Student Obstacles and Historical Obstacles to Foundational Concepts of Calculus
Robert Ely
Abstract
My dissertation contains two sections. This is the first, and larger, section, which comprises
a description of a study in which 240 university calculus students were given a questionnaire about
foundational calculus concepts: function, limit, continuity, and the real number line. Based on their
responses and some follow-up interviews they were categorized according to the epistemological
obstacles they displayed to these concepts. A cluster analysis revealed several clusters of commonly
co-occurring obstacles in students. A historical analysis shows that two of the three important
clusters were displayed by I. Newton and G. W. Leibniz respectively, suggesting significant parallels
between student conceptions and historical conceptions. This indicates that one key to
understanding these parallels lies in shared epistemological dispositions, such as the disposition toward
smooth motion or toward algebraic simplicity. These dispositions support, but do not strictly
constrain, the multiple connected conceptions observed in the study.
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Student Obstacles and Historical Obstacles to Foundational Concepts of Calculus
After seeing a proof that 0.99999… = 1, one of Anna Sierpinska’s students, “Ewa,” objected:
“Arithmetically or algebraically, it’s all right, but in reality… It will be close to one but will not equal
one. There will be a slight, very slight difference, but a difference all the same… It’s like the
asymptote of a hyperbola: they will never touch each other… The difference is getting smaller and
smaller but it will not turn into zero…” (1987, p. 378)
Although Ewa admitted that the proof was mathematically correct, this did nothing to convince her
about what happens “in reality.” Her intuition was that the number 0.9999… is not really a static
number at all, but rather an infinite process that never finishes. Perhaps the number itself is moving
along the number line and approaching 1. Either way, the process is dynamic and temporal: the
number 1 is “never” reached. In fact, her belief was so strong that when a mathematical proof
contradicted it, she rejected the proof rather than changing her intuition.
Ewa’s views are not unusual among calculus students. As we shall see, research in undergraduate
mathematics education has repeatedly shown students to have similar obstacles to the concepts of
limits, real numbers, and functions, and that these obstacles are not going away.
A closer look at Ewa’s comments reveals two things that help us understand the obstacles she is
experiencing. First of all, she illustrates the dynamic quality of the numerical limit process by
describing the dynamic quality of a completely different graphical process. This strongly suggests that
Ewa’s conception of the real numbers is fundamentally connected to her conception of the limit as a
temporal process that never attains its goal, which in turn appears to be connected to a continuous
motion-based graphical conception of functions.
Second of all, Ewa’s combination of dynamic conceptions suggests some parallels with the way
Newton himself may have thought of these concepts. Functions (although the term did not exist in
Newton’s day) were smooth curves drawn in space, and limits were continuous approaches along
such curves. The real number line was a continuous background on which points could be placed
and moved dynamically.
Is this parallel between a student’s conceptions and historical conceptions really significant, or are
am I being overly imaginative? Well, we know this parallel is not totally trivial, because there were
certainly other ways that mathematicians thought about these foundational calculus concepts. For
example, two hundred years after Newton, Weierstrass treatment of these same concepts were
devoid of any intuitions about geometry or motion whatsoever. For him a function was an arbitrary
uni-valued correspondence between two sets, the real numbers formed such a set with a particular
metric on it, and a limit was an abstract string of symbols containing an algebraic criterion for
closeness.
Although educational researchers have noticed parallels between student conceptions and historical
conceptions of calculus (Sfard, 1991; Lakoff & Nunez, 2000; Cornu, 1991; Kaput, 1994), most of
calculus curriculum and pedagogy fails to capitalize on or even notice such parallels. With the
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exception of a few calculus textbooks (e.g. Hahn, 1998), history is relegated to anecdotes in the
margins about quirky mathematicians. Few calculus teachers appear to really agree with Poincare
that “the task of the educator is to make the child’s spirit pass again where its forefathers have gone,
moving rapidly through certain stages but suppressing none of them.” (Bressoud, 1994, p. vii)
On the other hand, perhaps this lack of attention is warranted. In fact, the same educational
researchers warn us against blithely building curriculum and pedagogy around the observation of
such parallels between student thinking and historical thinking. Says Anna Sfard, “The deliberately
guided process of reconstruction may not follow the meandering path of those who were the first
travelers through an untrodden area.” (1994, p. 195)
In order to understand if (and perhaps how) to tread in the footsteps of these first travelers, we
must begin to investigate what these parallels are between student thinking and historical thinking
about calculus, and why they might or might not exist. In addition, rather than just observing these
parallels haphazardly, we should test whether or not these historical conceptions, connections, and
progressions occur consistently within students, and why.
Ewa’s comments suggest that the key to the matter is in the connected nature of conceptions about
these calculus concepts. If we really want to investigate the parallels in calculus between history and
cognition, we have to determine if student conceptions are connected to one another in the ways
that we see such conceptions connected in history.
By focusing on the connections among student conceptions, we are taking what Michele Artigue
terms a “horizontal” approach to the question of student thinking (1999). While most research
takes on a “vertical” approach, examining the hierarchical structures of mathematical concepts, in
this paper I follow Artigue’s suggestion to focus on horizontal relationships between concepts
instead.
Theoretical Background
The purpose of this theory section is to prepare such a connected approach to the research of
student thinking in calculus. First, I briefly discuss a philosophy of mathematics that allows us to
seriously investigate a deep connection between student thinking and historical thinking. Then I
describe a systematic set of categories of student thinking, called epistemological obstacles, within
the particular research domain of the foundational concepts of calculus (limit, function, real number
line, and continuity). In the last part I discuss how the results of studying student thinking within
these categories should give us an insight into the connections between the historical development
and the cognitive development of mathematics.
What is mathematics?
According to Romberg (1992), mathematics has been, and still is, predominantly considered to be an
absolute objective body of knowledge. Davis & Hersh (1981) more precisely claim that most people
are intuitively mathematical Platonists. This view holds that mathematical objects are real things,
unchanging over time, place, and culture. Two plus two equals four, no matter where you live in the
universe. Such facts are discovered, not invented. Thus merely by reflecting perceptively enough,
with enough time and perseverance, one could discover all of mathematics. The reason
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mathematics can be used to describe the real world is that the world is orderly, not because we
induce our mathematical concepts from it.
In the late 19th Century, when mathematical results began to arise that were unintuitive and even
counter-intuitive, such as non-Euclidean geometry and set theory, Platonism ran aground. By the
mid-1900s, and even until today, most mathematicians, if pressed on the matter, would instead claim
a formalistic view of mathematics. According to Davis & Hersh, “most writers on the subject seem
to agree that the typical working mathematician is a Platonist on weekdays and a formalist on
Sundays.” (1981, p. 321) Formalists believe that mathematics is just a formal game played with
meaningless symbols. The game is this: decide collectively and more or less arbitrarily on a set of
terms and axioms to “believe,” making this set as small as possible, then prove theorems from this
according to a set of rules. If the axioms, rules, or theorems happen to describe the outside world,
then that is perhaps a nice coincidence.
The problem with both of these views from our perspective is that neither helps us to understand
the connection between students’ conceptions of mathematics and historical development. For
example, it is striking how late in the historical development of calculus—200 years after the
fundamental theorem of calculus was developed—that the current definition of limit appeared, the
concept that fundamentally undergirds all of calculus. Formalism cannot explain why the logically
first would be chronologically last, and Platonism cannot explain why intuitive but incorrect
mathematics was developed with great confidence along the way. A philosophy that treats
mathematicians simply as being objectively but unwittingly “wrong” for such a long time does
nothing to help us understand why they thought this way.
We instead turn to a third, socio-historic view. One part of it is the mathematical philosophy of
fallibilism. Adapting some of Karl Popper’s ideas from the philosophy of science, Imre Lakatos
wrote a series of articles called Proofs and Refutations (1976). These claimed that mathematical truth
must be understood in conjunction with its history and with the cultures that produce it.
Mathematics is fallible, developing not through a cumulative advancement of rigorous proofs, but
rather through an iterative process of constant criticism and correction. Proofs are simply
explanations that happen to convince the community of researchers at a given time, that give
credibility to the results. The standards for this credibility change over place and time.
This view helps us understand something like the bizarre development of the limit concept
(described later in this paper). The intuitive limit definition was finally overhauled when a bulk of
critical evidence was amassed against it. But even more compelling, it was only by the early 1800s
that this bulk of critical evidence was actually considered “evidence” against the informal limit
concept. What counted as a counter-example or a rigorous proof depended on the beliefs,
assumptions, and outlooks of the culture that was doing the examining. Is it just a coincidence that
the Enlightenment’s unprecedented faith in man and his reason—which allowed mathematicians to
work with infinity for the first time, and thus develop calculus—achieved its mathematical crisis
(Fourier - 1807), philosophical crisis (Kant - 1781), and political crisis (French Revolution - 17891799) in a period of less than three decades? Our philosophy of mathematics must include an
understanding that mathematical truths, and the criteria for their acceptance, depend on culture,
time, and place.
This dependence on culture can be viewed weakly or strongly. The weak view is that our attention is
constrained by our history and culture, but not necessarily our conclusions. In other words, we will
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never develop a particular mathematical idea unless or until our culture provides a reason to focus
on or entertain a particular mathematical notion or question. But once this impetus is provided, the
societal influence is minimal and our mathematical conclusions are more or less logically derived.
This view still largely reconciles with Platonism and formalism. The strong view is that not only is
our looking constrained by history and culture, but so is what we see when we look. Our
mathematical conclusions, just like our sensory observations, are filtered through the concepts that
we use. The strong view reconciles best with the theories of learning I will use, described in the
following section.
One objection that can be raised to this strong view is the “two plus two still equals four
everywhere” objection. It is hard to deny that while many aspects of mathematical truth are culturedependent, many are not. Lakoff & Nunez (2000) provide a nice answer, by claiming that all of our
conceptions, including mathematical ones, are built up metaphorically one by one, grounded in our
bodies and in our physical perception of the world. In all cultures and times humans have two legs,
ten fingers, heartbeats, etc., so we should expect many of our mathematical concepts, especially basic
ones, to be the same. On the other hand, the larger our metaphorical structures become, the more
room there is for cultural variation and the more chances there are for particular environmental
factors to form and guide our concepts and conclusions. This theory of embodied cognition also
begins to suggest why historical development of concepts and individual development of concepts
would often parallel each other, because the metaphorical structure of a concept is the same whether
it is a historical concept or a student’s.
The most complete theoretical account of the relationships between historical development and
cognitive development is furnished by Jean Piaget in his final book, co-written with Roland Garcia,
Psychogenesis and the History of Science (1983/1989). The authors claim:
“…[our] goal is not to set up correspondences between historical and psychogenetic sequences in
terms of content, but rather to show that the mechanisms mediating transitions from one historical
period to the next are analogous to those mediating the transition from one psychogenetic stage to the
next.” (p. 28)
The transitional mechanisms of which they speak include the transitions from intra-object (analysis
of objects) to inter-object (analysis of relations between objects or transformations of objects) to
trans-object (making a structure of objects, which can in turn be analyzed as an object) levels of
analysis. Theoretically, if there are conceptions (structures) found in students and in history alike,
this is due to similar mechanisms governing the creation of these structures. Of course, it is much
easier to study the structures themselves and to conjecture about the transitional mechanisms; the
examples the authors provide are along these lines.
In order to investigate this question about relationships between historical structures and student
structures, we turn to the existing research about student thinking in calculus. Only with a catalogue
of student conceptions and mechanisms can we begin to conjecture about these relationships. In
the next section I describe findings about student conceptions of the foundational elements of
calculus, seen through the lenses of several relevant theories of learning in mathematics.
Student thinking about the elements of calculus
By “elements of calculus” I mean the building blocks with which the main ideas of calculus are
constructed. These foundational concepts include a) the real number line, b) functions, and c) limits—
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concepts used to construct main ideas such as derivatives and integrals. I also include here d)
continuity as a foundational concept of calculus. What I refer to here is global continuity, continuity
of a function on its whole domain, not at one point. Although this kind of continuity is not
foundational in the sense that it is not required for the logical construction of derivatives or
integrals, it is nonetheless a concept that students often erroneously conflate with other notions, but
that they are expected to know before taking calculus. Continuity is an important characteristic of
many physical phenomena and informs many important calculus theorems such as the intermediate
value theorem. These four foundational concepts are considered prerequisite to the main ideas of
calculus, and are usually taught first in a calculus class or are expected to be already known.
Most theories of student thinking in higher mathematics (APOS theory (Dubinsky & McDonald,
2001), mathematics as metaphor (Lakoff & Nunez, 2000), and so forth) address these foundational
concepts. Rather than discuss how these different theories treat all of these concepts, for
simplicity’s sake I will highlight four of these important theories of student thinking, one for each of
these four concepts. For each concept I have chosen a theory that has produced significant insights
about it. These four theories represent most of the influential research in undergraduate
mathematics education.
But before describing these theories in detail, I wish to describe a fifth theory of student thinking—
the theory of epistemological obstacles. This overarching theory organizes the others, in the sense
that each other theory can be seen as outlining and describing some important epistemological
obstacles. In this main section of my review, I will first describe the theory of epistemological
obstacles. Then I will describe each of the other theories, how it applies to one of the foundational
concepts, and which epistemological obstacles it outlines and describes for this foundational
concept. Some of these obstacles I use extensively in my own research; these I will describe in
detail. Finally, many of these epistemological obstacles are also found in the history of mathematics,
so I will supplement the descriptions of these obstacles with examples drawn both from student
work and from mathematics history.
Epistemological Obstacles
The theory of epistemological obstacles has its roots in the work of Bachelard (1938), and was
developed with respect to mathematics education in the research of Guy Brousseau (1983, 1997).
The idea of mathematical epistemological obstacles fits more broadly into Brousseau’s theory of
didactical situations in mathematics, as a kind of difficulty encountered in classroom situations even
when teachers and students are adequately performing their respective jobs. He describes a general
obstacle to learning as follows:
“Errors are not only the effect of ignorance, of uncertainty, of chance, as espoused by empiricist or
behaviourist learning theories, but the effect of a previous piece of knowledge which was interesting
and successful, but which is now revealed as false or simply unadapted. Errors of this type are not
erratic and unexpected, they constitute obstacles. As much in the teacher’s functioning as in that of
the student, the error is a component of the meaning of the acquired piece of knowledge.”
(Brousseau, 1997, p. 82)
So an obstacle to learning is not simply a lack of knowledge, but is rather a piece of knowledge that
both interferes with and is part of further learning. This idea is clearly Piagetian. The student enters
a problem situation not as a blank slate, but rather with functioning knowledge that he or she may
need to reconstruct. An obstacle is a specific form of Piagetian adaptation; an obstacle is often quite
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resilient, withstanding some contradictions and sometimes even the establishment of a better piece
of knowledge. (Duroux, 1982)
The word “epistemological” was originally used by Bachelard to indicate a connection with a
learner’s broad beliefs about what constitutes knowledge. In Brousseau’s work, the term
“epistemological” came to simply signify the fact that the obstacle’s role in the construction of the
concept is structural, part of the knowledge itself, not accidental. These two meanings of the word
“epistemological” still figure into the research on epistemological obstacles. For instance, Sierpinska
(1992) includes epistemological obstacles that operate on several levels, one of which deals with
beliefs or worldviews about what mathematics and knowledge are, another of which deals with
specific understandings of a particular concept. In this study, I use only the latter level for
epistemological obstacles, and refer to things operating on the former level as paradigms or
worldviews.
An epistemological obstacle is a kind of obstacle that is inherent to the concept itself, rather than
one caused by, say, a particular way the student has been taught, which Brousseau calls a didactical
obstacle. For Brousseau, the path to the desired knowledge necessarily goes through the given
epistemological obstacle. For this reason, epistemological obstacles are also found in the history of
mathematics, in the form of particular misconceptions along the path of the historical development
of certain mathematical concepts.
An example of an epistemological obstacle is the observed inability of both students and historical
mathematicians alike to quantify change algebraically. Kaput (1994) notes that the Greeks viewed
change as a quality to be noticed rather than a quantity to be mathematically described. Boyer notes
that “the quantities entering into Diophantine equations are constants rather than variables, and this
is true also of Hindu and Arabic algebra.” (1959, p. 72) Students experience this obstacle as well:
“I am struck by a parallel between certain aspects of Greek mathematics and aspects of our students’
mathematical condition as they traditionally enter the study of calculus. Students are primarily
(discrete) arithmetic creatures, whose understanding of algebraic literals is as ‘unknowns’ rather than
as variables…” (Kaput, 1994, p. 85)
It makes sense that students first understand a letter in an equation as signifying an unknown
quantity that must be solved for. With this perspective, they begin to understand an equation as an
equivalence relation, solved by performing the same operations on both sides. Only with this notion
of equivalence and this facility of manipulation can they understand and work with an algebraic
equation relating multiple variables. But in the meanwhile there is still the conception of the letters
as unknowns rather than variables that must be surmounted, the very conception that enabled them
to get this far. The obstacle is a necessary piece of knowledge, but now, almost paradoxically, it
stands in the way of further knowledge.
Two immediate objections can be raised, which I will address in turn. (A) How do we know if a
particular misconception is really inherent to the acquisition of the given concept? Might it rather
be, for instance, a vestige of the way the concept has always been taught and which no one has
found a better way to teach? (B) Is the history of mathematics a faithful source for epistemological
obstacles? Why should I believe that, say, Leonhard Euler, whose language, culture, and
mathematical background was so different from my own, would have encountered some of the
same obstacles that I have?
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Objection A seems virtually unanswerable. It is impossible to really prove that a given obstacle is
cognitively inherent to a concept. The most we could reasonably learn from an empirical study is
that the obstacle is common and persistent. In identifying epistemological obstacles in the literature
I therefore soften this requirement of necessity, an approach also used by Sierpinska (1988, 1992). I
could require that it at least be possible to make a theoretical case for the necessity of the obstacle in
learning the concept, but this presupposes too much as well. We do not actually know enough
about the concepts in question (particularly the real number line) to make such strong claims about
their precise genetic decompositions. One of the goals of this study is in fact to determine how
particular obstacles fit into the construction of these concepts.
