Journal of Mathematical Behavior 30 (2011) 93–114
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior
journal homepage: www.elsevier.com/locate/jmathb
Reinventing the formal definition of limit: The case of Amy and Mike
Craig Swinyard ∗
Department of Mathematics, University of Portland, 5000 N. Willamette Blvd., Portland, OR 97203, United States
a r t i c l e
i n f o
Available online 22 February 2011
Keywords:
Defining
Limit
Calculus
Reinvention
Realistic mathematics education
Undergraduate mathematics education
a b s t r a c t
Relatively little is known about how students come to reason coherently about the formal
definition of limit. While some have conjectured how students might think about limits formally, there is insufficient empirical evidence of students making sense of the conventional
ε–ı definition. This paper provides a detailed account of a teaching experiment designed to
produce such empirical data. In a ten-week teaching experiment, two students, neither of
whom had previously seen the conventional ε–ı definition of limit, reinvented a formal definition of limit capturing the intended meaning of the conventional definition. This paper
focuses on the evolution of the students’ definition, and serves not only as an existence
proof that students can reinvent a coherent definition of limit, but also as an illustration of
how students might reason as they reinvent such a definition.
© 2011 Elsevier Inc. All rights reserved.
1. Introduction
In response to high failure rates nationwide in introductory Calculus courses (Tall, 1992), a great deal of research (e.g.,
Bezuidenhout, 2001; Carlson, Larsen, & Jacobs, 2001; Davis & Vinner, 1986; Monaghan, 1991; Orton, 1983; Tall & Vinner,
1981; Williams, 1991) was conducted in the 1980s and 1990s to explore students’ difficulties in understanding introductory
calculus concepts. Such research has provided valuable insight into common misconceptions students possess regarding the
concepts of function, limit, continuity, derivative, and integral. Summarily, the research has focused almost exclusively on
first-year Calculus students, and has found that students’ understanding during an introductory calculus course is heavily
procedural, and lacks the type of conceptual depth for which educators hope (Ferrini-Mundy & Lauten, 1993; Zandieh, 2000).
For students endeavoring to major in disciplines other than mathematics, procedural understanding of calculus concepts
may be sufficient. However, for those majoring in mathematics, a more abstract, conceptual viewpoint is likely necessary
for the transition to upper division courses.
Many researchers (e.g., Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limit
plays as a central concept in calculus and analysis. Cornu (1991) observes that the concept of limit “holds a central position
which permeates the whole of mathematical analysis” (p. 153). As students proceed to more formal, rigorous mathematics,
the formal definition of limit1 serves as a foundational tool. Continuity (both point-wise and uniform), derivatives, integrals,
and Taylor series approximations are just a few of the topics studied in an undergraduate analysis course that build upon
coherent understanding of the formal definition of limit. While extensive research has been conducted on students’ informal
understanding of limit (Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995; Monaghan, 1991), little is known about how students
beyond the first year of calculus formalize their intuitive notions of limit to a depth required for success in upper-division
∗ Tel.: +1 503 943 8557/504 1253; fax: +1 503 943 7801.
E-mail address: [email protected]
1
The conventional ε–ı definition of limit is as follows:limf (x) = L provided: for every ε > 0, there exists a ı > 0, such that 0 < |x − a| < ı → |f(x) − L| < ε.
x→a
0732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmathb.2011.01.001
94
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
courses. While some have conjectured how students might think about limits formally (e.g., Cottrill et al., 1996), there is
insufficient empirical evidence of students making sense of the conventional ε–ı definition.
Freudenthal (1973), the pioneer of Realistic Mathematics Education (RME), argues that learners are constantly in the
process of actively organizing their world via the construction of what is known in RME as “emergent models.” Constructing
definitions and reinventing concepts and theorems are two types of mathematical activity that supports students in organizing and formalizing their intuitive notions. Research conducted by Zandieh and Rasmussen (2010) and Larsen (2009)
has detailed how such activity can support students in developing the type of advanced mathematical thinking that is
desired in upper-division courses. Specifically, Zandieh and Rasmussen’s work has focused on how students might construct and use geometric definitions in the classroom setting as they transition “from informal to more formal ways of
reasoning” (p. 57), and work by Larsen had depicted how students might actively reinvent algebraic concepts like group and
isomorphism.
The purpose of this paper is two-fold. First, whereas Larsen (2009) has provided evidence of students reinventing concepts
in upper-division mathematics, this paper contributes to RME by supplying an existence proof that students can reinvent
definitions at the upper-division level via a process of specifying their robust concept images (Tall & Vinner, 1981) of limit.
Second, this paper provides the reader a detailed account of the first of two teaching experiments designed to produce
empirical data of how students might specifically reinvent the formal ε–ı definition of limit. In a ten-week teaching experiment, two students, neither of whom had previously seen the conventional ε–ı definition of limit, reinvented a formal
definition of limit capturing the intended meaning of the conventional definition. This paper focuses on the evolution of the
students’ definition, and serves as an illustration of how students might reason about limit as they reinvent such a definition.
To provide appropriate context for the reader, I begin by situating this study within the existing literature on students’
understanding of limit and their use of mathematical definitions. Following this review of the literature, I briefly detail the
theoretical perspectives which framed the study, as well as the methodological design and analytic approaches that were
employed. I then trace the evolution of the two students’ precise definition of limit, highlighting six different phases of their
reinvention.
2. Relevant literature
2.1. Students’ understanding of the formal definition of limit
The majority of the research base on limit addresses students’ informal understandings of limit, focused largely on
the misconceptions students possess. In contrast, there is a paucity of research that describes students’ understanding of
the formal definition of limit. While some have suggested pedagogical approaches for teaching students about the formal
definition of limit (Gass, 1992; Steinmetz, 1977), not much is known about how students might reason about the formal
definition as they make sense of it. The research that does exist suggests that students, for a variety of reasons, struggle to
understand the formal definition of limit when it is presented to them (Cornu, 1991; Cottrill et al., 1996; Larsen, 2001; Tall,
1992; Tall & Vinner, 1981; Vinner, 1991; Williams, 1991). For instance, in studies conducted by Vinner (1991) and Cottrill et al.
(1996), only one student was able to provide a formal definition indicating “reasonably deep understanding of the concept”
(Vinner, 1991, p. 78), and that student had spent significant time with the limit concept. One explanation for students’
difficulties with formal treatments of limit is that their algebraic preparation prior to calculus often fails to prepare them
sufficiently for studying the limit concept formally (Cornu, 1991; Cottrill et al., 1996; Ervynck, 1981). Another explanation
is that students struggle to grasp the definition’s sophisticated notation and quantification structure (Cottrill et al., 1996;
Dubinsky, Elterman, & Gong, 1988; Tall & Vinner, 1981). Yet another explanation for calculus students’ difficulties with the
formal definition of limit is that they lack the perspective necessary for appreciating the formal definition’s mathematical
role. Dorier (1995) argues that historically “less formalized tools were used to solve most of the problems [related to limits],
while the ‘!-"-definition’ was conceived for solving more sophisticated problems and for unifying all of them” (p. 177).
Cornu adds, “[T]his unencapsulated pinnacle of difficulty occurs at the very beginning of a course on limits presented to a
naïve student. No wonder they find it hard!” (p. 163).
Research has established that students have difficulty understanding the formal definition of limit when it is presented
to them. What is unclear, however, is how students might successfully make sense of this sophisticated definition. Research
by Cottrill et al. (1996) has attempted to address this question. Cottrill et al. provide a genetic decomposition of the limit
concept. Based partially on data collected during their study, this seven-step conjecture (Fig. 1) describes how students
might come to understand the limit concept, both informally and formally.
Unfortunately, students in the study did not show sufficient evidence of reasoning at more sophisticated levels, and thus,
the latter steps of the genetic decomposition (i.e., Steps 5–7) are based on conjecture as opposed to empirical data.2
The genetic decomposition offered by Cottrill et al. (1996) provides noteworthy evidence of how students reason about
the informal process of finding limits (Steps 1–3 of their genetic decomposition); however, there is a dearth of empirical
data tracing the evolution of students’ thinking to the point of coherently reasoning about the formal process of validating
limits. Cottrill et al. call for research that “could contribute to the design of effective instruction that may help students learn
2
For discussion of the conceptual differences between the initial and latter steps of the genetic decomposition, see Swinyard and Larsen (2010).
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
1.
2.
3.
4.
5.
6.
7.
95
The action of evaluating f at a single point x that is considered to be close to, or even
equal to, a.
The action of evaluating the function f at a few points, each successive point closer to a
than was the previous point.
Construction of a coordinated schema as follows.
(a) Interiorization of the action of Step 2 to construct a domain process in which x
approaches a.
(b) Construction of a range process in which y approaches L.
(c) Coordination of (a), (b) via f. That is, the function f is applied to the process of x
approaching a to obtain the process of f(x) approaching L.
Perform actions on the limit concept by talking about, for example, limits of
combinations of functions. In this way, the schema of Step 3 is encapsulated to become
an object.
Reconstruct the processes of Step 3(c) in terms of intervals and inequalities. This is done
by introducing numerical estimates of the closeness of approach, in symbols,
0 < x − a < δ and f ( x ) − L < ε .
Apply a quantification schema to connect the reconstructed process of the previous step
to obtain the formal definition of limit.
A completed ε-δ conception applied to a specific situation.
Fig. 1. Refined Genetic Decomposition of Limit (Cottrill et al., 1996).
what is universally agreed to be the very difficult concept of limit” (p. 190). The research reported here is meant to do just
that – offer insight into how students might reason about the limit concept as they transition to engaging with the concept
more formally.
2.2. Students’ use of mathematical definitions
Research by Zaslavsky and Shir (2005) indicates that students use differing strategies when defining mathematical concepts, depending upon the type of mathematical concept being defined. In a study conducted with four high-level 12th-grade
students, Zaslavsky and Shir examined students’ conceptions of four mathematical definitions: square, isosceles triangle,
increasing function, and local maximum. Students demonstrated two different types of reasoning in defining the concepts.
