Problem-Solving Preferences of Mathematics Teachers: Focusing on Symmetry

ROZA LEIKIN

PROBLEM-SOLVING PREFERENCES OF MATHEMATICSTEACHERS: FOCUSING ON SYMMETRY

ABSTRACT. The aim of the study presented in this paper was to explore factors thatinfluence teachers’ problem-solving preferences in the process of (a) solving a problem,(b) explaining it to a peer, (c) liking it, and (d) teaching it. About 170 mathematics teacherstook part in the different stages of the study. A special mathematical activity was designedto examine factors that influence teachers’ problem-solving preferences and to developteachers’ preferences concerning whether to use symmetry when solving the problems.It was implemented and explored in an in-service program for professional developmentof high-school mathematics teachers. As a result, three interrelated factors that influ-ence teachers’ problem-solving preferences were identified: (i) Two patterns in teachers’problem-solving behavior, i.e., teachers’ tendency to apply a stereotypical solution to aproblem and teachers’ tendency to act according to problem-solving beliefs, (ii) the wayin which teachers characterize a problem-solving strategy, (iii) teachers’ familiarity witha particular problem-solving strategy and a mathematical topic to which the problembelongs. Findings were related to teachers’ developing thinking in solving problems andusing them with their students. The activity examined in this paper may serve as a model forprofessional development of mathematic teachers and be useful for different professionaldevelopment programs.

KEY WORDS: mathematics teachers, problem solving, professional development,symmetry

PROLOGUE

PART I: I Can SOLVE the Problem That Way but This Way I CanUNDERSTAND it

My friend’s son (Ron) asked me to help him solve the followingproblem:

Problem 1: Of all the triangles with a given side s and given area A,which one has the minimal perimeter? (See Figure 1)

This problem from a high-level Israeli matriculation examination issupposed, according to the curriculum, to be solved using calculus tools.Ron was one of the best pupils in his class, yet he had an unusual diffi-culty when solving this problem on his own. During our meeting, wediscussed which elements of the triangle could be chosen as a variable

Journal of Mathematics Teacher Education 6: 297–329, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1. Problem 1.

for an appropriate function and what is the most convenient choice forsolution. Eventually Ron solved this problem easily by performing a well-known algorithm (i.e., writing a function, finding its derivative, etc.). Atthis stage, I suggested solving this problem using symmetry. He had notstudied symmetry in school but solved the problem based on symmetryconsiderations with my guidance (see Appendix, Card 1, Solution 2).

Although he said he “enjoyed this solution and it was not difficult atall”, Ron’s reaction was unusual. He became sad, and then after someminutes of silence he said:

Why don’t they teach us to do it this way? I can solve the problem that way [using calculus],but this way I can understand the solution. I can see it, I can feel it, and the result makessense.

My decision to conduct this study was motivated by Ron’s “Why” and bymy conviction that using symmetry in problem solving could help teachersteach mathematics for understanding.

PART II: Teachers’ RELUCTANCE TO USE SYMMETRY when SolvingProblems

In this study, symmetry in its general sense was considered as an inter-disciplinary concept within mathematics. I first conducted a pilot study inwhich about 40 high school and junior high school mathematics teacherswith different teaching experiences took part. The purpose of this pilotinvestigation was to get a first impression about whether mathematicsteachers use symmetry when solving problems. “Thinking symmetry” waspresented to the teachers as a way of thought and as a problem-solvingstrategy that was an alternative to those based on different traditionalschool mathematics tools and that belonged to different mathematicaltopics. The teachers were asked to solve Problem 2 (see below), which isa well-known example of the use of symmetry in mathematical problem-solving. This problem is similar to Problem 1, which may be considereda special case of Problem 2. Eight high school teachers were interviewedindividually, six junior high school teachers were observed while working

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in pairs and twenty-two teachers (working in either senior high schoolor junior high school) solved the problem in small groups. Additionallyfour preservice teachers were interviewed before teaching Problem 2 andwere observed when presenting pupils with this problem and managing itssolution in microteaching settings.

Problem 2: Let l be a straight line and A and B two points on the sameside of the line. Find a point C on line l, such that the sumof its distances from point A and from point B is minimal.(See Figure 2)

Problem 2 (as in the case of Problem 1) is usually included in Israelisecondary mathematics textbooks in the topics of Calculus and AnalyticGeometry without any reference to the concept of symmetry. Thus, notsurprisingly, most of the teachers (38 out of 40) did not use symmetrywhen solving the problem. Moreover, when the teachers were presentedwith a “symmetry solution” they expressed uncertainty as to whether thissolution was admissible. I found that all the teachers demonstrated twopatterns of problem-solving behavior: (a) applying a stereotypical solutionto a problem, and (b) acting according to problem-solving beliefs.

Applying a stereotypical solution: Such teachers’ problem-solvingbehavior is based on an automatic connection between problems of aparticular type and a particular problem-solving strategy. Application ofa stereotypical solution relies mainly on teachers’ previous mathematicalexperience. For most of high school teachers, using the derivative of anappropriate function served as a stereotypical solution when finding aminimal distance. Secondary school teachers made a connection betweenconcepts of “minimal distance” and “perpendicular to a line”.

Acting according to problem-solving beliefs: When teachers approached aproblem based on their feeling of “what school mathematics is” or “what

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is good for their pupils” their mathematical behavior was attributed tothe teachers’ problem-solving beliefs. Usually this kind of behavior wasobserved despite the facts that the teachers knew how to solve a problemusing symmetry and that their pupils seemed comfortable with solutionsbased on symmetry. In these cases the teachers often made comments like“it [the solution based on symmetry] is beautiful but doesn’t prove” or that“this is not a solution, it is only an illustration”. Teachers’ beliefs regardingtheir pupils were expressed when they found solutions based on symmetry“good for a teacher but not good for a pupil”. In many cases when actingaccording to their problem-solving beliefs, the teachers tended to apply astereotypical solution.

