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Students’ Conceptions of aMathematical Definition

Orit Zaslavsky and Karni ShirTechnion—Israel Institute of Technology, Haifa

This article deals with 12th-grade students’ conceptions of a mathematical definition.Their conceptions of a definition were revealed through individual and group activi-ties in which they were asked to consider a number of possible definitions of four math-ematical concepts: two geometric and two analytic. Data consisted of written responsesto questionnaires and transcriptions of videotaped group discussions. The findings pointto three types of students’ arguments: mathematical, communicative, and figurative.In addition, two types of reasoning were identified surrounding the contemplation ofalternative definitions: for the geometric concepts, the dominant type of reasoning wasa definition-based reasoning; for the analytic concepts, the dominant type was anexample-based reasoning. Students’ conceptions of a definition are described in termsof the features and roles they attribute to a mathematical definition.

Key words: Beliefs, Calculus/analysis, Conceptual knowledge, Geometry, Higherorder thinking, Reasoning, Secondary mathematics

The mathematical term definition is one of a handful of meta-mathematical marker terms(others include axiom, theorem, proof, lemma, proposition, corollary), terms which serveto indicate the purported status and function of various elements of written mathematics.(Pimm, 1993, pp. 261–262)

Definitions are considered fundamental in mathematics and mathematics education.The notion of a mathematical definition includes roles, as well as features—someimperative and some optional (see Figure 1). The main roles attributed to definitionsinclude: (1) introducing the objects of a theory and capturing the essence of aconcept by conveying its characterizing properties (Mariotti & Fischbein, 1997;Pimm, 1993; Rissland, 1978); (2) constituting fundamental components for conceptformation (Klausmeier & Feldman, 1975; Sowder, 1980; Vinner, 1991; Wilson,1990); (3) establishing the foundation for proofs and problem solving (Moore,1994; Weber, 2002); and (4) creating uniformity in the meaning of concepts, whichallows us to communicate mathematical ideas more easily (Borasi, 1992).

Alternative definitions of a concept are equivalent statements that differ alongtheir optional features (van Dormolen & Zaslavsky, 2003). The distinction betweenimperative and optional features is discussed and illustrated later. Based on theirviews of the roles and imperative or preferable features of a definition, people formideas and preferences as to what constitutes a “good” or “acceptable” definitionamong a set of equivalent statements. This position concurs with Wilson’s (1990)

Journal for Research in Mathematics Education2005, Vol. 36, No. 4, 317–346

Copyright © 2005 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

assertion that “although we frequently use definitions, we rarely focus on thenature of definitions. There is little agreement on what constitutes a good defini-tion” (p. 33). Thus, one part of our study examines students’ views and preferencesregarding the roles and features of a mathematical definition.

When dealing with alternative definitions, we must take into account students’concept image and personal concept definition of the defined concept (Vinner,1991). By a concept image we refer to “something non-verbal associated in our mindwith the concept name” (p. 68), such as a visual representation, a collection ofimpressions or experiences, or a mental picture. Clearly, a student’s conception ofa mathematical concept has bearing on what he or she will accept as a definition;a student is not likely to accept a definition that does not concur with the concep-tion that he or she holds of the defined concept.

Therefore, in an attempt to address the issues noted above, the overall purposeof the study reported herewith was to investigate ways in which dealing with alter-native definitions of given mathematical concepts, and related statements describingthese concepts, may (1) elicit students’ existing conceptions of a mathematical defi-nition and of the defined concept; (2) support students’ reasoning; and (3) stimu-late the refinement of these conceptions.

THEORETICAL PERSPECTIVES

The notion of a mathematical definition develops over the schooling years mostlythrough examples of definitions of many specific mathematical concepts. Studentsrarely discuss nonexamples of a mathematical definition, nor do they deal explic-itly with the imperative features of a definition, let alone come across a formal defi-nition of the notion of a mathematical definition. In this sense, the concept of defi-nition is to some extent similar to the concept of proof. However, although manystudies address students’ or teachers’ conceptions of proof (e.g., Fischbein, 1982;Hoyles, 1997; Knuth, 2002), few studies investigate students’ conceptions of a math-ematical definition.

318 Students’ Conceptions of a Mathematical Definition

The Notion of aMathematical Definition

Roles Features

Imperative Optional

Figure 1. Elements of the notion of a mathamatical definition

Students usually interact with examples and nonexamples of a concept hand inhand with the definition of that concept, thus constructing their conception of theconcept (Wilson, 1990). Over time, by accumulating this kind of experiencesurrounding different concepts, students form their conception of a mathematicaldefinition, which consists of the roles and (imperative and optional) features of adefinition. Students are commonly exposed to just one (usually a textbook) defin-ition of a given concept. They rarely encounter alternative definitions, namely, avariety of examples of definitions for one concept, nor do they explicitly deal withnonexamples of a mathematical definition.

A nonexample of a mathematical definition of a particular concept could be astatement (describing this concept) that is not equivalent to its commonly accepteddefinition. For example, a common definition of an increasing function is: A func-tion with a domain D is an increasing function on D if x1 > x2 ⇒ f(x1) > f(x2), forany x1, x2 ∈ D. With respect to this definition, a nonexample of a definition of anincreasing function could be: A function with a domain D is an increasing func-tion on D if for any x ∈ D, f ′(x) > 0. To realize that these two statements arenonequivalent, one must employ logical considerations (similar to ways describedin Leikin & Winicki-Landman, 2000, for other concepts). Clearly, if a student holdsthe latter definition as a personal definition, he or she will not consider it as a nonex-ample of a definition. It follows that a personal concept definition may be either amathematical definition or a nonexample of a mathematical definition, dependingon the student’s understanding. Hence, in dealing with examples and nonexamplesof a mathematical definition, personal concept definitions play a critical role (Tall& Vinner, 1981; Vinner, 1994).

In addition to inadequate personal concept definitions, nonexamples of a math-ematical definition also rely on imperative features of a definition (van Dormolen& Zaslavsky, 2003). Thus, if a statement is equivalent to an acceptable definitionbut does not satisfy one of the imperative features of a definition, it constitutesanother kind of nonexample of a mathematical definition.1 The imperative featuresrelate to the following requirements: a mathematical definition must be noncon-tradicting (i.e., all conditions of a definition should coexist) and unambiguous (i.e.,its meaning should be uniquely interpreted). In addition, there are some features ofa mathematical definition that are imperative only when applicable: A mathemat-ical definition must be invariant under change of representation; and it should alsobe hierarchical, that is, it should be based on basic or previously defined concepts,in a noncircular manner. Clearly, a statement describing a concept that does notfulfill the imperative requirements of a definition is a nonexample of a definition.In order to identify such a statement as a nonexample, one must hold these imper-ative conditions as part of his or her notion of a mathematical definition.

319Orit Zaslavsky and Karni Shir

1 We restrict our discussion of nonexamples of a definition to statements describing the concept underconsideration, which are either nonequivalent to an acceptable definition of the concept or do not satisfysome imperative features of a definition. We deliberately exclude the discussion of other kinds of nonex-amples of a mathematical definition that we do not regard relevant to our context, such as a mathemat-ical object (e.g., a drawing) that is not a declarative or procedural description of any concept.

In addition to the imperative features mentioned on the previous page, there arefeatures for which there is no agreement on whether they are imperative. The mostnotable example of such a controversial feature is the requirement that a mathe-matical definition be minimal. A definition is considered minimal if it is econom-ical, with no superfluous unnecessary conditions or information. That is, a minimaldefinition should consist only of information that is strictly necessary for identi-fying the defined concept. For example, defining a rectangle as a quadrangle withfour right angles is not a minimal definition, since it is enough to require that therebe three right angles. Although some (e.g., Borasi, 1987, 1992; Hershkowitz,1990; Vinner, 1991; Winicki-Landman & Leikin, 2000) claim that minimality isimperative, others (e.g., de Villiers, 1998; Pimm, 1993; van Dormolen & Zaslavsky,2003) recognize the role of context with respect to the minimality criterion andaccept, to a certain extent, some redundant definitions. Thus, what may seem as anonexample of a definition to one may be an acceptable definition to anotherperson, depending on varying conceptions of what is imperative in general, and onviews of the ultimate need for a definition to be minimal, in particular.

