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Journal of Mathematical Behavior 31 (2012) 331– 343

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The Journal of Mathematical Behavior

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etaphor as a possible pathway to more formal understanding of theefinition of sequence convergence

aul Christian Dawkins ∗

orthern Illinois University, Mathematical Sciences, Watson Hall 320, DeKalb, IL 60115-2888, USA

r t i c l e i n f o

eywords:eal analysisequence convergenceefiningealistic Mathematics Educationransition to advanced mathematicalhinking

a b s t r a c t

This study presents how the introduction of a metaphor for sequence convergence con-stituted an experientially real context in which an undergraduate real analysis studentdeveloped a property-based definition of sequence convergence. I use elements fromZandieh and Rasmussen’s (2010) Defining as a Mathematical Activity framework to tracethe transformation of the student’s conception from a non-standard, personal concept defi-nition rooted in the metaphor to a concept definition for sequence convergence compatiblewith the standard definition. This account of the development of the definition of sequenceconvergence differs from prior research in the sense that it began neither with examples orvisual notions, nor with the statement of the formal definition. This study contributes to theRealistic Mathematics Education literature as it documents a student’s progression throughthe definition-of and definition-for stages of mathematical activity in an interactive lectureclassroom context.

© 2012 Elsevier Inc. All rights reserved.

. Introduction

For some time now, a growing body of mathematics education literature has identified that students struggle withdvanced mathematical thinking both in the complexity and abstractness of advanced mathematical topics and the difficul-ies posed by the shift from less formal to more formal modes of reasoning. However, systematic descriptions of the meansnd stages by which students transition from informal reasoning to formal reasoning are still forthcoming. To promote thisind of “mathematizing” by which informal experiences and observations are systematized, abstracted, and generalized, theathematician Hans Freudenthal (1973, 1991) developed the Realistic Mathematics Education (RME) approach to research

nd instructional design. RME shifts the focus of mathematics instruction away from the outputs of mathematical thoughte.g. algorithms, problem solutions, definitions, theorems, and proofs) to the mathematical activities that produce thoseutputs (e.g. algorithmising, developing paradimactic solutions, defining, conjecturing, and validating). Many researchers inndergraduate mathematics education (Gravemeijer & Doorman, 1999; Larsen & Zandieh, 2008; Rasmussen & Blumenfeld,007; Rasmussen, Zandieh, King, & Teppo, 2005; Zandieh & Rasmussen, 2010) have adopted the RME standpoint because

t emphasizes the processes by which students might transition into more formal mathematical reasoning and advancedathematical thinking. These studies leveraged conceptual domains such as physics, geometry, and analytic expressions to

reate experientially real settings in which students can develop more formal lines of reasoning. The scope of this approachtill needs further exploration, as many abstract mathematical topics do not immediately suggest such conceptual domainsn which students can begin.

∗ Tel.: +1 815 357 6755.E-mail address: [email protected]

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332 P.C. Dawkins / Journal of Mathematical Behavior 31 (2012) 331– 343

The present study documents one student’s learning about the definition of sequence convergence in the context of afirst-semester, undergraduate real analysis course. In this classroom, a mathematical metaphor for sequence convergencebecame integrated into classroom dialogue and became crucial for the student’s developing understanding. This metaphorprovided an experientially real setting in which it appears at least some students developed their understanding of theformal definition of sequence convergence. Using elements from the Defining as a Mathematical Activity (DMA) framework(Zandieh & Rasmussen, 2010), I identify the stages by which one such student transitioned from less formal reasoning withinthe mathematical metaphorical context into more formal and standardized reasoning within the mathematical context. Itriangulate the student’s verbal explanations regarding sequence convergence and his lines of reasoning over time in solvingmathematical tasks to identify his concept image and concept definition. I discuss the pitfalls and eventual successes thatthis metaphor entailed for this individual.

2. Relevant literature

The notion of limit stands squarely in the center of advanced mathematical thinking because many of the conceptsdefined throughout advanced calculus and analysis are defined in terms of limits (e.g. derivative, integral, Taylor series, etc.).Accordingly, much previous mathematics education research in the context of calculus and real analysis focused upon theteaching and learning of limits and the overwhelming consensus is that limits are subtle, complex, and quite difficult forstudents to coordinate according to the standardized mathematical practice (Cornu, 1991; Cottrill et al., 1996; Oehrtman,2009; Przenioslo, 2004; Szydlik, 2000; Tall & Vinner, 1981; Williams, 1991).

2.1. Transitioning from less formal to more formal limit reasoning

Students often learn about limits in two relatively distinct stages: (1) they are conceptually and computationally intro-duced to limits at the level of calculus, (2) then later a subset of those students go on to learn limits in the more rigorousproof-based setting of real analysis. Even when the ε–ı definition appears in calculus, it is not generally used to solve prob-lems or prove further results. Thus real analysis often marks a strong transition from less formal and computational limitreasoning into more formal and deductive approaches to limits. Mathematics education research repeatedly documents thedifficulties students have with transitioning to formal reasoning using mathematical definitions (Alcock & Simpson, 2002,2004; Edwards & Ward, 2008; Vinner, 1991). Vinner (1991) explores the many ways in which students might organize theirthinking about a given concept drawing in different ways from the concept definition (CD), the set of words used to define aconcept, and their concept image (CI), the set of all images, connections, procedures, examples, etc. connected to that concept.Less formal reasoning generally depends more strongly upon the CI rather than the CD, whereas a hallmark of formal proof-based reasoning is that its final form depends solely upon the CD or other statements that have been rigorously deducedfrom it. Though students may or may not have memorized the standard CD for a concept, many students hold a functionallyor expressly different definition, which Tall and Vinner (1981) called their personal concept definition (PCD). For example,while a student’s PCD may share the phrase “For any ε > 0. . .” with the standardized CD, the student may understand “For asingle ε > 0. . .” rather than the intended “For each and every ε > 0. . .”.