In summary, the pilot investigation revealed a conflict between pupils’preferences for using symmetry when solving problems and teachers’competence, or willingness, to satisfy these preferences. Consequently themain study had two principle purposes (a) to examine factors that influ-ence teachers’ preferences of a particular way of solving a problem and(b) to explore the possibility of developing teachers’ preferences for usingsymmetry when solving problems.

THEORETICAL BACKGROUND

Research on problem-solving in general and teaching problem-solving inparticular has been the focus of mathematics educators for many years.One of the directions that the US National Academy of Education (1999)emphasizes as being of great importance is how teachers can promotestudents’ understanding of various problem-solving methods. The currentstudy stems from an assumption that, if teachers are to be able to promotemeaningful problem-solving activity, they have themselves to be fluent inmathematical problem-solving. This problem-solving expertise is seen tobe strongly connected to teachers’ content knowledge (e.g., Polya, 1963;Silver & Marshall, 1990; Yerushalmy, Chazan & Gordon, 1990). Oneof the valuable components of this knowledge, which is connected toproblem-solving expertise, is solving problems in different ways.

Research on teaching mathematics has found that to teach for under-standing the teachers need deep and connected knowledge of the relatedsubject matter (see, for example, Chazan, 2000; Askew, 2001). Accordingto Askew (2001), a “connectionist orientation” distinguishes some highlyeffective teachers from others. Thus, solving problems in different waysis especially important as it may develop connectedness of one’s mathe-matical knowledge (NCTM, 2000). Ma (1999), in her study on teachers’mathematical understanding, demonstrated that for Chinese elementary


teachers who have deep conceptual understanding “solving one problemin several ways [yiti duojie] . . . seemed to be an important indicator ofability to do mathematics” (Ma, 1999, p. 140). Moreover, these teachersconsidered it one of the ways in which they could “improve them-selves”. Dhombres (1993) suggested that providing two different proofsfor one particular theorem opens different routes for solvers in theirmathematical knowledge, each of which may be available when appro-priate. Schoenfeld (1985) points out that awareness of opportunity to solveproblems in different ways helps students not to give up when solvingproblems.

However, teachers usually do not solve problems in different waysthemselves or in their classrooms (Schoenfeld, 1988). Moreover, some-times teachers do not accept students’ solutions that are different fromthose that they had taught in the mathematical lessons. This leads todevelopment of students’ beliefs that, for any mathematical problem, thereis only one acceptable solution, to be achieved by applying one particularproblem-solving strategy. Such teachers’ instructional behavior is rootedin their content knowledge and in their beliefs (Cooney, 2001; Calderhead,1996; Schoenfeld 2000; Sullivan & Mousley, 2001; Thompson, 1992).Thomson (1992) contends that teachers’ beliefs, knowledge, and prefer-ences concerning the discipline of mathematics constitute their conceptionof the nature of mathematics. Scheffler (1965) distinguishes betweenknowing and believing by claiming that knowing has “prepositional andprocedural nature” whereas believing is “construable as solely proposi-tional” (p. 15, ibid.). Cooney (2001) adds an additional condition toScheffler’s definition of beliefs as “abstract things, in the nature of ahabit or readiness” that express “a disposition to act in a certain wayunder certain circumstances” (Scheffler, 1965, p. 76). Cooney claims thatverbal proclamations of beliefs must be followed by actions, which areconsistent with these beliefs and provide evidence of the beliefs. In thisstudy, teachers’ statements regarding a particular problem-solving strategythat were substantiated either by applying this strategy or by reluctance touse or accept it were considered as expression of teachers’ beliefs.

Considering problem-solving-in-different-ways as an important pieceof teachers’ instructional behavior, this study examines the possibilityof influencing teachers’ preferences for varying their problem-solvingstrategies by presenting them with different solutions of mathematicalproblems. One of the presented strategies was always symmetry-based.Symmetry was chosen as a mathematical context for this study forseveral reasons. One the one hand, many mathematicians and mathematicseducators stress the important role that symmetry has in problem-solving

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in various branches of mathematics (e.g., Dreyfus & Eisenberg, 1990;Polya, 1973; Polya, 1981; Schoenfeld, 1985; Yaglom, 1962; Weyl, 1952).Solutions of mathematical problems using symmetry are often elegant andreveal the essence of the problem conditions. On the other hand, in theIsraeli secondary school mathematics curriculum, symmetry is mentionedjust once, in connection with the quadratic function. Dreyfus and Eisen-berg (1990) and my pilot investigation demonstrated that teachers inIsrael do not usually use symmetry when solving mathematical problems.Consequently one of the aims of the study was finding a way to advanceteachers’ use of symmetry in problem solving.

In the framework of this study a special mathematical activity “Solvingproblems in two different ways” was designed and implemented within anin-service professional development project for senior high school mathe-matics teachers (for details see Zaslavsky & Leikin, 1999). The goals anddesign of the activity were aligned with what Cooney (1994), Comiti andBall (1996), and Ball (1997) suggest as essential: providing constructiveand reflective opportunities to deepen teachers’ mathematical knowledgein general and their problem-solving expertise in particular. This studyembodied a reflective component in which teachers explicitly consideredthe possibility of the implications of their own learning experiences fortheir teaching (Cooney & Krainer, 1996, p. 1162). Teachers’ reflection-in-action (Schön, 1983) and teachers’ reflection-on-action (Jaworski, 1994)were fundamental parts of the mathematical activity that was designedfor the purposes of this study (as depicted in Figure 3 in the Methodo-logy section). Teachers’ reflective activities comprised various aspects,such as pedagogical considerations, implications for students’ learning,and concerns regarding teachers’ own mathematical practice. In this way,these reflective activities served both research and learning purposes.Furthermore, all the research instruments in this study were constructed toenhance reflection by the participating teachers, in the belief that reflectionis a key issue in teachers’ learning.