Our aim in this study was neither to determine correctness of nor to try to reacha universal agreement on the definition of a definition, but rather to elicit theunderlying considerations and perspectives that are brought forth in activitiesdealing with examples and nonexamples of definitions. Thus, within the frameworkof our study we designed an environment that stimulated discussions surroundingexamples and nonexamples of definitions of various mathematical concepts (similarto Borasi, 1992; Lakatos, 1976). This environment allowed us to identify students’conceptions of a mathematical definition. It also revealed how the general notionof a definition and specific concept definitions may interact, leading to moreprofound understanding of these constructs.

THE STUDY

The study reported here is part of a larger study addressing students’ and teachers’conceptions of mathematical definitions (other portions of the study are reportedin Shir & Zaslavsky, 2001, 2002). This part of the study examined how students’conceptions of a mathematical definition, in general, and their personal definitionsof specific concepts, are both reflected and developed through activities that elicitconsideration of alternative ways to define a mathematical concept.

Participants

Four 12th-grade students participated in this part of the study: Erez, Mike, Omer,and Yoav (pseudonyms). The students studied mathematics according to the mostextended curriculum offered in senior-high schools in Israel (called a 5-unit levelof mathematics). The reason for selecting top-level students in mathematics wasto allow an investigation of definitions of a wide variety of mathematical concepts,of various degrees of complexity and subtlety, for which a rather broad and sound


mathematical background was needed. They were selected on the basis of their artic-ulateness and willingness to devote time to the research activities. The group wasdiverse in terms of their achievement in mathematics within this top-level strand.Two were average students (Erez and Yoav) and two were outstanding students(Mike and Omer) who took additional extra-curricular undergraduate courses inmathematics at a nearby university.

Procedure

The study examined the students’ conceptions of the mathematical definitionsof four concepts: a square, an isosceles triangle, an increasing function, and a localmaximum point of a function. Based on a preliminary investigation with teachersand students, four questionnaires for students were designed, all sharing a similarstructure, although differing with respect to the specific concept at question. Foreach concept there was a sequence of three consecutive sessions. At the firstsession, each student replied individually to a written questionnaire. At the secondsession, which took between 1 1/2 to 2 1/2 hours, the students completed the samequestionnaire as before, only as a group with no interference from the researcher.This session was videotaped and transcribed. In the third and last session, thestudents were asked to reply again to the same written questionnaire individually.Altogether there were 12 sessions (three for each concept). In order to allow thestudents to devote as much time as they found necessary to work on the task, therewas no time limit set for any of the sessions.

Research Instruments

As discussed earlier, our study addressed several perspectives and considerationsassociated with the nature of mathematical definitions. The design of the researchinstruments took into account the potential of this wealth to enhance genuine math-ematical discourse (as recommended by the NCTM, 2000, and implemented byBorasi, 1992, and Lampert, 1990). The students’ conceptions of a mathematical defi-nition were examined through group discussions and written questionnaires thatelicited their reasoning about and justifications for accepting or rejecting specificstatements as possible definitions of a certain mathematical concept. As mentionedabove, the group discussions were stimulated by four questionnaires of a similarstructure, one for each concept (see Appendixes A, B, C, and D). Each question-naire consisted of seven or eight statements describing its focal concept. For eachstatement, the participants were asked to determine whether they would accept itas a definition of the described concept. In addition, they were asked to choose thestatement that they preferred as a definition of that concept.

In selecting the specific mathematical concepts for our research instruments, itwas important to include some that are considered straightforward and some thatare more complicated. The two geometric concepts—a square and an isoscelestriangle—were rather simple and familiar concepts for which there was no doubtregarding how the students perceived them. The other two—an increasing function


and a local maximum point of a function—were subtle analytic concepts of whichstudents have been found to have different concept-images (e.g., Rasslan & Vinner,1998; Vinner, 1994). Consequently, for each analytic concept the collection of state-ments included, in addition to its common textbook definition, some statements thatare equivalent to it and others that are not equivalent to it. The latter statements weredesigned as nonexamples of a definition, each reflecting a commonly held concep-tion of the focal concept (see Appendixes A and C). On the other hand, the collec-tion of statements designed for each geometric concept included only statementsthat were equivalent to each other. We assumed that it would be unlikely forstudents to select a statement that is not equivalent to their clearly accepted defin-ition. Since we did not anticipate any differences in the commonly accepted defi-nitions of these simple geometric concepts, we used these concepts as a trigger todeal with more general features of a definition. Thus, for each geometric conceptthe equivalent statements varied along a number of possible features of a defini-tion (see Appendixes B and D).

For the purpose of the study, we chose to address three central features that inour preliminary investigation proved to have the potential of stimulating reasoningabout the roles and features of a definition. The first feature, which we discussedearlier, had to do with the minimality of a definition. In terms of this feature, somestatements describing the two geometric concepts were minimal and some were not.

The second feature had to do with the form of presentation: A definition can beeither procedural—by genesis, or structural—by a common property (Leron,1988; Pimm, 1993; Rissland, 1978). For example, what follows is a procedural defi-nition of a square: A square is an object that can be constructed as follows—drawa segment; from each edge erect a perpendicular to the segment, in the samelength as the segment (both in the same direction). Connect the other 2 edges ofthe perpendiculars by a segment. The 4 segments form a quadrangle that is a square.On the other hand, a structural definition of a square relies on a property of the object(e.g., a square is a quadrangle, in which all sides are equal and all angles are 90°)or on a common property of the points that constitute the object (e.g., a square isthe locus of points for which the sum of any point’s distances from two vertical linesis a positive constant). The questionnaires for the geometric concepts contained bothprocedural and structural statements.

The third feature we addressed was the hierarchical nature of definitions. Asdiscussed by van Dormolen and Zaslavsky (2003), some concepts are related to eachother in a hierarchical manner. In such cases, the definition of a concept may behierarchical, that is, based on a more general and previously defined concept. Webuilt on the work of de Villiers (1994, 1998) and distinguished between levels ofhierarchy (Shir & Zaslavsky, 2001, 2002). For example, the following threeconcepts are hierarchical in nature: a polygon, a triangle, and an isosceles triangle.Thus, we can define an isosceles triangle as a triangle that has two equal sides. Wecan go one step back and define it as a polygon with three sides, two of which areequal. The further back we go, the higher the degree of hierarchy.


Data Analysis

In general, the methodology that guided us in our data analysis follows a quali-tative paradigm according to which the research findings are obtained as a resultof an iterative inductive process of data analysis (Denzin & Lincoln, 1994). In ourcase, the findings were obtained by classifying and organizing the documentedwritten responses and group discussions into categories that were not predetermined(Goetz & LeCompte, 1984; Guba & Lincoln, 1981).

The data consisted of students’ written responses to four different question-naires—each administered twice individually and once as a group report, and of thefour group discussions that were videotaped and transcribed. Students’ answersincluded an assertion as to whether they accepted the statement as a definition ofthe focal concept, followed by a written response, composed of one or more justi-fications supporting the assertion. Accordingly, every response was divided intojustification-units, each consisting of one justification. Altogether, students’ writtenresponses included 497 justification-units: 336 justifying the acceptance of a state-ment as a possible definition and 161 justifying the rejection of a statement as apossible definition.

