Experiences with Symbolism
Running head: EXPERIENCES WITH SYBOLISM
Middle School Students’ Experiences with Symbolism
Jeonglim Chae
University of Georgia
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Experiences with Symbolism
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Introduction to the study
Background
Whenever we think something and express or communicate about it, we need
some tools. In doing algebra, symbols provide one such tool with which we can think of
and communicate about our thoughts and ideas. Not only are symbols a tool for
representation, they have also played a critical role in developing algebra. If we consider
generality as what makes algebra most different from arithmetic, then the beginning of
algebra is historically traced back to ancient Mesopotamia and Egypt. In spite of almost
four thousand years of history of algebra, the history of symbols had not begun until the
16th century. It was Vieta who used symbols purposefully and systematically after some
mathematical symbols (e.g. , , ) were introduced with letters used for unknowns
(Kline, 1972). Before then, algebraic ideas were stated rhetorically, and special words,
abbreviations, and number symbols were used as notations. Since Vieta, algebra has
rapidly developed from a science of generalized numerical computations, to a science of
universal computations and then into a science of abstract structures thanks to symbolism
(Sfard, 1995).
Even though symbolism made it possible to study abstract structures in algebra by
expressing complicated mathematical ideas succinctly, symbolism is one of the major
difficulties for young students in learning algebra. Hiebert et al. (1997) explained that the
difficulties in dealing with symbols as a learning tool were attributed to the fact that
“meaning is not inherent” in symbols (p. 55). They insisted that meaning is not attached
to symbols automatically and without meaning symbols could not be used effectively. So
students should construct meaning for and with symbols as they actively use them. The
Experiences with Symbolism
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National Council of Teachers of Mathematics’ (2000) Algebra Standard also encouraged
using symbols as a tool to represent and analyze mathematical situations and structures in
all grade levels. In particular, students in Grades 6 – 8 are recommended to have
… extensive experience in interpreting relationships among quantities in a variety
of problem contexts before they can work meaningfully with variables and
symbolic expressions. An understanding of the meanings and uses of variables
develops gradually as students create and use symbolic expressions and relate
them to verbal, tabular, and graphical representations. Relationships among
quantities can often be expressed symbolically in more than one way, providing
opportunities for students to examine the equivalence of various algebraic
expressions (p. 225-226).
In this recommendation, NCTM put emphasis on using problem contexts to help students
develop meaning for symbols and appreciate quantitative relationships.
In line with the issues mentioned above, the present study is intended to provide
insight into students’ experiences with symbolism. In particular, the educational purpose
of this study is to inform mathematics educators of how students construct meaning for
algebraic symbols and learn mathematical concepts with symbols so that mathematics
educators can enhance students’ learning of mathematics with symbols.
Research questions
In the abstract development of algebra with systematic symbolism, Wheeler
(1989) argued that abstract algebra sacrificed the implicit meanings for its applicability
unlike rhetorical and syncopated algebra. For instance, Diophantus, as a syncopated
Experiences with Symbolism
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algebraist, created numeral expressions like 10 – x and 10 + x and multiplied them to get
100 – x2 as if they were numbers like 2 and 3 to solve his word problem. As he denoted
the letter x as an unknown but fixed value in the context of the problem, he could keep
the meaning of x for its applicability explicitly. However, he might have not obtained the
notion of variables, which abstract algebra achieved (Sfard & Linchevski, 1994). As
Kieran (1992) elaborated, symbolic language made algebra more powerful and applicable
by eliminating “many of the distinctions that the vernacular preserves” and inducing the
essences (p. 394). However, the powerful yet decontextualized language brought
difficulties for young students who were beginning to learn algebra:
Thus, the cognitive demands placed on algebra students included, on the one hand,
treating symbolic representations, which have little or no semantic content, as
mathematical objects and operating upon these objects with processes that usually
do not yield numerical solutions, and, on the other hand, modifying their former
interpretations of certain symbols and beginning to represent the relationships of
word-problem situations with operations that are often the inverse of those that
they used almost automatically for solving similar problems in arithmetic (Kieran,
1992, p. 394).
In fact, some researchers (see Kieran, 1992) have studied students’ difficulties in
manipulating symbols as mathematical objects and modifying their interpretations of
symbols. Also some studies (e.g. Stacey & MacGregor, 1997) were conducted to
investigate how meaning for symbols could be developed. Hiebert and Carpenter (1992)
reviewed such literature and summarized that making meaning for symbols could develop
in two ways: through connections between symbols and other representational forms or
Experiences with Symbolism
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through connections within the system of symbols. Then they analyzed that the former
way mainly served a public function of symbols as “recording what is already known” for
communication and the latter served a private function as “organizing and manipulating
ideas” (p. 73-74). This analysis drew my attention and provoked my curiosity about how
early algebra students begin to make sense of symbols, which we as adults with
mathematical knowledge take for granted.
My initial curiosity included sporadic questions like: how do young students
interpret mathematical symbols?; in what ways do they use symbols?; do they feel the
need for symbols?; what do they want to represent with symbols?; in what ways do their
understanding of symbols affect learning mathematical concepts?; and so on. Inspired by
these questions, the present study will investigate how middle school students develop
algebraic reasoning with algebraic symbols while doing mathematical activities. The
following questions will guide this study:
1. How do students make sense of algebraic notations in relation to other forms
of representation throughout mathematical activities?
2.
How do students’ mathematical concepts form and develop as they use the
algebraic notations throughout mathematical activities?
I presume that students’ prevalent experiences with algebraic symbolism occur in
classroom learning situations where the learning experience includes the teacher’s
lectures, reading mathematics books, doing hands-on activities, observing how the
teacher and other students use symbols, and discussions with other students. So I will
mainly focus on mathematical activities in the classroom setting in the present study. In
addition, it is most likely that the learning experiences begin with introducing algebraic
Experiences with Symbolism
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symbols and students try to make sense of them throughout the following activities. So
when students are introduced to algebraic symbols for the first time, they do not yet have
their own meanings for the symbols. In order to differentiate symbols before and after
students develop their meanings for them, I will use the term, ‘algebraic notations’,
before their meaning development.