OBJECTIVES AND RESEARCH QUESTIONS

The aim of the study presented in this paper was to explore factors thatinfluence teachers’ preferences when choosing a particular way to solvea mathematical problem. The possibility of developing teachers’ prefer-ences to solve problems using symmetry was examined and the changesin teachers’ problem-solving preferences were analyzed. Following theobjectives of the study, two main research questions were considered:


1. What factors influence teachers’ problem-solving preferences?2. How does the designed activity change teachers’ problem-solving

preferences?

METHODOLOGY

The study was carried out in two stages. In the first stage of the study, themain characteristics of different problem-solving strategies, as perceivedby the teachers, were identified and research instruments were constructedand examined. During the second stage of the study the research instru-ments were used to seek data to answer the two research questions.

POPULATION

One hundred experienced (at least 5 years of teaching) high school mathe-matics teachers participated on a one-time basis in the first stage of thestudy. Each of these teachers took part in a two-hour learning-researchsession in a group of 20 teachers; thus five sessions were organized at thisstage of the study. Teachers worked in pairs according to the Exchange-of-Knowledge method (as presented in the next section) and were askedto complete a Response Questionnaire at the end of the session. None ofthese teachers participated in the second stage of the study.

Thirty-three experienced (at least 5 years of teaching) high schoolmathematics teachers participated in the second stage of the study. Theteachers were experienced in teaching such topics as Calculus, AnalyticGeometry, Advanced Algebra (i.e., exponential and logarithmic equa-tions, complex numbers, number sequences, mathematics induction, andcombinatorics). Some of these teachers also had experience in teaching3D-Geometry, and Vectors. The teachers took part in four two-hour work-shops managed according to the Exchange-of-Knowledge method once aweek for a month. These workshops were incorporated at the end of thesecond year of a five-year professional development program, in whichthese teachers participated voluntarily (for details of the program seeZaslavsky & Leikin, 1999). These teachers completed Characterizationand Preferences questionnaires as described below.

Note here, that the teachers who took part in the second stage ofstudy continued their participation in the professional-development projectduring two years following the study. Thus, as can be seen in the ResearchInstruments section, these teachers provided supplementary data for thestudy.

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THE ACTIVITY

The teachers who participated in the study took part in a special mathe-matical activity “Solving problems in two different ways”. The activityexplicitly emphasized importance of two elements of Jaworski’s (1992)teaching triad: (i) mathematical challenge that involves stimulating mathe-matical thought and inquiry, and (ii) the management of learning thatconcerns the creation of a learning environment and establishes classroomvalues and expectations (Jaworski, 1992; Zaslavsky & Leikin, 1999).Therefore, solving-problem-in-different-ways was presented as an explicitmathematical challenge for the teachers and the teachers worked insmall-group learning settings that incorporated worked-out examples.Teachers were informed that symmetry was chosen for these activities toenable solving problems in different ways in different mathematical topicsbecause of its interdisciplinary nature within mathematics.

Firstly, the activity was based on the learning method Exchange-of-Knowledge in Pairs (Leikin, 1997; Leikin & Zaslavsky, 1999). The methodcombined the advantages of cooperative small-group settings with thoseof learning from worked-out examples. Based on research findings thatsome small-group learning methods have been shown to increase learners’communications (Good et al., 1992; Leikin & Zaslavsky, 1997; Webb,1991), small-group learning settings for the high school mathematicsteachers were used as a research tool to create and observe teachers’ inter-actions in the course of problem-solving. Noddings’ (1985) analysis ofsmall groups as a setting for research on mathematical problem-solvingand use of “pair problem-solving” in other studies in the past (e.g., Schoen-feld, 1985) showed “pair problem-solving” to be an effective tool to evokericher problem-solving procedures. As these studies suggested, learnersworking in pairs are far more verbal than learners working alone, and theirsolutions are often more elaborate and advanced. The learning setting usedin the study allowed teachers to choose between different ways to solutions– one of which was based on symmetry (see Appendix 1) – in order tosolve problems individually and in order to explain problems to the otherteacher.

Secondly, research in mathematics education demonstrated that the useof worked-out examples facilitates the acquisition of knowledge requiredfor problem-solving in mathematics and science (Ward & Sweller, 1990;Zhu & Simon, 1987). Consequently, learning materials built for the studyincluded worked-out examples (see Appendix 1) to alleviate the pressurethat may be present when teachers participate in educational research.The same learning setting (as presented below) was used in the first


and the second stages of the study accompanied by different types ofquestionnaires.

Working cardsFor the purpose of the study, eight special working cards were designedby the researcher, each consisting of three parts (for two examples, seeAppendix 1).

Part I presents a worked-out example of a problem solved in two ways,one of which is based on the use of symmetry.

Part II includes a similar problem that can be solved in either of thesetwo ways.

Part III includes a different problem that can be solved by means ofsymmetry.

Different cards included problems from different topics from thesecondary mathematics curriculum: geometry, algebra, combinatorics,calculus, and complex numbers. The alternative solutions presented onthe cards were based on different types of symmetry: (i) geometric, i.e.,connected to geometric transformations of geometric figures: (ii) alge-braic, i.e., connected to permutations of algebraic symbols; and (iii)logical, i.e., connected to symmetry in proofs (for details see Leikin,Berman & Zaslavsky, 1998). Thus overall the activity presented theteachers with different ways of solving problems.