For geometric concepts, they used definition-based reasoning, meaning justifications for a particular definition were based
on features or roles of mathematical definitions that the students deemed important (2005, p. 327). For instance, one student rejected a proposed statement as a definition of square because he felt the definition was too procedural (i.e., the
definition was a set of instructions on how to construct a square). This same student accepted a different statement as a definition of square because the statement was minimal (i.e., it included no superfluous conditions). In contrast, students used
example-based reasoning for analytic concepts, wherein students’ justifications for a particular definition relied on examples
and non-examples to convince themselves or others regarding a statement about the concept (p. 326). Zaslavsky and Shir
report that students viewed the classification of examples and non-examples of a concept as one of the central purposes of
mathematical definitions, commenting that “the students pointed to its power in ‘refuting functions”’ (p. 334).
Dealing with the rather straightforward geometric concepts allowed the students to focus on the notion of a definition,
whereas dealing with the more subtle analytic concepts led them to a process of monster-barring (Lakatos, 1976),
wherein the students iteratively modified their definition to better reflect the concept image they held (p. 328).
In sum, examples and non-examples served as powerful tools for students as they constructed and reasoned about analytic
definitions. This raises the possibility that students might gain similar traction from examples and non-examples as they
formally define limit.
Research by Zaslavsky and Shir (2005) provides evidence of how students reason about mathematical definitions. A related
issue is how and when students choose to use mathematical definitions. Research by Tall and Vinner (1981) suggests that
formal definitions may not be suitable starting points for students’ explorations of a concept. Tall and Vinner describe concept
image as the total cognitive structure that is associated with a concept, including all of the mental pictures and associated
properties and processes. Concept definition refers to whatever description is given to specify a concept when asked. Vinner
(1991) claims that in non-technical contexts, it is unnatural for students to consult definitions because conversations in
such contexts are dominated by one’s concept image. There is an inherent expectation at the collegiate level, however, that
as students engage in technical contexts “the concept image will be formed by means of the concept definition and will be
completely controlled by it” (p. 71). Vinner reasons, though, that even in technical contexts, students’ concept formation
is dominated by their concept image: “It is hard to train a cognitive system to act against its nature and to force it to
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C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
consult definitions” (p. 72). Harel (2004) agrees, noting that “although formal concept definitions may be the ultimate goal
of mathematics instruction, in the absence of rich and flexible corresponding concept images, students are unable to retain
these concept definitions for a long period of time.” Thus, it seems students would be more likely to use a robust concept
image of limit to make sense of the formal definition, rather than develop their concept image of limit based on the formal
definition. Such mathematical activity would be consistent with what Zandieh and Rasmussen (2010) classify as situational
activity. Building on the work of Gravemeijer (1999) and Tall and Vinner (1981), Zandieh and Rasmussen developed a defining
as a mathematical activity (DMA) framework to capture how learners might construct and use definitions in the classroom
setting as they transition “from informal to more formal ways of reasoning” (p. 57). Central to Zandieh and Rasmussen’s work
was describing the relationships between students’ concept images and concept definitions as they were engaged at each of
the four levels of mathematical activity espoused by Gravemeijer (1999) – situational, referential, general, and formal. The first
level of activity (situational) in Zandieh and Rasmussen’s DMA framework is characterized by students creating a concept
(or formal) definition based on a robust concept image. Zandieh and Rasmussen observed that the situational activity of
constructing a definition can subsequently support higher-level mathematical activity (i.e., referential, general, and formal
level activity), wherein concept images (and in turn, concept definitions) of higher-level concepts are constructed. Larsen’s
account (2009) helps to explicate the general level of activity in the DMA framework. The work presented in this paper helps
to further elucidate the situational level of activity.
3. Theoretical perspectives
The purpose of my research was to model student reasoning about limit in the context of reinventing a formal definition
of the concept. In particular, I attempted to describe what challenges the participating students experienced in their reinvention efforts, and how those challenges were resolved. Given the goals of the study, the epistemological stance of radical
constructivism (RC) (von Glasersfeld, 1995) served as a useful perspective in terms of research design. One central tenet of
RC is that “knowledge is not passively received either through the senses or by way of communication, but is actively built
up by the cognizing subject” (von Glasersfeld, 1995, p. 51). The idea that sophisticated mathematical understandings are
constructed via active engagement by the learner was central to my research. Instructional tasks were designed to encourage
the participating students to actively build upon their intuitive understandings of limit by conjecturing and refuting formulations of the definition in an iterative manner. In this sense, the students’ interactions were reminiscent of mathematical
conversations described by Lakatos (1976).
In addition to the overarching perspective of RC, the perspective of developmental research guided the instructional design
for my study. According to Gravemeijer (1998), the goal of developmental research is “to design instructional activities that
(a) link up with the informal situated knowledge of the students, and (b) enable them to develop more sophisticated, abstract,
formal knowledge, while (c) complying with the basic principle of intellectual autonomy” (p. 279). A well-established RME
heuristic commonly associated with developmental research is guided reinvention, which has been employed in numerous content areas of postsecondary mathematics education (see Larsen, 2009; Marrongelle & Rasmussen, 2006). Guided
reinvention is described by Gravemeijer, Cobb, Bowers, and Whitenack (2000) as “a process by which students formalize
their informal understandings and intuitions” (p. 237). An important aspect of this process is the identification of plausible instructional starting points that avoid what Freudenthal refers to as an antididactic inversion – that is, to avoid using
mature, conventional symbolizations of mathematical concepts as starting points for instruction. In an effort to avoid such
an inversion, I attempted to create an environment intended to mimic important aspects of the mathematical setting that
Cauchy and Weierstrass experienced. Neither of the mathematicians’ definitions was a reformulation or an interpretation of
the traditional formal definition. On the contrary, these mathematicians constructed their respective definitions in response
to an inherent need to classify functional behavior. I felt, then, that I might learn a great deal about students’ reasoning about
the limit concept if I engaged them in activities designed to foster their reinvention of the formal definition. In this sense,
the research reported here was unique – while other studies (e.g., Larsen, 2001; Fernandez, 2004) have sought to describe
how students reason about limit as they interpret the conventional ε–ı definition of limit, my research focused on how students reason about limit in the context of reinventing a definition which captures the intended meaning of the conventional
ε–ı definition. To ensure such a setting, students selected for the study had no prior experience with the conventional ε–ı
definition.
4. Background
4.1. Data collection methods
With the aim of gathering information about how an initial participant pool reasoned about limits informally, I conducted a
task-based Informal Limit Reasoning Survey with twelve undergraduate students from a large, urban university in the Pacific
Northwest region of the United States. All of the survey participants were students in two or more of the courses forming
a three-quarter introductory Calculus sequence I taught during the 2006–2007 academic year. Four of the twelve students
were selected for two teaching experiments. Each teaching experiment consisted of ten 60–100 min paired sessions and one
30–60 min individual exit interview. The paired sessions were conducted in a classroom, with the students responding to
instructional tasks on the blackboard in the front of the room. Only the participating students, researcher, and a research
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
97
assistant were present for each session. Each session was generally separated by a span of 6–10 days, allowing time for
ongoing analysis between sessions and subsequent construction of appropriate instructional activities based on the ongoing
analysis. All sessions, including the individual exit interviews, were videotaped by a research assistant. These twenty-two
videotaped sessions were the primary source of data for the study.
4.2. Teaching experiment participants
The four students selected for the two teaching experiments were chosen on the basis of possessing robust informal
understanding of limit, as well as my estimation of their ability to work effectively in tandem to reinvent the definition
of limit. Evidence of these criteria existed in the students’ responses to the task-based survey, as well as in their written
work throughout the three-term introductory Calculus sequence I taught. Additionally, a criterion for selection was that
participants have no previous experience with the conventional ε–ı formal definition of limit. Students studied the idea
of limit only informally during the three-term introductory sequence. The concept was introduced as a means by which
one might determine the height (i.e., y-value) a function “intends to reach” as x-values approach a particular point x = a.
Tabular inspection and the method of “zooming-in” on graphical representations to identify limit candidates were an initial
focus of discussion. Subsequently, students were taught methods of algebraic manipulation that would allow them to use
direct substitution to determine the limit of a function. Emphasis was also placed on distinguishing between graphical
examples and non-examples of limits. The conventional formal definition was not presented in class, and only appeared in
an appendix of the course textbook (Stewart, 2001). Student responses to the survey and their observed activity during the
teaching experiments further confirms that they were actively engaged in reinventing a definition.
The first pair of students3 (Amy and Mike) both earned A’s during all three terms of the introductory Calculus sequence.
At the time of the teaching experiment, Amy was a linguistics major in her mid-twenties; Mike was a nineteen year old
mathematics major who had just finished his freshmen year.
4.3. Teaching experiment tasks
In both teaching experiments, the central task was for the students to generate a precise definition of limit that captured
the intended meaning of the conventional ε–ı definition. Instructional activities were primarily focused on discussing limits
in a graphical setting, in hopes that the absence of analytic expressions might support the enrichment of the visual aspects
of the students’ respective concept images. Tasks included the students generating prototypical examples of limit, which
subsequently served as sources of motivation for the students as they attempted to precisely characterize what it means for
a function to have a limit. The majority of each teaching experiment, then, comprised a period of iterative refinement for the
students; as they attempted to characterize limit precisely, the examples and non-examples of limit that they encountered
created cognitive conflict for them, which they sought to resolve by refining their characterization. To this extent, the task
design was inspired by the proofs and refutations design heuristic adapted by Larsen and Zandieh (2007) based on Lakatos’
(1976) framework for historical mathematical discovery.
4.4. Analysis of data
Radical constructivism guided my analysis of student reasoning. How one interprets the tasks he/she is presented is
necessarily dependent upon one’s prior experiences. As the students engaged with the instructional tasks, their observable actions and behaviors provided evidence of how they might be interpreting said tasks. In a manner consistent with
Steffe and Thompson’s (2000) description of modeling students’ interpretations, I compared my models of the students’
interpretations with those targeted in instruction so that research findings could be cast as inferences about student reasoning given interpretations of instructional tasks. Thus, data analysis was a cyclic process in which hypotheses about
students’ reasoning were generated, reflected upon, and then refined until increasingly stable and viable hypotheses
emerged.