Thus the first research question is about how algebraic notations become symbols
to students as students do mathematical activities in the classroom setting. Specifically, I
will investigate it through how they relate algebraic notations to other forms of
representation such as narrative, tabular and graphical representations. Moreover, as
students have experiences about symbolism, their new mathematical concepts will be
created or previous ones may change or develop. With the second question, I will
investigate the development of students’ mathematical concepts throughout mathematical
activities. I will explain the research questions in detail in the next section.
Theoretical perspectives
This section includes theoretical perspectives with which the present study will be
guided. The first part is allotted to my perspectives on learning mathematics in general,
and the second part is to show briefly what and how theoretical perspectives on
symbolism are to be applied for the present study.
Perspectives on learning
Perspectives on learning include statements of subjectivity that will inform and
affect all the activities of the present study in general. In particular, this section mainly
Experiences with Symbolism
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includes my current personal beliefs formed through learning and teaching mathematics
and studying mathematics education as a graduate student. Since the beliefs will provide
lenses and constraints with which I design the present study and interpret all possible
phenomena the study will bring, I believe that it is worthwhile to state here. My
perspectives on learning are quite parallel to what radical constructivists assume as
underlying principles: (1) knowledge is not passively received but built up by the
cognizing subject, and (2) the function of cognition is adaptive and serves the
organization of the experiential world, not the discovery of ontological reality (von
Glasersfeld, 1995, p. 18). First of all, I believe that learners are cognizing agents so that
they do not receive or absorb what others speak about or try to deliver. For example, I
had learned mathematics via lectures and fortunately most of my mathematics teachers
helped me understand mathematical concepts. When I discussed or worked on problems
with my classmates who had shown similar mathematical abilities and performances, I
could find differences in how we understood a certain topic and strategies we used. If we
were receivers or absorbers of knowledge, we should barely find differences in
knowledge or our understanding of it. I think the differences came from what we did in
our own mind. Secondly, learners construct their own knowledge. Here construction
means neither creative invention or discovery like what professional mathematicians do
nor isolated construction without help from others. Rather construction means making
sense of and organizing cognitively what learners have experienced. Therefore, thirdly
learning occurs to learners through their experiences. What I mean by experiences is
mostly the learners’ interaction with their environment, and examples of experiences
related to learning include learning activities, reading books, communicating with others,
Experiences with Symbolism
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listening to teachers, reflecting and so on. Through the interactions, learners may
experience that sometimes they can make sense of the interactions within their existing
cognitive structure and therefore the structure becomes more supported, or sometimes
they feel a necessity of reorganizing the cognitive structure to make sense of the
interactions. I consider both as learning. Finally, I believe that knowledge is ultimately
personal. In fact, mathematical knowledge definitely has social and cultural aspects not
only because mathematics has been built historically and culturally throughout a long
period of time but also because an individual learner’s mathematical experiences include
his or her social interaction. In spite of the social aspects of mathematical knowledge, it
is ultimately the learners who make choices of whether the developed knowledge will be
included meaningfully into their knowledge structures and whether the knowledge will be
actively used for their mathematical activities like solving problems.
To connect my beliefs about learning with the present study, Kaput’s (1991)
epistemology with three worlds seems helpful. He identified the material world, the
subjective world, and the consensual world. I interpret the material world as a collection
of objects that a learner will learn about, or construct his or her knowledge about. The
subjective world seems the place where an individual learner constructs his or her
knowledge and to which the resulting knowledge or cognitive structure belongs. In fact,
Kaput (1991) contrasted the two worlds as the former for “what is experienced as
physical” and the latter for “what is experienced as mental” (p. 59). And the consensual
world includes consensual knowledge that a consensual community produces. For
example, mathematical concepts - as those as we assume mathematically competent
people accept - such as sum, fraction, and linear relationship belong to the consensual
Experiences with Symbolism
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world, but each student’s concepts of those are not necessarily the same as those in the
consensual world and they belong to his or her own subjective world. However, the
classification is not always clear or it would seem unnecessary providing that individuals
could not leave their own subjective worlds and so their experiences also stay there. That
is, individuals have experiences of the material world and the consensual world in their
own subjective world. So then everything is in the subjective world.
Nonetheless, I think the idea of three worlds would help me conduct this study in
understanding students’ learning. Algebraic notations that students will learn in class
have been already used among people who have made sense of them and participated in a
consensual community, even before students see the notations. That is, algebraic
notations (or symbols) are “experienced as physical” (Kaput, 1991, p. 59) by someone
else, and so I would say that they are not in each student’s subjective world yet but they
belong to the material world. Also we assume that some consensual meanings of
algebraic notations have been developed in the consensual world (I think that is why we
call them symbols in general without possessive case) so that people who have made
sense of them can communicate with them. [Here I put aside the issues such as whether
the consensual world, which all the members of the community reached an agreement on,
exists or whether three world exist exclusively.] Then when algebraic notations are
introduced to students, they begin to move into individual students’ subjective worlds as
students try to make sense of them. During the process, students will have a contact with
a consensual world through their teacher and resources and so the consensual world will
affect the process. For example, teachers will bring something that they believe are
consensual meanings of some algebraic notations in class and the so-called consensual
Experiences with Symbolism 10
meaning will affect students’ sense-making process. However, in this study I do not
intend to compare students’ subjective world about algebraic notations with any
consensual world. As I acknowledge students’ contact with a consensual world as their
learning experiences, I will try to model each individual student’s subjective world, and
look at how the individual students bring algebraic notations into their subjective world
and make them fit into their existing world or transform their existing world into a new
one. With this second-order model (Steffe & Wiegel, 1996), I aim to explain individual
students’ experience with algebraic notations while observing and interacting with the
students as the researcher.