The settingThe Exchange-of-knowledge method consists of two main stages:(i) preparation of “card experts”, and (ii) pairs of exchange-of-knowledge.

Preparation of “card experts”: Teachers worked in pairs. If there was anodd number of participants, then three teachers worked together, and therest worked in pairs. Equal numbers of pairs (including the group of threeif there was one) of teachers received two different working cards chosenfor a session. The teachers in a pair received the same working card andwere asked to read and understand the two methods of solution presentedin Part I of the card. After that the teachers were asked to solve a problemfrom Part II of the card in one of the two ways presented in Part I. Theteachers could do this individually or together with their partner, as theydesired. The process of deciding how to solve a new problem may beviewed as reflection-in-action. Finally, at this stage of work, the teacherswere requested to discuss their solutions of the problem from Part II in apair.

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Problems in Part III of the cards served as a reserve for the teachers whofinished two parts of the cards earlier than others. Usually the problems inthis part were both different from, and more difficult than, the two previousproblems.

Pairs of exchange-of-knowledge: After completion of the work with thiscard, the teachers were called “card experts” (experts in the field of thecard) and changed their peers for one who had a different card and workedin a new pair. In these new pairs the “card expert” teachers were requiredto choose one of the ways to solve the problem presented in Part I of theircard in order to explain the problem to the other teacher. The need to makea choice brought teachers into reflective analysis of the solutions. Afterexplaining solutions, teachers exchanged their cards and were asked tosolve problems from Part II of the cards that were new to them. Teacherscould communicate when solving the problems, ask questions and helpone another.

The whole group discussion: Each workshop ended with a whole groupdiscussion that took from 15 to 20 minutes. The teachers were asked toreflect on their participation in the activity. Usually during these discus-sions the teachers expressed their difficulties, suggestions, and concernsregarding mathematical and didactical issues of the sessions.

THE DATA COLLECTION AND ITS ANALYSIS

The research data were collected from several sources: research ques-tionnaires, videotaped pairs work and group discussion, and some othersupplementary data. This section describes data collection and the waysthe data were analyzed.

Questionnaires

Three different questionnaires were constructed for this study: at the firststage of the study – a response questionnaire, at the second stage of thestudy – a preference questionnaire and a characterization questionnaire.

The Response Questionnaire: The questionnaire was administered on aone-time basis to 100 high-school mathematics teachers at the first stageof the study. One of the purposes of the study was to explore factorsthat influence teachers’ preferences when choosing particular ways ofsolving a mathematical problem. Teachers’ views regarding the advan-tages and disadvantages of the different solution strategies were observed


in the course of the pilot study. To verify this observation and makeit more precise, the teachers were asked to list advantages and disad-vantages of the different solutions that were presented in the workingcards (see Appendix 2). Advantages and disadvantages listed by theteachers were categorized according to their similarity and the identifiedcategories were called “problem-solving-characteristics”. Each one of thecategories was defined as a characteristic if it was found in not fewerthan 30 to 100 response questionnaires. Reliability of the categorizationwas examined by inter-coder agreement between four experts. The expertswere asked to classify teachers’ responses according to the categoriesdefined by the researcher or to suggest new categories. A category wasaccepted if at least three of the four experts agreed about the classifi-cation. These problem-solving characteristics were used for constructionof the characterization questionnaire that was used in the second stageof the study. The identified characteristics represented teachers’ opin-ions on how particular ways to solution were (i) difficult (when teaching,solving, explaining, and understanding), (ii) interesting, (iii) conventional(i.e., regular, standard), (iv) convincing, (v) inspiring (i.e., developingcreative mathematical thinking), (vi) challenging (i.e., demanding creativemathematical thinking), (vii) beautiful.

The Characterization Questionnaire (see Appendix 3): Together with thepreference questionnaire (see Appendix 4), the characterization question-naire aimed to analyze the relationship between the ways teachers charac-terize different problem-solving strategies and their preferences regardingthe use of these strategies. The questionnaire consisted of propositionseach with a particular problem-solving characteristic that had been definedon the basis of teachers’ answers to the response questionnaire. In theresponse questionnaire, there was no consistency among the teachers asto which characteristics were associated with regular problem-solvingstrategies and which were associated with symmetry-based solutions.Thus, for each characteristic the teachers were asked to answer which ofthe two ways of solution fits the proposition better. Each of the problem-solving characteristics appeared in the response questionnaire as bothadvantages and disadvantages of the solutions. Thus, the teachers wereasked to report whether they consider a proposition an advantage or adisadvantage of a particular problem solving strategy. All the proposi-tions were formulated as positive statements to make the characterizationquestionnaire easy to use. When classified, all the responses in whichteachers claimed that a solution based on symmetry was more difficult,were considered equivalent to those that claimed that a solution based

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on calculus tools was easier. Similarly, teachers’ responses in which theystated that conventionality of a solution was an advantage were consideredidentical to those in which the teachers claimed that unconventionality ofa solution was its disadvantage.

The Preference Questionnaire (see Appendix 4): In this study teachers’problem-solving preferences are considered for the four differentprocesses in which teachers of mathematics are often involved: solvinga problem, explaining a problem’s solution, liking a solution, teaching tosolve a problem. The preference questionnaire was used for two purposes.Firstly, combined with the characterization questionnaire it aimed to showrelationships between the ways in which teachers characterize differentproblem-solving strategies and teachers’ preferences regarding the use ofthese strategies. Secondly, it aimed to follow changes in teachers’ prefer-ences regarding solution strategies in each of the processes listed above.The questionnaire was administered at the end of each of the four work-shops. Thus, each teacher answered these questionnaires four times. Onehundred and eleven questionnaires of each kind were received.

