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Journal of Mathematical Behavior 30 (2011) 93–114

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The Journal of Mathematical Behavior

journa l homepage: www.e lsev ier .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:DefiningLimitCalculusReinventionRealistic mathematics educationUndergraduate mathematics education

a b s t r a c t

Relatively little is known about how students come to reason coherently about the formaldefinition of limit. While some have conjectured how students might think about limits for-mally, there is insufficient empirical evidence of students making sense of the conventional!–ı definition. This paper provides a detailed account of a teaching experiment designed toproduce such empirical data. In a ten-week teaching experiment, two students, neither ofwhom had previously seen the conventional !–ı definition of limit, reinvented a formal def-inition of limit capturing the intended meaning of the conventional definition. This paperfocuses on the evolution of the students’ definition, and serves not only as an existenceproof that students can reinvent a coherent definition of limit, but also as an illustration ofhow 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 introductorycalculus concepts. Such research has provided valuable insight into common misconceptions students possess regarding theconcepts of function, limit, continuity, derivative, and integral. Summarily, the research has focused almost exclusively onfirst-year Calculus students, and has found that students’ understanding during an introductory calculus course is heavilyprocedural, 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 conceptsmay be sufficient. However, for those majoring in mathematics, a more abstract, conceptual viewpoint is likely necessaryfor the transition to upper division courses.

Many researchers (e.g., Artigue, 2000; Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995) have noted the vital role limitplays as a central concept in calculus and analysis. Cornu (1991) observes that the concept of limit “holds a central positionwhich 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 uponcoherent understanding of the formal definition of limit. While extensive research has been conducted on students’ informalunderstanding of limit (Bezuidenhout, 2001; Cornu, 1991; Dorier, 1995; Monaghan, 1991), little is known about how studentsbeyond 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 !–ı definition of limit is as follows:limx"a

f (x) = L provided: for every ! > 0, there exists a ı > 0, such that 0 < |x # a| < ı " |f(x) # L| < !.

0732-3123/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jmathb.2011.01.001

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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 isinsufficient empirical evidence of students making sense of the conventional !–ı definition.

Freudenthal (1973), the pioneer of Realistic Mathematics Education (RME), argues that learners are constantly in theprocess of actively organizing their world via the construction of what is known in RME as “emergent models.” Constructingdefinitions and reinventing concepts and theorems are two types of mathematical activity that supports students in orga-nizing 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 isdesired in upper-division courses. Specifically, Zandieh and Rasmussen’s work has focused on how students might con-struct and use geometric definitions in the classroom setting as they transition “from informal to more formal ways ofreasoning” (p. 57), and work by Larsen had depicted how students might actively reinvent algebraic concepts like group andisomorphism.

The purpose of this paper is two-fold. First, whereas Larsen (2009) has provided evidence of students reinventing conceptsin upper-division mathematics, this paper contributes to RME by supplying an existence proof that students can reinventdefinitions 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 produceempirical data of how students might specifically reinvent the formal !–ı definition of limit. In a ten-week teaching exper-iment, two students, neither of whom had previously seen the conventional !–ı definition of limit, reinvented a formaldefinition of limit capturing the intended meaning of the conventional definition. This paper focuses on the evolution of thestudents’ 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 thetheoretical perspectives which framed the study, as well as the methodological design and analytic approaches that wereemployed. I then trace the evolution of the two students’ precise definition of limit, highlighting six different phases of theirreinvention.

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 onthe misconceptions students possess. In contrast, there is a paucity of research that describes students’ understanding ofthe formal definition of limit. While some have suggested pedagogical approaches for teaching students about the formaldefinition of limit (Gass, 1992; Steinmetz, 1977), not much is known about how students might reason about the formaldefinition as they make sense of it. The research that does exist suggests that students, for a variety of reasons, struggle tounderstand 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 themsufficiently for studying the limit concept formally (Cornu, 1991; Cottrill et al., 1996; Ervynck, 1981). Another explanationis 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 theformal definition of limit is that they lack the perspective necessary for appreciating the formal definition’s mathematicalrole. 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 anaïve 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 presentedto them. What is unclear, however, is how students might successfully make sense of this sophisticated definition. Researchby Cottrill et al. (1996) has attempted to address this question. Cottrill et al. provide a genetic decomposition of the limitconcept. Based partially on data collected during their study, this seven-step conjecture (Fig. 1) describes how studentsmight 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 aboutthe informal process of finding limits (Steps 1–3 of their genetic decomposition); however, there is a dearth of empiricaldata tracing the evolution of students’ thinking to the point of coherently reasoning about the formal process of validatinglimits. 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 95

1. The action of evaluating f at a single point x that is considered to be close to, or even equal to, a.

2. The action of evaluating the function f at a few points, each successive point closer to athan was the previous point.

3. 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.4. 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.

5. 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,

!<"< ax0 and #<" Lxf )( .6. Apply a quantification schema to connect the reconstructed process of the previous step

to obtain the formal definition of limit.7. A completed !-" 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 justthat – offer insight into how students might reason about the limit concept as they transition to engaging with the conceptmore formally.

