Nonstandard Student Conceptions about Infinitesimal and Infinite Numbers
Robert Ely
Abstract
This is the second and smaller section of my dissertation, a case study of an undergraduate calculus student’s
alternate conceptions of the real number line. Interviews with the student reveal robust conceptions of the
real number line that include infinitesimal and infinite quantities and distances. Similarities between these
conceptions and those of G. W. Leibniz are discussed, illuminated by the formalization of infinitesimals in
Robinson’s nonstandard analysis. This formalization suggests that these conceptions form a framework for
the student that is not only cognitively consistent and powerful, but is also provably mathematically
consistent and powerful. This supports existing literature about the versatility and power of student alternate
conceptions, and provokes interesting questions about the relationship between the cognitive mechanisms
that govern a learner’s construction of consistent conceptions and the rules of mathematical proof that
govern the public construction of consistent mathematics.
1
Nonstandard Student Conceptions about Infinitesimal and Infinite Numbers
Researchers of student misconceptions in mathematics education have produced a great deal of useful
information for helping teachers understand their students’ thinking. These researchers’ beliefs about student
conceptions have an important effect on the types of misconceptions they are able to identify. If researchers
treat a misconception as a lack of knowledge that needs to be filled, or as an incorrect idea that needs to be
eradicated, then they may miss some of the important aspects of how that conception functions within the
student’s understanding. On the other hand, if researchers treat a misconception as a legitimate conception
which has a structure alternate to the standard one, then they are poised to better understand how this
alternate conception operates meaningfully for the student. This paper offers an extreme example of how
treating a student’s misconceptions as alternate conceptions can allow the researcher to see a meaningful
structure to the student’s conceptions that may have otherwise been overlooked, a structure reflected in a
powerful, consistent, and historically-prominent mathematical structure.
Theoretical Background
Over the course of the last 40 years, mathematics education researchers have accumulated a large body of
research about student misconceptions. During this time their outlook on misconceptions has undergone
two broad changes. The first general transition was to stop treating a misconception as a complete lack of
knowledge and to start treating it as a piece of knowledge, albeit an incorrect one (Leinhardt, et al., 1990).
Rather than viewing learners as empty vessels waiting to be filled, this perspective assumed that learners often
approach a subject with misconceptions arising from their experiences. Learning occurs when these
misconceptions are replaced by correct conceptions. This perspective therefore suggests that it is important
for researchers to learn more about these student conceptions, particularly what kinds of environmental and
pedagogical factors create or reinforce them. Many studies using this approach provided excellent
information not only about what misconceptions often occur, but also about what likely caused them, such as
instructional influences and pedagogy (e.g., Schwarzenberger & Tall, 1978), or overgeneralization and
inappropriate transfer (e.g., Matz, 1982).
2
The second important shift in thinking about misconceptions was to stop treating learning as a process of
replacing misconceptions with correct conceptions, but rather as a process of building knowledge from prior
understandings. This difference emerged from a general constructivist epistemology, which understands
prior knowledge to be always integrated within the new knowledge (Piaget & Garcia, 1989/1983). The crux
of the issue is that from a constructivist perspective, a learner only has cause to reconstruct her knowledge
when she experiences a perturbation, an experience or conception that does not fit with her current schema
(von Glasersfeld, 1995). Therefore, the intent of research is less about trying to eliminate causes of
misconceptions, but rather to understand the nature of these misconceptions, in order to give learners tasks
that induce perturbations leading to the restructuring of knowledge. While before the focus had been on
replacing knowledge, now the focus is on refining and reorganizing it (Smith, diSessa, & Roschelle, 1993).
Rather than describing the misconception from the expert’s point of view, now it is crucial to understand the
conception from the learner’s point of view (Lobato & Siebert, 2004). In fact, with this approach some
researchers use the term alternate conception rather than “misconception,” because there is no a priori reason to
consider a given conception to be wrong. In particular, from the learner’s perspective there is no reason to
think that a conception is wrong until it is perturbed. Rather than describing the conception as right or
wrong, it makes more sense to consider other aspects of the conception, such as its domain of viability.
The term “alternate conception” acknowledges that a misconception may be very sophisticated and contain
elements that can be built into powerful conceptions. Theoretically it allows us to look at a conception
freshly in terms of its various qualities, affordances, and limitations, rather than judging the conception as
incorrect. Nonetheless, a teacher still has classroom goals, target conceptions that he wants the students to
ultimately construct. In order to believe in these target conceptions, it seems that he must acknowledge that
alternate conceptions are by their very nature less desirable in some way. Even the mathematics teacher who
takes a strongly learner-oriented perspective must still recognize these alternate conceptions to be
“misconceptions,” because they are less generalizable, powerful, and flexible than the target conceptions he
wishes the students to construct. After all, in mathematics we can actually prove when one idea is correct and
another one is not. If we can prove that a particular conception is mathematically incorrect, at the end of the
day we must acknowledge that it really is a misconception from a mathematical perspective, even if we know
how it arose and how it functions with some local viability within a student’s constructed knowledge. Even
with a strongly constructivist approach, the term “alternate conception” begins to just sound like a
euphemism.
However, what if it were possible to show that students are able to display conceptions that are legitimately
alternate, and not mathematically incorrect? In other words, what if there were examples of student
conceptions that are completely different from the standard “target” conceptions, but yet it can be
mathematically proven that they are correct, coherent, and equally powerful? Such conceptions would fully
deserve the term “alternate conceptions” rather than “misconceptions,” for there is no mathematical or
cognitive reason to favor the standard conceptions over these alternate ones.
The existence of such legitimately alternate conceptions would have several profound implications for student
learning. First, such student conceptions would likely be extremely resilient and resistant to instruction.
3
Since these alternate mathematical systems are powerful, coherent, and self-consistent, it would likely be
extremely difficult to produce experiences for the student that cause a perturbation or cognitive conflict
leading to the reconstruction of the conceptions. Second, students have access to deep mathematical ideas
that are not explicitly taught to them, or are even incommensurable with the ones that are taught to them.
Third, there may be a strong relationship between the rules for mathematical consistency employed in the
greater mathematical community and the rules for cognitive consistency that govern a student’s construction
of knowledge.
In this paper I will show the existence of such a set of alternate conceptions held by a student about the real
number line. These alternate conceptions are also important pieces of a nonstandard model of analysis
developed by 20th-Century logicians. What this means is that this student’s conceptions support the
development of a model that is proven to be as mathematically correct, consistent, and powerful as the
substantively different version found in standard mathematics. As such, this paper is an existence proof for
legitimately alternate student conceptions. In order to make this case, I must first show that these
conceptions are important pieces of a provably mathematically-consistent system; this requires an overview of
the mathematics involved in this nonstandard model of real analysis, commonly studied in the branch of logic
known as model theory. This model will be described in light of the original conceptions of G. W. Leibniz
that inspired its creation.