Overall, the research procedure integrated a mathematical activity, inwhich the teachers were involved, with reflection on the activity in parti-cular and on their teaching practice in general. The need to choose differentways to solutions for different purposes required reflection-in-action andthe need to answer questionnaires required reflection-on-action. Solvingproblems in different ways was embedded in small-group learning settingsthat allowed the teachers to choose problem-solving strategies freely, andallowed the researcher to keep track of these choices (See Figure 3).

Videotapes

To analyze teachers’ problem solving strategies and factors that influenceteachers’ preferences for using a particular strategy when solving the prob-lems, four teachers were videotaped during each session: first as two pairsof “card experts” and then as two pairs of “exchange-of-knowledge”. Eachpair of “card experts” was given a different working card; the original pairswere then split at the “exchange-of-knowledge” stage so that members ofthe new pair were experts in different cards. In addition to the pair work,all the group discussions that concluded each of the meetings at the secondstage of the study were videotaped. The videotaped data were used mainlyto support numerical data from the questionnaires and qualitative data fromthe pilot study.


Figure 3. The setting.

Supplementary data

During the second stage of the study, two staff members observed interac-tions (which were not videotaped) between the teachers in two pairs andconducted written protocols of these observations to collect additional dataabout teachers’ problem-solving strategies and their preferences.

As mentioned earlier, most of the teachers who took part in thesecond stage of the study continued their participation in the professional-development project during two years following the study. The researchercontinued her work on the project as a staff member. Thus, some supple-mentary data were collected during these two following years from severalsources. First, the data were collected from reports of other staff memberswho conducted the workshops for these teachers and observed interestingepisodes connected with the use of symmetry in their workshops. Second,

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all the workshops in the project were videotaped and different episodesrelevant to the study could be observed from the videotapes. Third, theresearcher continued to communicate with the teachers during meetings,through e-mail messages and telephone calls, and the teachers shared withher their mathematical and pedagogical experiences related to the use ofsymmetry. Note that no systematic investigation took place in teachers’classrooms.

RESULTS OF THE STUDY

The prologue in this paper described briefly the main findings of the pilotstudy. This section presents results of the main study that included twostages: the first stage aimed at the construction of the research tools andthe second stage aimed at answering the research questions. The findings inthis section are from the second stage of the main study and are organizedin accordance with the research questions.

What factors influence teachers’ problem-solving preferences?

Three main factors that influence teachers’ problem-solving preferenceswere identified in this study:

i. The ways in which teachers characterize different problem-solvingstrategies;

ii. Patterns of teachers’ problem-solving behavior, i.e., teachers’ tend-ency to apply a stereotypical solution to a problem and teachers’tendency to act according to their problem-solving beliefs; and

iii. Teachers’ familiarity with the type of symmetry and with the mathe-matical topic to which the problem belongs.

All of these factors are interrelated and are connected to teachers’ mathe-matical and teaching experiences.

The ways in which teachers characterize different problem-solvingstrategies and their problem-solving preferences

Identification of problem-solving characteristicsAnalysis of the response questionnaire indicated that different problem-solving characteristics were attributed to different solution strategies,i.e., symmetry-based strategies and those that do not use symmetry. Forexample, sixty-two teachers (of one hundred) referred to difficulty of thesolution. Among them, thirty-two teachers thought that a solution based


on calculus tools was easier for a solver whereas thirty teachers founda solution based on symmetry easier. In addition, some of the problem-solving characteristics were considered by different teachers to be eitheradvantages or disadvantages. For example, 95 teachers referred to theconventionality of a solution. Sixty-one teachers found the calculus solu-tion more conventional. Fifty of these teachers claimed that this wasan advantage of the solution and 11 teachers thought that this was itsdisadvantage. Thirty-four teachers found a solution using symmetry lessconventional, 22 of them thinking that this was the advantage of thesolution and 12 teachers considering it its disadvantage.

Relationship between the ways in which teachers characterize differentproblem-solving strategies and teachers’ preferences towards thesestrategiesCorrelations between teachers’ responses on the preferences questionnaire(Appendix 4) and their responses on the characterization questionnaire(Appendix 3) were examined. As a result, the following relationshipsbetween teachers’ problem-solving preferences and the ways in which theteachers characterize the problems were found:1

• Teachers prefer to solve a problem using the way they consider easierto solve, easier to explain, easier to understand, or more convincing

• Teachers prefer to explain a problem using the way they considereasier to explain, more interesting, more convincing, more beautiful,or more challenging.

• Teachers like the way they consider more beautiful, easier to solve,easier to understand, or easier to explain.

• Teachers prefer to teach a problem using the way they consider moreconvincing.

Some of these relationships between teachers’ problem-solving prefer-ences and the ways in which the teachers characterize the problems (e.g.,preferring to solve a problem using the way they consider easier to solve)seem to be very natural. Some other results may be considered as unex-pected, like for example, teachers’ preferences regarding teaching. It couldbe desired that the teachers would choose for teaching not only solu-tions that they considered to be more convincing but also solutions theyconsidered to be challenging and beautiful. Note that no correlation wasfound between teachers’ preferences of different types and their perceptionof a solution as more likely to develop mathematical thinking.