There are several different aspects of this interplay between CI and CD. Alcock and Simpson (2002) categorized studentreasoning about sequences according to whether that reasoning depended primarily upon (sets of) examples (part of theirCI) or upon the formal definition itself (the CD). The authors point out that even though both forms of reasoning are nativewithin mathematical reasoning, the mathematical community favors the latter as more formal and desirable insomuch asany reasoning based upon the properties of the definition itself necessarily apply to every element of the category delineatedby those properties. Other studies have observed the power students’ visual understanding of sequence convergence canhold. Pinto and Tall (2002) described how one students’ development of a rich generic example (element of the CI) of aconvergent sequence supported both strong reasoning about the class of convergent sequences and understanding of theformal definition itself (CD). They contrasted the way this student used the graphical representation to give meaning to theformal definition and Dubinsky, Elterman, and Gong’s (1988) account of how students extract meaning from the definitionstatement itself. Alcock and Simpson (2004) observed that students in their study who depended upon visual reasoningtended to lack motivation for formalizing their reasoning because of the convincing power of their visual images.

Several researchers proposed theoretically or empirically developed decompositions of limit definitions for instruction tosupport students’ learning (Cory & Garofalo, 2011; Cottrill et al., 1996; Mamona-Downs, 2001; Przenioslo, 2005; Roh, 2008,2010a). The two primary tools employed for such decompositions are (1) research on common student misconceptions (limitcannot be reached, assumed monotonicity, non-uniqueness of the limit) and (2) analysis of the conceptual difficulties posedby the structure of the definitions themselves (parameter quantities, function concept, quantification, standard notions ofarbitrarily small quantities). Several such studies propose mathematical activities that intend to foster students’ understand-ing of different aspects of the formal definition. These activities make use of computer interfaces (Cory & Garofalo, 2011;Cottrill et al., 1996) or epsilon strips on the two dimensional graph of sequences (Przenioslo, 2005; Roh, 2008, 2010a, 2010b)
to provide students with an interactive environment to help conceptualize various aspects of the formal definitions of limits.
Alternatively to such approaches where students coordinate elements of their CI with the provided CD, some researchersinvited students to participate in developing a formal definition (CD) using their informal knowledge (CI). Swinyard (2008;Swinyard & Larsen, 2010) guided two pairs of calculus students, each working over the course of one semester, through

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P.C. Dawkins / Journal of Mathematical Behavior 31 (2012) 331– 343 333

ediscovery of the definition of the limit of a function as x goes to infinity and at a point. In both cases, the students beganith a set of examples of functions whose limits did and did not exist. They worked to create a definition that properly

ncluded all of the limits that did exist and excluded all those that did not. Their final definition in each case was fullyompatible with the standardized ε–ı definition.

To summarize, previous research has observed how students can make use of examples, graphical representations,omputer environments, and the statement of the CD to transition from less formal to more formal understanding of limitefinitions, both in the presence and absence of the standardized definition. However, in many cases students use exemplar-ased or visual reasoning in lieu of definition-based reasoning, which fails to facilitate student transition into the standardathematical practice of forming deductive proof based upon the statement of the definition (Alcock & Simpson, 2004).

hough these studies provide multiple detailed accounts of possible pathways to understanding of limit definitions, fewf them report how student understanding became functional for further mathematical activity or proof. How use of limitefinitions for proof and theory development influences student understanding of the limit definition itself is an openuestion.

. Theoretical framework

Realistic Mathematics Education represents the research tradition begun by Hans Freudenthal (1973, 1991) where math-matics is viewed as an inherently human activity. RME focuses upon the processes of mathematical cognition rather thanpon the end results of mathematical thought. Advanced mathematical thinking consists more of defining, theorizing,nd validating rather than simply definitions, theorems, and proofs. From this vantage, instruction must guide students toxperience mathematical processes because they are the psychological antecedents to standardized mathematics. To teachrganized, mathematical outputs without the antecedent psychological processes that produced them is to commit whatreudenthal deems the “antididactical inversion.”

Zandieh and Rasmussen (2010) developed a framework for analyzing student development of formal definition couchedithin the RME perspective of mathematics as a human activity. The Defining as a Mathematical Activity (DMA) framework

s built around a series of stages, originally described by Gravemeijer (1999), through which students move from less formalo more formal mathematical activity. In Zandieh and Rasmussen’s (2010) analysis, these four stages are distinguished byhe development of and use of students’ CI and CD (Tall & Vinner, 1981). The situational stage consists of students developing

CD from elements of their CI such as examples and properties of the category the definition delineates. The referentialctivity begins when students reason with their chosen CD with consistent reference and dependence upon elements ofheir CI, since the CD still stands as a model of that CI. During this stage, however, students begin to re-organize their CI inerms of their CD in such a way as to begin to invert the property–category relationship as described by Alcock and Simpson2002). In less formal mathematical activity, the exemplars of a category dominate reasoning about that category; in formal

athematical activity, categories are determined by their defining property, which then may supplant exemplar-basedeasoning. In the referential stage, a new form of reasoning emerges based upon the statement of the definition rather thann the objects it describes.

The general stage of mathematical activity is marked by the independence of the CD from the CI. Students’ reasoningith their CD may cease to display features native to the set of exemplars that formed the initial CI of the category. In this

tage, students are no longer creating a definition of their CI, but have developed a definition for more formal mathematicalctivity. The formal stage of mathematical activity comes when students make deductions about classes of objects basedolely on their CD and their mathematical understanding thereof without having to refer back to particular examples or theriginal elements of their CI that fostered the definition. Zandieh and Rasmussen (2010) note that students in their study areperating in the formal stage when they “use definitions as links in chains of reasoning without having to revisit or unpackhe meaning of these definitions” (p. 70). Reasoning without unpacking the CD often means students are using elements ofheir CI, but at this point the CI has been reorganized according to the CD such that the student’s activity properly reflects thetandardized definition. Fig. 1 portrays the interactions between the CD and CI through the various stages of the framework.

Though the final account of defining in this paper will differ from the exact pathway delineated in the DMA frameworkin ways explained in Section 7), my analysis maintains two common themes. First, I observe the distinction between

“definition-of” the CI (situational and referential stages) and a “definition-for” further mathematical activity (generalnd formal stages). Second, I explore the complex interactions and interdependencies between the CI, CD, and ongoingathematical activity as a student forms and uses a definition.Under what circumstances would a context of reasoning be “experientially real” for a student? I propose that a context

f reasoning is experientially real for a student when it affords them greater conceptual control than other possible contextsf reasoning. A context provides conceptual control to a learner when:

) they can quickly and efficiently process information in that context,) they can call upon a set of appropriate conceptual actions for solving problems and answering questions within that

context, and/or) they can independently assess the validity of their mathematical actions within that context.


Fig. 1. Stages of the DMA framework.