Based on the coding scheme developed in our preliminary study (Shir &Zaslavsky, 2001), we analyzed the justification-units according to the type ofreasons provided for acceptance or rejection of a statement. Our analysis yielded26 types of reasons (14 for acceptance and 12 for rejection), indicating three kindsof underlying considerations (the same as those identified in our earlier work (Shir& Zaslavsky, 2001)): mathematical, communicative, and figurative. We elaborateon these themes later in the article.

The group discussions were divided into episodes according to the central issuethat was discussed. Altogether there were 291 episodes focusing on 14 central issues(13 on task and one off task). The on-task issues were associated with variousfeatures and roles of definitions (e.g., questions regarding whether a definition mustbe minimal, whether it could be procedural, or to what extent it should be useful).Each episode was coded according to its central issue and its length of time. Thefrequencies of occurrences of each of the 13 issues were calculated by consideringan episode as a single occurrence of its central issue. In addition, the total lengthof time devoted to each issue was computed. These issues were further groupedaccording to the three underlying considerations noted above (mathematical,communicative, and figurative), similar to the way in which the written justifica-tions were classified, and correspondingly the frequencies and lengths of time ofeach type of consideration were obtained.

As we analyzed the kinds of justifications (written and oral) that the studentsemployed, we noticed differences between their justifications for the geometric andthe analytic concepts. This observation led to another analysis—a distinctionbetween justifications that were based on examples and those that were related togeneral characteristics of a mathematical definition with no reference to a specificexample. Finally, there were several cases in which a student switched from a


(written) decision to accept a statement to a (written) decision to reject it or viceversa, after reconsidering the statement or discussing it with others. These shiftswere documented and analyzed. The transcriptions of the group discussions providedadditional explanatory backing for these shifts.

FINDINGS

In this section, we describe in detail the kinds of reasoning that characterizedstudents’ responses and discussions surrounding the consideration of various state-ments as mathematical definitions. We compare the kinds of reasoning used acrossthe four focal concepts and examine the interrelations between students’ groupdiscussions and the shifts in their assertions. Through their reasoning, we identifystudents’ conceptions of a mathematical definition and their development of under-standings related to the notion of a mathematical definition as well as to the focalconcepts. We conclude this section by analyzing students’ shifts in their assertions,indicating modifications in their conception of a definition as well as in theirunderstandings of the focal concepts.

Students’ Mathematical, Communicative, and Figurative Considerations

We found that students employed two main types of considerations across all fourconcepts: mathematical and communicative. In addition, another kind of consid-eration was identified for the geometric concepts, which we term figurative.

Mathematical considerations are justifications involving logical concerns.According to this perspective, a statement was evaluated based mainly on itscorrectness, i.e., whether the condition specified in the statement is both a neces-sary and sufficient condition for the focal concept, according to the individual’sunderstanding of the concept. Mathematical consideration also includes referenceto both commonly agreed upon and controversial views of the imperative require-ments for a definition.

Communicative considerations are arguments that focus on the communicativenature of a definition. According to this perspective, a statement was evaluated basedmainly on its clarity and whether it is comprehensible and within reach to those whodeal with it. This type of consideration deals mainly with optional features of a defi-nition, which are mostly a matter of personal taste, and reflects the perceived roleof a definition.

The third category of arguments, which we call a figurative consideration, hasto do with the way the participants perceive a geometric object and its differentcomponents. People are exposed to squares and isosceles triangles rather frequentlyin real-life contexts, from early childhood. Thus, they are likely to conceptualizethe technical mathematical concept of a square, for example, making use of theireveryday concept of a square (Núñez, 2000), which appears without its diagonals.According to this perspective, there is a distinction between the parts that seem inte-gral to a geometric object (such as the sides and angles of a polygon) and those that


are often hidden (such as the diagonals of a polygon); the latent parts are notequally considered integral parts of the object. Arguments of this category focusedon the issue of whether it is legitimate to define a figural concept (Fischbein, 1993)by properties of its latent parts (e.g., congruence of its diagonals). Thus, thisperspective is characterized by a reluctance to accept statements that are based onlatent parts of geometric concepts.2

Table 1 presents the distribution of the justifications given by the students foracceptance or nonacceptance of a statement as a definition for the four concepts,classified by their underlying type of consideration (for further analysis of the argu-ments into the different subcategories comprising each type, see Shir & Zaslavsky2001, 2002). The four mathematical concepts presented in Table 1 are organizedaccording to the type of the focal concept: First the two geometric concepts and thenthe two analytic ones.

Table 1Distribution of Students’ Written Justifications by Concept and Their UnderlyingConsiderationsUnderlying Square Isosceles triangle Increasing function Local maximumconsideration (N = 152) (N = 150) (N = 111) (N = 84)Mathematical 39% 39% 84% 88%Communicative 59% 52% 13% 9%Figurative 2% 7% — —Other — 2% 3% 3%

As shown in Table 1, although all four questionnaires were of a similar structure,the distribution of types of considerations depended to a large extent on the typeof focal concept. For the analytic concepts, which were less familiar to the studentsand more complicated, students used mainly mathematical considerations(84%–89% of the justifications); for the more familiar geometric concepts, bothcommunicative and mathematical considerations were rather frequently employed(39%–59% of the justifications). Similar findings recurred in the group discussions.

As indicated in Table 2, the distribution of the number of episodes dealing withmathematical, communicative, and figurative issues is very similar between the twogeometric concepts and greatly resembles the distribution of written justifications forthese two concepts. A closer look at the time allocated to each kind of issue showsthat for these concepts the communication considerations were dominant (72%–78%of the time). Additionally, the figurative considerations took 10% of the time in eachof these cases. In fact, the relative “weight” of the figurative considerations is muchhigher, as it was applicable only to statements including latent parts (2 of the 8 state-ments describing a square and 2 of the 7 statements describing an isosceles triangle).


2 These three types of considerations are illustrated in more detail in Shir & Zaslavsky (2002).

Unlike the findings for the geometric concepts, for the analytic concepts most ofthe time was devoted to mathematical considerations. The case of the localmaximum was rather extreme (100% of the time involved mathematical consider-ations). The students spent this time trying, unsuccessfully, to reach an agreementregarding what a local maximum actually is. The participants were not clear on thefull scope and meaning of the concept and thus were not able to move on to consid-ering alternative definitions. In a less extreme way, this phenomenon recurred inthe discussions regarding the definition of an increasing function. In this case, mostof the mathematical discussions (i.e., 75% of the time involving mathematicalconsiderations) were spent on debating what an increasing function is (or is not).Only after resolving this matter and reaching a consensus were they able to moveon to considering ways of defining an increasing function, from both mathemat-ical and communicative aspects.

Students’ Example-Based and Definition-Based Reasoning

By example-based reasoning we refer to justifications that use examples toconvince one’s self or others regarding a certain assertion. Examples (or coun-terexamples) may be used to determine the boundaries of a concept or to supportor reject a conjecture (as in Rissland, 1991). The examples that the students in ourstudy used were mostly counterexamples of concepts, which served to support therejection of a statement as a possible definition of a certain concept. Examples ofconcepts or of concept-definitions were suggested only twice.

We turn to a written response illustrating how students employed example-basedreasoning. In order to support his rejection of statement (a) in Appendix A as a defi-nition of an increasing function, Mike used an example of the trigonometric func-tion tan x and argued that “I don’t accept it [statement (a)] as a definition [because,for example] f(x) = tan (x) is not an increasing function, yet at any point f ′(x) is posi-tive.” In this case, the function tan (x) is brought as a counterexample showing thata function satisfying the condition that its derivative is positive at any point of thedomain is not necessarily an increasing function.