2.2. Students’ use of mathematical definitions

Research by Zaslavsky and Shir (2005) indicates that students use differing strategies when defining mathematical con-cepts, depending upon the type of mathematical concept being defined. In a study conducted with four high-level 12th-gradestudents, 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 basedon features or roles of mathematical definitions that the students deemed important (2005, p. 327). For instance, one stu-dent rejected a proposed statement as a definition of square because he felt the definition was too procedural (i.e., thedefinition was a set of instructions on how to construct a square). This same student accepted a different statement as a def-inition of square because the statement was minimal (i.e., it included no superfluous conditions). In contrast, students usedexample-based reasoning for analytic concepts, wherein students’ justifications for a particular definition relied on examplesand non-examples to convince themselves or others regarding a statement about the concept (p. 326). Zaslavsky and Shirreport that students viewed the classification of examples and non-examples of a concept as one of the central purposes ofmathematical 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 analyticdefinitions. This raises the possibility that students might gain similar traction from examples and non-examples as theyformally define limit.

Research by Zaslavsky and Shir (2005) provides evidence of how students reason about mathematical definitions. A relatedissue is how and when students choose to use mathematical definitions. Research by Tall and Vinner (1981) suggests thatformal definitions may not be suitable starting points for students’ explorations of a concept. Tall and Vinner describe conceptimage as the total cognitive structure that is associated with a concept, including all of the mental pictures and associatedproperties 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 insuch contexts are dominated by one’s concept image. There is an inherent expectation at the collegiate level, however, thatas students engage in technical contexts “the concept image will be formed by means of the concept definition and will becompletely controlled by it” (p. 71). Vinner reasons, though, that even in technical contexts, students’ concept formationis dominated by their concept image: “It is hard to train a cognitive system to act against its nature and to force it to


consult definitions” (p. 72). Harel (2004) agrees, noting that “although formal concept definitions may be the ultimate goalof mathematics instruction, in the absence of rich and flexible corresponding concept images, students are unable to retainthese concept definitions for a long period of time.” Thus, it seems students would be more likely to use a robust conceptimage of limit to make sense of the formal definition, rather than develop their concept image of limit based on the formaldefinition. Such mathematical activity would be consistent with what Zandieh and Rasmussen (2010) classify as situationalactivity. Building on the work of Gravemeijer (1999) and Tall and Vinner (1981), Zandieh and Rasmussen developed a definingas a mathematical activity (DMA) framework to capture how learners might construct and use definitions in the classroomsetting as they transition “from informal to more formal ways of reasoning” (p. 57). Central to Zandieh and Rasmussen’s workwas describing the relationships between students’ concept images and concept definitions as they were engaged at each ofthe four levels of mathematical activity espoused by Gravemeijer (1999) – situational, referential, general, and formal. The firstlevel 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 ofconstructing a definition can subsequently support higher-level mathematical activity (i.e., referential, general, and formallevel activity), wherein concept images (and in turn, concept definitions) of higher-level concepts are constructed. Larsen’saccount (2009) helps to explicate the general level of activity in the DMA framework. The work presented in this paper helpsto 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 definitionof the concept. In particular, I attempted to describe what challenges the participating students experienced in their rein-vention efforts, and how those challenges were resolved. Given the goals of the study, the epistemological stance of radicalconstructivism (RC) (von Glasersfeld, 1995) served as a useful perspective in terms of research design. One central tenet ofRC is that “knowledge is not passively received either through the senses or by way of communication, but is actively builtup by the cognizing subject” (von Glasersfeld, 1995, p. 51). The idea that sophisticated mathematical understandings areconstructed via active engagement by the learner was central to my research. Instructional tasks were designed to encouragethe participating students to actively build upon their intuitive understandings of limit by conjecturing and refuting formu-lations of the definition in an iterative manner. In this sense, the students’ interactions were reminiscent of mathematicalconversations described by Lakatos (1976).

In addition to the overarching perspective of RC, the perspective of developmental research guided the instructional designfor 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 RMEheuristic commonly associated with developmental research is guided reinvention, which has been employed in numer-ous content areas of postsecondary mathematics education (see Larsen, 2009; Marrongelle & Rasmussen, 2006). Guidedreinvention is described by Gravemeijer, Cobb, Bowers, and Whitenack (2000) as “a process by which students formalizetheir informal understandings and intuitions” (p. 237). An important aspect of this process is the identification of plausi-ble instructional starting points that avoid what Freudenthal refers to as an antididactic inversion – that is, to avoid usingmature, conventional symbolizations of mathematical concepts as starting points for instruction. In an effort to avoid suchan inversion, I attempted to create an environment intended to mimic important aspects of the mathematical setting thatCauchy and Weierstrass experienced. Neither of the mathematicians’ definitions was a reformulation or an interpretation ofthe traditional formal definition. On the contrary, these mathematicians constructed their respective definitions in responseto an inherent need to classify functional behavior. I felt, then, that I might learn a great deal about students’ reasoning aboutthe 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 describehow students reason about limit as they interpret the conventional !–ı definition of limit, my research focused on how stu-dents reason about limit in the context of reinventing a definition which captures the intended meaning of the conventional!–ı definition. To ensure such a setting, students selected for the study had no prior experience with the conventional !–ıdefinition.