Learning and memory research indicates that people can process and recall information much more quickly, efficiently,and flexibly when they have a greater body of related knowledge or expertise (Kimball & Holyoak, 2000). I suggest twomain categories of contexts that could be experientially real for a student: 1) a context mediated by physical objects orsigns upon which appropriate mental actions can be performed or 2) a context in which students have a body of experiencethat facilitates ease of processing and provides basic intuition for assessing the validity of reasoning within the context.Children’s activity with manipulatives and college students’ analysis of analytical expressions (Rasmussen & Blumenfeld,2007) fall into both categories; physics and geometry fall primarily into the latter (Gravemeijer & Doorman, 1999; Larsen& Zandieh, 2008; Zandieh & Rasmussen, 2010). However, “experientially real” is a relative measure; mathematicians mayexperience equal or greater cognitive control reasoning about mathematical axioms, definitions, and theorems as withinalternative contexts. If students already have connected understanding and routines for action within the formal definitioncontext, then they would not be expected to pass through the stages of the DMA framework as their CD might afford asmuch conceptual control as their set of exemplars.

Accordingly, Zandieh and Rasmussen (2010) point out that the framework assumes that students operate in a relativelynew reasoning environment, meaning that they have not developed well-beaten pathways for modeling the original, lessformal context. The present study analyzes one undergraduate student developing the definition of sequence convergencein his first semester of real analysis. He previously completed a transition to proof course, but this was his first experiencedeveloping definitions of the nature and complexity of sequence convergence (multiple parameter quantities and multiplequantifiers) in a fully proof-oriented course. This learning context matches the intent of the DMA framework as an analyticaltool. In the words of Zandieh and Rasmussen (2010), the student in this study was “creating a new mathematical reality” ofrigorously defining sequence convergence and being able to reason deductively about all convergent sequences.

4. Methods

This study was conducted at a mid-sized university (25,000 students) in the USA. All data was gathered during one 15-week semester of undergraduate real analysis taught by a research mathematician. At this university, this course generallyincludes a proof-based development of real numbers, sequences, limits of functions, and continuity. At the time of datagathering, the professor had been teaching for over 10 years and had previously taught undergraduate analysis thrice. Shehas received multiple teaching awards and is widely regarded as an excellent and challenging instructor. The class met twiceweekly for 80 min and began with 23 students.

Data gathered includes field notes of class meetings, biweekly professor interviews, weekly student interviews witha small group of volunteers from the class (6 students), copies of student written class notes, and copies of interviewparticipant exams. Written field notes recorded all communication written on the blackboards, major aspects and key quotesfrom professor and student verbal communication, and physical gestures displayed in the discussion of course material. Allinterviews were audio-recorded and any written records of the interactions maintained. The student interview participantswere selected from among mathematics majors (to minimize the probability that study participants would withdraw fromthe course) who volunteered so as to represent a variety of final grades in the “Intro to Proofs” course that serves as apre-requisite to analysis at this university.

The original research questions targeted students’ recall and interpretation of classroom explanations and events. Inter-view protocols were developed in response to and as a reflection of classroom interchanges. Because the professor spentextended time creating and exploring definitions, the subsequent interviews investigated students’ interpretations of class-room explanations and activities and their understanding and use of particular definitions. The interview questioning often
moved from very open-ended to very particular so as to initially allow interview participants flexibility in their choices ofverbalization and representation. For example, the interview protocol may begin by asking a student, “What does it meanfor a sequence to converge to a point?” Later, students would be asked about particular aspects of the definition or the expla-nation thereof such as, “What does ‘arbitrary but fixed’ mean and why did [the professor] discuss it?” or “In the definition


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Fig. 2. The professor’s number line representation of the first sequence.

f convergence, what are �, K, and n?” Finally, a student might be asked to construct a proof that a particular sequenceonverges to a given limit. Many of the tasks used during interviews came directly from homework, test reviews, or fromhe tests themselves such that the mathematical activity did not extend beyond that in the classroom, but strongly reflectedt.

Interviews were transcribed and then coded according to the open coding method described by Strauss and Corbin1998). In line with the grounded theory approach, no a priori theoretical tools were brought to bear during initial analysis.

developed categories relating to particular mathematical topics, explanations or activities from class discussion, trendsn the professor’s pedagogy and any parallel trends in student mathematical activity, and meta-mathematical topics thatrose during class dialogue and interviews. Observations of the professor’s pedagogy in action and her explanations duringnterviews displayed a strong affinity for the principles of RME, though she has no direct connection to that tradition.

The current study developed from my investigation of student reasoning about and use of the metaphors the professoregularly introduced (Dawkins, 2009). The student Tidus featured in this study appeared as an instructive case because ofis consistent reference to the party metaphor both for communication and sense making about sequence convergence. Ioticed that his language use regarding the definition of sequence convergence shifted over time, and so I sought to use histterances to model his PCD at various stages. I then tested my accounts of his conceptions against his mathematical activityn exam and interview tasks. My models of his PCD properly accounted for his reasoning, both mathematically appropriatend inappropriate, at each stage. In the same way the professor “organically” displayed affinity for RME design heuristics fornstruction, my grounded theory account of Tidus’ mathematical activity strongly resembled aspects of the DMA framework.

will thus compare my account of Tidus’ learning to the trajectory laid out in the DMA.

. Instructional context

Beginning early in the semester, the professor discussed definitions in two types of contexts: metaphorical contextsnd mathematical contexts (standard representations such as numerals, the number line, algebraic expressions, etc.). Thelass serially discussed their reasoning within these two contexts using three distinct linguistic registers: the metaphori-al register, the intuitive register, and the formal-symbolic register. For example, to introduce the definition of sequenceonvergence, the professor presented the class with an intuitive register definition meant to represent common notions ofonvergence that said, “A sequence (an) converges to L if an gets closer and closer to L as n gets larger and larger.” However,he pointed out that in the case of (4 − 1/n), five is a possible limit of this sequence by this definition. Fig. 2 shows the numberine representation the professor used to display this sequence; she commonly used such number line diagrams to discussequences. As students proposed different ways to amend the intuitive register definition to be more specific, Zell told theeacher that “if you want to find a party, see where everyone is at.” The professor agreed that they needed to find “whereveryone is at,” and clearly no one is at 5.