Historical and Mathematical Background
Throughout the 1670s and 1680s, Gottfried Wilhelm Leibniz developed the infinitesimal calculus. His system
of calculus was largely detailed in personal correspondences, and its first complete exposition was found in
L’Hospital’s 1696 calculus textbook, Analyse des Infiniments Petits, pour l’Inteligence des Lignes Courbes.1 Leibniz’
system is fundamentally based on infinitesimal quantities, positive quantities which are smaller than any finite
number. Leibniz accomplished the basic operations of calculus by partitioning a curved line into an infinite
number of straight “sides” that are infinitesimal in length. A derivative is then the slope of one of these
straight sides, and an integral is an infinite sum of infinitesimal polygonal areas.
In order to calculate these derivatives and integrals, it was crucial for Leibniz to develop an algebra of
operations on infinitesimal, finite, and infinite numbers. Leibniz’ first published treatment in which he
employed this algebra appeared in 1684 (tr. Struik, 1969). In order for his calculus to work, Leibniz posited
that there exist infinitely many infinitesimal numbers, a belief opposed by a number of mathematicians of the
time who believed that there exist no infinitesimals or that there is only one (Mancosu, 1996). For Leibniz,
1
Mathematics historian Fred Rickey translates this as Analysis of the Little-Bitty-Guys for the Study of Curved
Lines.
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the continuum is infinitely divisible, containing no atomic unit. For example, if one raises an infinitesimal to
successively larger powers it will become smaller and smaller. In every neighborhood of a given number,
there is a microcosmic world of numbers that looks like the larger continuum. “Since the continuum is
divisible to infinity, any atom will be of infinite kinds, like a sort of world, and there will be worlds within
worlds to infinity” (Leibniz [1663-72], 1966; p. 241). Note that Leibniz did believe in fundamental indivisible
quantities called monads, but these entities are spiritual in nature, not physical.2
Leibniz’ algebra of operations on infinitesimal numbers included rules such as i×f is infinitesimal, f÷I is
infinitesimal, (f1+i1)/(f2+i2)  f1/f2, and f+i  f (here i indicates an infinitesimal number, f is a finite number,
and I is an infinite number). Although there are at least a dozen such rules and heuristics for working with
infinitesimals, the most important for calculus are the ones like the last two specified above which show how
to simplify difference quotients.
This method of calculus was employed and extended by most of the great continental mathematicians of the
18th Century, including the Bernoullis, L. Euler, and A. Legendre. During this time there were vehement
attacks on the idea of an infinitesimal quantity, notably by Bishop Berkeley in 1734. What was an
infinitesimal? Could one be produced and examined in the real world? Leibniz at times discussed an
infinitesimal as the final term in a sequence of numbers approaching zero, but this did not particularly clarify
matters. Historian H. Bos (1974) suggests that this vagueness was a crucial element which led to the rapid
development of calculus, but which also ultimately caused the system to be rejected on mathematical and
philosophical grounds. Even though the idea was vague, mathematicians still happily worked with
infinitesimals for two reasons: they were intuitive and they kept producing important results. An explosion of
mathematical activity made use of the new infinitesimal calculus in the 18th Century, producing a wealth of
discoveries in mechanics, calculus of variations, probability theory, astronomy, and more.
But in the beginning of the 1800s mathematicians began to encounter some unintuitive and even
contradictory results arising from a cavalier treatment of convergence, many of which surfaced in a discussion
of J. Fourier’s 1807 paper on trigonometric series (Bressoud, 1994). The informal reliance on the intuition of
infinitesimal quantities did nothing to resolve these debates, so mathematicians such as A. Cauchy and B.
Bolzano worked to develop a foundational system of limits that could be used to ground the behavior of
infinite series and functions. This culminated in the ε-δ definition of limit, ultimately formalized by K.
Weierstrass in the 1860s, marking the complete disappearance of infinitesimals from the foundations of real
analysis.
It was curious that infinitesimals, which supported a century of robust mathematical findings, could not be
formalized in such a way as to create a rigorous system of calculus. It seemed strange that these entities could
be used so intuitively to discover calculus-based results, and yet would have to be discarded when it came
time to check one’s proofs. The answer is that the development of mathematical logic was not advanced
2
For a more complete treatment of monads vs. infinitesimals in Leibniz’ metaphysics, see Ross, 1986.
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enough to allow for a rigorous treatment of infinitesimals until the work of Abraham Robinson in the 1960s.
In order to make sense of this work, some results and definitions from model theory must be described.
The logicians of the early 1900s followed D. Hilbert’s program to establish mathematical proof as a set of
rigorous formal operations that do not result in paradoxes or contradictions.3 Mathematicians such as G.
Frege and B. Russell developed a formal language in which mathematical statements could be written and
studied abstractly, and mathematical proof could be rigorously reduced to a set of syntactical rules governing
these mathematical statements. The goal was to define and study proof abstractly, and then to see if these
abstract proof rules actual governed what was true or not in the real mathematical worlds (models) that these
statements apply to.
Therefore, in order to do mathematical logic, we need three things. First, we must have a formal language
and some abstract formulas (statements) written in this formal language. The idea is that this language is
totally free from ambiguity. However, it is totally devoid of meaning as well. We want to see if these
formulas relate to each other in “real life” the way they do syntactically. In order for a formula to have
meaning, there must be some real mathematical world or model in which the formula applies. So the second
thing we need is a model, namely a set of actual mathematical objects and operations that this formula could
apply to. The final thing we need is an interpretation, something that tells us what each symbol in our
abstract mathematical language “means” in the particular mathematical model. Generally this interpretation is
presented with the model. Once a formula is interpreted in a particular model, it becomes a meaningful
mathematical statement whose truth-hood can be determined.
For instance, consider the formula vo R(vo , vo ) . This formula is just simply a string of symbols that has no
meaning on its own. However, suppose we take as our mathematical model the natural numbers {0, 1, 2, 3,
4, …} with the standard operations of arithmetic {+, -, ×, ÷, =}. Now, let us use the interpretation that
vo means “for every natural number n,” and R is the equality relation, that is, R(vo , vo ) means “n = n.”
Then our abstract meaningless formula, interpreted in our model, becomes the statement “For every natural
number n, n equals itself.” This statement clearly has some meaning, and it is also true.
However, suppose we examine the same exact formula, but we take as our model the set of all integers {…, 3, -2, -1, 0, 1, 2, 3, …} with the standard operations of arithmetic and of ordering {+, -, ×, ÷, =, <, >}.
Suppose we use as our interpretation that vo means “for every integer a,” and R is the “less-than” relation,
that is, R(vo , vo ) means “a < a.” Now our formula reads “For every integer a, a is less than a.” Just as in
the previous model, this statement is now meaningful, but in this model it is clearly untrue.
3
This research program was limited by the K. Godel’s development of the incompleteness theorems of 1931.
6
With these three important pieces—formulas in an abstract language, a mathematical model, and an
interpretation of the formulas that make them into meaningful statements in the mathematical model—we
can study the relationships among a collection of mathematical formulas. Consider T to be a set of formulas
in our abstract mathematical language and l to be some other formula in this language. A model of T is some
model where there is an interpretation that makes all of the formulas in T meaningful and true in the model.
Now we could examine if (a) l follows logically as a consequence of the formulas in T, or if (b) l contradicts T,
or if (c) l is independent of T. If (a), then this means that in every mathematical model of T, l is true as well. If
(b), then this means that l is false in every mathematical model of T. If (c), then there are some models of T
where l is true and some models of T where l is false.