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 atask-based Informal Limit Reasoning Survey with twelve undergraduate students from a large, urban university in the PacificNorthwest region of the United States. All of the survey participants were students in two or more of the courses forminga three-quarter introductory Calculus sequence I taught during the 2006–2007 academic year. Four of the twelve studentswere selected for two teaching experiments. Each teaching experiment consisted of ten 60–100 min paired sessions and one30–60 min individual exit interview. The paired sessions were conducted in a classroom, with the students responding toinstructional tasks on the blackboard in the front of the room. Only the participating students, researcher, and a research


assistant were present for each session. Each session was generally separated by a span of 6–10 days, allowing time forongoing analysis between sessions and subsequent construction of appropriate instructional activities based on the ongoinganalysis. All sessions, including the individual exit interviews, were videotaped by a research assistant. These twenty-twovideotaped 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 informalunderstanding of limit, as well as my estimation of their ability to work effectively in tandem to reinvent the definitionof limit. Evidence of these criteria existed in the students’ responses to the task-based survey, as well as in their writtenwork throughout the three-term introductory Calculus sequence I taught. Additionally, a criterion for selection was thatparticipants have no previous experience with the conventional !–ı formal definition of limit. Students studied the ideaof limit only informally during the three-term introductory sequence. The concept was introduced as a means by whichone 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 initialfocus of discussion. Subsequently, students were taught methods of algebraic manipulation that would allow them to usedirect substitution to determine the limit of a function. Emphasis was also placed on distinguishing between graphicalexamples and non-examples of limits. The conventional formal definition was not presented in class, and only appeared inan appendix of the course textbook (Stewart, 2001). Student responses to the survey and their observed activity during theteaching 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 oldmathematics 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 capturedthe intended meaning of the conventional !–ı definition. Instructional activities were primarily focused on discussing limitsin a graphical setting, in hopes that the absence of analytic expressions might support the enrichment of the visual aspectsof the students’ respective concept images. Tasks included the students generating prototypical examples of limit, whichsubsequently served as sources of motivation for the students as they attempted to precisely characterize what it means fora function to have a limit. The majority of each teaching experiment, then, comprised a period of iterative refinement for thestudents; as they attempted to characterize limit precisely, the examples and non-examples of limit that they encounteredcreated cognitive conflict for them, which they sought to resolve by refining their characterization. To this extent, the taskdesign 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 isnecessarily dependent upon one’s prior experiences. As the students engaged with the instructional tasks, their observ-able actions and behaviors provided evidence of how they might be interpreting said tasks. In a manner consistent withSteffe 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 rea-soning given interpretations of instructional tasks. Thus, data analysis was a cyclic process in which hypotheses aboutstudents’ reasoning were generated, reflected upon, and then refined until increasingly stable and viable hypothesesemerged.

The analysis of data for this study occurred at a variety of levels. As each teaching experiment was unfolding, I conductedan ongoing analysis, which informed decisions about subsequent sessions within the same teaching experiment. Ongoinganalysis included transcribing each session, paying particular attention to articulated thoughts that seemed to provide thestudents leverage, the voicing of concerns or perceived hurdles that needed to be overcome, and signs of/causes for progressand 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 tasksthat 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 pairof students. Post analysis consisted of reviewing the videos and transcriptions of all ten sessions, highlighting noteworthyexcerpts 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).


subsequently led to the construction of a lengthy description of each pair’s reinvention. These descriptions were foundationalin analysis, because they allowed me to view the evolution and growth of each pair of students’ reasoning over the courseof ten sessions.

Finally, following both teaching experiments, I conducted a retrospective analysis (Cobb, 2000), in which I analyzed theentire corpus of data at a deeper level than the preceding analyses. Retrospective analysis consisted of reviewing the postanalyses of both teaching experiments, comparing and contrasting student reasoning between the four students. Doing soled to a refinement of my description of thematic elements present in student reasoning. At all three levels of analysis, Iengaged in frequent discussions with other mathematics educators who had intimate knowledge of the study in an effortto 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 reinventionof the definition of limit unfolded in roughly six phases:

Phase 1: Getting informal ideas out on the table (Sessions 1–3)Phase 2: Initial x-first characterizations of limit (Sessions 3–4)Phase 3: Employment of a zooming metaphor (Sessions 4–5)Phase 4: Dissatisfaction with the infinite limiting process (Sessions 5–6)Phase 5: Characterizing limit at infinity (Sessions 7–8)Phase 6: 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 establishedconsensus about their informal ideas related to the limit concept. Tasks included discussing how one might establish thevalue 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:

Task 1 : What is limx"4

34

x + 3? Task 2 : What is limx"4

x2 # 2x # 8x # 4

?

These algebraic tasks prompted Amy and Mike to voice and negotiate fundamental elements of their concept image oflimit over the first three sessions. For example, in contrasting the two tasks, Mike noted that direct substitution allowsone to determine the value of a limit algebraically, and that if a function is not continuous at the limiting value, algebraicmanipulation 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 ordifferent?.

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 youcould 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 algebraicmanipulation alters a function containing a removable discontinuity into a continuous function that has the same limitat the limiting point. She further observed that computing the function value for a continuous function is a “shortcut” todetermining 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 theconventional !–N context.


Fig. 2. !–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 cutfor 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 lastweek 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 alsodiscussed 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 beendoing at other points close to it. . ..Built into the definition of the limit is that it’s an approximation of the behaviorof 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 thesecond case it doesn’t, but that’s not what we’re interested in, right? We’re just interested in what the relationship isaround it.