The professor then extended the metaphor asking, “How many terms make a party?” She provided another examplehat was defined as 5 − 1/n for the first million terms and 4 for every latter term. Fig. 3 displays the term-enumeration andumber line representations that the professor used to display this sequence. She posed the question, “How many peopleave to be at the party,” to which Locke responded “infinitely many.” The professor invited the students to talk for a while andhare their ideas about how they should form their formal definition. To bring the discussion together again, she provided aevised intuitive register definition that stated, “*1 A sequence converges to the real number L if we can make the terms of
he sequence stay as close to L as we wish by going far enough out in the sequence.”
She verbally mapped this statement into the metaphorical context saying “only finitely many guys can be outside theoom for you to have a party. What if you make the room smaller?. . . You would have more stragglers.” She then began to

Fig. 3. The professor’s term-enumeration and number line representations of the second sequence.


Table 1Sequence convergence in the various registers.

Linguistic register Sequence elements Proximity Sequence order “Cut off point”

Metaphorical People Location Time Eventually
Informal mathematical Terms Closeness Physical arrangement “Far enough out in the sequence”Formal-symbolic an ε Indices (n) K
ask the class how they could make various aspects of this informal definition “more rigorous.” She updated the previousdefinition, introducing some previously discussed notation, to the following:

“*2 (xn) converges to L if, given ε > 0, all terms after a certain term xK are in (L − ε, L + ε).”The professor ended the process of translation with the definition:“* A sequence (xn) converges to the real # L if given ε > 0, there exists K ∈ N such that ∀ n ≥ K, xn ∈ (L − ε, L + ε).”During later lectures, the professor began to refer to these terms outside the party as “stragglers.” Immediately after

completing the definition of sequence convergence, the class worked to prove that (3n/2n + 1) converges to 3/2, and theytranslated 3n/(2n + 1) ∈ (3/2 − ε, 3/2 + ε) into absolute value notation to say |3n/(2n + 1) − 3/2| < ε in line with their workseveral weeks earlier on epsilon neighborhoods. Their proof, once completed, only justified the claim in terms of absolutevalue notation. Though the phrase “given ε > 0” may be interpreted differently by students, the professor often portrayedthe standard meaning by an iterative shrinking of the value of epsilon. In addition, the professor explained at different timesthe standard notion that epsilon values are “fixed, but arbitrary.”

Thus the professor translated the intuitive register definition into the formal-symbolic register by replacing “stay as closeto L as we wish” with epsilon neighborhood conditions and “going as far out as we wish” with index terminology. However,for the next several weeks of class meetings that covered the section on sequences, the professor verbally referred to theparty metaphor though everything written on the board was either graphical, intuitive register, or formal-symbolic register(Table 1).

6. Results

Tidus consistently borrowed terminology from each of the three registers to communicate and solve problems aboutsequences. However, the nature of his PCD and CI shifted over time and displayed various traits of the metaphorical contextand the standard mathematical context. When possible, my account of elements of his conception of sequence convergencewill be validated against his reasoning on tasks during exams or interviews. This highlights the forms of reasoning hisunderstanding supported and failed to support. Four interviews and two exams included material related to sequenceconvergence. Any quotes will be marked with their source type and chronological order: I-1, T-1, I-2, I-3, I-4, or T-2.

6.1. Neighborhood as place

Tidus consistently referred to the epsilon neighborhood as a place or a set rather than a measure of closeness betweensequence terms and the limit value. He expressed his understanding of the definition of sequence convergence saying, “atsome point . . . eventually all of those numbers will be in four’s neighborhood” (I-1). He articulated that same definition latersaying, “For any epsilon that you pick, an infinite amount of terms will be in that epsilon neighborhood and a finite amountof terms will be outside” (I-2). This appeared on the first test (T-1) in that six different times he juxtaposed absolute valueand set-type expression such as |xn − 5| < .01/2 alongside xn ∈ (5 − .01/2, 5 + .01/2) or |an − L| < ε alongside L − ε < an < L + ε (thisdouble inequality might imply more quantitative relationships, it also might be interpreted as the term an being physicallybetween the bounds). Consistently thereafter when discussing sequence convergence, Tidus would talk about numbers orterms being “in the epsilon neighborhood” rather than using closeness or approximation language.

At this point, Tidus never referenced multiple values of epsilon regarding a given sequence. Though this difficulty has beennoted among students elsewhere (Przenioslo, 2005; Roh, 2010), this difficulty would naturally be exacerbated by viewing theepsilon neighborhood as a physical place in which case the size would be static. However, his single epsilon CD dominatedhis reasoning to the point that he built a limit argument that conflicted with his calculus-based limit knowledge. On theexam, he assert that “If (an) converges and 4 < an < 5 for all n, then lim(an) > 4” is false by the following argument (Fig. 4).

Tidus concluded “lim (an) = 4 and an is trapped when an = 4.5 and L = 4” (T-1). [It should be noted that his conclusions implyTidus meant ε = .6 rather than .06.] Tidus’ understanding of convergence at this point involved a single value for epsilon and“trapping” all of the values of the sequence in that epsilon neighborhood.

On that same exam, Tidus produced a mostly accurate proof that if a sequence converges to 5, then it cannot convergeto 5.01. He chose ε = .01/2 and pointed out that the epsilon neighborhoods were disjoint. Since it is impossible for the terms
of the sequence to be in the two neighborhoods at the same time, this produced a contradiction (T-1). Thus, Tidus’ image ofthe epsilon neighborhood as a place supported success with certain types of reasoning, and success on tasks in which thedistinction that numbers are either inside or outside of the neighborhood was important.


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.2. The cut-off point

Tidus came to focus closely upon the condition that the terms of the sequence must stay in the neighborhood after somearticular point. He spoke about this point in various ways:

“K represents how far n has to go on the number line to get into the epsilon neighborhood.” (I-1)“K is that last term where, that’s the last straggler and everything after this K are all in that neighborhood or in the partyas she puts it.” (I-2)“The hypothesis tells us that the sequence converges to 5, so at some point every other point will be in the epsilonneighborhood, and that’s what that K is. That point where there’s an infinite amount of terms in the epsilon neighborhoodeliminating the stragglers.” (I-3)

The index K came to signify that cut-off point for Tidus, but his language regarding the effect of this point rotated between:

the metaphorical language of “after” referencing time in the party metaphor, “eliminating the stragglers” which alsoreferenced the party metaphor language, “terms are trapped” within the epsilon neighborhood, “how far n has to go in thenumber line” citing spatial arrangement in the term-enumeration representation, andthe “infinite amount of terms” expression that Locke introduced on the first day of sequence limit discussion.