A given set of formulas T may have many different mathematical models. Many important findings of the
early 20th-Century logicians result from investigating these alternate mathematical models. Lowenheim and
Skolem found something interesting: there can be two different models which have apparent differences
from the outside, but from inside the models they look the same. In other words, all of the first-order logical
formulas that hold true about the objects in one model also hold true about the objects in the other model.
Yet there are more global statements that can be made about the models themselves that are true about one
of the models but are not true about the other model. In this way one can find substantively different
models, but where all of the same statements are true within each model.
The first such example, created by Skolem in 1920s, was a nonstandard model of arithmetic. The standard
model of arithmetic is the natural numbers N (with the operations of addition and multiplication). The
nonstandard model M developed by Skolem is the same as the standard model N, but with one notable
difference—it contains infinite numbers. In other words, M contains all of the standard counting numbers,
plus it has some numbers that are larger than any of these counting numbers. Within the two models, all
statements that hold in one model hold in the other: 2+2=4, every even number is divisible by 2, every
number has a successor, and so on. This means that both models are equally powerful and consistent, and
both could be used to prove statements in arithmetic. Yet from outside the models we can detect that the
models really are different; the nonstandard model contains infinite numbers. In other words, there is no way
to make a first-order statement that says “n is an infinite number.”
About forty years later, Abraham Robinson developed the second important nonstandard model of a
mathematical system. This was a nonstandard model of analysis, in which Leibniz’ infinitesimal calculus can
be formalized. First of all, by “analysis” I mean the set of formulas that characterize advanced calculus,
statements like “every Cauchy sequence converges” and “every absolutely continuous function is the integral
of its derivative.” The standard model of analysis is the set of real numbers. In order to create the
nonstandard model of analysis, also known as the nonstandard real numbers, Robinson used the same
compactness theorem employed in Skolem’s creation of nonstandard arithmetic. He created a model of
analysis that looks just like the regular real numbers (as far as first-order logic is concerned), but has a slight
difference: it contains infinitely small and infinitely large numbers!
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Not only did Robinson show that there are infinitesimal and infinite numbers in his model, he also formalized
how to understand these numbers in comparison to finite numbers, and how to operate with them. For
example, an infinitesimal number can be viewed as an infinite sequence of numbers that converges to 0
(actually an equivalence class of such sequences). Likewise an infinite number is an infinite sequence that
approaches infinity. The same set of operations concerning combinations of finite, infinite, and infinitesimal
numbers that worked in Leibniz’ informal system also held in the nonstandard real numbers. In tribute to
Leibniz, Robinson called an infinitesimal neighborhood of points a monad, although Leibniz had used the
term only for spiritual particles.
The most important of Robinson’s results is that with this formalization of infinitesimal and infinite
quantities, every first-order formula that holds true in the standard real numbers also holds true in the
nonstandard real numbers and vice versa. This is called the “transfer principle,” and it is has allowed for
some important mathematical discoveries. In particular, if a theorem is hard to prove using standard analysis,
it might be easy to prove using nonstandard analysis. Nonetheless, we know that the same theorem is true in
both models, so it does not matter which model was used to prove the result. For example, the proof of the
intermediate value theorem, which is surprisingly complicated in standard analysis, is almost trivial using
nonstandard analysis. Another example is the work of Albeverio et al. (1986), using nonstandard analysis to
develop new results about stochastic processes.
The nonstandard real numbers provide a satisfying vindication of Leibniz’ infinitesimal system. Robinson
proved the system to be as consistent and powerful as the standard version of analysis. This means that
calculus can be done using infinitesimals with a clean conscience, and in fact several calculus books have been
written that teach the subject using this approach (Keisler, 1986; Henle & Kleinberg, 1979).
In this section I have shown how 20th-Century logicians developed a mathematical basis for investigating
alternate models of important mathematical systems. In particular, mathematicians used the model of
nonstandard analysis intuitively and informally for over a hundred years, establishing powerful and important
calculus-based results. This alternate model of the nonstandard real numbers marks a powerful, coherent,
and mathematically correct mode of thought, yet one that is substantially different from the standard real
numbers taught in today’s classrooms. In the next sections, I will show how this alternate model appears in a
student’s thinking in spite of the goals of her instructors.
Methods
Data Collection
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As part of my larger dissertation study, 233 university calculus students completed a questionnaire on the first
day of their Calculus I or Calculus II class. Their responses were then used to catalogue their conceptions
about various calculus concepts. Six of these students, two young men and four young women, participated
in follow-up interviews. The main purpose of these interviews was to clarify their conceptions, in order to
refine the coding scheme for my larger study. Therefore I chose to interview students whose responses on
the original questionnaire were enigmatic or appeared inconsistent in some way. Each interview was about 30
minutes long, and was audio recorded.
Since each student I interviewed had different responses on the questionnaire to clarify, I used different
interview protocols for each interview. As is typical for a semi-structured interview technique, these
protocols were not intended to be rigid, and could be stepped away from freely in order to adapt to and
pursue the participant’ responses (Denzin, 1989). For the sake of simplicity I will describe here only the
interviews with one student, whose pseudonym for the sake of this study is “Sarah,” since she is the student
who most definitively displayed evidence of nonstandard conceptions about the real number line. The other
interviews in this study are described in the earlier paper in this dissertation. At the time of the interviews,
Sarah was taking Calculus II. She was a sophomore in college who had taken AP Calculus in high school,
which gave her credit at the university for Calculus I.
I began Sarah’s interview with Item 6 from the original questionnaire (see Figure 1), because the student
questionnaire responses had been surprising for this item, and because it was a good starting point for
investigating conceptions about the real number line.
Figure 1—Item 6 on the Calculus Conception Questionnaire
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.
The following are some of the follow-up questions I used in the interview:
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


Can you provide an example of two different points that are infinitely close to one another, but are
not touching?
What is between those two points?
Can you find a point that is infinitely close to both of those points?
Sarah’s responses to these questions surprised me and my prepared interview protocol was not detailed
enough to pursue them. I asked her to describe more about these infinitely small numbers she was using, and
how she would represent them. I then moved on to questions about the other questionnaire items, primarily
to clarify what she believed about whether certain graphs were functions or not. Near the end of the
interview I asked a few more questions about the real number line, following up on a function question that
used the word “irrational”:




In every space on the real number line, do you think there is an irrational number?
You don’t think irrational numbers can touch on the real number line…why?
Can you find two irrational numbers that don’t have any other rational numbers between them?
Are there any rational numbers in the space between 0.999… and 1? (She had already identified that
there was an infinitely small space between these two numbers.)
Because I found Sarah’s ideas about the real number line intriguing, and I wished to pursue them, I
performed another interview with her a week later. I prepared another interview protocol which started with
the same questions as above, but also included questions such as:



What would you get if you took this (infinitely small) number and squared it?
What would you get if you took 1 over this number?
If you took 1 over this other (infinitely small) number, how would that compare to 1 over this
one?