Perspectives on symbolism
In order to study students’ experience with symbolism, it is necessary to define
the term ‘symbols’ as opposed to algebraic notations and decide which symbols will be
the focus of the present study. Cobb’s (2000) broad definition of symbols seems to be a
good start and his definitions is:
…to denote any situation in which a concrete entity such as a mark on paper, an
icon on a computer screen, or an arrangement of physical materials is interpreted
as standing for or signifying something else (p. 17).
His definition is very broad so that it can be applied not only for symbols in mathematics
but also those in everyday life. As an example of symbol in everyday life, I could use
two distinct erasers pretending they were cars in order to explain a traffic accident that I
experienced. By moving the two erasers, I could explain how the accident happened.
Here the erasers were symbols since they were concrete entities in the accident situation
Experiences with Symbolism 11
and stood for cars. As an example of a mathematical symbol, the fractional numeral,
‘3/4’, could stand for a fair share of sharing 3 apples among 4 children, a ratio of 3 out of
4, or an operator as in 3/4 of something in various contexts. Even when removing a
specific context, ‘3/4’ still could refer to the abstract concept of fraction. Both the erasers
and 3/4 are legitimate symbols under the same definition, but they are very different. The
erasers have quite clear meaning both to me and to the person listening to my traffic
accident but lose the meaning as cars once leaving the accident context. Unlike the
erasers, the fraction ‘3/4’ has varying meanings in contexts and becomes an abstract
entity even without the context. Considering the research questions, the second example
is a definite interest of this study. So in order to define symbols in mathematics for this
study I modify Cobb’s (2000) definition as to denote a mathematical situation in which a
concrete entity written on paper, a board, or computer screen is interpreted as standing
for or signifying something mathematical.
Since symbols should stand for or signify something, it has to be noted that a
physical entity like 3/4 written on paper is not a symbol at all for a first grader who has
just learned counting. However, the fraction 3/4 is a symbol for those who have
meanings for it and it also has consensual meanings developed by the consensual
community. To differentiate these two, I will call such entities ‘notations’ for those who
do not have their own meaning as yet.
With the definition above, the term, symbols, will be used in this study to
encompass other terms such as symbolic representations and symbolic expressions. An
equation, y = 65x, has a string of symbols such as letters x and y, a numeral 65, and the
equality sign and each symbol stands for something mathematical. However, at the same
Experiences with Symbolism 12
time, the whole equation may also signify a situation in which a distance (y) changes in
relation to time (x) when driving at 65 mph. Then the equation, which may be considered
as a symbolic representation or a symbolic expression for some other purposes, is a
symbol to signify the changing situation. In fact, since the mathematical content of this
study is the representation of changing situations with variables, it is frequent that
equations should be considered as symbols. In addition to that, the differentiation
between symbols and other similar terms is unnecessary in order to highlight the
relationships that students make between something to signify and something signified in
this study.
Another noteworthy point in the definition of symbol is what symbols stand for or
signify. What is something mathematical? Literatures on symbols provided a seemingly
agreeable answer as symbols signifying process and symbols signifying concept. The
frequent case is that a learner is introduced initially to symbols as a vehicle to signify
process, gets familiar with the process, and conceives the symbols as an object carrying a
concept. Also researchers have tried to explain the cognitive shift from symbol as
process to symbol as concept. For example, Mason’s (1987) spiral model explained the
shift of symbol uses as:
from confidently manipulable objects/symbols,
through their use to gain a ‘sense of’ some idea involving a full range of imagery
but at an inarticulate level,
through a symbolic record of that sense,
to a confidently manipulable use of the new symbols,
and so on in a continuing spiral (p. 74-75)
Experiences with Symbolism 13
In particular, Mason not only explained the shift of symbol uses between symbol as
process and symbol as concept but also identified the continuous acquisition of new
symbols based on symbols that are used confidently. However, these studies seemed to
divide up symbol uses artificially and highlight the linear or hierarchical development of
symbol uses from those signifying a process to signifying a concept and from one symbol
to another. To me, the demarcation between process and concept is not clear and they
seem to grow together.
In contrast with the previous models, I believe that the notion of procept by Gray
and Tall (1991, cited in Tall et al., 2001, p5.) seemed to resolve the issues mentioned
above. They still used the notions of process and concept, but they focused on a powerful
way of using symbols to switch between process and concept from time to time rather
than focusing on symbol uses as progressing from one to the other. Tall et al. (2001)
elaborated the notion of procept as:
It [procept] is now seen mainly as a cognitive construct, in which the symbol can
act as a pivot, switching from a focus on process to compute or manipulate, to a
concept that may be thought about as a manipulable entity. We believe that
procepts are at the root of human ability to manipulate mathematical ideas in
arithmetic, algebra and other theories involving manipulable symbols. They
allow the biological brain to switch effortlessly from doing a process to thinking
about a concept in a minimal way (p. 5).
So procept enables a learner to conceive symbols not only as signifying a process (e.g.,
compute or manipulate) but also as a concept and to switch between them flexibly. Thus
“being able to think about the symbolism as an entity allows it to be manipulated itself, to
Experiences with Symbolism 14
think about mathematics in a compressed and manipulable way, moving easily between
process and concept” (Tall et al., 2001, p. 8).
