physics-related
tasks. This is especially striking given the fact that many of the students had little prior
programming experience. This property is not only extremely useful for research studies. Easy
learnability is also essential for applications of programming to subject matter instruction, since we
do not want students to get bogged down in the details of programming when trying to learn
about, for example, physics and mathematics.
The Boxer programming environment gets its named because, in Boxer, everything on the
computer display appears inside a box. Programs appear in boxes called “do-it” boxes, graphics
appears in “graphics” boxes, and variables appear in “data” boxes. Figure Chapter 8. -2 shows a
simple Boxer graphics program, with a graphics box on the left and a do-it box on the right. The
triangular object inside the graphics box is called a “sprite.” Boxer’s sprites, like Logo’s Turtles, are
the vehicles of all graphics programming. When given programming commands, sprites move
around the computer display within the confines of a graphics box. The do-it box to the right of
the graphics box contains a short program that tells the sprite to go forward 50 turtle steps, then
turn right 45 degrees, and finally go forward another 35 turtle steps.
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a-program
fd 50
rt 45
fd 35
Doit
Data
Figure Chapter 8. -2. A Boxer graphics program.
The program shown in Figure Chapter 8. -3 illustrates the use of variables. Variables take on a
particularly simple form in Boxer, they are just boxes whose contents can be changed by
programming instructions. The key programming command that is used for working with
variables is the “change” command. The program shown in Figure Chapter 8. -3 uses the change
command to change the contents of the variable named bob to 3, then to change the contents of
the variable hal to 5, and finally to change jane to the sum of bob and hal. Notice that, in the final
statement, a do-it box has been used to group the two variables to be added. Do-it boxes can be
used to group terms in a Boxer statement much as one would use parentheses in an algebraic
expression.
jane
bob
hal
8
3
5
Data
Data
Data
a-program
change bob 3
change hal 5
change jane
bob + hal
Doit
Doit
Figure Chapter 8. -3. Programming with variables in Boxer.
It is worth taking a moment to note the common syntax shared by the programming
commands used in these examples. To use most Boxer commands, we write a command name
followed by a list of arguments:
command-name <argument-1> <argument-2> …
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For example, both the fd and rt commands each take a single argument, a number that specifies
the number of steps to go forward or the number of degrees to turn. In contrast, the change
command takes two arguments, one that specifies the variable to be changed and another that tells
Boxer what new value to insert in this variable.
draw-square
repeat 4
fd 50
rt 90
Doit
Doit
Data
Figure Chapter 8. -4. A program to draw a square.
We will make quite frequent use of one additional command, repeat, which provides a means
of repeating a set of programming instructions. The two arguments to the repeat command
specify the number of repeat cycles and the set of instructions to be repeated. Figure Chapter 8. -4
shows a program that uses the repeat command to draw a square by repeating, four times, the
commands fd 50 and rt 90.
Finally, my study made use of the fact that students can define and call their own procedures.
To create a new procedure in Boxer, we write a set of instructions in a do-it box and then give that
box a name. This is illustrated by the program in Figure Chapter 8. -5. The procedure named
draw-pattern uses the named procedure draw-square to create a pattern of four tiled squares.
draw-pattern
draw-square
repeat 4
repeat 4
draw-square
rt 90
Doit
Doit
fd 50
rt 90
Doit
Doit
Data
Figure Chapter 8. -5. A program that uses a named procedure to draw a pattern.
These four programs illustrate most of what the reader will need to know to understand the
examples that follow. We will see that the combination of simple sprite-graphics, variable
programming with the change command, and the use of repeat were sufficient for the students in
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my study to do quite a lot of interesting physics programming. I will fill in additional details as
we proceed.
A Practice of Programming-Physics
I am now ready to introduce the reader to the practice of programming-physics that was
employed in this study. As I mentioned in Chapter 1, the particular practice that I used was
adapted from work done by the Boxer Research Group at UC Berkeley. In that research,
programming was used as part of physics courses that the Boxer Research Group designed for 6th
graders and high school students (diSessa, 1989; Sherin et al., 1993).
To illustrate this practice, I will present a few examples from the brief curriculum given to the
college students in this study. The programming-physics training was intermingled with the more
general instruction in Boxer programming. To start, the students were taught only how to create
the sort of sprite graphics programs shown in Figure Chapter 8. -2. Following this
instruction—and prior to any instruction in the use of variables—the students were asked to
perform their first physics-related tasks. In one of these first tasks, they were simply told to “make
a realistic simulation of the motion of a ball that is dropped from someone's hand.” In response,
they created programs like that shown in Figure Chapter 8. -6, which was made by two students
in this study, Anne and Clem.
drop
setup
fd 1 dot
fd 3 dot
fd 5 dot
fd 7 dot
fd 9 dot
fd 11 dot
fd 13 dot
fd 15 dot
fd 17 dot
fd 19 dot
fd 21 dot
fd 23 dot
fd 25 dot
fd 27 dot
fd 29 dot
fd 31 dot
fd 33 dot
fd 35 dot
fd 37 dot
fd 39 dot
fd 41 dot
Doit
Data
Figure Chapter 8. -6. Anne and Clem's simulation of a dropped ball without variables.
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Figure Chapter 8. -6 shows two boxes, a graphics box containing a sprite—the sprite, in this
case, is shaped like a ball—and a do-it box named drop, which contains the program itself. Anne
and Clem’s program begins by executing a setup command, which positions the sprite at the top
of the graphics box and orients it so it is prepared to move downward. Each of the remaining lines
in the program contains a fd command followed by the dot command. The dot command causes
the sprite to draw a dot at its current location. Thus we see that this program causes the sprite to
move forward in larger and larger steps, leaving a trail of dots in its wake. The sprite moves
forward one step and then leaves a dot, then three steps, then five steps, etc.
It is important to note how this sort of program was understood by students. All of the pairs
tended to assume that each of the movements of the sprite takes the same amount of time, an
assumption that was reinforced by later aspects of the curriculum. Given this assumption, Anne
and Clem’s program suggests that the ball is speeding up. Furthermore, note that the amount that
the sprite steps forward increases by two each time that it moves. Thus, not only is the speed
increasing, it is increasing at a constant rate for each time interval. Anne and Clem chose this
constant increase to reflect the fact that, neglecting air resistance, a dropped ball undergoes a
constant acceleration.
a
b
c
Data
counter
drive
0.000
setup
change counter 20
repeat 23
fd counter
dot
Data
Doit
repeat 40
change counter
counter - .5
Doit
fd counter
dot
Doit
Doit
Figure Chapter 8. -7. Using variables to simulate constant deceleration.
Of course, creating simulations in this manner, using only a sequence of forward commands,
would be quite tedious for students and not very illuminating. So, after making only a few
programs along the lines of Figure Chapter 8. -6, the students were taught to use variables and to
apply them to create simulations. An early application of variables to simulation programming is
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shown in Figure Chapter 8. -7. In this task, the students’ job was to write a program that causes
the car-shaped sprite to move from point (a) to point (b) at a constant speed, and then slow down
at a constant rate as it moves from point (b) to point (c).
In the student simulation shown in Figure Chapter 8. -7 this motion is achieved using a
variable named counter. The program, which is named drive, begins with a setup command,
which places the car at point (a) in preparation for the motion. Then, in the next line, a change
command is used to initialize the value of counter to 20. The remainder of the program consists
of two repeat commands, each of which causes a short program to repeat a fixed number of times.
The first repeat tells the sprite to, 23 times, move forward the amount in the counter variable and
make a dot. Since the value in counter doesn’t change during this repeat, the sprite moves forward
the same amount on each cycle of the repeat, leaving a trail of equally spaced dots. Thus, this first
repeat takes the car from point (a) to point (b) at a constant speed, as required by the task.
The second repeat statement is similar but with one important difference: In addition to a fd
and dot command, it also contains the line
change counter
counter - .5
Doit
which causes the value of counter to be decreased by .5 on each cycle of the repeat. Thus the car
steps forward in smaller and smaller increments as it moves from (b) to (c). In this way, the
students have simulated a constant deceleration, but without resort to a long string of forward
commands, as in Figure Chapter 8. -6.
The simple program in Figure Chapter 8. -7 embodies most of the key structures and
techniques needed to create a wide range of simulations. There is a variable, in this case named
counter, that corresponds to the velocity of the moving object. This variable is initialized at the
start of the program then, within a repeat statement, values are added or subtracted from this
variable to simulate an accelerated motion.
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a
b
c
Data
pos
vel
acc
850.000
0.000
-0.500
Data
Data
Data
tick
change vel
change pos
drive
setup
change
change
change
repeat
pos 0
acc 0
vel 20
23 tick
vel + acc
Doit
change acc -.5
repeat 40 tick
pos + vel
Doit
Doit
fd vel
dot
Doit
Data
Figure Chapter 8. -8. A simulation using the Tick Model.
After the students were given a few opportunities to write their own simulations using
variables, they were taught a standardized version of this structure called the “Tick Model.” The
simulation in Figure Chapter 8. -7, rewritten to employ the Tick Model, is shown in Figure
Chapter 8. -8. This new version of the simulation uses three variables, pos, vel, and acc, which
correspond to the position, velocity and acceleration of the object. Furthermore, each cycle of a
repeat is handled by a separate procedure named tick. Each time that this procedure is executed,
the value of acc is added onto vel, vel is added onto pos, and the sprite is told to move forward an
amount equal to vel.
Comparing Figure Chapter 8. -8 with Figure Chapter 8. -7 we see that, when the Tick Model
is used, the contents of drive are somewhat simplified. This new version of drive begins by
initializing all of the variables, then repeats the tick procedure 23 times. Since the value of acc is
zero during this portion of the program, the sprite just steps forward at a constant speed. Then,
following these first 23 ticks, the acceleration is changed to -.5 and tick repeated 40 more times.
During each cycle of this second repeat statement the value in vel is thus decreased by .5, and the
sprite slows to a halt.
Once the Tick Model was introduced, students were directed to use it in all of their
simulations. To facilitate this, their programming environment was modified so that, when a
particular function key was pressed, the three variables and the tick procedure appeared on the
display as shown in Figure Chapter 8. -9.
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pos
vel
acc
850.000
0.000
-0.500
Data
Data
Data
tick
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Figure Chapter 8. -9. The Tick Model.
The above examples should serve to capture the essence of the practice of programmingphysics used in this study. Now I want to take a moment to reflect on some of the noteworthy
features of this practice. To begin, notice that the goal of each task is the production of a
simulation of some particular motion. For example, one task in my study gave students only the
following instructions:
In this task, you must imagine that the block shown is attached to a spring so that it oscillates back and
forth across the surface of a frictionless table. Make a simulation of this motion.
As in this example, instructions were typically very brief; they usually just described a motion in
simple terms and then asked students to create a simulation of that motion. Note that it would be
possible to create a practice of programming-physics based on tasks with very different types of
goals. For example, it is possible to write programs to obtain answers to textbook-like questions.
In addition, the Tick Model forces a very specific structure onto students’ simulations: It
essentially uses the numerical method described by Feynman. Alternatively, for example, it is
possible to create simulations of motion with constant acceleration by using this equation, which
gives the position as a function of time:
S = So + vot + 12 at 2
A simulation based on this equation might work by employing a time variable that is constantly
incremented. On each cycle, the value of this time variable would be substituted into the above
position equation and then the sprite would simply be placed at the corresponding location. Such
an approach has the benefit of producing a simulation that is more accurate than that produced by
simulations based around the Tick Model. However, it relies on the existence of an analytic
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solution in which to substitute values for the time. In contrast, like Feynman’s numerical approach,
the Tick Model is highly general and can easily be applied in cases in which it is not easy to obtain
an analytic solution. A more extensive comparison of the Tick Model to alternatives can be found
in Sherin (1992).
Sample Programming Episodes
In the previous section, I presented a number of examples of finished programs, without any
discussion of how those programs came to be written. Now I want to move into a discussion of
the construction process, and I will present some brief descriptions of programming episodes, taken
from the work of the university students in my study. In addition to giving the reader a feel for
what the construction process was like, these accounts will also provide an opportunity for me to
describe more of the tasks given to students, as well as further examples of the programs they
created.
As preamble, I will begin by attempting to evoke the setting in which these programming
episodes occurred. At any time, a single pair of students was in the laboratory, and the students
sat together at a single computer, usually with one student typing at the keyboard and the other
operating the mouse. As compared with the algebra-physics sessions, there was thus somewhat less
of a “performance” aspect to the sessions, since the students sat facing a computer monitor, rather
than standing in front of a whiteboard. Two cameras were positioned quite close to the students,
one behind and one to the side. During the sessions, I sat off to the side and operated the cameras
or simply observed.
Tim and Steve: Simulating the drop
I will open with an account of Tim and Steve’s work on what proved to be one of the least
difficult simulation tasks. Recall that, before they were even introduced to the Tick Model,
students were asked to program a simulation of a ball that is released from rest and simply drops
to the ground. In response, Tim and Steve, like all of the students in the study, wrote a program
that consisted only of a string of fd commands (refer to Figure Chapter 8. -6).
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POS
VEL
ACC
-539.000
-98.000
-9.800
Data
Data
Data
TICK
change vel
DROP2
setup
change
change
change
repeat
pos 0
vel 0
acc -9.8
10 tick
Doit
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Data
Figure Chapter 8. -10. Tim and Steve's simulation of a dropped ball using the Tick Model. In this initial
version, the ball goes up instead of down.
Later in the study, all of the students were asked to redo this simulation using the Tick
Model. This turned out to be rather straightforward for all the pairs, including Tim and Steve,
who wrote the program shown Figure Chapter 8. -10 in only about 3 minutes. Here’s a
description of what they did in those 3 minutes: First, Tim placed the tick model box in their
working area on the display. (Recall that, to do this, all he had to do was press a single function
key.) Then, with virtually no comment or discussion, he typed the program shown in the box
named drop2. This program first executes setup and initializes the variable—pos and vel to zero
and acc to -9.8—then it repeats the tick procedure 10 times.
This program would have produced an acceptable simulation except for one hitch: It turns out
that when this simulation was run, the ball moved upward instead of downward. The reason is
that, in this case, it happens that the setup command positions the ball at the top of the screen
with the sprite facing downward.13 Presumably, Tim and Steve had assumed that the sprite was
facing upward and, since it was shaped like a ball in this task, there was no easy way for them to
determine the sprite’s orientation. However, once the simulation was run this problem was easy to
recognize and diagnose. Tim observed simply that “It went the wrong way,” and then proceeded
13
The behavior of the setup command varied across tasks.
283
to modify the program so that the acceleration was initialized to a positive value, rather than a
negative. With this change, the program performed appropriately.
POS
VEL
ACC
539.000
98.000
9.800
Data
Data
Data
TICK
change vel
DROP2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
10 tick
Doit
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Data
Figure Chapter 8. -11. Tim and Steve's corrected simulation.
There were not many interesting twists and turns in the story of this episode; the task was
completed in short order, and with only a minor hitch. Nonetheless, there are a number of
important points to be made about Tim and Steve’s work on this simulation. First, it is
worthwhile to comment on the fact that this task seemed to be almost trivial, a fact that is largely
due to the power of the programming structures that had been taught to the students. Notice how
simple drop2 is; all it has to do is initialize the variables and then repeat the tick procedure the
desired number of times, a technique that Tim and Steve had seen in a number of example
programs. As we will see, this initialize-and-then-repeat structure appeared in almost every
simulation program.
A second point illustrated by this episode is the importance of feedback from the graphical
display. The first time that Tim and Steve ran their program, the ball moved in the wrong
direction, a clear indication that their program needed to be changed. In this case, the error
indicated was not of a very deep nature; it was due to an idiosyncrasy in how the setup procedure
had been defined for the task. In general, however, feedback from the graphical display played an
important and fundamental role in the development of simulations. In some cases this feedback
indicated that radical changes were required, while, in others, this feedback simply led students to
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adjust some of the quantities in their programs in order to make their simulations more
aesthetically pleasing. For example, in writing their drop simulations, students usually adjusted
the value of the acceleration and the number of repeats so that the ball stopped near the bottom of
the graphics box.
Tim and Steve: Adding air resistance to the drop simulation
In this next example, we trace Tim and Steve’s work into a more advanced and somewhat
more difficult task. In this task, the students were asked to include air resistance in their
simulation of a dropped ball, the very simulation that we just discussed above. The instructions
given were brief, but they included a couple of important hints and suggestions:
(a) Make a new simulation of the DROP that includes air resistance. (Hint: Air resistance is bigger if
you're moving faster.)
(b) Try running your simulation for a very long time. (The ball might go off the bottom and come in
the top; that's okay.) If you've done your simulation correctly, the speed of the ball should eventually
reach a constant value. (The "terminal velocity.")
Recall that the algebra-physics subjects were also given a task that involved air resistance. In
that task, the students were essentially asked to find how the terminal velocity of an object
depends on its mass. This task was moderately difficult for students since, in general,
introductory physics instruction does not take up the topic of air resistance very often; students
usually solve one or two problems pertaining to air resistance in their early physics careers. For this
reason, the students in my algebra-physics study needed to invent their own expressions to
account for the resistive effect of the air. Most commonly, they invented a relation for the force
due to air resistance, such as:
Fair = kv
The programming task placed students in a similar predicament. On a sheet of paper, Tim
and Steve drew an object with an arrow acting down from the bottom and an arrow acting up
from the top (refer to Figure Chapter 8. -12). They knew that the force acting down was equal to
mg—the force of gravity—but, like the algebra-physics students, they did not know immediately
how to find the force of air resistance. In the following discussion, this issue is worked out:
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W=mg
Figure Chapter 8. -12. Tim and Steve's paper diagram for the air resistance task.
Tim
…
Tim
Steve
Tim
Steve
Tim
Steve
Tim
Steve
Anyway. Well, when you first start out there’s no air resistance. I mean there’s no - but as
you go faster the little v- little vector here starts growing. [g. arrow at top of diagram].
So we have to have the size, is bigger if you’re moving faster. (4.0) (…)
I don’t think we really have to worry about that. It just decreases.
The faster you’re moving the bigger the - so what is this. So this is - this is something
times velocity.
Right.
Right?
Yeah, okay. So it’s a function of velocity, right?
Yeah.
Okay. So what should we do? Just make it like one times - point one times V or
something?
The above discussion is very similar to discussions in which the algebra-physics students
constructed equations for the force of air resistance. Like their predecessors in this study, Tim and
Steve came to the conclusion that the force of air resistance should be a constant times the velocity
of the falling object. With this decided, they created a variable named res and modified their
program to compute a value for this variable:
drop2
setup
change
change
change
repeat
Doit
pos 0
vel 0
acc 9.8
10 tick
POS
VEL
ACC
0
0
9.8
Data
Data
Data
RES
Data
TICK
change Res
vel * .1
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
At this point I need to pause and note that Tim and Steve have just done something that we
have not yet seen: They have modified the content of the tick procedure. In contrast, in all of the
previous examples, the students did all of their programming in a separate procedure, which then
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called tick. This approach will not work here because lines added to the drop procedure above the
repeat command only affect the initial conditions of the simulation; they cannot dictate changes
that must occur during the time of the motion. By putting their new statement inside of the tick
procedure, Tim and Steve have constructed their simulation so that the value of res will be
recomputed at each moment of the motion.
Since this issue will play a fairly important role in future discussions, I want to say a little more
about how this simulation differs from the examples we have already examined. In the previous
examples that used the Tick Model, the value of the acceleration was constant throughout the
motion. However, in the present task and some others that we will see, the acceleration varies over
the time of the motion. Tim and Steve were quite clearly aware of this issue. They knew that the
effect of air resistance would grow, causing the acceleration of the object to decrease through time
until it was zero.
Steve
We have to alter the acceleration, huh. Like decrease it in- with time?
…
Tim
Yeah, basically it’s um, as this gets bigger [g. arrow at top of diagram], the acce - the
value of eight- nine point eight gets smaller. Until it reaches zero and there’s no
acceleration at all.
In its present state, Tim and Steve’ program computes a value for the variable res, but this
variable does not affect any other aspects of the simulation. Thus, the students still needed to
modify the tick procedure so that the size of the air resistance was somehow taken into account in
the acceleration. Tim and Steve were aware of this, and they knew that they wanted a statement
that began with “change acc … ” inside the tick procedure. However, as they set out to write this
line, they realized that it wasn’t clear to them exactly where to place it among the other lines in
tick:
Tim
No we have to (then account) in the acceleration. So we have to change the acceleration.
Do we change the acceleration before we change the velocity? [g. change vel line in tick]
Or do we change acceleration after we change velocity? Which also changes the
resistance. [g. change Res line] Oh, I’m getting confused. What’s the order in which you
change things?
(3.0)
Tim
You see what I’m sayin’?
Steve
Uh:::.
Tim
They all like change each other. (Around).
[They laugh.]
Tim
So it matters which order you do it in, right?
Steve
I think the velocity’s first, right?
Tim
I don’t know.
Steve
Cause the first velocity is-
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Tim
no air resistance.
Steve
Yeah, and then it’s just this acceleration.
Tim
Is that true?
Steve
Well, not really, huh. (1.0) I don’t know, I’m confused too.
(3.0)
Tim
Well you have to change resistance because you change the velocity, because (0.5) the
velocity (1.0) - the new velocity is dependent upon the acceleration which is dependent
upon the resistance. So we should change resistance, then acceleration, then velocity.
Right? Does that make sense, or am I just talking nonsense.
In the above discussion, Tim and Steve decided on a specific ordering for the various lines that
must appear in the tick procedure: res would be changed first, then acc, then vel, then pos. They
next modified tick to reflect this decision. This required only the addition of a single new line to
compute the value of the acceleration.
TICK
change Res
vel * .1
Doit
change acc
acc - res
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
There are a couple of important points to be made concerning the line that Tim and Steve
have added to compute the acceleration. First, it appears that they are treating res like an
acceleration, rather than a force. This may be intentional, but it is more likely that they have
simply not considered the issue carefully. In fact, students in the algebra-physics portion of my
study were often ambivalent as to whether expressions they had written for “air resistance” should
be treated as a force or an acceleration. I will say more about this in a moment.
Second, there is a major problem with the statement that Tim and Steve have just written. A
correct expression for the acceleration here would subtract some type of air resistance term from a
term associated with gravity:
change acc
grav - res
Doit
Furthermore, the term associated with gravity should remain constant throughout the simulation
since gravity, too a good approximation, applies a constant force to a falling object. It is possible
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that this is what Tim and Steve had in mind; but rather then having a constant gravity term, they
have used the variable named acc, and this variable changes as the simulation runs.
It turns out that this substitution causes some striking behavior. When Tim and Steve ran their
program, rather than just moving downward, the ball oscillated up and down. The students found
this quite amusing.
Steve
Oh my god. Oh no:::.
[They both laugh.]
Tim
[Runs the program again.]
Steve
But when you drop balls it’s not gonna do that though dude!
It took quite a while for Tim and Steve to correct this problem and ultimately get a working
simulation. In the process they considered many alternatives, including the possibility that it
would be necessary for the program to explicitly check for a condition which signals that terminal
velocity has been reached. Eventually, with a little help from me, they modified their acceleration
line to be:
change acc
mass * 9.8 - res
/ mass
Doit
Doit
If rendered in algebraic notation, this statement might be written as:
a=
mg − Fair
m
When this modification was made, the program appeared as shown in Figure Chapter 8. -13.
Tim and Steve ran this program, and were quite pleased and excited by the results. They could
tell that their program was working properly by watching how the values of certain variables
changed during the simulation. In particular, they noticed that the acceleration died away to zero
and that the velocity leveled off at a constant value, thus indicating that a constant velocity was
reached.
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drop2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
1090 tick
Doit
POS
VEL
ACC
RES
MASS
450.800
19.600
0.0000
98.000
10
Data
Data
Data
Data
Data
TICK
change Res
vel * 5
Doit
change acc
mass * 9.8 - res
/ mass
Doit
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
redisplay
dot
Doit
Data
Data
Figure Chapter 8. -13. Tim and Steve's final air resistance program.
Finally, when Tim and Steve told me that they were satisfied with their program, I asked
them to explain why their simulation reaches a terminal velocity. Tim explained:
change Res
vel * 5
Doit
change acc
mass * 9.8 - res
/ mass
Doit
Doit
Tim
Well it reaches terminal velocity when the resistance is equal to the weight. And, the
weight’s constant. (6.0) We - we had it multiplies by the acceleration [g. mass * 9.8] which
is the change in velocity which is a different acceleration than the one you use. (…)
(6.0)
Tim
Well, basically, as this - this [g. change res line] eventually gets close to this number [g.
mass * 9.8] and the difference [g. [mass * 9.8 - res]] becomes (next to nothing.) And once
that happens there’s no more change in velocity. Velocity becomes constant.
…
Tim
Well, whatever force vector’s going down, that um, cause it to, um, (3.0) accelerate. And,
as you - when you accelerate, the resistance gets bigger. So, it kinda like - so, every time
you accelerate a little more, the resistance gets a little bigger.
Bruce
Hm-mm.
Tim
So eventually, um, this grows and this grows and eventually it’ll reach a point where
they’re equal.
Tim’s explanation in the above passage is based around only two lines in the program, the two
lines they have added to the tick procedure to compute the resistance and acceleration. What he
tells us is that, as the ball falls, the value in the resistance variable grows. Furthermore, as it grows,
the value computed for the acceleration becomes less and less until, eventually, the resistance and
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gravity terms cancel and the acceleration is zero. “So eventually, um, this grows and this grows
and eventually it’ll reach a point where they’re equal.”
With Tim’s above explanation, this episode wound down to a conclusion. Before wrapping up
this chapter, I want to highlight a few of the more important points to be made concerning Tim
and Steve’s work in this episode. This will help to prepare the ground for the more thorough and
principled analyses in upcoming chapters.
The power of the Tick structure
To begin, let’s look at the structure of Tim and Steve’s simulation program. First, as in the
Drop simulation, the initialize-and-then-repeat structure is evident in this program. Furthermore,
in this episode we also learned that any quantities that change through the time of the simulation
must be recomputed inside the tick procedure. In particular, in cases in which the acceleration is
not constant, a new value of the acceleration must be computed on each cycle of the repeat.
program
setup
change pos #
change vel #
repeat ## tick
Doit
POS
VEL
ACC
450.800
19.600
0.0000
Data
Data
Data
TICK
change force ??
change acc
force / mass
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Figure Chapter 8. -14. A programming structure for handling cases of non-constant acceleration.
Now that we know how to deal with cases of non-constant acceleration, the power of this
technique becomes evident. Using the basic structure embodied in Tim and Steve’s simulation
and generalized in Figure Chapter 8. -14, we can create a wide range of simulations. The template
shown in this figure essentially uses the initialize-and-then-repeat structure: The position and
velocity are initialized, and then tick is repeated some number of times. In addition, two key lines
have been added at the top of the tick procedure; the force is computed and then substituted to
find the acceleration. Thus, if we know the force on an object and the initial values of position and
velocity, we can simulate its motion.
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This is precisely the power and generality that appealed to Feynman and Redish. Compare the
program produced by Tim and Steve with what is possible in algebra-physics. While some tasks
are comparatively easy for students in algebra-physics, such as computing the terminal velocity,
others are considerably more difficult. For example, it is harder to find how much time the ball
takes to reach terminal velocity, or how far the ball falls in a given time. To answer these questions
one must solve a differential equation, a task that ranges from difficult to extremely difficult,
depending on the precise function used for the force of air resistance, and the mathematical
expertise of the student.
In contrast, writing a simulation that includes air resistance is not that much harder than
simulating a drop without air resistance. Tim and Steve’s simulation only contains two additional
lines of programming. Furthermore, this simulation can be used to answer even those questions
that are so difficult in algebra-physics. For example, the distance that the ball falls in a given time
can simply be read out of the pos variable, as long as one knows how to deal with unit
considerations. Though the result is only approximate, it can easily be refined to any desired
accuracy, with only the time available as a limit.
Some similarities to algebra-physics
Another noteworthy feature of this episode is the fact that it contains construction and
interpretation incidents that are strongly reminiscent of those we encountered in algebra-physics.
The most obvious of these is Tim and Steve’s construction of an expression for air resistance.
Given the similarity of this construction incident to incidents in algebra-physics, it seems likely
that we will be able to explain this event much as we explained algebra-physics constructions. In
addition, notice that Tim’s final interpretation of the program is also reminiscent of PHYSICAL
CHANGE
interpretations from algebra-physics.
However, there were also cases in which similarities to algebra were actually the source of
problems for students. In fact, I will argue later that the difficulty that Tim and Steve had in
correcting their expression for acceleration was, in part, due to confusion traceable to overlap with
algebra-physics. As I will attempt to show, the relationship between programming statements and
algebraic expressions may sometimes be a little subtle.
Programming is tyrannical in enforcing ordering
Finally, recall that Tim and Steve essentially discussed what order the various quantities—res,
vel, and pos—should be changed. In programming, this seems like a very sensible question, and
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answering this question is essential to the completion of the task. However, as I will argue later,
not only is it not necessary to answer this question in algebra-physics, the question hardly even
makes sense. The general point here is that programming is tyrannical in forcing an ordering on
how we see the physical world. One of the most important tasks of the succeeding chapters will be
to explore the consequences of this enforced ordering.
A Preview of the Comparison
In the preceding sections, we took our first steps into programming-physics. The episodes
presented were designed to prepare the ground for the discussions and examples that follow, and
to start us toward working through some of the noteworthy features of programming-physics.
Ultimately, my goal is to complete the project, laid out in Chapter 1, of describing how the
understanding associated with programming-physics would differ from physics-as-understood-by
physicists.
This task—which constitutes the major objective of this research—will be taken up in the next
three chapters. I will begin in Chapter 9 by describing how the Theory of Forms and Devices can
be extended to describe programming-physics. This will be followed in Chapter 10 by the results
of a systematic analysis of the programming-physics data corpus in terms of forms and devices.
Finally, the analyses in Chapters 9 and 10 , the work of Part 1, and a few additional ingredients
will be drawn together to produce a comparison of programming and algebra-physics in Chapter
11.
To conclude the present chapter, I will present a brief overview of this comparison, and a little
of what I hope to capture.
The comparison based on the theory of forms and devices
The Theory of Forms and Devices will provide the major basis of my comparison and most of
the work of the succeeding chapters be will aimed in that direction. The first step in this portion
of the comparison will be to extend the theory to programming-physics. This will involve working
through whether and in what manner the known forms and devices can be applied, and also
determining where new forms and devices must be posited. As I hinted in the last episode, there
will be some fundamental similarities to build upon, and many of the forms and devices seen in
algebra-physics will have similar counterparts in programming-physics. Nonetheless, we will see
that programming-physics requires the use of some distinct forms and devices, and it does not
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draw on some of those that appeared in algebra-physics. Finding these differences will constitute
the core of my comparison of the two practices. In addition, as for algebra-physics, I will attempt
to measure the frequency with which these various knowledge elements appeared in
programming-physics.
I do not want to give away too much more of the results at this point, but I do want to at least
bring the reader up to my own jumping off point for this endeavor. The Theory of Forms and
Devices was, in part, designed to capture two informal conjectures concerning important
differences between programming-physics and algebra-physics. Ultimately, the theory allowed
these conjectures to be stated more formally, but here I will just briefly present these conjectures
in their original, informal terms.
Programming-physics privileges a different intuitive vocabulary. The first of these informal
conjectures is based in the notion that different symbol systems allow different “intuitive
conceptions” to be more or less explicitly represented, and that those that are directly supported
become privileged in the associated practice of physics. Because these are the intuitive conceptions
that can be represented within the symbolic practice, we must funnel our work through these
privileged conceptions in order to write new expressions. And when we look at existing
expressions, these intuitive conceptions are what we will see. The upshot of this is a more
prominent role for the privileged intuitions within physics understanding. My analysis in terms of
symbolic forms is, in part, an attempt to capture this “informal conjecture.”
This first informal conjecture also included some specific guesses concerning what sort of
intuitive conceptions would be supported by each of the symbolic practices. Some of these guesses
were based on the observation that, in contrast to algebraic statements, programming statements
are directed and rigidly ordered. In describing programming statements as “directed,” I am trying
to capture the fact that many programming statements force an asymmetry on the programmer.
To illustrate, compare the following algebra and programming statements:
F 1 = F2
change force1 force2
The algebra statement specifies a symmetric relation; it says that F2 is equal to F1, just as much as
it says that F1 is equal to F2. In contrast, the programming statement is an asymmetric, directed
relation; it says to put the value of force2 into force1.
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Furthermore, every line in a program must be ordered. There is substantial flexibility in how
algebraic relations can be arranged on a sheet of paper but, as Tim and Steve discovered,
programming enforces a strict ordering on the statements that appear.
Because of the directed and ordered properties of programming, I speculated that
programming would support some intuitions that are not well supported by algebraic notation.
For example, since causal intuitions require an ordering of the world’s influences and effects, I
hypothesized that these intuitions might be more strongly drawn upon in programming-physics.
Furthermore, I conjectured that intuitions based in symmetric relations, such as those relating to
balance and equilibrium, would appear less in programming physics. In the next chapters, these
informal conjectures will be made more precise and we will see the degree to which they are born
out.
Programs are easier to interpret than equations. The second informal conjecture that my analysis
attempted to capture was the notion that programs might be easier to interpret than equations.
This conjecture derives from the observation that programs can be “mentally run” to generate
motion phenomena (Sherin et al., 1993). The idea is that a programmer can do what a computer
does and step, line-by-line, through a program, a procedure that appears relatively
straightforward. In contrast, when I began this work, I was not aware of any obvious candidates
for a universal technique that allows the generation of an associated motion phenomenon from an
equation.
In the chapters that follow, we will investigate how easy or difficult it really was for students to
interpret the sort of programs that appeared in this study. As we will see, the analysis of the
devices involved in programming-physics can go toward answering this question. I will argue that,
while programming does involve some new devices that can be roughly associated with the mental
tracing of a program, the interpretation of a program is still a complex business, and it draws on
many of the same representational devices encountered in the analysis of algebra-physics.
Other fundamental differences
The form/device analysis, described just above, is suited to drawing out a certain class of
differences between programming-physics and algebra-physics. This class of differences relates to
how people interact with and see meaning in the symbolic expressions that are employed in each
practice. Of course, there are many profound differences that will not be captured by this analysis,
and many alternative ways of talking about the differences that I will uncover. For example, in
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working through the form/device analysis, there will be no place in which it is clearly appropriate
to comment on the simple fact that what a student does in each practice is very different. In
algebra-physics, a student manipulates equations to solve problems and derive new equations. In
contrast, in the practice of programming-physics employed in this study, a student writes
simulation programs. Furthermore, these differences in what one actually does in each practice are
tied to some fundamental differences between programming languages and algebraic notation.
Here I will mention just two very important differences:
Algebra has rules of manipulation. Algebraic statements can be manipulated using rules that act on
the symbols in the expressions to produce new, correct statements. For example, given an
equation, we can manipulate the equation to solve for an unknown. In contrast, programming
languages do not come equipped with rules for manipulating the symbols in a program to produce
new programs.
Programs can be run on computers. Programming statements can be typed into a computer and
then run to produce various types of output, including dynamic displays of motion that are
generated by the program.
I will, to a minor extent, comment on these sorts of differences when, in Chapter 11, I
compare programming-physics and algebra-physics. Many of the related issues will be collected
under the rubric of what I call “channels of feedback.” That discussion will be treated as largely
separate from my analysis in terms of forms and devices, but, in the process, I will show how both
sets of issues can be fit within a single, broader view.
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Chapter 9. The Construction and Interpretation of
Simulation Programs
In Part 1 of this document, we explored the structured and meaningful world of equation use in
physics. I argued that, for a physicist and physics student, a sheet of equations is not just a
collection of individual symbols, manipulated according to a set of mechanical rules. Rather, I
attempted to show that initiates in physics can see certain types of meaningful structure in these
symbolic displays, and that they possess a repertoire of interpretive strategies. My account of these
capabilities took the form of the positing of two specific types of knowledge, what I called
symbolic forms and representational devices.
The purpose of this chapter is to perform a similar analysis for programming-physics. As for a
page of equations, we want to know what sorts of structure an initiate in programming-physics
sees in a program that appears on a computer display or that is written on a sheet of paper. This
means adapting and applying the Theory of Forms and Devices to programming-physics. We will
see that, with no change to its basic outline, the theory can be applied directly to programming; in
fact, it turns out that there are even counterparts for many of the specific forms and devices that
we encountered in algebra-physics. However, there are some notable absences and some
programming-specific forms and devices for which there are no similar counterparts in algebraphysics. Our job here will be to see where and how similar forms and devices appear in
programming-physics, and where new and somewhat different elements must be posited.
Of course, in moving from algebra to programming we are setting out on less sturdy ground.