One possible complication with the softening of the necessity requirement is that an obstacle might
be common and persistent due to institutionalized teaching practices rather than epistemological
structure. In other words, it may now become difficult to tell didactical and epistemological
obstacles apart. On the other hand, if an obstacle is consistently experienced across time and place
due to how it is being taught, then this kind of teaching looks to be as much a part of our culture as
the concept itself. Therefore, it is not so easily eradicated as a didactical obstacle, which usually
refers to an almost accidental development due to an unusual teaching situation. We should treat it
as epistemological in nature.
Objection B about the connection between student obstacles and historical obstacles is part of what
this study is attempting to understand. If an obstacle is (virtually) necessary, then it should appear in
history and in students. Nonetheless, the differences between pedagogical situations and historical
mathematical ones must be addressed. For example, a student’s obstacle is only an obstacle because
there is a goal conception or “correct” conception in mind, which the teacher is seeking. In
contrast, at no point did the historical figures know that their current conception was not the
“correct” conception. The “final draft” version of a concept such as limit is not inherent to
mathematics itself, or dropped down from heaven. Any sense of inevitability to this historical
process is imposed only by our hindsight.
Realizing this actually makes the history of mathematics appear to be an excellent source for
identifying epistemological obstacles. After all, the current version of a given concept is formed by
such a historical process, meaning that the various misconceptions and realizations along the way are
imbedded in the concept as well. They form its context, its notation, its terminology, its story…—in
some sense they are the concept. It would be surprising if students, in learning such a concept, did
not encounter these same obstacles.
Thus, in identifying potential epistemological obstacles for this study, I use four necessary, not
sufficient, criteria, closely related to the criteria outlined by Duroux (1982):
a. An epistemological obstacle is a piece of knowledge that functions adequately in one domain
or among certain tasks.
b. This piece of knowledge fails to function adequately in a broader domain, or among a
broader set of tasks.
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c. The obstacle is common. Encountering the obstacle is an expected, perhaps necessary, step
in the acquisition of the new knowledge or conception. For the sake of this study, this
criterion usually amounts to the fact that the obstacle is prominent in the literature.
d. The obstacle is found in the history of mathematics. If the epistemological obstacle really is
part of the concept itself, then it should appear in the concept’s historical evolution.
The epistemological obstacles I propose in the following sections generally meet these four criteria.
However, a few are only hypothetical epistemological obstacles, because they fail to meet one of the
four criteria. Hopefully research on student thinking and mathematics history will establish them by
determining that they indeed meet criteria (c) or (d). Finally, some of the potential obstacles here are
based specifically on the work of researchers (e.g. Cornu, 1983; Sierpinska, 1988) who have applied
the theory of epistemological obstacles to the foundational concepts of calculus, especially to the
concept of limit. I will discuss their work and the work of other researchers of the foundational
concepts of calculus in the following sections.
Continuity and concept images
The theory of concept images emerged in 1981 with a paper of Tall and Vinner. Quite simply, a
concept image is the cognitive structure in a student’s mind that is associated with a particular concept.
The idea is that this image may be quite different from the formal definition of the concept. This
work in the 1980s (Davis & Vinner, 1986; Vinner & Dreyfus, 1989; Schwarzenberger & Tall, 1978)
is the first body of research actually conducted and read by mathematicians in higher education to
focus on the differences between cognitive structure and mathematical structure. The cognitive
structure includes both the way that the concepts are connected and organized in a person’s mind as
well as the elements that constitute these concepts. The mathematical structure includes the logical
dependences of the concepts and the technical definitions of the concepts.
Here is an example of how people consistently think of a concept and its connections to other
concepts in a manner that does not reflect the mathematical definition and connections. Although
Graph 1 in Figure 1 below is discontinuous (on the real number line), 15% of the students Tall and
Vinner surveyed considered it to be continuous, because it is “given by a single formula” (1981, p.
164). Additionally, 30% of students considered Graph 2 to be discontinuous although it is not.
Half of them said this because it is not given by a single formula and half because it is not smooth (it
has a sudden change in gradient, for instance).
1) f(x) = 1/x
2) f(x) = 0 (x<0)
= x (x>=0)
(sorry about the
bad drawing)
Figure 1
Clearly the concept images of continuity evoked in students’ minds are not the same as the
mathematical definition of continuity, either the formal definition (f(x) from D to R is continuous
if,  ε > 0 and  a  D,  δ > 0 s.t.  x  D, |x – a| < δ  |f(x) – f(a)| < ε) or the informal
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definition adequate for calculus (a function is continuous on R if its graph can be drawn without
lifting one’s pencil). So where did these misconceptions come from?
These particular concept images of continuity are good candidates for epistemological obstacles.
They are misconceptions and they are common, fulfilling criteria (b) and (c). Furthermore, these
obstacles figure prominently in the history of mathematics, so they fulfill criterion (d) as well.
Leonhard Euler considered Graph 1 above to be continuous for the same reason as the
aforementioned students—it is a function given by one analytic expression. Likewise, he would
have considered Graph 2 discontinuous for the same reason (and Euler’s conception of function
was broader than that of contemporaries such as d’Alembert, who did not consider something like
Graph 2 to represent a function at all (Kleiner, 1989)). The notion of discontinuity of the 18th
Century might be better thought of as discontiguity. A piecewise function, with one formula on one
domain and a different formula on another, was discontinuous even if the two pieces met. Thus
their view of function informed their view of continuity. We should not be surprised that Euler and
his contemporaries would share these obstacles with modern-day students, because they also share
the experience of working with primarily “nice” functions: polynomials, trigonometric functions, etc.
Within the context of such functions, when describing naturally smooth, simple physical behaviors,
these obstacles are perfectly adequate and functional, fulfilling criterion (a).
Thus the following two conceptions of continuity fulfill our four criteria of epistemological
obstacles:
smoothness—a graph or function is continuous only if it is smooth, that is if it and all of its
derivatives are continuous. In particular, it has no sharp corners or breaks.
single formula—a graph or function is continuous only if it is represented by a single analytic
formula.
The concept images research (Schwarzenberger & Tall, 1978; Davis & Vinner, 1986; Vinner &
Dreyfus, 1989; et al.) specifically addresses each of the four foundational concepts of calculus, and I
will discuss some of these findings in the following sections.
The real number line and mathematics as metaphor
Like the researchers of concept images, Lakoff and Nunez (2000) also believe that the cognitive
structure of mathematics is different from its logical structure. Rather than simply noting this
difference, however, they specify that this cognitive structure is metaphorical. A cognitive metaphor is
a way of understanding one thing in terms of another. In particular, it is a mapping from one
domain to another. Each element in the source domain corresponds to an element of the target
domain, and the target domain inherits the underlying structure of the source domain. For these
authors, all of mathematics (in fact, all human thinking) is a set of metaphors built on and drawn
from other metaphors.
An example of a metaphor in higher mathematics is the Basic Metaphor of Infinity (BMI). In fact,
the authors claim that all of higher mathematics uses this metaphor in various and numerous forms.
Although later in this section I dispute the strength of this claim, it is clear that the BMI is a very
important metaphor that allows us to productively think about infinity. The BMI is, quite simply,
that “processes that go on indefinitely are conceptualized as having an end and ultimate result.”
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(Lakoff & Nunez, 2000, p. 158) An infinite process inherits the structure of a finite process. A
finite process has a beginning, a clear way of progressing from one step to the next, and a final
resultant state. An infinite process also has a beginning and a clear way of progressing from one
step to the next, and in order to preserve the structure, it inherits a “final resultant state” too. As we
shall see, the BMI applies clearly to concepts like the limit of an infinite sequence—an unending
string of numbers is viewed as having a final limit value just like a finite sequence does.
If our concepts are drawn metaphorically from other concepts, and those are drawn metaphorically
from other concepts, where do our first concepts come from? For Lakoff and Nunez, these
metaphors have their fundamental source in our bodies and in our physical experience. This theory
of mathematical cognition, and of cognition in general, is known as embodied cognition. It is often
stated in opposition to information-processing theories, which assume that our minds have
inherently logical, mathematical, or computer-like structure.
One topic in higher mathematics to which Lakoff and Nunez apply the theory of mathematics as
metaphor is the real number line. They claim that learners naturally view the line, (or the plane, or
space) as continuous. “Space does not consist of objects. Rather it is the background setting that
objects are located in” (Lakoff & Nunez, 2000, p. 260). It is absolutely continuous, like a path traced
out by a moving point. However, in order to understand advanced calculus and the functions that
underlie it, one must develop the metaphor that the real number line is a set of numbers. One must
be able to think of a space (or line) as simply a set with certain properties, with nothing “spatial”
about it. What we call points are simply handy ways of representing elements of the set. It is
important to think this way in order to understand the way the rational numbers relate to the real
numbers. The real number line (the irrational numbers and the rational numbers) is the analytic
closure of the rational numbers, in the sense that it contains exactly the numbers that could be the
limit of a sequence of rational numbers. Spatial reasoning fails here—all it can tell one is that
between every two irrationals there is a rational, and between every two rationals there is an
irrational. This spatial fact misleadingly allows one to think of the rationals and irrationals as being
similar sets. Only a more abstract analytic approach allows one to realize that the analytic properties
of the sets are quite different.
This obstacle also appears to stand in the way of understanding a function as a mapping from one
set to another. I discuss this later as a hypothesis about the connections between different student
conceptions of limit, function, and the real number line.
This way of thinking about the real number line therefore seems to be another epistemological
obstacle. This reasoning prevents further understanding, so it meets criterion (b). On the other
hand, the continuum way of thinking about the real number line is not an arbitrary misconception,
but rather arises naturally from one’s physical experience in the world. Lakoff & Nunez (2000)
point out that our physical motions through space are intuitively continuous, that in moving from
one place to another we occupy every location in between, and that this conception is important in
allowing us to understand our own movement in the world. Along with other authors (Berlinski,
1995; Kaput, 1994) they notice that the Greeks thought about numbers not as counting units but as
magnitudes, like lengths marked on a measuring stick. This kind of thinking slowed the
development of discrete mathematics. Therefore, this obstacle meets criteria (a) and (d) too.
11
naturally continuous line—The real number line is a continuum. It is a background on which
points can be situated, not itself a set of points. This is an obstacle to understanding abstract
properties of the reals, and perhaps to gaining abstract conceptions of function and limit.
The naturally continuous line obstacle raises another real number line obstacle. In Euclid’s
geometry, all theorems are built up from a few axioms that are supposedly understood by everyone
to be true. Likewise, all definitions are in terms of a few basic words that are commonly understood
and have a common intuition—line and point. In fact, if “line” is visualized differently, we can
conceptualize different reasonable axiomatic systems of geometry, such as hyperbolic or elliptical
geometry. The interpretation of the undefinable words “line” and “point” are foundational to our
common understanding and discussion of mathematics. And “line” can be conceived of in different
ways, as underscored by the previous epistemological obstacle.
In fact, this intuitive understanding of “line” is based on an intuitive understanding of “point” that
Lakoff and Nunez claim must also be reconsidered metaphorically in order to progress
mathematically. According to them, our natural physical intuition is that a point is a very little disc,
with zero or infinitesimal diameter. For instance, to most people it makes sense to say that two
points are “touching.” If points on a line did not all touch each other, then there would be gaps in
the continuous line. But for a mathematician, two points cannot “touch”; either they are the same
point or different points.
Our intuition is based on the fact that two actual things have no gaps between them if they touch.
With this conception, even if one views the number line to be comprised of points, these are arrayed
like a string of pearls, each of which touches a pearl on either side. This leads to the assumption
that for each number there is a next number and a previous number. With this idea, for example, a
student may say that 0.999… is the number right before 1. Student responses show that this
misconception is relatively commonplace (Sierpinska, 1987).
Thus, this obstacle fulfills criteria (b) and (c). Due to people’s physical intuition about touching, this
is a bit of functional, reasonable knowledge, and so also fulfills criterion (a). I have not yet found a
good historical example of a mathematician explicitly displaying this obstacle. The best evidence for
its historical existence is the lateness with which the idea of an open set was introduced, in Cantor’s
work of the late 1800s. An open set of the real numbers has no largest member, just as any real
number does not have any numbers that are closest to it. Nonetheless, I do not know if it can
legitimately be said to fulfill criterion (d).
next number—The real number line is like a string of pearls. Each point touches a point next
to it on either side. This obstacle may prevent an understanding of the density of the real
numbers, and may interfere with intuition about closed and open sets.1
There is one more important potential epistemological obstacle to the real number line that I wish to
describe: infinitesimals. Although the concept of infinitesimal is found in history as an obstacle to
limits (Cornu, 1991), I categorize it as a real number obstacle for two reasons: a) it is an inclusion of
multiple extra numbers into the real numbers, and b) the concept of infinitesimal may not really be
1
A set is dense if between any two of its points you can always find another point in the set. For instance, the
integers are not dense, but the rational numbers are. An open set is one that does not contain any of the points on its
edge, such as the set of all the points strictly between 0 and 1 but not including 0 or 1.
12
an epistemological obstacle to limit. An epistemological obstacle is meant to be a stumbling block
that is necessary to understanding the concept. It is meant to be unavoidable and constitutive.
However, infinitesimals turn out to be an alternate, logically consistent, way of thinking about
calculus without really using limits.
Briefly, Leibniz’ calculus is a calculus of infinitesimals—measurable, manipulable quantities that are
smaller than any positive real number but greater than zero. Infinitesimals were heavily used in the
18th Century, especially on the continent, but experienced some philosophical criticism (Mancosu,
1996). As we shall see in the limit section ahead, they were more or less banished by Cauchy in
favor of the static conception of limits in the early 19th Century. Over 100 years later the logician
Abraham Robinson formulated the hyperreal numbers, which include the real numbers as well as
infinitesimal numbers and infinite numbers. He showed that any (first order) statement that is true
with respect to the real numbers will also be true with respect to the hyperreal numbers. In other
words, calculus works exactly the same way if you use infinitesimals or if you don’t use
infinitesimals.
Although a textbook was written using this nonstandard approach to calculus (Keisler, 1976), it never
gained widespread support. While part of this may be due to traditionalism, certainly part of it is
due to the difficulty of teaching students enough first-order logic to understand the hyperreal
numbers.
Lakoff and Nunez use the BMI to show how infinitesimals are conceptually constructed.
Robinson’s construction of an infinitesimal number appeals to the compactness theorem in logic,
which allows one to create a system in which the statements “x is smaller than 1,” “x is smaller than
½,” “x is smaller than ⅓,” “x is smaller than ¼,”… etc. are all true for a positive x. This process
creates an infinitesimal number from an infinite set of statements. The problem here is that while
Robinson logically created the infinitesimals this way, Leibniz’ intuitive conception of infinitesimals
has little to do with this infinite process. In fact, this intuitive conception of the infinitesimal, which
I believe many students share, is based on trying to avoid thinking about infinite processes. For
instance, by viewing the derivative to be the ratio of an infinitesimal change in y to an infinitesimal
change in x, one can avoid thinking about it as a process of this ratio as the distances become
smaller and smaller. The metaphysical nature of an infinitesimal is intuitive and algebraically it is
quite handy.
This potential obstacle meets criterion (d). It is plausible that it meets criterion (c), and certainly
there have been many students who are taught calculus using an infinitesimal approach. It meets
criterion (a) in the sense that it allowed for the development of calculus on the continent. But it is
not a necessary part of the understanding of the real numbers, nor is it clear that it meets criterion
(b), that it prevents further understanding of the real numbers. It could be that it forms an obstacle
to the formal definition of limit instead.
infinitesimals—There exist positive quantities smaller than any real number but greater than
zero. Two numbers may be infinitesimally close to one another. This “obstacle” may interfere with
the formal definition of limit and the real number line, but it also could lead to alternate and
adequate definitions of these concepts.
13
Limits and encapsulation
Probably the most difficult calculus concept for students to understand is the formal definition of
the limit: limx→af(x) = L if  ε > 0,  δ > 0 s.t.  x  D, |x – a| < δ  |f(x) – L| < ε. Although it
is traditionally the first of the definitions taught in an undergraduate calculus class, it was historically
the last of these definitions to be formulated, not acquiring its current form until the days of
Weierstrass (1866). For this reason, a traditional calculus class that attempts to organize its material
axiomatically (according to its logical structure) ends up unfortunately teaching the most
mathematically and cognitively advanced material first. Especially after concept image research
established that a student’s cognitive structure of calculus concepts is different from the logical
structure of these concepts (Tall & Vinner, 1981), researchers have dedicated extensive study to
these cognitive structures.
These researchers over the last 25 years (e.g. Artigue, 1992; Cornu, 1981; Sierpinska, 1987) have
devoted their attention to the concept of limit not only because this concept is so foundational to
calculus, but also because it marks most students’ first encounter with mathematical infinity. It is of
continuing interest because student misconceptions about it are so widespread, resilient, and
resistant to change (Williams, 1991). After briefly outlining some robust findings about student
misconceptions of limit, I will highlight the theory of encapsulation, which attempts to place these
misconceptions into a broader theory of student mathematical thinking. Relying largely on these
findings and on the theory of encapsulation, I formulate several epistemological obstacles to the
concept of limit, illustrated by examples from the history of mathematics.
Researchers (e.g. Tall & Vinner, 1981; Cornu, 1991; Przenioslo, 2004; Davis & Vinner, 1986;
Mamona-Downs, 2001; Szydlik, 2000; Williams, 1991, 2000; Grey & Tall, 1991; Monaghan, 1991;
Cottrill et al. 1996; Sierpinska, 1987) have identified a number of common misconceptions about
limit, including a) the limit is a bound which cannot be reached or exceeded, b) the limit is an
approximation, and c) the limit can be obtained by simply plugging in values at or near a given point.