Data Analysis
This paper is a case study of one student’s conceptions about the real number line. Since its purpose is to
provide insight into the larger issue of alternate mathematical systems found in student thinking, rather than
simply being focused on the case itself, it is an “instrumental” rather than “intrinsic” case study (Stake, 2000).
As such, any account of Sarah’s thinking must be understood with respect to how my own concerns and
conceptions inform the interview and my analysis of the interview. For example, although I was not
expecting conceptions relating to nonstandard analysis to emerge through the course of the interview, I was
ready to interpret Sarah’s emergent notation as a sign of such thinking based on the method of construction
of the nonstandard real numbers in a class I had taken two years previously. My understandings also
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provided a lens for analyzing the data, because I constantly had reference to both Leibniz’ and Robinson’s
conceptions to create categories of thought, and to compare and contrast within these categories.
My categories for coding the interview data were first developed through a process of open coding (Strauss &
Corbin, 1990). First I read Sarah’s utterances and classified them with respect to the particular mathematical
processes and objects they were in reference to. Then I formulated categories of Sarah’s beliefs, practices,
notations, and terminology for these mathematical processes and objects. These emerged in light of my own
mathematical and historical conceptions. In particular, most of these categories of beliefs, practices,
notations, and terminology are determined as historical ways in which thinking in terms of infinitesimals and
infinite numbers differs from thinking about finite numbers. Once these categories were able to make a
meaningful account for all of Sarah’s relevant utterances, I considered the framework to have reached
saturation. I then collapsed a few categories in which the information was too sparse to form conclusions.
The final categories of thought is highlighted in the results section. For example, one category is how an
infinitesimal number is represented; another is the belief about whether the number line contains atomic
(indivisible) elements.
Results
In this section, I classify Sarah’s responses by the various beliefs that she displays about the real number line,
illustrating these with pieces from the interviews. In these transcript pieces, “I” indicates the interviewer and
“S” indicates Sarah.
There exist numbers that are infinitely close together. Sarah answered “true” to Item 6b, that it is possible to find two
different real numbers that are infinitely close to one another. When I asked her for an example of two such
numbers, she replied “3.999999 repeating forever” and “4.” In fact, she believes that there are more numbers
that are infinitely close together as well:
I:
Okay. And can you find another number that's infinitely close to both of those numbers? Is
it possible to find another number that's infinitely close to both of those two?
S:
Yes because the repeating is you don't know how long it's repeating. 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.
I:
Okay. So how would you write down like the two different numbers that are like that? Or if
you can, I don't know.
S:
How would you write...?
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I:
How would you explain it? Like...
S:
You mean like... Well, you could just say like 3.99 whatever repeating forever to
infinity, and 4. And then in between that is like 3.999 to infinity plus [pause] an
infinitely small number.
I:
Plus an infinitely small number.
Sarah believes that the numbers in question are distinct entities, and that these entities are infinitely close
together, closer together than any finite representation can describe. It is important to recognize that
according to a standard model of analysis, her beliefs are quite incorrect; 3.999… is equal to 4. Most of
Sarah’s beliefs that follow are therefore completely false according to the standard mathematical system. Her
last utterance in this segment explicitly supports that she believes the following to be true as well:
There exist infinitesimal numbers and infinitesimal distances. When I asked Sarah what is between 0.999…[repeating]
and 1, this is when she began to develop the notation for expressing infinitesimal numbers and distances:
S:
If there was a way to express something like... I mean, this is not real at all... but like, zero
repeating forever and then 1, then I would say that. Like an infinitely minuscule...
hmm...
…
I:
Is there... are there any numbers between .9 repeating and 1?
S:
Um... I mean, this isn't really a number. [she writes 0.000…1] This isn't real so.
I:
Point 0 repeating with a 1?
S:
Yeah. But um... I don't know. I mean... There is numbers because no matter how small
you get it there's still going to be some kind of space and even in that tiny infinitely
small space you can still cut that into infinity too and put numbers in there. So I
would say yes but I don't know how to express that.
I:
So how to express a number that's in there.
S:
Yeah.
Here, early in the first interview, she develops an explicit notation as a way of expressing these infinitesimal
distances and numbers. At the beginning of the second interview, Sarah recaps her belief that between
0.999… and 1 there exists “an infinitely small space.” When I asked again if there are any numbers between
these two numbers, she said “infinitely small numbers.”
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Infinitesimal numbers can be written as infinite decimal expansions with extra decimals places at the end. Sarah becomes
more and more comfortable with this notation that she is inventing for infinitesimal numbers, writing
numbers like “3.999…with another nine after it” (which I will represent as 3.999…9) and “0.000…01.” She
uses this notation repeatedly during both interviews.
Infinitesimal numbers can be ordered. Sarah believes that infinitesimal numbers can be compared and ordered just
like regular numbers can:
S:
Oh, I would say that this number [0.000…1] would be infinitely close to zero.
I:
Oh, okay.
S:
So would this number [0.000…01].
I:
And so would that number. Now, how would they relate to each other compared to zero?
Like...like...
S:
Like oh....
I:
Where would you put both of these two numbers?
S:
This one would just be a little bigger than that one.
Considering infinitesimal numbers to be larger or smaller than other infinitesimal numbers is not a trivial
observation. For instance, the early 18th-Century mathematician Niewentijt believed that there is only one
infinitesimal number (Mancosu, 1996). Sarah’s ability to order infinitesimals is similar to her ability to
generate more infinitesimals and to operate with these numbers. Her ability to do this is shown below.
There are infinitely many infinitesimal numbers in a small space. Sarah confirmed not only that a few sporadic
infinitesimals exist in her world, but in fact infinitely many exist, packed into any given infinitely small space:
I:
Okay. Um, so now I ask you, are there, are there other numbers between here and here?
[points to 1 and .9 repeating] Like...
S:
Um, maybe an even smaller one, like...hahaha [writes 0.000…01]
I:
Okay...
S:
If you can do that.
I:
No, that, how many numbers are between here and here?
S:
An infinitely small, an infinite amount of infinitely small numbers.
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Sarah has illustrated her belief that there exist an infinite number of infinitesimal numbers by talking about a
little number line that looks just like the big number line once you zoom in on it enough. This indicates the
following conception:
There are infinitesimal worlds of numbers that look like scaled-down versions of the larger world. In particular, Sarah said
that all of these infinitely small numbers that live in this space “would look exactly like the big number line
except they wouldn't be called the same numbers.” Sarah appears to be metaphorically extending her image
of the full number line to describe what transpires in infinitesimal spaces.
The continuum is infinitely divisible. Sarah believes that the number line can be cut infinitely many times and there
will always be something left to cut. This means that no two numbers on the line are ever touching. For
instance, she says that Item 6a is false, that it is not possible to find two different numbers on the real number
line that are touching one another. When I asked why this is, she replied, “You can keep halving it, like
cutting it in half and cutting it in half. You can get closer and closer and closer but there's always gonna be
like a little space between it.”
Sarah also appears to believe that the continuum contains no atomic units, not even at the infinitesimal level:
I:
What did, what would you get if you took this number and squared it? .0 repeating
infinitely with a one after it.
S:
A smaller number.
I:
A smaller number, okay. Like, how, talk to me more, like, could you write it? Like how
would you, you know what I mean?