The students in this class adopted mathematically non-standard usage of “infinitely many terms” in place of “everyerm after some point”. However, when asked during class or interview whether an infinite number of terms in an epsiloneighborhood constituted convergence, most students said “No” providing an oscillating sequence counterexample. Tidusesponded to the true-false exam question “If every ε-neighborhood of 5 contains infinitely many terms of the sequencean), then (an) converges to 5,” with the counterexample 5 sin(n�/2) (T-1). The professor repeatedly challenged studentsho used the phrase “infinitely many terms” in this way, but they did not seem perturbed because they could adequately

ddress her questions. Tidus later extended the phrase to “infinite amount of terms will be in that epsilon neighborhood and finite amount of terms will be outside” (I-2), which is compatible with the conditions within the formal definition thoughot minimal.

On the first test, the professor included a true-false question that stated: “Suppose the sequence X converges to −1. For ∈ N, let XK denote the sequence obtained by changing the first K terms of X to the value K and leaving the rest of the termss they are. Then, for all K ∈ N, the sequence XK converges to −1.” The professor spent some time during class articulating theoint that a finite number of terms have no effect on the convergence of a sequence. Tidus’ image of sequence convergencehat focused on the existence of a point after which the terms must stay in the neighborhood allowed him to dismiss thehanging of K terms as irrelevant for convergence. After writing out an example sequence X and X3, he argued that thetatement is true because, “If you change the value of the stragglers and did [sic] not the value of the terms [in] the ε-eighborhood, the sequence will still converge to L” (T-1). He used party metaphor language to refer to the finitely-manyerms outside the neighborhood, and the metaphor possibly supported his understanding that their absence did not affectthe party.”

The notion that convergence means infinitely many terms in the neighborhood and finitely many outside became theenter of Tidus’ CI by the second interview. He articulated his definition as, “For any epsilon that you pick, an infinite amountf terms will be in that epsilon neighborhood and a finite amount of terms will be outside” (I-2). As stated, this definitions equivalent to the standard formal definition, however proofs most often used in undergraduate real analysis dependpon the structure of the formal definition stated in terms of indices. When I asked him about the roles of �, K and n inhe professor’s definition, Tidus was able to recite from memory the standard definition in the formal-symbolic register.
e explained his understanding of the roles of each element in the metaphorical register saying, “I think of K as that termhere that’s the last straggler and everything after this K are all in that neighborhood or in the party as she puts it” (I-2). In
his way, Tidus serially expressed his ideas about sequence convergence in all three registers, and borrowed from both theathematical and party contexts for reasoning about sequence limits.


6.3. Finding the K

Along with his focus on the cut-off point, Tidus also developed a sense that convergence hinged upon “finding” the K ratherthan its existence. The professor always modeled proofs that particular sequence converged by writing her “scratchwork”on a different blackboard on the side of the room. She explained that finding the particular K “that will work” is not partof the proof itself, but that the K must simply be “presented” in the proof. This mirrors what Raman (2003) describes asthe “public” and “private” aspects of proof. During an interview in which Tidus proved that ((n + 1)/n) converges to 1, hesaid, “For this proof I remember precisely you use your scratchwork and you look for the white tiger [a term for K takenfrom another classroom metaphor]. And then you show the white tiger in the proof.” In writing his proof, Tidus drew a linebetween his “scratchwork” and the proof itself, and did not declare in the proof that K > 1/ε as he did in the scratchwork.With the addition of declaring his value for K, his proof was correct. As I questioned his understanding of the proof itself, heexplained:

Tidus: So this is a K that we pick that when it gets to that point everything else is going to be in that epsilon neighborhood. . .I am going to pick that K where it’s past that point, that point when we get into the neighborhood. . .

I: Alright, so do you have any questions about that proof, why it works?Tidus: I looked at this proof a bunch of times and I really can’t understand what it’s trying to say. Like, I understand that

there’s this K and once you get past there it’s, all of the terms are in the epsilon neighborhood, but I don’t see howthis proves it. . . I see that we are picking an epsilon, which is arbitrary and we are finding when that n is greaterthan [1/ε] . . . so we are finding a K where that happens and I guess from the scratchwork we pick that K when thishappens. (I-2)

Tidus not only revealed a lack of sense of understanding of the string of inequalities he accurately produced, but hedisplayed his association between K and the scratchwork. He did not think that K needed to be declared at all in the proof,he just needed to find it behind the scenes. In another instance, we discussed his proof on the first exam that if a sequenceconverges to 5, it cannot converge to 5.01. He used proof by contradiction and assumed that the sequence converged to bothlimits. He declared that, “we are going to find this K1 where when that term gets past that, the sequence is going to be in itsrespective epsilon neighborhood.” When I asked him where K1 comes from, he answered, “the scratchwork” even though thespecific sequence was not provided (I-3). On his exam he had written scratchwork with a line separating it from the proof,but had only used that space to choose his value for epsilon and write out expressions for the two epsilon neighborhoodsused in the argument; K was nowhere mentioned in the scratchwork.

6.4. The emergence of the arbitrariness of epsilon

For some time, Tidus did not seem to reason at all about changing the value of epsilon relative to the same limit, butliterally would “pick any” and leave it static thereafter. He understood that he could pick a convenient epsilon and did so onseveral tasks on the first test. During the second interview (I-2), he said:

• “If you show that the [sequence] can’t go out [of the epsilon neighborhood], then it’s going to converge.”• “It has to converge to an L in any epsilon neighborhood.”• “There should be one number in the [sequence] where the [sequence] converges to that point, where all of the terms are

in that epsilon neighborhood.”

Though Tidus does mention that the condition must hold “for any epsilon neighborhood,” he also speaks in the first andthird quotes as if convergence to the limit is not a global property of the sequence, but it is something that the sequence is“going to do” relative to the “one number in the [sequence] where the [sequence] converges to that point.” Similarly duringthe third interview (I-3) he said, “We are finding a K1 where it is going to converge to 5.” Thus the focus of Tidus’ viewof convergence was on the bounding of infinitely many terms after K rather than upon the size of epsilon or the universalquantifier associated.

During the third interview, Tidus began to juxtapose his use of epsilon language with “closeness” language in the contextof the definition of Cauchy sequences. In the course of arguing why Cauchy sequences must be bounded, he said, “Wellif they are getting close to each other, at some point they are going to be in each other’s, I guess you could say epsilonneighborhood? Something like that because if they are getting closer to each other they can’t be going out anywhere else”(I-3). Tidus continued to use his informal register expression “at some point,” and “in each other’s. . . epsilon neighborhood,”but he used epsilon to express proximity as well.