Recall that the ability to see meaning in algebraic expressions is learned. Algebraic expressions do
not radiate meaning in a way that is obvious for any person that looks at them, and physics
students are certainly not born with the ability to recognize the type of structure that physicists see
in equations. Rather, equations were meaningful for the students in my study because they had
spent at least two semesters working with physics equations, listening to physics instructors talk
about these equations, and talking to other students with and about equations.
The situation in this examination of programming-physics is quite different. We do not have a
population of students that have spent several semesters in initiation into a practice. Instead, the
programming-physics students had only a very brief introduction to the programming of
simulations—between one and two hours. This observation has important implications for what
we can expect from this analysis and how we must understand the results. First, because their
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experience is limited, the programming-physics students will have less of an ability to see structure
in a program. Second, more than in the examination of algebra-physics, I will be limited in how
much I can claim to be describing a stable and uniform practice. Because their shared background
is limited, at least with regard to simulation programming, there may be more variation across
students. Furthermore, the behavior of individual students may be less stable through the time of
the study since I am observing them during the early stages of their learning of programmingphysics.
Nonetheless, for several reasons, the situation is not as problematic as one might think. As we
shall see, it is possible for students to borrow heavily from their experience in algebra-physics when
understanding simulation programs. This adds greatly to the possible structure that students can
see in programs. Furthermore, it is not the case that students received no instruction. All of the
students were taught to use a set of simple but very powerful programming structures, and ways
of understanding these structures were implied in the instruction. The result is that the students in
this study were actually able to see quite a lot of structure in the programs that they wrote.
On Line-Spanning Structure
We are now ready to begin the project of applying the Theory of Forms and Devices to
programming. The first step will be to identify the places where programs are constructed and
interpreted much like algebraic expressions, and to identify where construction and interpretation
are very different in programming. When this is done, we can focus our attention on the places of
divergence. In fact, I will argue that there is both substantial overlap and interesting divergence.
To begin this discussion, I need to first take a moment to reflect on the types of meaningful
structure that we observed in algebra-physics. Although this point was not emphasized, my
description of algebra-physics always focused on individual equations or portions of individual
equations. I talked about how a given (single) equation was constructed or how a given equation
was interpreted. This implicitly presumed that, for my subjects, there were not symbolic forms
that span equations—a position that I arrived at after some analysis of my data corpus.
Now, in turning to programming, we are faced with a similar issue of what level to focus our
attention for analysis. One way to approach this issue is to ask: What gets interpreted? Is an entire
program always interpreted? Are individual lines of programming interpreted? Or are there more
arbitrary portions of programs that are interpreted?
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In programming, I will indeed claim that there are forms that span lines of programming.
Then, what I hope to show is that it is in within-line structure that a great overlap between
programming and algebra appears. And the locus of substantial divergence will be the linespanning structure in programming.
The claim that there are no line-spanning symbolic forms in algebra-physics (as it appeared in
my data corpus) is a very significant claim, and one that I believe readers may find controversial.
The primary argument I will make in favor of this position is that, although students certainly do
associate meaning with the arrangement of algebraic expressions—and thus with line-spanning
structure—this is not the same sort of meaningful structure that is defined to be involved with
symbolic forms. Thus, my argument comes down to a clarification of what counts as a symbolic
form. I will explain just a little further here.
When working on an algebra-physics task, students often juxtapose multiple algebraic
expressions, and these statements are usually arranged in some manner. For example, it is
common to arrange statements so that they reflect the sequence in which the statements were
derived. Furthermore, it is certainly the case that this ordering is registered by students, in the
technical sense I introduced in Chapter 2. This is a plausible assertion because the ordering of
statements plays a role in the solution process. In addition, because it reflects the solution process,
the arrangement of statements is meaningful, in a certain sense, for students.
However, this is not necessarily the sort of meaningful structure that is associated with
symbolic forms. In forms, there are conceptual schemata that include physics-related entities (e.g.,
physical quantities) and relations among these entities. These schemata are then connected to
symbol patterns, which parse an arrangement of symbols in some manner. In this way, we can
conceptualize a physical circumstance in terms of the conceptual schemata, and then reflect this
conceptualization in an arrangement of symbols.
Though the arrangement of algebraic equations did carry meaning for the students in my
study, it was meaning of a different sort. For example, in order to explain how they solved a
problem, it was common for students to follow the lines they had written, explaining what they
had done in each step. Thus, the arrangement of lines can reflect the relationship between
components of a solution process—or even the relationships among components of an argument
or explanation. Although these are meaningful relationships, they are not relationships among
physics relevant entities.
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It is important to note that it need not have been the case that this was the only type of
meaningful line-spanning structure used by students. In fact, some of the specific line-spanning
symbolic forms that I will identify in programming could have had counterparts in my algebraphysics corpus. When I describe these programming forms, I will comment on where and whether
they could have had algebraic counterparts. However, when I do describe a hypothetical algebraic
counterpart, I think it will be clear to readers familiar with algebra-physics that the associated
behavior is somewhat unusual, and not typical of algebra-physics. Thus, I hope that looking at the
line-spanning forms in programming, and contrasting them with what is done in algebra, will help
to clarify the claim that there are no line-spanning forms in algebra-physics, as it is traditionally
practiced.
One important clue that there is really something different going on in the programmingphysics corpus will be some episodes in which students have long and explicit discussions
concerning how the lines in a program should be ordered. Furthermore, in these episodes,
students do not make their decisions based solely on the behavior of the simulation; they choose
the ordering to reflect a conceptualization of how things are in the world. In contrast, I did not see
extended discussions about how algebraic statements should be ordered. Instead, to the extent
that there was an ordering, this usually just fell out as part of the solution process.
It is not hard to make guesses about the origins of this difference. As I discussed at the end of
Chapter 8, one important difference between programming and algebraic notation is that
programming is rigid in enforcing an ordering on the lines that are written. Although people do
tend to arrange algebraic expressions in certain traditional and useful ways, programming is more
strict and absolute in requiring that statements be ordered, and that ordering is meaningful in one
particular way. Furthermore, there are even ways of arranging statements into groups and giving
those groups names.
Again, I believe that this is a controversial point. But I do hope that skeptical readers will
withhold final judgment until they see the type of meaningful structure that students associated
with programs. In discussing the forms and devices of programming in the upcoming chapters, I
will add to this argument as appropriate.
Where Programs Can Be Interpreted Like Equations
The point of the above discussion was to warn the reader that, in programming, line-spanning
structure is going to be important. While the entirety of a program or procedure is rarely
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interpreted at once, portions of a program containing multiple lines are often interpreted, and
forms can span these lines. As I will try to show, there are a variety of ways that portions of a
program are selected for interpretation, including groups of lines as well as individual statements.
However, this said, it does turn out that one of the most common sub-units of a program that
students interpret is an individual line of programming. Furthermore, it is here that the
interpretation of programs often resembles the interpretation of algebraic expressions. In this
section, I want to spend some time on the construction and interpretation of individual lines of
programming; in particular, I want to look at a range of cases in which single programming
statements are treated in a manner very similar to algebraic equations.
We already encountered some relevant examples when, in Chapter 8, we examined Tim and
Steve’s construction of a simulation of a dropped ball with air resistance. Recall that, using their
original simulation of a dropped ball as a starting point, Tim and Steve began by adding a single
line to compute the value of a resistance variable:
change Res
vel * .1
Doit
When writing this line they commented:
Tim
The faster you’re moving the bigger the - so what is this. So this is - this is something
times velocity.
Steve
Right.
Tim
Right?
Steve
Yeah, okay. So it’s a function of velocity, right?
Tim
Yeah.
Steve
Okay. So what should we do? Just make it like one times - point one times V or
something?
The point here is that this single line of programming was constructed and interpreted in a
manner that strongly resembles the construction and interpretation of similar expressions in
algebra-physics. We encountered the following examples in Chapter 3:
R = µv
Mark
So this has to depend on velocity. [g. R] That's all I'm saying. Your resistance - the
resistor force depends on the velocity of the object. The higher the velocity the bigger the
resistance.
_______________
Fair = kv
Bob
Okay, and it gets f-, it gets greater as the velocity increases because it's hitting more
atoms (0.5) of air.
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The account of Tim and Steve’s programming statement in terms of forms and devices closely
parallels my earlier accounts of these algebra-physics constructions. Let’s begin with the role
played by the DEPENDENCE form in these construction efforts. Recall that the dependence form is
associated with a symbol pattern that specifies only that a given symbol—in this case, the
velocity—appears somewhere in an expression.
DEPENDENCE
[…x…]
In his above statement Mark says “this has to depend on the velocity,” evidence that the
DEPENDENCE
form is engaged here. There is also evidence for DEPENDENCE in one of Steve’s
statements above; he says that the resistance is “a function of” velocity. As noted in Appendix D,
terminology that employs the phrase “function of” is strong evidence for the
DEPENDENCE
form.
The PROP+ form is also a key element in the construction of the algebraic expression for the
force of air resistance.
PROP+
…x… 
 … 
Evidence for PROP+ can be found in both Mark and Bob’s statements. Mark says “The higher the
velocity the bigger the resistance,” and Bob says “it gets greater as the velocity increases.”
Similarly, in constructing the programming version of this expression, Tim said “the faster you’re
moving the bigger the [resistance].”
Two additional forms are also involved in each of these constructions. First, the IDENTITY form
is implicated since these expressions are all commonly understood as specifying how to compute
the value of a single unknown quantity.
IDENTITY
x = […]
And COEFFICIENT is implicated in the writing of the factor that multiples the velocity in each of
these expressions.
COEFFICIENT
[x
]
There is actually a minor difference here between the programming and algebra cases that
bears mentioning. Notice that, in the programming case, Tim and Steve wrote a specific
numerical value for the coefficient, rather than a symbol like ‘C’ or ‘K.’ This turns out to be
indicative of a rather consistent difference between programming and algebra-physics. As I will
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discuss later, it was somewhat more common for the programming students to work with specific
values for quantities, rather than with those quantities represented by letters.
Finally, the representational device involved in Tim and Steve’s programming construction is a
straightforward counterpart of the device in the algebra-physics constructions. Note that each of
the above statements describes a change through the time of the motion. For example, Bob says
“it gets greater as the velocity increases.” Thus we can say that the PHYSICAL CHANGE device is
involved.
The existence of this particular overlap between programming-physics and algebra-physics is
not too surprising. There are lines of programming that are straightforwardly related to algebraic
expressions, and it is very plausible that these lines would be constructed and interpreted in a
manner very similar to that of algebra-physics expressions. These observations are not surprising
particularly given the fact that my student subjects had significant experience in algebra-physics
upon entry in the study.
DROP2
setup
change
change
change
repeat
Doit
pos 0
vel 0
acc 9.8
1090 tick
POS
VEL
ACC
RES
MASS
450.800
19.600
0.0000
98.000
10
Data
Data
Data
Data
Data
TICK
change Res
vel * 5
Doit
change acc
mass * 9.8 - res
/ mass
Doit
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Data
Figure Chapter 9. -1. Tim and Steve's simulation of a dropped ball with air resistance.
This sort of adaptation of algebra-physics knowledge can account for a surprising percentage
of the interesting work in programming-physics. For illustration, consider the final version of Tim
and Steve’s air resistance simulation, which is reproduced in Figure Chapter 9. -1. As I mentioned
in Chapter 8, this program is nearly identical to their simulation of a dropped ball without air
resistance, with the primary difference being the two added lines at the top of the tick procedure.
The first of these lines is the statement that computes the value of the resistance force, which I
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discussed just above. The second new line uses this resistance to compute the acceleration at each
moment of the drop.
change acc
mass * 9.8 - res
/ mass
Doit
Doit
This second line is also analogous to some algebraic expressions that I discussed in Part 1. Recall,
for example, the extended discussion in Chapter 3 of Mike and Karl’s
MOMENT
COMPETING TERMS/GENERIC
expression for the acceleration of a dropped ball under the influence of gravity and air
resistance:
a(t ) = −g +
f (v)
m
Clearly, there is a great similarity between this equation and Tim and Steve’s programming
statement.
Thus, it seems that some of the core elements of Tim and Steve’s work on this fairly difficult
simulation task can be accounted for with forms and devices that are straightforward counterparts
of those encountered in algebra-physics. Later, I will argue that there are some important aspects
of this and similar episodes that cannot be accounted for in this way. For instance, the placement
of these two lines inside the tick procedure, as well as their ordering within tick, needs some
explanation. Nonetheless, the importance of knowledge that carries over from algebra-physics is
very clear.
I want to take a moment to narrow down where, within programming, algebra-like
constructions and interpretations are localized. In the above examples, the interpreted statements
all used the change command. However, it turns out that algebra-like expressions are not limited
to change statements. More generally, any command that takes a numerical argument can have
one of these algebra-like expressions in the place of that argument. For example, the fd command
takes a single number as an argument, and that argument may take the form of an algebra-like
expression. Similarly, the first argument to repeat specifies the number of times to repeat a series
of commands. However, the vast majority of algebra-like expressions did appear as part of a
change command, as in the above examples.
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It is important to note that the first argument in a change statement must be a single variable,
it cannot be an expression. For this reason, the algebraic counterparts of statements involving the
change command are limited to expressions of the form
x=[…],
the class of expressions that can be interpreted in terms of the IDENTITY form. Notably absent
from this class of expressions are many that would often be interpreted in terms of
BALANCING ,
such as those that state the equality of forces. An example is the equating of forces in the Mass on
a Spring problem solved by the algebra students: kx = mg. This is not to say that we could not
devise a means of working expressions of this sort into simulation programs. For example, if
students were taught to use conditionals (e.g., “if” statements) then the expressions which specify
conditions might very well be of this form. I will say more about this in Chapter 11.
More examples of algebra-like interpretations: The Shoved Block simulation
We have seen that a student’s ability to see meaning in an algebraic expression can be
borrowed and applied to the understanding of a class of single programming lines, particularly
lines involving the change command. At this stage, I could proceed methodically through each of
the forms and devices that we encountered in algebra-physics, providing an example episode in
which these forms and devices are applied to single programming statements. But these episodes
would be so similar to their algebraic counterparts that such an exercise would do little more than
replicate the examples and arguments given in Chapters 3 and 4. For this reason, I will not provide
programming examples of every form and device that appeared in algebra-physics and which also
appears in programming-physics. However, I do want to present just a few more examples in
order to illustrate how these algebra-like interpretation and construction episodes fit within the
context of student work on a programming task.
In each of the examples I will present here students were working on a programming
counterpart of the “Shoved Block” problem. Recall that in the algebra-physics version of this task,
the subjects were asked to consider a situation in which a heavier and lighter block, both resting on
a table with friction, are each given a hard shove so that the blocks start off quickly but eventually
come to rest. The problem then was to figure out which block travels a greater distance, given that
they start off with the same initial velocity.
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In the corresponding programming task, the students were asked to program a simulation of a
the shoved block motion. Although these written instructions do not specify that the block is
always given the same initial velocity, this was clarified verbally for the students:
Imagine that a block is resting on a table with friction. Program a simulation for the case in which the
block is given a short, hard shove.
How does the motion differ if the block is heavier? Can you show this in your simulation? Modify
your simulation if necessary.
As a first step in this task, the students usually began by creating a simulation in which the
block simply slows down at a constant rate. Greg and Fred’s simulation, shown in Figure Chapter
9. -2, is a typical example. In the procedure named shove, their variables are initialized so that the
velocity is positive and the acceleration is negative, and then the tick procedure is repeated 150
times. Notice that this simulation program is nearly identical to the simulations of a drop we
encountered in Chapter 8, with the single exception that the acceleration is negative instead of
positive. In short, this is a straightforward application of the initialize-and-then-repeat structure
that the students had learned to implement.
Data
shove
setup
change
change
change
repeat
Doit
pos
vel
acc
150
0
12
-.08
tick
POS
VEL
ACC
894.000
-0.0000
-0.080
Data
Data
Data
TICK
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Figure Chapter 9. -2. Greg and Fred's simulation of a shoved block that slides to a halt.
But Greg and Fred’s simulation only addresses the first part of the task. In the second part,
the students were asked to modify their simulation to show how the motion would differ if the
block were heavier. As I discussed in Part 1 of this manuscript, it turns out that both blocks travel
the same distance. But, this answer was not obvious to the students. As in algebra-physics, most of
the students began with the notion that a heavier mass should slow down more quickly, and hence
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travel a shorter distance, because of the increased frictional force. Here, I will focus on programs
that implement this initial, not entirely correct, intuition.
Greg and Fred were no exception in this regard. They began with the algebraic expression for
the force due to friction F = µN, writing this equation on a sheet of paper. Then, by looking at this
equation, they concluded that the frictional force would be greater if the block was heavier.
Fred
So, increasing the mass is just gonna increase the frictional force.
Greg
[nods]
Fred
The resistance.
Greg
Uh. By- What? Linearly?
Fred
Sure.
Greg
Yeah. Linearly.
With this information in hand, they went on to modify their program to take into account the
value of the mass. After making a variable named mass, they altered the line that initializes the
value of the acceleration. In this new version, rather than simply placing a constant value into acc,
they computed a value based on the mass:
shove
setup
change
change
change
repeat
pos
vel
acc
150
0
12
-.08 - mass
tick
Doit
Greg
So we can say mass data box is equal, uh, to acceleration plus - or acceleration minus a
certain (1.0) number that we give it. The mass. Say, the mass or something.
Fred
You're just saying that we just, um, decrease the acceleration or reduce the magnitude of
the acceleration by a fixed amount.
Greg
Yeah. No, I'm just makin' it - tryin' to make it organized, so we know that it's because of
the mass, you know.
…
Fred
So basically you're saying this is an additional acceleration factor.
Greg
Hm-mm.
Fred
Provided by the increased mass.
Greg
Hm-mm. (1.0) Does that make sense? Kind of? So, if you make this bigger [g. mass
variable], our mass, it's gonna decrease it even more. That makes sense, right?
This is precisely the type of form-device construction act that I focused on in my examination
of algebra-physics, and two forms play a prominent role in the construction of this line. First, the
BASE +CHANGE
form is prominent. Recall that -.08 was the value of the acceleration that Greg and
Fred used in their original simulation. The increasing of the mass is seen as leading to a change
from this base value. Fred says: “… we just, um, decrease the acceleration or reduce the
magnitude of the acceleration by a fixed amount.” In addition, the
307
PROP+
form is implicated
here: “So, if you make this bigger [g. mass variable], our mass, it's gonna decrease it even more.”
Also notice that the CHANGING PARAMETERS device is clearly evident in this final statement.
There are a couple of points to be made about this construction episode. First notice that the
variable that Greg and Fred have named “mass” is really playing the role of the change in mass,
since a value of zero simply produces the base case. Greg and Fred realized this shortly after the
above episode:
Greg
Yeah. I'm tryin' to make it, like, as simple as he did it here, see [g. pos, vel, acc variables]
So if you just input a mass, then it'll just do it for you.
Fred
But then remember that this point oh eight was- it had a mass to begin with. So you're
doing, like, we're adding more weight [g. mass variable] on top of the first mass.
(1.0)
Greg
Uh:::, yeah. Well, that's what he wanted. A bigger mass, right? [g. instructions] [reads] If
the block is heavier.
Fred
Okay, so, that's what we'll pretend. That first we have a weight, and we're adding more
weight on top of it.
Following this realization, they altered this key line of programming to be in accordance with a
more standard interpretation of the mass variable:
change acc
- mass
Doit
A second interesting point concerning this episode relates to the role played by the equation
F=µN. Notice that, once Greg and Fred had written this equation, they could have essentially just
transcribed it into their program. For example, using the fact that the normal force is N=mg, they
could have written:
change force
- mu * mass * g
Doit
change acc
force / mass
Doit
Or simply:
change acc
- mu * g
Doit
Instead, rather than simply transcribing an equation into their program, they took a detour
through the realm of symbolic forms. First they interpreted the equation F=µN in terms of forms,
then they wrote a programming statement that was in accordance with those forms. This
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observation is interesting because it hints very strongly that forms are a central element of the
cognitive currency in both programming-physics and algebra-physics.
I will return to Greg and Fred’s work on this task shortly, but I want to digress for a moment
to point out that most of the other subject pairs took a very similar route in writing their shoved
block simulations. For example, Ned and David began by commenting that a heavier block
would slow down more quickly:
David
[reads] How does the motion differ if the block is heavier? Can you show this in your
simulation?
Ned
It's just more - it's (just) - the deceleration would be greater.
Then they wrote the following line to compute the acceleration from the mass:
change acc
mass / -100
Doit
David
Uh just the (st…) the acceleration is dependent on the mass. And the greater the mass,
the greater, the greater the acceleration is.
Bruce
Uh-huh.=
David
=To the left.
Bruce
Uh-huh. I see. The greater the acceleration is to the left. Right. (4.0) And where did the
divided by the slash minus a hundred come from?
Ned
That - we just chose that arbitrarily.
David
Actually, it's cause we, we used a hundred kilograms as a fir- as, as the first mass that we
used.
Bruce
Uh-huh.
David
And we, and we had that as being minus one acceleration in the first equation, so we just
used it.
Bruce
Hm-mm.
David
It didn't really mean anything.
Bruce
// [Points to screen.]
David
// I guess, uh, - (…)
Bruce
Go ahead.
David
I guess that could be like the coefficient of friction or something.
As in Greg and Fred’s work, the PROP+ form is implicated in Ned and David’s construction:
“And the greater the mass, the greater the acceleration is.” The DEPENDENCE form also appears
clearly in David’s first statement “the acceleration is dependent on the mass.” Interestingly, the
factor of -100 in the denominator seems to be understood in two different ways by Ned and
David. In some instances, it is treated as a COEFFICIENT that scales the mass in computing the
acceleration. In other instances, it appears to be understood as corresponding to the original value
they used for the mass. In that case, the
RATIO
form is implicated, and the acceleration is
computed from the ratio of the current mass and a reference value for the mass.
309
In writing their statement to compute the acceleration, Tim and Steve also employed such
reference values.
change acc
st_acc * enter_weight / st_wgt
Doit
In this statement, st_wgt and st_acc were the “standard weight” and “standard acceleration.” The
variable named enter_weight was the current value of the weight, which they presumed would be
entered each time by a user of their simulation. So, as in Ned and David’s statement, the ratio of
the current mass to a reference is important.
In all of the above episodes, the students were composing an expression that specified how the
acceleration of the shoved block depended on the mass. So, what we are seeing here is that there is
at least one specific niche in this programming-physics task where students engaged in some very
algebra-like construction and interpretation. Interestingly, it turns out that there was also a second
niche in this task where algebra-like expressions were written. This second niche arises because, if
the acceleration is altered and the initial velocity remains unchanged, then a different number of
ticks will be required to slow precisely to a halt. To account for this, Greg and Fred’s final shove
procedure appeared as follows:
shove
setup
change
change
change
change
pos 0
vel 12
mass .08
acc
- mass
Doit
repeat
150 - mass * 2
tick
Doit
Doit
In this new version of the program, the repeat statement has been modified so that the number of
cycles of the repeat will vary depending on the value of the mass. In particular, if the mass is
larger, then the number of repeats will be less. This is in agreement with the assumption that a
larger mass should experience a greater deceleration and thus stop more quickly. Greg and Fred
explained:
Fred
So, we have this idea that increasing the mass will, um, decrease the acceleration and
also decrease the number of steps it'll go. Right? Greg?
…
Greg
That was- that was our idea that - that, um, [g. a range on the path] a mass, if you have a
lighter mass it'll travel this distance. But if you have a heavier mass, [g. smaller range] it'll
travel, like, you know, less than that distance.
310
Bruce
I see.
Greg
With the same push. So that's why we thought that this repeat [g. 150 - mass * 2] is kind of
like a distance. So we tried to make it less than our initial, um, one fifty. [g. 150] So we
subtracted the mass [g. - mass * 2] a factor of the mass.
This is yet another example of a very algebra-like construction of a single programming statement.
Again, the main device active here is CHANGING PARAMETERS and the BASE -CHANGE , PROP+, and
COEFFICIENT
forms are implicated. Notice that the base value of 150 for the number of repeats
corresponds to the value used in their original simulation when the acceleration was -.08.
Once more, the big point here is that students constructed and interpreted certain
programming statements in a manner that was very similar to the construction and interpretation
of equations. This observation has many important implications for this research project. It means
that students can borrow from algebra-physics in their understanding of programming
expressions, and thus that we can have higher expectations for the ability of students to see
structure in programs. In addition, the observation that there is this area of overlap between
programming and algebra-physics can be seen as a first step in establishing that a programming
language can do some of the work of algebraic notation.
The Symbolic Forms of Programming-Physics
Now I will begin to discuss how an analysis of programming-physics in terms of symbolic
forms departs from what we observed in the analysis of algebra-physics. Thus far we have seen that
many of the forms in programming-physics knowledge are straightforward counterparts of the
forms of algebra-physics. However, even though many of the forms may be the same, the
frequency with which these forms appear may vary greatly between the two practices. This is one
type of difference that the form-based analysis can uncover, and such a frequency analysis will be
the job of the next chapter, when I report on the results of the systematic analysis of the
programming data corpus.
Instead, my task now is to describe the programming-physics forms for which there are no
obvious counterparts in algebra-physics. For the most part, these new forms involve a sensitivity to
line-spanning structuring in programming. First I will describe a new cluster of forms, the Process
Cluster. Then I will describe one additional form that did not appear in algebra-physics, SERIAL
DEPENDENCE,
which is an important line-spanning addition to the Dependence Cluster.
311
A new cluster: The ÒProcess ClusterÓ
When programs are being constructed, programming requires that the statements be arranged
in a sequence. This means that we will need to posit knowledge that accounts for the capability of
students to successfully order programming statements when constructing new programs. In my
analysis of algebra-physics, I asked questions such as: How did the students know to write ‘+’
instead of ‘×’? Now, in programming-physics, we need to be able to answer the question: How
did the students know to write that particular statement at that location in the program?
SEQUENTIAL PROCESSES
SETUP -LOOP
VARIATION
CONSTANCY
[
[
[
[
[
[
[
[
[
…
…
…
]
]
]
]
change a …
change b …
…
repeat …
]
…
change x …
change x …
…
]
]
]
]
Table Chapter 9. -1. Forms in the Process Cluster.
Just as in algebra-physics, the answer to this sort of question is frequently mundane.
Sometimes when a student writes an equation, it is reasonable to hypothesize that they are simply
drawing on a database of remembered equations. The same is true in programming: In some
cases, students know how to order the lines in a program because they are following, more or less
exactly, the arrangement they have seen in other programs. Of course, as in algebra-physics, there
are also cases that require truly novel constructions. For the most part, I will account for the ability
of students to produce novel arrangements of statements by positing a new cluster of forms that I
call the “Process Cluster.” The forms in this cluster are listed in Table Chapter 9. -1.
To get this discussion of Process forms started, I want to begin with an example of the type of
construction act that needs to be explained. Recall that, in Chapter 8, I described how Tim and
Steve produced a simulation of a dropped ball. Following their work on that simulation, they
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reworked their program to simulate a motion in a which a ball, rather than simply being dropped,
is tossed straight up and then falls back down. The first working version of this new simulation
appeared as follows:
drop
toss
setup
change
change
change
repeat
drop
change acc -9.8
repeat 10 tick
pos 0
vel 98
acc -9.8
10 tick
Doit
Doit
Execution of this program starts in the box named toss. Toss begins by executing setup to position
the ball at its starting point and orient it facing upward. Then each of the three variables are
initialized and tick is repeated 10 times, taking the ball up to the apex of its motion. Following
the repeat statement, there is a single line that calls a procedure named drop, which takes the ball
from its apex back down to the bottom of its motion.
One thing that needs explaining here is how Tim and Steve knew to order the lines in toss. To
begin, notice that the first five lines of this procedure, from setup to the repeat statement, are in
line with the initialize-and-then-repeat structure, which we have already encountered a number of
times. Just as this structure is familiar to readers, it was also familiar to Tim and Steve; they had
already seen several examples and created a few programs following this structure. Thus, I believe
that it is reasonable to hypothesize that the sequence of statements
setup
initialize variables
repeat ## tick
is a structure that Tim and Steve essentially remember; it is part of a known repertoire of program
structures. (I will have more to say about this structure in a moment.)
But in appending a call to drop at the bottom of the toss procedure, Tim and Steve are
departing from the initialize-and-then-repeat structure. How did Tim and Steve know to write
this particular line in this location? To an experienced programmer, this may seem like an almost
trivial move, hardly worthy of an explanation. But, I hope to show that working to describe the
knowledge involved here actually extracts some of the most important differences between
algebra-physics and programming-physics.
The key to the explanation here is to note that, in writing the program in this manner, Tim
have Steve have broken the motion into two segments, the upward motion to the apex and the
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downward motion. With this in mind, the toss procedure can be understood as consisting of two
sequenced parts that correspond to these two portions of the motion. This is illustrated in Figure
Chapter 9. -3.
toss
setup
change
change
change
repeat
pos 0
vel 98
acc -9.8
10 tick
drop
Ball goes up.
Ball comes down.
Doit
Figure Chapter 9. -3. Tim and Steve's Toss procedure broken into two parts.
When a procedure is seen in this manner, as consisting of two or more sequential pieces that
correspond to segments of a motion, I say that the SEQUENTIAL PROCESSES form is active. The
entities in the conceptual schema associated with SEQUENTIAL PROCESSES are two or more processes,
and the relationship that holds among these processes is simply that they are ordered.
Furthermore, the associated symbol pattern is:
SEQUENTIAL PROCESSES
[
[
[
…
…
…
]
]
]
In this notation, “[…]” stands for a single programming statement or a group of statements.
Thus, like other forms, when the SEQUENTIAL PROCESSES form is engaged it constrains what
students write when constructing new expressions. Furthermore, unlike the way that line-spanning
symbol arrangements are used in algebra, this arrangement follows a conceptualization of the
physical. In the present case, once Tim and Steve have understood the motion as consisting of two
parts—the upward motion and the drop—then the SEQUENTIAL PROCESSES form specifies that the
statements corresponding to these parts of the motion will be written one above the other, with an
order corresponding to the ordering of the processes. This is how I account for the fact that Tim
and Steve know to write “drop” at the bottom of the toss procedure.
As I hope to show in the next chapter, the SEQUENTIAL PROCESSES form was comparatively
common and accounted for many of the episodes in which students constructed truly novel
arrangements of statements. I will present one more example here to illustrate further. This
example relates to a simulation task that we have not yet encountered, the simulation of a mass
that oscillates horizontally on a spring. Early in their work on this task, when they were operating
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in a somewhat experimental mode, Adam and Jerry produced a simulation that did not make use
of the Tick Model structure:
repeat 50
change pos 50
change pos 0
change pos -50
Doit
Adam
We could have one program that makes it this way. [g. right] One - and then one that
makes it go that way [g. left] then this way then that way. [g. back and forth]
Their idea, in writing this short program, was to simulate an oscillation by sequentially placing the
block at three positions: +50, 0, and then -50. If this was done repeatedly, then the block would
appear to oscillate, albeit in a somewhat jumpy and discontinuous manner.14 I explain Adam and
Jerry’s sequencing of these three statements by appeal to the SEQUENTIAL PROCESSES form; the
motion is broken down into sequential sub-processes that correspond to ordered statements in the
program.
Notice that, except for ordering, SEQUENTIAL PROCESSES treats all of the processes involved as
equivalent. As far as the schematization associated with this form is concerned, none of the
processes plays a special role. In contrast, we can imagine that it would be possible to develop
standard ways of breaking a motion into sub-processes in which the sub-processes are nonequivalent.
One such breakdown, associated with the initialize-repeat structure, is what I call the SETUP LOOP
form. In SETUP -LOOP , the program is seen as decomposed into a portion that gets everything
ready for the motion—for example, positioning the sprite and initializing variables—followed by
a repeat statement that repeatedly executes a procedure that specifies what transpires at each
moment of the motion. To take a simple example, Tim and Steve’s drop simulation, which we
examined in Chapter 8, can be seen in terms of SETUP -LOOP (refer to Figure Chapter 9. -4). Note
that, since SETUP -LOOP breaks down a motion into sub-processes associated with program parts, it
may be thought of as a special case of SEQUENTIAL PROCESSES.
14
When implemented, this program actually produces no motion whatsoever. The sprite does not move
in the graphics box, only the value of the variable is changed, since the fd statement must be used to
cause the sprite to move. Adam and Jerry made this correction in later versions of their program.
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drop2
setup
change pos 0
change vel 0
change acc 9.8
Setup
repeat 10 tick
Loop
Doit
Figure Chapter 9. -4. The SETUP -LOOP structure in Tim and Steve's drop simulation.
The
SETUP -LOOP
form can be seen in the final interpretation of the drop simulation given by
almost every pair of students, though it is sometimes a little subtle and difficult to recognize.
Here are a few examples:
drop
setup
change
change
change
repeat
vel 0
acc 2
pos 0
22 tick
Doit
Clem
Change velocity, change [g.~ setup lines] - we initialize everything. [g.~ pos, vel, acc
variables] And then, um, and then we repeated the tick twenty two times. [g.m along
repeat line] [This last said with strong suggestion of "that's it."]
_______________
drop2
setup
change acc 2
change vel 0
repeat 23 tick
Doit
Ned
It's just pretty much the tick model. [g. the tick model.] We just started the acceleration [g.~ 2 in change acc 2] or, the acceleration's constant at two.
Bruce
Hm-mm.
Ned
That's just gravity. Velocity's [g.~ 0 in change vel 0] zero because you're letting it go [g.
release from top of graphics box], so the initial is zero.
Bruce
Hm-mm.
Ned
And we just repeated the tick. [g. repeated circling motion around tick box]
Bruce
Okay.
Ned
Twenty three just so it stops at the bottom. [g. along motion; g. bottom of graphics box.]
_______________
drop2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
10 tick
Doit
Tim
Um, we just did the regular setup.
Bruce
Hm-mm.
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Tim
To take us up there [g.m top of path] and then set position and velocity to zero. [g.m
change pos 0; change vel 0] And then we just have acceleration, nine point eight. [g.m
change acc 9.8] Which is so it’s gravity.
Bruce
Hm-mm.
Tim
And then we just repeat it ten times it just (0.5) falls. [g.m along path in g-box]
At first blush, it may seem like the above passages are little more than a simple reading of the lines
that appear in the various drop procedures. But, if we look closely, there are some clues that the
students are seeing the program in terms of SETUP -LOOP. First note the use of the words “initial”
and “initialize” in describing the lines above the repeat statement. This suggests an interpretation
of these lines that is in accord with SETUP -LOOP.
In addition, a more subtle—and perhaps more speculative—observation can be made
concerning the phrasing these students used in describing the repeat statements in these programs.
Tim said: “And then we just repeat it ten times it just falls.” And Ned said something quite
similar: “And we just repeated the tick.” Both of these statements suggest something obvious or
automatic about the repeating of tick at the end of these programs, a stance that I believe is closely
associated with the
SETUP -LOOP
form. We can think of the setup portion of the program as getting
things all ready to go—like winding up a spring or cocking the hammer on a pistol. Then, after
everything is ready, you just pull the trigger and the motion happens according to the parameters
you have defined. According to this stance, repeating tick is just like turning on the machine that
generates the motion from the initial conditions. This is the SETUP -LOOP stance: Get things ready
and then just let it run.
I want to make a side comment about
SEQUENTIAL PROCESSES
SETUP -LOOP
before proceeding. In my discussion of
above, I spoke of the initialize-and-then-repeat structure as if it was
something that students just remembered, like a remembered equation. In contrast, in this
discussion of
SETUP -LOOP,
I have been talking about this structure as the sort of meaningful pattern
associated with symbolic forms. Although this may seem confusing in the present context, it is
actually completely consistent with all of my earlier discussion of forms. The point is that, even
though the initialize-repeat structure may be “memorized” by students, this does not necessarily
mean that they are incapable of seeing it as meaningful. We encountered precisely the same
situation in algebra-physics. Any algebraic expression that a student writes from memory, such as
F=-kx, can also be interpreted by students and seen as meaningful in various ways.