Spectrum of outcomes
procedural
To DO
routine
mathematics
accurately
proceptual
To perform
mathematics
flexibly &
efficiently
To THINK
about
mathematics
symbolically
Procept
Process (es)
Procedures
Progress
Process
Procedures
Procedure
Sophistication
of development
Figure 1: A spectrum of performances in the carrying out of mathematical processes (Tall
et al., 2001, p.88)
Before reaching the proceptual level, students can do a specific computation by
knowing a specific procedure and then they become more flexible and efficient forming a
process out of multiple procedures (see Figure 1). To distinguish procedure and process,
they meant procedure as “a specific sequence of steps carried out a step at a time” and
Experiences with Symbolism 15
process as “in a more general sense any number of procedures which essentially have the
same effect” (Tall et al., 2001, p. 7). As an example, solving a linear equation, 3x + 2 =
17, is a process in general but solving this equation by subtracting 2 from both sides and
then dividing both sides by 3 is a procedure. If a student can solve the equation with the
procedure accurately and it is the only procedure available to him, he is at the procedural
level in the spectrum. However, when he has other procedure(s) and he can relate
procedures meaningfully, he is at the process level. So the ‘process’ here as a noun is
more than the sum of multiple procedures and more advanced than processing the task to
get a result.
By relating them to the SOLO (Structure of Observed Learning Outcome)
taxonomy (Biggs & Collis, 1982), each level in the procept model will be understood
better. The SOLO taxonomy has five levels: prestructural, unistructural, multistructural,
relational, and extended abstract. Olive (1991) used the taxonomy to analyze students’
performance of geometric tasks and so he explained that unistructural students can “use
one piece of information only in responding to the task” (p. 91). This level corresponds
to the procedural level in the procept model in that students in this level can do only one
procedure. As developing from the multistructural level to the relational level in the
SOLO taxonomy, it is said that students become able to relate multiple information.
These two levels correspond to the process level in the procept model where students
have multiple procedures and relate them to each other so that they develop their process.
Finally, the proceptual level is parallel to the extended abstract level in the SOLO
taxonomy where students can “derive a general principle from the integrated data that can
be applied to new situations” (Olive, 1991, p. 92). Thus, by looking at the structure of
Experiences with Symbolism 16
the students’ learning outcomes (relative to SOLO) I shall be able to locate students’
progress and developmental sophistication relative to the procept spectrum.
In the procept spectrum (see Figure 1), I interpret ‘progress’ as being fluent at
each level; for example, at the procedural level students become fluent at solving linear
equations and then they can solve them accurately with a procedure. As moving toward
the proceptual level, students develop their mathematical sophistication. However, the
procept model did not show explicitly how students make the development from a level
to the next.
As a parallel model to the procept model, APOS theory (Dubinsky & McDonald,
2001) explained students’ learning of any mathematical concepts by mental constructions
of actions, processes, objects, and schemas. Those with an action conception can
perform an operation with external stimuli. Then through repeating an action and
reflecting on it they can construct a process, which allows them to perform the action
without external stimuli. From a process, the students construct an object when they
consider the process “as a totality and realize that transformation can act on it” (p. 274).
Finally, a schema for a mathematical concept is constructed as a collection of the mental
constructions of actions, processes, and objects, and other schemas so that students can
access and use the cognitive structure to solve related mathematical problem situations.
When APOS theory is applied to students’ learning of symbolism, it is quite
similar to the procept model. Both suggest some mental constructions, which are
hierarchical, although not necessarily linear. Also the mental constructions in both
models correspond with each other. However, both seem to explain students’ learning of
Experiences with Symbolism 17
symbolism within symbols rather than paying more explicit attention to the relationship
with other forms of representation.
Since this study will investigate students’ symbolism in relation to other forms of
representation, adopting another model for explaining the relationship seems necessary.
In that sense, Kaput’s (1991) representation model seems helpful with the models
previously mentioned together.
Cognitive
Cog A
Cog B
Operation
(Deliberate)
Interpretation
or
(Passive)
Evocation
Outward Projection
(writing, speaking,
drawing, etc.)
A
B
Individual Acts of Reference
Shared
A
Shared Referential
Meaning
Shared
B
Figure 2. Kaput’s referential relationship (1991, p. 60)
Kaput (1991) explained the referential relationships between ‘notation A’ as a
representation (the term ‘notation’ was used as a collective term of representations.) and
‘referent B’. In his model (see Figure 2.), the bottom rectangle describes the relationship
between A and B as we consider it as consensual (e.g. the symbolic expression y = 3x + 2
[A] represents the line [B] with slope 3 and y-intercept 2). However, when considering
Experiences with Symbolism 18
individuals’ referential relationships, he insisted that the cognitive acts should be
included. The upper rectangle describes “acts of interpretation, mental operation, and
projection to a physical display” through which individuals make A and B related
referentially (p. 59). Then he emphasized that the acts happened in the subjective world
but the result of the mental acts would be shown in a material form of referent B.
In the present study, Kaput’s (1991) model will play the major theoretical
framework to explain students’ sense-making of algebraic notations in relation to other
representations, which is the first research question in this study. Since the mathematical
content of the present study is to represent changing situations with two variables,
considered other representations are narrative, tabular, and graphical representations.
Narrative
representation
Symbolic
representation
Tabular
representation
Graphical
representation
Figure 3: Referential relationship under this study
Among four forms of representation shown in Figure 3, six ways of referential
relationship will be investigated in this study. The other six ways will not be investigated
Experiences with Symbolism 19
directly, but they will be discussed somehow through symbolic representations. That is,
students’ referential relationship between tabular and graphical representations will be
implied by those between tabular and symbolic representations and between symbolic
and graphical representations. Since symbolism is the main focus in this study, symbolic
expression will be placed in ‘notation A’ in the diagram (Figure 2). The other
representations such as verbal, tabular, and graphical will play as referents in order to
explain how students make sense of symbolism. For instance, with this diagram it will be
explained how students interpret a narrative situation where two variables change
relatively, what mental operation they go through and how they relate the situation with
symbolic expression. Since the referential acts are bi-directional, the other direction will
be also explained.