The analysis of data for this study occurred at a variety of levels. As each teaching experiment was unfolding, I conducted
an ongoing analysis, which informed decisions about subsequent sessions within the same teaching experiment. Ongoing
analysis included transcribing each session, paying particular attention to articulated thoughts that seemed to provide the
students leverage, the voicing of concerns or perceived hurdles that needed to be overcome, and signs of/causes for progress
and revision. Ongoing analysis provided an opportunity to form ideas about how best to proceed in the following session.
Prior to each session, I composed a document which included central objectives of the upcoming session, anticipated tasks
that would be employed, as well as a rationale for each of those tasks.
Following the completion of each teaching experiment, I conducted a post analysis of the data generated by each pair
of students. Post analysis consisted of reviewing the videos and transcriptions of all ten sessions, highlighting noteworthy
excerpts that shed light on students’ reasoning and/or illustrated significant challenges or marked progress. This process
3
The interested reader can find background information about the second pair of students in Swinyard (2008).
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C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
subsequently led to the construction of a lengthy description of each pair’s reinvention. These descriptions were foundational
in analysis, because they allowed me to view the evolution and growth of each pair of students’ reasoning over the course
of ten sessions.
Finally, following both teaching experiments, I conducted a retrospective analysis (Cobb, 2000), in which I analyzed the
entire corpus of data at a deeper level than the preceding analyses. Retrospective analysis consisted of reviewing the post
analyses of both teaching experiments, comparing and contrasting student reasoning between the four students. Doing so
led to a refinement of my description of thematic elements present in student reasoning. At all three levels of analysis, I
engaged in frequent discussions with other mathematics educators who had intimate knowledge of the study in an effort
to reach consensus about the data.
5. Tracing the reinvention of a formal definition of limit: the case of Amy and Mike
The purpose of this paper is to provide a detailed account of the first teaching experiment. Amy and Mike’s reinvention
of the definition of limit unfolded in roughly six phases:
Phase 1:
Phase 2:
Phase 3:
Phase 4:
Phase 5:
Phase 6:
Getting informal ideas out on the table (Sessions 1–3)
Initial x-first characterizations of limit (Sessions 3–4)
Employment of a zooming metaphor (Sessions 4–5)
Dissatisfaction with the infinite limiting process (Sessions 5–6)
Characterizing limit at infinity (Sessions 7–8)
Using limit at infinity as a template to define limit at a point4 (Sessions 9–10)
In what follows, I describe the evolution of Amy and Mike’s definition of limit across these six phases.
5.1. Phase 1: Getting informal ideas out on the table (Sessions 1–3)
The first few sessions of the teaching experiment might best be described as a time when Amy and Mike established
consensus about their informal ideas related to the limit concept. Tasks included discussing how one might establish the
value of a limit in both continuous and discontinuous cases, given an algebraic representation of a function. For instance,
during the first session, Amy and Mike were provided the following two tasks:
3
Task 1 : What is lim x + 3?
x→4 4
x2 − 2x − 8
?
x−4
x→4
Task 2 : What is lim
These algebraic tasks prompted Amy and Mike to voice and negotiate fundamental elements of their concept image of
limit over the first three sessions. For example, in contrasting the two tasks, Mike noted that direct substitution allows
one to determine the value of a limit algebraically, and that if a function is not continuous at the limiting value, algebraic
manipulation can be employed to support the use of direct substitution.
Craig: In regards to those first two [tasks]. . ., how was your approach to determining each of those limits the same or
different?.
Mike: They’re the same, in the sense that you’re basically solving an equation for this particular value, 4. In [the second]
case you had to do more steps so that you could do that because you can’t divide by 0, so they were the same, just,
you had to use some algebra [in the second case] so that you solve for it. . ..You had to get it in the case where you
could plug in the limit and that would give you the value.
Craig: What do you mean by plug in the limit?
Mike: Or plug in what the value that x is approaching to the function. . .
Craig: And you can do that in the first case as well?
Mike: Yeah, we just plug in 4 [in the first case] and we got 6 and in [the second] case we did some algebra, plugged in 4, and
we got 6.
Amy agreed that algebraic manipulation can be used to support direct substitution. Specifically, Amy noted that algebraic
manipulation alters a function containing a removable discontinuity into a continuous function that has the same limit
at the limiting point. She further observed that computing the function value for a continuous function is a “shortcut” to
determining the limit of the function at the limiting point.
4
The phrase limit at a point and the word limit will be used interchangeably in this report. The phrase limit at infinity will always be used to describe the
conventional ε–N context.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
99
Fig. 2. ε–ı illustration of limit.
Craig: In each of case 1 and case 2, what is it that you exactly plug in to get 6?
Amy: In those cases I actually plugged in 4 because I felt like with what I knew about the function, either as it was in its
original form or after manipulating it algebraically, that plugging in 4 would be a safe thing to plug in. . .as a short cut
for looking at what happens around 4.
Craig: And what is it about those first two [cases], either as it was originally stated or once you altered it? You talked last
week about simplifying the function to, like, a simple form or whatever. . .
Amy: Into something continuous.
In addition to establishing direct substitution as a viable means of determining a limit algebraically, Amy and Mike also
discussed what a limit represents. On multiple occasions, reference was made to viewing the limit as an approximation,
along the lines of what Oehrtman (2004) describes.
Amy: We’re approximating what. . .we think the function would do precisely at a certain point, based off of what it’s been
doing at other points close to it. . ..Built into the definition of the limit is that it’s an approximation of the behavior
of the function. . .It’s what we know about the relationship between the input and output around that point. We say,
okay, well then we’re just going to project that onto this point as well and then state what our output would then be,
whether or not that relationship does in fact hold for that point. . ..In the first case it does hold for the point and in the
second case it doesn’t, but that’s not what we’re interested in, right? We’re just interested in what the relationship is
around it.
Amy and Mike’s initial discussions of what a limit represents helped establish intuitive notions that I later leveraged to
support them in reinventing a formal definition. In this sense, “getting informal ideas out on the table” not only helped Amy
and Mike to negotiate meaning, it also supported and directed my construction of instructional tasks for future sessions.
The central goal of the teaching experiment was to model Amy and Mike’s reasoning as they reinvented a formal definition
of limit. Engaging Amy and Mike in thought and discussion about limits in a graphical setting seemed important, given the
visual imagery that such a graphical setting stands to provide as a basis for the formal definition of limit. Indeed, one could
imagine students developing an approach consistent with the picture shown in Fig. 2 (Stewart, 2001), using geometric
language to produce a formal definition of limit.
These initial sessions, however, shed light on Amy and Mike’s substantial reliance on algebraic representations and
general reluctance to engage in graphical conversations, unless they had been assured that such graphs had originated from
an algebraic representation. For instance, in an effort to shift Amy and Mike’s attention away from algebraic representations,
I asked them to consider a scenario in which no algebraic representation for a function was provided, but rather a graphical
representation existed in and of itself. Amy’s comments below suggest that she did not separate graphical representations
of functions from the algebraic equations that define them.
Craig: [C]an a graph of a function exist in and of itself, separate from either coming from a table or coming from an equation
or formula?
Amy: I mean, how do you know it’s a graph of a function? How do you know it’s a graph at all? It’s just like, you know, like
a pretty picture?
Amy’s reliance on algebraic representations and general distrust for graphical representations is consistent with findings in
the research literature (Knuth, 2000). Following Amy’s comments, I attempted to convince both her and Mike that functions
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C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
Fig. 3. Amy and Mike’s examples of limit.
could exist without an algebraic representation if, for instance, such graphs were formed by collecting empirical data. It is
evident that Amy opposed such a possibility.
Amy: I don’t know Craig. I’m really having a hard time with this kind of like abstract, umm, immaculately-conceived graph. . ..
The perspective that Amy shared above, and with which Mike concurred during the third session, proved to be quite paralyzing, as I would later find, in that both students were extremely hesitant to engage in graphical conversations about limits;
for them, certainty about a function’s local behavior was based on the presence of algebraic formulas.
By the end of Phase 1, Amy and Mike had explicitly stated that while all three functional representations could provide
them with an idea for what the limit of a function might be (i.e., serve as a means by which to generate a limit candidate),
only algebraic representations would allow them to determine a limit with complete certainty (i.e., serve as a means of
validating such a candidate). Their strategy for describing the local behavior of a function could be summarized as follows:
For one to determine a limit with certainty, he/she must employ algebraic techniques (e.g., rationalizing the numerator,
factoring, applying L’Hospital’s Rule) to the algebraic representation so as to eventually use direct substitution. Further, it
was evident that they believed that a function could not exist without a defining algebraic representation, and that direct
substitution would always be possible with algebraic representations.
5.2. Phase 2: Initial x-first characterizations of limit (Sessions 3–4)
Throughout the first three sessions, my intent was to motivate the generation of a precise definition of limit by posing the
following question: In the absence of an algebraic representation of a function, how might one determine the existence of
a limit at a particular point? Unfortunately, Amy and Mike’s reliance on algebraic representations of functions made such a
question moot, as they could not conceive of a function not having an algebraic representation. With these thoughts in mind,
beginning in Session 4, tasks and activities were primarily focused on discussing limits in a graphical setting, in hopes that
the absence of analytic expressions might support the enrichment of the visual aspects of Amy and Mike’s respective concept
images. To assuage their concerns, I conceded that if it helped them to do so, they could imagine that the graphs we were
discussing had algebraic origins and accurately depicted those algebraic origins. As the teaching experiment progressed,
this concession appeared helpful – indeed, Amy and Mike’s outward reliance on algebraic representations and distrust of
graphical representations waned.
5.2.1. Generation of prototypical examples of limit
Tasks during the fourth session included generating prototypical examples of limit, in response to the following prompt.
Prompt: Please generate as many distinct examples of how a function could have a limit of 2 at x = 5. In other words,
what are the different scenarios in which a function could have a limit of 2 at x = 5?
In response to this task, Amy and Mike demonstrated a robust categorization of limits, not only drawing an example
of each of the three scenarios in which a real-valued function f can have a general limit, but also drawing an example of a
one-sided limit, as shown in Fig. 3.
In the excerpt below, Amy and Mike describe the graphs they generated.
Amy: Well, these are the first situations that I could come up with off the top of my head. This is a function where the limit
at x = 5 is 2 and also where the value of the function at x = 5 is 2. And this is one where the limit is 2 but the function is
undefined at 5. And this is one where the function is not undefined but the value of the function is. . .6, at 5, despite
the fact that the limit is still 2.