The
SETUP -LOOP
form sets up a distinction between what is inside the loop structure and what
is outside. It turns out that this boundary between the inside and outside of the loop is a very
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significant landmark for students, just as the line that separates the numerator from denominator
was a critical landmark in algebra-physics. This is an important enough point that I want to take a
moment just to present a pair of examples in which students’ sensitivity to this landmark is
evident. The first of these examples involves Greg and Fred’s work on the air resistance
simulation. The following program is taken from an intermediate stage of their work:
drop
setup
change
change
change
change
change
repeat
vel 0
air 0
mass 20
force mass * acc - vel
acc force / mass
sqrt
tick
1200 / acc
Doit
Doit
The feature of this program that is important for our current concerns is that the lines that
compute the force and acceleration are outside the loop—they are in the setup area—rather than
inside the repeat loop. For this reason, the force and acceleration will not be recomputed at each
point along the motion, as they should be to produce a correct simulation. Ultimately, Fred
caught on to this:
Fred
Now notice that this equation [g.m change force line] won’t do anything because it’s not
inside a loop. First of all, our velocity is zero. [g.m change vel 0]
…
Fred
None of this - none of this stuff is inside a loop. [g. down from top of drop box to repeat
line] Force times mass times acceleration minus vel. [g. along change force line]…And,
the whole repeating thing [g.O repeat line] doesn’t even touch what’s up here. [g. above
repeat]
Similarly, during their work on the spring simulation, Greg and Fred commented frequently
on whether various quantities should be computed inside or outside the repeat loop:
Fred
Yeah. So, whatever it is, if we were to give initial displacement it would be before a loop.
Greg
Uh-huh.
Fred
And inside a loop, position would change and - and acceleration would change. Hey that’s
pretty easy.
The point here is simply that comments of this sort were quite ubiquitous. The loop boundary was
an extremely significant landmark and it frequently played a role in student discussions.
Like
SETUP -LOOP,
the last two forms in this cluster also depend strongly on the landmark
associated with the loop boundary. In particular, these two forms involve registering the presence
of an individual programming statement as either inside or outside the loop. The key idea here is
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that, if you want a quantity to vary through the time of a motion, then that quantity must be
computed within the repeat loop. And, conversely, if you want the quantity to be constant, it
must be computed outside the loop. I call the corresponding forms VARIATION and CONSTANCY.
Greg and Fred’s statements above provide some first examples of episodes involving these
forms. In those episodes, note that Greg and Fred were sensitive to whether a given statement was
inside or outside the loop. For example, Fred said: “Now notice that this equation won’t do
anything because it’s not inside a loop.”
To further illustrate, let’s look at how Greg and Fred’s work on the Air Resistance task
proceeded. As I noted, the version of the Air Resistance simulation presented above has the
problem that the change force and change acc lines are outside the loop. Fred was aware of this
difficulty, and spent quite a while explaining to Greg why the lines must be moved inside the
loop:
Fred
No, we don’t change it there. Air is - the air term is gonna be something that changes
during the calculation or during the fall. Remember that you said it’s dependent on
velocity, right?
…
Fred
So it’s actually gonna change the force each point, each tick.
Greg
Hm-mm.
Fred
So we’re gonna have to put change something.
…
Fred
It- it won’t calculate velocity but I’m just saying that, well, it’s gonna repeat this thing a
couple of times, right? [g. repeat line] And it’s changing velocity in tick. [g. change vel line
in tick] But we have to say, some time inside the repeat [g. repeat line], that our frictional
force is changing. And that would change acceleration.
Fred’s argument here is that since the “air term” changes through the motion, it must be
recomputed at each point. Ultimately, Greg was convinced and they went on to modify their
simulation so it appeared as follows.
drop
setup
change
change
change
repeat
vel 0
air 0
mass 20
200
change force mass * acc - vel
change acc force / mass
tick
Doit
Doit
In this simulation program, the lines which compute the force and acceleration have now been
moved inside the loop—along with tick, they are part of the short procedure that is executed on
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each cycle of the repeat. This is an example of a construction effort that involves the VARIATION
form; because they want these quantities to vary, they know to compute them inside the loop.
Another observation that can be inserted here is that the above episode—and many of the
other passages I have thus far presented in this sub-section—are excerpts of extended and explicit
discussions concerning the location of lines in a program. The existence of these discussions
supports my point that, unlike algebra, it is important to posit symbolic forms to explain how
students construct the line ordering in their programs.
I want to digress to point out an important property of VARIATION and CONSTANCY: These
forms constrain where statements are written, but they do not, in most cases, uniquely specify
exactly where a statement should be placed in a program. This observation is parallel to points I
made during my discussion of algebra-physics. In that discussion we saw that not all features of an
algebraic expression are specified by the forms engaged in a construction effort. For example,
when we construct an expression from PROP+, or recognize an expression in terms of PROP+, it is
not significant whether the quantity in question is raised to the first, second, or third power.
Similarly, when programs are seen in terms of
VARIATION
and CONSTANCY, some aspects of the
ordering of statements are not relevant, all that matters is whether the lines are inside or outside
the loop.
Thus, what we are seeing is that, although programming enforces an ordering of all
statements, every detail of this ordering will not always be treated as meaningful. This is a
favorable situation, since the ordering of statements in a program is frequently not consequential,
even for the operation of the program. For example, it often does not matter how the initialization
statements in a simulation are ordered. Programming does force us to select an ordering for these
statements, but it is usually not appropriate to ascribe any particular meaning to the sequence
selected.
So, VARIATION and CONSTANCY, together with SEQUENTIAL PROCESSES and SETUP -LOOP,
constitute the four forms in the Process Cluster. Now I want to conclude this discussion of the
Process Cluster with a comment concerning the novel nature of these forms. Notice that, for the
sort of structure associated with this cluster, there is not much that can be borrowed from algebraphysics. Thus, much of the ability that these students have for recognizing this sort of structure
must have been learned during the few hours of the experimental sessions. For this reason, we are
most likely seeing only a fraction of the possible meaningful patterns that students could learn to
recognize at this level. If we developed a true practice of programming-physics, with many
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participants engaged over a significant duration of time, I believe that the participants would learn
to recognize a wider range of patterns of this sort.
More specifically, just as
PROCESSES
SETUP -LOOP
does away with the uniformity inherent in SEQUENTIAL
by breaking a process into non-equivalent sub-processes, it is possible that a true practice
would include a rich vocabulary of forms, each of which breaks a process into sub-processes in a
unique manner. I believe that this is one of the major ways that the vocabulary of forms would be
extended in a more complete programming-physics practice, at least one built roughly along the
lines we are exploring here.
I did see some hints of possible directions in which the sub-process vocabulary might be
extended. My guesses in this regard are quite speculative, and there was certainly not enough
uniformity across subjects for me to add any forms to the list I have already given, but it is worth
mentioning a single example just to illustrate what I have in mind. This example is taken from
Greg and Fred’s work on their simulation of a mass oscillating on a spring. During their work on
this task, these students spent a fair amount of time discussing an issue that was prominent for
many of the subjects: How does the mass get moving? After some discussion, Greg and Fred
decided that there were two different methods they could use to initiate the motion: Either
displace the block away from the origin and release it, or just give the block a slap.
Fred
… I think there’s two ways you can think of this.
Greg
Yeah.
Fred
That, somehow there’s gonna be something to start the motion okay.
Greg
Yeah.
Fred
And it’s either gonna to be a displacement of the - of the::: // whatever it is.
// Hm-mm.
Greg
Fred
The block.
Greg
Hm-mm.
Fred
Or it’s gonna be a slap, an applied force.
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go2
❝ Initial
Shove ❞
setup
change
change
change
change
change
change
vel 0
pos 0
k 5
mass 10
force 7
acc
- force / mass
Doit
tick
repeat 300
change acc
- k * pos / mass
Doit
change force - pos * k
tick
Doit
Doit
Figure Chapter 9. -5. Greg and Fred's Spring simulation that uses an "initial slap."
During the time they spent on this simulation, Greg and Fred created simulations
corresponding to each of these two possibilities. Figure Chapter 9. -5 shows a version of their
simulation that uses the latter method, what they came to call an “initial shove,” to initiate the
motion of the block. The lines associated with this initial shove are indicated in the figure. In the
first of these lines, a variable named force, which corresponds to the force of the applied slap, is set
to a particular value. Then, in the next statement, an acceleration is computed from this force.
Finally, there is a call to tick that essentially actuates this acceleration; one cycle of time goes by
with the acceleration set to the value determined in the previous line.
The structure employed in this program can be thought of as an extension to SETUP -LOOP.
Between the initialization lines and the repeat command, there is an intervening sub-process that
plays the role of initiating the motion—it gives the kick that gets the motion going. So, instead of
just SETUP -LOOP, we have a tripartite structure that might be called SETUP -INITIATE -LOOP. The
point here is that this SETUP -INITIATE -LOOP structure is just one of the many possible ways that the
vocabulary of sub-processes might be extended. I believe that if we were studying a true practice
of programming-physics, rather than just a mock-up in which students had only a couple hours of
experience, then the Process Cluster would be enriched to contain additional forms of this sort.
Before concluding this section, I want to briefly connect to some closely related work. Other
researchers have argued that programming knowledge includes what have been called “templates”
(Linn, Sloane, & Clancy, 1987; Mann, 1991) or “plans” (Ehrlich & Soloway, 1984; Soloway,
1986). These templates are schematic structures—like the ones I have been describing—that
constrain, at certain levels, the programs that experts and students write. For example, Soloway
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(1986) describes what he calls the “sentinel-controlled running-total loop plan.” This template
reads in one value after another, adding these values to a total, and terminating when a specific
“sentinel” value is encountered:
initialize
ask user for a value
if input is not the sentinel value then
add new value into running total
loop back to input
This template describes the structure of a program at a certain level and degree of specificity.
In the template literature, it is usually emphasized that templates can exist at many levels; they can
exist at the level of individual programming statements, or closer to the level of entire programs.
Furthermore larger structures can be filled in to varying degrees of specificity. Notice that, in
allowing templates to exist in such diversity, these researchers are defining “template” in a very
general way. This is not a difficulty, but it suggests that we should be able to identify subcategories of templates that do different sorts of jobs, and which are tied to differently to other
knowledge. In fact, Ehrlich and Soloway (1984) do describe two “main kinds” of plans, “Control
Flow Plans,” and “Variable Plans.”
My position here is that, at least to a first approximation, it is reasonable to view the Process
forms—and perhaps programming forms, in general—as a particular sub-category of template.
The templates in this sub-category specify programs at a certain level, and have a certain degree of
specificity. Note that the structures in the Process forms have only a small number of components,
each of which is quite generic. For example, the SETUP -LOOP form has just two parts, and the
SEQUENTIAL PROCESSES
form is even looser and simpler; it is just a sequence of equivalent ordered
components, with the content of these components unspecified. This is in contrast to the
comparative complexity and specificity of Soloway’s sentinel-controlled running-total loop plan.
There is another way to understand the special characteristics of this sub-category of template.
In the templates literature, it is often emphasized that templates are associated with goals or
functions. If you have some goal that you want to accomplish, then you use the template that
accomplishes that goal. In my account of forms, the story of use is somewhat different. We do
not pull out a form to serve a function; instead, when constructing a program, a form is activated
because of the meaning of the construct—the associated conceptual schema. Thus, Process forms
are a particular sub-category of template that correspond to how we conceptualize a process, and
thus allow the generative construction of simulations out of meaningful components.
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A new Dependence form:
SERIAL DEPENDENCE
Next, I will discuss another form that appeared in programming, but which I did not observe
in algebra-physics. Like the forms in the Process Cluster, this form is responsible for determining
how the lines in a program should be arranged; nonetheless, it is of a somewhat different sort. The
conceptual schema associated with this new form does not have processes as the fundamental
entities in its description. Rather it is more like the forms we encountered in algebra-physics in
being concerned with physical quantities and their relations.
I consider this new form, which I call SERIAL DEPENDENCE , to be an addition to the
Dependence Cluster. In my discussion of algebra-physics, I listed three forms in this cluster: NO
DEPENDENCE, DEPENDENCE,
and SOLE DEPENDENCE . Stated simply, the forms in this cluster are
concerned with whether one quantity or an expression depends on some other quantity. With
regard to symbol pattern, this means simply registering whether or not a given symbol appears, in
any manner, in an expression. As an example, I cited Mark’s statement that the force of air
resistance must depend on the velocity of the falling object:
R = µv
Mark
So this has to depend on velocity. [g. R] That's all I'm saying. Your resistance - the
resistor force depends on the velocity of the object. The higher the velocity the bigger the
resistance.
In programming-physics, the notion of dependence gets extended in an interesting way.
When writing simulations, students were often concerned with chains of dependent quantities,
such as d depends on c, which depends on b, which depends on a. Within a program, these chains
show up as the following symbol pattern:
change b
change c
SERIAL DEPENDENCE
change d
... a ...
... b ...
... c ...
...
In this programming symbol pattern, b is computed from a, c is computed from b, d is computed
from c, etc. This sequencing of linked dependence relations, which students wrote into and read
out of programs, is the meaningful pattern I call SERIAL DEPENDENCE
We already encountered an episode involving SERIAL DEPENDENCE in Chapter 8, when I gave an
extended description of Tim and Steve’s work on the Air Resistance simulation. Recall that, in
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that episode, Tim raised the issue of how the lines in their modified tick procedure should be
ordered. In the end, they arrived at the following result:
TICK
change Res
vel * .1
Doit
change acc
acc - res
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Tim
No we have to (then account) in the acceleration. So we have to change the acceleration.
Do we change the acceleration before we change the velocity? [g. change vel line in tick]
Or do we change acceleration after we change velocity? Which also changes the
resistance. [g. change Res line] Oh, I’m getting confused. What’s the order in which you
change things?
…
Tim
Well you have to change resistance because you change the velocity, because (0.5) the
velocity (1.0) - the new velocity is dependent upon the acceleration which is dependent
upon the resistance. So we should change resistance, then acceleration, then velocity.
Right? Does that make sense, or am I just talking nonsense.
The question that Tim and Steve were wrestling with was, given that a number of quantities must
be computed in the tick procedure, “What’s the order in which you change things?” What they
decided was to compute res (a quantity associated with the air resistance) from the current value
of the velocity. Then they got an acceleration from this value of res, and finally a new velocity and
position from the acceleration.
Tim’s assertion is that this ordering somehow actually reflects the structure of dependence in
this situation, “the new velocity is dependent upon the acceleration which is dependent upon the
resistance.” Of course, he does not provide much of an argument for this view in the above
passage. Nevertheless, this ordering does turn out to be in agreement with one important way of
conceptualizing the structure of Newtonian physics: Find the force using a “force law” and then
use F=ma to find the acceleration. I will have more to say about this structure in Chapter 11.
SERIAL DEPENDENCE issues arose frequently during student work on the Air Resistance
simulation, as in Tim and Steve’s discussion above, and also in the context of the Spring
simulation. Ned and David’s brief discussion below is the tail end of a typical passage.
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TICK
change vel
vel + acc
Doit
change pos
pos + vel
Doit
change force
-1 * pos
* k
Doit
Doit
change acc
force / mass
Doit
fd vel
dot
Doit
Ned
Does the force change because the acceleration changes?
David
The force changes ’cause X changes. And so that, and so=
Ned
=That causes acceleration to change.
In this example, Ned and David decided on a particular dependence structure; the position
determines the force which determines the acceleration. Again note that, like the Process forms,
SERIAL DEPENDENCE
constrains the arrangement of lines in a simulation program.
Similarly, the question of which quantities should determine which others played a central role
in Adam and Jerry’s work on the Spring simulation. Early on, Adam strongly argued that the
position was at the root of the dependence structure:
Adam
Well basically, the position is ba- um, the acceleration, everything,
Bruce
Hm-mm.
Adam
is based on where this is, the position, [g. pos in change acc line]
In accordance with this viewpoint, Adam and Jerry ultimately settled on the version of the tick
procedure shown below. (In this program, the variable k was elsewhere set to a negative value.)
TICK
change force
k * pos
Doit
change acc
force / mass
Doit
change vel
vel + acc
Doit
change pos
fd vel
dot
pos + vel
Doit
Doit
Finally, after they had produced a complete and working simulation, Adam and Jerry had a
number of discussions which involved recognition of SERIAL DEPENDENCE in this procedure. Here is
326
an extended excerpt where Adam once again argues for the primacy of position in determining
the other quantities:
Adam
And the position is the most important thing here. If we made it the other way around, we
don’t care about the acceleration, we care about where it is.
Jerry
Hm-mm.
Adam
See acceleration [g. change acc line] derives from the position. [g. change force line]
Jerry
We don’t know how acceleration changes like- the only th-
Adam
We wouldn’t know how position changes (…) acceleration I guess.
Jerry
Hm-mm.
Adam
That’s the best way. It’s easier just to think about it as position affecting acceleration.
Jerry
Hm-mm.
Adam
Than acceleration affecting position.
Jerry
Hm-mm.
Bruce
It’s easier to think of it that way?
Adam
Well it - because - I mean, what we’re // thinking of is that force depends on how far we pull
it. If we do it your way, we have to know how far - how the force depends on how fast you
pull it back. Or something. Something of that term.
// That’s what happens.
Jerry
Jerry
Hm-mm.
Adam
Yeah.
Jerry
This is like - this is like saying, okay, given a position, this is the acceleration. Whereas
the other way would be saying here is our acceleration, now where is it along the-
In this and the preceding passages, the students involved were engaged with a quite unusual set
of issues. The question under consideration—which quantities determine which others—is not a
question that arises very often in algebra-physics. In fact, as I mentioned in Chapter 8, such a
question may not even make sense within algebra-physics. But I will forestall a full discussion of
these issues until Chapter 11. My goal for this section has simply been to establish that the SERIAL
DEPENDENCE
form corresponds to a new type of meaningful pattern that my students learned to
see in their simulation programs and which, because it spans multiple statements, constrains the
arrangement of statements in novel constructions.
Representational Devices and the Interpretation of Simulation Programs
One of the original intuitions with which I began this work was that programs should be easier
to interpret than algebraic expressions. As I said in Chapter 8, this intuition is rooted in the notion
that programs can be “mentally run” to generate motion phenomena. If this is correct—if
interpreting a program always means mentally running it—then the interpretation of programs
may be, overall, a very different kind of thing than the interpretation of equations. Furthermore, if
327
there is a single, preferred mode for interpreting programs, this could mean that interpreting
programs is much easier than the interpretation of algebraic expressions.
However, we have already seen some evidence that this is not the case. Earlier in this chapter I
presented a number of examples in which individual programming statements were interpreted
much like the expressions of algebra-physics. For example, David’s interpretation of this
statement is very similar to the CHANGING PARAMETERS interpretations we encountered in algebraphysics.
change acc
mass / -100
Doit
David
Uh just the (st…) the acceleration is dependent on the mass. And the greater the mass,
the greater, the greater the acceleration is.
Furthermore, it turns out that the use of algebra-like devices even extends beyond the
interpretation of individual programming statements. In some cases, the devices of algebra-physics
were used to interpret groups of statements selected from a program. For illustration, consider
Ned and David’s Air Resistance simulation:
drop2
setup
change
change
change
change
change
TICK
change vel
pos 0
drag 0
mass 5.0
grav 2.0
force
mass * grav
Doit
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
change acc
change vel 0
repeat 700 tick
force - drag
/ mass
Doit
Doit
Doit
change drag
vel / 1.001
Doit
dot
Doit
The above program is the final version of the simulation that Ned and David produced, and it
correctly simulates the approach of the dropped object to terminal velocity. Like all of my
subjects, Ned and David were somewhat impressed when their simulation actually worked, even
though they were fairly clear on the physical principles involved. Thus it was natural for them to
try to talk through why this simulation has the property that it approaches a terminal velocity. The
final explanation, given by David, was based solely around two key lines from the program:
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change acc
force - drag
/ mass
Doit
Doit
change drag
vel / 1.001
Doit
David
Um, yeah, because at, at the terminal velocity, this value for vel [g. vel in change drag
line], the drag will equal the force. [g. drag, force, drag in change acc line] So then you
have no - no more acceleration. And so as - as uh - as velocity approaches this value [g.
1.001] times the force, [g. force],
Bruce
Hm-mm.
David
um, the acceleration decreases. And so, as it gets close, the acceleration decreases
slower and slower and slower until finally it reaches zero.
What David is saying here is that, as the velocity increases, the value of the drag variable will
increase until it equals the value of force. At that moment the first line returns zero and terminal
velocity has been reached. Furthermore, by looking at the second line, he is able to tell that this
cancellation will occur when the velocity is equal to 1.001 times the force.
There are two important points to be made about this passage. The first point is that, although
it involves multiple lines, it is really quite similar to the PHYSICAL CHANGE interpretations we
encountered in algebra-physics. As in those interpretations, David looks at a symbolic display and
talks through how the variables in that display vary through the time of the motion. The following
interpretation, which I originally presented in Chapter 4, is clearly of the same ilk:
∑F = mg - kv
Alan
And so I guess we were just assuming that this was [g. kv] de-accelerating - that we would
have gravity accelerating the ball and then as, as we got a certain velocity, [g. kv] this
would contribute to making accelerating closer - de-accelerate it and so we would just
have a function of G minus K V. And then we assumed that since we know that it reaches
terminal velocity, its acceleration's gonna be zero, we just assumed G minus K V would
equal zero.
A second point to be made about David’s interpretation is that, if his interpretation is at all
representative, then understanding a program may mean focusing on only a small piece. Keep in
mind that it really does matter what the rest of the program is; these two lines alone are not
sufficient to generate the correct behavior. But David’s explanation does not mention the other
lines in the program; he instead makes implicit presumptions about the behavior of these other
lines. In David’s explanation, the rest of the program fades into the background.
Really, I could have made this same point at the beginning of this chapter when I observed
that students frequently give interpretations of single lines in a program. Just as in David’s
explanation, those interpretations often required implicit assumptions about the rest of the
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program. But David’s interpretation makes it strikingly clear that students may try to understand
global properties of a program by looking only at a small piece.
Clearly, the interpretation of a program is not a wholly different and much easier task than the
interpretation of an equation. Individual and groups of statements are interpreted through devices
that are straightforward counterparts of the devices of algebra-physics. If anything, this might be
taken as suggesting that interpreting programs is a little trickier than interpreting equations, since
some cleverness must go into selecting the lines for focus. But there are a number of factors to
weigh, and I will defer the question of which, on balance, is “easier” until Chapter 11.
There are some more profound ways in which the interpreting of programs departs from the
interpreting of equations, and talking about these points of strong departure is the task of the
remainder of this section. What I will argue is that there are two critical and quite common
programming-physics devices for which there no algebraic counterparts, what I call TRACING and
RECURRENCE.
A new Narrative device:
TRACING
I have taken pains to argue that not all algebra-physics interpretations involve mentally
stepping through a program—what has sometimes been called “mental simulation” (e.g.,
Letovsky, Pinto, Lampert, Soloway, 1987). However, there is one common representational device
that can reasonably be considered to be a variety of mental running. Interpretations based on this
new Narrative device, which I call TRACING , take their narrative form from the line-by-line
structure of a program.
For a first example of a TRACING interpretation, I turn to Adam and Jerry’s work on the Spring
simulation. During my discussion of
SEQUENTIAL PROCESSES,
I described how Adam and Jerry
created a preliminary version of this simulation which worked by successively placing the block at
three distinct locations. After modifying this program to leave a trail of dots, Adam checked that
he had something sensible by tracing through what they had written:
repeat 50
change pos 50
dot
change pos 0
dot
change pos -50
dot
Doit
Adam
Make a dot right there. [g. g-box] Change to fifty. Make a dot. [g. a point to the right of
center in the g-box] Go back to here make a dot. [g. center] Go here, make a dot. [g. left]
And repeat.
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In this interpretation, Adam steps through the program one line at a time. As he does, he points
into the graphics box on the display, moving his hand as the block would move for each
statement. This is a TRACING interpretation, a narrative interpretation that follows the structure of
the written program.
To further clarify the nature of TRACING , it helps to distinguish it from PHYSICAL CHANGE , since
both of these devices involve a description of a phenomenon that follows through the time of the
motion. The difference enters in how these descriptions are related to the written expression. In
the case of PHYSICAL CHANGE , one or two lines in the program are selected, and then the
interpretation describes how quantities in the expression vary through time. In contrast, rather
than focusing on how the quantities in a single line vary through time, a TRACING interpretation
proceeds, line-by-line, through a program. Compare David’s interpretation of the Air Resistance
program described just above. The action in that interpretation concerns how the value of the drag
variable changes through time, with attention focused on two statements. But the action in
Adam’s
TRACING
interpretation flows from line to line in the procedure.
Interestingly, stepping through the lines of a program will not, in all cases, correspond to
stepping through the time of the motion. For illustration, consider Ned and David’s
interpretation of this tick procedure, which has been modified for the Spring simulation:
TICK
change vel
vel + acc
Doit
change pos
pos + vel
Doit
change force
pos * k
Doit
change acc
force / mass
Doit
fd vel
dot
Doit
David
See, cause if we start off with zero acceleration, velocity would be velocity. And it'll go
once. [g. moves finger short distance to right] Position'll be position plus velocity. Force
would- (0.5) position times K. And then acceleration'd be force divided by mass. And then,
it'll use that acc, right, the next time around?
In this passage, David proceeds through the lines in tick under the assumption that the
acceleration is zero. This is clearly an instance of the TRACING device; in fact, as noted in Appendix
E, TRACING interpretations frequently take the form of a series of statements separated by the
phrase “and then,” as in David’s statement above. The point I want to make about this
interpretation is that, although it follows the lines in the program, it is not precisely correct to
331
think about the sequence of operations as happening through time. In the physical system that this
simulation purports to model, all of these parameters are constantly changing; they do not change
one at a time as in this program. In essence, the act of programming forces the students to
explode each instant of the motion into a series of actions that happen through what I will call
“pseudo-time.” Similar issues arose in my discussion of SERIAL DEPENDENCE and I will discuss this
issue further in Chapter 11.
TRACING interpretations do not always follow at the level of individual lines. In some cases, a
program is parsed in chunks—perhaps as dictated by a Process form—and then the interpretation
steps through these chunks. Compare the following two examples which I originally presented in
my discussion of
SETUP -LOOP.
Tim mentions each line in his version of the Drop simulation, but
Clem, in a very simple TRACING interpretation, just says “we initialize everything” when stepping
past the “setup” portion of the program.
drop2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
10 tick
Doit
Tim
Um, we just did the regular setup.
Bruce
Hm-mm.
Tim
To take us up there [g.m top of path] and then set position and velocity to zero. [g.m
change pos 0; change vel 0] And then we just have acceleration, nine point eight. [g.m
change acc 9.8] Which is so it’s gravity.
Bruce
Hm-mm.
Tim
And then we just repeat it ten times it just (0.5) falls. [g.m along path in g-box]
_______________
drop
setup
change
change
change
repeat
vel 0
acc 2
pos 0
22 tick
Doit
Clem
Change velocity, change [g.~ setup lines] - we initialize everything. [g.~ pos, vel, acc
variables] And then, um, and then we repeated the tick twenty two times. [g.m along
repeat line] [This last said with strong suggestion of "that's it."]
The TRACING device is our first point of strong departure from the devices of algebra-physics.
It is closely related to a commonsense notion of mental running, a phenomenon that is likely to be
unique to programming. Furthermore, as I hope to show in the next chapter, TRACING is not an
inconsequential addition; it turns out to be one of the most common devices in programmingphysics.
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A new Static device: RECURRENCE
The second new device in programming-physics, which I call RECURRENCE, constitutes an
addition to the Static Class and is a new type of stance toward individual statements. A brief
review here of Static devices is warranted. Devices in the Static class include
SPECIFIC MOMENT ,
in
which the stance is taken that an expression is true only at some particular instant in a motion, and
GENERIC MOMENT ,
in which the stance is adopted that an expression is true at any moment. Mike
and Karl’s interpretation of the following expression from their work on the Air Resistance task is
an example of GENERIC MOMENT . In Chapter 4, I emphasized Mike’s use of the phrase “at any
time.”
a(t ) = −g +
Mike
f (v)
m
So, at least we can agree on this and we can start our problem from this scenario. [g.
Indicates the diagram with a sweeping gesture.] Right? Okay? So, at any time,, At
any time, the acceleration due to gravity is G, and the acceleration due to the
resistance force is F of V over M. [w. g + f(v)/m.]
Now, in contrast, consider Fred’s interpretation of an Air Resistance simulation. Notice that,
where Mike would have said “at any time,” Fred says “each tick.”
drop
setup
change
change
change
repeat
vel 0
air 0
mass 20
200
change force mass * acc - vel
change acc force / mass
tick
Doit
Doit
Fred
No, we don’t change it there. Air is - the air term is gonna be something that changes
during the calculation or during the fall. Remember that you said it’s dependent on
velocity, right?
…
Fred
So it’s actually gonna change the force each point, each tick.
Greg
Hm-mm.
Fred
So we’re gonna have to put change something.
The point of this example is that, rather than adopting the stance that an expression states what is
true at any time, Fred adopts the stance that the change force line states what transpires at each
moment along the motion. Given only this example, this may seem like a subtle and perhaps
inconsequential distinction. However, there are cases where this difference in stance actually has
very strong implications. Allow me to explain.
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Earlier I tried to show that it is common for programming statements involving the change
command to be constructed and interpreted in a manner very similar to certain classes of
equations. For example, the statement
change acc
mass * 9.8 - res
/ mass
Doit
Doit
can be understood very much like the corresponding equation:
a( t ) =
mg − Fair
m
However, this simple transcription between programming and algebraic notation does not always
work. Consider the following statement from the standard tick procedure along with Tim’s
interpretation:
change vel
vel + acc
Doit
Tim
So, every time - every tick it gains nine point eight. The acceleration increases the
velocity by nine point eight.
If we were to translate this line into algebraic notation in a straightforward manner, then we might
write v=v+a. Clearly there is something wrong with this translated expression, since we can cancel
the two velocity terms to obtain 0 = a.
The translation fails because each of the two vel’s that appear in the programming statement
does not refer to the same quantity. The vel in the second argument stands for the old value of the
velocity, the one produced the last time through the tick model, and the vel in the first argument
is a new value of the velocity that is to be computed from that old value. The result, as Tim tells us
in his RECURRENCE interpretation, is that the value in the velocity variable changes by 9.8. Thus, a
more correct translation into algebraic notation might look like: vnew = vold +a.
So, in this case, it is somewhat inappropriate to think of this programming expression as a
constraint among various quantities that holds at any instant. Instead it is best read as describing
what transpires on each tick, that the value of acc is added onto the velocity. This is an instance in
which the difference between RECURRENCE and GENERIC MOMENT is critical.
It was very common for RECURRENCE interpretations to be given for lines of the form
change x x+y, as in the change vel and change pos lines in tick. But, as we have already seen in
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Greg and Fred’s interpretation above—my first
RECURRENCE
example—there are some exceptions.
For further illustration, I will present an episode in which Ned and David invented a quite unique
solution for the Air Resistance task. Rather than computing the acceleration at each point by
finding the total force and dividing by the mass, they created a program in which the value of the
acceleration was scaled down on each iteration of the tick procedure. In the following passage,
note David’s use of the phrase “every time you repeat.”15
TICK
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
change acc
acc / drag
Doit
dot
Doit
David
What if it's like acc divided by two? Acc divided by two, acc divided by two, acc divided by
two, acc divided by two, acc divided by two.
Ned
What?
David
So, like you - so, it's not like - so that every time you repeat the acceleration's cut down by
a factor. Not like subtracted but it's cut down by a factor. (And) if it keeps going infinitely
many times, eventually it's gonna just keep on,, You know what I mean?
This is a construction involving RECURRENCE because the statement is understood as specifying
how, on each iteration, the new value of acc is computed from the previous value.
This close and subtle relation between GENERIC MOMENT and RECURRENCE was actually a cause
of some difficulty for students; in general, students were perhaps not as disposed to see
RECURRENCE
as they could profitably have been. For an example, we return to Tim and Steve’s
work on the Air Resistance simulation, which we have had reason to examine on numerous
occasions. Recall that, in my discussion of SERIAL DEPENDENCE , I presented an early version of Tim
and Steve’s modified tick procedure:
15
Interestingly, this simulation is in agreement with the assumption that the force of air resistance is a
linear function of velocity: F=-kv.
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TICK
change Res
vel * .1
Doit
change acc
acc - res
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
As I discussed in Chapter 8, there is a problem with the change acc statement in this procedure,
and the resulting simulation actually produces oscillations—the ball does not just fall, it moves up
and down. Tim and Steve were surprised and confused by this behavior, and they were never able
to produce an adequate explanation. After rewriting the change acc statement as
change acc
mass * acc - res
/ mass
Doit
Doit
they repeatedly gave
GENERIC MOMENT /COMPETING TERMS
interpretations of the following sort:
Tim
We just said, this is - the resistance is a vector going up.
Bruce
Uh-huh.
Tim
And then this is the - mass times acceleration is a vector going down. [g. mass * acc]
Bruce
I see.
Tim
And we subtracted it to get a force. The total for - the sum of the forces. Divided that by
mass to get acceleration.
Here, the mass * acc term is seen as corresponding to the force of gravity acting down, rather than
being associated with the previous value of the acceleration.
Of course, Tim and Steve’s error can be easily explained as a minor slip-up; it is almost certain
that they intend the acc in mass * acc to stand for the acceleration due to gravity. In fact, after
some pretty strong coaxing, they rewrote this line as:
change acc
mass * 9.8 - res
/ mass
Doit
Doit
and they seemed to think that this was in agreement with their original intent. Nonetheless their
lack of inclination to adopt a RECURRENCE stance had important implications here. If Tim and
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Steve could have productively adopted
RECURRENCE,
they would have been in a better position to
understand the strange behavior produced by their original simulation, and perhaps corrected the
difficulty.
This story has a very interesting postscript. After completing their work on the Air Resistance
task, Tim and Steve—like all students—were asked to create the simulation of a mass oscillating
on a spring. After reading the instructions for this new task, Tim and Steve realized that they had
a leg-up on this one, they already knew how to make a simulation that oscillates! They simply
copied their original oscillating air resistance simulation, modified only so that the oscillations
were horizontal rather than vertical.
tick
change res
vel * k
Doit
change acc
mass * acc - res
/ mass
Doit
Doit
change vel
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
This simulation worked quite well and these students were satisfied. Unfortunately, however, Tim
and Steve’s understanding of this simulation was questionable from the point of view of
Newtonian physics. In the following passage, Tim essentially gives a COMPETING TERMS
interpretation of the above statement for the acceleration:
Tim
Just like try - well you can just - basically this- this thing [g.m.O (mass*acc - res)/mass] is
just basically dealing with like (10.0) I guess, forces?
Bruce
Hm-mm.
Tim
You have two opposing, two different force vectors [g. points in opposite directions with
each hand], and they’re- and how they’re - how they’re (effect- changing) each other?
Bruce
Hm-mm. And what’re those - what are those two different forces?
(16.0)
Tim
One’s kinda the - (2.0) one has to do with I guess the mass of the ball or whatever this is.
Block. And like, (1.0) - when you like pull it out, [g. pulling block out to the right] it has um the spring wants to pull it back that way [g. left] so it starts moving and it gets - there’s this
force vector I guess (4.0),, It kinda has to do with like the, I don’t know, inertia? Of the
block? Once it starts moving? That way. I don’t know.
(4.0)
Bruce
Hm-mm.
Tim
And the other - other force is the force that the spring exerts on the block.
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Bruce
Hm-mm.
Tim
So there’s like the force of the block and the force of the spring.
In this passage it is clear that Tim understands the change acc line as computing the total force
and dividing by the mass. One of these forces is associated with the force exerted by the spring
and the other is associated with the “inertia” of the block. This is not a correct description of the
situation from the point of view of Newtonian mechanics. A physicist would say that there is only
one force on the block, the force exerted by the spring.
Although Tim’s interpretation is not really appropriate, it turns out that this simulation is
correct. Not only does it produce oscillations, it is also in agreement with the usual assumption
that the force exerted by a spring is linear in the position, though it is not a trivial exercise to see
that this is the case. But a first step toward a correct understanding of this unusual simulation
would be to adopt a RECURRENCE stance toward the change acc statement.
One possible conclusion we can draw from these episodes is that Tim and Steve are not very
strongly disposed to employ
RECURRENCE.
Conceivably, if they were more experienced in
programming-physics, Tim and Steve would be more aware that some statements require this
type of interpretation, and they might be able to come up with useful interpretations of even these
tricky expressions. Part of the problem here is almost certainly interference from all of their
experience with GENERIC MOMENT interpretation in algebra-physics. Expertise in programmingphysics might very well lead to an expansion in how frequently
RECURRENCE
is employed, and to
more flexible use.
Conclusion
We have now taken our first steps into the structured and meaningful world of programs. As
for algebra-physics, this step took the form of laying out the knowledge—forms and
devices—that is associated with recognizing a certain type of meaningful structure in programs.
Along the way, I frequently emphasized the caveat that these students have much less experience
with programming-physics than did their opposite numbers in algebra-physics. Nonetheless, it
turns out that they quickly learned to see a great deal of meaningful, physics-relevant structure in
programs. Part of the reason for this, I argued, is that much can be borrowed from algebraphysics, particularly for interpreting certain types of individual programming statements.