Many of these misconceptions appear to be didactical in nature. But the most significant and
common conception about limits that is most certainly epistemological in nature is that they are
dynamic rather than static, that they involve some kind of motion that happens in time and space.
This conception is so embedded in our practice and language (“the limit of f(x) approaches L as x goes
to a”) that it hardly seems right to call it a misconception. But the formal definition of limit has no
motion in it—it is a sophisticated logical connection between x-values that are close to some x0 and
y-values that are close to y0. The fact that the motion-based limit conception is common, reasonable,
and useful, but ultimately inadequate, makes it an epistemological obstacle. I will address this
obstacle in more detail later.
In trying to understand these misconceptions and why they exist, some researchers have examined
students’ underlying beliefs about mathematics and infinity (Szydlik, 2000; Sierpinska, 1987). In a
complex study, Sierpinska determined students’ different (a) epistemological obstacles to the limit
concept, (b) conceptions of infinity, and (c) attitudes toward mathematics. She claims that some
combinations of these three components are stable and some are not. For instance, one student in
the study, “Tom,” demonstrates a stable set of conceptions: a “heuristic dynamic” conception of
limit (0.999… never equals 1, “no matter how many nines you put”), a “potential actualist”
conception of infinity (“…unless it really goes to the very infinity, then it may be 1”), and a
14
“discursive empiricist” attitude toward mathematics (mathematics is memorized, not discussed;
when confronted with a proof that 0.999… = 1, he did not change his belief).
Tom’s conceptions of limit and of infinity are not unusual. The question “Does 0.999… = 1?” is
very common in limit research, and it often produces such a response, that 0.999… “never reaches”
1. The BMI can go some distance to explaining Tom’s reasoning. He is able to conceive of the
process of creating the number, but has not fully accepted the final resultant state of the infinite
process. On the other hand, he does not seem to fully reject the idea that infinity could ever exist or
be attained. The better explanation is perhaps that Tom does not really view 0.999… to be a single
number; rather he sees it only as the dynamic process that produces the number’s decimal
expansion. Since the process of writing the digits is never complete, the number (process) 0.999…
“never” equals 1 (object).
In 1991, Anna Sfard wrote an article called “On the dual nature of mathematical conceptions:
Reflections on processes and objects as different sides of the same coin.” Rarely has an article’s title
so lucidly outlined its theory. For several years researchers of mathematical reflective abstraction
(e.g. Dubinsky & Lewin, 1986) had discussed concepts in undergraduate mathematics as both
processes and objects. This duality led researchers to discuss a particular form of abstraction called
encapsulation, the act of changing a process into an object (Dubinsky, 1991). This form of abstraction
is important to mathematical development, because only when a process has been encapsulated into
an object can it in turn be operated on by other processes. For instance, a function might be
understood as a process that inputs numbers and outputs other numbers. But only if a function is
seen as a thing, and not just as a process, can one understand an operator such as a derivative as a
process that inputs functions and outputs functions. Sfard (1991) calls this same reflective transition
from process to object reification.
By a few years later, some of these researchers (Cottrill et al. 1996) had expanded the idea of
processes and objects to include actions and schemas, calling this broader theory APOS. The first
stages of a student’s abstraction involve performing actions without the student fully exerting
control or recognizing pattern. Once she reflects about the actions and notices regularities in them,
she can interiorize (Piaget, 1950) them into a process that she can perform and regulate internally
rather than externally. By interiorizing various actions into processes, and by subsequently
encapsulating processes into objects which can in turn be operated on, she forms a schema: a
“coherent collection of actions, processes, objects, and other schemas that are linked in some way.”
(Cottrill et al. 1996). Researchers of the last decade have applied this theory to undergraduate
mathematics topics of all kinds. (e.g. Asiala et al., 1996; Clark et al., 1997; Dubinsky & McDonald,
2001)
Cottrill et al. (1996) have used APOS to formulate a genetic decomposition of the concept of limit,
treating certain interiorizations and encapsulations as prerequisite to others. This research is very
practical in orientation. The genetic decomposition is used here less as a strong claim about student
cognition than as a set of guidelines for developing a curriculum. Likewise, the researchers used the
results of implementing this curriculum to refine the decomposition and thereby the curriculum also.
Their decomposition is somewhat detailed, so for our purposes I highlight two main parts. The first
is the encapsulation of the “approaching” process (actually two processes—one for y and one for x)
15
into the object of the limit. The second is the reconstruction of these processes in technical terms
(inequalities, intervals, ε’s, etc.) and the application of a quantification schema.
This decomposition reflects the obstacles Sierpinska used—the heuristic dynamic obstacle, the
heuristic static obstacle, and the obstacles of “false rigour.” The heuristic dynamic obstacle can be
thought of as viewing the limit in terms of a motion, a process of approaching, but not as a static
object that can be operated on. Once the motion process is encapsulated, a student may be able to
see the limit in static terms, either as an approximation or with a sense of arbitrary closeness. But
the student may not yet grasp the logical subtlety, and the importance of the logical subtlety, that for
each ε neighborhood on the y-axis we must be able to choose a δ neighborhood on the x-axis (any
will do) so that if a particular thing happens in the δ neighborhood another thing will happen in the
ε neighborhood rather than vice versa. The research shows that these are the two most substantial
obstacles to student understanding of limit. In particular, the dynamic conception of limit is very
resilient and difficult to change (Williams, 1991).
Let’s examine the dynamic obstacle a little more closely. The BMI is able to account for students
who have not yet conceived of or fully accepted a final resultant state for an infinite process. APOS
also explains students who have not yet encapsulated an infinite process into a single object. But
both of these theories seem to apply only to processes that have discrete steps or stages. For
instance, Cottrill et al.’s genetic decomposition describes the process of x approaching a as
evaluating a sequence of values along the graph of a function. But Williams’ (1991) study reveals
that the majority of students prefer thinking of the limit of a function in terms of movement along
the graph rather than plugging in a sequence of discrete points on the graph. In other words,
student intuitions about the limit of a function, with implications of smooth motion and continuity,
are different than student intuitions about taking the limit of a sequence or a stepwise process. Some
students conceive of the former in terms of the latter, but most do not.
The dynamic obstacle therefore appears to have two separate but possibly related parts: the stepwise
part and the continuous part. The stepwise obstacle prevents students from being able to accept the
final resultant state of an infinite process, because “you’ll never finish.” It also may cause them to
impose some sort of stepwise process on an otherwise continuous one, like in the case of Zeno’s
Achilles and the Tortoise paradox (Weller et al. 2004), where breaking up a continuous motion
artificially into an infinite number of steps may stymie the student. The continuous obstacle
prevents students from being able to think of the limit of a function in terms of proximities, which
appears to be prerequisite to understanding the formal limit definition.
dynamic (temporal) stepwise—The limit of a sequence or procedure, sometimes function,
requires infinitely many discrete steps, which are taken in time. Thus this process is never complete,
and the limit is not accepted as a legitimate object.
Although some researchers consider the misconception that a limit cannot be reached as a separate
misconception, I include it here as part of this obstacle. I do so because students who experience
this obstacle are stuck on the idea that an infinite process never actually ends, and thus the result is
never attained.
At its core, this obstacle indicates a skepticism about our ability to adequately deal with the infinite.
This same skepticism was maintained almost universally in ancient Greece and medieval Europe.
16
For example, Aristotle believed that mathematics could only deal with potential infinity, namely
processes that are finite at any time but can keep progressing as much as one wishes. Actual infinity,
on the other hand, is inconceivable. Like Aristotle, the Greeks (and the medieval scholastics)
“generally regarded infinity as an inadmissible concept. It is something boundless and
indeterminate” (Kline, 1980, p. 199).
But Lakoff & Nunez (2000) posit that the BMI is the metaphorical way of accepting actual infinity
into mathematics, and that this metaphorical acceptance is necessary to the development of calculus.
By accepting the “final” states of infinite processes as real things, the Enlightenment mathematicians
were willing to tread with hubris into the actually infinite.
One of Cornu’s (1991) four historical epistemological obstacles to the concept of limit, the Greeks’
failure to link geometry with numbers, is related to this obstacle. Mathematicians such as Eudoxus
and Archimedes used the method of exhaustion, claiming that they could carry on certain geometric
processes and constructions to whatever degree of precision necessary. Even though these
arguments are similar to limit arguments, the lack of algebraic representation marks the deeper lack
of a systematic universal approach to infinite processes.
Therefore, this misconception meets the criteria for an epistemological obstacle. It (d) is found
historically, (c) is common among researched students, (a) is firmly grounded in intuitions about
finite processes, and (b) it prevents understanding and even acceptance of the limit of a sequence.
The other dynamic obstacle to the limit concept is continuous, not discrete. It applies to the limit of
a function over the real number line rather than to the limit of an infinite stepwise process or
sequence. Unlike the other dynamic obstacle, this conception accepts the existence of the limit, but
views it as the resultant position or state of a continuous temporal motion.
dynamic (temporal) continuous—The limit of a function involves a continuous motion, one taken
in time. This prevents intuition about how y-values can be gotten arbitrarily close to L by restricting
the x-values.
As mentioned above, this obstacle fulfills criteria (a), (b), and (c) of an epistemological obstacle.
This obstacle also appears to be quite similar to the dynamic function obstacle in the following
section. The determining factor here may be that the student does not conceive of the function
itself as a mapping from x to y, but rather as a particular kind of nice drawable curve. In APOS
terms, the limit may be accepted as an object, but the function is still considered a process.
Newton held this combination of beliefs. For instance, Kleiner (2001) says that one of the most
important developments in calculus was when Euler (75 years after Newton) focused the subject on
functions rather than on curves. For Newton, a curve was a smooth path traced out by a moving
body; the notion of function did not exist. For Euler, a function was an analytic expression for y in
terms of x, a single object on which operations and processes could be performed.
Consider also Newton’s own (amazing) comments about limits, here explaining derivatives:
“By the ultimate ratio of evanescent qualities is to be understood the ratio of the quantities not before
they vanish, nor afterwards, but with which they vanish…. Those ultimate ratios with which quantities
vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities
17
decreasing without limit do always converge; and to which they approach nearer than by any given
difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum.”
(Edwards, 1979, p. 225)
Here we see how large a role motion takes in Newton’s conception of limit. We also see how vague
and muddy the concept is as a result. It is no wonder that the concept was attacked philosophically
for both its ambiguity and its hubris by contemporaries such as Bishop Berkeley (Kline, 1980). Only
in the days of Cauchy did the concept of limit begin to be defined with suitable rigor, and this by
replacing the intuitions of motion and time with the algebra of proximity. For these reasons, this
obstacle meets criterion (d) also.
intuitive static—The student can conceive of the limit is terms of arbitrary closeness, but does
not understand the logical dependences and notational structure found in the formal limit definition.
In the early 1800s, several mathematicians had noticed that calculus was on shaky ground. Impelled
by Fourier’s work, Cauchy recognized that the ambiguities found in the limit notion were no longer
just philosophical nuisances but were causing inconsistencies and roadblocks in analysis itself. The
limit needed to be considered the fundamental building block of calculus, and this building block
needed to be solid. In his Cours d’analyse of 1821, Cauchy proposed a definition of limit that
minimized intuition about movement and rigorized the troublesome infinitesimal. His notion is that
the variable quantity can be made to differ from the limit “by as little as one wishes” (Edwards,
1979, p. 310). The idea, in terms of functions, is that f(x) can be gotten as close to L as one would
like by choosing x sufficiently close to a.
Cauchy did not use the notation of ε, δ, intervals, and inequalities, though he did produce some
proofs that were essentially formal limit arguments. The formal algebraic definition of limit would
not be developed until the days of Weierstrass 40 years later. The main difference between the
conceptions of Cauchy and Weierstrass is that Cauchy did not fully recognize and understand the
importance of the placement of the quantifiers “for every” and “there exists” in the limit concept.
For this reason, Cauchy did not distinguish between pointwise and uniform continuity, and between
pointwise and uniform convergence (Kleiner, 2001). Weierstrass did recognize the importance of
these differences, and in preserving their precision he finally rid the concept of limit of all traces of
infinitesimals, geometric intuition, and dynamic language.
Cauchy reveals that this obstacle meets criterion (d) of an epistemological obstacle. Furthermore,
his errors are a testimony to the complexity of the logic in the formal limit definition. I wonder how
likely it is that a calculus student would be able to get beyond this obstacle before taking a class in
real analysis. This conception requires very sophisticated thinking, and a significant move toward
abstraction. This conception helps in understanding limits in many ways, particularly for strange
functions such as sin(1/x) near x = 0. So it meets criterion (a). It is still significant and troublesome
to overlook the logical subtlety in the definition, so this meets criterion (b) as well. Although the
limit research mentions this obstacle as occurring in students (Williams, 1991 and 2000), the static
conceptions described in these studies do not entirely suggest that the students had passed beyond
the two dynamic obstacles.
Finally, the static conception of limit suggests a hypothesis about the connection between limit,
function, and real number line that we should expect in our students. In order to have a static
conception of limit, even an informal one, one must be able to view a limit in terms of closeness,
18
not motion. Points that are close to each other along the x-axis must correspond to points that are
close on the y-axis. But this suggests a developing notion that a function is a mapping from one set
of points to another, not a motion traced out in space. This in turn suggests a conception of the real
number line as a set of points, not as a continuous backdrop. This will become clearer after the
function section of this paper, but at this point we can conjecture that students will have a static
conception of limit only with a formal conception of function and a “set of numbers” conception of
the real number line.
Functions and prototype theory
The concept of function is central in calculus and analysis, and is arguably what most distinguishes
modern from classical mathematics (Kleiner, 1989). Although the current definition of function—a
uni-valued mapping from one set to another—is simple, the concept itself is a complex mixture of
geometric, algebraic, and formal logical thinking. Even if students are explicitly taught the formal
definition of function, surprisingly many of them cannot produce this definition when asked (Vinner
1991), and even more rarely answer function questions in such a way that reflects understanding of
the formal definition (Vinner & Dreyfus, 1989). For instance, only 8% of college students and high
school mathematics teachers surveyed by Vinner & Dreyfus were able to produce a function that
took on different values for integers than for non-integers. On the other hand, half of the
respondents instead viewed a function to be a dependence relation, rule, or formula.
One theory that explains the consistency of these misconceptions is prototype theory. This general
theory of cognition emerged in the 1970s in opposition to the theories of information processing.
Information-processing theories generally view a category to be strictly defined by criteria, and that a
particular thing’s membership in the category is determined by whether it satisfies these criteria.
Prototype theory instead says that there are gradations of membership in a category, and that a
thing’s membership in a category is the degree to which the thing resembles one or more of that
category’s prototypes. For instance, while there are certain shades of color that almost everyone
considers green, there are other shades that some people consider green and some do not, or that
people consider green based on contextual elements like background colors, and so on.
Of course, there are still criteria involved, the criteria that determine the resemblance between a
thing and a category’s prototype. The difference is that these criteria are not strict, but are rather
holistic or metaphorical. A complex category such as “chair” has too many and too contextdependent dimensions for us to be able to produce an algorithm that determines membership. On
the other hand, treating the connection between a thing and a prototype as metaphorical allows for
this connection to be context-dependent, goal-oriented, or even rather capricious. The theory of
prototypes also explains why students might view mathematical definitions as fuzzy or statistical,
“holding most of the time,” rather than being strict and logical.
Prototype theory has been used several times to explain student conceptions of function. In
response to the question of how many different functions pass through a given pair of points in the
plane, over half of the high school students surveyed by Markovits (1982) said only one function,
drawing a line. More than half also responded that there were no functions passing through three
non-collinear points, and most of the rest drew a parabola. Although in a later study by Schwarz and
Hershkowitz (1999), more students responded correctly, they still appealed almost universally to
linear and quadratic functions. One student’s response is illuminating: after trying repeatedly to
draw a linear function through the three points, at one time even having drawn a legitimate
19
piecewise linear function, the student erased these attempts, drew a parabola, and said there was only
one possible function “because one cannot draw a linear function here” (1999, p. 379). These
examples support the conclusion that students may have one or two prototypes for function, the
line and the parabola.
Although these results show students to be hindered by their prototypes, the study also shows that
these prototypes help some students’ understanding of functions, by using the prototype as a
reference from which to begin new tasks or to give a frame of reference for other examples. In this
way prototypes can be seen as a type of epistemological obstacle, allowing for certain understandings
while at the same time preventing others. A person whose function concept has a line and a
parabola as prototypes may be able to quickly understand the “vertical line test” or the intuitive
picture of a derivative; on the other hand he/she may consider all functions to have similar
attributes like drawability, continuity, differentiability, a single simple formula, etc.
Prototype theory can also be seen as a way of building on the theory of mathematics as metaphor.
The theory of thinking as metaphor can explain how connections get made between one thing and
another, and how structure from one domain becomes imposed on another. But it does not address
the other half of the issue, which is how connections between concepts become qualified or severed
entirely. However, prototype theory can explain how certain concepts begin as being confounded
but gradually become appropriately dissociated from one another. For example, historically the first
prototypes of algebraic geometry were algebraically and graphically simple, based on simple and
regular physical phenomena. Therefore, all graphs were considered to be drawable, continuous,
differentiable, integrable, and given by one simple formula. But as the concept of function
developed, more and more examples of functions arose in the history of mathematics, and these
examples became prototypes for categories such as “continuous but not differentiable,” “function
but not given by a single formula,” and so on. In other words, these prototypes were able to sever
inappropriate metaphorical projections from one concept to another.
The following table gives a simple account of the development of the concept of function based on
the theory of mathematics as metaphor supplemented by prototype theory. Each time a new
metaphor appears, two concepts are combined into one single category with its own prototype.
When one category is dissociated into two different concepts, a new category (along with new
prototypes) is created containing objects that fit into one concept but not the other.