S:
I, I don't know how you would write it because it'd be kind of like exactly the same as that.
It would look the same as that, but it would have more zeroes.
I:
But it would have more zeroes?
S:
Mm-hm.
I:
How many more zeroes would it have?
S:
Um, I don't know. But it would be, it would be smaller.
I:
Okay, okay. Yeah. Um, so okay.
S:
Since this is infinitely small, it would be infinitely smaller than that.
I:
It would be infinitely smaller than that?
14
S:
Yeah.
I:
So it's like, so could you zoom in, like, tell me what you mean by infinitely smaller.
S:
Like, I still think, I mean, I think maybe you said this example, like if you start cutting like,
cutting something, you're never going to get it to be nothing. But it's going to be
infinitely smaller than that, like maybe like a miniscule little thing, you can't measure
it but it's not gone.
This indicates that infinitesimal numbers can also be divided into yet smaller numbers for Sarah. But it
suggests something else as well, that infinitesimal numbers can be operated on in order to produce other
infinitesimal numbers. For Sarah these numbers have an algebra of operations that can be used to produce
more infinitesimal (and infinite!) numbers.
Infinitesimal numbers can be operated upon like normal numbers. When asked to perform operations with these
infinitesimal numbers she has created, Sarah reveals that there is an algebra of operations that applies to these
numbers just as consistently as it applies to regular finite numbers:
I:
Um, now, where would this number [0.000…1] squared be?
S:
Closer to the zero, so...
I:
Like how close to the zero?
S:
I, I don't really know. Maybe like infinitely close to the zero.
I:
Okay.
S:
Because you're squaring infinity, so it's going to be, or, you're squaring this. So it's
going to be infinitely more closer.
I:
Oh, so could you zoom in then again infinitely much and then see that?
S:
Yeah, yeah.
I:
Okay.
S:
I mean you could never, but in theory, yeah.
I:
Okay.
S:
Like most of the, like this could never, you could never be able to tell like, calculate it if
you're talking about infinity and then infinity smaller, and then infinity smaller. But
like, if infinity goes on forever, there's gotta be something a little bit smaller. More
infinity.
15
This is important evidence for the systematicity of Sarah’s conceptual structure. She is able to create new
numbers from old ones, implying that by the standard laws of operations on numbers, the new numbers
would have to exist. In fact, not only can infinitesimal numbers be used to generate more infinitesimal
numbers, but they can also be used to generate infinite numbers.
There exist infinitely large numbers. In particular, these infinitely large numbers are reciprocals of infinitesimals.
This means that there are many of these numbers, and that they can be operated on in conjunction with the
infinitesimal numbers.
I:
Okay. Okay, let me ask you a different question. What is one over this number? [points
to 0.000…1]
S:
Oh yeah, this one doesn't, well this is kind of like the one, an infinitely small number, that,
that you, that we talked about before. If this [0.000…1] is smaller and smaller,
then...this [the quotient: 1 divided by 0.000…1] is just going to get bigger and bigger
and bigger and bigger.
…
I:
The quotient is going to get bigger and bigger and bigger. Okay. Um...what about one
over this number again here? [points to 0.000…01]
S:
Bigger infinity than that.
I:
Bigger infinity than that?
S:
Yeah, I know that's really bad, heh.
I:
No, you tell me what you think, you're telling me what you think, not what, not what you're
going to write down on your test next week.
S:
Hahahaha, yeah.
I:
So, a bigger infinity than that.
S:
Yeah.
I:
Um...
S:
I just think of it like, if it's infinity, like yeah, like, what I'm saying, it's kind of like not really
making sense, like I know it doesn't really sound right, but I feel like if there's an
infinity, then there has to be something bigger than the infinity, which is still infinity
or, you know, small infinity.
I:
Mm-hm.
S:
Hahaha.
16
This passage shows not only that infinite numbers exist for Sarah, but there is an entire range of these
numbers. There is not just simply one infinite number bigger than all of the rest, but the infinite numbers
keep going further and further as well.
S:
I think of it like, like, I think of it in a realistic way, like infinity is, is real, but like, the way
our brains can think, there has to be, we're using infinity to express something that's
too big, but like if you actually went through and started like, counting like little tiny
grains of sand, or something, like, eventually you're going to get a number if you
could count them all, but you can't.
I:
Mm-hm.
S:
So like, once you get a number, then the numbers go, then the numbers are...the
same as regular numbers, you know? But they're still, you can never do that, so that's
why it's called infinity.
I:
Okay, okay. Um, so would you say that you can have like, is it, would you say that it's true
that you could have infinity and infinity plus one and infinity plus two?
S:
Mm-hm.
I:
Okay.
S:
But it's still infinity, but you can have that.
Sarah believes that if you could get up to infinity, then the numbers would sort of start over again, just like the
regular numbers. This supports the notion that the number line not only has an infinite scalability downward
in size, but also an infinite scalability upward in size.
Infinitesimals can be rational. Although it is clear that she never thought about this particular issue before, I
asked her if there are any rational numbers between 0.999… and 1. After puzzling for a while, she replied:
S:
I know that there has to be a rational nu--my principles say there has to be a rational
number in there. Because it's a tiny tiny space and there's gotta be a rational nu... like the
tiniest little space... (pause) Because in infinity there's still an infinitely small space, and in
that infinitely small space... I mean, it's an infinitely small space, but it's infinity. So in
infinity there is rational numbers.
This piece of the interview is interesting, because it shows how Sarah resorts to her mathematical “principles”
when she is presented with a question about her model that she has never entertained before. In fact, she has
likely never explicitly entertained most of these questions before, especially given how early in the first
interview she created a notation to answer them. Nonetheless, she is able to appeal to her “mathematical
17
principles” in such a way that they provide her with coherent and consistent mathematical responses. In fact,
we can see how she persists in her principles and beliefs, even when she knows that they are “wrong.”
Sarah recognizes that her conceptions of the real number line are idiosyncratic and “incorrect.” Throughout both of the
interviews, Sarah continually provided disclaimers for her “crazy” thinking. She made statements such as,
“That could be totally wrong, but that’s just the way I think about it.” In one of the above segments, she
apologizes for saying that 1/0.000…01 would be a bigger infinity than 1/0.000…1 by saying, “Yeah, I know
that's really bad, heh.”
Sarah persists with her conceptions despite being shown contrary explanations. Even though Sarah believes that her
conceptions are incorrect, she still maintains them despite being shown the “correct” conceptions. At the
end of the first interview, she said that she had seen a proof in high school that 0.999… actually equals 1, but
she never believed the proof. She asked me why she was supposed to believe that 0.999…=1. I obliged with
a standard informal proof, multiplying both sides of the equation by 9 and subtracting. She still objected,
saying that if 0.999… really equals 1, then why do you even have 0.999…? “The only reason to have .9
repeating is to show that it's not 1!”
Then I tried to explain how there are different representations for the same number. I asked her if she
believed that 0.333… was equal to 1/3. To my surprise she said no, that she never really believed that either.
I:
Okay, the only reason you believe .3 repeating equals 1--or, equals 1/3, is because...
S:
It's what they make you memorize. But I don't know if I hadn't memorized this, I don't know
what...