Even though much can be borrowed from algebra, most of my time in this chapter was spent
describing the forms and devices in programming-physics for which there were no obvious
338
algebraic counterparts. In the forms arena, I argued that the big story was the addition of forms
involving patterns that span lines of programming. Most of these line-spanning forms were in a
new cluster—the Process Cluster. Also, we encountered a new type of Dependence form,
DEPENDENCE,
SERIAL
which involves chained dependence relations of multiple physical quantities.
The story for devices was a little shorter and less dramatic—there were only two new devices.
One of these, RECURRENCE, is a new type of stance toward single programming statements. The
other new device, TRACING , is an absolutely critical and extremely common new device that is
closely related to what we might call “mental running.”
With this chapter, I have laid out the basic elements in the form-device system of
programming-physics. We have seen where and how forms like those in algebra-physics can be
applied, and I have described the new forms that appear. Now, we are ready to further pin down
some of the details of this practice. In the next chapter, I will attempt to establish through a
systematic analysis that the Theory of Forms and Devices can adequately account for the
necessary phenomena in programming-physics—the subset of phenomena that they should
account for—much as I did in the systematic analysis of the algebra-physics corpus described in
Chapter 6. I will attempt to show, in this manner, that the theory works, that the lists of forms
and devices presented here are sufficient, and I will obtain specific counts indicating the relative
frequency of the various forms and devices.
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Chapter 10. Systematic Analysis of the Programming
Data Corpus
The broad strokes of my analysis of programming-physics are now in place. Because I gave a
number of examples—and simply because I showed how it could be done—I believe that the
discussion of the previous chapter makes it plausible that the Theory of Forms and Devices can be
applied to programming-physics. Now, as I did for algebra-physics, I will fill in the
methodological basis for the assertions made in that previous chapter. In particular, I will describe
how my account of the forms and devices of programming-physics was arrived at through a
process of systematically coding and recoding a large portion of my programming data corpus,
and I will present the detailed results of this systematic analysis.
To clarify, there are essentially two types of results of this systematic analysis. The first result
is simply the lists of forms and devices that are required to describe the data corpus. This “result”
was already presented in the previous chapter and now, in this chapter, I will say how I arrived at
these lists. The second type of result is the set of tallies of the specific forms and devices. These
tallies provide a summary of my data and a more detailed image of the practice of programmingphysics. Just as for algebra-physics—and with the same caveats—the tallying of specific forms and
devices paints a particular kind of picture of programming-physics knowledge. The key
presumption is that these tallies reflect the priority weightings of knowledge elements within the
form-device knowledge system.
Of course, one of the main reasons for producing the characterization of programming-physics
is to compare the set of numbers obtained to those obtained in the systematic analysis of algebraphysics. And, to a certain extent, I will do this comparison in the present chapter. As I present
various numerical results, I will try to provide some added perspective by weighing them against
the analogous values from algebra-physics. However, I will not go too far in drawing conclusions
and interpreting the differences observed. That job will be left primarily to the next chapter,
where I will pull together all of my various observations concerning programming-physics and I
will sum up the differences between programming-physics and algebra-physics. In contrast, the
main work of this chapter will be focused on methodological groundwork. I will first spend
significant time on describing what data I collected, how it was collected, and on my analysis
techniques. Then, when this is done, I will present the numerical results with some preliminary
comparisons to algebra-physics.
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As for the analysis in Chapter 6, there are some limits to the analysis I will present here and
most of the same caveats apply. The analysis process I will describe is highly heuristic, and was not
corroborated by additional coders. In addition, there are limits to what we can conclude due to
the small number of subjects involved and the limited task selection. Nevertheless, I believe that
that results allow for a revealing comparison of the two practices.
About the Data Corpus
Subjects
The participants in the programming portion of this study were all currently enrolled in
“Physics 7C” at U.C. Berkeley, a third semester introductory physics course for scientists and
engineers. This is the same course from which I drew students for the algebra pool and a brief
description of this and preceding courses can be found in Chapter 6. However, since the two
portions of the study were conducted during successive semesters—algebra in the fall and
programming in the spring—the programming pool subjects were actually drawn from a different
class of students taught by a different professor.
Recruitment in this phase of the study worked precisely as it did in the first phase. I went to a
Physics 7C lecture early in the semester and made a brief announcement asking for volunteers.
The students were told they would be paid $6 per hour instead of the $5 per hour paid to the
algebra subjects. I felt that this slight increase was necessary since, as I will discuss below, the
programming study required a somewhat greater time commitment from students. Thus, I
wanted additional inducement for students to stay in the study for its duration.
Once I had a list of volunteers, I selected names from the list at random and telephoned
students. As for the algebra-physics study, I scheduled 7 pairs hoping that at least 5 would finish
the study. It happened that all 7 pairs completed all of the programming tasks.
Of the 52 students that volunteered, 41 were men and 11 women. This high percentage of
male volunteers is not surprising given the fact that the class is predominantly male. The fourteen
students that participated in the study consisted of 12 men and 2 women, with 5 pairings of two
men and 2 pairings of a man and a women. Even though I had 52 total volunteers, the selection
and pairing of subjects was largely dictated by scheduling constraints. The students in Physics 7C
have very busy—and often very similar—schedules, and students needed to agree to a weekly 2hour block, not already occupied, in order to participate in the study.
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It is important to emphasize that I did not screen for programming background in any way;
enrollment in Physics 7C was the only requirement and I did not mention programming in my
recruitment announcement. It turned out that all of the participating students except one reported
that they had received some prior programming instruction, but the range of experience was great.
For example, Tim was a computer science major and stated that he knew how to program in
Basic, Pascal, Scheme, C, and C++. In contrast, Steve, who was Tim’s partner and had gone to the
same high school as Tim, reported that his only programming experience was some brief
instruction he had received in junior high school. Although some students reported experience
between these extremes—for instance, some had taken a single college programming
course—Steve’s experience was closer to the norm.
As I stated above, unlike the algebra subjects, all 7 programming pairs completed the entirety
of the tasks. Thus, I was faced with selecting 5 of these pairs for analysis. I could have chosen these
5 at random but I opted, instead, to make my work easier. It turned out that the speech of 2 of
the 7 pairs was much more difficult to understand, which posed problems during early attempts
at transcription. For that reason, I chose not to include these two pairs in the systematic analysis.
This is certainly not a “principled” maneuver, and language issues are actually important and
interesting for this work. However, given the exploratory nature of this research and the already
high burden of transcription and analysis, I believe that this choice was reasonable.
Tasks and Training
Students’ work progressed through a series of programming tasks and exercises, all presented
within the Boxer programming environment. As shown in Table Chapter 10. -1, the tasks were
divided into 4 types. First, the students were asked to complete a number of training exercises
that mixed instruction in Boxer programming with more specific training in the creation of
motion simulations. This instruction began with an introduction to sprite graphics in boxer—just
making sprites move around the display using the fd and rt commands. Early on, students were
also taught to use the repeat command and to use do-it boxes to make separate procedures.
Training Tasks and
Exercises
Main Body
“Quick” Tasks
Cross Training
3.5 hours
5.0 hours
0.5 hours
1.5 hours
Table Chapter 10. -1. The four main types of tasks in order of execution.
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In the next phase of the training, students were given their first simulation tasks. This included
simulating the motion of a car that accelerates as it moves in a straight line. In addition, students
were asked to simulate the motion of a ball that is dropped and a ball that is tossed straight up.
Since they had not yet been instructed in the use of variables in Boxer, the students were limited
to using a string of forward commands to simulate these motions. An example is Amy and
Clem’s first simulation of a dropped ball, which I presented in Figure 8-6.
Following these first simulation tasks, the students received some additional Boxer training,
mainly having to do with the use of variables. Then they were given some tasks which required
them to use variables to simulate motions, once again involving the motion of an accelerating car.
Finally, there was a brief tutorial on the Tick Model which included examples of its use. All
together, the training took about 3 or 4 hours for students to complete, which generally accounted
for the first 2 full sessions.
The main body of tasks, which immediately followed these training tasks and exercises, took
about 5 hours for students to complete and constituted roughly half of the total time they spent
in the study. The tasks in the main body were, at least in their statement, quite simple. Each of
these tasks just briefly described a motion and asked students to create a simulation of that
motion. For example, as I described earlier, one task asked students to simulate the motion of a
mass oscillating back and forth at the end of a spring:
In this task, you must imagine that the block shown is attached to a spring so that it oscillates back and
forth across the surface of a frictionless table.
(a) Make a simulation of this motion.
(b) If the force, mass, and spring constant don't appear in your simulation, then add them.
This task, though simple in specification, typically took students about 45 minutes to complete.
The study concluded with two more brief segments, the “Quick” tasks and “Cross Training.”
Recall that there were counterparts for each of these segments in the algebra study, but those tasks
were not analyzed and I did not comment on them very frequently. In the programming Quick
tasks, like their algebra counterparts, students were asked a number of short questions that were
designed to be answered briefly. For example, one question asked:
Some students say that the tick model is "obvious." What do you think?
The purpose of these questions was to allow a forum for an open discussion of some issues,
without disrupting the flow of the simulation tasks.
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Finally, the purpose of the Cross Training segment was to engage students in a slightly more
algebra-like practice of programming-physics. This meant adapting some of the programs they
had already written to obtain real numerical results. For example, one question asked students to
use their Toss simulation to find out how high a ball would go if it was thrown straight up at a
speed of 100 miles per hour. This type of task proved to be quite hard for students, in part
because of the difficulty of associating units with the numbers that appear in a simulation
program. Like the algebra Cross Training, the programming Cross Training tasks were not
analyzed and will not be discussed further in this document. The instructions for all of the tasks in
the Main Body, the Quick Tasks, and the Cross Training tasks are reproduced in Appendix C.
1. Shoved Block
Imagine that a block is resting on a table with friction. Program a simulation for the
case in which the block is given a short, hard shove.
How does the motion differ if the block is heavier? Can you show this in your
simulation? Modify your simulation if necessary.
2. Dropped Ball
The idea here is to make a realistic simulation of the motion of a ball that is dropped
from someone's hand.
Using the tick model, redo your simulation of a dropped ball.
3. Tossed Ball
Now we want to imagine a situation in which a ball is tossed exactly straight up and
then falls back down to where it started.
Using the tick model, redo your simulation of a ball tossed straight up.
4. Air Resistance
(a) Make a new simulation of the DROP that includes air resistance. (Hint: Air
resistance is bigger if you're moving faster.)
(b) Try running your simulation for a very long time. (The ball might go off the bottom
and come back in the top, that's okay.) If you've done you're simulation correctly, the
speed of the ball should eventually reach a constant value. (The "terminal velocity.")
5. More Air
Resistance
Your simulation of a ball dropped with air resistance is included in this box.
(a) How does the terminal velocity of a dropped ball depend upon the mass of the
ball? For example, what happens to the terminal velocity if the mass doubles? Can
you show this with your simulation? Modify your simulation if necessary.
(b) Suppose there was a wind blowing straight up. How would your simulation and the
terminal velocity be different?
(c) What if the wind is blowing straight down?
6. Mass on a
Spring
In this task, you must imagine that the block shown is attached to a spring so that it
oscillates back and forth across the surface of a frictionless table.
(a) Make a simulation of this motion.
(b) If the force, mass, and spring constant don't appear in your simulation, then add
them.
Table Chapter 10. -2. The 6 tasks selected for systematic analysis.
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Since it would have been prohibitively time-consuming to analyze the videotape data
associated with all of the tasks, only a subset was included in the systematic analysis. In narrowing
the tasks for analysis, my goal was to have a selection that could be seen as representing a relatively
consistent practice of programming-physics. This meant eliminating the Quick and Cross
Training tasks, since they deviated considerably from what I took to be the core of the practice.
Furthermore, it made sense to eliminate the training tasks, since the students were just being
introduced to programming-physics during this phase of their work. This left the Main Body as a
possible source of tasks and, indeed, the Main Body contains a collection of tasks that are of
roughly the same sort. Table Chapter 10. -2 shows the 6 tasks that were selected for inclusion in
the systematic analysis.
It is worth a moment to explore the correspondence between the programming and algebra
tasks included in the systematic analysis (refer to Table Chapter 10. -3). Students in both
conditions were given a version of the Shoved Block task and were asked to deal with the issue of
how the motion differs if the mass is increased. There were also tasks included in both analyses
pertaining to dropped and tossed balls. In the algebra condition, students were asked how high a
ball travels when it is tossed up with a given velocity. The programming students were asked to
simulate two related motions: a ball that is tossed straight up and a ball that is just dropped. In
addition, both conditions included two tasks pertaining to motion with air resistance, as well as a
task involving a mass on a spring.
Algebra
Programming
Shoved Block
Shoved Block
Vertical Pitch
Dropped Ball
Tossed Ball
Air Resistance
Air Resistance
Air Resistance with Wind
More Air Resistance
Mass on a Spring
Mass on a Spring
Stranded Skater
Buoyant Cube
Running in the Rain
Table Chapter 10. -3. The correspondence between the algebra and programming tasks included in the
systematic analysis.
However, there were no programming counterparts for the last three algebra-physics tasks
listed in Table Chapter 10. -3. This is partly because it is difficult to construct programming
analogs for these tasks. To take a case in point, it is not easy to use programming to deal with
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situations, such as the one in the Stranded Skater problem, that are usually handled using
conservation principles. In addition, it was also necessary to include fewer programming tasks
simply because these tasks took longer for the students to complete.
One final comment about Table Chapter 10. -3: The correspondences indicated in this table
are, in some ways, really quite weak. Although the corresponding tasks pertain to physical
phenomena that are similar in some respects, there are many ways in which the tasks and
phenomena under study are extremely different. For example, in the algebra version of the Mass
on a Spring task, the mass hangs motionless at the end of the spring. In contrast, in the
programming version of this task, the mass oscillates horizontally at the end of a spring. I will say
more about these differences in Chapter 11.
Thus, although the apparent correspondences in Table Chapter 10. -3 are indicative of a real
effort to maintain parallels, we must not forget that this simple table hides strong differences
between the algebra and programming tasks. Furthermore, there is really no way that I could have
forced the tasks in each practice to be identical. Programming-physics and algebra-physics tasks
are, by their nature, different and any attempt to alleviate these differences would alter the very
practices I am trying to study. I will discuss this issue further when I discuss the “matching of
conditions” at the end of this chapter.
Details of the experimental sessions
All of the programming sessions were conducted in a laboratory setting. Each pair sat at a
single computer and, typically, one student operated the mouse while the other typed at the
keyboard. Two cameras were used to capture student activity. One camera, positioned behind the
students, was raised high so that it could see over the students’ heads and record the computer
display. The second camera was positioned off to the side so that the students were visible in
profile. The resulting two images were mixed together into a single videotape image for later
viewing, with the profile image in a small inset in one corner. In this study and my pilot studies, I
found that this was an adequate technique for recording pairs of students while they worked at a
computer.
During the experimental sessions, students essentially progressed through a series of Boxer
boxes, each with its own written instructions. The students would open the box, read the
instructions, program the simulation, and then explain their work to me when they were done. I
tried not to interrupt the students very often while they were in the middle of programming a
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simulation; for the most part, I saved my questions until the simulation was completed. My
strategy in this regard was similar to that employed for the algebra study, with a single exception:
I was quick to help students over certain kinds of programming-related difficulties. This included
helping them deal with Boxer bugs and system crashes, and also intervening when they were
stymied by some of the nitty-gritty of programming syntax.
A typical session lasted between 1.5 and 2 hours and, as shown in Table Chapter 10. -5, all of
the pairs except for Greg and Fred came to the laboratory on 6 separate occasions. Although Greg
and Fred required 8 sessions to complete the tasks, I do not believe that this is indicative of any
systematic difference in their working style. Actually, across all of the pairs, there was fairly wide
variation in the time spent on individual tasks. For example, the time spent on the Air Resistance
task ranged between 25 minutes and 1.5 hours, with Greg and Fred taking about 50 minutes. This
variation just happened to balance out for the other pairs, so that the same number of sessions was
required for each.
# of Sessions
Total Hours
Analyzed Hours
Adam&Jerry
Ned&David
Amy&Clem
Tim&Steve
Greg&Fred
6
10.5
3.0
6
9.0
3.0
6
9.5
2.0
6
11.0
3.5
8
13.0
4.5
Totals
32
53.0
16.0
Table Chapter 10. -5. The time students spent in the experimental sessions.
The total time spent by students in the experimental sessions was about 53 hours, with roughly
18 hours spent on the training exercises, 25 hours on the Main Body, and the remaining 10 on the
Quick tasks and Cross Training. When these 53 hours are narrowed to include only work on the
tasks included in the systematic analysis, the result is about 16 hours of videotape. This is
somewhat more than the 11.5 hours that were analyzed in the algebra portion of the study.
Analysis Techniques
The next step here is to describe how, precisely, I went about analyzing the collection of
videotapes described above. As was the case in the algebra study, the analysis involved an iterative
coding and recoding of transcripts. In overview, I began by first viewing and transcribing all of the
videotapes corresponding to the Main Body of tasks—about 25 hours of videotape. Then, one at a
time, I selected individual episodes for analysis. (Recall that an “episode” is one pair’s work on one
task.) For each of these episodes, I polished the transcript, coded the episode, and refined my
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coding scheme as necessary. After 5 episodes were coded in this manner, I felt that the coding
scheme was stable enough to apply to a large portion of the data. At this point, I proceeded to
code the 30 episodes corresponding to student work on the six tasks listed in Table Chapter 10. 2. I began by viewing each of these episodes again and polishing all of the transcripts. Then three
full iterations of coding these 30 episodes and adjusting the coding scheme were required before a
stable coding was reached.
Now, in the remainder of this section, I will fill in the details of how the analysis was
conducted.
Transcription
The analysis of every episode was based on a transcription of that episode. For the
programming data corpus, creating transcripts that contained all of the needed detail required the
development of some specialized techniques. Not only did I need to record everything that the
students said, I needed to have information concerning what appeared on the computer display at
any time, and I needed to record what students were pointing to as they spoke. This was
accomplished, as it was in the algebra study, by building a transcript that consisted of two
components. In the first component, I recorded all student verbalizations and made brief notes
concerning changes made to the program. In the second, supplementary component, I kept track
of the successive states through which the program passed. To facilitate this tracking of the
program, I created a set of tools in the Boxer environment so that, as I read through the written
component of a transcript, I could essentially “play back” the series of changes that were made to
a program.
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POS
VEL
ACC
539.000
98.000
9.800
Data
Data
Data
TICK
change vel
DROP2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
10 tick
Doit
vel + acc
Doit
change pos
pos + vel
Doit
fd vel
dot
Doit
Data
Data
Figure Chapter 10. -6. Tim and Steve's simulation of a dropped ball.
My approach to the recording of gestures was analogous to that used for recording gestures in
the algebra study; where possible, I treated students gestures as indicating a particular item in the
display, such as a programming statement or variable. For various practical reasons, this was
slightly more difficult than in the algebra study. The most important difference was the simple
fact that the characters on the computer display were much smaller than the symbols written on
the whiteboard by the algebra subjects. In fact, a student’s fingertip could often span several
characters on the display! However, this difficulty was somewhat moderated by the fact that the
programming students did not always gesture with their hands; they frequently used the mouse
pointer or the typing cursor to indicate a specific location on the screen (refer to the notations
described in Appendix A). For example, in the following passage, Tim explains the simulation of
the dropped ball shown in Figure Chapter 10. -6, using the mouse pointer to indicate various
items on the display. Here, the “g.m” notation indicates a gesture employing the mouse pointer.
Tim
Um, we just did the regular setup.
Bruce
Hm-mm.
Tim
To take us up there [g.m top of dot trail in graphics box] and then set position
and velocity to zero. [g.m change pos 0; change vel 0] And then we just have
acceleration, nine point eight. [g.m change acc 9.8] Which is so it’s gravity.
Bruce
Hm-mm.
Tim
And then we just repeat it ten times it just (0.5) falls. [g.m along dot trail in
graphics box]
In this passage, Tim points back and forth between the trail of dots in the graphics box and
various lines in the program.
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Actually, the students were quite expressive in their use of the mouse pointer. They did not
only “point” with it, they employed a range of different gestures. For example, a circling motion
was often used to more clearly include and exclude portions of the display from the target of the
gesture. And sometimes students acted out motions by moving the mouse pointer over a graphics
box. Of course, the range of possible gestures was more limited than what students could do with
their hands, but this was still a quite powerful means of gesturing and one that is well-tuned to a
computer display.
The base-level coding scheme
Once an episode was transcribed, the goal was ultimately to code the resulting transcript in
terms of forms and devices. However, as in the algebra analysis, an intervening step was required
before the jump to forms and devices could be made. The problem is that the raw transcripts do
not tell us what events within the transcripts need to be coded. Thus, before coding the forms and
devices, there needed to be a prior step in which the relevant events were identified. In addition,
this preliminary coding provides some information concerning the character of the data corpus.
In general, the base-level analysis was geared toward identifying two types of events which
would then be coded in terms of forms and devices:
(1) A programming expression is constructed by a student.
(1) A student interprets an expression.
As for algebra, the identification of these two types of events requires two different kinds of
analyses. First, events of type (1) require an expression-centered analysis that focuses on coding
pieces of a program. And events of type (2) require an utterance-centered analysis. This latter
analysis takes utterances from the transcript and decides if they qualify as interpretations. I will
now, in turn, describe each of these two kinds of analyses.
Expression-centered coding: Identifying constructed expressions
The first part of the base-level coding is the expression-centered analysis. To begin, I want to
remind the reader of how this analysis worked in the algebra study. The idea there was to look at
every expression that a student wrote and decide if it was remembered, derived, or constructed.
However, before coding the expressions in this manner I had to decide what counted as “an
expression.” In the end, I decided to treat any strings of more than one symbol as an expression.
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This included equations as we usually understand them, as well as strings of symbols that did not
include an equal sign.
Thus, the very first step in adapting my analysis scheme to programming-physics is to decide
what counts as “an expression” in programming. It turns out that, although many adjustments will
be necessary to apply my analysis techniques to the programming data, this first issue is perhaps
the trickiest. In the case of algebra, I treated a page of equations as a set of individual expressions
to be coded separately. However, in Chapter 9 I argued that, unlike in algebra, programmingphysics involves symbolic forms that span statements. Furthermore, even more than in algebra, it
is clearly the case that a program is more than just a list of independent statements.
Nonetheless, it turned out that an analysis based primarily around individual programming
statements worked quite well for most circumstances. I looked at each individual programming
statement, such as change vel 25 or change acc force / mass and coded it as remembered, derived,
or constructed, just as I did for algebraic expressions. The only exception to this rule was when
statements were moved or copied as blocks. These cases were treated as single events to be coded.
So, the basic move here is to treat a program very much like I treated a blackboard full of
equations, coding each individual statement. However, as we will see, there is one very important
difference that is not captured by this gloss. In coding a statement as remembered, derived, or
constructed, the location of the statement within the program must be taken into account. Thus,
unlike algebra, I treated a program as a separate but ordered collection of statements.
In a sense, this approach is only an approximate technique for identifying expression-writing
events in programming. A more precise approach would be to identify clues in student activity
that bound expression-writing events. Nonetheless, I believe that the statement-oriented approach
I employed is a reasonable one. We have already seen that there is a great deal of interesting action
at the level of individual lines. Furthermore, we should realize that any sort of practical breakdown
is likely to be somewhat arbitrary. All I am really looking for here is a means of pinning codes to
the data in order to force some type of systematicity on the analysis, and to obtain some rough
measures of the frequency of various forms and devices. This statement-based approach does this
job in a way that is feasible to implement.
There are a few other fine points to be mentioned here concerning what should be counted as
an expression. First, it is sometimes the case that programming statements serve as arguments to
other programming statements. The primary case that appeared in my data corpus involved the
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repeat command. For example, consider this line taken from Greg and Fred’s Air Resistance
simulation:
repeat 200
change force mass * acc - vel
change acc force / mass
tick
Doit
In such cases, the repeat statement, treated as a whole, was considered to be an expression. In
addition, each individual statement inside the repeated procedure was also treated as an
expression to be coded.
Some other fine points have to do with modifications. Just as in algebra, when a statement was
modified, the resulting statement was treated as a new expression to be coded. Note that this
includes the moving of a statement to a new location within a program.
In addition, there are a few types of modifications and additions that I chose not to count as
expression-writing events:
•
The correction of “typos” or other simple errors in programming syntax.
•
Adjustments to numerical values that appear within a program. In many cases,
however, there were interpretive utterances associated with these adjustments (which
were coded).
•
The placing of the Tick Model. It turned out that the Tick Model was created
automatically as a first step in every student simulation.
With these last few exceptions, I have laid out the basic guidelines I used in identifying
expressions. Once the expressions were identified each one was coded as remembered, derived, or
constructed. Again, the analysis strategies here are very similar to those of the algebra study, but a
few comments on each of these categories are merited.
Remembered
A remembered expression was one that a student appeared to have just written “from
memory.” The evidence for coding an expression as remembered was essentially the same as that
used in the algebra study. If a complicated expression was produced without substantial effort or
comment, this constituted evidence that an expression was remembered. Furthermore, if I
recognized the expression as one that students were familiar with, this added weight to the
coding.
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There are just a few programming-specific comments to be made concerning this category.
First, in some cases, students used the “copy and paste” feature of the programming environment
to duplicate statements. The resulting expressions were coded as remembered.
Second, lines that are essentially transcribed versions of algebraic expressions were treated as
remembered, with a single exception that I will discuss below. For example, if a student wrote
change force
- k * pos
Doit
this was, in most cases, understood to be a transcribed version of the remembered equation F=-kx.
In addition, in some instances, students performed extended derivations on a sheet of paper and
then transcribed the result into their programs. These transcribed statements were also coded as
remembered.
The rationale for this is that, first, these transcribed statements do not require active
construction. The transcribing of algebraic expressions into programming is, in many cases,
relatively straightforward. In addition, I wanted to reserve the derived coding for expressions that
were written based on manipulations of existing programming statements (see below). Thus, in
coding these transcribed statements as remembered, I am using this as something of a catchall
category for some kinds of expressions that are not derived and do not involve active construction.
(Keep in mind that the primary goal of the base-level analysis is to identify constructed
expressions.)
Finally, some programming statements are no more than a one word call to a procedure such
as tick. These statements were usually coded as remembered since they required no active
construction on the part of the student. However, in some cases the placement of these simple
statements constituted a novel construction. I will say more about this in a moment.
Derived
In the algebra study, an expression was coded as derived when it was written by applying rulesof-manipulation to existing expressions. Since there are no formal rules-of-manipulation associated
with programming languages, it may seem that this coding should not apply here. However,
although it is somewhat rare, there are some programming events which I believe are aptly treated
as derived.
The most clear instance of derivation in programming is when algebraic manipulations are
applied to algebra-like expressions that appear within a programming statement. To illustrate,
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consider the following line which appeared in an early version of Ned and David’s Shoved Block
simulation:
repeat
30 /
tick
-1 * acc
Doit
Doit
Later in their work, the expression for the number of repeats was simplified, an event that was
coded as derived:
repeat
-30 / acc
tick
Doit
Furthermore, there were a number of very simple, almost trivial, changes that I treated as
derivations. One example of such an event was when the name of a variable was changed to
something that a student deemed more appropriate. In addition, a derived coding was given when
a numerical value that appeared in a program was replaced by a variable, as when
change force
-.3 * pos
Doit
was changed to
change force
- k * pos
Doit
Finally, and most interestingly, there were a few more dramatic instances of programmingspecific modifications in which programs were replaced by equivalent programs. For example,
Tim and Steve’s work on the Toss simulation eventually reached the following stage:
toss
drop
setup
change
change
change
repeat
drop
change acc -9.8
repeat 9 tick
pos 0
vel 98
acc -9.8
10 tick
Doit
Doit
I will tell this story in more detail later. For now, I just want to mention that Tim and Steve
eventually realized that the above simulation could be replaced with the following equivalent
program:
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toss
setup
change
change
change
repeat
pos 0
vel 98
acc -9.8
19 tick
Doit
Instances of this sort, in which programming statements were replaced by simplified but
equivalent alternatives, were coded as derived.
Constructed
Except for a few other cases that I will discuss in a moment, all of the remaining expressionwriting events fall into this last category. Constructed expressions were expressions that were
invented on the spot by students. I have argued that, in some cases, these constructions are much
like the expressions constructed by the algebra subjects. To illustrate this, in Chapter 9 I described
Tim and Steve’s construction of an expression for the force of air resistance
change Res
vel * .1
Doit
and Greg and Fred’s construction of an expression for the acceleration in the Shoved Block task:
change acc -.08 - mass
However, there is one very important, programming-specific issue to be dealt with here. Even
when a programming statement is a precise duplicate of a statement from an earlier task, the
choice of its location within the program may constitute an act of construction of the sort under
investigation here. It was essential to keep track of these events since, like all these construction
events, they must be accounted for in terms of forms and devices.
With regard to the ordering of statements, students had less experience to draw on, since their
experience from algebra-physics was not as directly relevant. Really, the arrangement of
statements in a program could only be coded as remembered if it followed the ordering of
statements in one of the programs that the students wrote earlier, or that they were shown during
their training. Nonetheless, it turned out that, in the vast majority of cases, the location of
statements could be treated as remembered. This is because almost all of the students’ programs
followed the same structure—the initialize-and-then-repeat structure that I have mentioned on
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several occasions. It is primarily when students departed from or added to this basic structure that
I treated the location as constructed.
I have one last comment on the constructed category. In general, when work was done on a
sheet of paper and then an expression was transcribed into a program, I coded the expression
writing-event as remembered. There was an important exception to this rule however. If a single
expression was constructed on a sheet of paper and then directly transcribed, without any prior
paper-based manipulations, I coded the expression as constructed. The rationale for this is that the
forms that could be seen in the programming expression are basically the same as those in the
algebra version of the expression. And, in fact, it was sometimes hard to tell whether what the
students wrote on the sheet of paper was supposed to be a programming statement or an algebraic
expression.
Marginal cases
As in algebra, there were a few marginal cases that were treated separately. In general, these are
expressions that, while not literally remembered, were also not worthy of being called “novel
constructions.” The two marginal cases here are completely analogous to their algebra-physics
counterparts and the reader is referred to Chapter 6 for a more complete discussion. First, as in
algebra, some expressions were coded as principle-based. This coding was much more rare in
programming-physics, however.
As we will see, codings of the second marginal case—value-assignments—were extremely
common in programming. Recall that in algebra-physics, value assignment statements were
expressions in which a single quantity is equated to a numerical value, such as vo =0. Similarly, value
assignment statements in programming set a variable to a specific value, as in change acc -9.8.
Although these marginal cases were kept track of separately in the base-level analysis, they
were not treated further in later stages since these events do not involve the construction of an
expression.
Summary of expression-centered coding
Table Chapter 10. -7 provides an overview of the expression-centered component of the baselevel analysis. In summary, after transcribing an episode, I identified the expressions that were
written in the episode. Each of the identified expressions was then coded as remembered, derived,
constructed or as one of the marginal cases. The main function of this component of the base-level
analysis was to identify constructed expressions, so that these could later be coded in terms of
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forms and devices. Any individual programming statement may be considered constructed because
of its content or its location within the program.
Expression-Centered
Coding Scheme
Remembered
Derived
Constructed
Marginal Cases:
Value Assignment
Principle-Based
Combinations
Table Chapter 10. -7. Summary of categories for coding programming expressions in the base-level analysis.
Utterance-centered coding: Identifying interpretations of equations
The second component of the base-level analysis was the utterance-centered coding. The
purpose of the utterance centered coding was to identify all of the utterances in an episode in
which a student “interpreted” all or part of a program. In some respects, this component of the
base-level analysis was easier to adapt for use in programming than was the expression-centered
coding. The expression-centered coding of the programming corpus was complicated by the fact
that it was difficult to know how to segment a program into “expressions.” In contrast, in the
utterance-centered coding, the goal is to pick out “utterances.” While the segmenting of a
programming transcript into utterances can certainly be a difficult task, it is largely the same task
in programming-physics that it was in algebra-physics.
In my analysis of the algebra corpus, my tactic for capturing all of the interpretive utterances
in an episode was to cast a wide net. Here I will use the same approach. Operationally, this means
taking every statement that referred to or incorporated any part of an expression. Each of the
following examples includes an interpretive utterance. Note how, in each sample passage, the
students point to some part of the program as they speak.
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shove
setup
change
change
change
change
pos 0
vel 12
mass .08
acc
- mass
Doit
repeat
150 - mass * 2
tick
Doit
Doit
Greg
… So that's why we thought that this repeat [g. 150 - mass * 2] is kind of like a
distance. So we tried to make it less than our initial, um, one fifty. [g. 150] So we
subtracted the mass [g. - mass * 2] a factor of the mass.
_______________
drop2
setup
change
change
change
repeat
pos 0
vel 0
acc 9.8
10 tick
Doit
Tim
Um, we just did the regular setup.
Bruce
Hm-mm.
Tim
To take us up there [g.m top of path] and then set position and velocity to zero. [g.m
change pos 0; change vel 0] And then we just have acceleration, nine point eight.
[g.m change acc 9.8] Which is so it’s gravity.
Bruce
Hm-mm.
Tim
And then we just repeat it ten times it just (0.5) falls. [g.m along path in g-box]
_______________
change acc
force - drag
/ mass
Doit
Doit
change drag
vel / 1.001
Doit
David
Um, yeah, because at, at the terminal velocity, this value for vel [g. vel in change
drag line], the drag will equal the force. [g. drag, force, drag in change acc
line] So then you have no - no more acceleration. And so as - as uh - as velocity
approaches this value [g. 1.001] times the force, [g. force],
Bruce
Hm-mm.
David
um, the acceleration decreases. And so, as it gets close, the acceleration decreases
slower and slower and slower until finally it reaches zero.
A sample base-level coding
Before moving on to the next phase of the analysis, I want to go through the base-level analysis
of one very brief episode. This episode is Tim and Steve’s work on the Drop simulation, and it
was one of the shortest episodes in my corpus, lasting only about 6 minutes. Tim and Steve’s
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finished simulation for this task was shown earlier in Figure Chapter 10. -6. This episode was also
discussed in Chapter 8.
The entire transcript for this episode is provided in Table Chapter 10. -9, annotated with the
appropriate base-level codings. Reading through this transcript, we see that the very first thing that
these students did was to press the key that creates the Tick Model box. Students nearly always
began work on a task by placing a Tick Model box in their working environment and, as I
specified above, this act was not coded in any way.
After placing the Tick Model box, Tim and Steve proceeded quickly to write all of the lines in
the drop2 procedure. The first statement in this procedure, setup, is a simple call to a procedure
that takes no arguments. Such one word calls are coded as remembered. Three initialization lines
follow this first statement, each of which is coded as a value assignment. Finally, the repeat
command at the end of the procedure is coded as remembered.
So, none of the expression-writing events in this episode were coded as constructed. The fact
that all of these lines were written quickly and with little discussion is good justification for these
codings. Furthermore, this program follows the initialize-and-then-repeat structure, so there is no
need to code any of the statements as constructed due to their location within the program.
Up to this point in the transcript, the episode is not very interesting, at least from the point of
view of forms and devices. In fact, Tim and Steve’s statements are little more than some partial
narration of what they were typing into the program. When Tim and Steve tried their program,
however, things got a little bit more exciting. As I described in Chapter 8, the sprite initially went
in the wrong direction—it moved upward instead of falling downward. To correct this problem,
the students had to initialize their acceleration variable to a negative value, rather than a positive
value. According to the specification layed out above, this modification does not get coded in any
way because Tim and Steve have only altered a numerical value that appears in a program.
However, as is common, there was an interpretive utterance associated with this modification.
After this modification, Tim and Steve were satisfied with their program and they explained
their simulation to me. The episode then concluded with two final interpretive utterances.
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[They make a tick model box and then look at their old work
silently.]
Tim
Tim
Steve
Tim
Steve
Tim
Steve
Tim
Tim
We need a drop two.
So. We should do setup first, right? [w. setup] Then we wanna-,,
(10.0) Set the position, [g. pos:()] Set the velocity. [g. vel:()]
Acceleration we can set to [g.m acc:()]
nine point eight?
To negative nine point eight I guess, huh.
Yeah.
That’s it, isn’t it? Then the tick? (Right?)
Yeah.
[w. change pos 0; change vel 0; change acc -9.8. ]
Repeat. [w. repeat |] How many times? (1.0) … Let’s say ten.
[w.mod repeat 10 tick] Tick.
tick model box
Uncoded.
setup
Remembered expression.
change pos 0
change vel 0
change acc -9.8
Value assignment.
repeat 10 tick
Remembered expression.
[Tim runs the simulation and the ball goes up.]
Tim
It went the wrong way. [laughs]
[T runs it a few more times.]
Steve Yeah, cause - oh, this is subtracting it. [g. change vel
[vel + acc]]
Tim
So this should be a positive in here. [w.mod change acc 9.8]
That’s kinda weird.