The following excerpts illustrate how students employed example-basedreasoning in their group discussions. In the first excerpt, the students debated the


Table 2Distribution of the Number and Lengths of Time (in minutes) of On-Task EpisodesAccording to the Three Types of Considerations

Local Increasing Isoscelesmaximum function triangle Square

Underlying Number Time Number Time Number Time Number Timeconsideration (N = 28) (48 m) (N = 44) (32 m) (N = 97) (93 m) (N = 109) (66 m)Mathematical 100% 100% 53% 79% 38% 16% 30% 16%Communicative — — 47% 21% 52% 72% 50% 78%Figurative — — — — 10% 12% 10% 6%

question of whether to accept a statement claiming that the derivative at a localmaximum must be zero (statement [a], Appendix C) as a definition of a localmaximum:Mike: Does someone oppose statement (a)?Omer: Yes [nods his head].Erez: One moment, why do you oppose (a)?

1 for x = 0Omer: Take . . . , for example, take f (x) = .

0 for x ≠ 0You’ll get this function [sketches the graph of his example].

Mike: It’s not a local maximum, in my opinion it’s definitely not.Omer: Of course it is. It’s even a global maximum.

Omer used his example in order to explain to his peers why he thought statement(a) cannot serve as a definition. He used the example to show that the statement wasnot a necessary condition for a local maximum. Omer’s example evoked disagree-ment between the members of the group with respect to their understanding (orconcept image) of the focal concept, that is, a local maximum.

In addition to example-based reasoning, another type of reasoning was identi-fied, which we term definition-based reasoning. By definition-based reasoning werefer to justifications that rely on various features or possible roles of a mathemat-ical definition. This type of reasoning was used to support both acceptance and rejec-tion of a statement as a definition.

We now turn to a few illustrations of how students used definition-basedreasoning in order to justify their assertions. Erez, for example, supported hisrejection of a procedural statement (statement [h], Appendix B) as a definition ofa square, by arguing that “a definition should not be given in the form of buildinginstructions.”

In the following excerpt, Erez used definition-based reasoning to try to convincehis peers to accept a statement describing a square by its diagonals’ properties (state-ment [d], Appendix B) as a definition of square:

Let’s see. What conditions should a definition satisfy? A definition should have no super-fluous details, it should be accurate and correct, and you should be able to check thingsaccording to it. I believe that this statement satisfies all of these requirements.

This excerpt reinforces our claim that the group discussions led the participants(in this case, Erez) to articulate what conditions they thought (mathematical) defi-nitions, in general, should satisfy, a typical characteristic of definition-basedreasoning.

Table 3 depicts the distribution of manifestations of example-based and defini-tion-based reasoning among the written justifications. As shown in Table 3, for allfour concepts, other types of reasoning were employed at the most in 22% of thejustifications. It is interesting to note that definition-based reasoning was the mostfrequently used type for all four concepts (in at least 54% of the justifications).Example-based reasoning was far more frequent for the analytic concepts


(24%–29%) than for the geometric concepts (0%–1%). A closer look only at thejustifications provided for rejection of a statement as definition of an analyticconcept indicates that 78% of those for the increasing function and 71% of thosefor a local maximum were example-based.

The differences in Table 3 between the analytic concepts and the geometric withrespect to the type of reasoning employed are closely connected to the differencespresented in Tables 1 and 2 with respect to the underlying considerations that wereapplied. Although all example-based justifications reflected only mathematicalconsiderations, the definition-based reasoning reflected all three types of consid-erations—mathematical, communicative, or figurative. In all three tables the distri-butions of the analytic concepts differ from those of the geometric concepts, yetthe distributions for the analytic concepts are very similar as are the distributionsfor the geometric concepts. Apparently, dealing with the rather straightforwardgeometric 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 ofmonster-barring (Lakatos, 1976), wherein the students iteratively modified theirdefinition to better reflect the concept image they held.

Students’ Conceptions of a Mathematical Definition

As mentioned earlier, students’ conceptions of a mathematical definition wereelicited through their written responses and their verbal reasoning. In the groupdiscussions, and with no interference of the researcher, students began to realizethat they had diverse opinions regarding the crucial conditions that a definition mustsatisfy. As a result, a number of questions in relation to the conception of a defin-ition arose implicitly, of which eight seemed particularly interesting. Due to thedesign of the specific tasks for each concept (as described earlier), some questionsoccurred in all the discussions (see vi and vii below), whereas others were more natu-rally related to the geometric concepts (see i–v below). We now elaborate on eachquestion and present excerpts exemplifying the different views expressed by thestudents with respect to these questions.

i. May there be superfluous conditions in a definition (i.e., must a definition beminimal)? In general, the issue of minimality elicited debate and discussion


Table 3The Distribution of Written Justifications According to Type of Reasoning


Type of reasoning (N = 84) (N = 111) (N = 150) (N = 152)Definition-based reasoning 55% 54% 97% 89%Example-based reasoning 29% 24% — 1%Other 16% 22% 3% 10%Total 100% 100% 100% 100%

surrounding its imperativeness. These discussions ended in an agreement that,although there might be cases in which a minimal definition is preferable, minimalityis not an imperative feature of mathematical definitions. The following excerpt, withreference to a statement describing a square in a “nonminimal” manner (statement[e], Appendix B), illustrates the group’s debate over whether it is legitimate for adefinition to have more details than necessary:Erez: [Referring to statement (e)] It’s correct, but it is not a definition.Yoav: It’s correct, and it is a definition.Erez: It has too many details.Yoav: Too many details, but it is still a definition.Omer: What do “too many details” have to do with that? Mike: In which definition here don’t you have too many details?Erez: [Referring to Omer’s question] Well . . . in fact . . . maybe it is.

We can see that Erez began rethinking the issue of minimality. As a result, later hewas willing to consider a nonminimal statement as a definition.

In the following excerpt, the group discussed once again the question of mini-mality, with reference to a nonminimal statement describing an Isosceles Triangle(statement [d], Appendix D):Yoav: I think that we should accept statement (d). [However] I don’t prefer it because it

has two [pairs of] equal elements [sides and angles].Erez: I also think that we should accept statement (d).Yoav: So why do you argue with me? [Turns to Omer] Omer, we don’t prefer it since

there are too many details, but it doesn’t mean that we should not accept it. Whatdoes it matter if there are equal angles or equal sides?

Note that they distinguished between the legitimacy of a nonminimal definition andtheir personal preferences regarding such definition.

ii. May a set of procedures for building a mathematical object serve as a basisfor its definition (i.e., may a definition be procedural)? The students were not willingto accept a procedural statement as a mathematical definition. The followingexcerpt conveys the group’s consensus regarding whether it is legitimate for a defi-nition to be procedural, with reference to the statement describing a square in aprocedural manner (statement [h], Appendix B).Erez: Statement (h) [a procedural definition of a square] is a guideline regarding how

to build a square.Mike: No way, (h) is too long.Yoav: It’s an instruction, it’s not—Erez: [Interrupts] It’s a description of how to construct a square. [Turns to Omer who

wrote the group report.] You should write that we don’t accept it [as definition ofa square].

Omer: So, [the answer] to our question if we are willing to accept it as a definition . . . isno, we don’t [accept it].