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The two patterns of teachers’ problem-solving behavior and theirproblem-solving preferencesTeachers’ mathematical and teaching experiences are mainly expressedin the two patterns of teachers’ problem-solving behavior, i.e., teachers’tendency to apply a stereotypical solution to a problem and teachers’ tend-ency to act according to their problem-solving beliefs. These patterns wereidentified first in the course of the pilot study, as described in the prologue.

Both in the pilot study and during the second stage of the main study,some teachers, despite knowing how to solve a problem using symmetry,were reluctant to use symmetry and even to accept these solutions,expressing their feelings of the mathematical or didactical insufficiencyof a solution. The teachers could explain that “The solution is not inthe curriculum”, “The solution will not be accepted by the examinationcommittee”, “The solution is only demonstration but not a proof”, and“The solution it is not so good for most of the students”. These and otherexpressions of teachers’ feelings about mathematical nature of solutionsand about its didactical value were followed by the teachers’ actions inthe form of rejection of a solution and by attempts to apply a stereotypicalsolution. In other cases, even if the teachers were not competent to solveproblems using symmetry, they persisted in approaching the problemsusing symmetry because they found this way of solution “being moreelegant”, “being challenging”, “developing mathematical thinking”, and“explaining the result of solutions”. In the cases when the teachers aban-doned a particular solution, despite the fact that they were competent tofind the solution, or when the teachers kept on unsuccessfully solving aproblem in a particular way, their actions were considered as providing anevidence of their beliefs (Cooney, 2001). Such mathematical behavior wasconsidered to be based on teachers’ problem-solving beliefs.

Most of the videotapes of the pair work that were recorded during thesecond stage of the main study include episodes that demonstrate the twopatterns of the teachers’ problem-solving behavior. For example, duringthe first workshop two teachers (Naomi and Shelly) who worked as apair of “card experts” with card 1 solved the problem from part II of thecard (see Appendix 1) first using symmetry and then using a derivative“to check the solution”. They found the symmetry solution “elegant andbeautiful”. Nevertheless, they were not certain regarding the propriety ofusing this solution in the classroom. At the end of the solution, Shelly said:“Do you think you may solve [the problem] this way with the students?Maybe it is not strong enough? What about exams?” In the same work-shop, the teachers in the second videotaped pair of “card experts” also


attempted to solve the problem from part two of Card 2 (see Appendix 1)by using symmetry. They were able to connect algebraic symmetry of theexpression and geometric symmetry of its graph. However, the teachersfelt uncertainty regarding the correctness of the solution and decided tosolve the problem in the regular way. The following excerpt from the tran-script relates to teachers’ solution of the problem from Part II of Card 2(Appendix 1).

Rasan: She stated [the researcher in the solution presented in Part I of thecard] that the graph is symmetrical with respect to y = x becauseyou may exchange x and y. You may change x with y and y withx and you will get the same value. Do you understand?

Simon: Yes.Rasan: O.K. There is no difference between x and y [in the first problem].

Here there is no difference between x and (–y).Simon: Yes, it is symmetrical according to x = –y.Rasan: Thus it is perpendicular to it.Simon: Are you sure? I do not know. . . . Is this correct? I think we should

go to the first way, a regular way.Rasan: Let’s solve it [using the derivative].

When these teachers changed their peers and worked in pairs of“exchange-of-knowledge”, Naomi worked with Rasan who named thesymmetry solution “philosophical”. In the following excerpt, Rasan andNaomi discuss the ways in which they solved Problem 2 on their cards:

Naomi: Which way did you do it?Rasan: I did it the first [standard] way. This way [symmetry-based] is a

little philosophical . . .

Naomi: This is the standard way. I chose this method [symmetry-based]and I did it both ways . . .

Rasan: Ah. You chose a problematic one . . . Sophistication.

Thus these two pairs of the teachers, from the beginning, tried to usesymmetry when solving the problems. Similar to the findings in the pilotstudy, at the second stage of the study some teachers, such as Naomi,sometimes used both solutions “to check themselves” or “to feel secure”.Other teachers, such as Rasan, who considered a symmetry solution “notmathematical enough” or “too philosophical”, rejected symmetry-basedsolutions and preferred to solve problems in the regular way.

During the third workshop of the second stage of the main study Rasanand Simon worked together again. Rasan solved a problem (Problem 3: seeFigure 4) using logical symmetry and presented it to Simon who persisted

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Figure 5. The triangle must be isosceles.

in finding a stereotypical solution to the problem. Despite Rasan’s enthu-siasm, Simon was reluctant to accept his solution. Simon’s reluctanceresulted in Rasan’s feeling that symmetry is “not mathematical enough”.Consequently, he expressed his belief that the symmetry solution is just“blah-blah” and joined Simon in his attempts to find a stereotypicalsolution as presented in the following episode.

Problem 3: Among all the triangles inscribed in a circle which one hasthe maximal area, assuming that such a triangle exists?(See Figure 4)

Rasan: Ah. Let’s do it this way. Let’s take one side. [Points to side AB,Figure 5]

Simon: Why? Why should we go this way? The angle . . .

Rasan: Let’s do it this way. Suppose that this is constant [points to thechosen side AB, Figure 5]. To have maximal area for the triangleit must be . . . this vertex [vertex C] must be here [Rasan drawsan isosceles triangle having the chosen side AB as a base. AC =BC].

Simon: O.K. it is. We have said this.Rasan: This is all. We have finished. Let’s go to another side.


Simon: This is the same idea [here Simon made a connection betweenRasan’s solution and other problems (see for example Problem 1in Part III of Card 1, Appendix 1) that were solved using logicalsymmetry in proofs].

Rasan: This is the same idea; that is why this [the triangle] is a symmet-rical from all the directions, in other words it is equilateral.[While saying this he pointed to the isosceles triangle shown inFigure 5. He did not draw a new triangle.]