Mike: And this is one where the limit’s 2 from the right side of the function as it approaches 5 from the right.
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Fig. 4. Removable discontinuity graph.
Amy and Mike agreed that while the jump discontinuity graph drawn by Mike would have a one-sided limit at x = 5, it would
not have a general limit. Subsequently, Amy and Mike equated “not limit” with the presence of a jump discontinuity.
The prototypical examples Amy and Mike generated subsequently served as sources of motivation as they attempted to
precisely characterize what it means for a function to have a limit. The rest of the teaching experiment, then, constituted
a period of iterative refinement for Amy and Mike; as they attempted to characterize limit precisely, the examples and
non-examples of limit that they encountered created cognitive conflict for them, which they sought to relieve by refining
their characterization. Throughout this refinement process, I encouraged Amy and Mike to incorporate explicit language in
their characterization of limit as they mulled over and wrestled with the essential characteristics and subtleties associated
with the concept.
5.2.2. Initial x-first characterizations
Evidence from the teaching experiment supports Larsen’s (2001) conjectures regarding the type of thinking students are
likely to employ in their initial forays into formal limit reasoning. Indeed, Amy and Mike showed a strong preference for
reasoning from what I refer to as an x-first perspective as they made initial attempts at precisely characterizing limit at a
point. By x-first, I mean that Amy and Mike focused their attention first on the inputs (x-values) and then on the corresponding
outputs (y-values), in a manner conducive to finding a limit candidate. Amy and Mike began employing an x-first perspective
as early as the end of the third session, in response to being asked how they would convince someone that the function in
Fig. 4 has a limit of 7 at x = 5.
Amy: I would be like, pick a point, any point. And. . .I’ll show you that for any x-value you can give me, I’ll give you a y-value
that, that as your x-values get closer to 5, my y-values get closer to 7.
The exhaustive process Amy describes is focused on the x-axis first, in a manner consistent with how Larsen (2001)
describes students’ informal understandings of limit. Mike displayed similar reasoning in response to a task asking him to
justify that the limit of another function at x = 0 was 2. His remarks towards the end of the third session indicated that, if
pressed, he would justify the existence of a limit graphically by first considering x-values.
Craig: Tell me about the process that you would go through in that case, Mike, to convince them that the limit is 2.
Mike: Well I would do as Amy did earlier and tell them, give me any x-value. . .as close as you can get to 0. I will plug it in
and I will give you a y-value that’s just about 2.
During the fourth session, as Amy and Mike discussed the prototypical graphical examples of limit they had generated at
the outset of the session, their initial characterization of limit was cast from this same x-first perspective (i.e., as a description
of what numeric value (L) the y-values get close to as x-values get closer to a).
Craig: Under what conditions would you say that the graph of a function has a limit of 2 at x = 5? What would have to be
true about that function? Or what would have to be true about that graph?
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Fig. 5. Jump discontinuity graph.
Amy: . . ..[W]hat would have to be true of the graph, like, would be that, umm, from both sides, as x-values get closer to 2,
y-values get closer to 5.5
Amy and Mike’s first characterization of limit, constructed during Session 4, could be summarized as follows:
Definition #1: f has a limit L at x = a provided as x-values get closer to a, y-values get closer to L. (Session 4)
It is worth noting that validating a candidate via the formal definition of limit relies on one’s ability to reverse his or her
thinking. Instead of initiating one’s exploration of functional behavior on the x-axis, a student must first focus his or her
attention on the y-axis. As Oehrtman, Carlson, and Thompson (2008) suggest: “In order to understand the definition of a
limit, a student must coordinate an entire interval of output values, imagine reversing the function process and determine
the corresponding region of input values” (p. 160). Thus, the process of validating a candidate requires a student to recognize
that his or her customary ritual of first considering input values is no longer appropriate. Instead the student must consider
first a range of output values around the candidate, project back to the x-axis, and subsequently determine whether an
interval around the limit value exists that will produce outputs within the pre-selected y-interval (see Fig. 2). This y-first
validation process is the process that is formalized via the definition of limit. As the teaching experiment progressed, Amy
and Mike’s inclination to reason from an x-first perspective posed a significant challenge to their efforts to reinvent a precise
definition of limit.
5.3. Phase 3: Employment of a zooming metaphor (Sessions 4–5)
Efforts to elicit a shift in Amy and Mike’s reasoning to a y-first perspective were unsuccessful during Session 4, despite
providing them with a graph of a jump discontinuity (shown in Fig. 5) intended to illustrate for them the insufficiency of
their initial x-first characterization.
Although both students had noted prior to Session 4 that the concept of limit describes local functional behavior, neither
of them had yet explicated local functional behavior in any concrete way. As the conversation continued, however, the jump
discontinuity graph in Fig. 5 provided fertile ground for articulating the subtleties involved in describing local functional
behavior. I pointed out to Amy and Mike that their initial definition of limit would incorrectly conclude that a function with
a jump discontinuity like the one in Fig. 5 has a limit, for as x-values get closer to x = 4, corresponding y-values get closer to,
say, 8. When asked if such a small jump (7.99–8.01) would constitute being close enough to 8 to lead one to conclude that a
limit exists, Amy employed, for the first time, a zooming metaphor to describe the graphical inspection one might undertake
to establish insufficient “closeness” geometrically.
Amy: Then I would say let’s just zoom in a lot more and all of a sudden [7.99 and 8.01] start to look pretty dang different.
Amy and Mike subsequently discussed how zooming in on a graph, in a manner consistent with how one might zoom on a
graphing calculator, would help determine the non-existence of a limit. In short, Amy and Mike equated the non-existence
of a limit at x = a with a jump discontinuity at x = a, and concluded that iteratively zooming-in on a calculator could help one
locate such jump discontinuities. Inherent in their discussion, however, was that graphical inspection via zooming could
only lead one to conclude that a limit does not exist.
5
Here, Amy inadvertently reversed her x-value and y-value, stating that as x-values get closer to 2, y-values get closer to 5, when, in fact, she meant
that as the x-values get closer to 5, y-values get closer to 2. Her hand gestures during this comment, as well as her subsequent reasoning, confirm that she
merely misspoke.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
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Amy: I feel like, uh, graphs maybe could help you see what a limit isn’t. Helps you rule out some things. But like, it can’t help
you conclusively state what a limit is.
Amy and Mike determined that establishing the existence of a limit would require the zooming process to be unending.
Craig:
Mike:
Craig:
Mike:
Craig:
Mike:
Craig:
Amy:
What would have to happen for that limit to actually be 8? Under what conditions would that limit be 8?
I would have to be able to zoom in infinitely. I can’t really comprehend what that would be but
Oh, like keep this process going forever, or whatever?
Yeah.
Like this zooming-in process?. . .Okay. So you’re saying that if I could do that forever.
Umm-hmm.
. . .Okay. Anything to add to that Amy?
I like it. Yeah. Do it forever and then I’ll be happy with the graph.
This discussion led to the following refinement of their initial definition of limit.
Definition #2: If you could zoom forever and always get closer to a and L, then you have a limit. (End of Session 4)
The employment of a zooming metaphor proved to be important in Amy and Mike’s reinvention process, for it caused
them to question whether an iterative zooming process could ever establish the existence of a limit. Amy and Mike’s concerns
regarding the completion of an infinite limiting process would become increasingly apparent as the experiment continued.
I discuss these concerns in my description of the fourth phase of the experiment. First, however, I will note that in response
to Amy and Mike’s initial x-first characterizations of limit, I attempted to leverage their use of a zooming metaphor to elicit a
shift to a y-first perspective. Although Amy and Mike used zooming as a metaphor for inspecting a function’s local behavior
during the fourth session, they were not specific about what it means to zoom. At the outset of the fifth session, I felt that
such explication might lead them to realize that determining the existence of a limit requires one to zoom along the y-axis,
not the x-axis. Through a sequence of questions, Amy and Mike discussed the effect that zooming along each axis has on a
function’s graph. In sum, Amy and Mike both consistently expressed the opinion that zooming along the y-axis pronounces
the existence of a vertical jump discontinuity, whereas zooming along the x-axis does not. Thus, it appeared, by the end
of the fifth session, that Amy and Mike were beginning to focus their attention on the y-axis. However, efforts during the
sixth session to elicit a shift to a y-first perspective by having Amy and Mike
! " explore a function with infinite oscillations
backfired. As they explored the graphical behavior around x = 0 of y = sin 1x + 5 on the calculator, they began by zooming
on the x-axis so as to get a better sense of the function’s behavior around the origin. Unfortunately, successive zooms on
the x-axis resulted in the calculator graphing the function in a manner suggestive of a jump discontinuity. This led them to
conclude that they had incorrectly assumed that to locate a jump discontinuity, one must zoom on the y-axis.
Amy: Yeah, I mean if I, if there is a vertical gap, there is a certain point at which zooming in on the x-axis is going to make
it show up.
Craig: . . ..If we kept the y-axis the way that it was and just zoomed in on the x-axis would we see that vertical jump, or no?
Amy: Yeah, we’d see it.
Thus, Amy and Mike’s x-first perspective persisted, despite attempts to leverage their zooming metaphor to elicit a shift
to a y-first perspective. The following formulation,6 constructed during Session 6, illustrates that even after much discussion
and refinement, Amy and Mike continued to define limit in an x-first fashion:
Definition #4: The limit L of a function at x = a exists if every time we look at the function more closely as we get
infinitely close to x = a, it bears out the same pattern of behavior, i.e., looks to be approaching some y-value L w/no
vertical gaps in the graph. (Session 6)
5.4. Phase 4: Dissatisfaction with the infinite limiting process (Sessions 5–6)
Reasoning from an x-first perspective was not the only challenge to Amy and Mike’s reinvention of a formal definition of
limit. Amy and Mike invested significant energy attempting to characterize what it means for x-values to get “infinitely close”
to a, and for y-values to get “infinitely close” to L along the x- and y-axes, respectively. Characterizing what they referred to as
“infinite closeness” was a non-trivial task for Amy and Mike. In Section 5.3, I noted that as the fourth session progressed, Amy
and Mike began employing a zooming metaphor to describe the act of inspecting the local behavior of the function in greater
detail. By “zooming in on the graph,” they believed they would eventually be able to detect jump discontinuities that would
prohibit the existence of a limit. It was apparent, however, that both students felt that establishing the existence of a limit
6
The reader will note that this discussion skips from Definition #2 to Definition #4. The evolution from Definition #2 to Definition #3 relates more
closely to Amy and Mike’s concern about whether an infinite process could be completed, and thus, is described in Section 5.4.