In particular, when explaining students’ “acts of interpretation, mental operation,
and projection to a physical display” in each referential relationship (see Figure 2), either
the procept model or APOS theory (whichever makes better sense with the data) will
inform me to analyze the data. While explaining students’ sense-making of symbolism, I
expect that a certain mathematical concept of students will emerge in the context. The
possible relevant concepts would be those of variables or rates. These concepts will be
selected as answering the first research question, and the analysis of them will be what
students’ concepts are and how they develop, if ever, through mathematical activities
under the second research question from my perspectives.
Experiences with Symbolism 20
Methodology
The present study is to be conducted within the activities of a NSF-funded
ongoing project, Coordinating Students’ and Teachers’ Algebraic Reasoning (CoSTAR).
The project purposely studies “teachers’ and students’ understandings of shared
classroom interactions and ways that teachers and students work together to shape the
teaching and learning of middle-school algebra” for three years (Izsak et al., 2002, p. 2).
Whereas CoSTAR investigates both teachers’ and students’ algebraic reasoning, the
present study is to be conducted from only the students’ perspectives, so that some
activities (such as classroom interactions) that will be approached through both teachers’
and students’ perspectives in CoSTAR, are to be interpreted only from the viewpoint of
the students in this study.
Participants in the present study are selected from students in Pierce Middle
School, a rural school that provides the research site for CoSTAR. Participating students
are in grade 7 in the academic year 2003-2004 and they begin to learn algebra with
materials from the Connected Mathematics Project (CMP; Lappan et al., 2002). Teachers
in Pierce Middle School select 6 units out of 8 in CMP material to teach, and the second
unit, “Variables and Patterns: Introducing Algebra”, is the mathematical content of this
study. The unit is composed of 5 investigations, but the last investigation, “Using a
Graphing Calculator”, is excluded in the study since it is mainly about how to draw a
graph and make a table on a calculator, not to use a graphing calculator as an
investigating tool for the mathematical topics. The four investigations under this study
are “Variables and Coordinate Graphs”, “Graphing Change”, “Analyzing Graphs and
Tables”, and “Patterns and Rules”. Through out these investigations, CMP asks students
Experiences with Symbolism 21
to (1) recognize problem situations in which two or more quantitative variables are
related to each other, (2) find patterns that help in predicting the values of a dependent
variable as values of other related independent variables change, (3) construct tables,
graphs, and simple symbolic expressions that describe patterns of change in variables,
and (4) solve problems and make decisions about variables using information given in
tables, graphs, and symbolic expressions (Lappan et al., 2002, p.4). In particular,
changing situations in this unit are introduced without the notion of functions, and one
situation where students organize bicycle tours is used as a context throughout the four
investigations.
Since CoSTAR launched its investigation in the spring of 2003, I have worked for
the case study of Ms. Susan Moseley including observing and taping her classes and her
student interviews. In Ms. Moseley’s class, three pairs of students have already
participated in the case study in CoSTAR. As avoiding overlap, four students are
selected for the present study based on Ms. Moseley’s recommendation. I mainly
consider students’ participation in class activities, their abilities for self-expression, and
compatibility with a partner to include them in this study. So two boys, Greg and Jeffrey,
and two girls, Angela and Peggy, are included.
With the background information aforementioned, the rest of the methodology
section will discuss methods of both data collection and data analysis. The data
collection section will include data collection methods and interview protocols, and the
data analysis section will include data analysis methods and an example of data analysis.
Experiences with Symbolism 22
Data collection
The data collection method in this study has two layers. The first layer is also a
part of CoSTAR data collection and it is to videotape the classroom activities on a daily
basis through out the unit. For each class, two project staff members capture the
classroom activities from two different angles. One with a stationary camera records the
teacher’s perspective, which contains teacher’s written work on the front board and the
whole class view, and the other with a mobile camera records students’ perspective,
which includes students’ written work on their seats and the group working view.
Once class taping is done, two views from both cameras are synchronized and
digitized through a video mixer. Therefore, we can see one view from the stationary
camera in the main view with the other view from the mobile camera in the small view on
top of the main view at the same time, or vice versa. In addition, lesson graphs (see
Appendix) are written for each class, which show how class activities go and what
mathematical issues emerge in the class along with the time stamps. Mixing two views
and writing lesson graphs are done on the same day of taping and these are shared
responsibilities with CoSTAR staff.
As the second layer of the data collection of this study, I conduct a series of
interviews with participating students. Based on observation of classroom activities and
lesson graphs, interview tasks are carefully chosen in order to answer the research
questions. In particular, watching classroom activities and reading lesson graphs over
time helps me determine what mathematical issues around algebraic notations this study
will investigate. So each interview aims to look at students’ sense-making of the targeted
algebraic notations and their concept development with them. However, as the
Experiences with Symbolism 23
interviewer, I would not limit the purpose of the interviews to understanding students’
experiences of symbolism but would sometimes ask probing questions relating to their
classroom activities. By doing so, I would rather extend their experiences by actively
interacting with participants so that they could possibly have learning experiences during
the interviews as a benefit of involvement with this study and I could also have dynamic
and rich data, which I could analyze for purpose of answering my second research
question concerning students’ mathematical concepts.
Each interview is also video-recorded with two cameras, likewise for description
of taping classroom activities. One stationary camera provides a whole picture of an
interview and the other captures a focused view of students’ work. During the interviews,
I interact with a pair of students; Angela and Jeffrey make one pair and Greg and Peggy
make the other. The main reason for pair interviewing is that students can reveal their
thoughts more clearly by communicating with each other rather than having one-to-one
interaction with the interviewer. They also can provoke each other’s thinking through
interactions and by reflecting on their own thinking relative to that of the other. However,
neither interactions between students nor their possible interactions with the interviewer
are the main focus of the analysis of this study. Rather each individual student’s thinking
as it relates to the research questions is to be the main focus of the analysis.