Table 2.1—Associations and dissociations of categories due to prototypes
Rough timeframe
early 1600s
Effects
new metaphor—‘curves are smooth continuous motions’
c. 1640
c. 1670
new metaphor—‘curves are algebraic expressions’
new metaphors—‘limits are continuous motions,’ analytic
(infinite) expressions are curves are differentiable, BMI, etc.
dissociation—‘algebraic’ is not the same as ‘analytic’
dissociation—functions need not be one single expression
dissociation—analytic expressions (functions) need not be
continuous or smooth motion
c. 1760
c. 1807
Prototypes
projectile and
planetary motion,
parametric curves
polynomials
Taylor’s series
vibrating strings
Fourier series
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c. 1840
c. 1860
late 1800s
new metaphor—functions are mappings from one domain to
another (numberline is a single object, not a continuous
measuring tool)
dissociation—functions are not curves
dissociation—continuous doesn’t mean differentiable
dissociation—differentiable is not smooth
characteristic
function of the
rationals
Weierstrass function
Present-day students share many of the erroneous connections between these concepts as well.
Consider the misconception that a function must be given by a single formula. 43% of the students
Vinner (1991) surveyed said directly that a function is given by rules or formulas. More tellingly, as
many as 81% of them responded to other questions in such a way that indicated they believed
functions were given only by rules or formulas.
Even though this kind of conception is incorrect, it is understandable if students have only a few
prototypes such as linear and quadratic function, and thus only have a category for functions given
by a single formula Vinner & Dreyfus (1989). As mentioned above, having these simple prototypes
is useful and understandable given how common the types of behavior described by them are
encountered naturally. For these reasons, this satisfies conditions (a) and (b) for epistemological
obstacles. In addition, because such a misconception is noted in studies by Tall & Vinner (1981),
Vinner (1991), and Sfard (1992) (in the latter specifically as an epistemological obstacle), it meets
condition (c).
In the middle of the 1700s, a debate developed about the solution to the differential equation
problem of the vibrating string. D’Alembert solved the problem in 1747, but only included for the
initial condition functions given by a single analytic expression (differentiable also). The following
year, Euler proposed a more general solution including as possibilities for the initial condition
function any freehand curve or functions defined piecewise as analytic expressions (Kleiner, 1989).
For what it’s worth, Doniel Bernoulli gave a solution in 1753 that allowed for arbitrary continuous
functions, functions that could be given by a trigonometric series. All three of these mathematicians
disagreed with one another about what could be functions, whether a function needed to be one
analytic expression or could be several. It would take 100 years before mathematicians would
consider arbitrary functions, or at least ones not given by analytic expressions at all. For these
reasons, the single formula misconception meets condition (d) of being an epistemological obstacle
as well.
single formula—A function is given by one rule or formula, usually simple. This perhaps
includes the belief that mathematical behavior should be “nice,” and a faith in the power of
symbolic operations (Sierpinska, 1992).
Another (mis)conception of function that appears in the history of calculus is that a function is a
motion traced out in time, and that this motion is continuous or smooth. Perhaps a better way to
say it is that when calculus was first being developed the objects of study were not functions as such,
but were geometric curves traced out by changing quantities. Rather than applying to functions,
Newton’s calculus was performed on fluents, “flowing Quantities” that change smoothly through
time (Schrader, 1994, p. 509). A fluxion was a rate of change or velocity of a fluent, and the ratio of
fluxions is what we call a derivative. For Newton, calculus was a way of understanding physical
motions and velocities; his mathematics addressed dynamic, continuous, real-world phenomena
(Kleiner, 1989).
21
A century after Newton’s Principia Mathematica, calculus was performed on “functions,” functions
that might not be directly tied to real-world phenomena. Nonetheless, as mentioned above,
mathematicians such as Euler considered piecewise-defined functions to be discontinuous (or more
appropriately, discontiguous) because they did not represent one single formula or kind of behavior.
Furthermore, Euler and his contemporaries would have viewed most graphs that we call
discontinuous to not even represent functions at all (Grattan-Guinness, 1970). Such graphs
represented random behavior not worth describing or understanding, so no one had any such
prototypes.
Therefore the conception that a function is a continuous or smooth motion meets criterion (d) of an
epistemological obstacle. It appears to be an obstacle, because it arose naturally for the scientists
who used mathematics to describe simple real-world phenomena. The obstacle seems based on the
general Enlightenment belief in a sensible clockwork world that operates according to
mathematically simple and pleasing rules. After all, discontinuous, discrete, pathological, and chaotic
phenomena occur in the world as well, but most such behavior has only been treated as
mathematically describable in the last 150 years. An example is the discontinuous fluid dynamic
behavior of compression thrust, first studied in the 1860s (Bochner, 1978). So historically this
obstacle meets criteria (a) and (b). It prevented further understanding of discontinuous and nontemporal phenomena, but it functioned adequately within the limited domain with which these
mathematicians were working.
But does the obstacle meet criteria (a), (b), and (c) for students as well? If it was this particular way
of viewing the world that led mathematicians to encounter this obstacle, then why should students
who do not share this worldview encounter the obstacle as well? If they do, might it rather be a
didactical obstacle, based instead on the types of examples they are taught and therefore use as
prototypes?
Encapsulation research suggests that this may still be an epistemological obstacle for students.
Encapsulation has usually been applied in this arena by claiming that students first understand a
function as a process that takes inputs and transforms them into outputs, then ultimately understand
a function as an object which itself can be operated on in the same way (Dubinsky & Harel, 1992;
Breidenbach et al., 1992). However, encapsulation can also be applied here by saying that students
first understand a function as the action or process of drawing its graph. The connection between
these two applications of encapsulation is related to the connection between a function’s graphical,
algebraic, and input/output representations. These complex connections have been studied closely
by Monk (1992), Carlson (1998), and Kaput (1992), among others. The point is that this latter
dynamic view of function may stand in the way of treating a function as an object. For this reason,
combined with the possibility that students may not consider discontinuous or non-temporal
functions to be legitimate, we consider the dynamic obstacle to functions to meet criterion (b) of an
epistemological obstacle.
But does this obstacle arise consistently among present-day students, meeting criterion (c)? One
mention of this in the research is Sierpinska’s discussion of the epistemological obstacle to function
that “the changes of a variable are changes in time.” (1992, p. 55) But she discusses it only briefly in
its historical context, not as experienced by students. Of course there are several observations of the
dynamic continuous obstacle to the limit concept, cited earlier in this manuscript, and it may be that
this obstacle is essentially the same as the dynamic function obstacle. Even though the dynamic
22
function obstacle is not explicitly addressed in the research as being experienced by students, it is
reasonable to suspect that students would encounter it, perhaps as a result of conflating the idea of a
function with the temporal act of drawing its graph. As mentioned earlier, many students do not
consider graphs with sharp corners or discontinuities to be functions (Vinner & Dreyfus, 1989),
which may be evidence for the appearance of this obstacle in student thinking:
dynamic (continuous or smooth)—A function is a curve traced out in space over time, and this
motion is continuous or smooth. This obstacle may prevent the understanding of a function as an
object that can itself be operated on.
Because of this obstacle to function, we consider the “goal” conception of function for calculus to
be more than just the recognition of the formal definition (that a function is any correspondence
from one set to another so that each element in the first set corresponds to exactly one element of
the second). The goal conception must also include the ability to recognize a function as an object
and perform processes on it, ultimately processes such as differentiation and integration.
The last important obstacle is the notion that a function requires change. In other words, a function
requires a strict dependence relation—when x changes, y changes also. Function research has
noticed consistent and pronounced difficulties with constant functions (Markovitz et al., 1986; Sfard,
1988; Sfard, 1992). Vinner & Dreyfus (1989, p. 364) observed that when asked for a function whose
values are all equal, many students replied “y = x.” A large number of students preferred the
dependence relation definition of function to the formal (correspondence) definition. It is
reasonable, in fact desirable, for this obstacle to arise, because it reflects student understanding of
independent versus dependent variables. This obstacle reflects the general tendency of students to
ignore degenerate cases. After all, degeneracy really means a lack of proximity to the “center” or
prototypes of the category. For these reasons the obstacle meets criteria (a), (b), and (c) for an
epistemological obstacle.
I have not found any historical examples of this obstacle, or even more generally the obstacle of
overlooking important but degenerate cases.
requires changes—A function is a dependence relation, so if x changes then y must also. In
other words, there are no constant functions.
Specific versions of research questions
By examining the research, I have established a set of epistemological obstacles and target
conceptions to use for interpreting student thinking about the elements of calculus. These can help
me address the following questions:



How common are the various student conceptions of the elements of calculus?
What consistent connections between these conceptions, if any, do students display? How
do these align with the ones observed historically?
What do these alignments and/or lack of alignments suggest about the meaningful parallels
between cognitive conceptual development and historical development?
23
Methods
Data Collection
Questionnaire
235 students at a large public Midwestern university completed the initial Calculus Concept
Questionnaire, 159 enrolled in a standard Calculus I class, 74 enrolled in a standard Calculus II class,
and 2 enrolled in graduate school in mathematics. I selected these students in order to allow for a
variety of different conceptions from different points in students’ learning of calculus. The
questionnaire was administered in the first two days of the fall 2005 semester for two reasons. First,
that was when the calculus discussion sections could most easily spare the time. Second, because it
was important that students respond according to their understanding rather than to recentlymemorized definitions, this timing minimized the effects of any recent instruction about the
definitions of the concepts of limit, function, etc. I instructed the students that many questions did
not have “right” answers, and that they should respond according to their own beliefs and
conceptualizations. I asked the students what their most recent mathematics classes had been,
where they had taken them, and what grades they got in those classes.
Three months later, I administered the same questionnaire to 8 of the same students. Those are the
only people who signed up for the follow-up questionnaire. This questionnaire was identical to the
first questionnaire, and was intended to provide data about possible ways in which the students’
conceptions might change over time.
The items on the questionnaire were either created by me or selected from other published studies.
All items can be found in Appendix 1. Items 1(i) and 3 are based on items used by Vinner &
Dreyfus (1989); item 1(i) also contains a graph used in Vinner (1991); item 2 is from Szydlik (2000);
item 4 is from Weller, et al. (2004); items 7a and 7c can be found with graphs in Tall & Vinner
(1981); item 9 is based on an item in Sfard (1992); item 13 is based on statements in Lakoff &
Nunez (2000); item 14 is based on an item in Williams (1991). I created the rest of the items.
Some of the items are multiple choice, with the selections informed by the research about the
selected epistemological obstacles. For most items, students are provided the opportunity to answer
a different choice than the ones presented and/or explain their reasoning. As I mentioned, these
comments and explanations are all taken into account when coding.
For each of the four calculus concepts, there is one self-reporting question for each concept, items
13 to 16. These four items are based on an item in the Williams (2000) study. Each response on
these items is intended to reflect a particular way of conceiving of the concept, most of which
represent one of the epistemological obstacles or the target conception. 3-5 statements are
presented about a given concept, and the student can respond to each statement on a scale between
1 (highly disagree) and 6 (highly agree). However, because students may have memorized the
“correct” answer to these items while still exhibiting obstacles, I also included a number of indirect
items as well.
The purpose of the initial questionnaire was primarily to determine which of the conceptions
(epistemological obstacles and target conceptions) selected for this study were being experienced by
the students. Therefore, it was necessary to devise and refine a method for attributing conceptions
24
to students based on their responses. I created this coding scheme based on my understanding of
the obstacles in the literature, and refined it based on student responses in a pilot study. But the
most important method for refining this coding scheme was through the interviews. I describe the
coding scheme in detail, and how it was refined as a result of the interviews, in the data analysis
section that follows.
Interviews
Soon after the initial questionnaire was administered, and the data were being coded, I interviewed
six of the respondents. These interviews were audio recorded and transcribed. Pseudonyms are
used in this paper when referring to these students. Along with informing the coding scheme, the
interviews provided examples of students experiencing different conceptions, and also helped to
provide understanding of the epistemological obstacles themselves, how they manifest themselves,
and how they interact.
Each interview participant was selected because his or her initial questionnaire contained a response
pattern that was both (a) difficult to code with my initial coding scheme and (b) displayed by
multiple participants. The purpose was not to determine the meaning of a few idiosyncratic student
responses, but rather to determine the meaning of a handful of recurring responses that my initial
coding scheme was unable to account for. The interviews were semi-structured. Each participant
was first asked to complete items from the questionnaire, and based on their responses they were
given follow-up questions and new tasks. This format allowed for freedom to respond to student
responses by asking questions that emerged as the interview proceeded.
For example, many more students were responding “true” to Item 6b than I had originally expected,
so one of my goals in the interviews was to determine the kinds of understandings that this response
indicated, particularly about infinitesimals. In most of the interviews, I asked the students the item,
then followed it with questions such as: “Can you give me an example of two different numbers that
are infinitely close to one another?”, “Can you give me an example of a number that is infinitely
close to both of those numbers?”, “How many numbers are between those two?”, etc.
Because each interview was tailored to investigate the conceptions of that purposefully-selected
student, the protocols for the interviews varied. They were also semi-structured, allowing the
interviewer to respond flexibly and follow up on student comments. An example of a protocol
question was to answer Item 1b on the Calculus Conception Questionnaire. Questions used to
follow up on this protocol were:



Why do you think that this graph does not represent a function?
What is something that could be done to this graph to make it into a function?
When you took this questionnaire in class, you answered that this was a function. Do you
have any ideas about why you answered differently this time?
I used a general grounded theory to guide the interviews and the qualitative interview coding
process. Because the main use of the interview data was to refine and verify the coding scheme for
the quantitative part of the study, I use Strauss’s view that grounded theory is verificational in nature
(Strauss & Corbin, 1994). The reason the interviews are important to the study is that identifying
and understanding obstacles is a complex meaning-filled undertaking. An epistemological obstacle
25
is not simply a task that a student cannot perform, or a bit of knowledge the student does not
possess, but is rather a way of thinking about a concept. This type of conception may have
particular item responses or task behaviors as symptoms; these responses and behaviors indicate the
obstacle with varying degrees of certainty, and only to the degree that we really understand the
obstacle. An interview is the form of data collection that best allows for this complexity to be
displayed and dynamically responded to in order to refine my coding of the obstacles.
Data Analysis
Coding of student conceptions
The following coding scheme details the method used to determine if a student was experiencing the
conceptions chosen for the study. For each conception, I briefly describe the conception and then
list the item responses that indicate the existence of the conception. These responses are
accompanied by an asterisk if they are particularly important indicators for the conception. Some of
the indicators changed through the course of my study based on student responses and interview
data. I describe these changes in the coding scheme in this section as well. In general, for a
student to be considered as displaying a given conception, he must have nearly all of the indicators
for that conception. More specifically, he must have all of the indicators except possibly one of the
important indicators or two of the unimportant indicators. Note that this requirement is stricter for
the conceptions with fewer indicators, such as the real number line conceptions; such cases are
specified below. If a student is a borderline case according to this coding procedure, then other
responses may be taken into account as weak indicators; these are also specified below. I always
looked carefully through the student responses for other written indicators that the student may or
may not have had the conception. Often this was the deciding factor in coding the conception to
the student.
There are several reasons why, for most conceptions, not all of the indicators are required for the
conception to be attributed. For instance, students will often misread an item or two on a
questionnaire. Additionally, even the best item still leaves some ambiguity about the thinking of the
respondents. Relying on multiple items to describe a concept diminishes the effects of measurement
error.
Nonetheless, this coding procedure for attributing epistemological obstacles to participants is still a
cautious one. Unless I could make a reasonably strong case that the participant was displaying one
of the obstacles to a given concept, the participant was coded as “other” for that concept. This
means that there are likely more false negatives than false positives in the data.
A reliability check was performed wherein another rater coded twenty of the student questionnaires.
The rater was chosen for his expertise with student conceptions in calculus. He was given only the
questionnaires and descriptions of the obstacles, and was asked to develop his own rubric for
attributing obstacles to students. Agreement of coding for a particular concept was considered to be
complete for a given student’s questionnaire responses if our coding was identical for that concept.
Agreement was considered to be 50% if one of us coded a particular obstacle for that concept while
the other coded nothing. Agreement was considered to be 0% if we coded different obstacles to the
student for the concept. With this definition of inter-rater reliability, the agreement was 87%, which
is well within generally accepted parameters for reliability of coding. The conceptions for a given
26
concept are meant to be exclusive to one another. For example, a student with an intuitive static
limit obstacle will, by definition, not have the dynamic continuous limit obstacle. For this reason, I
do not worry to assess these conceptions independently, but often use responses on a single item as
indicators for multiple conceptions within a given concept. However, one of the goals of this study
is to see if obstacles to different concepts are conceptually related to each other, or if they function
independently. Therefore conceptions of different concepts are assessed as independently of one
another as possible. In other words, if a particular item response serves as an indicator for the
smoothness continuity obstacle, it will not also serve as an indicator for the dynamic smooth
function obstacle.
There is one exception to this, and that is the relationship between the real number line concept and
the limit concept. We already know that these concepts are related to one another in some ways
(Schwarzenberger & Tall, 1978). Because these concepts are related, it is impossible to assess their
conceptions in completely independent ways. One item, Item 2, taken from Szydlik (2000), contains
responses that serve to indicate obstacles to both concepts. The data are discussed with these
dependences in mind.
The final coding scheme is represented in Table 3.1 for easy reference.
Continuity—smoothness. A graph or function is continuous only if it is smooth, that is, if it and all of
its derivatives are continuous. In particular, it has no sharp corners or breaks. Initially I used the
following as indicators: 1.ii.b, not 1.ii.c, 5.c“not continuous”, 7.b.“not continuous”, and 16.a*.
Through the course of coding it became evident that most students responded “not continuous” on
7.b without graphing the function, or based on graphing the function incorrectly, so I eliminated it
as an indicator for this obstacle.
Continuity—single formula. a graph or function is continuous only if it is represented by a single
analytic formula. Indicators are 5.a.“continuous”, 5.b.“not continuous”, 5.c.“continuous”,
7.a.“continuous”, 7.b.“not continuous”, and 16.c*.