Simon: Yes . . . [continues solving in this way, trying to find a function.]Rasan: Do you understand my idea?Simon: Yes [saying it automatically – does not listen, continues to

struggle with the stereotypical solution]. SilenceRasan: But this is “Blah-Blah”, not mathematics [joins Simon in his

solution]

This excerpt shows that Rasan used geometric symmetry when statingthat the triangle was isosceles and logical symmetry in proofs to explainwhy it must be an equilateral triangle. Nevertheless, Simon, who claimedhe understood Rasan’s approach, did not accept the solution. Moreover,he persisted in writing a function of two variables with which he didnot know how to proceed while, based on the fact that the triangle isisosceles as stated by Rasan, he could have found a function in one variableand approached the solution more straightforwardly. As noted before theexcerpt, Simon’s reluctance to use symmetry caused Rasan to doubt themathematical nature of his own solution. The transcript demonstrates, thatafter some minutes of silence he concluded [to himself]: “But this is blah-blah, this is not mathematics” and joined Simon in his attempt to solve theproblem using derivatives.

Familiarity with the type of symmetry and of the mathematics of theproblemThe familiarity with the type of symmetry and the mathematical topicof the problem were found to influence teachers’ problem-solving pref-erences. Use of geometric symmetry was the most familiar to the teachers,while algebraic symmetry was less familiar. During the first meeting ofthe second stage, most of the teachers reported that they never heard aboutthe concept of algebraic symmetry. Teachers differed in their familiaritywith geometric and algebraic symmetry and this affected their preferenceas to which type of symmetry they used. Thus, during the first session ofthe second stage of the main study when solving a problem that could besolved using geometric symmetry (see Appendix 1, Card 1, Part II), 60%

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of the teachers preferred to solve the problem using geometric symmetry.At the same workshop, only 14% of the teachers who may have solvedthe problem using algebraic symmetry (see Appendix 1, Card 2, Part II)preferred the algebraic symmetry solution. Over the course of the mainstudy, the teachers were involved in the discussion of a unifying approachto the concept of symmetry (Leikin et al., 2000), as they consideredapplications of algebraic symmetry in different mathematical topics. Inother words, the concept of algebraic symmetry became more familiarto these teachers. Thus at the last workshop almost an equal number ofteachers preferred using different kinds of symmetry: 64% of teachers whosolved systems of equations preferred solving the problems using algebraicsymmetry: 67% of the teachers who where asked to find the value of anintegral preferred doing this using geometric symmetry.

The mathematics topic to which a problem belongs also influencedteachers’ preferences for using symmetry when solving a problem. Forexample, during the third meeting, 45.5% of teachers preferred to usegeometric symmetry for the “section of a cube” problem whereas at thesame workshop 75% of teachers preferred to solve the combinatorialproblem using geometric symmetry.2

Changes in teachers’ problem-solving preferences

To follow the changes in teachers’ problem-solving preferences, thepercentage of teachers (i) who preferred a solution without the use ofsymmetry, (ii) who preferred two ways of solution, and (iii) who preferreda solution based on symmetry were calculated. This calculation was basedon the preference questionnaire, taking each type of preference separatelyfor each session. Table I presents the percentage of teachers who reportedthat they preferred symmetry-based solutions only and who preferred usingtwo different ways of solution at the first and at the last workshops. It alsopresents the total percentage of the teachers who used symmetry at the firstand at the last workshops.

Overall, the activity influenced teachers’ preferences for different typesof problem-solving with regard to the use of symmetry. Note that at thebeginning on the second stage of the main study the teachers did notfeel confident enough with using symmetry. Thus, about 36% of teacherswho reported that they preferred to solve a problem using symmetry werecurious to compare the two ways of solutions and the other teachersdecided to “check correctness of the symmetry-based solution using thetraditional one”. In this way the teachers became involved in solving theproblems in different ways and could experience the advantages of this


TABLE I

Percentage of teachers who preferred a symmetry-based solution

Preferences For solving For explaining In liking For teaching

a problem a problem to a peer

First Last First Last First Last First Last

meeting meeting meeting meeting meeting meeting meeting meeting

Symmetry-based

solution only 24.1% 65.4% 27.6% 46.2% 44.8% 80.8% 17.2% 38.5%

Two different

ways one of

which is based

on symmetry 13.8% 7.6% 17.2% 25.9% 3.4% 7.6% 34.5% 39.5%

Total 37.9% 73.0% 44.8% 72.1% 48.2% 88.4% 51.7% 77.0%

process. In the course of the intervention, teachers’ explanations for whythey prefer to use two ways of solving a problem changed. During the thirdand the fourth meetings they claimed that they “found it interesting to solvea problem in two different ways and to compare them”, and that now they“can feel why it is important to let students know different problem-solvingstrategies”. Note that at the end of the intervention there were teachers(about 30%) who still preferred standard solutions. These were mostlymathematics teachers who had long teaching experience. They expressedtheir unwillingness to change “clear and easier” ways of solution for thenew ones, which are not included in the secondary school curriculum. Thismay be attributed to inertia in the teachers’ tendency to use stereotypicalsolutions and in the teachers’ problem-solving beliefs. Additionally wemay conjecture here that the period of investigation was relatively short(four weeks) to develop these teachers’ readiness for change.

Interestingly, different types of problem-solving preferences changedin different manners. The percentage of teachers who liked symmetryincreased quickly (from 44.8% at the first session to 63.6% at the secondsession). The percentage of teachers who preferred to solve problems usingsymmetry only changed slowly at the beginning and then increased quickly(from 24.1% at the first session to 27% at the second session to 61% at thethird session). The manner in which teachers’ preferences to use symmetrywhen explaining a problem changed was similar to that in which teachers’preferences to use symmetry when teaching increased and was differentfrom those when using symmetry in solving problems.