[They run the program a few times and it looks okay. They call me over to
see.]
Bruce You wanna just explain what it does? How it works?
Tim
Um, we just did the regular setup.
Bruce Hm-mm.
Tim
To take us up there [g.m top of path] and then set
position and velocity to zero. [g.m change pos 0;
change vel 0] And then we just have acceleration,
nine point eight. [g.m change acc 9.8] Which is so it’s
gravity.
Bruce Hm-mm.
Tim
And then we just repeat it ten times it just (0.5) falls.
[g.m along path in g-box]
Bruce Hm-mm.
Tim
So, every time - every tick it gains nine point eight.
The acceleration increases the velocity by nine point
eight.
Bruce Hm-mm. Okay. Sounds good.
“cause…is subtracting
it.”
Interpretive utterance.
change acc 9.8
Uncoded.
“Um, we just did the
regular setup.…repeat
it ten times it just falls.”
Interpretive utterance.
“…every tick it gains
nine point eight …”
Interpretive utterance.
Table Chapter 10. -9. Complete transcript of Tim and Steve’s work on the Drop simulation. Base-level
codings are given in the right-hand column.
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Coding forms and devices
The base-level analysis identified two types of important events in the data, events in which an
expression was constructed and events in which students produced an interpretive utterance. After
the base-level analysis was complete, the next task was to code each of these important events in
terms of forms and devices. Just as in the algebra-physics analysis, the coding of events in terms of
forms and the coding in terms of devices each required me to perform two tasks simultaneously:
(1) define the lists of forms and devices and (2) code each individual event in terms of the forms
and devices that are on these lists.
Principles for defining the set of forms
Principles for defining the set of devices
Coverage of the data corpus
Coverage of the data corpus
Overall parsimony of the set
Overall parsimony of the set
Inclusion of a symbol pattern
Close relation to the structure of interpretive
utterances
Symbol patterns only preserve distinctions
consequential for interpretive behavior
Restriction to reliably codable distinctions and
allowance for ambiguity
Impenetrability
Continuity with prior knowledge
Continuity with prior knowledge
Approximation to expertise
Approximation to expertise
Table Chapter 10. -11. Summary of the heuristic principles used to guide in defining the lists of forms and
devices.
Evidence for recognizing forms in specific
events
Evidence for recognizing devices in specific
events
Corresponding symbol pattern
Specific verbal clues
Specific verbal clues
Corresponding inferences
Related graphical concomitants
Related graphical concomitants
Corresponding symbolic conventions
Global as well as local evidence
Global as well as local evidence
Table Chapter 10. -12. Summary of the types of evidence used to recognize forms and devices in particular
events.
Although the resulting lists were different, the procedure employed here was essentially the
same as that used for the analysis of the algebra corpus. I iteratively coded and recoded the data
until the lists of forms and devices and individual codings were stable. This process began with
lists of forms and devices derived from a pilot study. Then I coded 5 focus episodes, refining my
categories as I went. Finally, I took three full passes at all of the episodes included in the
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systematic analysis. As for the algebra analysis, this whole process relied on a set of heuristic
principles and some specific rules for recognizing individual forms and devices. The principles
used in defining the lists of forms and devices are summarized in Table Chapter 10. -11. In
addition, Table Chapter 10. -12 has a summary of the types of evidence used for recognizing
forms and devices in specific events. The material summarized in these tables is described in more
detail in Chapter 6, and the specific rules for recognizing individual forms and devices are given in
Appendices D and E.
On the Òsubstitution and execution machineryÓ
Before proceeding, I need to add one small addendum concerning the coding of forms. Recall
that in the analysis of the algebra corpus, some of the events identified by the base-level analysis
could not be coded in terms of forms. In some cases, I argued, another class of resources did the
work that I might otherwise attribute to forms. The idea was that students come to their learning
of physics with a number of well-developed skills relating to the use of symbolic expressions.
Among these skills are the ability to manipulate expressions to derive new expressions, and the
ability to substitute numbers into expressions. I argued that some interpretations given by students
seemed to draw on elements of this “substitution machinery” rather than on symbolic forms.
This same substitution machinery can play a role in producing interpretations of programming
expressions, especially in the case of the “algebra-like” programming statements discussed in
Chapter 9. Furthermore, in the case of programming, this category needs to be extended to
include some general “execution machinery.” The point is that the students in my study quickly
developed the ability to “mentally” execute certain programming instructions. For example,
students looking at the simple sprite graphics program:
fd 50
rt 90
fd 30
…
could mentally step through this program, showing with their hands how the sprite would move as
each instruction was executed, first moving forward, then turning right, and then moving forward
again. Such behavior does not require inferences based on symbolic forms. Instead, it draws on
specific knowledge concerning each statement that appears in the program. Thus, events of this
sort are not assigned any coding in terms of forms.
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Coding the sample episode
Before presenting the results of the analysis, I want to briefly show how the sample episode
discussed above was coded in terms of forms and devices. Since there were no constructed
expressions in the sample episode, only the three interpretive utterances need to be coded. In the
first of these interpretive utterances, Steve points to the line change vel vel + acc and says: “this is
subtracting it.” I coded this utterance as involving the BASE ±CHANGE form, with the idea that the
acc in vel+acc is seen as applying a change to the base value in vel. The device associated with this
utterance is RECURRENCE since the “subtracting” described by Steve transpires on each tick of the
program:
Steve
Yeah, cause - oh, this is subtracting it. [g. change vel
[vel + acc]]
Tim
So this should be a positive in here. [w.mod change acc 9.8]
That’s kinda weird.
“cause…this is
subtracting it.”
Form: BASE±CHANGE
Device: RECURRENCE
The evidence for these codings here are actually quite weak; in fact, this is a case of especially
weak local evidence. In truth, there are many other ways of interpreting Steve’s statement. This is
where the “Global as well as local evidence” heuristic comes in. Even though there is not much
evidence in this specific event, I saw many other examples where more extensive, but similar,
interpretations were given of this same programming statement. Thus, I was able to use those
other examples to help in coding this event. In fact, as we shall see in a moment, the last
interpretation in this very episode can help in this manner.
Both the second and third interpretive utterances in this excerpt showed up as examples in
Chapter 9, so we have a head start on analyzing them. The first of these was a TRACING
interpretation:
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Tim
Bruce
Tim
To take us up there [g.m top of path] and then set
position and velocity to zero. [g.m change pos 0;
change vel 0] And then we just have acceleration,
nine point eight. [g.m change acc 9.8] Which is so it’s
gravity.
Hm-mm.
And then we just repeat it ten times it just (0.5) falls.
[g.m along path in g-box]
“Um, we just did the
regular setup.…repeat
it ten times it just falls.”
Form: SETUP-LOOP
Device: TRACING
To see that the device here is TRACING , note that Tim’s utterance follows the structure of the
program, with statements separated by the phrase “and then.” And, in Chapter 9, I argued that
the
SETUP -LOOP
form is behind this interpretation, since Tim is associating portions of the
program with two sub-processes, one that prepares for the motion, and one in which the object
“just falls.”
The episode concluded with this third and final interpretive utterance:
So, every time - every tick it gains nine point eight.
The acceleration increases the velocity by nine point
eight.
Tim
“…every tick it gains
nine point eight …”
Form: BASE+CHANGE
Device: RECURRENCE
As I argued in Chapter 9, the device here is RECURRENCE, since Tim is announcing what transpires
on each Tick of the program. The form associated with this event is BASE ±CHANGE . As in the first
interpretive utterance in this episode, the original velocity in the line change vel vel + acc is seen as
a base onto which the acceleration is added. Tim’s use of the word “gains” here is evidence for this
coding.
Summary of Results
Base-level results
I am now ready to present the results of the systematic analysis of the programming corpus.
As I said at the start of this chapter, I will provide some perspective by comparing these results to
those obtained for algebra-physics. However, I will not go much beyond a simple comparison of
the raw numerical values. Instead, my careful thinking about the implications of any differences
will be reserved for the next chapter.
First, I will summarize the outcome of the base-level analysis. As shown in Table Chapter 10. 13, the expression-centered component of the base-level analysis identified a total of 246
364
expression-writing events. Of these 246 events, 70 involved the modification of existing
expressions, and 13 were events in which an existing statement was moved to a different location
in the program.
Expressions “from scratch”
Modifications
Moves
Total expression-writing events
163
70
13
246
Table Chapter 10. -13. Expressions found in the base-level analysis.
Table Chapter 10. -15 contains the results of coding these 246 events as remembered, derived,
constructed, or as belonging to one of the marginal cases. Note that the codings do not total to
246 since, as in the algebra analysis, events may receive multiple codings.
Remembered
Derived
Constructed
Marginal Cases
Total
Codings
With Other
Codings
93
12
61
88
11
2
12
5
Table Chapter 10. -15. Base-level coding of expressions.
Since the purpose of this component of the base-level analysis is to identify constructed
expressions, the highlighted row in Table Chapter 10. -15 merits special attention. It is important
to keep in mind that statements coded as constructed may have received this coding because of the
location of the statement, because of how the statement itself was constructed, or for both reasons.
Of the 61 events coded as constructed, 24 were coded this way because of location, 25 because of
the statement itself, and 2 because of both.
Table Chapter 10. -17 has a breakdown by type of the 88 marginal cases. Notice that the great
majority of these were “value assignment” statements. In fact, these 83 statements constitute a
significant fraction—right around one-third—of the 246 total expression-writing events.
Total
Codings
With Other
Codings
83
5
0
5
Value Assignment
Principle-based
Table Chapter 10. -17. Breakdown of the marginal cases.
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The results of the second component of the base-level analysis, the utterance-centered
component, are fairly simple to report since this component only identifies interpretive utterances,
it does not code them further in any way. As shown in Table Chapter 10. -19, a total of 195
interpretive utterances were identified in the programming corpus. When this number is
combined with the 61 expression-construction episodes, the result is 256 total events that must be
coded in terms of forms and devices.
Interpretive Utterances
Constructed Expressions
Total Events
195
61
256
Table Chapter 10. -19. Programming corpus events coded in terms of forms and devices.
Now I want to take a moment to compare some of these values to those obtained in the
algebra analysis. First, notice that the 246 total expression-writing events is less than half the total
from the algebra corpus, which was 547. Of course, such a result is not necessarily very meaningful
in itself, since the students in each condition spent a different amount of time engaged in a
different number of tasks. In fact, the programming students did do fewer total tasks—6 instead
of 8—but the total time spent by the programming students on the coded tasks was 16.0 hours,
which is more than the 11.3 hours spent by the algebra students. Although a comparison based in
expressions/hour or expressions/task would produce different results, in either case, the “rate” for
programming is somewhat lower.
Interpretive Utterances
Constructed Expressions
Total Events
144
72
219
Table Chapter 10. -21. Algebra corpus events coded in terms of forms and devices.
The comparison gets a little more interesting if we look at the number of expressionconstruction and interpretation events. Comparing Table Chapter 10. -19 and Table Chapter 10. 21, we see that, although the number of expression-writing events in programming was much
lower, the total number of events to be coded was actually a little higher in programming, 256 as
compared to 219. Again, it is not completely clear how to compare these values appropriately. The
number of programming construction and interpretation events is much higher in comparison to
the total number of expressions written, but the number of events per hour is slightly lower
(16/hour versus 19/hour).
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Whether we choose to normalize by expression, by task, or by hour, the above results paint an
image of a particular class of differences between the programming and algebra-physics corpuses:
The programming students wrote fewer expressions, but interacted meaningfully with
(constructed or interpreted) a larger fraction of these expressions than the algebra subjects.
Furthermore, these two effects balanced out to a certain extent, so that the number of
construction and interpretation events per hour were in the same ballpark.
Assuming, for the moment, that this result reflects a real difference between programmingphysics and algebra-physics, what can we make of this observation? There could be a number of
reasons for the differences observed here. Some of these reasons probably have to do with the fact
that the algebra subjects were quite familiar with the practice involved while the programming
subjects were learning a new practice. Given this difference in familiarity, one would expect the
programming subjects to proceed more slowly, and thus to generate fewer equations per hour.
Furthermore, since they have less of a body of experience to draw on, it is not surprising that they
needed to invent more of the expressions that they wrote, and that they were more likely to
discuss and interpret expressions.
But I also think that these results point to some real differences—differences that would
remain even if the students were equally familiar with the two practices. Recall that in the algebraphysics corpus, many of the total expressions that students wrote were produced by manipulating
expressions that were already written on the whiteboard. In fact, of the 547 expressions in the
algebra-physics corpus, 241 were produced only by manipulating existing expressions (i.e., they
were coded as derived and received no other codings). This is not surprising. In caricature, the
practice of algebra-physics involves first composing expressions, then pushing those expressions
through intermediate states, and finally doing something with the final expression obtained. The
“intermediate” expressions do not need to be constructed and are often not commented upon. In
contrast, programming-physics does not require the generation of these intermediate expressions.
In programming-physics, you compose the simulation and run it, then the computer does the rest.
There’s no generating of additional expressions, just the running of the original ones. Thus, you
might expect more expressions in algebra-physics, and you might expect students to interact
meaningfully with a higher percentage of programming expressions.
However, even if the numbers presented in the above tables are presumed to appropriately
describe my data corpus, there are reasons to doubt that these results somehow describe
programming-physics as it would exist outside my study. The results of the base-level analysis by
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subject pair are shown in Table Chapter 10. -22. Notice that, as for algebra-physics, there is
substantial variation across pairs. This observation casts some doubt on whether the rough
generalizations above concerning differences between algebra and programming are, in any way,
universal.
N&D
Expressions
Interpretive Utterances
Constructed Expressions
Interpretations+Constructions
47
38
12
50
G&F
50
52
11
63
A&C
36
18
5
23
T&S
44
35
9
44
A&J
69
52
24
76
Totals
246
195
61
256
Table Chapter 10. -22. Results of the base-level analysis by subject pair.
There are a few other interesting comments to be made concerning Table Chapter 10. -22.
Recall that, in the algebra analysis, the number of interpretation and construction events were not
very strongly correlated with the total number of expressions. Here, more of the variation across
subjects in interpretations (r=.83) and constructions (r=.99) appears to be explainable by
differences in the total number of expressions written (as compared to r=.07 and r=.11 for
algebra). Interestingly, these observations, taken as a whole, can be seen to agree with the rough
differences between algebra and programming described above. If a large component of algebraphysics is the writing of expressions, mostly derived, that are somewhat independent of
interpretation and construction activity, then it makes sense that the number of interpretations
and constructions would not track closely with the total expressions. Similarly, because
programming does not include this component, this could explain the observation that there is
more tracking of construction and interpretation with the total number of expressions.
It is also worth commenting that the strong negative correlation between interpretive
utterances and construction events, observed in algebra, does not show up in Table Chapter 10. 22. Instead, there is a weaker, positive correlation (r=.75). I believe that more data is needed
before we can determine whether there are any interesting generalizations to be made concerning
differences across individuals, particularly due to the fact that any such differences would be
partly obscured by the fact that all of my data is for pairs of students.
I will conclude this discussion of the base-level results with the same point I made in
summarizing the base-level results of the algebra analysis: I believe that the results of this section
can be taken as suggesting that the construction and interpretation of expressions are not rare in
simulation programming. As I commented above, the number of construction and interpretation
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events per hour was just a little less here than in the algebra corpus. In the algebra corpus there
were 219 events spread over 11.3 hours and the programming corpus had 256 events spread over
16.0 hours. Furthermore, there were many more of these important events in programming as
compared to the total number of expressions written.
Results from the coding of symbolic forms
The results of coding the 256 events identified in the base-level analysis in terms of symbolic
forms are given in Table Chapter 10. -24, with results listed both by cluster and individual form.
The values in the “Count” column are the raw number of times that each form was coded. The
numbers in the “percent” column were obtained by dividing these raw counts by the total counts
of all forms. The reader can refer to Appendix D for a brief descriptions of each form.
As I discussed in earlier sections of this chapter, it was not appropriate to code some of the
identified events in terms of forms. I argued that other resources were implicated in these events,
particularly what I called the “substitution and execution machinery.” Of the 256 total events, 27
fell into this auxiliary category.
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Count
Percent
CANCELING
62
35
17
0
10
18%
10%
5%
0%
3%
Proportionality
PROP+
PROP-
63
45
13
18%
13%
4%
COEFFICIENT
RATIO
CANCELING(B)
1
2
Terms are Amount
PARTS -OF -A-WHOLE
Competing Terms
COMPETING TERMS
OPPOSITION
BALANCING
BASE ± CHANGE
WHOLE - PART
SAME AMOUNT
Count
Percent
81
43
5
6
27
24%
13%
2%
1%
8%
SCALING
38
29
9
11%
9%
3%
1%
0%
Multiplication
EXTENSIVE / INTENSIVE
4
4
1%
1%
36
11%
EXTENSIVE •EXTENSIVE
0
0%
0
37
0
0
0%
11%
0%
0%
Process
VARIATION
56
29
9
13
16%
9%
3%
4%
CONSTANCY
5
1%
0
1
0%
0%
Dependence
DEPENDENCE
NO DEPENDENCE
SOLE DEPENDENCE
SERIAL DEPENDENCE
Coefficient
SEQUENT. PROCESSES
SETUP -LOOP
Other
DYING AWAY
NO EVIDENCE
Table Chapter 10. -24. Results for the systematic coding of the programming corpus in terms of symbolic forms.
To begin discussion of these results, I want to examine the relative frequency of forms by
cluster. In Figure Chapter 10. -7, the breakdown of codings by cluster is displayed in a pie chart.
Glancing at this chart, none of the clusters stand out as a clear winner. The Dependence
Cluster—including
SERIAL DEPENDENCE —is
the most common with 24% of all codings. At the
next level are the Proportionality, Competing Terms, and Process clusters, with 18%, 18%, and
16% respectively. In the third echelon are the Coefficient and Terms are Amounts clusters, both
with 11%. Finally, the Multiplication Cluster received the fewest number of codings, only about
1%.
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Terms are Amounts
11%
Multiplication
1%
Dependence
24%
Coefficient
11%
Process
16%
Propotionality
18%
Competing Terms
18%
Figure Chapter 10. -7. Percentage of form codings by cluster.
In many respects, the distribution revealed in Figure Chapter 10. -7 is quite similar to the
distribution observed in algebra-physics. Table Chapter 10. -26 provides a head-to-head
comparison of the relative frequency of the clusters in algebra-physics and programming-physics.
Notice that, in both cases, there are essentially three tiers of clusters. First, in both algebra and
programming, the Proportionality, Competing Terms, and Dependence clusters are the most
common. Then there is a middle tier with Terms are Amounts and Coefficient. And, finally, the
Multiplication Cluster is at the bottom in both practices. Programming also has the Process
Cluster, which falls between the top two tiers.
Algebra
Proportionality
Competing Terms
Dependence
Terms are Amounts
Coefficient
Multiplication
Process
28%
27%
23%
14%
5%
2%
-
Programming
18%
18%
24%
11%
11%
1%
16%
Table Chapter 10. -26. Comparison of form frequencies by cluster.
I believe that this similarity in distribution probably has a lot to do with a fact I have discussed
often, that individual programming statements can be interpreted much like algebraic expressions.
In fact, the similarity revealed in Table Chapter 10. -26 is accentuated if we throw out all codings
of the programming-specific forms—the Process forms and SERIAL DEPENDENCE . When any
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codings of these forms are simply neglected, then the results are as shown in Table Chapter 10. 28, which reveals an even greater similarity between the programming and algebra distribution.
Algebra
Proportionality
Competing Terms
Dependence
Terms are Amounts
Coefficient
Multiplication
28%
27%
23%
14%
5%
2%
Programming
24%
24%
21%
14%
15%
2%
Table Chapter 10. -28. Comparison of form frequencies by cluster with codings of programming-specific forms
neglected.
Let’s take a moment to think about what this observation means. Note that, in casting out
SERIAL DEPENDENCE
and the Process forms, I have eliminated all of the forms that have to do with
line-spanning structure. Thus, Table Chapter 10. -28 suggests that the type of meaningful
structure that students see within programming statements is very similar to what they see within
algebraic expressions. This observation will have a number of implications for the discussion in
Chapter 11.
The most noticeable difference revealed in Table Chapter 10. -28 is the relative frequency of
the Coefficient forms. This, in some degree, likely has to do with idiosyncrasies of the specific
tasks I selected to include in the systematic analysis. However, I believe that this difference also
reflects an important feature of programming-physics. While working on the programmingphysics tasks, the students often spent a large amount of time “tuning” their programs; that is,
they would go through many cycles of running their program, and adjusting the various numerical
parameters that appeared in the program. Frequently, these numerical parameters were
coefficients, such as the one that appears in an expression for the force due to a spring: F= -kx.
Thus, because of the prevalence of this tuning activity, extra attention was focused on coefficients.
The comparison by cluster is summed up by the two bar graphs shown in Figure Chapter 10. 9 and Figure Chapter 10. -11, both of which display the number of counts by cluster. Notice that
the display in Figure Chapter 10. -11 uses light and dark bars to distinguish counts associated
with the programming-specific forms. These two figures primarily serve to highlight the
differences I have already mentioned. First, without the programming-specific forms, the
distributions have a rough overall similarity, with the Coefficient Cluster as the major exception.
In addition, looking at these figures highlights the fact that the Process forms were not
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particularly rare in programming, there was a significant number of codings of these forms as
compared to forms in the other clusters. Finally, Figure Chapter 10. -11 makes clear the
importance of SERIAL DEPENDENCE . SERIAL DEPENDENCE adds a non-negligible amount to the
Dependence Cluster, making that cluster the most common in the programming-physics corpus.
80
70
Algebra
Forms
60
Counts
50
40
30
20
10
00
Competing
terms
Proportionality
Dependence
Terms are
Amounts
Coefficient
Multiplication
Other
Figure Chapter 10. -9. Algebra forms by cluster.
80
Serial
Dependenc
70
Programming
Forms
60
Counts
50
40
30
20
10
0
Competing
Terms
Proportionality
Dependence
Terms are
Amounts
Coefficient
Multiplication
Process
Figure Chapter 10. -11. Programming forms by cluster. Light portions of bars indicate counts of programmingspecific forms.
I will now turn my attention from clusters to a discussion of individual forms. As in algebraphysics, a few forms account for many of the codings, though the distribution is a little more
uniform in programming. Table Chapter 10. -30 and Table Chapter 10. -32 show the most
common forms in programming and algebra. In programming, the PROP+ and DEPENDENCE forms
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leads the way, followed by
COMPETING TERMS
and BASE ±CHANGE . The story was precisely the same
in algebra, with the exception that BALANCING appears in the number four slot, between COMPETING
TERMS
and BASE ±CHANGE . Of course, this is a very significant exception—it is a very important
observation that one of the most common forms in algebra-physics did not appear at all in
programming-physics. This observation will play an important role in the discussion in Chapter
11.
Count
45
43
35
37
29
29
27
PROP+
DEPENDENCE
COMPETING TERMS
BASE ±CHANGE
SEQUENTIAL PROCESSES
COEFFICIENT
SERIAL DEPENDENCE
Percent
13%
13%
11%
11%
9%
9%
8%
Table Chapter 10. -30. The most common forms in programming-physics. All other forms received below 5%
of codings.
Count
52
40
30
28
23
PROP+
DEPENDENCE
COMPETING TERMS
BALANCING
BASE ±CHANGE
Percent
19%
15%
11%
10%
8%
Table Chapter 10. -32. The most common forms in algebra-physics. All other forms received below 6% of
codings.
It is also important to notice that Table Chapter 10. -30 contains two of the programmingspecific forms, SEQUENTIAL PROCESSES and SERIAL DEPENDENCE . This suggests that these new forms
are not inconsequential additions. The analysis of the results by cluster pointed to a similar
conclusion.
In summary, the distribution of forms in the programming-physics corpus seems to share a
common core with the algebra-physics corpus. PROP+, DEPENDENCE, and COMPETING TERMS were
again very common individual forms, and the associated clusters were very common in the coding
of both corpuses. The most interesting differences were associated with forms that appeared only
in one practice or the other. The Process forms and SERIAL DEPENDENCE appeared only in
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programming, and with a reasonable regularity. Furthermore, the absence from programmingphysics of one of the most common forms—BALANCING —constitutes a significant observation.
N&D
G&F
A&C
Competing Terms
Proportionalit
Dependence
Terms are Amounts
Coefficient
Multiplication
Process
13
19
11
6
11
2
6
15
15
24
13
13
5
5
2
4
16
Totals
68
96
T&S
A&J
Totals
7
23
11
8
6
7
1
6
6
13
36
8
7
1
21
62
63
81
37
38
4
56
23
62
92
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Table Chapter 10. -34. Form cluster codings broken down by subject pair.
The above discussion, to a certain extent, presumed not only that the numerical frequencies
described my data corpus, but that the overall average of my corpus was somehow representative
of programming-physics practice. As usual, I will break down my frequencies in order to help
estimate the extent to which this latter assumption is merited. Table Chapter 10. -34, which gives
the results for form cluster codings broken down by subject pair, shows substantial variation across
pairs. This suggests the usual caveats: My results may depend sensitively on the particular subjects
that participated in my study, and, because of individual differences, it may not make sense to
look for a universal distribution.
As in the algebra corpus, however, it is possible to find some weak suggestions of across-pair
uniformities. For example, the Multiplication Cluster received less than 3% of the codings for
every pair. In addition, the three least common clusters overall—Terms are Amounts, Coefficient,
and Multiplication—always received less than 28% of the total codings. In contrast, the three
most common clusters—Competing Terms, Proportionality, and Dependence—always received
more than 50% for all pairs. These observations do not eliminate the importance of the observed
differences, but they do hint at a degree of similarity.
Finally, the results by task are given in Table Chapter 10. -35. There is substantial variation
across tasks, which again suggests that my overall results may depend sensitively on the particular
tasks included in the study. I will comment on this issue further later in the chapter.
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Shove
Drop
Toss
Air
Competing Terms
Proportionalit
Dependence
Terms are Amounts
Coefficient
Multiplication
Process
24
15
17
10
4
2
10
9
15
Totals
72
13
9
137
More Air
42
25
30
4
21
3
Spring
8
9
20
10
7
54
Totals
12
5
16
3
20
62
63
81
37
38
4
56
56
341
Table Chapter 10. -35. Form cluster codings broken down by task.
Results from the coding of representational devices
The results for the systematic coding of representational devices are summarized in Table
Chapter 10. -37. (Refer to Appendix E for a brief description of each device.) In addition, the
results collected by class are shown in a pie chart in Figure Chapter 10. -13. Looking at this chart,
it is clear that Narrative devices dominate. Furthermore, not only is the Narrative class the most
common overall, the three most common individual devices are members of this class: CHANGING
PARAMETERS , TRACING ,
and PHYSICAL CHANGE .
Narrative
CHANGING
PARAMETERS
CHANGING SITUATIONS
PHYSICAL CHANGE
TRACING
Special Case
SPECIFIC VALUE
LIMITING CASE
RESTRICTED VALUE
RELATIVE VALUES
Count
Percent
195
73
63%
24%
18
44
60
6%
14%
19%
GENERIC MOMENT
44
19
6
14%
6%
2%
CONSERVATION
14
5
5%
2%
Other
Static
SPECIFIC MOMENT
STEADY STATE
STATIC FORCES
ACCOUNTING
RECURRENCE
NO EVIDENCE
Count
Percent
67
2
22%
1%
8
0
13
3%
0%
4%
0
4
41
0%
1%
13%
1
1
0%
Table Chapter 10. -37. Results for the systematic coding of the programming-corpus in terms of
representational devices.
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Special Case
14%
Static
22%
Narrative
63%
Figure Chapter 10. -13. Percentage of device codings by class.
If we compare the results summarized in Figure Chapter 10. -13 to the results by class for
algebra-physics, we find the same basic distribution (refer to Table Chapter 10. -39). In both
cases, Narrative devices were the most common, followed by Static and then Special Case devices.
However, in programming, the prevalence of the Narrative Class appears to be somewhat
accentuated, mostly at the expense of the Static Class. Although, as I will discuss in a moment,
this difference may be an artifact of my task selection or the variation across individuals, I will
argue in Chapter 11 that this diminishing in the use of Static devices is precisely what one might
expect given the differing nature of algebra-physics and programming-physics.
Algebra
Narrative
Static
Special Case
54%
35%
11%
Programming
63%
22%
14%
Table Chapter 10. -39. Comparison of device frequencies by class.
The comparison by class is further illustrated by the bar graphs in Figure Chapter 10. -15 and
Figure Chapter 10. -17, both of which display the raw number of counts by class. An examination
of these two figures clearly illustrates the significant role played by the two new devices,
TRACING
and RECURRENCE. In fact, the prevalence of the Narrative Class is largely due to TRACING ; it
accounts for about 30% of all Narrative codings. And the majority of Static codings—about
60%—were associated with
RECURRENCE.
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140
Algebra
Devices
120
Counts
100
80
60
40
20
0
Narrative
Static
Special Case
Figure Chapter 10. -15. Algebra devices by class.
200
Programming
180
Devices
Tracing
160
140
Counts
120
100
80
60
Recurrence
40
20
0
Narrative
Static
Special Case
Figure Chapter 10. -17. Programming devices by class. Light portions of bars indicate counts of programmingspecific forms.
As in my analysis of forms, it is interesting to look at what happens if we throw out the devices
that are associated with line-spanning structure. This only means eliminating TRACING , since that is
the only device that is uniquely associated with line-spanning structure. When counts associated
with the TRACING device are thrown out, then we obtain the results shown in Table Chapter 10. 41.
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Algebra
Narrative
Static
Special Case
54%
35%
11%
Programming
54%
27%
18%
Table Chapter 10. -41. Comparison with tracing eliminated.
A first observation to make about this table is that the distribution has gotten closer to the
distribution observed in algebra-physics. But this table allows another important observation that is
somewhat more surprising. Although Table Chapter 10. -41 shows approximately equal
percentages for Narrative devices, it seems to indicate a shift from Static to Special Case devices in
programming. This is partly a reflection of the diminishing of Static interpretations for the reasons
mentioned above. However, I believe it is also suggestive of another important feature of
programming-physics. As I will discuss in the next chapter, programming forces students to
embed expressions in the particular—in order to run a program, you need to make selections
concerning the numerical values that will be in the expressions that you have written, and you need
to make certain other choices concerning the programming context in which the expressions are to
be embedded. These features of programming may tend to make Special Case interpretations
more prevalent.
In summary, the basic ranking of device classes in the programming-physics corpus was
substantially similar to that obtained for algebra-physics: The Narrative Class was most common,
followed by the Static Class and the Special Case Class. However, the results for programming
were somewhat skewed away from the Static Class to the two other classes, especially to Narrative
devices. In addition, the two new programming-specific devices were both quite common.
To conclude this section, I must again note that the above frequency differences between
programming and algebra might be an artifact of the task selection or the variation across
individuals. In order to help calibrate this effect, it is helpful to break down these results by subject
pair and task. The results for device class codings by subject pair are given in Table Chapter 10. 43. This table indicates some fairly large differences across pairs, both in the total number of
codings and their distribution. For example, Adam and Jerry had more than five times as many
Narrative as Static codings, while the other pairs had closer to 2 times as many Narrative codings
as either other class. But, again, there are some uniformities. For example, the Narrative Class was
the most common for all subjects and it always received over 49% of the total codings. In
contrast, the Special Case Class never received more than 27% of the codings for a pair.
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N&D
Narrative
Static
Special Case
Totals
G&F
33
16
18
67
A&C
48
20
7
75
13
5
5
23
T&S
A&J
33
15
8
56
Totals
68
12
6
86
195
68
44
307
Table Chapter 10. -43. Device class codings broken down by subject pair.
Table Chapter 10. -45 has the results broken down by simulation task. Again we see the usual
strong variation across task, both in number and distribution. However, there is some indication
that the prevalence of the Narrative Class may be reflected across tasks.
Shove
Drop
Narrative
Static
Special Case
41
7
10
3
3
Totals
58
6
Toss
Air
18
More Air
Spring
Totals
2
55
36
15
38
3
8
40
19
9
195
68
44
20
106
49
68
307
Table Chapter 10. -45. Device class codings broken down by task.
Methodological Issues
I will now close this chapter by reflecting on the types of conclusions that we can draw from all
the numbers presented in this chapter and in Chapter 6. There are a number of issues to be
addressed and caveats to have in mind, and I want to have a discussion of these issues behind us
before presenting my more interpretive comparisons in the next chapter.
Just as the results of Chapter 6 can be taken to provide a “description” of algebra-physics,
understood in multiple senses, the results presented in this chapter furnish a description of
programming-physics. There are several components of this description:
1. The very idea that elements like symbolic forms and representational devices exist.
1. The lists of specific forms and devices needed to account for my data.
1. The frequency of individual forms and devices.
Of course, the first two “results” were essentially presented in previous chapters. However, in
describing the systematic analysis techniques in this chapter, I have supplied some of the
methodological basis underlying the assertions of those preceding chapters.
Much went into the building of the lists of forms and devices, and attaching forms and
devices to events in the data corpus. This was a highly heuristic process—I did not describe an
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exact and rigorous technique for extracting the coding categories from the data. Furthermore, the
coding of individual episodes may be questioned since I did not develop a scheme so that my
codings could be corroborated by a second coder. For these reasons, the analysis performed here
must be understood to be somewhat preliminary.
However, these are not the issues that I want to consider in this section. For the moment, I
want to assume that my results in some way describe the data corpus that I collected. Even if we
accept this assumption, a host of difficult issues arise. In particular, I want to think about issues
that cluster roughly around this question: In what sense can I be said to be describing practices of
algebra-physics or programming-physics that have some existence outside of the limits of this
study? Most of these issues have been commented on elsewhere. Here, I draw them together and
summarize my position.
Issues relating to subjects
First, there are a few issues relating to the subjects that participated in my study. The subjects
in both pools were drawn from a population of students at a specific stage in their learning of
physics. They were not expert physicists, but they were also certainly not complete novices.
Instead, they were at an intermediate stage, at the tail end of the introductory sequence. As I
discussed in Chapter 6, the decision to look at students in this intermediate range was a principled
choice; I wanted to be able to say something about the nature of expertise, as well as to see some
of the phenomena associated with learning difficulties.
But, because I only have information from this one time-slice, any projections up and down
the learning timeline must be understood to be speculative. Most importantly, throughout this
work I have implied that the form-device knowledge system continues to exist into expertise,
albeit with many refinements. But this speculation must ultimately be verified with new data
involving expert physicists. It may be that we cannot understand my results as describing expert
physics practice, not even in its broad strokes.
Of course, I do believe that the form-device system continues to play an important role in
expertise. I should note, however, that some refinement of the methods employed in this study
would likely be required to see an expert’s form-device system in action. In particular, the tasks
used in this study would probably be very easy for experts; they could probably solve them in a
rote and mechanical manner. To really see an expert’s form-device knowledge at work, we would
need to have tasks that challenge the experts, at least a little.
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It is also worth commenting on the fact that, for both pools, the number of subjects was
relatively small. Nonetheless, this is not a “small N” study in the usual sense. Notice that, I did
not try to characterize individual students or pairs of students in any way. For example, an
alternative study might have given the algebra and programming subjects identical pre-tests and
post-tests. Then I could have made claims that sounded like: “8 of the 10 programming subjects
did X on the post-test, while only 4 of the 10 algebra subjects did X.” Claims of this sort rely on
producing characterizations of individual students. But I did not make any claims of this sort.
Instead, my data was always aggregated across the subjects in each pool. Furthermore, for each
pool, over 200 individual events were examined and coded.
Nonetheless, even though I aggregated my data before drawing conclusions, the observed
variation across subject pairs—and small number of subjects—could be reason for concern. First,
because there appear to be sizable variations across pairs, it is less likely that my overall sample of
construction and interpretation events reflects the overall distribution in the practice.
Second—and more profoundly—the existence of these differences casts doubt on the whole
program of looking for a universal set of weightings. I have always presumed that individual
differences would exist; but, in performing this analysis, I was hoping that there were rough
universals to be captured. The variation across subjects is reason to worry that this might not be
the case. Although these worries are somewhat moderated by the observation of some weak
similarities, these problems cannot be completely avoided and, where necessary, I have tried to
temper my claims appropriately.
There are also some issues relating to the subject population that are specific to programmingphysics. As I have commented at numerous times, it is of great relevance to my examination of
programming-physics that the students came to the study with experience in algebra-physics.
Thus, the results described here are almost certainly not the same as what we would find if I had
examined students that were introduced to physics solely through a programming-based practice.
The reasons for choosing to study experienced physics students in the programming portion
of the study were largely pragmatic. For one thing, it would simply have been too time
consuming to create, from scratch, a population of true programming-physics initiates.