Continuity—connected. A graph or function is globally continuous if it can be drawn without lifting the
pencil. This understanding is satisfactory for calculus, provided students also understand the simple
connection between local and global continuity. Thus, the indicators are all the most correct
responses for the purposes of a calculus course: 1.ii.c, not 1.ii.a, 5.b.“continuous”, 5.c.“continuous”,
7.a.“not continuous”, 7.b.“continuous”, and 16.b*.
Real Number Line—naturally continuous line. The real number line is a continuum, a background on
which objects can be situated, not itself a set of points. Because this is indicated only by a student
responding 13.c, I consider it too unreliable to be reported on in this study.
Real number line—next number. The real number line is like a string of pearls; each point touches a
point next to it on either side. Indicators began as 2.b, 6.a.”true”, and 13.a, but several student
responses led me to remove 2.b as an indicator, because it is technically consistent to believe 2.a or
2.c and yet still be experiencing this obstacle. Both indicators are required for a student to be
considered as having this conception.
Real number line—infinitesimals. There exist positive quantities and distances smaller than any real
number. Initially, the indicators were 2.b, 6.b.“true”, and not 13.a, but I also included 6.a.”false”
27
after an interview and some reflection. It is clear to me now that a belief in infinitesimals precludes
the belief that two numbers could be adjacent. All of these indicators are required in order to be
coded for this conception.
Real number line—set of numbers. The target conception in standard calculus, that the real number line
is simply a set of (finite) numbers with certain properties. Indicators are not 2.b, 6.a.“false”,
6.b.“false”, and 13.b*. Note that 2.c is still an admissible answer, because it is not clear that
infinitesimals are an obstacle to this conception.
Limit—dynamic (temporal) stepwise. The limit of a sequence, procedure, or function is not a legitimate
object, because it requires infinitely many steps to be taken in time, which is never complete.
Indicators are 2.c, issues with the question itself in item 4, 12.a, and 14.b. Other weaker indicators
are 12.b and 14.a; these responses do not specifically confirm or deny this obstacle. Any student
specifying in writing that an infinite process could not be completed was coded as experiencing this
obstacle.
Limit—dynamic continuous. The limit of a function involves a continuous motion, often one taken in
time. Indicators are 11.”no”, the reason for 11.”no” being that the function is discontinuous or
doesn’t move right, and 14.a or 14.b. Not answering 17.c with certainty was treated as a weak
indicator as well. Note that a student who answers “yes” on item 11 is never coded as experiencing
this obstacle. It is possible to experience both this obstacle and the dynamic (temporal) stepwise
limit obstacle at the same time.
Limit—intuitive static. The student (i) can conceive of the limit in terms of arbitrary closeness, but (ii)
does not understand the logical dependences and notational structure found in the formal limit
definition. Strong indicators are 11“yes”*, something like “the values get arbitrarily close to 0” on
item 11*, and not 17.c with certainty*; weak indicators began as 2.a, “no balls in Bin A” on item 4, not
8.c, not 12.a, and 14.c. Since this obstacle requires both the positive conception and the
misconception, the indicators are more or less “weighted” so that half of the total weight reflects on
each part of this obstacle. Because so few students answer item 4 correctly, I changed the indicator
for this item to be no issues with the question itself instead. The indicator is still a weak one, but a
student with the intuitive static obstacle should be able to visualize the completion of an infinite
process, even if she visualizes it incorrectly.
Limit—formal. The student (i) does not have dynamic obstacles to limit, and (ii) understands the
logical implications and notations in the ε, δ limit definition. When coding, indicators are
“weighted” equally between these two aspects. The two very strong indicators are 14.d* and 17.c*
with certainty > 3. Weak indicators are 2.a, 4.“no balls in Bin A”, 8.b, 11.”yes”, and 12.c.
Function—single formula. A function is given by one rule or formula, usually simple. Indicators are
5.a.“function” (hereafter “f”), 5.b.“not a function” (hereafter “nf”), 5.c.f, 7.a.f, 7.b.nf, 7.c.nf, 9.”no”,
15.b*. Interviews revealed that a few students answered 15.b because of the “one y value for each x
value” phrase, overlooking the single formula part, so I made it a weaker indicator, looked closely at
the other responses if a student favored 15.b highly.
A variant of this is “generally formulaic”: A function involves formulas, possibly more than one.
Indicators are the same, except without 5.a. Even with these seven indicators there still is not
enough information to be sure that a student is experiencing this obstacle, especially since he might
28
strongly reject 15.b because of the word “single.” Therefore, I do not consider this obstacle in the
section of this paper about relationships between conceptions.
Function—dynamic. A student has a dynamic function obstacle if she is experiencing either of the
following two obstacles:
dynamic (continuous). A function is a continuous curve traced out in space. Indicators are not 1.i.a,
1.i.b, 1.i.c, 5.b.f, 5.c.f, 7.a.nf, 7.c.nf, 15.d. It became clear through the interviews that some students
had memorized the “vertical line test,” and responded accordingly even though they also
experienced this obstacle. Therefore, they responded that 1.b was not a function, while still
displaying this obstacle. Similarly, some students responded 15.d without displaying this obstacle,
because this response included the vertical line test. Therefore I downplayed the importance of the
response 15.d in the final coding, and I still coded students with this obstacle even if they did not
display 2 or 3 of the indicators.
dynamic (smooth). This is the same as the previous obstacle, but the motion must be smooth, not just
continuous. Initially the indicators were the same, but with not 1.i.c, and the addition of 7.b.nf.
However, when I discovered some students’ difficulty with correctly graphing the function in 7.b, I
removed this indicator. I added and removed indicators frequently here, trying to account for
certain patterns of student responses, and finally settled on this combination, realizing that (a) some
students believe functions can be discontinuous, but must have no cusps, and (b) some students
believe a function must pass the horizontal line test.
Function—requires changes. A function is a strict dependence relation, meaning that if x changes then y
must also. A student must have both of these indicators: 3.“no”, 15.c. Through the course of the
study it became clear that most people who answer “no” on item 3 are misinterpreting the word
“values.” While this indicates a limited concept of function, it does not necessarily indicate this
particular obstacle. Therefore, I added the requirement that a student must answer “no” and
explicitly show an understanding of the word “value” in order to be considered for this obstacle.
Some students with this obstacle may have therefore been overlooked if they did not explain their
answer on 3 adequately.
Function—object and arbitrary uni-valued correspondence. This is the target conception: a function is a univalued mapping from one set to another, possibly arbitrary, undrawable, or non-formulaic. It is also
an object that can itself be operated on, although the questionnaire items have less bearing on this
aspect of the concept. The indicators began as 3.“yes” (with a suitable example), 7.c.f, 9.“yes”,
10.“infinitely many”, 15.a*. I quickly decided to include with item 10 that students need to answer
in a way that indicates understanding rather than guessing. I also ignored 7.c if students did not
understand the word “rational.” Finally, an interview confirmed that some students read item 9
differently than it was intended by Sfard (1992), thinking that if you repeat the random r process for
the same input x twice, you might get two different y-values. Therefore, I considered this item only
as a weak indicator.
29
Table 3.1—Coding scheme for Calculus Conception Questionnaire items and conceptions
Conception
Continuity
—smoothness
—single formula
—connected
Real Number Line
—next number
—infinitesimals
—set of numbers
Limit
—dynamic (temporal) stepwise
—dynamic continuous
—intuitive static
Indicators³
1.ii.b; not 1.ii.c; 5.c.nc; 16.a¹
5.a.c; 5.b.nc; 5.c.c; 7.a.c; 7.b.nc; 16.c¹
1.ii.c; not 1.ii.a; 5.b.c; 5.c.c; 7.a.nc; 7.b.c; 16.b¹
6.a.T¹; 13.a¹
2.b¹; 6.a.F¹; 6.b.T¹; not 13.a¹
not 2.b; 6.a.F; 6.b.F; 13.b¹
2.c; issues with the question asked in 4; 12.a; 14.b; 12.b²; 14.a²
11.no; mention of discontinuity in 11; 14.a; 14.b; not 17.c w/ certainty²
11.yes¹; 11.closeness¹; not 17.c w/ certainty¹; no issues with the question
asked in 4²; not 8.c²; not 12.a²; 14.c²
14.d¹; 17.c w/ certainty¹; 2.a²; 4.Bin A empty²; 8.b²; 11.yes²; 12.c²
—formal
Function
—single formula
5.a.f; 5.c.f; 7.a.f; 7.b.nf; 7.c.nf; 9.no; 15.b¹; 5.b.nf²
—dynamic (continuous)
1.i.a.nf; 1.i.b.f; 1.i.c.f; 5.b.f; 5.c.f; 7.a.nf; 7.c.nf; 15.d
—dynamic (smooth)
1.i.a.nf; 1.i.b.f; 1.i.c.nf; 5.b.f; 5.c.f; 7.a.nf; 7.c.nf; 15.d
—requires changes
3.no and evidence of not misinterpreting “values”¹; 15.c¹
—uni-valued correspondence
3.yes w/ example; 7.c.f; 10. “infinitely many” w/ a reason; 15.a¹; 9.yes²
¹ especially strong indicator
² especially weak indicator
³ “c” and “nc” mean “continuous” and “not continuous;” “f” and “nf” mean “function” and “not a function”
Relationships between conceptions
In order to determine the relationships between the conceptions, I use the data analysis technique of
cluster analysis. Cluster analysis is a method of organizing objects into groups and subgroups
according to a set of precise similarity criteria (Kachigan, 1982). In my case, the objects being
organized are the student conceptions. I chose this method for several reasons. First, it fits
naturally with my research question about how these student conceptions form groups. Second, it is
a descriptive, not analytic, method. Unlike factor analysis, it employs no a priori assumptions about
the underlying distributions of the studied phenomena. Since my study is exploratory in nature,
rather than verificational, this technique works well to produce the most significant possible
groupings. Finally, it is a method of displaying complex sets of data in a simple format that is easy
to read and interpret.
Two important decisions need to be made when displaying a cluster analysis: how to determine
“distance” between the objects (here, student conceptions) and the method of determining between
which points the distances are measured for determining clusters (Kaufman & Rousseeuw, 1990).
Once these precise similarity criteria are determined, the cluster analysis performs an iterated
procedure: beginning with each conception in its own cluster, it combines the two closest clusters
into one cluster, then combines the next two closest and repeats this until there is only one cluster.
A dendrogram of this process, found in the results section, shows all of these clusters and how they
are agglomerated at each step of the process, forming a taxonomy derived from the data.
The data used in the cluster analysis is binary: each student either has or does not have each
conception. Of the various ways of determining distance between conceptions, the metric that is
30
most appropriate for this study is based on conditional probability (Sokal & Sneath, 1963). To
determine the proximity of Conceptions A and B, calculate how much a student’s probability of
having Conception A increases by assuming that he/she has Conception B also. Technically, one
averages this calculation between P(B|A) and P(A|B). This method of determining proximity is
more meaningful than other available metrics for this study, because it is not readily distorted by low
variance in the data. For instance, because most students did not have most conceptions, a standard
Pearson’s correlation produced artificially small correlation coefficients due to the low variance.
To determine how the clustering algorithm creates linkages between clusters, I used the most
common method—average linkage clustering. At each step of the iteration, this method combines
two clusters by minimizing the average of all the distances between pairs of conceptions in each
cluster.
The cluster analysis methodology does not produce any information about the statistical significance
of the relationships it describes, because it does not make any assumptions about the structure or
distribution underlying the data. In this study, the strength of the relationships between conceptions
is also discussed in addition to the cluster analysis. This is done in terms of odds ratios, a statistic
very closely related to the conditional probability metric chosen for the cluster analysis. The
reliability of the odds ratios can be low for small n, i.e., a conception displayed by only a few subjects
may appear distorted in the cluster analysis due to variance in the sampling process. In addition,
these conditional probabilities are asymmetrical with respect to measurement error. If a student
displaying Conceptions A and B is miscoded, this will have a much larger effect on the measured
probabilities than if a student displaying one or neither of the conceptions is miscoded.
In the second half of the following section, I display the results from the cluster analysis, describing
ways in which these same conceptions did or did not cluster historically, and discussing the meaning
of the relationships found in the clusters.
Results
This section is divided into two parts. In Part A, I address only the individual conceptions that the
calculus students have about the four elements of calculus—continuity, the real number line, limits,
and functions. I describe what percentages of students have the various conceptions, and whether
this is influenced by their calculus background. I also describe some of the conceptions based on
the interview data, and the implications for their status as epistemological obstacles. When student
responses to individual items are discussed, these items are includes as figures. All other items may
be found in the appendix. In Part B, I address the observed relationships between these student
conceptions, the significance of these relationships, and how the relationships manifested
themselves historically.
31
Response Patterns and Obstacle Frequencies Among Calculus Students
Conceptions of Continuity
Table 4.1—Conceptions of continuity
Item
1a is discontinuous
1b is continuous
1c is continuous
5a is continuous
5b is continuous
5c is continuous
7a is discontinuous
7b is continuous
7c is discontinuous
16a is preferred
16b is preferred
16c is preferred
Continuity
Conceptions
Target conception—
Connected
Smoothness
Single formula
Other
Total
All Calc
Students
Students with
Calc
backgrounds
Students
without Calc
backgrounds
%
89
74
71
89
93
72
56
53
87
43
53
9
N
207
174
168
208
218
170
131
126
202
100
126
20
%
89
73
71
89
93
69
61
56
87
44
56
4
n
173
141
138
172
181
133
119
109
169
86
108
7
%
89
83
69
89
89
92
33
39
86
33
39
33
n
32
30
25
32
32
33
12
14
31
12
14
12
52
120
54
105
39
14
27
7
15
62
17
34
233
29
3
13
57
6
26
194
14
31
17
5
11
6
36
Smoothness. Students with some calculus experience are twice as likely to display the smoothness
obstacle as students who have not taken any calculus. For example, only 69% of students with
calculus backgrounds considered graph 5c (y = |x|) to be continuous, but nearly all of the students
(92%) without calculus backgrounds responded this way. Additionally, 2 of the 8 students who took
the follow-up questionnaire displayed the target conception (connectedness) at the beginning of the
semester but displayed the smoothness obstacle at the end of the semester. This appears to be
largely due to the conflation of the concept of continuity with the newly-learned concept of
differentiability.
Some of the interviewed subjects betrayed this kind of confusion about continuity. For instance,
when asked which of the three graphs in Item 1 were continuous, Toni said the following:
Toni:
Int:
Toni:
…I know B is continuous. C is cusps. But I should know if it's continuous. I believe it's
continuous but there's something with limits and things that I'm thinking of. I don't know.
You're uncertain about C. Can you say what you're kind of debating between?
I remember something about cusps and I just can't remember if that was in reference to
whether it was continuous or... I think it's actually whether it can be defined at those points.
Something to that effect, but it's... we did that a couple weeks ago, and while I should
remember, I don't.
32
Int:
Toni:
Int:
Toni:
That's okay, this isn't a quiz. Now you responded very quickly that B was continuous. Why is
that? Can you explain that?
There's no breaks in the graph or any points where it's undefined.
Okay. But...
But I suppose that would make C... theoretically, that would make C discontinuous as well
seeing as how cusps are undefined…
Although at the beginning of the semester Toni answered that graph 1c was continuous, now in the
middle of the semester she concluded that graph 1c is probably discontinuous, because “cusps are
undefined.” She admitted that her thinking about this was muddled by what she did in her Calculus
I class a couple weeks prior, and she could not remember now what it has to do with “something
with limits and things.”
The fact that this conception is so easily altered by recent classroom experience indicates that some
instances of this obstacle are didactical rather than epistemological.
Single formula. The single formula obstacle becomes significantly less prevalent as calculus experience
increases. While 31% of students with no calculus background display the single formula obstacle to
continuity, only 3% of students who have seen some calculus have the obstacle. Furthermore,
students in Calculus I were more likely to display this obstacle (10%) than were the students in
Calculus II (2%).
Connected. Over half of the calculus students have the target conception of continuity,
connectedness. Students with a calculus background are more likely to have this conception,
although this may also be related to the fact that the more calculus experience a student has, the
better she is at graphing. For example, students in Calculus II were, on average, 23% more likely
than Calculus I students to correctly identify functions as being continuous when the graph was not
provided (Items 7a and 7b).
Conceptions of the Real Number Line
Although the entirety of the questionnaire and all of the response patterns can be found in the
appendix and in Figure 1, some of the specific items and responses are reproduced here for ease of
reference. Table 2 shows the frequencies of the responses on the items that specifically addressed
the real number line (Items 2, 6, and 13) as well as the frequencies of the three different conceptions
of the real number line selected for this study.
Figure 1—Two items about the real number line
2. What is between 0.99999… [the nines repeat] and 1?
(a)
(b)
(c)
(d)
Nothing, because 0.99999… = 1.
An infinitely small distance, because 0.99999… is just less than 1.
You can’t really answer because 0.99999… keeps going forever and never finishes.
[If none of the above answers adequately reflects what you think, mark (d) and write your
own answer.]
33
6. True or false:
a)
T
F
It is possible to choose two different points on the real number line that are
touching one another.
b)
T
F
It is possible to choose two different points on the real number line that are
infinitely close to one another.
Table 4.2—Responses about and conceptions of the real number line
Item Response
2a
2b
2c
6a is true
6b is true
13a is best
13b is best
13c is best
Real Number Line
Conceptions
Target conception—Set
of numbers
Next number
Infinitesimals
Other
Total
All Calc
Students
Students with
Calc
backgrounds
Students
without Calc
backgrounds
%
12
56
31
11
83
27
31
43
n
29
131
73
26
195
63
73
100
%
12
56
31
13
82
29
29
42
n
24
108
61
25
159
56
57
82
%
14
53
33
3
97
14
36
50
n
5
19
12
1
35
5
13
18
15
35
15
29
17
6
6
31
48
13
73
112
233
7
29
49
13
57
95
194
0
42
42
0
15
15
36
Closeness. Calculus students’ misconceptions about the meaning of “closeness” in the real numbers
are common and robust. This idea is not new (e.g. Nuñez, et al., 1999), but it is strongly supported
by this study. While only 11% of the surveyed students answered “true” for 6a, 83% answered
“true” for 6b. In particular, nearly all of the students new to calculus (97%) answered “true” for 6b.