Amy and Mike’s initial discussions of what a limit represents helped establish intuitive notions that I later leveraged tosupport them in reinventing a formal definition. In this sense, “getting informal ideas out on the table” not only helped Amyand 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 definitionof limit. Engaging Amy and Mike in thought and discussion about limits in a graphical setting seemed important, given thevisual imagery that such a graphical setting stands to provide as a basis for the formal definition of limit. Indeed, one couldimagine students developing an approach consistent with the picture shown in Fig. 2 (Stewart, 2001), using geometriclanguage to produce a formal definition of limit.

These initial sessions, however, shed light on Amy and Mike’s substantial reliance on algebraic representations andgeneral reluctance to engage in graphical conversations, unless they had been assured that such graphs had originated froman 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 graphicalrepresentation existed in and of itself. Amy’s comments below suggest that she did not separate graphical representationsof 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 equationor 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, likea pretty picture?

Amy’s reliance on algebraic representations and general distrust for graphical representations is consistent with findings inthe research literature (Knuth, 2000). Following Amy’s comments, I attempted to convince both her and Mike that functions


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 isevident 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 para-lyzing, 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 providethem 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 ofvalidating 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, itwas evident that they believed that a function could not exist without a defining algebraic representation, and that directsubstitution 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 thefollowing question: In the absence of an algebraic representation of a function, how might one determine the existence ofa limit at a particular point? Unfortunately, Amy and Mike’s reliance on algebraic representations of functions made such aquestion 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 thatthe absence of analytic expressions might support the enrichment of the visual aspects of Amy and Mike’s respective conceptimages. To assuage their concerns, I conceded that if it helped them to do so, they could imagine that the graphs we werediscussing 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 ofgraphical representations waned.

5.2.1. Generation of prototypical examples of limitTasks 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 exampleof each of the three scenarios in which a real-valued function f can have a general limit, but also drawing an example of aone-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 limitat 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 isundefined at 5. And this is one where the function is not undefined but the value of the function is. . .6, at 5, despitethe 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.


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 wouldnot 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 toprecisely characterize what it means for a function to have a limit. The rest of the teaching experiment, then, constituteda period of iterative refinement for Amy and Mike; as they attempted to characterize limit precisely, the examples andnon-examples of limit that they encountered created cognitive conflict for them, which they sought to relieve by refiningtheir characterization. Throughout this refinement process, I encouraged Amy and Mike to incorporate explicit language intheir characterization of limit as they mulled over and wrestled with the essential characteristics and subtleties associatedwith the concept.

5.2.2. Initial x-first characterizationsEvidence 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 forreasoning from what I refer to as an x-first perspective as they made initial attempts at precisely characterizing limit at apoint. By x-first, I mean that Amy and Mike focused their attention first on the inputs (x-values) and then on the correspondingoutputs (y-values), in a manner conducive to finding a limit candidate. Amy and Mike began employing an x-first perspectiveas early as the end of the third session, in response to being asked how they would convince someone that the function inFig. 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-valuethat, 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 tojustify that the limit of another function at x = 0 was 2. His remarks towards the end of the third session indicated that, ifpressed, 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 atthe outset of the session, their initial characterization of limit was cast from this same x-first perspective (i.e., as a descriptionof 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 betrue about that function? Or what would have to be true about that graph?


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 herthinking. Instead of initiating one’s exploration of functional behavior on the x-axis, a student must first focus his or herattention on the y-axis. As Oehrtman, Carlson, and Thompson (2008) suggest: “In order to understand the definition of alimit, a student must coordinate an entire interval of output values, imagine reversing the function process and determinethe corresponding region of input values” (p. 160). Thus, the process of validating a candidate requires a student to recognizethat his or her customary ritual of first considering input values is no longer appropriate. Instead the student must considerfirst a range of output values around the candidate, project back to the x-axis, and subsequently determine whether aninterval around the limit value exists that will produce outputs within the pre-selected y-interval (see Fig. 2). This y-firstvalidation process is the process that is formalized via the definition of limit. As the teaching experiment progressed, Amyand Mike’s inclination to reason from an x-first perspective posed a significant challenge to their efforts to reinvent a precisedefinition 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, despiteproviding them with a graph of a jump discontinuity (shown in Fig. 5) intended to illustrate for them the insufficiency oftheir initial x-first characterization.

Although both students had noted prior to Session 4 that the concept of limit describes local functional behavior, neitherof them had yet explicated local functional behavior in any concrete way. As the conversation continued, however, the jumpdiscontinuity graph in Fig. 5 provided fertile ground for articulating the subtleties involved in describing local functionalbehavior. I pointed out to Amy and Mike that their initial definition of limit would incorrectly conclude that a function witha 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 alimit exists, Amy employed, for the first time, a zooming metaphor to describe the graphical inspection one might undertaketo 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 agraphing calculator, would help determine the non-existence of a limit. In short, Amy and Mike equated the non-existenceof a limit at x = a with a jump discontinuity at x = a, and concluded that iteratively zooming-in on a calculator could help onelocate such jump discontinuities. Inherent in their discussion, however, was that graphical inspection via zooming couldonly 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 meantthat 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 shemerely misspoke.