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would require them to zoom in an infinite number of times. The very real problem of attempting to mathematize a physical
process that is unending would be an ongoing concern for both of them, particularly Amy, as the experiment progressed.
For example, when asked at the end of the fifth session to characterize what it means graphically for a function to have a
limit L at x = a, Amy insisted on capitalizing the word forever, highlighting her concern.
Definition #3: A function has a limit L at a when zooming in FOREVER both horizontally and vertically yields no gaps
that have length > 0 AND that it looks like it approaches a finite number L. (Session 5)
At the outset of Session 6, Amy spoke at length about the fundamental issues that were problematic for her and Mike as
they attempted to define limit. Foremost among Amy’s concerns was the seemingly impossible task of describing how one
might carry out an infinite process.
Amy: I don’t know, it seems like we keep dancing around some kind of concept that we have to talk about in a series of
analogies or hypothetical situations, you know? Like . . .the hypothetical situation in which you are doing something
forever. . ..I guess like the first thing that leaps to mind for me is that we’re trying to parse out what we mean by, by
these impossible processes. . ..
Craig: And you’re saying impossible there why?
Amy: Because you can’t zoom in forever.
As the conversation continued, Amy, in a manner consistent with the reasoning she communicated during Session 5, once
again noted that the best one could do is establish when a limit fails to exist.
Amy: [Y]ou can’t do something an infinite number of times. . ..[W]e keep getting back to. . .we can’t prove it, we can only
not be disproven through, um, yeah, we only cannot be disproven. . ..
Craig: And each time that you’re zooming, either. . .that gap shows up or doesn’t, is what you’re saying. Is that, is that what
you mean by we can only not be disproven?
Amy: Yeah, yeah, exactly. That we can, you know, we can have our assertion that the limit at x = a is 6. Um, but through the
methods that we’ve been talking about, all we can do is you know just like, you know, grind through endless iterations
until we get tired of it and like I give up. And you know like all you can do is find. . .the level of examination which
disproves your idea but you can’t ever get to where you can conclusively prove it through the methods we’ve been
discussing.
It was evident, then, that Amy was cognizant that describing the completion of an infinite process would not be possible.
After Amy and Mike initially defined limit in terms of closeness along the x- and y-axes, I challenged them to describe more
precisely what they meant by x getting close to a, and y getting close to L, respectively. In response, they appeared to believe
this challenge would be resolved by defining infinite closeness. Tension then arose as they recognized that the construct they
were attempting to define was nonsensical. Indeed, the dilemma of characterizing the completion of an infinite process
continued to paralyze Amy and Mike’s reinvention efforts during Session 6.
Amy: I have a hard time getting too worked up over the language about what it means to zoom and what we’re looking for
when we zoom when we have lurking in the back this presupposition that whatever that means to zoom,. . .we have
to repeat that process an infinite number of times.
By the end of the sixth session, Amy and Mike’s reinvention efforts had reached a point of diminishing returns. Two
central challenges persisted: 1) Amy and Mike’s inclination to reason from an x-first perspective; and, 2) Their inability to
adequately characterize the infinite limiting process. These challenges motivated a pedagogical shift at the outset of the
seventh session.
5.5. Phase 5: Characterizing limit at infinity (Sessions 7–8)
In response to Amy and Mike’s struggles to characterize the infinite limiting process and their disinclination to assume a
y-first perspective, I shifted their attention to defining limit at infinity, anticipating, for multiple reasons, that their efforts to
characterize and formalize limit at infinity might provide necessary support for defining limit at a point. First, I felt that limit
at infinity would be a less cognitively taxing context in which to characterize what Amy and Mike meant by x-values getting
infinitely close to a, and y-values getting infinitely close to L. In the context of limit at infinity, instead of having to contemplate
and coordinate the troublesome notion of infinite closeness on both the x- and y-axes, as is necessary in the context of limit
at a point, the students would need only resolve issues of closeness on one of the two axes, as the conventional ε–N picture
in Fig. 6 (Stewart, 2001) illustrates.
Further, it is worth noting that the axis along which characterizing closeness is necessary in describing limit at infinity
(the y-axis) is the same axis to which I was hoping to shift the students’ attention. Thus, I theorized that limit at infinity was
also a context supportive of shifting the students’ attention to the y-axis – I believed the students would be unlikely to focus
their attention first upon inspecting functional behavior for x-values closer to a particular limit point (a) on the x-axis (as
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
105
Fig. 6. Conventional ε–N illustration of limit at infinity.
they had done in their attempts to define limit at a point) since limits at infinity require one to instead imagine functional
behavior for x-values increasing without bound. Finally, I anticipated that reinventing the definition of limit at infinity might
support the students in defining limit at a point, as the two definitions are similar structurally:
Limit at infinity: lim f (x) = L provided: for every ε > 0, there exists an N such that x > N → |f(x) − L| < ε
x→∞
Limit at a point: limf (x) = L provided: for every ε > 0, there exists a ı > 0, such that 0 < |x − a| < ı → |f(x) − L| < ε
x→a
For these reasons, reinventing the definition of limit at infinity was the central focus of the fifth phase of the teaching
experiment.
At the outset of Session 7, Amy and Mike first generated prototypical examples of limit at infinity in response to the
following prompt.
Prompt: Please generate (draw) as many distinct examples of how a function f could have a limit of 4 as x → ∞. In
other words, what are the different scenarios in which a function could have a limit of 4 as x → ∞?
In response to the task, Amy and Mike drew the examples shown in Fig. 7.
As was the case with limit at a point, Amy and Mike’s efforts to define limit at infinity stalled as they used a variety of
vague, and mathematically invalid, characterizations to describe the functional behavior shown in the graphs in Fig. 7.
“On the interval (b, ∞) the function needs to approach some finite value L.”
“Some interval (b, ∞) on which as x-values increase, their corresponding y-values get closer to L”
“As x gets larger, the distance |L − y| between L and your corresponding y-values continues to decrease”
As the seventh session neared its conclusion, and as Amy and Mike continued in vain to try and pin down what they meant by
“infinitely close,” I encouraged them to consolidate their conditions into one, concise definition. Amy responded as follows:
Amy: Yeah, so. . .there needs to be some interval from a to ∞ where the function is continuous and, um, where the maximum
distance between the y-values and L show a pattern of decreasing as x increases.
Next, Amy began to write what she had just said aloud. Before completing the transcription, though, she stopped writing
and verbalized a significant observation.
Fig. 7. Examples of limit at infinity.
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Fig. 8. Dampened sinusoidal function with Close = 1.
Amy: Is this going to be enough? I mean, because what if, if we leave it at that, then isn’t that true for like, for L. . .like bigger
than or equal to 4?7 You know? Like doesn’t that make it true for like, every single,
Mike: Hmm? Say it again.
Amy: So, okay, we need some interval from a to ∞ on which f is continuous and the maximum distances between y-values
and some finite number L show a pattern of decreasing as x increases. . ..[W]hat I’m having trouble with is just, is
this [definition] specific enough to, like, to L being 4? I mean, isn’t this thing that we just said also true for 5 and 6
and 9.2, because. . .on that interval, f is continuous, and the maximum distances between y-values. . .and 10 are also
decreasing.
This conversation marked a watershed moment in the teaching experiment. Amy recognized that their definition was not
specific enough to a particular y-value, 4, and that while the distance between the y-values of the dampened sinusoidal
function and 4 did show a pattern of decreasing, that was also true for the distance between y-values and other potential
limits greater than 4. Amy’s observation motivated Amy and Mike to refine their definition of limit at infinity in such a way
so as to eliminate all other potential limit candidates not equal to 4.
In an effort to capitalize on Amy’s observation, I encouraged them to first consider what it would mean for the function
to merely be close to a proposed limit L. Since Amy had spontaneously introduced absolute value notation earlier during the
seventh session, I thought defining closeness might lead Amy and Mike to think about progressively restrictive definitions
of closeness, and that they might subsequently shrink y-bands around the limit L and use absolute value statements to
notate those increasingly restrictive definitions. In this way, I thought they might be able to adequately operationalize the
troublesome notion of infinite closeness.
Craig: You’re saying your definition. . ., you don’t want it to be such that someone could conclude. . .this limit could be
anything other than 4.
Amy: Yeah.
Craig: . . .[S]o let’s just say we want to maybe not show that 4 is the limit, but at least that this function gets close to 4.
Because you guys were saying it’s got to get how close to 4 to be the limit?
Amy: Infinite.
Craig: Infinitely close, but let’s – infinitely is kind of tough, so let’s back off of infinitely for a second. Let’s say we just want
to be close to 4. If we were able to show that this thing gets within 1 of 4, then that would keep the limit from being,
say, 6 or 2.
Amy: Mm-hmm.
Craig: Does that make sense? So. . .instead of having a limit of 4, let’s say, let’s describe what it would mean for this function
to get within 1 of 4. How would you write that out?.
Amy: Well, I feel like it would be useful to talk about it being, being bounded. . ..[W]hat if we were to say that there is some
y-value that this function will. . .never get bigger than and it will never get smaller than, you know?
Craig: So if we wanted to. . .show that this function is within 1 of 4, what would your bounds be, then, Amy?
Amy: Uh, within 1, well, I guess that then it would be between 3 and 5, or would it be between 3.5 and 4.5? I guess 3 and 5.
Following this discussion, Amy drew a vertical line from the dampened sinusoidal function down to the x-axis, indicating
a point past which the function would always be within the y-interval (3, 5). She then, for the first time, drew horizontal
bounds around L on the y-axis at y = 3 and y = 5 to indicate the interval in which f would fall within 1 of L (Fig. 8).8
As the conversation continued, it was evident in Mike’s subsequent comments that he realized that being within 1 of L
was a far cry from being “infinitely close,” which is the precision he desired. His remarks below suggest that he had played
7
Amy’s choice of the number 4, here, is in reference to the dampened sinusoidal function shown in Fig. 7, and Amy and Mike’s conversations about that
function having a limit of 4 as x → ∞.