In order to follow the data collection schedule in CoSTAR, I have already
conducted all interviews with the students. Retrospectively, interviews had two phases
according to the time line. The first phase of interviews were conducted in the fall of
2003 when students were taught the unit, “Variables and Patterns”, in class. Each pair of
students had five 50 to 60 minutes interviews, one each week from early November to
Experiences with Symbolism 24
mid December in 2003. By the end of 2003, ten interviews were all synchronized and
digitized and the interview graphs were completed. Interview graphs have the same
format as lesson graphs, showing how the interview proceeds, along with the
interviewer’s notes.
After reviewing the ten interviews, I planned to have the second phase of
interviews with the same pairs in February 2004 in order to pursue issues that were not
answered in the fall of 2003 due to the time constraint and the interviewer’s lack of
research experience. I proposed to interview each pair of students three times and
finished six interviews by mid February 2004. Whereas the interview schedules in the
first phase were constrained by both the school activities and the CoSTAR data collection
schedule, the second phase interviews were not so constrained by students’ learning in
class. Thus, the six interviews were conducted in two weeks.
In addition, while the intent of the first phase of ten interviews was to investigate
students’ thinking as focusing on emerging issues following the classroom activities, the
second phase of six interviews was to understand how students relate algebraic notations
to the other forms of representation in order to make sense of algebraic notation for
changing situations. Thus the second phase of interviews targeted the research questions
more directly as supported by the first phase of interviews.
The purpose of the interviews in the first phase was to see how students relate
different forms of representation to make sense of algebraic notations used in classroom
activities and how their related mathematical concepts develop. The specific goals of the
interviews were:
Experiences with Symbolism 25

Students are to describe a situation with two variables, to identify variables in
a situation, to identify which variable is independent and which is dependent,
and to plot the graph of two variables.

Students are to make a table and/or a graph after reading a narrative of a
changing situation, to interpret data given in a table and a graph, and to compare
tabular, graphical and narrative representations of the situation.

Students are to search for patterns of change in a graph and a table, to describe
a situation with verbal rules, and to predict a change.

Students are to show their understanding of the relationship between rate, time,
and distance, to identify and represent rates in a table and graph, and to express
patterns in symbols.
In order to pursue the goals, I developed the interview protocol based on my own analysis
of the unit, “Variables and Patterns”, and emerged issues from observing students’
classroom activities. Below, I provide the interview protocol containing sets of prototype
questions under eight themes.
Themes
1. Variables
2. Independent vs.
dependent variables
Interview questions



What does variable mean to you?
Create examples of variables.
Create situations where two variables change relative
to each other.
 What do independent and dependent mean to you?
 How do you tell which variable is independent and
which is dependent variable in the changing
situation?
Experiences with Symbolism 26


3. Representing changing
situations








4. Connecting data points
on a graph





5. Distance vs. steepness of
segment connecting two
data points on a graph



How do you represent a changing situation?
Can you generate a data set of your changing
situation based on your invented story?
Given a table, can you draw a graph?
How do you determine if a given data set is
cumulative or not?
Given a table or a graph, can you make up a story?
Can you discern a pattern in a table or a graph?
What does the scale mean to you in a graph?
How does the scale help you understand the graph?
Does connecting coordinate points always make
sense on a graph?
When do you decide to connect the data points on a
graph?
Can you think of various ways to connect data
points? How are they different from each other?
Given a certain way to connect data points, what does
that way indicate to you about the data?
Where do you see the most increase/decrease in a
given graph? How about the least increase/decrease?
Does connecting data points help you see the
increase/decrease? How?
What do you mean by the ‘distance between dots’,
‘(diagonal) space between dots’, or ‘biggest jump’
[theses are students’ verbatim in class discussion]?
What does the steepness mean to you? How is it
different from the ‘distance between dots’?
How do you compare steepness of two different line
segments? Can you measure the steepness?
Can you draw a line segment of the same steepness
with a given line segment? How about steeper one,
or less steep one?


6. Constant rate of change
What does the constant rate of change mean to you?
How can you use the constant rate of change to
answer some questions?
 How do a graph and a table show the constant rate of
change?
7. Slope of line segment


What does the slope mean to you?
How do you determine the slope of a line segment?
Experiences with Symbolism 27


8. Symbolic representation






What does a general rule mean to you?
How can you use symbols to represent the general
rule?
How does the formula tell you the story?
Can you generate a table with a given formula?
Can you draw a graph with a given formula?
How does a formula show independent and
dependent variables of the changing situation?
How does a formula show a constant rate of change?
How does a formula show a slope of a line segment?
While the first phase of interviews addressed most of the questions in the protocol,
the second phase of interviews focused on several themes only. That is, I pursued
questions of how students use symbols to represent changing situations and how they
make sense of symbolic representation with respect to the other modes of representation.
Some interview questions were adapted from the CPM unit being studied. The daily
interview protocol in the second phase of interviews is shown below. The day numbers
begin at Day 6 to be consecutive with the 5 days of interviews for each pair conducted
during the first phase.
Experiences with Symbolism 28
Day 6
1. What do you think a rule means?
2. Suppose you are asked to baby-sit your younger sister or brother and your
parents would pay you $3 per hour. Can you find a rule for the total payment?
Can you use symbols to express the rule as an equation?
3. Suppose you join a book club and you have to pay $10 membership fee and $4
per book purchased. Can you find a rule for the total cost? Can you use
symbols to express the rule as an equation?
4. This table shows the relationship between the number of people on a club
picnic and the cost of lunches. Can you find a rule for the cost? Can you use
symbols to express the rule as an equation? Using the equation, can you find
the lunch cost for 25 people? How many people could eat lunch if they had
paid $89.25?
The
number
of
people
Cost in
dollars
1
4.25
2
3
4
5
6
7
8
9
8.50 12.75 17.00 21.25 25.50 29.75 34.00 38.25
5. This graph shows the relationship between the number of concert tickets and
the total cost. Can you find a rule for the total cost? Can you use symbols to
express the rule as an equation?