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 helpyou 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: What would have to happen for that limit to actually be 8? Under what conditions would that limit be 8?Mike: I would have to be able to zoom in infinitely. I can’t really comprehend what that would be butCraig: Oh, like keep this process going forever, or whatever?Mike: Yeah.Craig: Like this zooming-in process?. . .Okay. So you’re saying that if I could do that forever.Mike: Umm-hmm.Craig: . . .Okay. Anything to add to that Amy?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 causedthem to question whether an iterative zooming process could ever establish the existence of a limit. Amy and Mike’s concernsregarding 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 responseto Amy and Mike’s initial x-first characterizations of limit, I attempted to leverage their use of a zooming metaphor to elicit ashift to a y-first perspective. Although Amy and Mike used zooming as a metaphor for inspecting a function’s local behaviorduring the fourth session, they were not specific about what it means to zoom. At the outset of the fifth session, I felt thatsuch 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 afunction’s graph. In sum, Amy and Mike both consistently expressed the opinion that zooming along the y-axis pronouncesthe existence of a vertical jump discontinuity, whereas zooming along the x-axis does not. Thus, it appeared, by the endof the fifth session, that Amy and Mike were beginning to focus their attention on the y-axis. However, efforts during thesixth session to elicit a shift to a y-first perspective by having Amy and Mike explore a function with infinite oscillationsbackfired. 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 onthe x-axis resulted in the calculator graphing the function in a manner suggestive of a jump discontinuity. This led them toconclude 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 makeit 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 shiftto a y-first perspective. The following formulation,6 constructed during Session 6, illustrates that even after much discussionand 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 getinfinitely close to x = a, it bears out the same pattern of behavior, i.e., looks to be approaching some y-value L w/novertical 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 oflimit. 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, Amyand Mike began employing a zooming metaphor to describe the act of inspecting the local behavior of the function in greaterdetail. By “zooming in on the graph,” they believed they would eventually be able to detect jump discontinuities that wouldprohibit 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 moreclosely to Amy and Mike’s concern about whether an infinite process could be completed, and thus, is described in Section 5.4.


would require them to zoom in an infinite number of times. The very real problem of attempting to mathematize a physicalprocess 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 alimit 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 gapsthat 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 asthey attempted to define limit. Foremost among Amy’s concerns was the seemingly impossible task of describing how onemight 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 ofanalogies or hypothetical situations, you know? Like . . .the hypothetical situation in which you are doing somethingforever. . ..I guess like the first thing that leaps to mind for me is that we’re trying to parse out what we mean by, bythese 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, onceagain 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 onlynot 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 whatyou 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 themethods that we’ve been talking about, all we can do is you know just like, you know, grind through endless iterationsuntil we get tired of it and like I give up. And you know like all you can do is find. . .the level of examination whichdisproves your idea but you can’t ever get to where you can conclusively prove it through the methods we’ve beendiscussing.

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 moreprecisely what they meant by x getting close to a, and y getting close to L, respectively. In response, they appeared to believethis challenge would be resolved by defining infinite closeness. Tension then arose as they recognized that the construct theywere attempting to define was nonsensical. Indeed, the dilemma of characterizing the completion of an infinite processcontinued 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 forwhen we zoom when we have lurking in the back this presupposition that whatever that means to zoom,. . .we haveto 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. Twocentral challenges persisted: 1) Amy and Mike’s inclination to reason from an x-first perspective; and, 2) Their inability toadequately characterize the infinite limiting process. These challenges motivated a pedagogical shift at the outset of theseventh 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 ay-first perspective, I shifted their attention to defining limit at infinity, anticipating, for multiple reasons, that their efforts tocharacterize and formalize limit at infinity might provide necessary support for defining limit at a point. First, I felt that limitat infinity would be a less cognitively taxing context in which to characterize what Amy and Mike meant by x-values gettinginfinitely close to a, and y-values getting infinitely close to L. In the context of limit at infinity, instead of having to contemplateand coordinate the troublesome notion of infinite closeness on both the x- and y-axes, as is necessary in the context of limitat a point, the students would need only resolve issues of closeness on one of the two axes, as the conventional !–N picturein 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 wasalso a context supportive of shifting the students’ attention to the y-axis – I believed the students would be unlikely to focustheir attention first upon inspecting functional behavior for x-values closer to a particular limit point (a) on the x-axis (as


Fig. 6. Conventional !–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 functionalbehavior for x-values increasing without bound. Finally, I anticipated that reinventing the definition of limit at infinity mightsupport the students in defining limit at a point, as the two definitions are similar structurally:

Limit at infinity: limx"$

f (x) = L provided: for every ! > 0, there exists an N such that x > N " |f(x) # L| < !

Limit at a point: limx"a

f (x) = L provided: for every ! > 0, there exists a ı > 0, such that 0 < |x # a| < ı " |f(x) # L| < !

For these reasons, reinventing the definition of limit at infinity was the central focus of the fifth phase of the teachingexperiment.

At the outset of Session 7, Amy and Mike first generated prototypical examples of limit at infinity in response to thefollowing prompt.

Prompt: Please generate (draw) as many distinct examples of how a function f could have a limit of 4 as x " $. Inother 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 maximumdistance 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 writingand verbalized a significant observation.

Fig. 7. Examples of limit at infinity.


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 biggerthan 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, isthis [definition] specific enough to, like, to L being 4? I mean, isn’t this thing that we just said also true for 5 and 6and 9.2, because. . .on that interval, f is continuous, and the maximum distances between y-values. . .and 10 are alsodecreasing.