Since the vast majority of students believe that it is possible to find two real numbers that are
infinitely close together, it is also not surprising that only 12% of the students believed that 0.999…
was equal to 1 (Item 2).
Next number. About 7% of the students responded in a way consistent with the belief that the
numbers on the line are right next to one another, like a string of pearls, and that they touch each
other, with nothing between. For example, Amanda replied in her interview that 6a is true. Later,
she provides the example of 2.999… and 3 as being infinitely close to each other. When asked
“What is between those numbers?” she replied: “There’s like nothing, because they’re so close to
each other that it’s, like, they’re like the next…”
Infinitesimals. The infinitesimals obstacle encompasses a number of different conceptions. We have
seen that most students believe it possible to find two numbers that are infinitely close to one
34
another. This belief alone does not indicate a belief in infinitesimals. For instance, some students
respond that 0.999… is infinitely close to 1 because it gets closer and closer to 1. One student
likened this to asymptotic movement, when she was asked why she believed 6b to be true:
Amanda: ‘cause when you say that I’m thinking like um, sort of, um, like, an asymptote, where like
coming from both sides [gestures with hands moving together and upwards], sort of, so if
there’s like an asymptote at like x or whatever number, and then there’s like coming at it
from both sides…
This kind of response does not signify a belief in the existence of distances and numbers that are
smaller than any real number, but rather the belief that the real numbers can be dynamic objects, i.e.
“0.999… is approaching 1.
However, 31% of the students replied to Items 6a, 6b, 2, and 13 in such a way that indicates a belief
in the actual existence of infinitesimal numbers and distances. The interviews illuminate this, but
they also reveal that the infinitesimals obstacle is not just a single conception but rather includes
multiple conceptions.
i. One way of thinking about infinitesimals shown in the interviews appears to avoid the issue of
infinity completely. When asked about Item 6, Toni replied:
Int:
Toni:
Int:
Toni:
Int:
Toni:
…
Int:
Toni:
Int:
Toni:
So if there's some amount of distance between the two, can you give me an example of two
numbers that are infinitely close to each other?
Like 1.001, 1.002, or something to that effect.
Okay, like, so those two, that would be, is that a good example? 1.001...
You can get a lot closer, but it's... there's a ridiculous amount of zeroes in there, they're just
gonna be getting... there's that much less distance between them.
Okay. So can you find like a number that's infinitely close to both of these two numbers?
Sure. You can always go in between them, I suppose.
Okay. So let me ask you another couple questions about this. These two numbers [1.001 and
1.002] are infinitely close together. Is that a fair characterization of what you're saying?
Yeah.
Okay. What about these two numbers: 1.1 and 1.2. Would you say that those are infinitely
close together?
They're close, I don't know if I'd say infinitely close. Because in the number world it has to
be ridiculously small numbers to be infinitely close.
Toni confirms this reasoning later when asked about Item 2:
Int:
Toni:
Int:
Toni:
Int:
Toni:
Would you say... How would you answer if I ask what is between .9999 but not repeating, and 1?
.0001.
Oh, okay, you'd answer the number. The distance between them is .0001. If you had to choose one of
these answers [points to Item 2], would any of those fit for this question?
For that question I'd say B.
B fits because they're infinitely close.
Right.
For Toni, “infinitely” close means “very” close, and in a quantifiable way. Some distance between
0.1 and 0.001 demarks the boundary between “close” and “infinitely close.” When asked later about
0.999…, she claimed that the number was infinitely close to 1, just as 0.999 was. This confirms that
Toni does believe in infinitesimals—her notion is static rather than dynamic, and she believes that
35
the number line is full of infinitely small numbers and distances—but infinitesimal distances are not
qualitatively different from finite distances. In particular, a finite number of finite operations can
result in an infinitesimal distance.
ii. Don’s belief in infinitesimals appeals to the metaphor of “zooming in,” a common metaphor for
describing infinitesimals (Keisler, 1976). His example of two infinitely close numbers is 0 and 1/∞.
He says that they are “approaching each other infinitely close.” When asked if either number is
moving, he says:
Don:
Uh, kind of like as a graph. As you, like, it's not moving but as you look at it closer and
closer. As the magnifying zooms in, kinda.
When asked about the degree of zooming-in, he responds, “Well, I guess only if you zoomed in to
infinity you would be able to tell that they are different points. So that would not be practically
possible but theoretically possible.” Don later confirms that the same was true for 0.999…. This
number is infinitely close to 1 because it would be necessary to zoom in infinitely much to see the
difference, not because it is moving toward 1.
iii. Sarah displays an understanding of the real number line with infinitesimals that is highly
developed and internally consistent. Her conception is very similar to that of Leibniz (Baron, 1987),
with an algebra of operations that can be performed on infinitesimal, finite, and infinite numbers.
Her intuitive construction of infinitesimal numbers resembles the technique formalized by Abraham
Robinson in 1959 (1996). A detailed account of her conception as a non-standard model appears in
a later chapter of this document.
Sarah claimed that 3.999… and 4 were examples of two infinitely close numbers. When asked about
a number infinitely close to both of these, she wrote 3.999… with another 9 after it. Her reasoning
was “If it's repeating for infinity there's always going to be one more to infinity. You can always add
1 to infinity. So you can always add another 9 to the infinity of 9s that are coming.” Other different
but legitimate numbers she mentioned were 0.999… with two nines after it and 0.000… with a 1
after the infinite string of zeroes.
Sarah’s conception of infinitesimals involves the creation of infinitesimal decimal numbers that can
be operated on in the same way as finite decimals. For instance, they can be squared to get a smaller
infinitesimal, or taken the reciprocal of to get an infinite number, and so on. Her idea hinges on the
notion that it is possible to add more to the end of an infinite string. She continually apologizes for
the “wrong”-ness and “crazy”-ness of her conception, although her conception appears to be
internally consistent and mathematically coherent.
Set of numbers. 15% students have the target conception of the real number line, and this does not
change based on the students’ calculus experience. Similarly, more calculus background has little
effect on a student’s likelihood of answering Item 2 correctly.
Conceptions of Limits
Table 3 shows calculus student response categories for some of the items pertaining to limits, as well
as the proportions of students experiencing each obstacle. A few particular patterns are noted
below.
36
Figure 2—Some items about limits
4. Suppose you put two tennis balls numbered 1 and 2 in Bin A and then move ball 1 to Bin B. Then
you put balls 3 and 4 into Bin A and move ball 2 to Bin B. Then you put balls 5 and 6 into Bin A and
move ball 3 to Bin B. And so on infinitely. How many balls are in Bin A after you are done? Why?
17. Which of the following, if true, is enough to guarantee that lim f ( x ) = L.
x a
(a)
(b)
(c)
(d)
For every b, there is a positive c so that if |x – a| < b, then |f(x) – L| < c.
There is a positive b and a positive c so that |x – a| < c and |f(x) – L| < b.
For every positive b, there is a c so that |f(x) – L| < b whenever |x – a| < c.
There is a b, so that for every positive c, |f(x) – L| < b as long as |x – a| < c.
How certain are you?
Just guessing
1 2
3
4
5
6
7
8
Very certain
9 10
Table 4.3—Some responses about and conceptions of limits
Item
4 correct (“no balls left”)
4 objecting to infinity
8 function
8 number
8 process
8 equation
11 “yes”
11 “no, never ends”
11 “no, not cont.”
12a
12b
12c
All Calc
Students
Students with
Calc
backgrounds
Students
without Calc
backgrounds
%
1
18
28
33
27
14
49
7
15
19
37
37
%
1
18
25
36
29
12
51
6
18
20
39
34
%
3
17
42
17
14
25
36
11
3
17
33
50
n
3
41
65
77
63
33
116
16
36
44
87
87
n
2
34
49
69
57
23
98
12
35
38
75
65
n
1
6
15
6
5
9
13
4
1
6
12
18
37
Table 4.3 (cont’d) — Some responses about and conceptions of limits
All Calc
Students
14a* preferred
14b* preferred
14c* preferred
14d* preferred
17c (correct)
17c, confidence > 3
17c, confidence > 5
Epistemological
Obstacle
Limit
Target conception—
Formal
Dynamic stepwise
Dynamic continuous
Intuitive static
Other
Total
Students with
Calc
backgrounds
Students
without Calc
backgrounds
46
34
24
7
40
6
1
107
80
56
16
94
14
3
50
33
26
6
37
97
63
50
12
72
25
47
14
6
56
9
17
5
2
20
0
1
1
1
0
0
25
31
11
33
58
73
25
76
233
24
34
11
32
46
65
21
62
194
33
22
8
36
12
8
3
13
36
* The totals for Item 14 add up to more than 100% because students sometimes favored more than one response
equally.
Dynamic Stepwise. 25% of the students displayed the dynamic stepwise conception of limit, objecting
to the idea that an infinite process can ever be completed. An example of a response that indicates
this conception is an explicit objection to the statement “and so on infinitely” found in Item 4.
Student questionnaire responses include statements such as “you can never ‘be done’”, and
“unknown, if you continue infinitely you'll never be done.” There is some evidence that this
conception may not actually operate as an obstacle to an advanced conception of limit. For
instance, almost all (5 out of 6) of the very few students who answered Item 17 correctly with a
confidence greater than 3 also displayed the dynamic stepwise conception of limit. This indicates
that although a student can have an ontological rejection of the idea of infinite processes, he or she
can still have a sophisticated operational ability with formal limits and with mathematical logic. This
is supported also by the relationships between this conception and the other conceptions described
in the following part of this results section.
Dynamic Continuous. About 31% of the students conceptualized limits in terms of continuous motion
along a curve (dynamic continuous). Although fewer students with calculus backgrounds answered
Item 11 incorrectly, the ones who did answer that the function has no limit are much more likely
(18% compared to 3%) to specify that this was because the function is discontinuous. Many
students responded explicitly that the limit did not exist because the function was discontinuous.
Some elaborated on their questionnaires with comments such as:
“No, it leaps back and forth and has no definite end.”
“No, there are jumps and not defined points.”
Some students’ comments also indicated confusion with the converse: a function is discontinuous at
a point if it has no limit at that point. For instance, one respondent said that the function has “many
38
different limits,” and one interviewee said, “Doesn’t it have to be continuous to have a limit? or
wait, I can’t remember…”
Intuitive Static. On average, 11% (n=25) of students have this obstacle, making it the most advanced
limit conception observed with any regularity among calculus students. Calculus II students are
more likely (18%) than Calculus I students (7%) to have this obstacle. Although this conception is
somewhat uncommon, around half of the students were able to answer Item 11 correctly, signifying
a partial operational knowledge of the limit in terms of closeness.
Formal. Only one calculus student appeared to have a firm formal conception of the limit. In fact,
none of the calculus students simultaneously answered Item 17 correctly with a confidence greater
than 3 and favored response “d” on Item 14.
Conceptions of Functions
Table 4 shows calculus student response categories for some of the items about functions, as well as
the proportions of students experiencing each obstacle. A few particular patterns are noted below.
Table 4.4 — Some responses about and conceptions of functions
Item
1a function
1b not a function
1c function
3 with correct example
5a not a function
5b function
5c function
7a function
7b function
7c function
9 is a function
10 correct with some explanation
15a preferred
15b preferred
15c preferred
15d preferred
15e preferred
All Calc
Students
Students with
Calc
backgrounds
Students
without Calc
backgrounds
%
82
89
84
45
81
97
93
90
77
37
43
17
31
36
19
15
31
%
82
90
88
50
81
98
92
90
81
39
44
14
29
39
19
15
29
%
81
81
67
25
78
89
97
86
61
31
36
3
33
25
19
14
44
n
193
207
198
106
188
227
218
211
182
88
102
7
72
85
45
35
72
n
160
175
170
97
158
190
179
175
158
75
86
7
56
75
37
29
56
n
29
29
24
9
28
32
35
31
22
11
13
8
12
9
7
5
16
39
Table 4.4 (cont’d) — Some responses about and conceptions of functions
Function
Target conception—Arbitrary univalued correspondence
Single formula
Generally formulaic
Dynamic--continuous
-smooth
Requires changes
Other
Total
All Calc
Students
Students with
Calc
backgrounds
Students
without Calc
backgrounds
6
13
6
12
3
1
5
29
14
9
5
1
50
12
67
32
20
12
3
118
233
5
30
14
9
5
1
49
9
58
27
17
10
2
95
194
8
22
11
8
3
3
61
3
8
4
3
1
1
22
36
Single formula and generally formulaic. Although only 5% of the students required a function to be given
by a single formula in order to be considered a function, many students tended to consider functions
to require some kind of formulaic element. For this reason, I coded 29% of the respondents as
having the conception that a function must be given by formulas, but any number of formulas. For
example, the following is one interviewee’s response when asked about Item 7:
Int:
Ryan:
You said c is not a function but these two [points to 7a and 7b] are. Why? What’s the
difference?
um, that’s just, um, defining rational and irrational numbers. There’s no, um, it’s almost like
there’s no set formula for it. You can’t…there’s like no hard numbers. You’re plugging in
and applying, like, a concept to it, instead of, like, multiplying it by two. You know, the
concept of it being rational or irrational.
Other studies have noted this formulaic approach to function (see the theory section), but in this
study it appeared with less prevalence. For instance, 43% of these students responded correctly
(“yes”) to Item 9: “Consider the following procedure: for each x-value, let r(x) be a random number
between 1 and 10. Is r(x) a function?” Some of the students who answered “no” thought that the
value for a given x was not set; if you asked what the function value was for a given x, you would get
a different answer each time, meaning that the process is not single-valued. The question on which
this is based, administered to a control group of undergraduates by Sfard (1992), was answered
correctly by 17%.
Dynamic. Only 14% of the students conceptualized functions as being given by motions traced out
along continuous (or smooth) graphs. I attributed this conception conservatively for a few reasons.
For one, some students who favored response “d” on Item 15 did so primarily because it included
the phrase about the “vertical line test.” One interviewee said:
Don:
Int:
D is how I would visualize it graphically, although I would probably visualize it slightly
different maybe. So the vertical line test, so you would have your function and then
(unintelligible) and if it hits it in more than one spot then it's not a function. So I would... I
guess I wouldn't maybe visualize it as this last part.
You mean the “moving particle” thing, that doesn't really matter one way or the other to
you?
40
Don:
I don't know, I might not visualize it as that, but I definitely would see the vertical line test
and that makes sense to me.
In addition, more students answered Item 1 correctly than I anticipated. Only 61%, 65%, and 52%
of the Israeli undergraduates surveyed by Vinner and Dreyfus (1989) answered correctly the
problems on which Items 1a, 1b, and 1c were based respectively.
Requires changes. Only 3 students have the obstacle that a function requires change. The student
responses for Item 3 were similar to the responses obtained by Vinner & Dreyfus (1989). While
these authors showed that many students appeal to a “dependence relation” definition of function,
my findings indicate that few students display such a strict dependence relation conception that it
acts as an epistemological obstacle to the function concept. In my coding scheme, a student must
respond appropriately not only to Item 3 but also to Item 15c in order to be considered to have this
obstacle. It is notable that this obstacle, which I almost dismissed because it did not appear
prominently in history, also does not seem to appear prominently among students.
Arbitrary uni-valued correspondence. 6% of the students responded in such a way indicating an abstract
notion of function.
Other or mixed conceptions. It is disappointing that the majority of students fell into the “other”
category for conceptions about functions. Although some of this finding might be due to an overly
cautious coding scheme, certainly some of it can also be attributed to an incomplete catalog of
obstacles to the function concept used in this study. Another possibility is that students may spend
more of their time with varied conceptions of function, conceptions that are, say, quite situationally
dependent, rather than being stuck on particular epistemological obstacles.
Relationships Between Conceptions
Figure 2 contains a graphical display of the cluster analysis for the conceptions. This dendrogram
starts with all of the conceptions in separate clusters on the left, and shows how they cluster
together as the algorithm iterates. The algorithm stops when all of the conceptions are in one
cluster, on the right. For example, Continuity (connected) and Limit (intuitive static) cluster
together at the 13th step of the process.
41
Although this method does not provide, a priori, a step after which the algorithm has determined
the most salient clusters, in this case we notice two natural steps of clustering. At Steps 19 and 20,
four clusters emerge. Each of these contains sub-clusters appearing in Steps 13-16, denoted by
indentation in the following list:
Cluster 1—smooth motion-based
Continuity - smoothness
Function – dynamic smooth
Real Number Line – next number
Limit – dynamic continuous
Cluster 2—formulaic static
Continuity – single formula
Real Number Line - infinitesimals
Function – single formula
Cluster 3—target conceptions
Limit – formal
Function — object and correspondence
Real Number Line – set of numbers
Function – requires changes
Continuity – connected
Limit – intuitive static
Cluster 4 – ??
Limit – dynamic stepwise
Function – dynamic continuous
Smooth-Motion-Based Cluster
The conceptions in this cluster are precisely the ones that rely on a smooth-motion-based
understanding of the four elements of calculus. Functions are given by a smooth (infinitely
differentiable) graph, and such graphs are the only ones considered continuous. These two
conceptions are the closest in the cluster. A limit is achieved by approaching a point on such a
graph by using a smooth motion. Motion along the real number line is possible by moving from
one point to the next without skipping any.