Toward the end of the third interview, he wrote down a form of the formal-symbolic definition that began, “Let ε > 0. . .”
and then appended to the end of his explanation, “and it should work for every epsilon” (I-3). This was the first time that Tidushas specifically referenced the generality or arbitrariness of epsilon in conjunction with the formal definition. When probedabout the difference between “Given ε > 0. . .” and “For every � > 0. . .”, Tidus said “I don’t see how it could be different. . .‘Cause either way you are going to have to pick. . . an epsilon. For every you are saying you are taking any epsilon, that K is

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oing to be different for every epsilon. . . For a different epsilon there will exist a different K” (I-3). This was also the firstime during interviews that Tidus expressed any awareness of this “dependence” between the quantities.

During Tidus’ final interview regarding sequences (I-4), he worked on review problems for his upcoming exam andonsidered the question,

“(a) If (xn) is a sequence and xK is a term of the sequence such that |xn − xK| < .1 for all n ≥ K, then which of the following,f any, are correct?

i) (xn) is boundedii) (xn) is Cauchyii) (xn) has a Cauchy subsequence

b) Let (xn) be a sequence such that for every ε > 0, |xn − xm| < ε for all m, n ∈ N. Then which of the following, if any, are correct?

i) (xn) is Cauchyi) (xn) is a constant sequence”

idus originally indicated that the sequence in part (a) must converge and thus be Cauchy, but then expressed that he had realization and said, “Oh, but it’s not for every epsilon. . . there could be an epsilon where it doesn’t work” (I-4). Abouthe latter question, he noted, “It’s not picking a certain term where this happens. . . that could only happen if it’s a constantequence. . . cause it doesn’t matter what term in the constant sequence you pick for it to happen, it’s going to happenverywhere” (I-4). When pressed to prove why every term of the second sequence must be equal, Tidus could not producehe proper argument, but his image of what must happen if there was not a “certain term” led him to believe the sequence

ust be constant. He argued, “because here it doesn’t say. . . that you have to pick a certain term in the sequence wherehis happens. . . [which] like the convergent ones [allows] stragglers.” In both cases, the tasks required Tidus to reasonbout the definition of convergence or the definition of Cauchy in terms of multiple values of epsilon at the same time, ande was successful in reaching accurate conclusions. It appeared in the latter case that his cut-off term notion helped himnderstand the consequences of its absence. However, had he not understood that the condition holds for every epsilon,hen an alternating sequence satisfies the definition for a single epsilon without a “certain term.”

. Discussion

The professor in this study introduced into the classroom multiple contexts students could use to reason about the math-matical concepts of real analysis. In the context of sequence convergence, these contexts included the party metaphoricalontext and the abstract mathematical context represented in terms of number line diagrams, term-enumeration expres-ions, algebraic expressions, and propositional statements. She made verbal mappings between the contexts by translatingefinitions into the registers associated with each context. At the first interview, Tidus could not remember the formal-ymbolic register definition of sequence convergence, but could accurately recall the conditions of convergence via thearty metaphor. Tidus more easily recalled the propositional definition within the party metaphor, and used the metaphoror sense making. This indicates he experienced more conceptual control in the metaphorical context than in the mathemat-cal context, meaning that the party metaphor was experientially real for him. Other participants in this study such as Lockearely made use of the party metaphor, but reasoned quickly and efficiently in the mathematical context. For such students,he mathematical context is just as “experientially real” because it afforded as much if not more conceptual control than the

etaphorical context.Tidus’ conceptual control within the party metaphor allowed him to quickly create a condition for sequence conver-

ence that acted as his PCD. He blended his knowledge of sequences and elements of the party metaphor to create a newathematical reality in which he could speak and reason about the class of convergent functions according to properties of

onvergence rather than example sequences that converge. This mathematical reality segued elements of the party metaphorneighborhoods, time, cut-off points) into mathematical objects (epsilon neighborhoods, indexed ordering, K parameters)sed to define and construct proofs about convergent sequences.

.1. Establishing a property-based PCD

When Tidus’ classmate introduced the party metaphor and the professor elaborated it to portray most of the key elementsf sequence convergence, this verbal exploration introduced an experientially real setting from which Tidus formed a PCD:

sequence converges if all of the terms after some point are within an epsilon neighborhood of the limit. Though therofessor (and some other students) viewed the party story as a contextual embedding of the mathematical definition,idus appeared to construct an understanding of the definition rooted in the metaphorical context. The professor guided the
tudents through a dialogue in which they refined various possible definitions before they ended with a statement in theormal-symbolic register. Tidus’s explanations and use of the mathematical objects in the formal-symbolic definition revealhat he attributed properties to them indicative of their party metaphorical embodiment. The metaphor was the teacher’s
odel of the standardized CD, but for Tidus the definition was a model of (or definition-of) the party metaphorical condition.


During the initial interview (I-1) with Tidus, he was able to identify how each of the key mathematical quantities in theformal-symbolic definition related to the party metaphor: “K represents how far n has to go on the number line to get intothe epsilon neighborhood.” Tidus was still working to develop a PCD using his CI, particularly using the party metaphor. Hisreasoning reflected aspects of the party metaphor as evidenced by language such as “eventually” referencing time insteadof indices and “in four’s neighborhood” as if the epsilon neighborhood were a physical place where four resides. This step ofestablishing a PCD in an informal context closely reflects the situational stage of the DMA framework.

7.2. Applying the newly established PCD

By the time I interviewed Tidus after the exam, his primary PCD had become stable to the extent that he confidentlyprovided the definition “For any epsilon that you pick, an infinite amount of terms will be in that epsilon neighborhoodand a finite amount of terms will be outside.” This definition is devoid of any linguistic reference to the party metaphor. Heconfidently reproduced the formal-symbolic definition provided in class. However, when I asked him how he rememberedthat formal definition, he revisited the party context saying, “I think of K as that term where that’s the last straggler andeverything after this K are all in that neighborhood or in the party as she puts it” and similarly explained the purpose of nand epsilon referencing the party metaphorical context. It is important to note here that Tidus recognizes the informalityof party language and understands the expectation to adopt the more formal register to which most of the class dialoguehad transitioned. However, his explanations and performance on the exam still reflected his non-standard PCD influencedby the party metaphor.