It is interesting to note that the two procedural statements, one describing a square(statement [h], Appendix B) and the other describing an isosceles triangle (state-


ment [g], Appendix D), were the only statements on which all four participantsagreed immediately. For both geometric concepts, the students unanimously rejectedthe procedural definitions and argued that guidelines for constructing an object couldnot serve as a definition. Because procedural definitions are not very common inhigh school mathematics textbooks, it is likely that the participants had never comeacross a procedural definition before. Indeed, in a follow-up discussion with the fourstudents, they each claimed that they had never seen a procedural definition in math-ematics. Thus, a procedural definition was obviously not part of their conceptimage of a definition. Similar findings were reported with respect to mathematicsteachers (Shir & Zaslavsky, 2001). Thus, the inclination to reject procedural defi-nitions is connected to common classroom practice.

iii. May any concept serve as a basis for a definition of another mathematicalconcept? According to the students, mathematical definitions should be based, whenpossible, on simple, familiar, and clear concepts. The students shared the opinionthat there is no need to go back to concepts that are too basic. However, they differedin their views regarding which concepts should be considered too basic. Statements(a) and (b) in Appendix B, which are both hierarchical statements of level 1,elicited discussion regarding the legitimacy of basing a definition of a square onthat of a rhombus (statement [a]) or of a rectangle (statement [b]): Erez: I don’t accept statements (a) and (b).Yoav: Me neither. Omer: Why not?Erez: Because you need to know what a rhombus is, and you need to know what a

rectangle is.Omer: So what?Erez: It’s not acceptable to base a definition on other concepts [that you may not know

or may not have defined yet].Yoav: Actually, a square is a special rectangle or a special rhombus, so you can define

it using those concepts.Erez: There’s no doubt that it is true; indeed it is correct. But a definition, according to

its nature, should be based on the lowest base.Yoav: You can say [it’s a] polygon.Erez: Yes, a polygon with four sides.

A similar debate occurred when the students discussed statements describing anisosceles triangle. The following excerpt deals with the question of what theythought an “overly basic” concept would be, with reference to statements describingan isosceles triangle as a closed path of connected lines (statement [f], Appendix D):Yoav: A three-sided polygon is still normal, but a closed path of connected lines . . . this

is [overly basic].Erez: [Interrupts] We learned what a closed path of connected lines is before we learned

about polygons.…………..

Erez: If you accept a polygon, you must also accept a closed path of connected lines.Mike: I don’t accept it [statement (f)].


Yoav: [Replies to Erez] No, I don’t have to accept both. I can accept just the one that Iconsider appropriate.

Mike: Erez, a definition should not go down to the most basic level. This is why you havepreceding definitions. If you want to define a concept you shouldn’t go down tothe base. You’ll get an extremely long definition.

Although the students agreed in principle that a mathematical definition shouldbe based on basic concepts, they were never able to agree on what a “basic concept”is, nor on what should be considered a concept that is “overly basic.”

iv. May any “correct” statement that is a necessary and sufficient condition ofa concept serve as a definition? According to the students, in order to qualify as apossible definition of a certain concept, a statement must be “correct” in that it mustbe a necessary and sufficient condition for the concept. However, they maintainedthat being “correct” was not enough for the statement to be accepted as a defini-tion. The group agreed, for example, that the statement describing a square as a paral-lelogram satisfying a certain condition (statement [c], Appendix B) was a correctstatement, yet they did not agree on the acceptance of it as a definition:Yoav: We don’t accept it [statement (c)]. It doesn’t mean that it’s not correct. We just

don’t accept it.Erez: It’s obviously correct; all the statements are correct, so should we accept them all

as a definition?!Omer: If it’s correct, why can’t it be a definition?Erez: Is statement (h) correct? Is it correct? Is statement (h) correct or isn’t it? [Refers

to their previous consensus not to accept statement (h), the procedural statement,as a definition]

Mike: Statement (h) is correct.Erez: [Turns to Omer] So why didn’t you accept statement (h) as a definition?Mike: [Turns to Omer] Yes, why didn’t you?Erez: [Turns to Omer] Do you have an answer to your question? Omer: Because it’s not abstract.Erez: This statement [refers to statement (c)] is also correct, but I’m not willing to accept

it, just like I wasn’t willing to accept the other one [statement (h)] although it’salso correct.

Since Erez gave an example of a correct statement that the others did not acceptas a definition, he managed to convince them all that the correctness of a statementis not enough for it to serve as a mathematical definition. Although the students didnot address the issue of equivalence directly, from their reasoning it is clear thattheir position was that the criterion of equivalence is insufficient for acceptance ofa definition.

v. May properties of any part of a geometric object serve as a basis for its defi-nition? In considering a statement describing a square by certain properties of itsdiagonals (statement [d], Appendix B), the group debated over the legitimacy ofdefining a geometric concept (e.g., a square) by properties of its latent parts:Omer: Yoav, draw a square.Yoav: [Sketches a square] Okay.


Omer: Erez, [now you] draw a square.Erez: [Looks at Yoav’s square] You mean he didn’t draw it with the diagonals? Omer: Exactly.Erez: What’s the connection?Omer: When you have a square you should refer to its sides, not to the connections

between them.Erez: Why?Yoav: Maybe this definition is a bit more complicated, but there’s no more to it.Omer: I am willing to accept it but it’s not my preference.Mike: When you draw a square you see this [“draws” a square with his finger on the table],

like he [Omer] told you, you see sides, you don’t see diagonals that are perpen-dicular. A square is first of all 4 sides. You don’t refer to the diagonals. The diag-onals are a property of the square.

Erez: So equal sides and right angles, they are also properties.Omer: No, they are a definition.Mike: Right, sides are sides. They build the square. Sides and angles build the square.

Diagonals don’t build the square.

Mike did not agree to accept statement (d) as a definition, whereas Omer, Erezand Yoav agreed to accept it, although they preferred other statements. Theydecided to leave it at that point and go back to it later, at which time Erez managedto convince Mike to accept statement (d), on the grounds that it was correct and reliedon a basic concept (a quadrangle).

When dealing with a similar statement (statement [c], Appendix D) describingan isosceles triangle based on properties of its latent parts (the medians, in this case),Omer and Yoav began to change their minds. In fact, they shifted from agreementwith Erez to agreement with Mike’s standpoint. That is, they now felt that it maybe a problem to use properties of latent parts in a definition.Omer: It’s an indirect property of an isosceles triangle.Erez: What do you mean by indirect property? Sides and angles are also indirect prop-

erties of an isosceles triangle, aren’t they?Yoav: No, they aren’t an indirect property, because . . .Erez: [Interrupts] You need to draw the medians in addition [to the figure itself], and

that’s why I only accept it but don’t prefer it. But it’s still a definition.Omer: No, no, no, no. This is a property that just happens to exist. When you say isosceles

triangle, it’s a triangle with two equal sides. It’s irrelevant that other things areinside of it.

Erez: In a circle, for example, the radius is also a part that’s inside of it.Omer: What?Erez: In a circle I also didn’t draw the radius, although I can define it this way [based

on the radius]. In the same way, in a triangle I didn’t draw the medians, but I canstill define it this way [based on the medians]. What bothers you is that they describeit with something that you can’t see.

Omer: Sort of . . . it’s an indirect property.

This is an example of an issue on which the four students were not able to reachan agreement. Although Erez accepted the statement as a definition, the others did


not, because they thought a definition should not be based on “indirect properties,”that is, on properties of latent parts of the figure.

According to the theory of figural concepts (Fischbein, 1993; Marriotti &Fischbein, 1997), geometric concepts have a dual nature and are composed of bothfigural and conceptual aspects. Although these two aspects should interact harmo-niously, often one of the two aspects appears independently. In our case, thesquare and the isosceles triangle have figural characteristics that are derived fromtheir shape. These figural characteristics include equal sides and right angles inthe case of the square and two equal sides and two equal angles in the case of theisosceles triangle. Since the figural prototypes of a square and isosceles triangledo not include their latent parts, presenting students with a definition that is basedon properties of their latent parts seemed to evoke a conflict. We must keep in mindthat

the drawing is not the geometrical figure itself, but a graphical or a concrete, materialembodiment of it. . . . The geometrical figure itself is only the corresponding idea thatis the abstract, idealized, purified figural entity, strictly determined by its definition.(Fischbein, 1993, p. 149)

Confronting people with a possible new definition of a figural concept that they havealready constructed in their minds creates a reversed process, in which the defini-tion is judged by the extent to which it corresponds with the figural aspects of theconcept. The above students’ discussions are manifestations of this phenomenon.

vi. Can there be more than one definition of a mathematical concept? At first allfour students agreed that a mathematical concept may have several (equivalent) defi-nitions. Later on, in the course of their subsequent discussions, one of them beganto question this assertion. The following excerpt, taken from the first activity(surrounding different statements describing an increasing function), demonstratestheir initial standpoint. The students discussed the question of which of two equiv-alent statements (statements [b] and [d], Appendix A) to accept as a definition:Erez: We can decide that both are definitions. A definition is not necessarily one specific

thing. Who says a definition is unique? Maybe we can define . . .Yoav: [Interrupts] You can define this in both ways.Mike: Truthfully, it’s better to have them both. This one [Statement (d)] for those who

know Hebrew and that one [Statement (b)] for those who don’t.3

Omer: Both of them are valid. . . . This way we can use whichever we want, accordingto what we need.