This conversation marked a watershed moment in the teaching experiment. Amy recognized that their definition was notspecific enough to a particular y-value, 4, and that while the distance between the y-values of the dampened sinusoidalfunction and 4 did show a pattern of decreasing, that was also true for the distance between y-values and other potentiallimits greater than 4. Amy’s observation motivated Amy and Mike to refine their definition of limit at infinity in such a wayso 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 functionto merely be close to a proposed limit L. Since Amy had spontaneously introduced absolute value notation earlier during theseventh session, I thought defining closeness might lead Amy and Mike to think about progressively restrictive definitionsof closeness, and that they might subsequently shrink y-bands around the limit L and use absolute value statements tonotate those increasingly restrictive definitions. In this way, I thought they might be able to adequately operationalize thetroublesome 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 beanything 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 wantto 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, indicatinga point past which the function would always be within the y-interval (3, 5). She then, for the first time, drew horizontalbounds 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 Lwas 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 thatfunction 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 onthe graph, which she had drawn previously. Amy’s spoken reasoning was such, however, that it was evident that this inaccuracy was merely an oversighton her part.


Fig. 9. Dampened sinusoidal function with Close =!.

out in his head the process of defining closeness along the y-axis with increasing strictness and had recognized the need toeliminate all possibilities but y = 4.

Craig: Well, the way that we’ve drawn this here, would you agree that – you said that this function does what? It gets withinMike: 1.Craig: of 4. And I asked you, is that enough for the limit to be 4? And you said what?Mike: No.Craig: Okay, so being within 1 of 4 isn’t enough. How close do you need to be to 4? One’s pretty close.Mike: No, it’s not. You need to be infinitely close to 4.Craig: So 1’s not enough?Mike: 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 iswithin 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 intervalon the y-axis. In response, I asked Mike what the bounds around y = 4 would be for the function to lie within ! of 4.

Craig: If I wanted to be within ! of 4, then what would my bounds be? Can you draw them?Mike: Yup. So you’d have 4 ! and 3 !.

After drawing the new bounds (shown in Fig. 9), Mike noted that yet more limit candidates had been eliminated fromconsideration.

Mike: [W]e know the limit isn’t 5 anymore, because it’s bounded by 4! and 3!.

Defining closeness in an increasingly restrictive manner appeared to support Mike in developing a keen sense of thelimiting process. In fact, his subsequent comments suggest that he recognized that he and Amy could adequately characterizelimit at infinity by eliminating all limit candidates other than L via a process of choosing progressively tighter bounds aroundL.

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 secondchallenge – mathematizing the unending physical process they were attempting to describe. Mike’s observation, “We cankeep 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. Whilehe may not have been applying the standard of mathematical rigor that would encapsulate the limiting process, he hadreinvented 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 Lwith increasingly tighter bounds. However, Amy’s perspective at this point in the teaching experiment appeared to differin a subtle, yet significant, way from the perspective held by Mike. Below, in describing what would constitute proof thata function’s limit is 4, Amy appears to have reflected upon prior conversations, wherein close had been defined as beingwithin 1, or !, 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 enoughx-values, you will find a point after which that function stays within those bounds.


The phrase “as close to 4 as you want” suggests a perspective distinct from that employed by Mike. Amy’s comments suggestthat a limit’s existence relies not on satisfying every definition of closeness, but rather any arbitrarily chosen definitionof closeness. Albeit loosely phrased, Amy’s description of limit at infinity contained the fundamental elements of the con-ventional definition of limit at infinity. Subsequently, she wrote the following revised definition of limit at infinity on theboard.

Limit at infinity: It is possible to make bounds arbitrarily close to 4 and by taking large enough x-values we will findan 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 forAmy, “infinitely close” was an abstraction that could be operationalized via the notion of arbitrarily close. I contend, then, thatAmy’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 precisenotation, resulted in a refined articulation of limit at infinity capturing the intended meaning of the conventional !–Ndefinition.

Final articulation of limit at infinity: limx"$

f (x) = L provided for any arbitrarily small positive number ", by taking

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 firstchallenge 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 possibleexplanations for why such a phenomenon occurred. One possible explanation for the students adopting a y-first perspectivewas 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 whatit means for a function to be within those error tolerances of a pre-determined y-value, L. Furthermore, this process wasmotivated by the students’ desire to eliminate y-values they knew were not the limit. Hence, the students’ adoption of a y-firstperspective could be partially explained by particular design details of the defining tasks in which they were engaged–thetasks resulted in them engaging in mathematical activities that focused their attention on the y-axis. A second possibleexplanation for the students’ adoption of a y-first perspective was that they became aware that their prior characterizationsof limit at infinity were deficient. In particular, as they began employing a y-first perspective, they were able to rule outnon-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) seeminglyprovided necessary support for them to successfully refine their definition of limit at a point. At the outset of the ninthsession, I provided Amy and Mike with the evolving articulations of limit at infinity they had constructed during Sessions 7and 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 todecrease.”Revised articulation: “It is possible to make bounds arbitrarily close to 4 and by taking large enough x-values we willfind an interval (a, $) on which f(x) is within those bounds.”Final articulation: lim

x"$f (x) = L “provided for any arbitrarily small positive number ", by taking sufficiently large

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 definitionof 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 getinfinitely close to x = a, it bears out the same pattern of behavior, i.e., looks to be approaching some y value L w/nogaps 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’relaughing. Amy, why are you laughing?

Amy: ‘Cause it’s so unwieldy. Convoluted.