Tidus used his definition with some success and with some misdirection on the exam (T-1), but he was constructing a CI ofsequence convergence through application of his PCD. Tidus’ reasoning at this point still referenced the party metaphor in thesense that his PCD (infinitely many in and finitely many out) represented his understanding of the conditions of convergencewithin the party context. Though he used this definition in the informal register to reason within the mathematical context,attributes from the party situation still dominated Tidus’ treatment of the objects in the CD. Tidus continued to think of theepsilon neighborhood as a place rather than as some measure of proximity, in the same way that the building housing aparty is a place. He explained that epsilon is, “how big your party is going to be or how small your party is going to be” (I-2).In the same way that the size of a building cannot change, Tidus had trouble reasoning about multiple values of epsilon inreference to the same limit. His argument upon the first exam that the sequence an = 4.5 converged to 4 when epsilon is .6conflicts with his calculus-based knowledge of sequence convergence. This means he must have been reasoning accordingto his PCD at that time which only required a single value for epsilon. Though non-standard, his PCD rooted in the partymetaphor supported his successful construction of a proof that any sequence that converges to 5 cannot also converge to5.01. The proof depended upon the fact that terms in the tail of the sequence cannot be in two neighborhoods or parties atonce and his party-based PCD afforded such reasoning. Tidus’ reorganization of his CI in light of his PCD, which still reflectedthe party metaphor, closely mirrors the referential stage of defining described in the DMA framework.

In contrast to the referential stage in which the PCD remains static, Tidus’ PCD underwent some modification in light ofthe process of proving that particular sequences converged. Tidus began to focus on finding a K for the given epsilon. Hestrongly associated the finding of that K with the process called “scratchwork.” The professor modeled this type of proofdirectly after the class agreed upon a formal definition for sequence convergence. She emphasized that finding the K shouldnot be part of the proof, but should be found in the scratchwork before being presented in the proof. Tidus studied thisprocess enough that during the second interview he could accurately produce such a proof. However, he readily admittedthat he did not understand how the proof satisfied the definition, or one might say that he could not assimilate the proofinto his PCD schema. The proof consists of strings of inequalities and absolute values that all establish proximity betweensequence terms and the limit. The absence of proximity from his PCD and his associated CI of sequence convergence hinderedhis proof comprehension. Tidus focused in on the selection of the K as a crucial aspect of the proof because it was the finalstep in the scratchwork (K < 1/�). It appears as though the process of producing this proof using scratchwork became itselfa paradigm of sequence convergence that Tidus accommodated into his PCD and CI.

7.3. Integrating proximity and quantification into the PCD

By the third interview (I-3), the class had already discussed the definition of Cauchy sequences and established theequivalence between the class of Cauchy sequences and convergent sequences (this course only considered sequences onthe Real number line). Tidus made use of this fact freely in interviews and on tests, and seemed comfortable with theequivalence. When Tidus tried to articulate in a more formal register how Cauchy described “terms getting closer to eachother,” he said “at some point they are going to be in each other’s, I guess you could say epsilon neighborhood” (I-3). Thisnew use of epsilon as a measure of proximity rather than the size of the party marks the first evidence that Tidus had movedinto the general stage of defining. His statement implied that both terms could have an epsilon neighborhood, which is notcompatible with the party metaphorical context.

It was during that same interview that Tidus first articulated that the conditions of the definition “should work for everyepsilon” (I-3). He also explained that K depends upon epsilon and thus “for a different epsilon there will exist a differentK.” Thus Tidus’ PCD and connected CI developed in two key ways. He now perceived a functional dependence of K uponepsilon forming a new relationship between the objects in the CD and he applied a universal quantifier to epsilon bringing

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is PCD into close conformity with the standardized meaning of the CD. These developments indicate that Tidus’ PCD wasow independent of the party metaphor and constituted a more general definition-for further mathematical activity. Tidus’hifts in understanding strongly mirror the general stage of defining in the DMA framework in which both the CD and CI areeorganized to stand on their own apart from the context in which they originated. However, Tidus’ PCD underwent furtherevision as he sought to accommodate new concepts and activities into his PCD (Cauchy criterion, reasoning about multiplepsilons for a single limit) not directly outlined in the general stage of defining.

.4. Reasoning about the relationships between the mathematical objects in the CD

During the final interview (I-4), Tidus was able to accurately ascertain (though not prove) the consequences of changinghe definitions of sequence convergence and Cauchy sequences. He pointed out that just because the conditions of sequenceonvergence are satisfied for a single epsilon does not mean it converges because “there could be an epsilon where it doesn’tork.” He argued that the removal of K from the definition of a Cauchy sequence means the sequence must be constant

ecause the presence of K allowed for “stragglers” and without K “it’s going to happen everywhere.” He used the termstragglers” from the metaphorical register, but his use of the term indicates that he attributed to those objects characteristicsndicative of the mathematical context. Tidus’ use of metaphorical language to refer to elements of the mathematical contextlongside his lines of reasoning that would not be afforded by the party metaphorical context strongly indicate that his PCDnd his CI of the formal definition of sequence convergence was independent from the party metaphorical context uponhich he previously relied.

As is described in the formal stage of the DMA framework, Tidus reasoned about the class delineated by a definitionithout having to unpack it’s meaning. On the second exam over sequences (T-2), he argued that the product of Cauchy

equences must be Cauchy because, “If both (an) and (bn) converge to a limit, then certainly (an)(bn) converges and hence isauchy.” He constructed a valid chain of argument about convergent sequences without having to unpack the definition ofhat class. He argued that:

) Cauchy sequences are convergent, so both of the given sequences converge.) The product of convergent sequences is convergent.) Convergent sequences are Cauchy, so the product sequence must be Cauchy.

Tidus displayed a rich understanding of the objects in the CD and their relationships and a CI of convergent sequencesore compatible with standard meanings within the CD.

. Conclusions

The party metaphor provided Tidus with an experientially real context in which he could reason about the definition ofequence convergence. His lines of reasoning that matched the party context but not the formal definition or his example-ased knowledge reveal the central role that the metaphor played in the initial development of his CI and PCD within theathematical context of reasoning (infinitely many terms in an epsilon neighborhood and finitely many outside). I triangulatey assessment of Tidus’ PCD in that it properly accounts for the tasks upon which he was successful and unsuccessful. The

nterviews conducted cannot perfectly ascertain how and why Tidus shifted his CI and CD over time, but the use of epsilons a measure of proximity first appeared in the context of the definition of Cauchy sequences. It appears a likely hypothesishat this alternate definition, which in real analysis is equivalent to convergence, helped lead Tidus to shift his use of thearameter epsilon because the party metaphor did not facilitate reasoning about the Cauchy definition.