Furthermore, there are some reasons that this approach is not too problematic. First, my analysis
was always tightly focused around the external representations, I only looked at what people said
while actually looking at and pointing to particular programming expressions. This means that my
analysis is only revealing where and how understanding gets connected up to programming
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expressions; it is not giving a full picture of my students’ physical intuition or other physics-related
knowledge. This is what we want. Of course, because these students have significant experience in
seeing structure in algebraic expressions, there is likely still some skewing of my programmingphysics results in the direction of algebra-physics. But this just means that any differences
uncovered here are likely, if anything, to be exaggerated in a study of true programming-physics
initiates.
Issues relating to tasks
There are also a number of sensitive issues relating to my selection of tasks. In this chapter and
Chapter 6, we saw that there was a rather larger variation in codings across the tasks included in
the study. This implies that if I had included different tasks, I would have obtained different
frequencies for specific forms and devices. Clearly what we really want is a selection of tasks that
are somehow representative of the practice we are trying to study. But, although this sounds
sensible, it is a very hard goal to reach in practice. What are the boundaries of the practice we
should study? What are the dimensions along which we should categorize tasks in order to judge
representativeness?
Suppose for example, that we chose to categorize tasks and judge representativeness according
to the physical concepts that the tasks traditionally involve. In that case, we would want a selection
of tasks having to do with the concept of force, a selection having to do with momentum, etc. But,
such a principled selection could easily be very skewed in the forms that are engaged. Even if we
choose some tasks having to do with forces, we could easily choose “too many” or “too few” tasks
that involve balanced forces.
Probably the best way to select tasks would be based on a post-hoc categorization. For
example, now that I have an idea of how forms and devices tend to line up with tasks, I could go
through a physics textbook and do an analysis of the form and device distribution as realized in
the problems in a textbook. Then I could select tasks that mirror this distribution and iterate my
study. If I got the same results as for the textbook analysis, then I could know that I was getting
somewhere. Of course, all of this relies on my knowing what the forms and devices are and how to
associate them with tasks, things that I did not know when I began this study.
This issue of what tasks are representative is even more sticky for programming-physics, since
there is no independently existing practice that I can try to mirror. My selection of tasks can only
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be based on guesses concerning what a practice based on programming would look like, and a set
of specific orientations concerning how best to use programming in instruction.
This problem of task selection is one of the biggest difficulties in drawing conclusions from
the frequency results. However, there are some moderating factors. Most importantly, when we
ultimately set out to draw conclusions, we will have more than just the quantitative results to lean
on; we will also have all of the specific examples I presented of forms and devices, as well as our
understanding of what these elements are and how they are tied to the representational forms.
These other resources can help us substantially in determining the plausibility of potential
conclusions.
The problem of ÒcausesÓ
Throughout this work, I have usually been careful to frame my study as a comparison of two
practices, rather than as a comparison of two symbol systems. Nonetheless, this work is intended,
in some sense, to be about symbol systems, and this is reflected in my choice of names for the two
practices, programming-physics and algebra-physics. However, there remains the possibility that
the symbol systems themselves are somehow irrelevant to all of the results I have obtained here. In
other words, even though I have found interesting differences in the forms and devices in
programming-physics and algebra-physics, how can I know that these differences are due to
differing “properties” of the two symbol systems? In asking this question we implicitly take the
view that symbol-use is one aspect of a larger practice; thus, any differences that we see between
two practices may potentially be better described as traceable to other elements of the practices.
And there is one very important example to be considered here. Programming-physics, as I
have constructed it, involves teaching a particular “difference” model, the “tick model.” Thus, one
could argue that any differences that I see could be due to the fact that programming-physics
teaches a difference model, and algebra-physics teaches something like a constraint-based model.
My primary response to this critique is that this issue, as posed, contains assumption that I do
not believe hold. Namely, in posing this question we assume that these models exist separately
from the symbol system involved; or, at least, that a variety of models can be taught in
conjunction with a given symbol system. Although there is certainly much looseness in the physics
practices that a symbol system can be embedded in, there are also severe limitations on how a
given system can be used naturally.
384
In fact, I tried for several months to construct a difference model practice of physics that
employed algebra, and did not come near to success. It is possible to employ a difference model
using something like algebraic notation, and what Feynman does in his introductory textbook is
an example. But, amongst the reasons that Feynman quickly abandons this approach, is that the
method he plays out is tedious and would be largely impractical to employ in many tasks over a
semester. Furthermore, note that to use algebraic notation to state difference relations, we have to
modify or at least add to traditional algebraic notation.
The point here is that the external representations we use, especially central ones like algebraic
notation, are inextricably tied up with the other elements of a practice that is built around the
representation. Another way to say this is that when “external representation” is suitably defined,
then it has properties that constrain the surrounding physics practice. I believe that those
constraints are critical in defining the practice, and that’s why I focus my attention there.
Furthermore, I have tried to structure my research so that it gets at these points of constraint. I
did not look at all statements that students made, only statements made while pointing to or
writing expressions. Thus, even though these constraints certainly leave room for much “play” in
the practices that we build, I hope to have gotten at the nature of the guiding influence imposed
by the external representation.
Final comments on the comparison: The matching of conditions
The comparison I am attempting to perform in this study is a difficult one. In most studies
that involve comparisons, such as “treatment” studies, one gives two different treatments, and
then compares subjects’ performance on the same tasks. Here, I am comparing subjects engaged in
different tasks, of a substantially different nature. Thus, some rethinking of what constitutes a
useful comparison is required. How much should we force conditions to be the same, when these
differences are just what we are interested in?
Broadly, I have uncovered two classes of assertions that I want to make that may be
problematic. The first has to do with representativeness. I would like to be able to argue that my
descriptions of algebra-physics and programming-physics are somehow representative of these
practices as they exist (or could exist) outside of my study. This does not require that I match the
two conditions; on the contrary, it requires that I somehow ensure that I am looking at a
representative sample of the practices in their true or natural forms.
385
The second class of problems has to do with assertions about causes. I would like to be able to
trace differences that I observe in the practices to factors closely related to symbol-use. This
requirement does suggest a need to “match” the two groups in certain ways. For example, I would
like to be able to exclude the possibility that the observed differences are due to differences in the
two populations. To the extent that the populations are closely matched, this possibility can be
more plausibly denied.
Conclusions
With all of the above results, and keeping all of the above caveats in mind, what can we
conclude? First, in a sense, the systematic analysis has put the totality of my viewpoint to test by
employing it to describe the data corpus. In the process, we did encounter a few marginal cases,
such as those involving the “Substitution and Execution Machinery,” which could not be
accounted for in terms of forms. However, as I argued in Chapter 6, the existence of these
marginal cases does not necessarily signal the breakdown of the theory, since my viewpoint
already presumes a certain degree of complexity. The form-device system is just one amongst
many resources and we should not expect this resource to account for all phenomena, even all of
the phenomena in a limited territory. On the contrary, I believe that it is impressive that the formdevice theory can do as much work as it does. And now we have seen evidence that this is true not
only of algebra-physics, but also of programming-physics.
Related to this first point is the observation that the interpretation of expressions and the
construction of novel expressions are not rare events, both in programming-physics and in algebraphysics. Since this observation should survive all of our worries about tasks and subjects—with the
sole caveat that, as discussed in Chapter 6, my prompts may have tended to over-encourage
interpretation—we can be relatively sure that I am not studying a collection of marginal
phenomena. This observation is extremely important with regard to drawing conclusions from
these results. Even if the systematic analyses of the two corpuses found very strong and reliable
differences, we could not draw very interesting conclusions from these differences if the
phenomena involved were relatively rare. We want to be able to presume that these phenomena are
common enough that an analysis of them tells us something important about the two practices.
Now let us consider some of the particular differences uncovered by the analysis of
programming-physics. We saw that, not only were there differences in the frequency of individual
forms, there were differences in the very forms that appeared. There were some new forms in
386
programming-physics—the Process forms and
SERIAL DEPENDENCE.
Since these new forms were
common across a variety of programming-tasks, and since they did not appear at all in my
algebra-physics corpus, there is probably a result here that we can count on.
Similarly, there were also some symbolic forms that appeared only in algebra-physics.
Probably the most important instance here is the BALANCING form, the fourth most common form
in algebra-physics. I believe this is another result we can count on and draw conclusions from.
In addition, there are some results pertaining to devices that I believe are relatively sound. In
particular, the analysis of the programming corpus revealed two new devices in programmingphysics: TRACING and RECURRENCE. Again, because these were common in the programming
corpus and did not appear at all in the algebra corpus, I believe it is appropriate to use this result in
what follows.
The above results, which are primarily based on forms and devices that appeared only in one
corpus or the other, are the main results I will build on. I believe that results of this sort—results
that depend only on the lists of forms and devices—are particularly plausible because it is often
easy to see how they follow from differences in the representations involved. However, in the next
chapter I will also want to argue for some differences between programming-physics and algebraphysics that are more speculative because they depend on some detailed aspects of the frequencies
obtained by the two analyses. For example, I will base part of my comparison on the observation
that, when line-spanning structure was ignored, the distribution of programming forms and
devices was close to that of algebra. Although this observation is on shaky ground because of
across-task and across-pair variations, the close similarities in these distributions is quite plausible.
This is especially true given my observation, in Chapter 9, that the construction and interpretation
of individual programming statements can be very similar to that of equations.
In addition, I will also make use of the observation that there seemed to be a general shift
from Static devices to the other two classes in programming. Again, this observation is based on
more shaky aspects of my frequency results. However, because this result is generally plausible
and—as I will argue—it is consistent with other results, I will take this shift in frequency as
suggestive of real differences between programming and algebra.
Those are the basic outlines of the comparison revealed by the systematic analysis of the two
corpuses. Now, in the next chapter, I will take these results as a given and move on to a more
interpretive and speculative comparison of programming-physics and algebra-physics.
387
Chapter 11. A Comparison of Algebra-Physics and
Programming-Physics
It is now time to, once again, step back from the forest of details and get some perspective. The
big picture I have been arguing for looks like this: Students come to their learning of physics with
quite a lot of relevant cognitive resources. As I discussed in Chapter 5, some of these resources
have to do with getting along in the physical world. In addition, there are some relevant cognitive
resources that have more schoolish origins, such as resources that develop during mathematics
instruction.
So, students have these previously existing cognitive resources. Then, when they start learning
physics, the existing resources get refined and new resources develop. In particular, this research
has been concerned with discussing a set of resources that are associated with seeing a certain type
of meaning in physics expressions. This set of resources is the form-device knowledge system.
This new knowledge system develops to do some very particular jobs. In a sense, the formdevice system acts like a way-station between the previously existing cognitive resources and
symbolic expressions. If you want to express an idea in terms of an external symbolic expression,
then that idea must feed through forms and devices. In addition, the form-device system allows
us to see a certain type of meaning in existing expressions.
Because of its special role as intermediary and connection to symbolic expressions, an
examination of the form-device system tells us a great deal about how the use of a particular
practice of symbol use influences a student’s developing understanding of physics. First, because it
is the link to pre-existing cognitive resources, an examination of the form-device system tells us
something about how these resources are refined and adapted during physics learning involving a
particular symbol system. In addition, the specific composition of the form-device system is itself
of interest. I have argued that, once developed, the form-device system should be considered to
be part of “physical intuition.”
An examination of the form-device system is also of importance from the point-of-view of
improving physics instruction. As physics educators, one of the main difficulties that we
face—and one of our most important goals—is to help students to use symbolic expressions with
understanding. We want connections to existing resources to be made during symbol use and we
want a student’s physical intuition to be refined. So the properties of the two symbolic practices
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revealed by this analysis are of special importance. We would like to know: How easy is it to use
the expressions in a practice “with understanding?”
The task of the present chapter is to draw together all of the preceding analyses to produce my
final comparison of programming-physics and algebra-physics. Contributing to this comparison,
we have in hand an account of the forms and devices in each of these two practices, as well as some
more or less reliable numerical values for the frequencies of these elements in each practice. In
what follows, I will begin by drawing conclusions based on my analyses of symbolic forms, and
then I will turn to representational devices.
In addition, following the discussions of forms and devices, I will do a little additional work.
Since the analysis of forms and devices cannot capture all of the interesting differences between
algebra-physics and programming-physics, I will build some additional theory to capture some of
these differences. This will be embodied in a discussion of what I call “channels of feedback,” and
we will come to see the form-device system as just one of these multiple “channels.”
Before setting out, I want to say something about the tactic I will employ here for drawing
conclusions. Recall that, in my presentation of both systematic analyses—Chapters 6 and 10—I
pointed out that there are a number of reasons to question my particular quantitative results.
Among the most important of these reasons were the observations of variation across individuals
and tasks. Though, for these same reasons, some caution is required in this chapter, my intent here
will be to try and draw as many speculative conclusions as possible. To do this, my approach will
be, in general, to begin with differences between algebra-physics and programming-physics, as
suggested by the systematic analyses. Then, I will attempt to back up these results, where
appropriate, with arguments for their plausibility. These arguments will be based in the qualitative
observations presented in earlier chapters, as well as some more straightforward observations
concerning each of the representational systems and how they are typically used.
A Comparison Based In Symbolic Forms
Recall that, as I discussed at the end of Chapter 8, the analysis in terms of symbolic forms was
designed partly to capture a number of informal hypotheses. The basic notion was that
programming tends to privilege a somewhat different “intuitive vocabulary.” More specifically, I
hypothesized that some important differences would follow from the fact that, in contrast to
algebraic expressions, programming statements are ordered and directed. For example, since causal
intuitions require an ordering of the world’s influences and effects, I hypothesized that these
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intuitions might be more strongly drawn upon in programming-physics. Furthermore, I
conjectured that intuitions based in symmetric relations, such as those relating to balance and
equilibrium, would appear less in programming-physics.
The Theory of Forms and Devices has provided us with a means of making these informal
hypotheses more precise. Given that theory, these informal hypotheses can be recast as questions
about the symbolic forms involved in each practice, as well as questions about how existing
intuitive resources might be refined through symbol use. Furthermore, with these more precise
specifications and the preceding results in hand, we are in a position to discuss whether some of
the specific hypotheses are born out. Is algebra-physics more about balance and equilibrium? Can
we identify any sense in which programming is more causal?
Let us turn to the results and see what conclusions we can begin to draw. The basic outlines of
the differences revealed in the preceding chapters are not too difficult to state succinctly. I argued
that programming-physics involves several new forms—a new dependence form called SERIAL
DEPENDENCE,
and a new cluster of forms that I called the Process Cluster. The existence of these
forms constitutes a substantial departure from algebra-physics. However, we saw that when these
forms were excluded from the analysis, a substrate of similarity was revealed. In particular, the
frequencies of forms, collected by cluster, were very similar (refer to Table Chapter 10. -28). I
argued in Chapter 10 that this similarity was quite plausible, given my qualitative observations.
Algebra
Proportionality
Competing Terms
Dependence
Terms are Amounts
Coefficient
Multiplication
28%
27%
23%
14%
5%
2%
Programming
24%
24%
21%
14%
15%
2%
Table Chapter 11. -47. Comparison of form frequencies by cluster with codings of programming-specific forms
neglected.
The major exception to the similarity revealed by the restricted analysis was a boost in the
frequency of programming’s Coefficient Cluster. Although this boost may just be an artifact of
individual variation or the limited task selection, I believe that it is indicative of a real difference
between programming-physics and algebra-physics. In the previous chapter, I maintained that this
difference has to do with the fact that the “tuning” of numerical values was a major activity in
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programming-physics. In fact, I will argue later in this chapter that one of the most important
features of programming-physics is that it forces students to deal with specific numerical values.
Neglecting for a moment this boost in the Coefficient Cluster, we are left with the observation
of a fundamental similarity in the forms of programming and algebra, at least at the level of
clusters. There is a quite simple and important way to interpret this observation. As I have
discussed, each of the programming-specific forms has to do with line-spanning structure. Thus,
in excluding these forms from the analysis, we are essentially examining the types of meaningful
structure that can be supported by individual programming statements. This seems to indicate
that individual programming statements and individual algebraic statements can actually support
very similar symbolic forms and that they seem to engender these forms with roughly similar
frequency. Given this observation, it may seem that we are moving toward a negative result on my
hypotheses concerning differences between algebra and programming.
Unbalanced BALANCING
But there are a number of reasons that this result is misleading. One reason is because, as I will
argue shortly, some of the evidence for my informal hypotheses is hiding in line-spanning
structure, which is excluded from Table Chapter 10. -28. More subtly, in collecting the results by
cluster, we miss many of the interesting differences that are associated with the individual forms
within clusters.
One fact that is totally hidden when the results are collected by cluster is that some of the
forms that appeared in algebra-physics, even some that were quite common, did not appear at all
in programming-physics. For example, the
PARTS -OF -A-WHOLE
form received no codings in the
programming analysis. In this case, I believe that the absence of this form has much to do with the
selection of programming tasks. It turns out that the majority of the algebra codings of this form
were associated with a single task—the Running in the Rain task—and there was no analog of this
task in the programming corpus. Furthermore, it is not too hard to imagine a case in which
programming expressions could be interpreted in terms of
PARTS -OF -A-WHOLE.
For example, we
can imagine a program in which there is an expression for the total mass of a compound object:
change Mtotal M1 + M2.
However, I believe that the absence of some forms in programming is indicative of real and
important differences between programming and algebra. Most notable amongst these absences is
the lack of any codings for the BALANCING form. One reason to think that this constitutes a real
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difference between these two practices is that BALANCING was quite prominent in algebra-physics: It
was the fourth most common form overall, alone accounting for 10% of all codings, and it
appeared regularly across a wide variety of tasks.
Another reason to believe that this result points to a real difference between programming and
algebra is that it is a plausible result, given some simple observations concerning the nature of these
two practices. In particular, situations involving BALANCING turn out to be somewhat easier to deal
with in algebra-physics than situations that do not involve BALANCING . For example, when we have
balanced forces or balanced momenta, then we can just set the quantities equal and solve, in the
process avoiding differential equations. Because of the simplification that results, the practice of
algebra-physics tends to seek out situations that can be understood in terms of
BALANCING .
In
contrast, BALANCING situations are not especially easy or natural in programming-physics.
To illustrate this point, I want to think about the spring-related tasks that appeared in each
portion of this study. Recall that I asked the students in the algebra pool to solve a problem in
which a mass hangs motionless at the end of a spring. In this case, there are two forces in balance,
the force due to gravity acting downward and the force of the spring acting up. Thus we can write
Fs = Fg
and, with this expression written, the path to a solution is relatively straightforward. Of course, it
is possible to use algebraic notation to solve problems involving a block that oscillates at the end of
a spring, but to fully derive the time dependence of the block’s position requires solving a
differential equation, or that we remember the solution to that equation. For this reason, it is
common in algebra-physics instruction to seek out the easier case in which the block is motionless
and the forces are in balance.
This is not to say that situations involving balance and equilibrium are not fundamental and
important phenomena in their own right. The physics of balance is not a poor man’s physics and I
do not mean to imply that the use of algebra constrains physics to the study of unimportant
phenomena. In fact, there is reason for us to have special interest in static phenomena—cases in
which all objects in a system are at rest relative to each other. For example, these cases happen to
be very important to engineers and architects, who need to design buildings and bridges that do
not move (i.e., that do not fall over).
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The situation in programming is quite different. To begin, notice that there is no way to write
symmetric BALANCING relations like Fs = Fg , at least within the practice of programming-physics
that I employed in my study. In particular, the programming statement
change Fs Fg
is not equivalent to Fs = Fg . Rather than stating a symmetric constraint, this programming
statement specifies a directed action: Assign the value in Fg to the variable Fs.
Furthermore, it is not even clear what it would mean to use programming to solve a problem
involving the static case. We could write a program that has, at its heart, a single line that
computes how far the spring is stretched:
change pos
mass * g / k
Doit
Then our program could use the result of this computation to simply position a graphical object at
the corresponding location on the display. But this would not be a very interesting program and it
presumes that we have already used algebraic methods to derive the relation embodied in the
above statement.
Furthermore, the oscillating motion is very easy in programming; you just type the expression
for the force in the Tick Model and then let the simulation run. So, unlike the algebra case, there
is no incentive to seek out a situation involving BALANCING ; if anything, all of the motivating factors
suggest that we do otherwise. Since dynamic simulations are easy and interesting in programming,
we might as well do them.
So there is reason to believe that algebra-physics really does tend to more frequently draw on
notions of BALANCING . We could attempt to modify our practices so that such differences in
tendencies were moderated. For example, we could expand our practice of programming-physics
to include “conditional” statements, such as:
if
force1 = force2
do-something
Doit
Doit
This statement says to execute the procedure named do-something if the variable force1 happens
to equal the variable force2. If we allowed these conditional statements in our practice of
programming, there would be a niche for symmetric equality relations that might lend themselves
to BALANCING interpretations.
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However, I believe that even in modified practices of this sort,
BALANCING
will still play a less
prominent role than it does in algebra-physics. This is true, if for no other reason, because we can
deal with time-varying situations so easily. There is simply no reason to lean as heavily on cases in
which influences happen to balance.
Ordering and processes
So, BALANCING is a symbolic form that is very common in algebra-physics, but which did not
appear at all in programming-physics. Now I want to turn this discussion to the forms that
appeared only in programming. Recall that, in my original “informal hypotheses,” I conjectured
that there would be important differences between programming and algebra traceable to the fact
that the lines in programs are strongly ordered. In part, the effects of this strong ordering are
reflected in the existence of the Process Cluster, a whole new cluster of forms that are unique to
programming-physics. The existence of this new cluster is extremely important; as I will argue, it
suggests that a whole new wing of intuitive knowledge is picked up and strongly supported by
programming.
The points here are so important that I want to support them with a careful discussion of an
example episode. This episode involves Tim and Steve’s work on the Toss simulation, which we
briefly encountered in Chapter 9. Here I want to go through this episode in some detail, telling
the whole story so that I can extract the relevant morals.
Like all of the students, Tim and Steve proceeded through a standard progression of tasks,
creating their Drop simulation just prior to embarking on the simulation of a tossed ball. Upon
starting work on the Toss, they commented that the second portion of the motion, during which
the ball drops from its apex to the ground, would be the same as the motion produced by their
Drop simulation.
Tim
It’s a toss. [g. tossing motion]
Steve
All the way up and back down.
Tim
Yeah. So the um - well the drop’s gonna be the same. Right?
Steve
Yeah.
Tim
As it was before.
This observation, that the second half of the Toss motion would be identical to the motion in the
Drop simulation, was common among my subjects.
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Following this initial discussion, Tim and Steve copied their work from the Drop simulation
and pasted it into their current working environment. At that point, they had the following
procedures:
toss
drop
setup
setup
change
change
change
repeat
Doit
pos 0
vel 0
acc 9.8
10 tick
Doit
Next they worked to fill out the toss procedure and adapt drop to be appropriate for its current
use. The result was as follows:
toss
drop
setup
change
change
change
repeat
drop
change acc 9.8
repeat 10 tick
pos 0
vel 98
acc -9.8
10 tick
Doit
Doit
Recall that, in Chapter 9, I argued that the SEQUENTIAL PROCESSES form was involved in Tim
and Steve’s construction of this toss procedure. The story I told there was that these students saw
the simulation as broken into two parts, a part that takes the ball up and then a call to drop, which
is supposed to take the ball down (refer to Figure Chapter 11. -1).
toss
setup
change
change
change
repeat
pos 0
vel 98
acc -9.8
10 tick
drop
Ball goes up.
Ball comes down.
Doit
Figure Chapter 11. -1. Tim and Steve's toss procedure seen in terms of SEQUENTIAL PROCESSES.
When Tim and Steve executed this simulation, the sprite began by moving upward in the
graphics box, slowing down along the way. This part was as expected. But after the sprite stopped
at its apex, it began to move upwards again, accelerating off the top of the graphics box. Looking
at the graphical display, it was obvious to the students that something was amiss:
Steve
Oh, shoot.
Tim
What happened? Setup. [Runs]
Steve
It kept going the same way.
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The reason for this incorrect behavior is the change in the sign of acc at the apex of the
motion. To explain: The ball/sprite starts off at the bottom of the graphics box, pointing upwards.
This means that the downward direction is essentially defined to be negative. Thus, when the
program starts, the ball’s initially positive velocity causes it to move upward, and the initially
negative acceleration decreases this velocity, making the velocity smaller and smaller until it is zero
at the apex. At this point, for the ball to come down, we would need the velocity to become
negative so that the ball moves in the negative direction. This requires that the acceleration
continue to be negative. If the acceleration is changed to have a positive value at this point, as in
Tim and Steve’s simulation, this makes the velocity become positive again, and the ball continues
upward.
Without making any comments whatsoever, Tim and Steve immediately modified their drop
procedure so that it made the acceleration -9.8, instead of 9.8. Thus, their program looked as
follows:
drop
toss
setup
change
change
change
repeat
drop
pos 0
vel 98
acc -9.8
10 tick
change acc -9.8
repeat 10 tick
Doit
Doit
This program, when run, produced behavior that was essentially correct and Tim and Steve were
largely satisfied. The only feature of their program that bothered them was that the motion
appeared to end slightly below its starting point. For that reason, they modified their drop
procedure so that tick was only repeated 9 times, rather than 10.
There are some interesting points to made about Tim and Steve’s simulation as it stands at
this point in time. First, because the change statement in drop now changes the acceleration
variable to -9.8—the value it already contains—this statement has no effect and can be safely
deleted. Furthermore, when this statement is eliminated, the above program essentially contains
two sequential repeat statements, both of which repeatedly execute the same tick procedure.
These two statements can thus be combined into a single longer repeat statement. Tim and Steve
recognized these points soon after completing their successful simulation:
Steve
So we don’t even need to have that change, huh? [g. change acc line in drop]
Tim
That can’t be right.
Steve
Cause it’s still the same.
Tim
Then why does it go back down?
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Steve
I don’t’ know.
Tim
Oh, cause we have an initial velocity right. It starts out positive [g. moves finger up along
path], it just keeps getting more and more negative.
Steve
Oh. So you could just repeat that nineteen times. [g. repeat line in toss]
Tim
Yeah. [w.mod repeat 19 tick in toss]
After this exchange, Tim and Steve acted on these observations to produce this final version of the
simulation, which generates identical behavior to their previous simulation.
toss
setup
change
change
change
repeat
pos 0
vel 98
acc -9.8
19 tick
Doit
After confirming that this new version of the simulation worked, Tim commented:
Tim
We don’t need this. [g. drop procedure] That’s right, cause gravity’s constant.
Now I want to take some time to reflect on what happened in the above episode. Initially,
Tim and Steve had a program that broke the motion into two parts, a part that takes the ball up to
the apex and a part that takes the ball back down. Then this program was transformed into a new
version that treated the motion as a single whole. These two views of a toss are really quite
dramatically different, but is there a sense in which either version is preferable? Have Tim and
Steve learned anything important here?
Fg = mg
F g = mg F = ma
F g = ma
mg
a=
m
a=g
Figure Chapter 11. -2. A free-body diagram for a tossed ball which shows all of the forces acting on the ball.
This diagram is valid at every point along the motion.
To begin to answer these questions, let us first look at these issues from the point of view of a
physicist. Figure Chapter 11. -2 shows a free-body diagram for the tossed ball in flight. The only
force acting on the ball is the force of gravity, Fg =mg. This force can be used to compute the
acceleration of the ball using the relation F=ma to yield, a=g, as shown in the figure.
The important thing to note about this figure is that it applies to every instant along the
motion. It can just as correctly be taken to describe a moment when the ball is on the way up, or
when it is on the way down. This means that the same local physical mechanism generates the
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entirety of the motion. There is always a constant acceleration downward, which causes the ball to
slow as it moves upward, then to speed up in the downward direction. Thus, there is a deep sense
in which the motion is the same at every point, and it is not “physically natural” to break the
motion into two parts.
This does not mean that it is incorrect to talk about the motion as having two parts; there may
very well be occasions in which this point of view is helpful and illuminating. However, if a student
does learn to see the toss motion as a single whole, then they may be on the way to learning
something profound. The power and subtlety of the formal physics approach here is that it reveals
a common, underlying mechanism that generates the entire motion.
Now let’s think about where Tim and Steve stand with respect to these issues. Initially, it is
not obvious to them that the toss motion could be treated as a single uniform process. This may
not seem too surprising, given the fact that it is subtle to recognize the underlying uniformity. On
the other hand, Tim and Steve have almost certainly received direct and fairly extensive
instruction relating to this issue. The fact that the acceleration is constant and downward
throughout the motion of any projectile is something that is repeated frequently in introductory
physics courses. That Tim and Steve have received such instruction is somewhat confirmed by
Tim’s final statement: “That’s right, cause gravity’s constant.” Nonetheless, in their first pass at
this simulation, Tim and Steve thought that they needed to alter the sign of the acceleration at the
apex. Did they just forget or was there still some lingering sense in which they didn’t really “get
it?” If so, why wasn’t the algebra-based instruction more successful?
The major point here is that programming forces you to engage with these issues in a way that
algebra does not. The question of how and whether to slice up the motion has clear and direct
implications for the program; you have to decide whether to have one loop or two. In algebraphysics, as it is usually practiced, the relevant differences are not so clearly manifested as
differences in the external representation. Given this observation, it is not too surprising that
algebra-physics instruction leaves some undiscovered intuitive cobwebs in this territory, and that
the act of programming tends to expose these cobwebs.
In fact, programming even does a little better than helping students to engage with these
issues, it actually helps them in working them through and getting them straight. Notice that, in
the above episode, Tim and Steve almost seemed to rediscover the fact that a toss can be treated
as a single uniform process. When their first program did not work, they were forced to make the
acceleration a constant throughout. Then, once this change was made, a brief inspection of their
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program revealed that the two parts of their program could be combined. Thus, the act of
programming a simulation directly supported these students in coming to see a toss as a single,
uniform process.
Let’s take a moment to think about an analogous situation we encountered in algebra-physics.
Recall that when working on the Shoved Block task, students frequently expressed two competing
intuitions concerning which block would travel farther. One intuition was that the heavier block
should travel less far because it experiences a greater frictional force; the second intuition was that
the heavier block should actually travel farther, since it has a greater momentum and is thus
“harder to stop.” Working with algebraic notation and traditional solution methods, the students
were able to resolve this conflict in their intuitions and determine that the heavier and lighter
block travel the same distance. In this way, the use of algebraic notation helps students in
straightening out issues relating to certain types of intuitive knowledge.
The above programming episode is analogous. The writing of a simulation helped Tim and
Steve in dealing with issues relating to a certain type of intuitive knowledge. Thus, the moral of
the story is the same: The activity of symbol use acts to restructure various parts of physical
intuition. Furthermore the key point here is that programming is more likely to have effects on
intuitions having to do with how the world is broken up into processes; programming picks up a
whole new wing of existing intuitive resources.
Tim and Steve’s work on the Toss simulation was not at all anomalous. In fact, three of the
four other pairs explicitly discussed the issue of whether the simulation should involve one or two
loops. Furthermore, nearly the same issue, in one form or another, arose for all pairs while working
on the Spring simulation. As for the Toss simulation, a single tick procedure and repeat loop can
be used to generate the entirety of the Spring simulation.
But this was far from obvious for the students in my study; pairs frequently discussed various
ways of breaking an oscillation into parts. For example, in the following passage, Ned suggests
breaking an oscillation into three separate pieces: (1) a piece during which the block moves from
the center to its far right location, (2) a piece that takes the block from the far right to the far left,
and (3) a piece that take the block from the far left back to the center. These three pieces would
then be repeated to produce multiple oscillations. Thus, Ned’s suggested solution involved three
repeat loops contained within a broader repeat structure. David countered that this complicated
approach was not necessary, arguing that “it should be able to do it on its own:”
Ned
Okay, repeat it however many times it takes to go here. [g. center to right] And then
change all these again. [g. all the change lines in spring] Inside this program. Repeat it so
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it goes back here. [g. right to left of g-box] Then change 'em again. [g. spring procedure]
And repeat it so it goes back here. [g. left to center] And then repeat the whole thing. [g.
circle around whole screen]
David
Right, I don't think we'll need to. It should be able to do it on its own. [g. block in center]
See, cause if we start off with zero acceleration, velocity would be velocity. And it'll go
once. [g. moves finger short distance to right] Position'll be position plus velocity. Force
would- (0.5) position times K. And then acceleration'd be force divided by mass. And then,
it'll use that acc, right, the next time around?
Ultimately, David prevailed in his argument, and the students created a simulation that
employed a single repeat loop.
spring
TICK
setup
change
change
change
change
change
change
repeat
change vel
pos 0.0
vel 10.0
acc 0.0
mass 20.0
force 0.0
k 1.5
20 tick
Doit
vel + acc
Doit
change pos
pos + vel
Doit
change force
pos * k
Doit
change acc
force / mass
Doit
fd vel
dot
Doit
This simulation essentially uses the relation F = -kx to compute the force at each instant, and then
that force is used to compute the acceleration. Thus, as in the motion of a projectile, there is a
sense in which the same local physical mechanism generates the entirety of the motion.
When Ned and David ran this simulation, it worked quite nicely, producing clearly oscillatory
behavior. But, interestingly, Ned and David were a little surprised that their simulation worked:
David
Dude, how did we do this?
Ned
I don't know. [laughs]
[They both watch their simulation for a moment.]
Ned
Seriously, I don't know. I- I was expecting this to just go that far. [g. motion from center to
right]
Ned and David are certainly accomplished students, and we might think that they should know
very well that the same forces and mechanism apply at each point along the spring’s motion. And,
in fact, they did know enough to create a simulation with a single loop on their first attempt. But
the result was still a little surprising to them; there were still relevant intuitions for them to work
through. It seems that Ned and David (and especially Ned) were still strongly inclined toward
seeing an oscillation as composed of several parts, at least in certain circumstances. Writing the
Spring simulation forced them to begin to sort out these intuitions.
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An important question arises at this point. We have seen that programming forces students to
engage more directly with a certain class of issues, namely issues relating to how motions should
most appropriately be sliced up into parts. But it may not be the case that students enter physics
learning with conceptual difficulties—or conceptualizations, of any sort—in this territory. Instead,
it is possible that in giving students a language that allows the breaking of motions in parts, we are
actually generating new conceptual difficulties. If this is the case, then it may not be very
important to deal with issues relating to processes; rather, these issues might be better avoided.
But, whether or not students enter instructions with conceptualizations relating to how the
world is carved into processes, it may nonetheless be useful for instruction to deal closely with
these issues. In fact, I laid the basis for such an argument when I described how the carving into
processes can relate, in some cases, to recognizing a single underlying mechanism.
In addition, at least one researcher, Kenneth Forbus, has hypothesized a central role for
processes in the physical reasoning of experts and novices. In his extensive manuscript entitled
Qualitative Process Theory, Forbus introduces two terms (and corresponding hypotheses
concerning intuitive knowledge) that are relevant to this discussion (Forbus, 1984). The first term
that he introduces, following Hayes (1979, 1984) is the concept of an object “history.” A history is
an object-centered description of a motion, bounded in space in time. It focuses on a particular
object and tells us how the various parameters associated with the object, such as its location, vary
over some finite time interval. Furthermore, for Forbus and Hayes, reasoning about the physical
world is largely about fitting together these histories. Hayes states: “Reasoning about dynamics
resembles a process of fitting together a jigsaw of historical pieces in an attempt to fill out
spacetime … .”
The second relevant term introduced by Forbus is “process.” Simplistically, processes act to
cause the changes that happen in a history—they make the history happen. The point is that
histories are, in a sense, only descriptions of what happens in the physical world. Processes provide a
certain type of explanation of these described events.
Forbus’ terms can easily be used to talk about the issues we have been dealing with here. The
question of whether a toss is treated as having one part or two can be phrased as whether we see it
as consisting of one or more histories. And our observation that the motions in the toss and spring
tasks can each be generated by a single underlying mechanism can be described, in Forbus’ terms,
by saying that a single process drives the entirety of these motions.
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The big point here is that Forbus is essentially claiming that histories and processes play a
major role in how people understand the physical world prior to any physics instruction. If we
accept Forbus’ claim, then this makes it less likely that the conceptual difficulties observed here
are purely an artifact of the use of programming, and it increases the apparent importance of
dealing with related issues in physics instruction. And my argument here is that programming is
better suited for engaging these issues and helping students to get them straight.
Directedness and causation
Another of my informal hypotheses was that there would be consequences of the fact that
programming employs directed statements, rather than symmetric equality relations. This was
born out partly by the fact that BALANCING did not appear in programming. But another element
of my original hypotheses was that this directed aspect of programming statements would lead to
a greater role for causal intuitions in programming. Let us consider the extent to which this
hypothesis is supported by the results.