I:
You don't know if you would have thought that.
S:
Yeah. I don't even know why that is what that is because I don't know how to divide 1 by 3. So...
She clarified in her further statements that she was comfortable with dividing 1 by 2 and getting 0.5, because
she could see that 5/10 is the same as 1/2. This was reasonable, because this was “a nice number,” as
opposed to the infinite repeating decimal expansion for 1/3.
I must point out that the entire second interview took place after I showed her these explanations and proofs.
Clearly they did not make much difference in her thinking. For instance, here is part of the very beginning of
the second interview:
18
I:
Like, the question I asked last time was what is between .9 repeating and one?
S:
Right.
I:
How would you answer that?
S:
Um...an infinitely small space. If, if I were to, well now I know that they equal each
other, I know that that's right, but...
I:
But does that, is that how you think about that?
S:
I think about a little, an infinitely small space.
Beliefs about school mathematics. Sarah’s ability to know the “correct” mathematical results but still not believe or
understand them is part of her view of what doing mathematics in school is all about:
S:
…But like, a lot of the things that I think, like, about math are just rules and
memorizing rules that someone made up, and that's why I think this is important to
have, because what math students do, and this is what I do too--studying for a math test is
just like memorizing the rules, doesn't matter why there's rules there, you just
memorize it so you can use the formula on your test and get an A. You know? But it's
important for kids to, like, think of the concepts of it. That's why this is good for me, see,
because I don't know the concepts…
School mathematics includes memorizing the rules and not worrying about why they exist or what the
concepts behind them are.
Sarah’s beliefs about mathematics support the conclusion that she is maintaining an underlying set of
intuitions that is remaining unaltered in spite of her classroom mathematics. Sarah assured me that she did
not learn her ideas about infinitesimal and infinite numbers from any of her classrooms, but that it was her
own way of making sense of things. The extreme resiliency of these conceptions to contradicting proofs,
explanations, and years of instruction indicates that her system is robust and coherent.
This last point is worth highlighting. Her system is coherent, and this explains why, when she explores her
conceptual structure in these interviews, she never encounters a perturbation along the way that indicates an
inconsistency in her structure. She generates new numbers from old, following a logical system of operations
supported by a sense of mathematical closure. She generates a notation for infinitesimals, which never fails to
express the quantities she is inventing. She extends metaphors for the full number line to the infinitesimal
scale and to the infinite scale. This includes extending the belief that in any space there must be a rational
number, so this must hold for infinitesimal spaces too. All of these are mathematically consistent properties
that she extends into her world of infinitesimal and infinite numbers, and they never break down or lead to a
19
contradiction for her. The mathematical coherence of the structure makes it resistant to change, even in spite
of the contradicting mathematics that she memorizes in order to get an A on the test.
Concluding remarks about Sarah’s real number system. It has been my goal to detail Sarah’s conceptions of
infinitesimals and the real number line in such a way that the reader can see (a) the coherence of her
conceptions, (b) the alternate nature of these conceptions in comparison to the standard real numbers, and (c)
the striking similarities between her conceptions and those of Leibniz and Robinson. A summary of the
similarities between Leibniz’ informal system of calculus, Robinson’s formalization of nonstandard analysis,
and Sarah’s number system is provided in Figure 2. Sarah’s version contains nearly all of the distinguishing
foundational features of Leibniz’ system, and as such is quite similar to Robinson’s as well.
20
Figure 2—Summary of the similarities between Sarah’s system, Leibniz’ system, and the system of nonstandard analysis
Leibniz’ foundational system (c. 1690)
Nonstandard analysis (c. 1970)
Sarah’s system (c. 2005)
There exist infinitesimal, finite, and infinite numbers.
Same. Nonstandard (infinitesimal and
infinite) numbers are generated by logically
“completing” the real numbers.
Same.
There are infinitely many infinitesimal and infinite
numbers. For instance, squaring an infinitesimal
number produces a smaller infinitesimal; squaring an
infinite number produces a larger infinite number. This
can be done infinitely. (Here Leibniz disagrees with
Niewentijt (Mancosu, 1996)) Any number, even an
infinitesimal, can be divided infinitely many times.
Same. More precisely, the nonstandard real
numbers are a field extension of the reals,
and are thus closed under the arithmetical
operations, which gives rise to infinitely
many infinitesimal and infinite numbers.
An infinitesimal is the inverse of an infinite
numbers
Same exactly as Leibniz. One can keep dividing
any length into infinitely smaller and smaller
segments. Squaring an infinitesimal produces a
smaller infinitesimal. The reciprocal of an
infinitesimal is an infinite number, and vice
versa.
An infinitesimal is not strictly defined or represented
(other than as, say, dx); an infinite series has an
infinitesimal term at its end.
An infinitesimal can be precisely
represented by an equivalence class of
sequences of positive rational numbers {an}
that converges to 0.
An infinitesimal can be represented by a decimal
expansion that has digits extending past the
“infinityth” decimal place. For instance,
0.0000…(infinite 0s) with a 1 at the end.
One can find two different numbers infinitely close to
one another. In particular, f + i  f (f is a finite
number, i is an infinitesimal).
Same. Any nonstandard number is
infinitely close to but not necessarily equal
to its standard part (n  s(n)).
Same. For instance, 0.999… is infinitely close to
1, and 0.999…9, which is a different number, is
infinitely close to both.
If one zooms in enough on part of a thing, one will see
that this part looks like a microcosm of the things itself.
There are worlds within worlds. This can only be done
finitely in the real world, but in the mathematical one it
can be done infinitely, or at least it is a “useful fiction”
to pretend that it can. On the other hand, monads are
indivisible and spiritual units, as different from
A monad is defined to be a set of all
numbers infinitely close to one another.
This is sort of a microcosm of the standard
real numbers. (Note that this differs
dramatically from Leibniz’s use of the word
“monad.”)
Same, at least for the number line. Zooming in
infinitely much on the number 1 reveals the
distance between the infinitely close numbers in
the box above this one. Zooming in on one of
the hash-marks on a ruler reveals a miniature
version of the original ruler. Likewise, if one
zooms in infinitely on a point on the number
21
infinitesimals (which are infinitely divisible) as the
limitlessness of God is from mere infinite quantities.
line, one sees a miniature number line, “but they
won’t be called the same numbers.”
Rules: i  f is inf’l; I  f is infinite; (f1+i1)/(f2+i2)  f1/f2; Same.
I/i could be infinite or finite or inf’l; etc. etc.
Same (I didn’t ask about I/i).
I have not found any reference to rationality of
infinitesimal numbers by Leibniz.
Infinitesimals are, in some sense, all rational
numbers.
There are rational and irrationals in every small
space, including infinitesimally small spaces.
Abandoned at the beginning of the 19th Century for
philosophical reasons, in favor of the formal limit.
Established in 1966 (Robinson) as logically
equivalent to the standard real numbers.
She refers to as her own strange way of thinking.
22
The primary difference between Sarah’s system and the other two is that it is unclear whether Sarah is willing
to treat infinitesimals entirely as static objects. Her intuitions put more emphasis on the infinite process of
making or representing the infinitesimals; for instance, she alludes to the 9’s “going forever” in 3.999….