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 shiftingtheir focus to reinventing the definition of limit at infinity. Specifically, at the end of Session 8, they expressed awareness thattheir definition of limit at infinity adequately addressed the prototypical examples of limit at infinity they had previouslygenerated. 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, appearedto contrast their current definition of limit at a point in a manner that served to motivate them to make further revisions totheir definition of limit at a point. Further, this presentation of their two definitions set the stage for them to subsequentlymake use of some of the structure and notation in their precise definition of limit at infinity as they refined their definitionof limit at a point.

Prototypical examples had served Amy and Mike as tools for constructing their definition of limit at infinity during Phase5 of the experiment. With this in mind, prior to asking them to begin making revisions to their definition of limit at a pointduring 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 char-acterizing limit at a point. Almost immediately, Amy and Mike began making spontaneous use of their precise definition oflimit 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 alongthe 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 getsinfinitely close to a on the x-axis”. However, now that they had precisely described arbitrary closeness along the y-axis in thecontext of a limit at infinity, they had a foundation from which to work as they returned to the task of describing arbitrarycloseness along the x-axis in the context of a finite limit at a point. As the conversation continued, it became evident thatMike also saw the potential for using their definition of limit at infinity as a structural template for precisely defining limitat a point.

Amy: [W]e’re interested in minimizing the, the distance between x and a.Mike: So basically we’re doing the same thing. We’re making that interval on the x-axis instead of the y-axis. . ..Amy: Same thing as what?Mike: 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.


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 Amyand Mike made use of their definition of limit at infinity as a model for characterizing arbitrary closeness along the x-axiswith absolute value statements.

The evolution of Amy and Mike’s definition of limit throughout the ninth session was not always fluid. Indeed, theirattention was devoted to multiple aspects of the concept simultaneously. In particular, their focus rapidly shifted back andforth between characterizing behavior along the x- and y-axes. Perhaps not surprisingly, Amy and Mike initially showedmore comfort in precisely describing closeness along the y-axis, a notion to which they had needed to attend during theseventh and eighth sessions when they defined limit at infinity. Their familiarity with describing closeness along the y-axiswas 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 itwould 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 saysomething 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 theinequality statement |L # f(x)| % ". After Amy reintroduced the notion of an arbitrarily small number ", and noted that “forany arbitrarily small number ", you can find an x-value that will satisfy that inequality,” Mike wrote a revised definitionthat included the same notation for characterizing closeness on the y-axis that they had used in their definition of limit atinfinity.

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 useof the notion of arbitrary closeness, a notion fundamental to Amy and Mike’s success in reinventing the definition of limit atinfinity. Hence, some of the reasoning captured in Amy and Mike’s definition of limit at infinity was beginning to make itsway into their evolving definition of limit at a point. As Amy and Mike discussed how best to describe closeness along thex-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 anx-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.


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 toillustrate increasingly tighter bounds around a, suggesting that he understood the fundamental difference between limit atinfinity and limit at a point, and that he was incorporating ideas from their definition of the former to precisely specify thelatter. Interestingly, Amy appeared unaware that they could use an absolute value inequality statement to represent infinitecloseness 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 xarbitrarily 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 aninequality statement with absolute value notation, but also, in a manner consistent with how they first operationalizedcloseness in the context of limit at infinity, suggested a corresponding variable # to represent closeness. As Amy made thissuggestion, Mike wrote “i.e., |x # a| < $” next to the phrase “arbitrarily close to a” in their definition. Amy offered no reluctanceto the introduction of this notation, and from that point forth, closeness along the x-axis was quantified in terms of $. Thissuggestion 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 ", Amyand Mike arrived at their final articulation of limit at a point.

Definition #9: limx"a

f (x) = L provided that: given any arbitrarily small # ", we can find an (a ± $) such that |L # f(x)| % "

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. Inaddition to offering insight into how students might reason as they reinvent such a definition, Amy and Mike’s experiencealso 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 isunique in that the students who participated in the teaching experiment were posed the challenge of reinventing the formaldefinition. Larsen (2009), as well as Zandieh and Rasmussen (2010), has reported the potential students have for reinventingformal definitions of other mathematical concepts. However, I know of no other studies that have traced students’ reinventionof the formal definition of limit. The result of Amy and Mike’s efforts is indeed noteworthy – both students had neither seennor were aware of the formal definition of limit, yet they were ultimately able to characterize limit in a manner synonymousto that of the conventional !–ı definition. Amy and Mike’s end product (Definition #9 in Section 5.6) even captures thecomplex quantification structure and subtleties of the !–ı definition. Amy and Mike’s successful characterization of limitestablishes reinvention as a possible avenue to developing coherent understanding of the formal !–ı definition. Evidence ofthis was apparent in Amy’s reasoning during the tenth session. Despite twelve days having passed between Amy and Mike’ssuccessful characterization of limit during Session 9 and the tenth session, Amy nevertheless was able to coherently reasonabout what it would mean for a function f to have a limit L at x = a. In the following excerpt, the function being discussed isshown 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, interms 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 valuesof x within that interval around a, all of their corresponding values of f(x), umm, produce, uh, distances from L thatare smaller than that " that you give me. . ..[W]hat we can say based off of this formulation is that with the possible


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 withinthat 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 for-mal definition of limit have the potential to reinvent it through engagement in purposefully designed tasks. This is notto say that students can necessarily reinvent such a complicated idea without support–the reinvention principle is notintended to suggest that students should reinvent everything for themselves. Intentional, carefully planned guidance by theresearcher/teacher does not have to be at odds with the goal of guided reinvention, which is for students to feel ownershipof, and responsibility for, the mathematics they learn. To be clear, Amy and Mike’s reinvention efforts were scaffolded insignificant ways – as the researcher, I intervened on multiple occasions to guide them towards paths I felt might be pro-ductive. Directing them to center their discussions around graphical representations, shifting their focus to reinventing adefinition of limit at infinity, and purposely engaging them in conversation designed to elicit a shift to a y-first perspectivewere all substantive interventions on my part as the researcher. The key, though, was that Amy and Mike took ownershipof the iterative process of constructing a precise definition of limit, and in so doing, developed sophisticated understandingof what is a complex mathematical idea.