8
The reader will note that the horizontal bounds Amy drew are not such that the function f is always within 1 unit of L = 4 beyond the vertical line seen on
the graph, which she had drawn previously. Amy’s spoken reasoning was such, however, that it was evident that this inaccuracy was merely an oversight
on her part.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
107
Fig. 9. Dampened sinusoidal function with Close = ½.
out in his head the process of defining closeness along the y-axis with increasing strictness and had recognized the need to
eliminate all possibilities but y = 4.
Craig:
Mike:
Craig:
Mike:
Craig:
Mike:
Craig:
Mike:
Well, the way that we’ve drawn this here, would you agree that – you said that this function does what? It gets within
1.
of 4. And I asked you, is that enough for the limit to be 4? And you said what?
No.
Okay, so being within 1 of 4 isn’t enough. How close do you need to be to 4? One’s pretty close.
No, it’s not. You need to be infinitely close to 4.
So 1’s not enough?
No. . ..The interval, like, I mean, the interval needs to be pretty much, like, 4. It needs to be from as close as you can
below 4, as close as you can above 4, infinitely, and. . .you’ll see that the limit. . .is then 4 because the function is
within those bounds, of infinitely close to 4 and 4, as you go towards ∞.
As Mike described the “infinitely” small interval around y = 4, he used his fingers to denote an increasingly smaller interval
on the y-axis. In response, I asked Mike what the bounds around y = 4 would be for the function to lie within ½ of 4.
Craig: If I wanted to be within ½ of 4, then what would my bounds be? Can you draw them?
Mike: Yup. So you’d have 4 ½ and 3 ½.
After drawing the new bounds (shown in Fig. 9), Mike noted that yet more limit candidates had been eliminated from
consideration.
Mike: [W]e know the limit isn’t 5 anymore, because it’s bounded by 4½ and 3½.
Defining closeness in an increasingly restrictive manner appeared to support Mike in developing a keen sense of the
limiting process. In fact, his subsequent comments suggest that he recognized that he and Amy could adequately characterize
limit at infinity by eliminating all limit candidates other than L via a process of choosing progressively tighter bounds around
L.
Mike: And we can keep doing that.
Craig: What do you mean we can keep doing that?
Mike: We can keep making our bounds closer and closer to 4, and the function will keep lying within those new bounds
that we make.
Defining closeness in an iteratively restrictive fashion appeared to move Amy and Mike closer to resolving the second
challenge – mathematizing the unending physical process they were attempting to describe. Mike’s observation, “We can
keep making our bounds closer and closer to 4, and the function will keep lying within those new bounds that we make,”
suggests that he had arrived at what he viewed as an adequate process by which to validate the existence of a limit. While
he may not have been applying the standard of mathematical rigor that would encapsulate the limiting process, he had
reinvented a process for establishing the existence of a limit that he believed would always work.
Similar to Mike, Amy also claimed that defining limit at infinity relied on bounding the function f around the limit L
with increasingly tighter bounds. However, Amy’s perspective at this point in the teaching experiment appeared to differ
in a subtle, yet significant, way from the perspective held by Mike. Below, in describing what would constitute proof that
a function’s limit is 4, Amy appears to have reflected upon prior conversations, wherein close had been defined as being
within 1, or ½, of 4, and to have projected those ideas to capture any infinitesimal definition of closeness.
Amy: [Y]ou would say that you could make the bounds as close to 4 as you want. And you – as long as you take big enough
x-values, you will find a point after which that function stays within those bounds.
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The phrase “as close to 4 as you want” suggests a perspective distinct from that employed by Mike. Amy’s comments suggest
that a limit’s existence relies not on satisfying every definition of closeness, but rather any arbitrarily chosen definition
of closeness. Albeit loosely phrased, Amy’s description of limit at infinity contained the fundamental elements of the conventional definition of limit at infinity. Subsequently, she wrote the following revised definition of limit at infinity on the
board.
Limit at infinity: It is possible to make bounds arbitrarily close to 4 and by taking large enough x-values we will find
an interval (a, ∞) on which f(x) is within those bounds.
Amy’s choice of the words “arbitrarily close” is significant – in short, their use allowed her to encapsulate the limiting process,
thus refining their definition of limit at infinity in a manner that resolved the second challenge they experienced. At least for
Amy, “infinitely close” was an abstraction that could be operationalized via the notion of arbitrarily close. I contend, then, that
Amy’s employment of what I refer to as an arbitrary closeness perspective allowed her to encapsulate the limiting process,
thus supporting her in fully mathematizing the physical process she was trying to describe.
During Session 8, Amy’s arbitrary closeness perspective, as well as her and Mike’s efforts to incorporate more precise
notation, resulted in a refined articulation of limit at infinity capturing the intended meaning of the conventional ε–N
definition.
Final articulation of limit at infinity: lim f (x) = L provided for any arbitrarily small positive number ", by taking
x→∞
sufficiently large values of x, we can find an interval (a, ∞) such that for all x in (a, ∞), |L − f(x)| ≤ ". (Session 8)
The reader might have noted that in the process of resolving the second challenge, Amy and Mike resolved the first
challenge as well – that is, the context of limit at infinity appeared to support them in adopting a y-first perspective.
Indeed, the students’ final definition of limit at infinity is expressed from a y-first perspective. There are a couple of possible
explanations for why such a phenomenon occurred. One possible explanation for the students adopting a y-first perspective
was the iterative nature of defining closeness in a context (limit at infinity) designed to deemphasize the x-axis. Specifically,
I repeatedly gave the students specific error tolerances along the y-axis and asked them how they might characterize what
it means for a function to be within those error tolerances of a pre-determined y-value, L. Furthermore, this process was
motivated by the students’ desire to eliminate y-values they knew were not the limit. Hence, the students’ adoption of a y-first
perspective could be partially explained by particular design details of the defining tasks in which they were engaged–the
tasks resulted in them engaging in mathematical activities that focused their attention on the y-axis. A second possible
explanation for the students’ adoption of a y-first perspective was that they became aware that their prior characterizations
of limit at infinity were deficient. In particular, as they began employing a y-first perspective, they were able to rule out
non-examples that they had not been able to eliminate with their x-first formulations of the definition.
5.6. Phase 6: Using limit at infinity as a template to define limit at a point (Sessions 9–10)
Prior to the ninth session, Amy and Mike’s definition of limit at a point was neither precise nor mathematically valid.
However, reinventing the definition of limit at infinity during the previous two sessions (Sessions 7 and 8) seemingly
provided necessary support for them to successfully refine their definition of limit at a point. At the outset of the ninth
session, I provided Amy and Mike with the evolving articulations of limit at infinity they had constructed during Sessions 7
and 8, pointing out the difference in specificity between one of their earlier articulations (which was mathematically invalid)
and their final articulation.
Earlier articulation: “As x gets larger, the distance |L − y| between L and your corresponding y-values continues to
decrease.”
Revised articulation: “It is possible to make bounds arbitrarily close to 4 and by taking large enough x-values we will
find an interval (a, ∞) on which f(x) is within those bounds.”
Final articulation: lim f (x) = L “provided for any arbitrarily small positive number ", by taking sufficiently large
x→∞
values of x, we can find an interval (a, ∞) such that for all x in (a, ∞), |L − f(x)| ≤ ".”
Having presented the three articulations of limit at infinity shown above, I next wrote Amy and Mike’s most recent definition
of limit at a point, constructed during Session 6, on the board.
Definition #4: The limit L of a function at x = a exists if every time we look at the function more closely as we get
infinitely close to x = a, it bears out the same pattern of behavior, i.e., looks to be approaching some y value L w/no
gaps in the graph. (Session 6)
The excerpt that follows suggests that Amy noted the contrast in specificity and precision between their two definitions.
Craig: So this was a couple weeks ago, this was the most, umm, recent articulation we had for limit at a point. Now you’re
laughing. Amy, why are you laughing?
Amy: ‘Cause it’s so unwieldy. Convoluted.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
109
Fig. 10. Prototypical examples of limit.
It appears, then, that following their initial attempts to define limit at a point, Amy and Mike may have benefited by shifting
their focus to reinventing the definition of limit at infinity. Specifically, at the end of Session 8, they expressed awareness that
their definition of limit at infinity adequately addressed the prototypical examples of limit at infinity they had previously
generated. This was a success they had not experienced during Sessions 4–6 with their characterizations of limit at a point.
At the outset of Session 9, the rigor and mathematical power of Amy and Mike’s definition of limit at infinity, then, appeared
to contrast their current definition of limit at a point in a manner that served to motivate them to make further revisions to
their definition of limit at a point. Further, this presentation of their two definitions set the stage for them to subsequently
make use of some of the structure and notation in their precise definition of limit at infinity as they refined their definition
of limit at a point.
Prototypical examples had served Amy and Mike as tools for constructing their definition of limit at infinity during Phase
5 of the experiment. With this in mind, prior to asking them to begin making revisions to their definition of limit at a point
during Session 9, I encouraged them to regenerate the different ways a function could have a finite limit L at x = a.
Prompt: Draw the different scenarios in which a function f could have a finite limit L at x = a.
In response, Mike drew a function continuous at x = a, a function with a removable discontinuity at x = a with f(a) undefined,
and, finally, a function with a removable discontinuity at x = a with f(a) defined elsewhere, as shown in Fig. 10.
Following Amy and Mike’s regeneration of prototypical examples, I charged them with the task of more precisely characterizing limit at a point. Almost immediately, Amy and Mike began making spontaneous use of their precise definition of
limit at infinity. It appears to have been important that their final articulation of limit at infinity was written on the board,
for Amy was looking at it as she said the following:
Amy: I wonder if it would be useful to invoke some of the same language that we used in uh, that definition. . ..Like along
the lines of umm, say as you take x-values wherein umm, the absolute value of the distance between x and a, umm,
gets arbitrarily small, y gets arbitrarily close to L.