Although Leibniz and Robinson describe infinitesimals in a similar way as the result of an infinite sequence or
series, they also often treat them as static entities, as suggested by notations like i and dx. Another way the
systems may differ is that it is not clear that Sarah had built any of the structure of calculus upon her real
number conceptions, while the other two systems were of course used precisely for this purpose.
Nonetheless, the similarities between the three systems are striking and robust. Without an understanding of
Leibniz’ system of infinitesimals, or of Robinson’s formalization of this system, it would be tempting to
dismiss Sarah’s ideas as being quirky but stubbornly wrong. With this understanding it becomes obvious that
Sarah has actually developed a coherent and self-consistent framework and notation for thinking about
infinitesimal quantities. Not only has this system never yet contradicted itself mathematically for Sarah, but it
has been proven by Robinson that it is possible for her to continue pursuing it indefinitely without it
contradicting itself. In fact, she is working within an alternate mathematical world that has been proven to be
equally powerful as the standard system that has no infinitesimal or infinite numbers.
Discussion
My purpose so far has been to display examples of a student’s legitimately alternate conceptions. In this
section I wish to explore some of the implications of the existence of such alternative conceptions.
In the theoretical framework section at the beginning of the paper I described how even from a constructivist
learner-oriented perspective the term “alternate conception” still generally operates as a euphemism for
“misconception.” Since the teacher brings a set of intended conceptions to the mathematics classroom,
conceptions that are alternate to these are generally less powerful, consistent, generalizable, or viable; they are
really misconceptions. However, Sarah’s conceptions are legitimately alternate conceptions, and are not
misconceptions in any way. Mathematically they are not misconceptions, because they are provably
mathematically correct, consistent, and part of a system that is equally powerful as its standard counterpart.
Cognitively they are not misconceptions, because they have survived intact this long, not experiencing
perturbation from contrary proofs or classroom instruction.
23
The existence of this legitimately alternate set of conceptions is important for four reasons. First of all, the
consistency of this set of conceptions provides us with a different perspective on student thinking about
infinity and infinite processes. Some of the student conceptions about infinity that are commonly labeled by
teachers and researchers as misconceptions may arise from the fact that students are appealing to a different
mathematical model than the one they are ostensibly being taught in school. Take for example the “Tennis
Ball Problem,” which is presented by Weller, et al. (2004):
Suppose you put two tennis balls numbered 1 and 2 in Bin A and then move ball 1 to Bin B, then put
balls 3 and 4 in Bin A and move Ball 2 to Bin B, the put balls 5 and 6 into Bin A and move 3 to Bin
B, and so on without end. How many balls are in Bin A when you are done? (p. 741-2)
Think about this for a minute before reading ahead; it is an interesting problem. Nearly all of the students
these authors interviewed answered this problem incorrectly, as did 99% of the students I asked (results in my
other dissertation paper). Most said that there were infinitely many balls remaining, or that half of the balls
remained. The correct answer is that no balls remain in Bin A, because for each ball k, the ball was removed
at step k of the process.
The authors apply an APOS analysis to this problem, claiming that the reason students answer this problem
incorrectly is that they are not able to encapsulate the infinite process involved. In other words, they are unable
to move from a process conception of the problem to an object conception of the problem; they could not
“imagine a resulting transcendent object” produced by the infinite process.
The findings about the existence of nonstandard models in student conceptions suggest that there may be
another explanation for these students’ “incorrect” answers. They may be appealing to a nonstandard model
of arithmetic (or of the real numbers) in order to answer this question. In an informal discussion I had with a
group of undergraduate students about this problem, many students seemed to indicate that they thought that
even though all of the numbered balls had been removed from the bin, there were a bunch of balls too big to
have numbers that were still in there. It is not that these students were unable to imagine a resulting
transcendent object; rather the resulting object they imagined contained infinite numbers! In other words,
students answering the Tennis Ball Problem incorrectly may be encapsulating the process into an object
perfectly well. It is just that their resulting object may be difficult for researchers to recognize, because it
appeals to a nonstandard model of arithmetic like Skolem’s, which contains actual infinite numbers.
Most mathematicians and mathematics educators would agree that a crucial part of gaining skill in higher
mathematics is becoming comfortable in talking about the infinite. The students who intuitively use a
nonstandard model of arithmetic or analysis in fact may be more comfortable working with the infinite than
other students, because their universes contain actual infinite objects. The presence of these nonstandard
models in their thinking might actually indicate a great deal of mathematical sophistication and maturity.
24
The second reason that the existence of a legitimately alternate set of conceptions is important is a simple
one: it provides a definitive argument against a “blame the student” mentality of teaching. We have all met
teachers who tend to use this kind of reasoning, blaming a student’s misconceptions on the fact that he or she
does not work hard enough, does not pay attention well enough, or simply does not have a mathematical
enough mind. Perhaps some misconceptions can be blamed on students, but this paper shows the existence
of “wrong” student conceptions that could not be construed in any way as the student’s fault. Sarah’s
thinking is mathematically sound, and this can be proved. Furthermore, by persisting in this line of
reasoning, there is no mathematical reason that she would ultimately reach a contradiction or establish a weak
and ungeneralizable system of thought. Instead, she will develop a system which is just as mathematically
sound and defensible as the standard system.
The third reason that it is important is that it underscores the fact that mathematics is not a monolithic field
with only one possible “correct” configuration. Infinitesimals were able to provide just as sound of a
mathematical foundation for advanced calculus as δs and εs did. It was not mathematical necessity but a
choice on the part of the mathematicians of the early 19th Century that changed the course of mathematical
history. Mathematics has its alternate realities, its choices that could have gone different ways due to various
historical factors. It should not surprise us when these alternate realities are reflected in the thinking of our
students while they are in the process of forming their own mathematical realities.
The final reason this research is important is that it marks a new contribution to the field of misconception
research, one that supports a constructivist interpretation of conceptions. One corollary of the view that
students learn by accommodating perturbations is that students do not simply learn the mathematics that they
are taught. If an idea from the classroom conflicts with a student’s prior knowledge, then the classroom idea
may be rejected or merely overlaid upon the underlying conceptions in some superficial way. Sarah
specifically said that mathematics for her was a matter of memorizing the correct things and reproducing
them on the test to get an A. Even after seeing many different explanations and proofs of the fact that
0.999… equals 1, she persisted in believing that the two numbers are different. For to accept that fact, she
would have to go against the rules of mathematical consistency and coherence that she used to ground her
intuitions about infinitesimal numbers.
A constructivist framework claims that there are mechanisms for cognitive consistency at work that govern
the construction of student knowledge. Because inconsistencies in a schema allow for perturbations,
cognition is a locally self-regulating system. Thinking can be thought of as a game with rules, and these rules
govern the construction of consistent cognitive structures. “Consistency, in maintaining semantic links and in
avoiding contradictions, is an indispensable condition of what I would call our ‘rational game’” (von
Glasersfeld, 1990).