Since there is no symbolic form that I labeled the “cause” form, the connection of causal
intuitions to my analysis is not immediately obvious. Among the forms that I identified, the ones
that I believe are most closely related to causal relations are the Dependence forms. A little
explanation is required here to make this relationship between dependence and causation clear.
First note that causal relations are asymmetric in the sense that when I say that X causes Y, this
is not equivalent to saying that Y causes X. Furthermore, there is a family of relations that are
asymmetric in this sense. Dependence is perhaps the weakest of these relations; when we say that
one quantity depends on another, all we are saying is that changing the value of the second
quantity will alter the value of the first quantity, all other things being equal. A slightly stronger
asymmetric relation between quantities is what I call “determining.” If quantity X determines
quantity Y, this means that knowing the value of X (perhaps together with some other quantities
that we know about) is enough to uniquely specify the value of Y. Finally, following Horwich
(1987), causal relations can be understood to be like determining relations, but with an extra
“ingredient” added, such as a requirement of time-ordering.
The point of the above story is not to provide a rational accounting of dependence and
causation. Instead, I simply want to argue that observations pertaining to the Dependence forms
are relevant to what I informally called “causal intuitions” in my original conjectures. More
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specifically, I want to claim that “Dependence” can be taken as a name for a weaker and more
encompassing category of relations that includes causal relations.
So, at least tentatively, we can rephrase the questions here as follows: Are there differences
between programming-physics and algebra-physics relating to Dependence forms? Did these
forms appear more often in programming or was the sense of dependence somehow stronger?
To answer these questions, first recall that, when only forms relating to within-line structure
were included in the programming analysis, the distribution obtained was very similar to that
obtained for algebra-physics. To some extent this indicates that the mere fact that programming
statements are directed does not have much of an effect. Although it precludes seeing symmetric
relations like BALANCING , directedness does not seem to increase how frequently dependence
relations are seen, as compared with the equations of algebra. Furthermore, whether or not we
choose to trust a detailed comparison of the frequency results, we have the observation that
DEPENDENCE
was one of the most common forms in algebra, and it appeared across a variety of
tasks.
This is an extremely important observation. Although algebraic equations are always, in a
certain sense, symmetric, it does not seem to be overly difficult for students—at least the
relatively advanced students in this study—to read asymmetries into these statements. For
example, although the equal sign in F=-kx specifies a symmetric relation, it is not especially
difficult for students to read this asymmetrically as “force is dependent on position.”
Of course, there may be nuance here that I am missing. It may be, for example, that there
really are separate symbolic forms associated with dependence, determining, and causation. If this
were the case, then an analysis that included these distinctions might reveal important differences
in how individual algebraic and programming statements are understood. However, I looked hard
for this variety and could not justify any such diversity, given the analysis principles I laid out in
Chapters 6 and 10. What was most telling was that students employed an extremely limited
vocabulary in talking about the relations I subsumed under DEPENDENCE. In almost all cases
students used one of a small set of stock phrases, such as “depends on” and “is a function of”
(refer to Appendix D).
This null result is quite interesting: It is an important observation that it was not noticeably
more difficult for my students to see an equal sign as specifying an asymmetric relation. However,
this null result is not the bottom line. I still want to argue that there is, in fact, an important sense
in which programming-physics draws more strongly on causal intuitions. To see this, we need to
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now include the programming forms that were associated with line-spanning structure. In
particular, remember that there was a new form added to the Dependence Cluster that I called
SERIAL DEPENDENCE .
When this form is included in the analysis, then the Dependence Cluster
becomes proportionally more common in programming than it was in algebra. (Refer to Figure
10-5 and Figure 10-6). So what can we make of this? Does this mean that programming is more
causal or, at least, that it draws more strongly on this family of asymmetric quantity relations?
What I want to argue here is that the existence of the SERIAL DEPENDENCE form points to a
sense in which programming-physics really is more causal. To begin this argument, I will say a
little about some relevant episodes that I touched on in Chapter 9. Recall that issues relating to
SERIAL DEPENDENCE
came up frequently when students were working on the Spring simulation.
Ned and David were quite typical in this regard. From their experience with algebra-physics, they
essentially knew that they had available two “relations for force”—F=-kx and F=ma—and they
knew that they would probably need to use both of these relations in their program. However, in
order to use these relations in a program, Ned and David needed to decide how they should be
ordered. The way they phrased the issue for themselves was to essentially ask: Should we use the
position to find the force using F = -kx and then the force to find the acceleration via F=ma, or
should we do this the other way around, finding acceleration and then position? Here’s the tail
end of their discussion:
Ned
Does the force change because the acceleration changes?
David
The force changes cause X changes. And so that, and so=
Ned
That causes acceleration to change.
So, ultimately, Ned and David decided to use the position to find the force and then the force to
find the acceleration. This was reflected in the final version of their tick procedure:
TICK
change vel
vel + acc
Doit
change pos
pos + vel
Doit
change force
-1 * pos
Doit
Doit
change acc
force / mass
Doit
fd vel
dot
Doit
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* k
Nearly the same issues arose for most pairs while they worked on the Spring simulation. For
example, as we saw in Chapter 9, Adam and Jerry came to see all of the various changes as flowing
from changes in the position of the oscillating block:
TICK
change force
k * pos
Doit
change acc
force / mass
Doit
change vel
vel + acc
Doit
change pos
fd vel
dot
pos + vel
Doit
Doit
Adam
Well basically, the position is ba- um, the acceleration, everything,
Bruce
Hm-mm.
Adam
is based on where this is, the position, [g. pos in change acc line]
…
Adam
And the position is the most important thing here. If we made it the other way around, we
don’t care about the acceleration, we care about where it is.
Jerry
Hm-mm.
Adam
See acceleration [g. change acc line] derives from the position. [g. change force line]
What we are seeing here is that programming forces students to engage with another unique
set of issues—issues pertaining to which quantities change first and which quantities determine
which other quantities. The importance of this observation is driven home if we think about
algebra-physics. When working within algebra-physics, we typically adopt the stance that many
quantities are changing constantly and simultaneously. For example, when we write F = ma, we do
not need to think about whether force or acceleration changes first. In fact, since these and other
quantities are changing constantly, this question does not even make sense within algebra-physics.
In contrast, programming forces you to take changes that happen simultaneously and break them
up into ordered operations. And, in doing so, you have to decide which operation happens first.
This is a step toward an ordered, causal world.
The point here is that, when line-spanning structure is included, it appears that dependence
structure plays a stronger role in programming than in algebra-physics. When we just looked
within individual programming statements, the directedness of these statements did not appear to
be especially consequential for seeing asymmetric relations. However, when these directed
operations are placed within a larger ordered process, then it appears that there are interesting
consequences: In total, Dependence forms are coded more frequently, and new kinds of
405
dependence structure are seen. Another way to say this is that, although the directedness of
individual programming statements may not be especially consequential, it is consequential that
programs involve a series of directed statements. I have argued that students can learn to see chains
of dependent quantities in a series of such directed statements, and simultaneous events get spread
out and ordered in a way that they do not in algebra.
Are causal intuitions problematic for physics learning?
A number of interesting questions arise at this point. Suppose we accept that programming
tends to draw on causal intuitions in a certain manner, and that it forces programmers to break up
simultaneous changes into sequential operations. Then we have to ask: Are these properties of
programming desirable from the point of view of physics instruction?
One reason we might be prompted to ask this question is that there are actually reasons to
believe that the use of causal intuitions in physics may be dangerous or even incorrect. (See diSessa
& Sherin, 1995, for a somewhat extended version of this discussion.) To illustrate this point, let us
think about the equation F=ma. In my discussion above I described how, when programming the
Spring simulation, students were forced to think about which quantity changes first, force or
acceleration. And, in all cases, the students decided that changes in force should lead to changes in
acceleration, a decision that I believe is appropriate.
However, an algebra-physics traditionalist might argue that it is incorrect to say that forces
cause accelerations. The equation F=ma is symmetric in this regard; it doesn’t say that forces cause
accelerations any more than it says that accelerations cause forces. Thus, in introducing causation
or other asymmetric relations here, we may be adding information that is not really part of a
“correct” physical description of the world. Furthermore, adding information may be dangerous
and could ultimately confuse students as they try to learn physics.
There a number of responses to this objection. First, we must keep in mind that a concern with
how someone understands physics is not the same thing as a concern with what is correct physics.
Even the best physicists may sometimes think of forces as causing accelerations, even if they
somehow know that this is not strictly correct. If it is the case that physicists think in this way,
then it may not be a problem to teach students to think causally.
Furthermore, I believe that it is immensely plausible that physicists draw on causal and related
intuitions. One reason that I believe that it is plausible is that I can think of many important uses
for causal intuitions. For example, notice that, although it may not be right to say that forces cause
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accelerations, it is not really true that there is no asymmetry in the relation between the concepts
of force and acceleration. At the least, there is an ontological asymmetry—forces and acceleration
are two different sorts of entities.
In addition, students do not always keep this ontological asymmetry straight. One way that
this problem manifests itself is in misinterpretations of the equation F=ma. Sometimes students
look at F=ma and see it as a statement that two forces are in balance. (Recall Alan’s inappropriate
BALANCING
interpretation I discussed in Chapter 4). This is incorrect because only the F in this
equation is directly associated with a force or forces that exist in the world. Thus, there is an
asymmetry here that causal intuitions could help students to keep straight: If you have in mind
that forces cause accelerations, then you are less likely to misinterpret this equation. This is the sort
of place that causal/dependence intuitions are useful. Instead of writing F=ma and seeing it in
terms of BALANCING , it is likely better for students to write a=F/m and recognize the
DEPENDENCE
form, with acceleration dependent on force. (See diSessa, 1993, for a similar argument.)
Here’s another way to think about these issues. To a physicist, the equations that are taught in
an introductory physics course do not constitute a haphazard collection; there is a structure to
Newtonian theory, and equations play different roles within that structure. For example, in
talking about Ned and David’s work on the Spring simulation, I said that they thought of F=-kx
and F=ma as two “relations for force.” But, to a physicist, these relations play very different roles
within the structure of physical theory. The equation F=-kx is a “force law;” it tells you how to
compute the force in a particular class of situations. In this case, it tells you how to find the force
exerted by a spring. In contrast, F=ma is the central dynamic constraint relation of Newtonian
physics; it tells you how to find the acceleration once you know the total force acting on an object.
Throughout the experimental sessions I saw evidence that this simple structure—force laws
that provide forces to plug into F=ma—was not evident to the students in my study. Once again,
this is a place that programming (and causal intuitions) could help. In reflecting on how F=-kx and
F=ma should be ordered within a program, students were engaging with relevant issues. Notice
how the two relations were ordered by students in the above Spring simulations; they first
computed the force and then used the resulting force to compute an acceleration. In an important
sense, this ordering reflects the structure of Newtonian physics.
Of course, the same structure could be reflected in the ordering of algebraic statements. For
example, we could encourage students to wonder whether it makes more sense to write:
407
F = − kx
F
a=
m
or
F = ma
−F
x=
k
But I did not see students using line ordering in this way, and I believe it is not generally done in
algebra-physics. Certainly, I did not observe students engaging in extended discussions
concerning how algebraic statements should be arranged, as they did in programming. I believe
that it is fairly clear why this is the case. First, to the extent that students might choose to be
careful in ordering the above two expressions, they would not want to order them to reflect some
physical relations among the quantities; instead, they would more likely want the ordering to
reflect a path from what is given in the problem to what they want to find. Furthermore, algebra
does not necessitate clear decisions—of any sort—about ordering, to the degree that
programming does. Students can just write F=-kx and F=ma, then manipulate as necessary. For
example, a student could observe that, since these two relations involve force, it is possible to
eliminate force and write -kx=ma.
F = ma
F = − kx
− kx = ma
In this case, there is not necessarily any ordering between these two “expressions for force.”
Before wrapping up this discussion, I want to say a little about the fact that programming
forces you to take instants and smear them out in time. Even more than the use of causal
intuitions in physics, this ordering of what are actually simultaneous changes may be thought to be
questionable from the point of view of standard “correct” physics. At the least, we must certainly
worry that students will be confused and will think that these things really do, in some sense,
happen one after the other, in real time.
My argument that this smearing out of instants is not a problem takes the same form as my
argument that it is acceptable to teach students to draw on causal intuitions. Even though it may
not be strictly correct to think of these simultaneous changes as ordered, it is, in many cases,
extremely useful and powerful. I call this strategy “reasoning through pseudo-time.” I do not want
to argue extensively for this here, but I will note that I am not alone in believing that this smearing
through pseudo-time is a powerful resource for reasoning about physical systems, and one that
experts routinely employ. Most notably, deKleer and Brown (1984) introduced the term
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“mythical time” for what I call pseudo-time, and their accounts of how people reason about
physical systems involve reasoning through mythical time.
In summary, I did not uncover evidence that individual programming statements are more
likely to be given cause-related interpretations than individual equations. However, when linespanning structure is included in the analysis of programming, then there is evidence that
programming tends to more strongly draw on causal intuitions. Questions of “what quantities
change first” and “what quantities determine which others” are issues that come up naturally and
persistently in programming physics, and they are issues that are strongly related to causal
reasoning.
A Comparison Based In Representational Devices
Now it is time for this discussion to turn to a comparison of the representational devices of
programming and algebra. My exposition here will be somewhat less extensive than my
comparison of the forms in each practice, in part because there is significant overlap in the issues
involved. Each of the practices—its forms, devices, and notational systems—is an integrally
related whole.
In this section I will begin by discussing how the results of the device analysis indicate a shift
toward what I will call “physical” devices. Then I will address the implications of the extremely
important fact that, much more than algebra, programming forces an embedding of expressions
in the particulars of a physical circumstance. Lastly, I will conclude with some final musings on the
topic of whether programs are easier to interpret than equations.
A shift to time-varying or ÒphysicalÓ devices
At the level of classes, the most significant difference revealed by the systematic analysis of the
programming corpus was a shift from the Static Class to the Narrative Class. In comparison to the
algebra-physics corpus, Static devices were proportionally less common and Narrative devices
more common. As I have noted, this shift might only be an artifact of individual variation and the
limited task selection. However, I believe that this is a plausible result, in part because it is
consistent with the image of differences that I have been building in this chapter. In particular, this
shift can be understood to be part and parcel of the changes described in my discussion of forms:
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In moving from algebra-physics to programming-physics there was less of a focus on situations of
equilibrium, and more of a focus on time-varying systems.
But, although these observations are closely tied, we can still learn more and further
understand these differences by looking closely at individual devices. First, recall that the
prevalence of the Narrative Class in programming had a lot to do with
TRACING ,
the
programming-specific Narrative device. TRACING was quite common in programming, accounting
for 20% of all device codings and nearly a third of Narrative codings. Furthermore, when the
TRACING
device was eliminated from the analysis, we saw that the increased prevalence of
Narrative devices in programming disappeared.
This last result seems to suggest that the only difference in the Narrative devices of
programming was the addition of TRACING . However, I believe there are some other important
differences. Table Chapter 11. -1 displays the results relating to Narrative devices for both algebra
and programming, with the TRACING device excluded. A close examination of these tables suggests
that the PHYSICAL CHANGE device was proportionally more common in programming-physics, as
compared to CHANGING PARAMETERS and CHANGING SITUATIONS
Algebra
Narrative
CHANGING PARAMETERS
CHANGING SITUATIONS
PHYSICAL CHANGE
Programming
Count
Percent
128
79
21
28
54%
33%
9%
12%
Narrative
CHANGING PARAMETERS
CHANGING SITUATIONS
PHYSICAL CHANGE
Count
Percent
135
73
18
44
54%
29%
7%
18%
Table Chapter 11. -1. Narrative Devices in algebra and programming.
Taken together, the addition of TRACING and the increased frequency of PHYSICAL CHANGE can
be seen as part of a systematic difference in the nature of Narrative devices in programming.
Recall how the various devices in this cluster differ. In PHYSICAL CHANGE , the narrated changes
follow the motion under study. For example, if we are examining a ball falling under the influence
of air resistance, then we might narrate how the velocity and the air resistance force change as the
ball drops. In contrast, the changes narrated in a CHANGING PARAMETERS interpretation are of a
different sort. Rather than describing changes that occur through the time of the motion under
study, a CHANGING PARAMETERS interpretation describes changes that could be understood to be
occurring across instances of a motion. For example, again considering the case of a ball dropped
with air resistance, a CHANGING PARAMETERS interpretation might describe how the motion would
410
differ if the ball was heavier. Similarly, the CHANGING SITUATIONS device also does not describe
changes that occur through the time of a motion. Instead, interpretations involving this device
compare motions in physically distinct circumstances.
Given these descriptions of the three Narrative devices of algebra-physics, we can see that there
is a sense in which TRACING and PHYSICAL CHANGE go together: Both of these devices involve
changes through the time (or pseudo-time) of the motion under study. What this suggests is that
programming tends to have more interpretations that follow the time of a motion. Furthermore,
as we have just seen, this is true not only because of the addition of a substantial number of
TRACING
interpretations, it may also be reflected in an increase in codings of PHYSICAL CHANGE .
Thus, what we are seeing here is a general shift toward what I call “physical” narratives—narratives
that follow the motion under study.
This shift toward time-varying devices was also manifested as a decrease in Static
interpretations in my programming corpus. The decrease in Static devices is part and parcel of the
shift away from a focus on equilibrium situations; it generally reflects the fact that programmingphysics tends to deal with time-varying situations, and the fact that programming tends to
support “physical” narrations. Interestingly, it turns out that the shift toward time-varying
interpretations is even reflected in individual Static interpretations that appear in programming.
Recall that there was a significant addition to the Static Class, the RECURRENCE device, which
accounted for 40 of the 67 codings of Static devices in programming. Thus, to the extent that the
RECURRENCE
device is unlike other Static devices, Static interpretations in programming will differ
from those of algebra-physics.
In fact, there may be a sense in which RECURRENCE is somewhat unique among Static devices.
Recall that the RECURRENCE device is closely related to GENERIC MOMENT . While GENERIC MOMENT is
associated with the stance that an equation is true at any time, RECURRENCE interpretations involve
adopting the stance that a relation tells you what transpires on each tick or at each instant. Here are
two examples reproduced from Chapter 9:
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drop
setup
change
change
change
repeat
vel 0
air 0
mass 20
200
change force mass * acc - vel
change acc force / mass
tick
Doit
Doit
Fred
No, we don’t change it there. Air is - the air term is gonna be something that changes
during the calculation or during the fall. Remember that you said it’s dependent on
velocity, right?
…
Fred
So it’s actually gonna change the force each point, each tick.
Greg
Hm-mm.
Fred
So we’re gonna have to put change something.
_______________
change vel
vel + acc
Doit
Tim
So, every time - every tick it gains nine point eight. The acceleration increases the
velocity by nine point eight.
I do believe that the RECURRENCE device belongs in the Static Class. It is certainly not like
devices in the Narrative Class, which involve describing how a particular parameter changes with
some imagined process. Instead, like other Static devices, it is a particular way of projecting an
expression into a moment in a motion. However, there is a sense in which RECURRENCE
interpretations are not “static.” When we see an expression in terms of RECURRENCE, we are seeing
the expression as describing a local event, some localized change. Thus, the dynamic nature of
programming even shows up in a change in how expressions are projected into moments—we see
statements as little local stories about change, rather than as static relations that hold.
Programming and embedding in particulars
Recall that the full systematic analyses of programming and algebra found that Special Case
devices were roughly as common in each practice. However, when devices associated with seeing
line-spanning structured were excluded from the programming-analysis (i.e., the
TRACING
device),
the frequencies suggested that the Special Case Class was somewhat more common in
programming (refer to Table Chapter 10. -41). Notice that, in Table Chapter 10. -41, Special
Case is not only increased relative to Static interpretations, it is also proportionally greater as
compared to Narrative interpretations.
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Algebra
Narrative
Static
Special Case
54%
35%
11%
Programming
54%
27%
18%
Table Chapter 11. -2. Comparison of devices with TRACING eliminated.
This is another result that I believe is quite plausible given the differing natures of
programming and algebra. In fact, this difference can be seen to reflect one of the most basic
differences between programming-physics and algebra-physics. (See Sherin et al., 1993, for a
similar argument.) In algebra-physics, it is possible to use equations to solve problems in which the
physical situation is only weakly specified. For example, in the Mass on a Spring task that I gave to
my algebra subjects, the students were asked to determine how the amount of stretch depends on
the mass of the object and the spring constant. However, they were not given any specific values
for these parameters. Instead, they simply derived the equation x=mg/k and then discussed the
properties of this expression. In this way, they were able to answer the question without specifying
the numerical particulars of the physical situation.
Furthermore, you can use algebraic expressions in a manner that is even further removed from
specific physical circumstances. In many cases, we can use equations to answer questions without
even having any physical circumstance in mind. For example, it is possible to use Maxwell’s
equations (a set of very general relations that describe phenomena relating to electricity and
magnetism) to derive the fact that waves can propagate in electric and magnetic fields. In that
case, we are deriving a very general feature of electric and magnetic phenomena; we do not need
to be talking about any particular physical circumstance.
In contrast, programming requires us to embed statements in particulars. For example, when
creating their Spring simulations, the students in my study were forced to invent numerical values
for all of the parameters that appeared in their programs. If they did not invent these numerical
values, then they could not run their programs.
This is a simple explanation of the increased prevalence of Special Case interpretations in
programming. Because programming already requires you to deal with equations in a particular
numerical regime, Special Case interpretations may be more natural.
There is still more to this beyond the fact that programming requires you to pick specific
values. When you write a statement in programming, that statement must be embedded within a
program, and it must be embedded in a specific manner. You cannot have the programming
413
statement change acc force / mass written just anywhere on the computer display, it must be
embedded within a program, doing a particular job. This cannot help but suggest that the
statement be interpreted in a specific manner.
Are programs easier to interpret than equations?
So, what’s the bottom line here? I originally began this work with the hypothesis that programs
would be easier to interpret because they can be “mentally run” to generate motion phenomena.
Was this born out in the data? Are programs really easier to interpret than equations?
The answer to this question is: Yes and no. Given the observations I have presented, there are
reasons to conclude that programs are easier to interpret, and reasons to conclude that they are
trickier to interpret. And it’s a little hard to draw conclusions concerning how these factors balance
out. I did find that my programming subjects interacted meaningfully with (constructed or
interpreted) a larger percentage of the expressions that they wrote, a result that could be taken as
suggesting that programs are easier to interpret than equations. But, as I discussed in Chapter 10,
there are a number of other possible causes for this result, such as the fact that algebra-physics
tends to include a large number of derived expressions.
Since there is no way to draw a definitive conclusion concerning which type of expressions are
easier to interpret, I will instead go through and discuss the various factors on each side. To begin,
an important observation that we must keep in mind is that the interpretation of programs was not
always very different than the interpretation of algebraic expressions. In many cases, individual
programming statements and collections of statements were interpreted much like their algebraic
counterparts, with the rest of the program fading into the background. This observation seems to
suggest that programs are no easier to interpret, and maybe even a little harder, since the lines to
interpret in a program must first be selected. In fact, there was often some cleverness involved in
the isolation of a few programming statements for the purposes of interpretation.
However, on the other side of this debate is the observation that the backdrop provided by the
rest of a program and the programming context may help by providing an orientation to the lines
that are focused on. This is essentially the point I made when discussing Special Case devices just
above. The fact that programming statements are always embedded within a program in a
particular manner provides a leg-up on interpreting those statements. Less has to be done to invent
a stance, at least once the program is written, since the interpretation is already partly suggested
by the role that the statement plays within the program.
414
This observation is actually a corollary of a more fundamental point. In general, interpreting
any specific symbolic expression will be easier when there is more support lying around in the
context, and a person thus has to do less to invent an interpretive framing. Furthermore, this is
true not only of individual expressions, but also of classes of expressions: Any class of symbolic
expressions will be easier to interpret to the extent that it is embedded in practices that provide
natural modes of interpretation.
Yet another way to say this is to state that, to some extent, the ease of interpretation of
symbolic expressions depends on the existence of naturally available and useful representational
devices. And there actually is some evidence that programming has an edge in this regard. First,
the fact that the interpretation of a programming statement can be framed by the running
program is an example of this sort of natural support. Because of the fact that programs can be
run, there is a naturally available representational device—the TRACING device. Although this was
not at all the only device in programming, it accounted for 20% of all interpretations, a quite
healthy percentage.
Furthermore, the fact that programs can be run may actually help to make other devices more
natural. Most notably, we saw that PHYSICAL CHANGE interpretations were more common in
programming-physics than algebra-physics. This may be partly because the orientation provided
by the running program provided natural support for this mode of interpretation.
In conclusion, there are points on both sides of this debate. In some respects the interpretation
of programs is not altogether different than the interpretation of equations, and some cleverness is
required in selecting lines and pushing other lines into the background. However, because
programming statements are embedded in particulars there is more support for interpretation.
Furthermore, there are some naturally available and very powerful representational devices that
simulation programmers can fall back on in interpreting programs.
Channels of Feedback
Throughout this chapter and this entire manuscript, I have been focusing on getting at a very
specific class of differences between programming-physics and algebra-physics. I have been
interested in showing how physics understanding will be altered in a certain fundamental sense,
and, in particular, I have been interested in showing how a different “conceptual understanding”
of physics will flow from the specific notational systems employed in each practice.
415
Of course, there are differences between programming-physics and algebra-physics that are
not captured by this analysis. As I have mentioned, I could have paid more attention to what each
of these notational systems allows. For example, algebraic notation allows you to manipulate
expressions to derive new expressions, and programs can be run to generate dynamic displays.
These differences in what is allowed are extremely important for several reasons. First, they in part
dictate what you can do with each of the notational systems. But, even more important from our
point of view, they will also have effects on students’ understanding. This sort of second-order
effect on understanding is not directly captured by my form-device analysis.
Although I will not come close to giving these other issues their due, I do not want to ignore
them altogether. The fact that programs can be run to produce dynamic displays is such an
important feature of programming-physics that it simply must be addressed, if only briefly. And,
in fact, it turns out that there are some comments that I can make without straying too far from
the theoretical framework I have laid out in this document.
To get at these additional issues, I am going to take a very particular rhetorical approach.
Rather than building an orthogonal body of theory, I am going to take the theoretical work that I
have already done—the Theory of Forms and Devices—and I will embed it in a somewhat
broader view. At the heart of this broader view is a new notion that I call “channels of feedback.”
Numerical Channel
Form-Device Channel
Numerical Channel
Form-Device Channel
Graphical Channel
Figure Chapter 11. -3. Channels of feedback in programming and algebra.
To explain channels of feedback I want to begin with an image of two students (refer to
Figure Chapter 11. -3). One of these students is looking at a sheet of equations that he generated
on the way to solving a problem. The other student is looking at a simulation program that she
just wrote. Now I want to ask the same question about both of these two students: How will each
of them decide if what they have just done is correct or “makes sense?”
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Actually, both of these students have a number of possible approaches they can take in
deciding if they are satisfied with what they have done, and each of these alternative approaches
involves a different “channel of feedback.” Let’s begin with the algebra student. One way that the
algebra student can draw conclusions concerning the sensibility of his work is to plug some
numbers into one of the equations, and then attempt to judge the reasonableness of the numerical
result produced. We saw an example of this behavior all the way back in Chapter 2. In that
chapter, I described an episode in which Jon and Ella were working on a problem in which a ball is
thrown straight upwards at 100 mph. In their work, they ultimately derived the final expression:
vo2
=x
2a
Then, once they had this expression, they plugged in values for vo and a, finding that the ball
reached an altitude of 336 feet. With this result in hand, they mused for a while on whether it
seemed reasonable. In the end they decided that 336 feet was at least the “right order of
magnitude.”
Jon
Ella
Jon
A hundred miles an hour is,, (1.0) If you were in a car going a hundred miles an hour,
[laughs] straight up into the air.
straight up into the air, would you go up a football field? (0.5) I have no idea. Well it doesn’t
sound - it’s on the right order of magnitude maybe.
I call this channel of feedback the “numerical” channel. One way to judge the acceptability of
expressions is to plug in numbers and then judge the reasonableness of those numbers, a procedure
that essentially gives indirect evidence as to whether the expression itself is also reasonable. The
thing to notice about the numerical channel is that it allows judgments to be made concerning the
sensibility of an expression without any attention being paid to the actual structure the expression.
Of course, there are other ways for algebra students to judge the sensibility of their
expressions. In fact, a large part of my work has been to show that students do have the ability to
judge expressions by looking directly at their structure. Here is an example we have encountered
on numerous occasions, including as early as Chapter 2. After deriving a final expression for the
Mass on a Spring problem, Jim commented:
x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k] And that if you have a stiffer spring,
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then you're position x is gonna decrease. [g. Uses fingers to indicate the gap between the
mass and ceiling] That why it's in the denominator. So, the answer makes sense.
So, Jim did not need to plug in any numbers here, he was able to judge the sensibility of the
expression itself. This, of course, is the sort of behavior that I accounted for in terms of forms and
devices. I call this channel the “form-device” channel.
Thus, we can now start to see where my earlier work fits within this larger picture that I am
building. Forms and devices are the knowledge associated with one of several channels by which
students get feedback on the reasonableness of their work with symbolic expressions. And, more
specifically, it’s the channel associated with seeing meaning in the structure of expressions.
These same two channels—numerical and form-device—are also available to the programming
student in Figure Chapter 11. -3. The preceding chapters can be taken as an argument that forms
and devices play a role in how programs are understood. In addition, the students in my study
did often look at the values in their variables in order to decide if their simulations were correct.
However, there is some subtlety here. In writing their programs, my subjects generally were not
careful to work with meaningful units, so there was no easy way for them to draw on their world
experience to judge the reasonableness of the values produced. Still it was possible for them to
make rough judgments given the relative size of other values that appeared in the simulation. And,
in some cases, it was enough to see that a value was in a certain qualitative range, such as greater
than zero or less than zero.
Now, the central observation here is that programming has an important channel of feedback
that is not available in algebra-physics, what I call the “graphical” channel. This is just the point
that I began this discussion with, that programs can be run to generate dynamic feedback in the
form of a moving graphical display. This constitutes a channel of feedback because, when a
program is run, a student can look at the display and judge if the simulated motion appears
reasonable.
Furthermore, programming’s graphical channel has a number of very nice properties. One of
these properties is that, in certain respects, looking at the dynamic graphical display is a lot like
looking at a real motion in the physical world. This means that, to judge whether these displays
are correct, a student can draw on some of the same capabilities that they use in recognizing and
judging the plausibility of physical phenomena. Compare this to the form-device channel, which
requires a substantial amount of specialized learning.
However, judging the reasonableness of what one sees in the graphical display is not always as
easy as it might first seem. For example, when the motion generated does not fit in a graphics box
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in Boxer, then the motion “wraps.” This means that if a moving object disappears off the top of
the graphics box, it will reappear at the bottom. On some occasions, this can be quite confusing.
Furthermore, there are difficulties caused by the fact that simulation programs are discrete. And
when these two problems combine—wrapping and discreteness—the situation can become very
difficult. Suppose that a program is causing the sprite to jump in discrete hops that are many
times the size of a graphics box. In that case, the sprite’s location within the graphics box on any
particular jump may appear nearly random.
In fact, this is precisely what happened when students worked on the Air Resistance
simulation, since that task required students to run their simulation for an extended period in
order to see the approach to terminal velocity. Thus, in that case, the dynamic display was not
useful for judging the correctness of the simulation. Instead, students had to watch how the
numerical values of various parameters changed through the time of the motion.
Thus, some learning is required in order to interpret what one sees in a graphics box, and to
know when it is useful to look closely at the graphical display. In practice, only when the
simulation produces smooth and uniform motion that is largely contained within the graphics box
can students draw on their existing world resources to judge the motion. Still the fact that, in
some cases, the motion in a dynamic display can be judged much like a motion in the physical
world is one of the great strengths of programming-physics.
Finally, I just want to add one other piece to this image of channels of feedback. We can think
of algebra and programming as each having some generative machinery. In algebra-physics you can
manipulate existing expressions to generate new expressions, and you can substitute numbers to
obtain results. In programming-physics you can run programs to generate a simulated motion in a
dynamic display. So, one way to think of the broader image I am building here is to imagine a
core of generative machinery with feedback at different places.
The only point I want to make here is that these differences in the generative machinery that
exists in each practice are hugely important—they are at the heart of many of the major
dichotomies in what is allowed by each notational system. And the implications of these
differences are not thoroughly captured by my analysis. For an analysis of these implications we
must look to other researchers and to future work.
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The Bottom Line
The goal of this entire project has been to say something about how the understanding
produced by programming-physics instruction would differ from that produced by algebraphysics instruction. To begin to get at this, in Chapter 1 I contrasted “direct” and “indirect”
effects of symbol use, and I stated that I was going to be concerned with the direct effects on
understanding that follow from the use of a symbol system. Furthermore, my focus throughout
this work has been specifically on effects that flow from the nature of the notational systems
themselves.
In this work, I have actually discussed two related types of residual effects of symbol use on
the knowledge of individuals. First, I have argued that existing cognitive resources are adapted by
and for symbol use. Here this means that certain of the cognitive resources that students bring
with them to physics learning develop during instruction. For example, as I discussed extensively
in Chapter 5, I believe that the sense-of-mechanism is altered through the use of symbolic
expressions. The fundamental hypothesis there was that resources that get exercised are reinforced
and thus play a more central role in expert intuition.
The second type of residual effect is the development of new knowledge. Here, I have been
principally concerned with the development of one particular new resource, the form-device
system. A key point about this knowledge system is that, although it is inextricably tied to symbol
use, it is not especially “formal” or “abstract” knowledge. Rather, this cognitive resource can be
thought of as possessing the character of physical intuition. And I believe that once this new
knowledge system has developed, it should be considered to be a new component of an expert’s
“physical intuition.”
Altogether, an analysis of the tuning of existing resources and of the new form-device system
tells us about the character of expert physical intuition. With this in mind, what final conclusions
can we draw? Are there any high-level generalizations to be made about how the physical intuition
of a programming-physics students will be different?
First, an important preliminary comment must be made. Since forms and devices are
inextricably tied to symbolic experience, there is a sense in which this entire component of
physical intuition—the form-device system—is fundamentally different in each of the two
practices. The “physical intuition” of programming-physics is rooted in a somewhat different class
of experiences. (Refer to Chapter 7 for a more thorough discussion of related issues.)
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But, there are generalizations that can be made at a higher level. Given the discussion in this
chapter, I believe that it is appropriate to describe algebra-physics as a physics of balance and
equilibrium, and programming-physics as a physics of processes and causation. Algebra-physics really
trains students to seek out equilibria in the world. Programming encourages students to look for
time-varying phenomena, and supports certain types of causal explanations as well as the
segmenting of the world into processes.
It is important to keep in mind that these are differences in degree only. Within algebraphysics, for example, it is certainly possible for students to talk about processes, even though these
accounts may not be as directly reflected in representational forms. Nonetheless, these are still
important and fundamental differences—these high-level generalizations reflect real differences in
the physical intuition that will result from learning physics through the practice of algebra-physics,
or through the practice of programming-physics.
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Chapter 12. Summary and Implications
To college and high school students taking their first physics course, it may appear as though the
world of the physicist is populated largely by symbols and equations. In fact, much student
activity in introductory physics does involve manipulating equations and solving textbook
problems. Given the current emphasis in educational reform on “teaching for understanding,” this
might seem to be an undesirable situation. Is time spent by students in this abstract world of
symbols time well-spent? Time spent in this manner might help students to become better
problem-solvers, but does it have any deep import for students’ “conceptual understanding” of
physics?
As we saw in Chapter 1, for physicist Richard Feynman, the answer to this last question is a
resounding “yes.” Recall that Feynman expressed the intuition that a full appreciation of the
“beauty of nature” requires some experience of the physicist’s world of symbols. To know physics
as a physicist knows it, a student must have the experience of writing the laws of physics as
symbolic expressions, and manipulating expressions to get from one expression to another.
In this manuscript, I have adopted this view and attempted to give it precision, given the
vocabulary and tools of cognitive science. I have tried to establish that for students—at least for
moderately advanced students—a page of expressions is experienced as highly structured and
meaningful, with meaning that is not purely formal and divorced from interpreting the world.
Furthermore, I argued that, because symbolic experience has this character, it can contribute in a
substantial way to the development of expert physical intuition.
But an examination of the traditional symbolic world of physics, which is based in algebraic
notation, was only half of this project. As a way of getting at the inter-relatedness of knowledge
and external representations, I compared the standard practice of algebra-physics with a practice
of physics symbol use based around an alternative symbol system: a programming language.
Throughout this work I adopted the stance that a programming language could, in a strong sense,
replace algebraic notation and function as a bona fide symbol system for physics. Then I worked
to determine, empirically, how a student’s physical intuition would differ if they learned physics
through this alternative practice. The analysis uncovered some interesting differences, both in
detail and character, as well as some surprising similarities.