Prior to defining limit at infinity, Amy and Mike had experienced difficulty describing precisely what they meant by “x gets
infinitely close to a on the x-axis”. However, now that they had precisely described arbitrary closeness along the y-axis in the
context of a limit at infinity, they had a foundation from which to work as they returned to the task of describing arbitrary
closeness along the x-axis in the context of a finite limit at a point. As the conversation continued, it became evident that
Mike also saw the potential for using their definition of limit at infinity as a structural template for precisely defining limit
at a point.
Amy:
Mike:
Amy:
Mike:
[W]e’re interested in minimizing the, the distance between x and a.
So basically we’re doing the same thing. We’re making that interval on the x-axis instead of the y-axis. . ..
Same thing as what?
Like how we had up here, for the infinite limit, we had this interval getting closer to a value. And we now want that
interval to be here.
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C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
Fig. 11. Removable discontinuity graph with demarcated x-interval.
As he said “And we now want that interval to be here,” Mike pointed to x = a on the x-axis. It is evident, then, that both Amy
and Mike made use of their definition of limit at infinity as a model for characterizing arbitrary closeness along the x-axis
with absolute value statements.
The evolution of Amy and Mike’s definition of limit throughout the ninth session was not always fluid. Indeed, their
attention was devoted to multiple aspects of the concept simultaneously. In particular, their focus rapidly shifted back and
forth between characterizing behavior along the x- and y-axes. Perhaps not surprisingly, Amy and Mike initially showed
more comfort in precisely describing closeness along the y-axis, a notion to which they had needed to attend during the
seventh and eighth sessions when they defined limit at infinity. Their familiarity with describing closeness along the y-axis
was evident in the first refinement they made to their definition during Session 9.
Definition #5: As x gets arbitrarily close to a, |L − f(x)| gets arbitrarily small.
Following this refinement, Amy again made use of their definition of limit at infinity to more precisely describe what it
would mean for |L − f(x)| to get arbitrarily small.
Amy: And so if you wanted to phrase that in the similar way that we phrased the earlier one, you know, you could say
something along the lines of by, by taking values of x sufficiently close to a, you could satisfy an inequality like that,
wherein this distance is smaller than any small number " you can think of.
In the excerpt above, “the earlier one” refers to their definition of limit at infinity, and “an inequality like that” refers to the
inequality statement |L − f(x)| ≤ ". After Amy reintroduced the notion of an arbitrarily small number ", and noted that “for
any arbitrarily small number ", you can find an x-value that will satisfy that inequality,” Mike wrote a revised definition
that included the same notation for characterizing closeness on the y-axis that they had used in their definition of limit at
infinity.
Definition #6: For any arbitrarily small # # you can find an x-value that satisfies |L − f(x)| ≤ ".
The reader will note that Amy and Mike’s sixth definition was from a y-first perspective. Also, Definition #6 included the use
of the notion of arbitrary closeness, a notion fundamental to Amy and Mike’s success in reinventing the definition of limit at
infinity. Hence, some of the reasoning captured in Amy and Mike’s definition of limit at infinity was beginning to make its
way into their evolving definition of limit at a point. As Amy and Mike discussed how best to describe closeness along the
x-axis, Mike once again utilized their definition of limit at infinity.
Amy: [Referring to Definition #6] [T]hat doesn’t seem to get at the idea that like x is getting closer to a. . ..You could find an
x-value that’s just like way out there or something like that, you know?
Craig: So you, you’re wanting to talk about x-values. . .
Amy: At or around a.
Mike: Yeah, I feel like this, this is the same thing we had for our other interval for limits at infinity so I feel like we want to,
we can just switch some things or change some things because our interval is now to a specific point.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
111
As Mike said “because our interval is now to a specific point,” he looked at the graph shown in Fig. 11, and used his hands to
illustrate increasingly tighter bounds around a, suggesting that he understood the fundamental difference between limit at
infinity and limit at a point, and that he was incorporating ideas from their definition of the former to precisely specify the
latter. Interestingly, Amy appeared unaware that they could use an absolute value inequality statement to represent infinite
closeness on the x-axis, even though they had done so for the y-axis.
Amy: [S]o I’m missing the chunk about being around a. How do we put that in?
Craig: So now you’ve pinned down what it means to be arbitrarily close to.
Amy: Close to L, yeah, but, but now I’m missing the bit about, about x being around a.
As they thought how best to describe x being near a, Mike refined their articulation:
Definition #7: For any arbitrarily small # " we can find a value of x arbitrarily close to a such that |L − f(x)| ≤ ".
As I read their definition back to them aloud, Amy recognized how to characterize arbitrary closeness along the x-axis.
Craig: Okay, now, so you’re saying [a limit exists] provided for any arbitrarily small number ", we can find a value of x
arbitrarily close to a, so not way out there, but close to a such that this [inequality holds].
Amy: Yeah. I mean,. . .I guess there’s no reason why if we wanted to we could put, put the. . .x being arbitrarily close to a in,
you know, couch it in the terms of a similar inequality wherein, wherein there’s an arbitrarily small number # that is,
you know, such that x − a is less than.
The excerpt above again suggests that Amy made a connection to their definition of limit at infinity and saw the opportunity,
as Mike had earlier, for parallel structure in their definition of limit at a point. Here, Amy not only suggested using an
inequality statement with absolute value notation, but also, in a manner consistent with how they first operationalized
closeness in the context of limit at infinity, suggested a corresponding variable # to represent closeness. As Amy made this
suggestion, Mike wrote “i.e., |x − a| < $” next to the phrase “arbitrarily close to a” in their definition. Amy offered no reluctance
to the introduction of this notation, and from that point forth, closeness along the x-axis was quantified in terms of $. This
suggestion led to a further refinement of their definition.
Definition #8: For any arbitrarily small # " we can find a value of x arbitrarily close to a, i.e. |x − a| < $, such that
|L − f(x)| ≤ ". Note: $ is an arbitrarily small #.
After discussing the number of x-values around a for which the inequality |L − f(x)| ≤ " must hold for each choice of ", Amy
and Mike arrived at their final articulation of limit at a point.
Definition #9: limf (x) = L provided that: given any arbitrarily small # ", we can find an (a ± $) such that |L − f(x)| ≤ "
x→a
for all x in that interval except possibly x = a.
6. Summary and concluding thoughts
The purpose of this report was to provide a detailed account of the evolution of two students’ definition of limit. In
addition to offering insight into how students might reason as they reinvent such a definition, Amy and Mike’s experience
also serves as an existence proof that students can reinvent a coherent definition of limit. While other studies (e.g., Fernandez,
2004; Larsen, 2001) have sought to describe how students interpret the formal definition of limit, the study reported here is
unique in that the students who participated in the teaching experiment were posed the challenge of reinventing the formal
definition. Larsen (2009), as well as Zandieh and Rasmussen (2010), has reported the potential students have for reinventing
formal definitions of other mathematical concepts. However, I know of no other studies that have traced students’ reinvention
of the formal definition of limit. The result of Amy and Mike’s efforts is indeed noteworthy – both students had neither seen
nor were aware of the formal definition of limit, yet they were ultimately able to characterize limit in a manner synonymous
to that of the conventional ε–ı definition. Amy and Mike’s end product (Definition #9 in Section 5.6) even captures the
complex quantification structure and subtleties of the ε–ı definition. Amy and Mike’s successful characterization of limit
establishes reinvention as a possible avenue to developing coherent understanding of the formal ε–ı definition. Evidence of
this was apparent in Amy’s reasoning during the tenth session. Despite twelve days having passed between Amy and Mike’s
successful characterization of limit during Session 9 and the tenth session, Amy nevertheless was able to coherently reason
about what it would mean for a function f to have a limit L at x = a. In the following excerpt, the function being discussed is
shown in Fig. 12.
Craig: So,. . .I pick a progressively smaller number for " and hand it to you, then what do you do with that in terms of the, in
terms of the picture I guess?
Amy: I can show you the interval around a for which it’s true that all the corresponding values of f(x) with, that for all values
of x within that interval around a, all of their corresponding values of f(x), umm, produce, uh, distances from L that
are smaller than that " that you give me. . ..[W]hat we can say based off of this formulation is that with the possible
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C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
Fig. 12. Generalized removable discontinuity graph.
exception of f(x) at a specifically, it is true that for every single value, every single of that infinite set of x-values within
that interval, that they are going to correspond to heights that are within that distance of L.
The reasoning Amy demonstrates in the preceding excerpt suggests that students who have never encountered the formal definition of limit have the potential to reinvent it through engagement in purposefully designed tasks. This is not
to say that students can necessarily reinvent such a complicated idea without support–the reinvention principle is not
intended to suggest that students should reinvent everything for themselves. Intentional, carefully planned guidance by the
researcher/teacher does not have to be at odds with the goal of guided reinvention, which is for students to feel ownership
of, and responsibility for, the mathematics they learn. To be clear, Amy and Mike’s reinvention efforts were scaffolded in
significant ways – as the researcher, I intervened on multiple occasions to guide them towards paths I felt might be productive. Directing them to center their discussions around graphical representations, shifting their focus to reinventing a
definition of limit at infinity, and purposely engaging them in conversation designed to elicit a shift to a y-first perspective
were all substantive interventions on my part as the researcher. The key, though, was that Amy and Mike took ownership
of the iterative process of constructing a precise definition of limit, and in so doing, developed sophisticated understanding
of what is a complex mathematical idea.
Amy and Mike’s reinvention efforts also offer some pedagogical lessons. During Phase 6 of the experiment, there was
ample evidence to suggest that reinventing the definition of limit at infinity provided Amy and Mike important support for
reinventing the definition of limit at a point. Indeed, Amy and Mike used their definition of limit at infinity as a structural
template to aid their reinvention efforts–on multiple occasions during the ninth session, they compared their precise definition of limit at infinity with their vague characterizations of limit at a point and subsequently evoked some of the same
notation and language they had used for the former to refine their definition of the latter. Examples of this include the use
of absolute value notation to characterize closeness, first on the y-axis and then eventually on the x-axis. Also, the y-first
structure of their definition of limit at infinity appeared to guide them in shifting their characterization of limit at a point
from an x-first perspective to a y-first perspective. It is noteworthy that from the outset of Session 9, Amy and Mike both
expressed sentiments indicating they were pleased with the precision of their definition of limit at infinity. The success
they had realized in defining limit at infinity appeared to raise their expectations of what they were capable of in regards to
precisely characterizing limit at a point. Pedagogically, this is noteworthy, as it suggests that students attempting to reinvent
and/or understand the definition of limit at a point may be well-served by first coming to understand the definition of limit
at infinity, a seemingly less cognitively complex concept. An inspection of Amy and Mike’s characterizations of limit at a
point indeed suggests that reinventing limit at infinity during Sessions 7 and 8 had a positive effect on Amy and Mike’s
efforts to precisely characterize limit at a point. Fig. 13 captures the key formulations in the evolution of their definition.