42
These four conceptions all rely on a common assumption that mathematical phenomena exhibit
smooth temporal motion. The fact that this same intuition supports the construction of all four
conceptions indicates that the relationships are epistemological in nature. However, the conceptions
are not simply expressions of one big epistemological obstacle, else the linkages between them
would be much stronger. Barring measurement error, the odds ratios in that case would
theoretically be infinite. Table 4.5 shows that the direct odds ratios between pairs of these four
conceptions are generally between 1 and 2, indicating moderately strong, but not overwhelmingly
strong relationships between the conceptions.
Table 4.5—Direct odds ratios for pairs of conceptions in the smooth-motion cluster
Limit dynamic
continuous
(n=73, 31%)
Continuity
smoothness
(n=62, 26%)
Function
dynamic smooth
(n=12, 5%)
Real Number
Line next number
(n=13, 6%)
Limit dynamic
1.37
1.37
1.53
continuous (n=73, 31%)
Continuity smoothness
1.40
1.63
1.18
(n=62, 26%)
Function dynamic
1.59
1.99
0.00
smooth (n=12, 5%)
Real Number Line
1.90
1.24
0.00
next number (n=13, 6%)
The odds ratios are read as follows: an entry of 1.40 in the table in position (Column C, Row R) means that a student is
1.4 times more likely to have Conception R if he has Conception C than if he or she does not have Conception C. A
direct odds ratio greater than 1 indicates a positive relationship between the two conceptions.
The conceptions in this cluster are too strongly related to ignore, but they are not related strongly
enough to be epistemological obstacles to each other. For these reasons, and because there is an
obvious intuition underlying all four of them, I posit that these conceptions are related by a
common epistemological disposition. The epistemological disposition toward smooth motion is a
‘disposition’ because it is an underlying tendency to construct similar conceptions for different
mathematical concepts. The use of the word “epistemological” here does not mean that this is a
disposition about how knowledge is constructed. Rather I use the term to indicate its structural role
in the creation of the particular conceptions, and to remind the reader of its development from the
theory of epistemological obstacles.
An epistemological disposition is a conceptual preference that supports, but does not compel, a
student’s construction of certain related conceptions of various mathematical concepts. Thus it
explains how a person might establish similar conceptions of different mathematical concepts, but
also leaves room for stronger influences, such as didactical experiences and epistemological
obstacles, to create (or prevent the creation of) different conceptions.
If this underlying factor I am calling an epistemological disposition is actually epistemological in
nature, one would expect to find evidence of it historically, just as researchers found the
epistemological obstacles occurring historically. Here, the historical connection is clear—this cluster
of four smooth-motion-based conceptions describes Isaac Newton’s conceptions about these four
elements of calculus.
Newton based his discoveries about integral and differential calculus in a strong physical intuition,
appealing to an underlying belief in the continuity and differentiability of the simultaneously
43
changing phenomena. He specifically avoided a discrete, static, incremental notion of limit, instead
grounding his new calculus concepts in time and motion. For instance, the “fluxion” of a variable is
an “instantaneous speed defined with respect to an independent, conventional dimension of time
and on the geometrical model of the line-segment” (Whiteside, 1964, p. x). Newton’s work with the
differential calculus explicitly relied on describing velocities of moving points in space, each
coordinate of which was implicitly defined as a smooth function of an underlying time variable
(Edwards, 1979).
The three smooth-motion-based conceptions for limit, function, and continuity were chosen for this
study in the first place with the specific historical example of Newton in mind. However, I am not
certain that the real number line–next number obstacle specifically applies to Newton, although my
research about this is inconclusive. Certainly Newton’s tendency to describe moving points in space
indicates a “naturally continuous line” conception, but these two conceptions of the real number
line may not be mutually exclusive. Recall that the naturally continuous line obstacle is based only
on one questionnaire item, likely compromising its validity. However, when I do perform a cluster
analysis that includes this conception, it ends up in the smooth-motion cluster, included at a late step
of the algorithm.
Static Formulaic Cluster
This cluster includes the conceptions that a function is given by a single formula, and that a graph is
continuous if it is given by a single formula. It also includes a belief in infinitesimals. As I
mentioned earlier, although there appear to be numerous different conceptions and metaphors for
infinitesimals, all of them agree that infinitely small numbers and distances exist as static quantities
(rather than being evanescent or not existing).
As in the case of Cluster 2, Table 4.6 shows that the relationships between the conceptions in this
cluster are too strong to be ignored, but too weak for the conceptions to be seen as expressions of
the same epistemological obstacle.
Table 4.6—Direct odds ratios for pairs of conceptions in the static formulaic cluster
Continuity single
formula (n=17, 7%)
Continuity single formula
(n=17, 7%)
Function single formula
(n=12, 5%)
Real Number Line
infinitesimals (n=73, 31%)
Function single
formula (n=12, 5%)
2.48
2.56
1.58
Real Number Line
infinitesimals (n=73, 31%)
1.97
1.11
1.08
Like the epistemological disposition of smooth motion accounts for the existence of Cluster 1, an
epistemological disposition toward the static formulaic may provide an account for this cluster. The
function and continuity obstacles indicate that a simple single formula underlies any mathematical
phenomenon, and that this formula provides the algebraic unity required for the phenomenon to be
continuous. For students, the relationship of these conceptions with infinitesimals is less clear, but
historically the single formula conceptions coincide very clearly with the infinitesimals conception.
For the continental mathematicians who used them, infinitesimals provided a static foundation for a
formula-based calculus.
44
This cluster, like the previous one, is historically prominent—it describes the conceptions of
Gottfried Wilhelm Leibniz and his immediate circle of continental mathematicians, including
L’Hospital and the Bernoullis. Leibniz was actually the first to use the term “function,” generally
referring in the 1670s and 1680s to a property derived from an original curve (such as its tangent or
subnormal). During his correspondence in the late 1690s with Johann Bernoulli, the term took on
the meaning that it would retain throughout the early 1700s on the continent—a quantity
represented by single algebraic expression in terms of one variable (Kleiner, 1987) All such
quantities were continuous by virtue of their representation by one unified formula.
Leibniz’s conception of infinitesimals is well-known and documented (e.g. Mancosu, 1996; Jolley,
1995). Briefly, Leibniz used the notation dx to refer to the infinitesimal differential of a quantity,
building differential and integral calculus with this same common notation. With this notation, he
was able to work with infinitesimal, finite, and infinite quantities formulaically, and according to a set
of operational rules. Most of his notation (notably dy/dx) is used to this day, because of its
efficiency for representing both integral and differential calculus phenomena, and also for its
suggestive power (for instance, the chain rule amounts to canceling fractions). Leibniz’s conception
is static in nature—infinitesimal quantities really exist (mathematically); they can be ignored in
comparison to finite things, but they are not in themselves changing or disappearing.
Leibniz (and his continental contemporaries) had an epistemological disposition toward the static
formulaic. It operated strongly enough to give him an algebraic framework for developing the
principles, foundations, and notations of calculus, but not so strongly that it prevented him from
working comfortably with dynamic phenomena and even developing coherent laws of motion
(Roberts, 2003).
It is no surprise that no conception of limit appears in Cluster 2. As opposed to Newton’s, Leibniz’
treatment of calculus avoids the concept of limit almost entirely:
In regards to the calculus itself, discrete infinitesimal differences of geometric variables played the
central role in Leibniz’ approach, while Newton’s fundamental concept was the fluxion or time rate of
change, based on intuitive ideas of continuous motion. As a consequence, Leibniz’ notation and
terminology effectively disguises the limit concept, which by contrast is fairly explicit in Newton’s
calculus. (Edwards, 1979, p. 266)
Even though no particular limit conception appears in this cluster, this of course does not mean that
students with an epistemological disposition toward the static formulaic have no conception of limit
at all. It means that such students are more likely to appeal to infinitesimals as a foundation for
interpreting questions of infinity, making them less likely to have a strong commitment to a
particular view of limits in order to interpret these same questions.
Target Conceptions Cluster
The main conceptions in this cluster are the target conceptions for function, continuity, and the real
number line, and the “intuitive static” obstacle to the limit concept. This last obstacle is the most
advanced conception about limits actually displayed by more than one calculus student (11%, n =
25).
45
The appearance of the other two conceptions in this cluster, the target conception for limit and the
“requires changes” obstacle to function, should be interpreted cautiously. The n for both of these is
only 3, and in the case of the former, 2 of the students displaying the conception were the 2 graduate
students in the study. Although it makes sense that at least the formal limit conception should
belong in this cluster, I hesitate to conclude that from these data.
This cluster is not particularly surprising. Its four main members are the most advanced
conceptions of limit, function, continuity, and the real number line displayed by calculus students.
Because these conceptions are the most abstract of the ones studied, one conjecture about this
cluster’s meaning is that there exists an epistemological disposition toward mathematical abstraction.
On the other hand, this cluster may simply describe features of the particular sample of students
studied, that there was a group of advanced students who “maxed out” the assessment. This would
indicate connections between the conceptions, but only trivial ones: the conceptions are all target
conceptions and a number of students have reached the targets.
Cluster 4
This cluster seems to contain the leftovers, the dynamic stepwise conception of limit and the
dynamic continuous conception of function. This latter conception is rather specific—functions
must be continuous, but they do not need to be smooth. The average odds ratio between these
conceptions is 1.27, indicating a positive but weak relationship. The noteworthy observation is not
that there is some underlying epistemological disposition here, but rather that these conceptions are
not in the other clusters. In particular, the dynamic stepwise conception is operating rather
independently of the other clusters, for instance, relating positively to the target conceptions for
function and the real number line but negatively to the target conception for continuity. As pointed
out in the preceding part, this obstacle appears to relate strongly to an advanced operational ability
with limit, which challenges whether it is really an epistemological obstacle after all.
Discussion
In the results section above I describe two kinds of findings about student conceptions of limits,
functions, continuity, and the real number line. The first set of findings is about the epistemological
obstacles themselves, offering a more detailed look at what these obstacles mean and how
commonly they occur among students with and without calculus backgrounds. These findings have
some implications for the theory of epistemological obstacles itself, which I will explore in this
section. The second set of findings is about the relationships between the epistemological obstacles.
The way these obstacles form clusters in student thinking suggests underlying mechanisms that I call
“epistemological dispositions,” found also in the thinking of the two independent developers of
calculus, Newton and Leibniz. In this section, I will also explore the implications of this new
theoretical idea, which suggests a broader framework for categorizing cognitive elements that are
epistemological in nature.
46
Epistemological obstacles
When I began this study, I used a looser interpretation of an epistemological obstacle than some
other researchers have. Now I will explore the ramifications of this choice for my study, and discuss
how my ideas about epistemological obstacles have changed due to the results of this study.
An epistemological obstacle is a piece of knowledge that both interferes with and is part of further
learning. In addition, for Brousseau, an epistemological obstacle is a misconception that every
learner must encounter in the course of learning the given concept. Based on the precedent of other
researchers, particularly Sierpinska (1987, 1992), I have chosen instead to treat an epistemological
obstacle as a misconception that learners commonly encounter in the course of learning the given
concept. I wanted to retain the idea that an epistemological obstacle is an important part of the
structure of the target knowledge, rather than being a vestige of some classroom practice (i.e.
didactical). I only weaken the definition because I do not think that we know enough about the
concepts of limit, function, continuity, and (especially) the real number line to make such strong
claims about what misconceptions every student must encounter as he/she learns. Let me explain in
more detail.
Figure 3—Concept decompositions using epistemological obstacles
Diagram A
Diagram B
Diagram C
Epist.
Obst. I
Epist.
Obst. I
Epist.
Obst. II
Epist.
Obst. II
Target
Conception
Epist.
Obst. I
Epist.
Obst. II
Target
Conception
Target
Conception
According to the Brousseauian treatment of epistemological obstacles, a concept can be described
by either the first or second diagram in Figure 3. Since each obstacle to a concept is a necessary
conception along the way, a prerequisite structure, or genetic decomposition, can be described for
the concept. Diagram A shows such a genetic decomposition in which each obstacle must be
encountered, in its turn, before the target conception can be achieved. In this case, Epistemological
Obstacle I is actually an obstacle to Epistemological Obstacle II, which is an obstacle to the Target
Conception. An example of this kind of progression appears in my study: the limit dynamic
continuous conception is an obstacle to the limit intuitive static conception, which in turn is an
obstacle to the target formal conception of limit. Diagram B is also a possible Brousseauian
configuration for epistemological obstacles to a concept. In this case, Epistemological Obstacles I
and II must both be encountered before the target conception is learned, but the order of
surmounting these obstacles may vary. It is important to remember that each conception
47
incorporates the prior knowledge, rather than obliterating it. Therefore each conception is
characterized by the tree above it; this tree describes its history and its composition.
Weakening the definition of epistemological obstacle allows also for the kind of structure found in
Diagram C. Here, Epistemological Obstacles I and II are both obstacles to the target conception,
however neither of them is strictly necessary for its construction. Some students may encounter
Epistemological Obstacle I, some Epistemological Obstacle II, some may encounter both, and some
may encounter neither. Notice what this means for the target conception—it is not just one
conception any longer. If Student X encounters Obstacle I before gaining the target conception,
and Student Y encounters Obstacle II, then the target conceptions these two students display will be
different from one another, because they have different histories. They are comprised of different
structures. This is why there are multiple nodes within the circle that designates the target
conception.
But why should I bother with the little circle in Diagram C? If Student X and Student Y’s target
conceptions are different because they have different compositions, then why do I even want to call
them the same thing? The answer is that these different target conceptions may function the same
way nearly all of the time. These different conceptions may be operationally the same with regards
to all of the tasks and activities that the students will likely encounter in their lives as students.
There is likely no a priori reason to prefer one of these two operationally similar conceptions to the
other.
The data in this study support the fact that, while structures like Diagram A and B exist for
particular concepts, structures more like Diagram C also exist. For example, this study reveals that
many students display the smoothness obstacle to continuity after they display the target conception
(connectedness), due to a confusion brought on by the introduction of the concept of derivative.
Therefore, there may be at least two different routes to the target conception:
(a) …--> smoothness --> connectedness
(b) … --> connectedness --> smoothness --> connectedness
This same target conception may have two different histories for two different students, even if
much of this difference may be the effect of instructional practice. Furthermore, each of these
routes may also have other variations, such as the appearance in various places of the single formula
obstacle, or of other obstacles I may have overlooked in this study. Why do I then call the target
conception just one name, even though it really encompasses at least two conceptions with different
conceptual histories? The reason is that students with these two different concept histories clearly
respond the same way to general tasks and discourses involving continuity, such as the questionnaire
items. Only with a much finer instrument, a detailed interview, did I detect the difference in these
conceptions and their histories.
The fact that the conception actually describes multiple similar conceptions is true not only for the
target conceptions, but also for the epistemological obstacles themselves. For example, the
conception of infinitesimals is an epistemological obstacle that appears to be a unitary conception
when working with coarse tasks and discourses (such as a few questionnaire items), but reveals itself
to be a collection of operationally and linguistically similar conceptions when probed more carefully.
The interviews revealed at least three different kinds of conceptions of infinitesimals. Here the
language usage appears to be a key feature. Toni’s use of the word “infinitely” sounds like anyone
48
else’s use of the word, until she is specifically asked to provide an example of two infinitely close
numbers, and she says “1.001 and 1.002.” The flexibility of language allows multiple people to talk
about something functionally while still having slightly different conceptions of the thing. The more
precise and technical the discourse, the more similar the conceptions must be in order to avoid
conversational conflict.
I began by weakening the requirement for an epistemological obstacle to only be a common, not a
necessary, misconception. This change apparently implies that an epistemological obstacle (or a
target conception) should be actually viewed as a set of operationally and linguistically similar
conceptions, each of which may have a different history. What this means is that a level of
discursive precision must be chosen when determining epistemological obstacles, because at a fine
level of precision these different conceptions and their histories are distinguishable, but at a coarser
level they are not. In fact, at a coarser level yet, even the epistemological obstacles cannot be
distinguished from one another. If a particular group of people only ever talks about limits very
generally, the members of the group will still be able to functionally communicate and probably not
even notice if they actually have different obstacles to the limit concept. This may be one reason
these epistemological obstacles are so persistent and stubborn in the classroom, because most
classroom tasks and discourses do not address concepts such as limit and function with a precision
fine enough to even reveal the differences in the conceptions of the participants.
Along with choosing a level of discursive precision, another choice must be made when identifying
an epistemological obstacle: how strongly must it operate? Since an epistemological obstacle is a
conception that operates strongly in the structure of the target knowledge, but not always with the
strength of necessity, then we are placing the idea of epistemological obstacle on a spectrum rather
than considering it to be an absolute category. This raises the question: if epistemological obstacles
are on the strong end of the spectrum, then what is on the weak side?
But there is a second question that follows naturally from this reconceptualization of epistemological
obstacles. In Diagram C of Figure 3, a student may encounter one, or both, or neither of
Epistemological Obstacles I and II on the way to constructing the target conception. Since the
strength of necessity is not at work here, what is it that leads a student to go one route or another
along the way?
The answer to both of these questions is found in epistemological dispositions.
Epistemological Dispositions
I began this paper by conjecturing that there were some parallels between student thinking and
historical thinking about the elements of calculus. This is supported by existing research (Sfard,
1991; Kaput, 1994), and can be meaningfully interpreted by the theories outlined by Piaget & Garcia
(1983/1989) and by Lakoff & Nunez (2000). This research has generally looked for “vertical”
relationships (prerequisite structures) rather than “horizontal” ones (relationships between
conceptions across numerous concepts). By treating conceptions as epistemological obstacles, the
results of this study suggest that (a) these horizontal relationships do exist for students and (b) there
are parallels between these relationships and the ones found historically, particularly ones displayed
by Newton and Leibniz. That some relationships between conceptions exist is not surprising. What
is surprising is that these clusters can be characterized so intuitively using the idea of epistemological
dispositions. After all, it might have turned out that there is no rhyme or reason at all to the
49
combinations of conceptions about functions, limits, continuity, and the real number line that
students display. But what I have found is that there is a very intuitive explanation for these
combinations of conceptions. These clusters appear because there are epistemological dispositions
at work, conceptual preferences like smooth motion and static formulas that support, but do not
compel, learners’ constructions of related conceptions for multiple mathematical concepts.