As the discussions progressed, and they dealt with other concepts as well as withthe requirements of a mathematical definition, Omer began doubting the possibilityof accepting more than one definition. He then tended to accept as definition onlythe “best” statement—the one he thought fulfilled all the imperative requirements—


3 What Mike actually meant was that Statement (d) is a more verbal one, thus, it requires a bettercommand of their spoken language (in this case, Hebrew), whereas Statement (b) relied mainly onsymbols.

and treated all other equivalent statements as theorems. Toward the very end of theirdiscussions, Erez and Yoav managed to convince Omer that it is possible, after all,to have several definitions to a mathematical concept. As Erez concluded, “I’m surethat a number of [possible] definitions were brought to the ‘definition committee,’and they said ‘we accept this and that, but prefer this and that.’ ”

vii. What are the roles of a definition? Throughout the group discussions, mainlyin the form of what we described as definition-based reasoning, the studentsreferred to several roles they attributed to mathematical definitions. One particularrole of a mathematical definition, namely, its role in classification of examples andnonexamples of a concept, came up in several cases during the group discussion.For example, when dealing with the definition of an increasing function (statement[b], Appendix A), the students pointed to its power in “refuting functions,” that is,in identifying nonexamples of an increasing function. In contrast, they referred toits limitation in proving that a certain function is an increasing function. Thisdistinction was enhanced by the necessity that the conditions of the definition besatisfied by every two points. Clearly, it seemed easier for them to find a pair ofpoints and show that they did not satisfy the required condition than to prove thatany two points did.

The recurring reference to the role of definitions was reflected also when thestudents dealt with the different ways of defining a square (see Appendix B).Yoav’s explanation for why he thought that statement (h) is not a definition wasthis: “It’s not a definition because you can’t check if something is a square accordingto this statement.” Similarly, Erez, who argued that statement (d) can serve as a defi-nition in contrast to statement (h), said,

According to (h), if you get a certain polygon, you can’t tell if it’s a square, butaccording to (d) you can. This, first of all, disqualifies statement (h) and gives you areason to accept (d).

In addition to the role of a definition in classifying mathematical objects, thestudents referred to other roles as well. In particular, they considered the useful-ness of definitions for proving and problem solving and their contribution to under-standing the meaning of mathematical concepts. For example, when dealing withdifferent statements describing an increasing function, the students debated whichof the two statements to prefer, the one describing an increasing function includingmany mathematical symbols (statement [b], Appendix A) or the verbal one withno symbols at all (statement [d], Appendix A). The argument they brought up infavor of statement (b) was that the inclusion of symbols in the definition would prob-ably be helpful in handling formulas and making inferences in the future. On theother hand, they claimed that statement (d) sheds light on the meaning and essenceof the concept, thus is helpful in explaining the meaning of an increasing function.It should be noted that in the context of our study, we were not interested in deter-mining which roles are more important than others. The main issue is that these argu-ments were raised and these roles seemed relevant to the students in supporting theirpreferences.


Students’ Shifts in Their Standpoints and Understandings

As illustrated above, the group discussions elicited much debate and exchangeof opinions among the four students, in the course of which they tried to convinceeach other to reach an agreement. As a result, there were several cases in which astudent switched from a decision to accept a statement to a decision to reject it orvice versa. These shifts are an indication of the meaningful argumentation that tookplace and of the mutual influence between the participants.

As shown in Table 4, every shift in a decision of an individual student regardinga certain statement from one stage to the following stage was counted. Thenumber of shifts per student was either 4 (Erez, Yoav, and Omer) or 5 (Mike).The number of shifts per statement was between 0 (for statements [b] and [f]) to4 (for statements [c] and [e]). In addition, there were other shifts back and forthamong the three stages above that did not appear in the written responses but weredocumented in the transcripts. Generally, the shifts portray individual differencesbetween the students. The distribution of the number of shifts per statement indi-cates the extent to which the various statements lend themselves to evokingdiscussion and musing regarding the subtleties of a mathematical definition orthe defined concept.

Table 4 presents the students’ decisions regarding the different statementsdescribing an increasing function at three stages of the study (see Table 4, note a).A similar analysis was carried out for the other three concepts. Table 5 summarizesthe distribution of number of shifts in students’ assertions per student according tothe four focal concepts. It is interesting to note that every student shifted his asser-tion at least once for each concept. Additionally, for each concept there were at least10 shifts reflecting a change in viewpoint or understanding.

Table 4 Distribution of the Students’ Written Assertions Regarding the Acceptance (+) orRejection (–) of a Statement as Definition of an Increasing Function at Different Stages ofthe Studya

Statement Erez Mike Yoav Omer No. of S1 S2 S3 S1 S2 S3 S1 S2 S3 S1 S2 S3 shifts

(a) + – – + – – + – – + – – 4(b) + + + + + + + + + + + + 0(c) + – – + – – + – – + – – 4(d) + + + + + + + + + + + + 0(e) + – – + – – + – – + – – 4(f) – – – – – – – – – – – – 0(g) + – – – – – + – – – – – 2(h) – – – + – – + – – + – – 3

No. of shifts 4 4 5 4 17a The same questionnaire was given at three different stages: (S1) the first time, prior to group discus-

sions, individually; (S2) the second time, at the end of the group discussion, collectively; (S3) the lasttime, a week after the group discussion, individually.


A closer look at students’ justifications (both written and oral) indicated two mainkinds of shifts: (1) shifts reflecting (either temporary or stable) changes in thestudents’ conceptions of a mathematical definition; (2) shifts reflecting (eithertemporary or stable) changes in the way students understood the focal mathemat-ical concept. Table 6 (p. 340) presents the distribution of type of shifts per concept.As indicated in Table 6, the shifts reflecting changes in a conception of a mathe-matical definition occurred only in the cases of the simple geometric concepts.Furthermore, the shifts in assertions regarding the more complex analytic conceptsreflected mostly changes in the understanding of the focal concept.

Shifts reflecting changes in students’ conceptions of a mathematical definition. Asshown in Table 6, shifts reflecting changes in students’ conceptions of a mathematicaldefinition occurred only for the two simple and familiar geometric concepts. Thechanges had to do with the issue of latent parts of a figure, the minimality of a defi-nition, and the kind of basic concepts on which a definition should rely.

With respect to the latent parts of a figure, Omer and Yoav initially accepted defi-nitions involving latent parts (e.g., the diagonals of a square or the medians of atriangle), but as a result of the group discussions they changed their minds anddecided to reject such statements as definitions. Mike changed his mind back andforth, from rejection to acceptance, and then back to rejection of statements basedon latent parts as definitions. The issue of what concepts may be considered basicenough but not too basic to use in a definition also caused several shifts. Forexample, Yoav and Mike were willing at first to accept a parallelogram or rhombusas part of a definition of a square but then changed their minds and decided thatthese were too “advanced” to serve as a basis. Other shifts were evoked by similardisagreement regarding whether a polygon could be a basis for the definition of anisosceles triangle.