Research, to date, has provided little insight into what is entailed in coming to understand the formal definition of
limit. Amy and Mike’s story offers insights into the issues students might struggle with, and subsequently reconcile, as
they attempt to make sense of this complicated concept. Specifically, the research reported here suggests that coherent
reasoning about the formal definition of limit likely requires students to make a couple of important cognitive shifts. First,
while an x-first perspective positively supports students as they develop skills related to finding limit candidates, I argue
that such a perspective is unlikely to support them in validating limit candidates. Evidence from the teaching experiment
suggests that Amy and Mike’s success in reinventing a formal definition of limit was contingent upon their adoption of a
y-first perspective. An instructional intervention that de-emphasized the x-axis and focused the students’ attention on the
y-axis resulted in a flurry of productive lines of reasoning for the students. Second, evidence suggests that Amy and Mike’s
attempts to characterize the infinite limiting process were supported by defining the notion of closeness in an iteratively
restrictive manner. Doing so appeared to provide them a suitable alternative to the nonsensical notion of infinite closeness.
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
Definition #1:
Definition #2:
Definition #3:
Definition #4:
Definition #5:
Definition #6:
Definition #7:
Definition #8:
Definition #9:
113
f has a limit L at x=a provided as x-values get closer to a, y-values get
closer to L. (Session 4)
If you could zoom forever and always get closer to a and L, then you
have a limit. (End of Session 4)
A function has a limit L at a when zooming in FOREVER both
horizontally and vertically yields no gaps that have length > 0 AND that
it looks like it approaches a finite number L. (Session 5)
The limit L of a function at x=a exists if every time we look at the
function more closely as we get infinitely close to x=a, it bears out the
same pattern of behavior, i.e., looks to be approaching some y value L
w/no gaps in the graph. (Session 6)
As x gets arbitrarily close to a, |L-f(x)| gets arbitrarily small. (Session 9)
For any arbitrarily small # λ you can find an x-value that satisfies |Lf(x)|≤λ. (Session 9)
For any arbitrarily small # λ we can find a value of x arbitrarily close to a
such that |L-f(x)|≤λ. (Session 9)
For any arbitrarily small # λ we can find a value of x arbitrarily close to
a, i.e. |x-a|<θ, such that |L-f(x)|≤λ. Note: θ is an arbitrarily small #.
(Session 9)
lim f ( x) = L provided that: given any arbitrarily small # λ, we can find
x? a
an (a±θ) such that |L-f(x)| ≤ λ for all x in that interval except possibly
x=a. (Session 9 – Final Definition)
Fig. 13. Amy and Mike’s evolving definition of limit.
Amy’s adoption of an arbitrary closeness perspective subsequently gave them the necessary mathematical rigor needed to
adequately mathematize the physical process they were attempting to describe. This sophisticated perspective ultimately
led Amy and Mike to a definition of limit capturing the intended meaning of the conventional ε–ı definition.
One might be tempted to infer from the findings reported here that inducing a cognitive shift in students from an x-first
and dynamic perspective of limit to a y-first and arbitrary closeness perspective subsequently places greater value on the
latter and devalues the former. I would argue to the contrary, however, as evidence from the teaching experiment suggests
that it is the ability to employ both perspectives flexibly that allows someone to develop a rich and robust understanding
of the limit concept and its formal definition. Reasoning from an x-first and dynamic perspective can support someone
in developing a sense for the essence of the limit concept – i.e., that limit describes the local behavior of a function as
its independent variable approaches a particular x-value. Meanwhile, as evidence from the teaching experiment suggests,
reasoning from a y-first and arbitrary closeness perspective likely supports learners in understanding the intricacies and
subtleties of the formal definition of limit.
On a related note, Amy and Mike’s adoption of a y-first perspective appeared to facilitate a subsequent recognition of the
distinction between finding limit candidates and subsequently validating those candidates. Their focus on the y-axis appeared
to foreground the presence of a y-value, L, about which they were constructing progressively tighter bounds. This led them
to wonder explicitly why a particular L was the focus of their graphical exploration, which, in turn, appeared to spur their
recognition that the definition they were constructing presupposes the existence of a limit candidate. Thus, evidence suggests
that a relationship may exist between a student adopting a y-first perspective and distinguishing between the actions of
finding and validating limits. The ability to distinguish these two appears to be at least partially supported by the experience
of contemplating the subtleties inherent to the limit concept while attempting to formulate a precise characterization of it.
Further, recognizing the distinction between finding and validating limit candidates appears to support students in coming
to see a need for a rigorous definition. This suggests, at least in the case of limit, that the necessity principle set forth by Harel
(2001) might be addressed as students are in the process of constructing a precise definition for the concept, rather than prior
to their engagement in guided reinvention. Put another way, the activity of attempting to construct a precise definition of
limit might simultaneously increase learners’ recognition of the need for such formality, and thereby constitute a medium
propitious for the emergence of a necessity principle of sorts.
References
Artigue, M. (2000). Teaching and learning calculus: What can be learned from education research and curricular changes in France? In E. Dubinsky, A.
Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 1–15). Providence, RI: American mathematical society.
Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and
Technology, 32(4), 487–500.
Carlson, M., Larsen, S., & Jacobs, S. (2001). An investigation of covariational reasoning and its role in learning the concepts of limit and accumulation. In
Proceedings of the Psychology of Mathematics Education North American Chapter, 23 Snowbird, UT, (pp. 145–153).
114
C. Swinyard / Journal of Mathematical Behavior 30 (2011) 93–114
Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In R. Lesh, & A. E. Kelly (Eds.), Research design in mathematics and science
education (pp. 307–333). Hillsdale, NJ: Erlbaum.
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Dordrecht, The Netherlands: Kluwer Academic Publishers.
Cottrill, J., Dubinsky, E., Nichols, D., Schwinngendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated
process schema. Journal of Mathematical Behavior, 15, 167–192.
Davis, R., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281–303.
Dorier, J. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29, 175–197.
Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification. For the Learning of Mathematics – An International Journal of
Mathematics Education, 8, 44–51.
Ervynck, G. (1981). Conceptual difficulties for first year university students in the acquisition of limit of a function. In Proceedings of the Psychology of
Mathematics Education, 5 Grenoble, France, (pp. 330–333).
Fernandez, E. (2004). The students’ take on the epsilon-delta definition of a limit. Primus, 14(1), 43–54.
Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and learning calculus. In P. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp.
155–176). New York: Macmillan.
Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: Reidel.
Gass, F. (1992). Limits via graphing technology. Primus, 2(1), 9–15.
Gravemeijer, K. (1998). Developmental research as a research method. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A
search for identity (ICMI study publication) (Book 2, pp. 277–297). Dordrecht, The Netherlands: Kluwer.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel, & K. McClain (Eds.),
Symbolizing and communicating in mathematics classrooms (pp. 225–273). Erlbaum, Mahwah, NJ.
Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zazkis (Eds.), The
learning and teaching of number theory (pp. 185–212). Dordrecht, The Netherlands: Kluwer.
Harel, G. (2004). A perspective on “concept image and concept definition in mathematics with particular reference to limits and continuity.”. In T. Carpenter,
J. Dossey, & L. Koehler (Eds.), Classics in mathematics education research (p. 98).
Knuth, E. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500–507.
Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press.
Larsen, S. (2001). Understanding the formal definition of limit. Unpublished manuscript, Arizona State University.
Larsen, S. (2009). Reinventing the concepts of group and isomorphism. Journal of Mathematical Behavior, 28(2–3), 119–137.
Larsen, S., & Zandieh, M. (2007). Proofs and refutations in the undergraduate mathematics classroom. Educational Studies in Mathematics, 67(3), 205–216.
Marrongelle, K., & Rasmussen, C. (2006). Pedagogical content tools: Integrating student reasoning and mathematics in instruction. Journal for Research in
Mathematics Education, 37(5), 388–420.
Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20–24.
Oehrtman, M. (2004). Approximation as a foundation for understanding limit concepts. In Proceedings of the Psychology of Mathematics Education North
American Chapter, 23 Snowbird, UT, (pp. 95–102).
Oehrtman, M. C., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ understanding of function.
In M. P. Carlson, & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics. Washington, DC: Mathematical
Association of America.
Orton, A. (1983). Students’ understanding of integration. Educational Studies in Mathematics, 14, 1–18.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh, & A. E. Kelly (Eds.),
Research design in mathematics and science education (pp. 267–307). Erlbaum: Hillsdale, NJ.
Steinmetz, A. (1977). The limit can be understood. MATYC Journal, 11(2), 114–121.
Stewart, J. (2001). Calculus: Concepts and contexts (2nd ed.). Australia: Brooks/Cole.
Swinyard, C. (2008). Students’ reasoning about the concept of limit in the context of reinventing the formal definition. Unpublished doctoral dissertation, Portland
State University.
Swinyard, C., & Larsen, S. (2010). What Does it Mean to Understand the Formal Definition of Limit?: Insights Gained from Engaging Students in Reinvention.
Journal for Research in Mathematics Education. Manuscript submitted for publication.
Tall, D. O. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of research on
mathematics teaching and learning (pp. 495–511). New York: Macmillan.
Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies
in Mathematics, 12, 151–169.
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Boston:
Kluwer.
von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: Falmer Press.
Williams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219–236.
Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput
(Eds.), Research in collegiate mathematics education IV (pp. 103–127). Providence, RI: American Mathematical Society.
Zandieh, M., & Rasmussen, C. (2010). Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of
reasoning. Journal of Mathematical Behavior.
Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.