Amy and Mike’s reinvention efforts also offer some pedagogical lessons. During Phase 6 of the experiment, there wasample evidence to suggest that reinventing the definition of limit at infinity provided Amy and Mike important support forreinventing the definition of limit at a point. Indeed, Amy and Mike used their definition of limit at infinity as a structuraltemplate to aid their reinvention efforts–on multiple occasions during the ninth session, they compared their precise defi-nition of limit at infinity with their vague characterizations of limit at a point and subsequently evoked some of the samenotation and language they had used for the former to refine their definition of the latter. Examples of this include the useof absolute value notation to characterize closeness, first on the y-axis and then eventually on the x-axis. Also, the y-firststructure of their definition of limit at infinity appeared to guide them in shifting their characterization of limit at a pointfrom an x-first perspective to a y-first perspective. It is noteworthy that from the outset of Session 9, Amy and Mike bothexpressed sentiments indicating they were pleased with the precision of their definition of limit at infinity. The successthey had realized in defining limit at infinity appeared to raise their expectations of what they were capable of in regards toprecisely characterizing limit at a point. Pedagogically, this is noteworthy, as it suggests that students attempting to reinventand/or understand the definition of limit at a point may be well-served by first coming to understand the definition of limitat infinity, a seemingly less cognitively complex concept. An inspection of Amy and Mike’s characterizations of limit at apoint indeed suggests that reinventing limit at infinity during Sessions 7 and 8 had a positive effect on Amy and Mike’sefforts 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 oflimit. Amy and Mike’s story offers insights into the issues students might struggle with, and subsequently reconcile, asthey attempt to make sense of this complicated concept. Specifically, the research reported here suggests that coherentreasoning 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 arguethat such a perspective is unlikely to support them in validating limit candidates. Evidence from the teaching experimentsuggests that Amy and Mike’s success in reinventing a formal definition of limit was contingent upon their adoption of ay-first perspective. An instructional intervention that de-emphasized the x-axis and focused the students’ attention on they-axis resulted in a flurry of productive lines of reasoning for the students. Second, evidence suggests that Amy and Mike’sattempts to characterize the infinite limiting process were supported by defining the notion of closeness in an iterativelyrestrictive manner. Doing so appeared to provide them a suitable alternative to the nonsensical notion of infinite closeness.


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)

Definition #2: If you could zoom forever and always get closer to a and L, then you have a limit. (End of Session 4)

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)

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 Lw/no gaps in the graph. (Session 6)

Definition #5: As x gets arbitrarily close to a, |L-f(x)| gets arbitrarily small. (Session 9) Definition #6: For any arbitrarily small # # you can find an x-value that satisfies |L-

f(x)|$#. (Session 9)Definition #7: For any arbitrarily small # # we can find a value of x arbitrarily close to a

such that |L-f(x)|$#. (Session 9) 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 #. (Session 9)

Definition #9: Lxfax

=?

)(lim provided that: given any arbitrarily small # #, we can find

an (a±%) such that |L-f(x)| $ # 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 toadequately mathematize the physical process they were attempting to describe. This sophisticated perspective ultimatelyled Amy and Mike to a definition of limit capturing the intended meaning of the conventional !–ı definition.

One might be tempted to infer from the findings reported here that inducing a cognitive shift in students from an x-firstand dynamic perspective of limit to a y-first and arbitrary closeness perspective subsequently places greater value on thelatter and devalues the former. I would argue to the contrary, however, as evidence from the teaching experiment suggeststhat it is the ability to employ both perspectives flexibly that allows someone to develop a rich and robust understandingof the limit concept and its formal definition. Reasoning from an x-first and dynamic perspective can support someonein developing a sense for the essence of the limit concept – i.e., that limit describes the local behavior of a function asits 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 andsubtleties 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 thedistinction between finding limit candidates and subsequently validating those candidates. Their focus on the y-axis appearedto foreground the presence of a y-value, L, about which they were constructing progressively tighter bounds. This led themto wonder explicitly why a particular L was the focus of their graphical exploration, which, in turn, appeared to spur theirrecognition that the definition they were constructing presupposes the existence of a limit candidate. Thus, evidence suggeststhat a relationship may exist between a student adopting a y-first perspective and distinguishing between the actions offinding and validating limits. The ability to distinguish these two appears to be at least partially supported by the experienceof 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 comingto 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 priorto their engagement in guided reinvention. Put another way, the activity of attempting to construct a precise definition oflimit might simultaneously increase learners’ recognition of the need for such formality, and thereby constitute a mediumpropitious for the emergence of a necessity principle of sorts.

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