25
Since Sarah’s structure is reflected so strongly by an actual nonstandard mathematical structure, this paper
suggests that there is a strong relationship between the mechanisms for cognitive consistency and the rules
for mathematical consistency. This is a huge question that I believe should be at the heart of mathematics
education inquiry: what is the relationship between the internal cognitive rules that govern the construction of
(locally) consistent sets of conceptions and our external socio-mathematical rules that govern the construction
of consistent mathematics? Perhaps (a) our culture’s criteria for mathematical consistency and coherence are
an externalization of our innate internal cognitive criteria for the consistency and coherence of constructed
knowledge. Or perhaps (b) our personal cognitive mechanisms for constructing consistent knowledge are an
internalization of the criteria for mathematical consistency (more generally, and internalization of the norms
of consistency in our discursive social practices). Or maybe (c) our internal cognitive mechanisms and our
public mathematical rules influence one another in complex and iterative ways.
This question is not a new one, and our further attempts to answer it should be informed by the more general
debate in educational psychology about if individuals’ patterns of thinking inform socio-cultural structures
(e.g. Chomsky, 1967), if socio-cultural structures inform individuals’ patterns of thinking (e.g. Hutchins,
1993), or if there is a complex interaction between the two (e.g. Piaget & Garcia, 1989/1983). If we are able
to more fully answer this question, we will also be answering why the thread of mathematical consistency can
unite the thinking of a 17th-Century mathematician and a present-day student.
This is only one of several areas of future inquiry that present themselves based on this paper. Two (simpler)
ones are certainly worthy of note. One direction for further research is clearly to look for other examples of
alternate student conceptions of infinite and infinitesimal numbers. One limitation of this case study is that it
naturally evokes the question of whether this is a highly unusual and isolated example, or if there are other
students who hold similar conceptions. Other different kinds of nonstandard models should also be
investigated. In particular, the Tennis Ball Problem should provide a rich and promising context for
investigations into whether students appeal to nonstandard models of arithmetic in their understandings of
infinity.
Another area where further research is needed is to see whether students who believe in infinitesimals are
able to actually build a system of calculus using these conceptions as a basis. For example, Sarah did not
connect her concept of a derivative to her ideas about infinitesimals. Could these connections be made and
supported? And conversely, are students who have a robust understanding of derivatives and integrals
involving images of infinitesimal slopes and rectangles more inclined to think of the number line using
infinitesimals?
Both of these lines of inquiry are important in order to understand the implications of this research for the
teaching of calculus and higher mathematics. Without more research about these connections, for example, I
am hesitant to endorse a textbook that takes an infinitesimal approach to calculus (Keisler, 1986; Henle &
Kleinberg, 1979). Perhaps the foundational treatment of infinitesimals in such a textbook could provide a
26
rigorous conceptual base for students like Sarah, but it is still unknown how many students like Sarah are out
there.
This study marks an existence proof for alternate student conceptions that are pieces of provably
mathematically-consistent systems. It shows how allowing for student misconceptions to be legitimate viable
alternate conceptions can lead the researcher to see meaningful connections between mathematical structures
and cognitive structures that otherwise would have been overlooked. Because this case study provides an
example of truly alternate conceptions, ones that are part of a system in no way inferior to the standard
system, it is a valuable contribution to the field of inquiry about students’ mathematical misconceptions. It
raises an important question about the relationship of mathematical consistency and cognitive consistency,
and it provides strong evidence that paying attention to alternate student conceptions is crucial to
understanding the teaching and learning of mathematics.
References
Albeverio, S., Fenstad, J. E., Hoegh-Krohn, R., & Lindstrøm, T. (1986). Nonstandard Methods in Stochastic Analysis and
Mathematical Physics. Academic Press.
Baron, M. E. (1987). The Origins of the Infinitesimal Calculus. Mineola, NY: Dover.
Bos, H. J. M. (1974). Differentials, higher-order differentials, and the derivative in Leibnizian calculus. Archive for History
of Exact Sciences, 14, pp. 1-90.
Bressoud, D. (1994). A Radical Approach to Real Analysis. Mathematical Association of America.
Brousseau, G. (1997). Theory of didactical situations in mathematics : didactique des mathématiques, 1970-1990. Dordrecht:
Kluwer Academic Publishers.
Chomsky, N. (1967). Review of Skinner's Verbal Behavior. In Jakobovits & Miron (eds.), Readings in the Psychology of
Language. Englewood Cliffs, NJ: Prentice-Hall.
Denzin, N. K. (1989). The research act: A theoretical introduction to sociological methods (3rd ed.). Englewood Cliff, NJ: Prentice
Hall.
Hutchins, E. (1993). Learning to navigate. In S. Chaiklin and J. Lave (Eds.), Understanding Practice: Perspectives on activity and
context (pp. 35-63). Cambridge: Cambridge University Press.
Keisler, H. J. (1986). Elementary Calculus: An Infinitesimal Approach. Boston, MA: PWS.
Kleiner, I. (2001). History of the infinitely small and the infinitely large in calculus. Educational Studies in Mathematics, 48,
137-174.
Knobloch, E. (1994). The infinite in Leibniz’s mathematics: The historiographical method of comprehension in context.
In K. Gavroglu et al. (Eds.), Trends in the Historiography of Science, Dordrecht, pp. 265-278.
27
Leibniz, G. W. (1663-72). Sämtliche Schriften und Briefen. Edited by the Akademie der Wissenschaften zu Berlin, Series VI
(Philosophische Schriften), Vol. 2:1663-72., 1966. Berlin: Akademie-Verlag.
Leibniz, G. W. (1684). Nova methodus pro maximis et minimis. Translation in Struik, D. J., 1969. A Source Book in
Mathematics, 1200-1800. Harvard Univ. Press: 271-81.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching.
Review of Educational Research, 60, 1-64.
Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer. The Journal of Mathematical
Behavior, 21(1), 87-116.
Mancosu, P. (1996). Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. New York: Oxford
University Press.
McCloskey, M. (1983). Naive theories of motion. In D. Gentner & A. L. Stevens (Eds.), Mental models (pp. 299-324).
Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Piaget, J., & Garcia, R. (1989). Psychogenesis and the history of science. New York: Columbia University Press.
Robinson, A. (1996). Non-standard Analysis. Princeton: Princeton University Press.
Ross, G. M. (1986). Are there real infinitesimals in Leibniz’s metaphysics? In A. Lamarra (Ed.), L’Infinito in Leibniz:
Problemi e Terminologia, pp. 125-141.
Schwarzenberger, R. L. E., & Tall, D. O. (1978). Conflicts in the learning of real numbers and limit. Mathematics Teaching,
82, 44-49.
Strauss, A. & Corbin, J. (1990). Basics of Qualitative Research: Grounded theory procedures and techniques. Sage Publications.
Stake, R. E. (2000). Case studies. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of Qualitative Research (2nd ed., pp.
435-454). Thousand Oaks, CA: Sage.
Tall, D. O. (1980). Intuitive infinitesimals in the calculus. Poster presented at the Fourth International Congress on
Mathematical Education, Berkeley.
von Glasersfeld, E. (1990). An exposition of constructivism: Why some like it radical. In R. B. Davis, C. A. Maher & N.
Noddings (Eds.), Constructivist Views on the Teaching and Learning of Mathematics (pp. 19-29). Reston, VA: NCTM.