.1. Metaphor and language as a pathway to formal reasoning

The fidelity with which Tidus’ intermediate conceptions of sequence convergence mirrored aspects of the professor’snstruction provide strong insight into the value and pitfalls of providing metaphors as experientially real settings fortudents to reason about formal mathematical definitions. In the first several weeks after the definition was introduced,idus still could not be said to have a robust or complete understanding of the definition of sequence convergence nor howo use it in proving (his PCD was not sufficiently compatible with the standardized CD and standard interpretations thereof).is reasoning still referred to, and was bound by, the limits of the party metaphor. However, in the course of his ongoingathematical activity he was able to form a conception more compatible with standardized interpretations. In contrast,

he interview participant Cyan was able to use the party metaphor more flexibly to reason in the mathematical contextithout borrowing the limitations of the metaphor. As noted previously, Locke displayed enough conceptual control within

he mathematical context that he rarely cited class metaphors at all.In light of the diversity among the CIs and PCDs among the students, it is important that the class flexibly rotated between

nd translated across the three linguistic registers. Different students in the class used some of the same language from thelassroom registers while reasoning about mental objects that displayed differing characteristics (“For any ε > 0. . .” meaningingle epsilons or quantified epsilons). Tidus also used the same language over time to refer to sequence terms or parametersn the definition to which he attributed differing attributes. Oehrtman (2009) points out that strong metaphorical reasoning


Table 2The interplay between Tidus’ PCD and ongoing mathematical activity.

Tidus’ personal concept definition and concept image Tidus’ mathematical activity

A sequence converges if. . .1) . . .all of the terms after some point are within an epsilonneighborhood of the limit.

The party metaphor criterion for convergence of a sequence.

2) . . .we are able to identify the time after which all of the terms arewithin an epsilon neighborhood of the limit.

Proving that given sequences converge by finding the K in the“scratchwork”

3) . . .we are able to identify the time after which all of the terms arecloser to the limit than distance epsilon.

Defining Cauchy sequences using epsilon as a bound of proximity.

4) . . .for any given ε > 0 we can find a K such that for all terms xn beyondthe K, |xn − L| < ε.

Establishing a functional dependence of K upon epsilon andapplying a universal quantifier to epsilon

occurs when both domains of reasoning influence one another. The professor elaborated the party metaphor using herunderstanding of the definition of sequence convergence. Tidus on the other hand reasoned about sequences in terms ofthe party metaphor. He revealed this by defaulting to party language to explain his understanding of mathematical objects.Even in the symbolic context, Tidus consistently translated absolute value expression that were less compatible with his CIwith inequality or set notation that matched his conception of epsilon neighborhoods as locations.

Tidus understood that the class privileged more formal registers over the informality of the party register, but it tookmore time for him to transform his understanding of the CD. As Tidus became more able to reason with the CI and PCD hehad built in the mathematical context, he attributed party metaphor qualities to mathematical objects less and less. Overtime, the translation between registers actually reflected a fundamental shift in the nature of the elements of his CI to whichthe terms and phrases referred. His definition shifted from a definition-of the party context to a definition-for reasoningwithin the mathematical context. The party metaphor provided Tidus with an essential tool for creating a property-basedPCD of sequence convergence that later conformed to the standard CD.

8.2. Learning limit definitions through formal theoretical development

Another primary contribution this study makes to previous limit definition literature is the role that proof and theorydevelopment play in understanding the definition of sequence convergence. Tidus created a property-based definition usingthe party metaphor, but he refined the definition through using it. Table 2 displays the different mathematical activitiesthat influenced the shifts in Tidus’ PCD and CI over time. In particular, he encapsulated his initial PCD (infinitely many in,finitely many out) into “finding the K” by trying to understand proofs of convergence of particular sequences. He appearedto develop a sense of epsilon as a measure of proximity in trying to understand the definition of Cauchy sequences. It isnot fully clear what motivated Tidus to recognize the dependence of K upon epsilon or to apply the universal quantifier toepsilon, but both occurred during his ongoing mathematical activity. It is also seems reasonable to assert that encapsulationof Tidus’ earlier PCD’s facilitated introduction of the quantifier (“it should work for every epsilon”). Rather than segueinghis informal limit-based activity or example-based knowledge, Tidus refined his PCD by reading and producing proofs andlearning related formal concepts.

This is also the point upon which my analysis of Tidus’ development of a formal definition diverges from that describedin the DMA framework. While Zandieh and Rasmussen’s (2010) trajectory indicates that students develop a CD during thesituational and general stages, Tidus’ PCD underwent several more transformations as he applied his definition to his ongoingmathematical activity. If the trajectory of the DMA framework were viewed as an arc of development, Tidus appeared tofollow more of a zig-zag in which his CI and PCD were repeatedly being reorganized in light of one another and as theyaccommodated new proofs and concepts.

8.3. RME processes in an interactive lecture classroom

While the bulk of research in RME focuses on lower levels of instruction, there is a growing body of work exploring guidedreinvention in undergraduate education (Gravemeijer & Doorman, 1999; Larsen & Zandieh, 2008; Swinyard, 2008; Zandieh &Rasmussen, 2010). Much of this work involves fundamentally shifting the classroom environment toward an inquiry-basedapproach. The class described in the present study stands in between these accounts of guided reinvention in the interviewsetting or an inquiry-based environment and accounts from traditional lecture classrooms (Weber, 2004). The presentclassroom placed much more emphasis on dialogue and student contributions than traditional Definition–Theorem–Proofinstruction, but most all of the dialogue was guided and filtered by the instructor such that it might best be describedas a highly interactive lecture. This account of Tidus’ process of definition development broadens the scope of classroomenvironments in which students might be reasonably expected to traverse through the mathematizing process of defining.
Moreover, this account contributes to the validity of the constructs within the DMA framework as a tool for describing thestages that students traverse as they mathematize a less formal, experientially real setting to develop formal, mathematicaldefinitions.

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cknowledgments

I would like to thank Alan Zollman, Helen Khoury, Mike Oehrtman, and the insightful reviewers for their feedback onarlier versions of this article.

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