80
70
Total cost
60
50
40
30
20
10
0
0
5
10
15
N um ber of tickets
20
25
Experiences with Symbolism 29
Day 7
1. Given the equation, d = 8t, can you make up a story? What does d, 8, and t,
represent in your story respectively?
2. Make a table that shows the distance traveled for every hour up to 5 hours.
3. Can you make a table that shows the distance traveled for every half hour up to
5 hours? How do the two tables differ? What do they have in common?
4. Draw a graph of the equation, d = 8t.
5. Make up a story with an equation C = 2n + 5 and draw a graph of it.
Day 8
Sidney started a table to help the partners determine their cost for the bike tour.
Number of
customers
1
2
3
4
5
6
7
8
9
10
Bike rental
$30
60
90
120
150
180
210
240
270
300
Food and camp
costs
125
250
375
500
625
750
875
1000
1125
1250
Van rental
700
700
700
700
700
700
700
700
700
700
1. Can you find a rule for the total cost for any number of customers? Can you
write the rule with symbols?
2. Theo’s father has a van he will let the students use at no charge. Then write a
new rule for the total cost for any number of customers with symbols.
3. If the partners require customers to bring their own bikes, write a new equation
for the total cost.
Experiences with Symbolism 30
Data analysis
Video-recorded interviews are to be the primary data set along with students’
written work in this study. The method to analyze the video data is informed by iterative
videotape analyses (Lesh & Lehrer, 2000), which have several interpretation cycles. The
first interpretation cycle includes debriefing, which includes brief notes and feedback
from both the interviewer and possibly some project staff and these are saved as word
documents right after each interview. In fact, unlike the first phase of interviews, Dr.
Olive observed the six interviews in the second phase while operating the camera
capturing students’ written work and he provided feedback after each interview. The
second interpretation cycle includes producing interview graphs, which have the same
form with the lesson graphs, while playing videotapes of interviews before conducting
the next interview session. The third interpretation cycle includes writing transcripts for
selected interviews. Since the second phase of interviews was more directly oriented
toward the research questions, they are to be transcribed completely. However, selective
transcripts for the first phase of interviews are planned. The last interpretation cycle is to
analyze all the interviews across students. Unlike the three previous interpretation cycles,
the fourth cycle is to be repeated to produce interpretations of students’ activities from
the theoretical perspectives of the present study. In order to answer the research
questions, supplementary data sets, such as students’ homework, notebook, written quiz
and video clips showing their participation and discussion in classroom activities, are also
to be included whenever necessary.
Experiences with Symbolism 31
In the rest of this section, I will present an initial analysis of a part of interview as
an example of data analysis. The analysis is about Greg based on the interview on
February 12, 2004. Interview tasks are shown in the Day 7 interview protocol in the
previous section. The analysis here includes the interview tasks related to the equation d
= 8t. I will mainly describe Greg’s referential relationships from the equation to a table
and a graph in terms of how he interpreted the equation, what operations he did in order
to relate the equation to a table or a graph and how he projected his interpretations and
operations into a table and a graph based on Kaput’s (1991) model (see Figure 2). Also I
will describe his concepts of variables and rates through a description of his referential
relationships.
Before I asked Greg to generate a table from the equation d = 8t, Peggy made up a
story of traveling 8 miles per hour with d for the traveled distance and t for the time
traveled. She said, “You’re traveling from Georgia to Washington, D. C and you’re
traveling 8 miles per hour. And you have different time intervals for when you stop.”
Greg agreed on Peggy’s story as saying, “I think it’s just like what she said”. Then, when
asked to make a table, Greg made a table with two columns and put distance on the left
and time on the right (see Figure 4 (a)). He wrote down the times first from 1 hour to 5
hours by a half hour interval but missing .5 hour and wrote 8 in the distance column for 1
hour. Then he stopped writing distances and added .5 hour in the time column. Next he
filled the distance column by adding 4 to each proceeding values, starting from 0. He
described what he did as saying;
Experiences with Symbolism 32
“I did my time and I went by 30 minutes intervals in all. And I went all
the way to five and [for] each [half hour] I added 4 each time. And each
hour it was 8 and you added 8 to each hour, like 4 then 8 and 12 and 16”.
(a) Greg’s table
(b) Peggy’s table
Figure 4: Students’ tables for the equation
So apparently Greg interpreted 8 in the equation as 8 miles for every hour and he could
add 8 miles repeatedly. In addition, he used the relation symbolized by the equation to
generate 4 miles for every half hour, which I considered as evidence of his concept of a
constant rate. When projecting his interpretations and operations into a table, he did so
without any hesitation. Unlike the convention, he put the distance column in the left of
the table, but he still wrote the time column first and then filled in the distance column.
Greg indicated that he regarded distance as the dependent variable and time as the
independent variable when he said, “I think the time is the independent variable because
it doesn’t like rely on anything else to have it change.” Moreover, he recognized the
changing patterns in both variables as saying, “they both change at a set interval”. That is,
he was able to recognize linear patterns in both variables and the relationship between
two variables, which I think will be critical when he has a situation with uneven intervals
for either variable.
Experiences with Symbolism 33
When I asked Greg to explain differences between his table and Peggy’s, he
considered both were basically the same. I think this showed Greg’s understanding of
rate and patterns in changing situations at the same time. Below is the excerpt from the
interview.
I: Do you see the difference between these two (pointing to the two tables
in Figure 4)?
G: Yes.
I: OK, would you tell me?
G: The intervals are different, the time and distance on each one.
I: The intervals are different. I mean the intervals in the time, right? How
about the intervals in the distance?
G: Not really because all you’re doing is just breaking it down to where
it’s half go with the…
I: OK, what do you mean by breaking down?
G: Like 4 is half of 8 so you just divide it by 2 and it’s just because it’s
half hour that you do that.