At the beginning, Erez held the conception that a definition must be minimal. Hechanged his mind following the group’s discussion, however, and from then on, wasconsistently willing to accept as definitions statements that were not necessarilyminimal. We see that in the cases associated with the latent parts and the basicconcepts, the discussions led some students to more restricted concept images of adefinition, whereas in the case of minimality, Erez’s concept image of a definitionexpanded.

Shifts reflecting changes in the way that students understood the focal mathe-matical concept. As mentioned above, the vast majority of shifts reflectingchanges in students’ understanding of the defined concepts occurred with respectto the two analytic concepts. For example, Erez’s shift (see the first row of Table4) regarding statement (a) (Appendix A), indicates that at first he thought anincreasing function is a function for which at any point of its domain the deriv-ative is positive. This is not surprising, given the overuse of this criterion to deter-mine the subdomains in which a (differentiable) function increases. Only later didErez become aware that this condition is insufficient for an increasing function.Omer convinced him of this by providing as an example the trigonometric func-


tion f(x) = tan x that is not an increasing function, although the derivative is posi-tive at any point of its domain. At first, Erez thought that tan x was also an increasingfunction. However, by applying the definition that they agreed on (statement [b],Appendix A), he realized that it was not an increasing function. Thus, Erez refinedhis understanding of the notion of an increasing function, and consequently (rightly)changed his decision regarding the acceptance of statement (a) as a definition.

In another case, Mike changed his mind regarding statement (a) (Appendix C),from accepting to rejecting it as a definition of a local maximum. This change wastriggered by the group discussion, in the course of which Mike realized that therewere several kinds of examples of special points of a function that he did not considerexamples of a local maximum although they satisfied statement (c). For instance,he did not consider as a local maximum a point where the function was not contin-uous, even if it was the “highest” point in the neighborhood (see example 2, Figure2). Once he became aware of such cases, he drew conclusions from them thatappeared consistent with his understanding of a local maximum. Consequently, he(rightly) decided to reject statement (c) as a definition, albeit for wrong reasons.


Figure 2. Examples of local maximum that students suggested

Example 1 Example 2 Example 3

In both cases, the students’ shifts from acceptance to rejection of a statement asa definition of the focal concept was due to a change in their understanding of thatconcept. Erez reduced his example space of increasing functions by excluding exam-ples such as f(x) = tan x and adjusted his personal concept definition accordingly.Mike became aware of a gap between his personal concept definition and hisexisting concept image. Similar to Erez, he adjusted his concept definition, althoughhe did not seem to change his concept image.

There was a single case in which the group discussions led one student to makea shift that was not in a “mathematically desired” direction. When dealing withdifferent statements describing a local maximum, Mike, Erez, and Yoav (wrongly)convinced Omer that continuity is a necessary condition for a local maximum.Consequently, he changed his mind and shifted from acceptance of statement (b)

(Appendix C) to rejection of it. Only after rethinking about this issue on his ownduring the following week did Omer shift back to his original decision, which wasmathematically sound to begin with.

DISCUSSION AND CONCLUSIONS

Our study examined four students’ conceptions of a central metamathematicalconcept—a mathematical definition.4 Their conceptions were elicited indirectly,mostly through their discussions surrounding alternative (acceptable or unaccept-able) definitions of four specific mathematical concepts. Although the roles andfeatures of a mathematical definition are not commonly addressed in an explicit wayin K–12 grades (as confirmed by the participating students), in the context of thesediscussions the students referred spontaneously to most of these characteristics. Thisin itself is a manifestation of an indirect learning mechanism that supports the devel-opment of students’ conceptions of a definition. Throughout the group discussions,we identified many instances of students’ insights and turning points in their under-standing of the concepts under discussion as well as in their understanding of thegeneral notion of a definition. Thus, our findings illustrate both students’ concep-tions of a mathematical definition at a given moment and the changing nature oftheir conceptions. Figure 3 is an attempt to capture the dynamics involved indealing with mathematical definitions.

As shown in our findings, the mere need to consider an alternative definition ofa specific concept evoked interactions between students’ concept images and theirpersonal definitions of the defined concept. These interactions led to refinementsof students’ understandings of the defined concept. Once they reached a commonunderstanding of the concept, their conceptions of a mathematical definition wereevoked, challenged, and often modified. In the course of these deep musings, theyemployed, to a large extent, example-based reasoning, mainly as a vehicle to refinetheir understandings of the more subtle defined concepts (increasing function andlocal maximum), and definition-based reasoning when dealing with the morestraightforward concepts (square and isosceles triangle).

Students’ views of the features of a mathematical definition resonated with thoseof the mathematical community with respect to the tendency to use minimal defi-nitions when possible and appropriate. Students were less consistent with the math-ematical community with respect to the illegitimacy they attributed to proceduraldefinitions and to those that are based on properties of a concept’s latent parts. Inaddition, they did not fully appreciate the arbitrariness of choice of a mathematicaldefinition, i.e., although from a mathematical point of view any one of a set of equiv-alent statements may serve as a definition, students were reluctant to accept someof the equivalent statements as a definition.


4 In a related study, we obtained similar results with respect to students’ conceptions of the roles andfeatures of mathematical definitions through written questionnaires administered to a larger populationof students (work in progress).

The students’ views of the roles of a mathematical definition were reflected to alarge extent in the students’ communicative considerations. The students regardeda mathematical definition as a communicative device (similar to what Borasi,1992, emphasizes), and thus their expectation of a definition was that it be easilycomprehended. It is interesting to note that the students referred to most of the rolesof mathematical definitions that are considered significant by the community ofmathematics educators. The need to state their preferences led them to evaluatedifferent roles in order to decide which of them was the most important. Thetension between coherence and usefulness was reflected, for example, in the debatesurrounding the need for a definition to be minimal, since in most cases minimaldefinitions are most useful for proving, although minimality is often sacrificed inorder to increase the coherence of a definition.

From a pedagogical point of view, the activity wherein students were asked toconsider a number of possible definitions of different mathematical concepts is apowerful learning environment. The task design and setting created a rich and stim-ulating learning environment, in which the learners were motivated to interact mean-ingfully without researcher interference (contrary to the way Borasi, 1992, workedwith her students). The researcher’s main role was to construct the initial task, whichled to many new questions that were raised and discussed by the learners. This envi-ronment elicited a genuine need to convince one’s self and each other, and conse-quently, to articulate well supported explanations and arguments (in the spirit of


The Notion of aMathematical Definition

ASpecificConcept

AlternativeDefinitions

PersonalDefinition

ConceptImage

Figure 3. The dynamics of dealing with alternative definitions

Yackel, 2000; Yackel & Cobb, 1996). In addition to the cognitive value of argu-mentation, these kinds of discussions move students away from treating theresearcher (or teacher) as the sole authority on what is right or wrong in mathematicstoward relying on their own sound reasoning (as advocated in NCTM, 2000).Moreover, by presenting students with the need to constantly generate relevantexamples (including counterexamples) to support their own claims, such activitiessurrounding subtle mathematical concepts have the potential to create an engaging,natural site for students’ to develop example-based reasoning. Generating exam-ples is an important cognitive activity, as the ability to generate examples as neededis one of the distinctions between novices and experts and serves a main cognitivetool for experts (Rissland, 1991). In addition, dealing with examples and coun-terexamples often presents learners with a cognitive conflict, the resolution ofwhich may lead to modification of their knowledge (Lakatos, 1976; Peled &Zaslavsky, 1997).