In this final chapter, I will begin with a summary of the thesis, then I will conclude with a
discussion of the various instructional implications of this work. In addition to laying the
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groundwork for future application of this work in instruction, this discussion of instructional
implications will also provide an opportunity to revisit some of the central themes from a new
angle.
The Thesis, In Review
The chief goal of this work was the comparison of the knowledge associated with
programming-physics and algebra-physics. Understood in the most general possible terms, this
comparison constitutes a huge project. Thus, although I attempted to structure this work so that I
could make quite broad and interesting claims, the study was of necessity focused and limited in
several important respects. First, I concerned myself only with what I called the “direct” effects of
symbol use in physics, the changes in an individual’s knowledge that occur through actual
participation in a symbolic practice. Furthermore, I was primarily interested in direct effects that
are more or less traceable to features of the notational systems themselves, rather than to what
these notational systems allow. For example, I did not do much to study the effects that flow
from programming’s generation of a dynamic display, except to the extent that this played out in
how students interacted with programming notations.
In addition, to perform the comparison at the center of this work, I framed my task in a very
particular manner, as a program in cognitive science. In general terms, the project was thus to
build a model, in the manner of cognitive science, of the knowledge of individuals associated with
each practice. Again, the degree to which I actually attained this very general goal was limited in
certain respects. In point of fact, I only looked closely at a particular cognitive resource associated
with seeing a certain type of meaning in symbolic expressions. The argument was that this
particular knowledge system was a good place to examine how the deep nature of understanding
would differ if we change symbolic practices.
The model of this cognitive resource—the form-device system—constitutes the central result
of this research and was the key to getting at the goals of this inquiry. Because it is so central to
this work, I will take the time here to present a moderately extensive summary of my account of
the knowledge comprising this model and how that knowledge develops:
Registrations. First I introduced the term “registration,” borrowed from Roschelle (1991), which
refers to how people “carve up their sensory experience into parts.” Each type of registration is a
feature of symbolic expressions to which physicists are sensitive in the sense that these structures
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are consequential for their behavior. I listed several kinds of registrations that are relevant to
symbol use in physics: individual symbols, numbers, whole equations, groups of symbols, and
symbol patterns.
A vocabulary of symbolic forms. Next I introduced the first of the two major theoretical
constructs in this work, what I called “symbolic forms.” A symbolic form consists of two
components: (1) a symbol pattern and (2) a conceptual schema. Taken together the set of forms
constitutes a conceptual vocabulary in terms of which expressions are composed and understood.
The forms in algebra-physics fell into six groups that I called “clusters:” Proportionality,
Competing Terms, Dependence, Terms are Amounts, Coefficient, and Multiplication. The forms
in programming-physics fell into these same six clusters as well as into a seventh cluster, the
Process Cluster.
Of these clusters, the three most common in both programming and algebra were
Proportionality, Competing Term, and Dependence. I will now say a little about each of these
clusters to help remind the reader of the sort of claims I made and examples I provided. First, the
Proportionality Cluster is associated with sensitivity to a particular aspect of the structure of
expressions, whether a given symbol lies above or below the division sign in a ratio. The most basic
form in this cluster is PROP+, in which the symbol appears above the ratio line and the expression is
thus directly proportional to the quantity in question.
PROP+
…x… 
 … 
PROP+ was the most common form in both my algebra and programming data, and it was
employed over a wide variety of circumstances. One common circumstance in which it appeared
was in interpreting derived relations in algebra-physics:
v=
Alan
mg
k
So now, since the mass is in the numerator, we can just say velocity terminal is
proportional to mass [w. Vt∝m] and that would explain why the steel ball reaches a faster
terminal velocity than the paper ball.
The next most common cluster was the Competing Terms Cluster. Forms in this cluster are
associated with seeing the terms in an expression as influences in competition. Two of the forms in
this cluster were among the most common individual forms in algebra-physics,
and BALANCING .
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COMPETING TERMS
COMPETING TERMS
±
BALANCING
=
± …
The BALANCING form, for example, appeared frequently in student work on the Buoyant Block
task:
Fb = Fg
Jack
Um, so we know the mass going - the force down is M G and that has to be balanced by the
force of the wa=
Jim
=the displaced water
Finally, the third most common cluster was Dependence. Forms in this cluster are associated
with sensitivity to a very basic aspect of the structure of expressions, the mere presence or absence
of a symbol. Both the DEPENDENCE and NO DEPENDENCE forms played a role in Mike and Karl’s
construction of an expression for the coefficient of friction:
µ
µ = µ1 + C 2
m
Karl
Well yeah maybe you could consider the frictional force as having two components. One
that goes to zero and the other one that's constant. So that one component would be
dependent on the weight. And the other component would be independent of the weight.
In arguing for the existence of symbolic forms in Chapter 3, I attempted to establish that, by
positing forms, we can account for certain phenomena that are not well captured by other existing
theories of the knowledge employed in physics problem solving. Among these phenomena,
perhaps the most important was the apparent ability of students to construct novel expressions
from scratch. Other important phenomena related to the details of how students write and
interpret expressions. For example, one interesting observation was the “qualitative equivalence” of
certain expressions for students. This was illustrated by Mike and Karl’s inability to choose
between two possible expressions for the spring force:
F = kx ; F = 1 2 kx 2
Karl
Okay, now, qualitatively, both K X and half K X squared do come out to be the same
answer because as (…) looking at it qualitatively, both half - both half K X squared and K
X, um, you know, increase as X increases.
The phenomenon of qualitative equivalence was also illustrated by students’ assertions that it was
not possible to easily decide between the expression v = vo + at and other incorrect alternatives.
Mark, for example, thought that the following expression might be correct:
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v = vo + 12 at 2
Mark
Cause we have initial velocity [w. circles vo term] plus if you have an acceleration over a
certain time. [w. circles 1/2at2] Yeah, I think that's right.
Symbolic forms account for these observations by specifying the particular aspects of expression
structure that are significant for students.
Forms also provided the major categories for comparing programming-physics and algebraphysics. I developed a method for systematically associating forms with events in each data
corpus, and then I obtained results for the frequency with which each type of form appeared. In
both practices, there were no clear winners amongst the various clusters, but a few individual forms
accounted for a large percentage of the codings. In algebra-physics, the most common forms were
PROP+, DEPENDENCE, COMPETING TERMS,
and BALANCING . In programming-physics, this ranking was
essentially duplicated, but with BALANCING replaced by BASE ±CHANGE .
Some of the most important differences between programming-physics and algebra-physics
were reflected in the forms that appeared exclusively in one practice or the other. Notably, the
BALANCING
form, one of the most common forms in algebra-physics, did not appear at all in
programming-physics. There were also a number of new and quite common forms in
programming, including a whole new cluster of forms, the Process Cluster. Process forms involve
seeing a program as a series of ordered processes. In addition, there was a new form added to the
Dependence Cluster, the
SERIAL DEPENDENCE
form. This form involves recognizing chains of
dependent quantities in programs, and it is tied to the ordered and directed properties of
programming statements.
Representational devices and the interpretation of symbolic expressions. In addition to symbolic
forms, I argued that physics initiates possess a complex repertoire of interpretive strategies that I
called “representational devices.” Representational devices are somewhat closer to observable
aspects of behavior than symbolic forms and, in fact, I maintained that they are directly reflected
in the structure of interpretive utterances. For this reason, the project to identify representational
devices involved, in part, the collecting and categorizing of interpretive utterances.
The observed devices fell into three classes: Narrative, Static, and Special Case. In Narrative
devices, some quantities or other aspect of the circumstance are imagined to vary and the result of
this variation is described. The most common Narrative device—and the most common device
overall in both programming and algebra—was
CHANGING PARAMETERS.
In this device a particular
quantity that appears in an expression is selected to vary while all others are held fixed:
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x=
Jim
mg
k
Okay, and this makes sort of sense because you figure that, as you have a more massive
block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in
diagram] which is what this is showing. [g.~ m then k]
In the second class of devices, the Static Class, an equation is projected into an instant in the
motion under consideration. For example, in the SPECIFIC MOMENT device, a student adopts the
stance that an expression is true at one particular instant:
Cv 2 = mg
Ella
The terminal velocity is just when the - I guess the kind of frictional force from the air
resistance equals the gravitational force?
Finally, in Special Case devices a student considers a special case out of the range of possible
circumstances under which an expression might hold. For example, in the LIMITING CASE device, a
student considers a circumstance in which one quantity takes on an extreme value, such as zero or
infinity:
Ae −v
Jack
So, you could say the probability if you're traveling at infinite velocity is very extremely
small.
In addition to allowing me to enumerate these interpretive strategies, the project of
categorizing interpretive utterances also provided insight into the nature of these utterances. I
argued that it was appropriate to characterize these interpretations as involving “embedding”
rather than correspondence. This meant that the equation was set within a context or framework
that was larger than any listing of the entailments of the equation. And I also observed that the
“meaning” embodied in these utterances was not a simple function of the arrangement of marks
and, instead, was strongly context dependent. For example, we saw that the interpretations that
students gave were sometimes directly related to the question they had been asked.
As part of my comparison of programming-physics and algebra-physics, I compiled a list of
the devices observed in each practice and measured their frequency of use. The analysis revealed
two devices in programming that did not appear in algebra. One of these new devices, TRACING ,
was the second most common device overall in programming. TRACING interpretations are closest
to what we might think of as the “mental running” of a programming. The other new
programming device, the
RECURRENCE
device, was an addition to the static class in which a
statement is seen as describing what transpires at a particular instant of a motion. RECURRENCE
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interpretations were frequently—though not always—associated with true recurrence relations,
such as the line that computes velocity in the Tick Model: change vel vel+acc.
In both practices, devices in the Narrative Class were most common, with Static devices
second and Special Case devices appearing only rarely. I argued that this result is quite interesting
because, while Special Case and Static interpretations bear some relation to relatively explicit
aspects of practice, Narrative devices seem to be altogether tacit. Students may occasionally be
taught that it is a useful heuristic to “consider a special case,” but they are rarely or never explicitly
taught that it is useful to select one quantity in an expression and imagine that it varies.
Although the clusters were ranked the same way in each practice, the distribution of devices
was nonetheless somewhat different. The results suggested a shift from Static devices to the other
two classes in programming, a shift that can be seen as relating to the general shift from static to
time-varying situations. In addition, there was a shift in the distribution of devices within the
Narrative class toward what I call “physical devices”—devices that embed the expression in a
process that follows the motion under study.
The refinement of physical intuition. In addition to describing the form-device system, I
speculated, mostly in Chapter 5, about the origins of this knowledge during physics learning. I
argued that this cognitive resource develops out of existing resources, including intuitive physics
knowledge and knowledge relating to the solving of algebra problems, and that these previously
existing resources continue to exist and to be refined. Furthermore, I took the stance that “expert
intuition” consists of these old resources— especially the sense-of-mechanism—somewhat refined
for use in expert practice, as well as some new resources, including symbolic forms. I described
how the refinement of the sense-of-mechanism and the development of symbolic forms could, in
part, be traced to experiences of symbol use.
Taken together, registrations, forms, and devices, as well as speculations about the
development of this knowledge, constitute what I have called The Theory of Forms and Devices.
I must again emphasize that, perhaps even more than specific results concerning differences
between algebra and programming, this framework is the central result of this work. For that
reason, I will conclude this summary with a couple of more general comments about the
framework.
First, the mere fact that the phenomena I have identified exist deserves extra emphasis. The
central claim of this work is that students—at least the moderately advanced students in this
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study—have the ability to see a certain type of meaningful structure in expressions. Equations
aren’t only memorized, and they aren’t only associated with physical principles; the arrangement
of symbols in an expression “says something” to students. For this reason, students can build new
expressions from scratch. And, because the experience of symbol use is meaningful, this adds to
the plausibility of deep effects on the nature of physics understanding.
This last observation takes us to the most important point of all: The differences between
programming-physics and algebra-physics captured here constitute deep and fundamental
differences in individual knowledge. When students enter instruction, existing knowledge is
refined and adapted for symbol use, and new knowledge develops that is inextricably rooted in
experience with symbols, knowledge that I have argued should be considered a component of an
expert’s “physical intuition.” Thus, the real nature of developing physics intuition will be different
if you learn physics through one practice of symbol use or another, through algebra or
programming. In the first case, the result is an intuition suited to seeing a physics of balance and
equilibrium; in the other, the result is an intuition suited to a physics of processes and causation.
Though these must be understood to be differences of degree, they are nonetheless deep and
fundamental differences.
Instructional Implications
As the final piece of this dissertation, I will reiterate some of my major themes from a
specifically instructional orientation. Throughout this work I have occasionally made comments
concerning instructional implications, but here I will explicitly take up this focus and draw these
observations together. This discussion will include implications for traditional physics instruction,
as well as some final comments concerning what we can conclude about the possibilities for
instruction based on programming-physics.
Implications for traditional algebra-physics instruction
A full half of this manuscript was devoted to producing a description of algebra-physics that
could be compared with the account of programming-physics. But this description of algebraphysics is not only useful for what it tells us about programming, it is also useful in its own right.
In fact, with my account in hand, we are in a position to think about some of the most important
problems faced by algebra-physics instruction. Among these problems is the observation that
students often appear to write and manipulate equations without understanding. Almost certainly,
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it would be worthwhile to work toward helping students to use equations with understanding, at
least more often than they do currently. However, this will not be an attainable goal if we do not
begin with some knowledge of what it means to use equations “with understanding.”
In this manuscript I have essentially described some aspects of meaningful symbol use in
physics, which can now give us more of an idea of the sort of behavior we are trying to attain. I
have pointed to some phenomena of interest, so we know what it looks like and where it happens.
I have described interpretations of equations; so now we know that there may be no single way of
associating meaning with equations, and that interpretation is a complex and context sensitive.
And I have characterized one knowledge system that allows students to see meaning in equations.
When interpretation fails
This work can also help us to be recognize a new class of student difficulties. Now that we
know where to look for understanding of equations, we can be ready to identify cases where this
understanding goes awry. This has not been a major focus of this work, but I did, on many
occasions, observe students giving weak or incorrect interpretations of equations. There were a few
examples of this behavior in Chapter 4, when I discussed what I called inconclusive and
inappropriate interpretations. In an inconclusive interpretation, a student sets out to produce a
certain type of interpretation of an expression, but the interpretation fails to produce a useful
result. An example I gave is the following excerpt taken from Jon and Ella’s work on the Running
in the Rain task, in which they try to interpret their expression for the total rain. The problem with
this interpretation is that, when they imagine that vR increases, this implies that one of the two
terms increases, but the other decreases:


h
w

#
drops


+
=
wx
vd2 v R 
 A⋅s  
 v d 1+ 2

vR


Ella
Okay, V R is bigger. Then (10.0),, This is gonna get smaller, [w. arrow pointing to second
term] but this is gonna get bigger. [w. arrow pointing to first term] I think.
Jon
Yeah, I think - that gets smaller [g. second term] cause V walking is smaller // than V
running.
In contrast, in an inappropriate interpretation, a student makes a statement that is, strictly
speaking, incorrect. These inappropriate interpretations included Alan’s BALANCING interpretation
from the Shoved Block task, which I have mentioned on numerous occasions. This interpretation
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is incorrect because, among other things, it implies the existence of two horizontal forces where
there is only one:
µmg=ma
Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no
longer moving because the force frictional is equal to the force it received.
The fact that the interpretation of an expression can be difficult and problematic is extremely
important from the point of view of instruction. Throughout this work I have primarily
emphasized the positive—I have tried to draw attention to the fact that students do have some
resources for using expressions with understanding. But, once we are clear on the fact that these
resources exist, we should begin to look for limitations in these resources, as they are possessed by
students.
Because limitations in students’ form-device knowledge is an important topic, and because I
have not emphasized this topic in the other chapters of this manuscript, I want to take a moment
to discuss one persistent difficulty that I observed. This difficulty has to do with how students
understood (and didn’t understand) the equation v = vo +at. This equation is interesting from the
point of view of this work for a number of reasons. To begin with, this is one of the first equations
that students learn in an introductory physics course.
In addition, to even the most mildly expert physicist, the expression v = vo +at is extremely
simple and straightforward. Here’s why: This equation gives the velocity as a function of time for
motion with constant acceleration. The first term, vo , is the initial velocity of the object, the
velocity it has at time t=0, and the second term adds on the amount that the velocity changes in a
time t. Furthermore, this change is pretty easy to compute since acceleration is defined to be the
change per unit time of the velocity and, in this case, the acceleration is presumed to be constant.
Thus, we just multiply the acceleration by the time to find out how much has been added to the
velocity. Another way to understand this is to think of the acceleration as the rate at which the
velocity changes. To get the total change, we multiply this rate by the time to give at.
As we have already seen in Chapter 3, this equation is not quite so straightforward for
students, even the moderately advanced students in my study. While working on the Vertical
Toss problem, students often struggled to regenerate this expression, sometimes writing
expressions that were incorrect. For example, Mark and Roger wrote the following equation and
commented:
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v = vo + 12 at 2
Cause we have initial velocity [w. circles vo term] plus if you have an acceleration over a
certain time. [w. circles 1/2at2] Yeah, I think that's right.
Mark
As I argued in Chapter 3, this statement suggests that Mark is interpreting this expression up to
the level of the BASE +CHANGE form. He knows that there should be a term corresponding to the
initial velocity and a term that describes how the velocity changes over time due to the
acceleration. However, since his specification is only at the level of BASE +CHANGE (plus some
DEPENDENCE
on acceleration and time), he is not capable of stating precisely what should be in the
second term. Instead, he must resort to just plugging in a piece of a remembered expression that
happens to include acceleration and time.
This is not meaningless symbol use, and the use of BASE +CHANGE here is not incorrect. Rather,
the problem is that Mark and Roger’s understanding is much weaker than it could be. To a
physicist, the precise form of the at term is obvious, given the definitions of velocity and
acceleration. Certainly we would like students to approach the point where this expression makes
as much sense to them as it does to a physicist. In terms of my framework, this means
understanding at in terms of the INTENSIVE •EXTENSIVE form, or in terms of the related rate•time
template. (Refer to Chapter 6 for a discussion of this template.)
Mark and Roger’s situation was entirely typical. At best, students were only able to give
BASE +CHANGE
interpretations of v = vo +at. Furthermore, when pressed, every student said this
expression would have to be remembered or derived from some other remembered equation, such
as the following expression for how the position varies with time in cases of constant acceleration:
x = xo + vot + 12 at 2
Jack
Well you have to remember something to know how to approach the problem, it's just a
question of which um equation you remember of motion. Um, given (1.5) Um given, if you
remember this one [g. equation] you can pretty much derive them all.
Because I felt that my subjects’ inability to see v = vo +at as obvious was quite interesting, I
asked a number of related questions during the Quick Task component of the study. One of these
related questions was the following, rather bald, prompt:
Some students say that the equation v = vo +at is “obvious.” What do you think?
Given what I observed during the earlier portions of the study, student responses to this question
were not too surprising. For example, here is a portion of Mike and Karl’s comments.
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Mike
Well yeah it is obvious because, well velocity will equal to V naught if it's not being
disturbed. But if it's being acc- If there's a f- acceleration action on it, then uh- and that's
constant, you know then velocity will be decreasing as time goes on. Or increasing,
whatever it works, I mean whichever it does. So, it's like whatever it is and then plus a
correction on the acceleration. So yeah it makes sense. It's obvious, yes it is.
…
Karl
What's obvious to me is that you have the final velocity is obviously going to be equal to
the initial velocity plus however much, however faster it gets. That's what's obvious to me.
What's not necessarily positively obvious is that the amount, the amount that it gets
faster is the acceleration times the time.
Here, Mike and Karl do seem to agree that v = vo +at is “obvious,” but only in a certain limited
sense. Mike says it is obvious because there should be a “V naught if it's not being disturbed” plus
a “correction” that has to do with the fact that it’s accelerating over time. This is an interpretation
at the level of BASE +CHANGE . But Karl is very clear in stating that, even with this interpretation, the
precise form of the second term is not entirely obvious: “What's not necessarily positively obvious
is that the amount, the amount that it gets faster is the acceleration times the time.”
These observations become even more interesting in light of student answers to another
question that I asked during the Quick Tasks:
5.
Imagine that we’ve got a pile of sand and that, each second, R grams of sand are added to the pile.
Initially, the pile has P grams in it. Write an expression for the mass of the sand after t seconds.
The idea here is that there is a pile of sand whose total mass grows linearly with time at a rate R,
starting with an initial mass of P. Thus, the correct answer to this task is the expression M = P+Rt.
Notice that this expression has precisely the same structure as v = vo +at, both involve an initial
value and a term that grows with time at a fixed, linear rate.
Now, the interesting observation is that students found the Pile of Sand question to be utterly
trivial. In every case, including Mike and Karl, the pair of students read the question, then one of
them picked up a marker and wrote an expression equivalent to M = P+Rt, without a single
comment. Furthermore, they generally seemed perplexed that I had even asked such a simple
question.
The point is that it appears that these students have nearly all the resources they need to see v
= vo +at as trivial and obvious, just as it is obvious to a physicist. But, somehow these resources are
not getting engaged here; students do not seem inclined to understand v = vo +at in the same way
that they understand M = P+Rt. For students, there is some mystery left in v = vo +at.
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Reforming algebra-physics instruction
What’s the moral of this story for algebra-physics instruction? Well, we could take this as just
another example of how students do not really understand physics expressions, even the simplest
ones. Then, the moral might be that work with equations is never going to be a good way for
students to develop a “conceptual understanding” of physics.
But this is more than a story about failure. First of all, although their understanding may not
have been up to the standards of what we ultimately desire for instruction, the students in my
study did have things to say about the meaning of v = vo +at. They did not only write it from
memory, they also understood it as expressing a base plus a “correction” due to the acceleration.
Furthermore, consider the observation that the Pile of Sand task was very easy for students. This
(along with the rest of my observations) constitutes evidence that students do have many of the
resources that are required to have a more thorough understanding of v = vo +at. It just seems that,
for some reason, these resources are not being engaged in this case. I therefore believe that the
moral of the v = vo +at story is this: Although students may fail to use expressions with as much
understanding as we would like, they possess many of the necessary resources; thus, there may be
some plausible steps we can take to get these resources engaged more frequently and at the right
times.
Even if students possess many of the necessary forms and devices, getting these elements
engaged in the right places may not be an easy instructional goal. The form-device system is not
just an unstructured collection of elements; these elements are tied to each other, to other
cognitive resources, and they are tuned to be cued in certain contexts and not others. Thus,
learning related to the form-device system is not simply a matter of learning individual forms and
devices. A big part of what students must learn is to adopt particular stances to individual physics
expressions such as v = vo +at, and this learning is likely to be tied up with other difficult
conceptual issues. For example, it may not be easy for students to learn to treat acceleration as a
“rate.”
But, presuming that physics instructors accept the task of getting students to understand
expressions in a more useful and appropriate manner, what should they actually do? How,
specifically, should instruction be changed?
First, note that the observations presented here really can help, since just being able to
recognize and understand the problems to be addressed is important. Once we have understood
that there is a problem with how students understand v = vo +at and we have undertaken to
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improve that understanding, then it is at least possible that there are some relatively
straightforward steps we can take. For example, I did not try this in my study, but it seems
plausible that pointing out the parallel between M = P+Rt and v = vo +at could have been a useful
component of instruction designed to help Mike and Karl better understand the latter expression.
In addition, knowing about specific forms and devices may help. We could, for example,
consider the interpretive strategies embodied in representational devices to be a target of
instruction. Of course, it almost certainly will not be sufficient to just tell students about
representational devices; in the long run, the design of new instructional strategies that takes my
theoretical results into account is going to require careful thought and experimentation. However,
there are some reasonable and modest first steps that are not at the level of “just tell them.” One
way that students learn is by watching instructors as they “model” appropriate behavior. For
example, physics instructors frequently model how to solve various types of physics problems by
standing at a blackboard and working through some problems while students simply watch. In a
similar manner—perhaps as part of modeling problem solving—instructors could strive to model
the interpretation of expressions. This means simply that, in addition to writing and manipulating
equations while students watch, they can point to equations and say something about what those
equations mean. Of course, many instructors probably do this to some extent already, but a little
more frequency and emphasis would likely help.
Ultimately, if future research confirms the observations presented here, we will probably want
to do better than just an increased attention to modeling this behavior; we will want to redesign
instruction to take into account the theory and observations I presented. A possibility that I believe
deserves serious consideration is the creation of a new course in mathematical modeling, which
would precede physics instruction in the curriculum. The purpose of this course would be to teach
students to invent simple types of mathematical models, and to use equations to express the
content of these models, not necessarily within the domain of physics. Although one of the aims
of this course would be to help students to develop the requisite form and device knowledge, this
does not mean that the forms and devices described in this document would be explicitly
mentioned in the course, or even that there would be a recognizable one-to-one mapping between
particular knowledge elements and course material. Instead, a successful course would likely focus
on various heuristic strategies and formally recognized categories, such as the distinction between
linear and quadratic relationships.
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The underlying presumption here is that a mathematical modeling course would give students
a leg-up when they got to physics instruction. For a student with some math modeling
background, each physics equation would not present a new and individual challenge; instead, the
student would be able to see physics expressions as instances of particular types of models. For
example, Mike and Karl would learn to see v = vo +at as an instance of a model with linear growth
in time.
From a certain perspective, this is really what physics is all about, building a certain class of
models of the physical world. When we worry about student misconceptions and related
qualitative understanding, we are essentially worrying about the particulars of certain particular
models, albeit some very important ones. But, it is open to question whether this should be
considered to be the most important target of physics instruction.
In fact, I am not alone in arguing that we should think of learning to build mathematical
models of the physical world as a central goal of physics instruction. For example, Hestenes
(1987) adopts a similar view, arguing that the structure of the models inherent in physical
principles and equations should be made explicit for students. Furthermore, this view can be seen
as part of a more general movement that takes models, not necessarily formulated as equations, as
the targets of physics instruction (e.g., White, 1993a, 1993b; White & Frederiksen, in press). My
contribution to these movements is twofold: (1) the proposal to teach a course in mathematical
modeling, understood more generally, prior to physics instruction, and (2) some new theoretical
resources for understanding the goals of a course in mathematical modeling, namely the learning
of forms and devices.
Some final thoughts on ÒConceptual PhysicsÓ
As instructors, we can see ourselves as faced with a choice: either make student symbol use
more meaningful, or use methods that do not spend as much time with equations. In proposing
the teaching of mathematical modeling, I am implicitly choosing the former alternative. However,
as I discussed in Chapter 1, there are approaches to physics instruction that have made the
opposite choice. These “Conceptual Physics” approaches do not use equations or, at least, do not
use them very much. The rational behind these alternative approaches is clear: Do not spend as
much time on symbol use. Instead, take time to attack the difficult conceptual issues of physics.
In Chapter 7, I discussed how the understanding produced by Conceptual Physics instruction
would differ from more traditional, equation-based instruction. Here, I will focus on summarizing
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some instructional considerations. First, with regard to tuning physical intuition and addressing
certain types of qualitative understanding, Conceptual Physics instruction may well have some
advantages. Recall that, in traditional physics instruction, physical intuition develops to support
and complement symbolic methods. Thus, when the “crutch” of symbol use is removed, a more
thoroughly functional physical intuition could result, as opposed to a physical intuition that just
“fills in the gaps” in a symbolic practice.
Of course, something is lost if we eliminate symbol use from instruction. The most obvious
losses have to do with the power of symbolic methods. Equations are compact and precise
statements of physical laws that are relatively easy to remember, and that can be manipulated to
derive new expressions. In addition, the use of equations is very powerful for obtaining numerical
results.
But there would also be some more subtle losses. I argued that an expert’s physical intuition
includes a component that is rooted in symbolic experiences. This symbolic component of
physical intuition is a real part of an expert’s physical sense. If physics is taught without much
symbol use then this component of physical intuition will simply be lost.
There is no escaping the fact that Conceptual Physics instruction teaches something different
than more traditional instruction. It leads to students having different capabilities and results in a
divergent sense of the physical world. But this observation that Conceptual Physics instruction is
not just an alternative route to the same end does not necessarily indicate a problem. I believe that
it is acceptable for us to be flexible in our instructional goals; learning about qualitative features of
the world has value, just as does learning to mathematize the world. Furthermore, Conceptual
Physics instruction likely has substantial value as a way to prepare students for more traditional,
algebra-physics, instruction.
This viewpoint really lies at the heart of the orientation that drives this work. My purpose in
arguing that understanding will be altered in a fundamental respect is not to deflect us from
designing alternative approaches; in fact, my purpose is quite the opposite. I want to free
educators, especially those with some domain expertise, to redesign subject matter for instruction.
Toward that end, I have tried to develop theoretical resources that can help us to understand the
consequences of alterations to subject matter, like that embodied in Conceptual Physics.
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Prospects for programming-physics
I have been especially interested in understanding the strengths and limitation of a particular
proposal, the use of programming in physics instructions. In this regard, this work is still very
preliminary, though I hope to have made some progress. First, I have tried to at least give the
reader a taste for a true practice of programming-physics: We have seen what the activities could
be like, at least within a limited domain.
In addition, I have begun to outline the deep ways in which programming-physics is a
different physics than algebra-physics. I argued that each practice tends to focus on different sorts
of phenomena and to draw on different aspects of physical intuition. The bottom line here was
my characterization of programming-physics as “a physics of processes and causation,” and
algebra-physics as “a physics of balance and equilibrium.” These descriptions do not necessarily
argue for one instructional practice or the other, but they do help us to understand the
consequences of each approach.
Even with only my descriptive comparisons in hand, there may be some real arguments to be
made for or against the use of each practice. A big issue here is the use of expressions “with
understanding.” Although I have taken pains to argue that students do have a surprising degree of
ability for seeing meaning in expressions, this does not mean that students do not frequently fall
into meaningless symbol manipulations, particularly early in the instructional cycle. Thus,
programming-physics might have an important advantage if it turned out that it was easier to use
programming expressions with understanding. I did not draw strong conclusions concerning this
issue, but we did encounter some reasons to believe that programming statements might be easier
to interpret. This boiled down to the observation that the programming context provides support
for the interpretation of expressions. Specifically, this support derived from the fact that
programming statements are always embedded in a program that includes many specifics of the
circumstance under study, and from the fact that the programming context may include more
naturally available devices.
But, many of the important pros and cons of programming-based instruction were not wellcaptured by my analysis. Chief among the elements left out by my examination was some analysis
of what is allowed by each symbol system. With programming, a student can tackle some
phenomena — particularly time-varying phenomena — that are not easily examined with algebra.
Furthermore, the same, core numerical methods can be applied to a wide range of these
phenomena, thus tending to emphasize and make plain the core simplicity of Newtonian theory.
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In addition, as I discussed in Chapter 11, the use of a programming language allows the
generation of a dynamic display, which provides a very powerful channel of feedback.
Of course there is a flip-side to this story; there are activities and analyses allowed by algebra
that are not allowed by programming. Some phenomena are more easily tackled in algebraphysics. And algebra allows you to manipulate expressions to derive new expressions.
It seems that both programming-physics and algebra-physics have their strong points. Thus,
one conclusion that we can draw is that optimal instruction would draw on both of these practices,
mixed together in a manner that emphasizes their strengths. Given this conclusion, what are some
strategies for instruction? How might we draw on these two practices, would we teach them
together? Or would it be best to teach one first and then the other?
There is certainly no definitive answer to this question, but I do want to emphasize one
strategy that is especially promising. I believe that programming-physics could provide a useful
entry into physics learning, especially for younger students who do not know algebra or are not
yet adept in its use. In fact, the Boxer Research Group at UC Berkeley has implemented
programming-physics instruction with some younger students, including a course for sixth graders
and a pre-physics course for high school students (diSessa, 1989; Sherin et al., 1993). In both
courses, students were able to engage readily in the construction of simulations. Even the sixth
grade students created many simulation programs and these programs did useful work for them,
functioning as precise statements of models of physical phenomena.
To give a feel for how this worked, I will tell a brief story from our high school course. (This
story is more fully recounted in diSessa and Sherin, 1995.) Our high school physics course was
preceded by a one month introduction to Boxer programming. Then, on the first day of the
physics course, the teacher conducted an open discussion of a number of example motion
phenomena. Included in these phenomena was the motion of a ball that is held above the ground
and then simply released. Next, on the second day, the students were asked to program
simulations of some of the motion phenomena. They were not given much prompting, other than
what had already come out in the discussions of the previous day. Nonetheless, the students did
not complain and they went right to work. Over the course of this single classroom session, quite a
variety of interesting simulations were produced, including a number of distinct simulations of the
dropped ball motion.
To make a complicated story simple, the students essentially produced three types of
simulations of this motion: (a) a program consisting only of a series of forward commands and
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with no particular pattern in the step increments, (b) a simulation in which the amount that the
sprite steps forward on each iteration is multiplied by a constant factor, and (c) a simulation in
which a constant value is added to the amount that the sprite steps. Examples of each of these
three types of simulation are shown in Figure Chapter 12. -1.
ho
fd
fd
fd
fd
fd
fd
fd
fd
fd
fd
fd
fd
fd
fd
(a)
Doit
1 dot
3 dot
5 dot
7 dot
9 dot
11 dot
13 dot
14 dot
15 dot
16 dot
17 dot
18 dot
19 dot
20 dot
n
pace
D1
112.455
20
21
Data
Data
Data
go
go
repeat 18
repeat
fd
n * 1.3
dot
fd
D1
dot
Doit
Doit
Doit
change n
pace
change D1
n * 1.3
(b)
D1 + 1
Doit
Doit
Doit
Doit
(c)
Doit
Doit
Figure Chapter 12. -1. Examples of the three types of simulations of a dropped ball.
Over the next few weeks, the students worked on better understanding each of these
simulations and, with experimentation, they slowly made progress toward deciding which of the
models more properly describes the physical world. In this way, programming provided a way for
students to do some real physics in the very first days of instruction. They developed their own
models and then expressed them in a formal language. Furthermore, once these programs were
written, they functioned in future classroom discussions as precise statements of the associated
models. Thus, programming can provide an entry into physics that does not use algebraic
notation, but which still preserves many of the benefits of symbol use, such as the ability to make
precise and formal statements of models. For this reason, programming has great potential as a
way for younger students, even sixth graders or younger, to learn physics within a practice that
includes many of the important properties of instruction based on algebraic notation.
The Symbolic Basis of Physical Intuition
As humans, we populate our environment with our own inventions. The very nature of our
experience in the world depends on what we have put there, as well as socially and culturally
determined ways of interacting. How can this not leave an indelible stamp on an individual’s
knowledge? Think of some basic examples. We know much about the artifacts in the world — we
know how to operate our computers, how to drive our cars, and how to play the piano. Clearly,
there is no doubt that our minds are cluttered with knowledge relating to human inventions.
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The only question is how deep this all goes. Are there only effects at the level of details? Is our
knowledge of human artifacts aptly described as just a database of facts added on to a more pure
and timeless core? Or, are there more deep effects that get down to the character of thought? Do
the very categories that we have for understanding the world depend on our inventions?
In the case of some artifacts, it seems unlikely that there are deep and far-reaching effects on
the character of human taught. For example, consider a particular invention, the automobile. How
does the existence of cars effect what people know? Well, there are some obvious additions to the
knowledge of individuals. For example, many people know the details necessary to operate a car.
But there are likely some more subtle effects. For example, we may in part understand some other
domains through metaphor to our experience with cars. Just a moment ago I looked at my old
computer and thought: “Boy that computer has really got a lot of miles on it.” And there are
probably some even more subtle effects. Because there are cars, our experience is altered since we
can and do travel to a greater variety of locations in the world.
So it seems that the existence of cars may have many effects on what we know. But, while
these effects may be spread through many aspects of our knowledge, they likely do not constitute
any major change in the character of knowledge. It seems relatively certain that, in a world
without cars, the basic nature of human knowledge and thought processes would be quite similar,
at least in domains that are not closely related to cars themselves.
But researchers have posited more profound effects for some other types of artifacts. The class
of human artifacts I have been calling external representations seems especially likely to have deep
and far-reaching influence (for some reasons discussed in Chapter 7). In this regard, perhaps the
most highly touted artifact is natural language. To some researchers, natural language is much
more than a tool for communication, it is the very basis of thought. Recall, for example,
Vygotsky’s statement, cited in Chapter 1, that natural language allows an infant to “divide his
formally undifferentiated thought (Vygotsky, 1934/86, p. 219).” In fact, for Vygotsky, language
plays a special role in giving human thought its unique character. Furthermore, as I discussed
extensively in Chapter 1, there has been a wide range of speculation concerning the effects of
literacy on human knowledge (Goody and Watt, 1968; Olson, 1994).
In this work, I did not look at spoken or written language, I examined th