Then I asked both students to draw a graph for the equation, but both seemed to
draw their graphs based on their tables. Greg put the time on the x-axis with the scale
of .5 hour and the distance on y-axis with the scale of 2 miles. Then he plotted data
points correctly. However, Greg wrote labeling numbers on the grid square rather than
on the grid line as in Figure 5. When he explained about his graph, he knew which grid
line he meant by a certain number. So it did not cause any trouble at this point.
Experiences with Symbolism 34
Since I assumed that Greg drew the graph from his table, I asked him whether he
could draw a graph from the equation without looking at his table. He said yes right
away and explained, “You can go by 8 each time for distance and for each hour you have
to set, you would have to set how many hours, like it was 5. You would do 5 down here,
then you could figure out, just plot it each time.”
Figure 5: Greg’s graph of d = 8t
When drawing a graph from the equation, his interpretation and operation were the same
as when making a table. He seemed to set an hour time interval for his convenience so
that he could plot the data points easily. However, when he plotted the data points, he
confused himself by his way of labeling unlike the previous graph. Then his graph
looked different from what he meant. Eventually, this led to a discussion between Greg
and Peggy when Peggy tried to interpolate data points for half hours in Greg’s graph. In
listening to what Peggy said about his graph, Greg reflected on his way of labeling the
axes. So when asked what would be a better way to put the numbers on the axes so as to
avoid confusion, Greg rewrote the numbers on the grid lines.
As my analysis in terms of Kaput’s model (see Figure 2), Greg apparently
interpreted the equation d = 8t as a relationship between distance and time. That is, he
interpreted the equation as the situation where the distance changed according to the time
Experiences with Symbolism 35
at the rate of 8 miles per hour. The important thing here is that he conceived the letters of
d and t as variables with the relation, not as simply naming distance and time. Thus as
his cognitive operation, he was able to specify time first and figure out the matching
distance with the relation of 8 miles per hour. Moreover, since he conceived time as the
independent variable, he was able to specify time flexibly either one-hour or half-hour
intervals. When he chose a one-hour interval, he could apply the relation of 8 miles per
hour to find out the matching distances for each hour by adding another 8 miles
repeatedly as he moved to the next hour. When he chose a half-hour interval, he kept the
relation but he was able to “break it down” as dividing 8 by 2 since it is a half-hour
interval. I infer that by using his concept of a constant rate, he was able to consider ‘8
miles per hour’ and ‘4 miles per half-hour’ as the same rate. This flexibility showed that
Greg was able to conceive the symbol ‘d = 8t’ as signifying a process and a concept at the
same time. Thus it could be evidence that Greg reached a proceptual level (Tall et al.,
2001) or constructed an object (Dubinsky & McDonald, 2001) for the equation d = 8t.
Projecting from the equation into a table seemed very natural to Greg as he put his
specified times and the matching distances in the two columns. Also, when projecting
from the equation into a graph, he seemed to have no problem in plotting the pairs of
specified time and the distance but his way of labeling axes caused some confusion.
In summary, this analysis highlighted three elements in Greg’s referential
relationship (Kaput, 1991) from the equation d = 8t into a table and a graph. He was
successful in interpreting the referent equation as the relation of two variables, specifying
the variables, and projecting the results of these cognitive operations into a table.
However, his projection into a graph indicated that a graphical representation was not as
Experiences with Symbolism 36
intuitive as a tabular representation, even when accomplishing deliberate interpretation of
the equation and cognitive operation from the interpretation. That is, when projecting
into a table, Greg focused on his operation of specifying variables and simply put
matching values in the same row of the table. He did not need to worry about the
convention of where to put these values. However, when projecting into a graph, his
cognitive operation had to include the conventions of graphical representations such as
appropriate scales and numerical labels on both axes. The conventions seemed not quite
intuitive to Greg so that his ways caused some confusion in communicating with Peggy.
Nonetheless, when explaining how to plot, Greg put his pencil at a value on the x-axis,
went up by matching the appropriate y-value, turned his pencil to the left, and went
horizontally to the y-axis, physically demonstrating where his pencil turned as the point
to be plotted in the coordinate system. Then while actually plotting, he began at a value
on the x-axis, went up by the matching y-value, and put a point without drawing the
horizontal line. Apparently, he seemed to visually coordinate his position with the y-axis.
I considered that this physical projection was the result of the fundamentally same
cognitive operation as he used for a tabular representation except for the conventions of a
graphical representation. In further analyses, Greg’s success will be contrasted with other
students’ difficulties in order to address my research questions concerning students’
experiences with symbolism.
Experiences with Symbolism 37
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Experiences with Symbolism 40
Appendix: Example of interview graph
Interview with Peggy & Greg on Feb 12 2004
Time
Description
8:13:27
Interview begins with providing an equation
“d = 8t” and asking Peggy to read the
equation. She reads, “d equals to 8 times t.”
Then I ask them to make up a story. Peggy
tells that d can be a distance traveled at 8
miles per hour. She clarifies that d stands for
a distance, 8 means 8 miles per hour, and t
stands for the time. Greg agrees as saying that
he thought as Peggy told.
8:15:10
Then with the equation I ask them to make a
table for every hour up to 5 hours. Greg
makes a table for every half hour and with
two columns of distance and time. He sets up
the time first and writes the distance for the
time.
Peggy makes a table for every half hour first
and changes into for every one hour.
8:18:05
I ask Peggy to explain why she changed into
every hour. Peggy tells that she simply forgot
and remembered my direction. For the
distances, Peggy tells that she added 8 for
each hour since the speed is 8mph. Then I ask
about the distance for hour 4 and Peggy
realizes her mistake and corrects it.
* Lesson graph has the same format with interview graph.
Comment
Interview tasks are to
generate other forms of
representation from
symbolic representation.
Greg makes his table
with a half hour interval,
which is my next
question.