The group discussions that developed as an outcome of the request to consider alter-native definitions can be viewed as a Lakatosian discourse (Lakatos, 1976) in thesense that they fostered gradual refinement of the students’ understanding of thenotion of a mathematical definition as well as of the defined concepts. Students’ shifts(see Tables 5 and 6) convey this process by capturing the specific understandingsthat were refined and by revealing the steps in which this process developed(although, as mentioned earlier, the refinement was not always in the “desired” direc-tion). From a mathematical point of view, the group activity involved challengingmathematics (Jaworski, 1992) for the students. The activity fostered students’ aware-


Table 5 Distribution of the Number of Shifts in Students’ Written Assertions Regarding theVarious Statements for All Four ConceptsThe concept Omer Yoav Mike Erez Total no. of shiftsSquare 2 2 5 1 10Isosceles triangle 3 4 4 1 12Increasing function 4 5 4 4 17Local maximum 6 2 1 3 12Total no. of shifts 15 13 14 9 51

Table 6Distribution of Number of Shifts in Viewpoints According to the Concept and Type of Shift


A shift reflecting: (N = 12) (N = 17) (N = 12) (N = 10)Change in the conception of amathematical definition — — 10 9Change in the understanding of the focal concept 9 17 — 1Other 3 — 2 —

ness and appreciation of several critical roles and characteristics of mathematical defi-nitions and allowed them to engage in a genuine mathematical debate.

Mathematics is often regarded as a discipline in which there is always one correctanswer for every (solvable) problem (Schoenfeld, 1989), with no room for subjec-tive views. This type of open activity (which in a way is similar to the approachpresented by Zaslavsky, 1995) may be helpful in conveying a different view of math-ematics—as a humanistic discipline, wherein results are socially constructed orrejected and are driven by personal values (Borasi, 1992; Brown, 1982; Kleiner &Avital, 1984). For each of the four concepts there was a subset of statements equiv-alent to its common and familiar definition; this fact contributed to the humanistic-discipline perspective by allowing for personal preferences, not just issues relatedto correctness, to play a legitimate role in determining what to accept.

Finally, the ability to evaluate mathematical results and processes has beenrecognized as indication of higher-order thinking (Resnick, 1987). The highest vanHiele level of geometric understanding (Mayberry, 1983) involves the ability tocompare deductive systems and theories. The highest level in Bloom’s taxonomy(Bloom, 1956) deals with the evaluation of mathematical entities. Although it mayseem rather implausible to incorporate in school mathematics situations that requireevaluation of such a nature, the tasks that served our study proved powerful in moti-vating the learners to make personal judgments, to weigh and compare the meritsof various potential definitions, and to evaluate them in light of their peers’ ideas.

We conclude by pointing to some further directions for research. We investigatedstudents’ conceptions of a mathematical definition in a context that was free of theneed to actually use definitions. Other aspects of students’ understanding of defi-nitions might be elicited in various contexts of use, such as problem solving thatrequires the use of definitions. In addition, we investigated a limited number ofconcepts, for which there is a general agreement within the mathematical commu-nity with respect to their meanings. The two straightforward ones were bothgeometric, whereas the two more complicated ones were both analytic. It may beworthwhile to examine a broader scope of concepts, including concepts for whichthere is not a consensus regarding their meaning (e.g., a trapezoid), or geometricconcepts that are more complicated and analytic concepts that are straightforward.

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Authors

Orit Zaslavsky, Department of Education in Technology and Science, Technion, Haifa 32000, Israel;[email protected]

Karni Shir, Department of Education in Technology and Science, Technion, Haifa 32000, Israel;[email protected]


APPENDIX A

The Statements in the Increasing Function Questionnaire

A function f: D → ! is called an Increasing Function on D if:(a) for any in the domain (i.e., x ∈ D), f ′(x) > 0.(b)a for any x1, x2 ∈ D, x1 > x2 ⇒ f(x1) > f(x2).(c) for any x ∈ D the function f is pointwise increasing

(f is called pointwise increasing at x = c if f ′(c) > 0). (d)b the values of y = f(x) increase as the values of x (x ∈ D) increase.(e) for any point on the function’s graph the tangent slope is positive.(f) for any x ∈ D, its values (f(x)) are positive.(g) for any x1, x2 that both belong to an interval where f is continuous

x1 > x2 ⇒ f(x1) > f(x2).(h) for any x ∈ D the function is pointwise increasing (whereas a function f is

called pointwise increasing at x = x0 if f(x0 – h) < f(x0 ) < f(x0 + h)). a The textbook definition (according to the textbook used by the participants).b Equivalent to the textbook definition.

APPENDIX B

The Statements in the Square Questionnaire by Characterizing Features

A Square is: Characterizing featuresLevel of

Minimal Type hierarchy(a) a rhombus with a right angle. Yes Structural 1(b) a rectangle with four equal sides. No Structural 1(c) a parallelogram with diagonals which

are equal, and perpendicular. Yes Structural 2(d) a quadrangle with diagonals that

are equal, perpendicular, and bisect each other. Yes Structural 3

(e)a a quadrangle in which all sides are equal and all angles are 90º. No Structural 3

(f) a polygon with four equal sides and four equal angles. Yes Structural 4


APPENDIX B (continued)

A Square is: Characterizing featuresLevel of

Minimal Type hierarchy(g) the locus of points for which the sum

of their distances from two given per- Not pendicular lines is a positive constant. Yes Structural applicable

(h) an object that can be constructed (in the Euclidean Plane) as follows: Draw a segment; from each edge erect a perpendicular to the segment, in the same length as the segment (both in the same direction). Connect the other 2 edges of the perpendic-ulars by a segment. The 4 segments Not form a quadrangle that is a square. Yes Procedural applicable

a The textbook definition (according to the textbook used by the participants).

APPENDIX C

The Statements in the Local Maximum Point of a Function Questionnaire

A point (x0, y0) is called a Local Maximum Point of a function f if: (a) f ′(x0) = 0, and there exists a neighborhood of x0 such that x < x0 ⇒ f ′(x) > 0

and x > x0 ⇒ f ′(x) < 0, for every x in this neighborhood.(b)a there exists a neighborhood of x0 in which y0 = f(x0) is the largest value of

f.(c) there exists a neighborhood of x0 in which left to (x0) the function f increases

and right to x0 the function f decreases.(d) f ′(x0) = 0 and f ″ (x0) < 0.(e) f is continuous at x0, and there exists a neighborhood of x0 such that

x < x0 ⇒ f ′(x) > 0 and x > x0 ⇒ f ′(x) < 0, for every x in this neighborhood.(f) there exists a neighborhood of x0 that includes points, on both sides of x0, for

which the values of f are smaller than y0 = f(x0).(g)b there exists a neighborhood of x0 in which the value f(x0) is larger than f(x)

for any x, x ≠ x0, in this neighborhood.

a Equivalent to the textbook definition.b The textbook definition (according to the textbook used by the participants).


APPENDIX D

The Statements in the Isosceles Triangle Questionnaire by Characterizing Features

An Isosceles Triangle is: Characterizing featuresLevel of

Minimal Type hierarchy(a)a a triangle with two equal length sides. Yes Structural 1(b) a triangle with an angle bisector that

is also a median and an altitude. No Structural 1(c) a triangle with two equal length

medians. Yes Structural 1(d) a triangle with two equal length sides

and two equal angles. No Structural 1(e) a polygon with three sides, two of

which are equal length. Yes Structural 2(f) a closed path of connected line

segments in the plane which does not cut across itself, with three sides, two of which are equal length. Yes Structural 3

(g) a geometric object that can be con-structed as follows: Draw a segment; from one edge draw another non-collinear segment equal length to the first one. Connect the two different edges of the two segments by a seg-ment. These 3 segments form a Not triangle that is an isosceles triangle. Yes Procedural applicable

a The textbook definition (according to the textbook used by the participants).


students’ conceptions of a mathematical definition€¦ · to indicate the purported status and...

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