Not only does the idea of epistemological dispositions provide an intuitive explanation for these
observed connections, it also answers one of the questions raised in the above discussion of
epistemological obstacles. In our reconceptualization of epistemological obstacles, an
epistemological obstacle may not mark the only path to the target conception, but one of multiple
paths; what mechanism then leads a learner to experience one epistemological obstacle over
another? The answer is found in the epistemological dispositions the learner brings to bear upon
the general domain of learning. For instance, the smoothness obstacle is a common, but not
necessary, obstacle to the target conception of continuity. A student is more likely to experience this
particular epistemological obstacle if her thinking is supported by an epistemological disposition
toward smooth motion, and less likely to experience the obstacle if her thinking is supported by a
different epistemological disposition.
Another surprising result is that two of the clusters observed in the study are precisely the two
clusters displayed by the two independent developers of calculus. This supports the conclusion that
the mechanism at work really is epistemological in nature: the two epistemological dispositions that
supported the development of calculus are two dispositions that still support student thinking. Even
though these epistemological dispositions are weak enough to be didactically mutable and context
dependent, they are still strong enough to provide some structure to the concepts of limit, function,
continuity, and the real number line that goes beyond a student’s experience in any particular
instructional setting. It is not surprising that this structure continues to appear after 300 years, for
we have inherited much of the milieu of metaphors, notations, language, and activities through
which they were originally developed.
In this study I have characterized this epistemological structure with two different mechanisms—
epistemological obstacles and epistemological dispositions. These work at different breadths of
domain and different strengths. Epistemological obstacles are narrow in domain, specific to a
particular concept, and they operate strongly, in the sense that they are both commonly encountered
and tenaciously held by students learning the concept. Epistemological dispositions, on the other
hand, operate on a broad domain, but relatively weakly. A disposition indicates a propensity, but
not a compulsion, to bring a particular approach to bear on a relatively broad variety of concepts.
An epistemological disposition influences which epistemological obstacles to a concept a given
student will encounter. Because these two mechanisms have varying strengths and breadths, I
propose a two-dimensional spectrum of conception types, found in Figure 4. It is worth
conjecturing a bit about other kinds of mechanisms that might operate epistemologically, but at
different strengths and breadths.
50
Figure 4—Types of epistemological mechanism based on strength and breadth
Breadth of domain
______________________________________
Narrow
Weaker
Broad
Concept definition
or notation
Epistemological
disposition
Epistemological
obstacle
Paradigm
Strength of
effect
(necessity)
Stronger
In the lower right part of the figure, one can find paradigms (Kuhn, 1962). These are conceptions
that appear to operate strongly, and on a broad enough domain to be often termed “worldviews.”
Paradigms, or paradigm-like conceptions, have been shown to form a strong constraint on student
and historical thinking alike (Sierpinska, 1992; Posner, et al., 1982). In the opposite corner, we find
notations and definitions, which are necessarily narrow in domain and specific to particular
concepts. They also appear to operate rather weakly, suggesting certain conceptions and operations
(such as the chain rule as a process of “canceling fractions” in dy/dx × dx/dt), but are often
overridden by particular pedagogical experiences, or by epistemological obstacles (Tall & Vinner,
1981). All four of these categories of conception show parallels between students and history.
Furthermore, all four of these categories are epistemological rather than didactical in nature, in the
sense that they are all expressions of something at work that is part of the knowledge itself.
Pedagogical Implications
The fact that these different epistemological dispositions are at work in the development of calculus,
not just by Newton and Leibniz, but by their successors, gives us some insight about why the subject
is challenging to students. Throughout the 18th Century, continental mathematicians used the
calculus tools and conceptions that Leibniz developed, continuing to appeal to an epistemological
disposition toward the static and algebraic. Mathematicians such as D’Alembert, Euler, and D.
Bernoulli debated about what kinds of algebraic expressions were functions, and what kinds of
algebraic criteria these must satisfy in order to be considered continuous (Kleiner, 1989). The
documents produced during this time became generally more and more analytic, and by the end of
the century Lagrange, in an effort to improve upon the rigor of calculus, wrote entirely using
formulas and never appealing to geometric intuition (Grabiner, 1996). On the other hand, British
mathematicians built on Newton’s approach to calculus, with mathematicians such as Taylor and
MacLaurin using powerful geometric techniques to develop many of the same ideas as their
continental counterparts. Although these two different traditions influenced each other during the
century, it was really not until the 1800s that an awkward but rigorous hybrid developed to shore up
the foundations of calculus. Current calculus textbooks reflect this dual history of the subject. The
notations they use for derivatives and integrals (e.g. dx, ∫) are almost, but not quite, entirely Leibniz’,
notations that strongly suggest algebraic operations and the intuition of infinitesimal quantities. Yet
they go to great lengths to avoid discussing these concepts in terms of the infinitesimals for which
51
the notations were invented, favoring instead a rigorous treatment of the limit—a concept grounded
in and supported by an epistemological disposition toward smooth motion. Today’s students are
learning concepts, and using words and notations, grounded in different metaphors and supported
by different epistemological dispositions; of course they are confused by this casual amalgam of
approaches.
This study suggests that a more deliberate and historically informed approach would help students
establish consistent and powerful conceptions. A more appropriate instructional approach would
begin building a set of metaphorically reinforcing calculus conceptions using one epistemological
disposition. The static formulaic disposition is probably best because it supports the Leibnizian
notation used in calculus. An approach of this type would follow Leibniz’ injunction to think of a
curve as a polygon with infinitely small sides. This intuition supports the informal use of
infinitesimals to determine derivatives (the slope of one of the curve’s infinitesimal sides) and
integrals (the sum of infinitely many such polygons with infinitesimal widths). Working inductively
in this way will give rise to a non-rigorous heuristic algebra of operations on infinitesimals, and
establishing a set of algebraic rules and procedures for integration and differentiation. With
infinitesimals as the foundation, the standard calculus notations are natural and suggestive of
properties and rules such as the chain rule, integration by parts, and substitution. They also provide
clear intuition about common applications, such as related rates and volumes of rotation. This
calculus of algebraic operations can also be used to work with Taylor’s series, discussing
approximations of curves and initially ignoring issues of convergence. The fundamental theorem
should be addressed by focusing on the graphical and algebraic relationships between graphs, their
derivatives, and their anti-derivatives. These relationships should be informed by the relationships
between sums and differences of numerical sequences, the place from which Leibniz drew much of
his calculus intuition (Berlinski, 1995). Studying this relationship carefully supports the input/output
conception of function, which provides a metaphor for understanding derivatives and antiderivatives as operators on functions.
At this point, the course should be problematized. What assumptions are we making about curves
in order for these relationships to hold, and what other kinds of curves exist? What are continuity
and differentiability, and when are they important? Do infinite series and areas always behave like
finite ones, or do we have to treat them more carefully? Before students construct formal
conceptions of limit, function, continuity, and the real number line, they must have some indication
of why their intuitive models (epistemological obstacles) for these concepts are limited and
problematic. Ideas about activities that would challenge student conceptions might be drawn from
the activities that challenged such conceptions in the history of mathematics, such as the “crisis” of
convergence and continuity in 1807 brought on by a paper of Fourier (Bressoud, 1994).
Limitations and Areas for Further Study
This study raises many questions, some of which are due to the particular methods and analysis
techniques used in the study, others suggesting avenues for further research.
My decision to introduce the idea of epistemological disposition was based on the meaning of the
data found in the clusters of conceptions, and therefore depends on this clustering being reliable.
The reliability of the clusters in turn strongly depends on the accuracy of the coding of the data.
This is especially the case for the conceptions that were displayed by small numbers of students (1217); a coding error for one of these students’ questionnaires could change the data significantly.
52
Although the inter-rater reliability check showed 87% agreement, it is still possible that a key
student’s paper was miscoded, or that the information gathered on the questionnaire was insufficient
to account for the student’s particular conceptual understanding.
Determining how students think based on their questionnaire answers is significantly more difficult
than simply determining what students can or cannot do. Several questionnaire items have ended up
providing different kinds of information than I had originally planned, these differences appearing
sometimes in the interviews and sometimes in the questionnaire responses. My ability to
appropriately understand and interpret these answers is based on my understanding of the concepts
and the literature, which is by no means complete. One way to address these concerns would be to
have multiple extra reviewers examine the coding, rather than just the one. Even though it would
have been impractical in my case, another way to increase the certainty of the results would be to
conduct this study with many more students. This would reduce the effects of any inaccuratelycoded data, as well as reducing the effects of sampling.
One question for further study is related to the findings about student thinking of infinitesimals.
The fact that the infinitesimals conception fell into the same cluster of conceptions as the single
formula conceptions of function and continuity indicates an epistemological disposition shared by
students and Leibniz alike. But how similar are these students’ conceptions of infinitesimals to the
conception held by Leibniz? We know that Leibniz used his intuition of infinitesimals in order to
build the ideas of calculus—can these students do this as well? Does their belief in infinitesimals
actually influence the way they understand derivatives and integrals, or how they conceptualize
Leibnizian notation?
A final area for further study is to determine whether this idea of epistemological disposition (or
more generally the spectrum in Figure 4) is a useful way of characterizing conceptual mechanisms
that are epistemological in nature. Is the new idea of epistemological dispositions actually useful, in
the sense that there are other examples of such dispositions than the two that I found? Or are my
findings particular to calculus precisely because it was developed by two different approaches?
Conclusion
The goal of this study was to explore student conceptions of the foundational elements of calculus
and how these conceptions related to historical conceptions. The idea was not just to see if
particular historical conceptions appeared in student thinking, but to see if the connections among
these conceptions appeared in student thinking as well. This marks a new approach to research
about student thinking, a horizontal rather than vertical approach. We cannot fully understand a
person’s thinking by just examining which conceptions come before other conceptions; we must
also study which conceptions are held at the same time as other conceptions. A person’s knowledge
should be understood as a network spanning multiple topics, rather than just having one isolated
strand for each topic.
By studying the relationships between conceptions, this study reveals these connections to be at least
partially due to epistemological dispositions, which support the establishment of particular related
conceptions. Since these epistemological dispositions are conceptual preferences that affect thinking
about numerous different topics, this explains some of the parallels between student thinking and
historical thinking—we are bringing the same conceptual approaches to bear that our predecessors
did.
53
By noticing the existence of these epistemological dispositions, this study helps to create a
framework for which factors that influence mathematical knowledge are part of the inherent
structure of the knowledge itself and which factors are due to other influences, such as classroom
practice. Among the influences that are part of calculus’ inherent structure are epistemological
obstacles on the one hand, acting strongly but narrowly in domain, and epistemological dispositions
on the other, acting weakly but across a broad domain. Hopefully such a framework will help us
determine the best way to incorporate the history of mathematics into our teaching.
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56
Appendix
Calculus Conception Questionnaire
[The spacing of the items is condensed in comparison to the original.]





Name:
E-mail:
What are the two most recent math classes that you have taken? Where did you take these
classes?
What grades did you receive in those two classes?
Are you willing to be interviewed in the future about your answers on this questionnaire?
I am using this study to better understand how you think. So please do not
worry about trying to get the “right” answer on every question. Answer according to
the way that you understand the concepts that are being discussed.
If you don’t understand what a question is asking, ask me about it or write “I
don’t understand.” Thank you again for your participation!
-Rob Ely
Wisconsin Center for Educational Research
1. Consider the following graphs:
i. List all of the graphs, if any, that represent functions.
ii. List all of the graphs, if any, that are continuous.
2. What is between 0.99999… [the nines repeat] and 1?
(a)
(b)
(c)
(d)
Nothing, because 0.99999… = 1.
An infinitely small distance, because 0.99999… is just less than 1.
You can’t really answer because 0.99999… keeps going forever and never finishes.
[If none of the above answers adequately reflects what you think, mark (d) and write your
own answer.]
57
3. Is there a function whose values are all equal to each other? If so, give an example. If not,
briefly explain why not.
4. Suppose you put two tennis balls numbered 1 and 2 in Bin A and then move ball 1 to Bin B.
Then you put balls 3 and 4 into Bin A and move ball 2 to Bin B. Then you put balls 5 and 6 into
Bin A and move ball 3 to Bin B. And so on infinitely. How many balls are in Bin A after you are
done? Why?
5. For each of the following, circle whether y is a function or not a function of x (on the domain for
which it is defined). Also circle if it is continuous or not continuous.
6. True or false:
T
F
It is possible to choose two different points on the real number line that are touching
one another.
T
F
It is possible to choose two different points on the real number line that are infinitely
close to one another.
58
7. For each of the following, circle whether y is a function or not a function of x (on the domain for
which it is defined). Also circle if it is continuous or not continuous.
8. What type of thing is in the box below?
(a)
(b)
(c)
(d)
a function
a number
a process
an equation
59
9. Consider the following procedure: for each x-value, let r(x) be a random number between 1 and
10. Is r(x) a function?
10. How many functions can be created from the set {1, 2, 3, 4, 5, …} to the set {0, 1}? Why?
11. Does the lim f ( x) exist? Why or why not?
x 
12. One day your friend Zeno says to you, “An arrow shot at a target will never reach the target.
Because in order to reach the target, it must first go halfway to the target. And in order to reach the
target from this point, it must first go half of the remaining distance to the target. And in order to
reach the target from here, it must first go half the distance from this point to the target. And so on
forever!”
What do you reply?
a)
“Technically the arrow will never reach the target, just like ½ + ¼ + 1/8 + 1/16 … will
never reach 1. But since the arrow gets so close to the target, we just say that it hits the target.”
b)
“This is sort of a paradox. According to math, the arrow will never reach the target. But in
real life, of course the arrow hits the target. Sometimes math just fails to describe the real world.”
c)
“Of course the arrow will hit the target. Each ‘step’ of the process you described takes half
as long as the previous step. So the time intervals add up to one unit of time, just like the distance
intervals add up to the whole distance—because ½ + ¼ + 1/8 + 1/16 + … = 1.”
d)
[If none of these responses adequately reflects your thinking, mark (d) and write your own
response.]
60
13. Rate each statement according to how well it describes how you think about the real number
line. Also put a checkmark next to the statement that reflects your thinking the best.
If none of these statements adequately describes how you think about the real number line, you may
write your own description at the bottom of the page.
Strongly
disagree
(or don’t
understand)
a)
The real number line is a set of points all in a
row. It’s sort of like a string of pearls that are all
touching. There is no space between one number and
the one right next to it.
b)
The real number line is just a set of numbers.
It is just an illusion that these numbers comprise a
smooth continuous line. You can never find the
“next” number on the line.
c)
The real number line is not itself made up of
numbers or points. Rather it a smooth continuous
background on which numbers or points can be
situated, sort of like a chalkboard is a smooth
background on which points or curves can be drawn.
Strongly
agree
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
14. Suppose lim f ( x ) = -1.
x 2
Rate each statement according to how well it describes how you think about the limit of a function.
Also put a checkmark next to the statement that reflects your thinking the best.
If none of these statements adequately describes how you think about limits, you may write your
own description at the bottom of the page.
Strongly
disagree
(or don’t
understand)
a)
The best way to picture a limit of a function is
in terms of movement along the graph. Here I picture
x moving closer and closer to 2 while the point on the
graph moves along the graph closer and closer to -1.
1
2
Strongly
agree
3
4
5
6
61
b)
The best way to picture a limit of a function is
in terms of a movement that is bounded. You can
keep moving toward the limit of -1 but you never
actually get to it, even though you may get extremely
close.
c)
The best way to picture a limit of a function is
in terms of closeness. I think of it like this: for x
values that are close to 2, the f(x) values will be close
to -1.
d)
The best way to picture a limit of a function is
in terms of closeness, but I think of it more
algebraically. For any small distance d, I can give you
an interval on the x-axis containing 2, where each xvalue in this interval (except possibly x = 2)
corresponds to an f(x) value within d of -1.
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
15. Rate each statement according to how well it describes how you think about functions. Also put
a checkmark next to the statement that reflects your thinking the best.
If none of these statements adequately describes how you think about functions, you may write your
own description at the bottom of the page.
Strongly
disagree
(or don’t
understand)
a)
A function is any correspondence from Set A
to Set B so that each value in Set A corresponds to at
most one value in Set B.
b)
A function is a single formula with variables
and constants that has only one y value for each x
value.
c)
A function is a dependence relationship
between x and y. If the x-variable changes, the yvariable must change also, and in a way that depends
on the x-variable.
d)
A function is a curve drawn on the coordinate
plane that passes the vertical line test. A good way to
think about it is as a path traced out by a moving
particle.
e)
A function is an input/output process. It
inputs a value of x, processes this value according to
some rule, and then produces a value for y.
Strongly
agree
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
62
16. Rate each statement according to how well it describes how you think about continuous
functions. Also put a checkmark next to the statement that reflects your thinking the best.
If none of these statements adequately describes how you think about continuity, you may write
your own description at the bottom of the page.
Strongly
disagree
(or don’t
understand)
a)
A function y=f(x) is continuous provided that
it describes a smooth motion with no sharp corners or
breaks.
b)
A function y=f(x) is continuous provided that
you can draw its graph without picking your pencil off
the paper.
c)
A function y=f(x) is continuous provided that
it is given by one formula or rule.
Strongly
agree
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
17. Which of the following, if true, is enough to guarantee that lim f ( x ) = L.
x a
(a)
(b)
(c)
(d)
For every b, there is a positive c so that if |x – a| < b, then |f(x) – L| < c.
There is a positive b and a positive c so that |x – a| < c and |f(x) – L| < b.
For every positive b, there is a c so that |f(x) – L| < b whenever |x – a| < c.
There is a b, so that for every positive c, |f(x) – L| < b as long as |x – a| < c.
How certain are you?
Just guessing
1 2 3
4
5
6
7
8
Very certain
9 10