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Videotaped whole group discussions support the numerical data thatwere obtained from questionnaires. For example, at the end of the thirdworkshop Alina and Dorit felt that symmetry had become an essential partof their mathematics.

Alina: At the beginning it was difficult [to use symmetry]. Today I seethat it is easier to work with symmetry than it was at the previousmeetings. We learned to think in this way. Now I feel this is apart of my mathematics. Today I am convinced that symmetry isa strong problem-solving tool.

Dorit: It [symmetry] is easy to understand. It is connected to manytopics. It can be used instead of algebraic manipulations, insteadof derivative. You can use it instead of congruence and trigono-metry. You just need to see symmetry.

The written questionnaires, the observers’ protocols and the videotapesshow that during the study more than 70% of teachers started to solve prob-lems using symmetry, as in the example of Rasan in the excerpt presentedearlier.

SUMMARY AND DISCUSSION

The study presented in this paper aimed to explore factors that influ-ence mathematics teachers’ problem-solving preferences when solving aproblem, when explaining it to a peer, in liking it and when teaching it.For the purpose of the study a special professional-development mathe-matical activity “Solving Problems in Two Different Ways” was designed,focusing on the concept of symmetry. Using different strategies to solve aproblem and then reflecting on the mathematics being used was an integralpart of teachers’ learning (Romberg & Collins, 2000). In sum, the proposedactivity provided an effective research and professional-development back-ground that was based on emphasizing a reflective component of anin-service program and explicit consideration of the implications of theteachers’ own learning experiences for their teaching practice. Three mainfactors that influence teachers’ problem-solving preferences were identi-fied in the study: (i) Two patterns of teachers’ problem-solving behavior,i.e., teachers’ tendency to apply a stereotypical solution to a problem, andteachers’ tendency to act according to their problem-solving beliefs; (ii)The way in which the teachers characterize a problem-solving strategy;(iii) Teachers’ familiarity with the type of symmetry and with the mathe-matical topic to which the problem belongs. Each of these factors is based


on the teachers’ mathematical and teaching experience and all of thefactors are interrelated.

Teachers’ tendency to apply a stereotypical solution was considered inthis study to relate mainly to teachers’ mathematical knowledge. Basedon their personal mathematical experience and on their teaching experi-ence, the teachers connected a particular mathematical topic to a particularproblem-solving strategy. For example, most of the teachers who partici-pated in this study approached maxim-minima problems either with aderivative or by constructing a perpendicular. Then this strategy seemedmore “standard”, “conventional” and “acceptable”. When explaining whythey chose not to use symmetry when solving the problems, the teachersreferred to students’ need for a conventional problem-solving procedure.As Ball (1992) point out, “typically, students experience mathematics as aseries of rules to be memorized and followed. Speed and accuracy are whatcount; justification and reasonableness play little role” (p. 84). So, ourteachers, when discussing different ways of solving problems in generaland of using symmetry in particular, often asked: “Will they accept thissolution if a student uses it at the matriculation examination?” Our argu-ment that any mathematically correct solution must be accepted was notconvincing. The use of a stereotypical procedure, at least at the beginningof the intervention, seemed more acceptable.

In this study, teachers’ mathematical behavior that was based ontheir feeling of “what school mathematics is” or “what is good for thestudents” was attributed to teachers’ problem-solving beliefs. Interviewswith teachers in the course of the pilot study and videotapes of the secondpart of the main study demonstrated that teachers’ problem-solving beliefsstrongly influenced their mathematical performance and their preferencesfor using different problem-solving tools. As was demonstrated in thepaper, some teachers did not use symmetry because they did not believethat it was mathematical enough, whereas others did not use symmetrybecause they believed that “the use of symmetry could be good for teachersbut not good for their students”.

The other important factor that influenced teachers’ problem-solvingpreferences was the way the teachers characterize a problem-solvingstrategy. Figure 6 depicts correlations between different types of problem-solving strategies and different problem-solving characteristics. Severalproblem-solving characteristics that were identified at the first stage of themain study as frequently mentioned, i.e., conventionality of the solution,difficulty in teaching, and inspiration, were not found to be significantlyrelated to any type of problem-solving preference. Some of the problem-

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Figure 6. Correlations between the way the teachers characterize a particularproblem-solving solution and teachers’ preferences regarding this solution.

solving characteristics seem to have a strong relationship to the teachers’preferences, the others seem to have little influence. For example, difficultyof the explanation of the solution was found to be related to the three typesof preferences with high levels of significance (see Figure 6). Teachers’opinion that a solution is more convincing also related to three types ofproblem-solving preferences. At the same time, teachers’ opinion that asolution is interesting or challenging related only to teachers’ preferenceswhen explaining a solution to a peer.

Modes in which problem-solving characteristics are related to teachers’problem-solving preferences sometimes disclosed very natural connec-tions. For example, teachers chose to solve a problem in a way that theyconsidered “easier to solve”, to explain a problem in a way that was“easier to explain”, and liked a strategy that was “more beautiful” intheir opinion. Understandably, teachers chose to solve a problem usinga strategy that seemed to be easier to explain and to understand. Teacherschose to explain a solution to their peer in a way that they found to bemore interesting, more beautiful, more challenging and more convincing.However, the finding that there was no significant correlation betweenteachers’ problem-solving preferences regarding a particular strategy andtheir opinion that the strategy is more inspiring (i.e., may better developstudents’ mathematical reasoning) was one of the disappointing aspectsof this study. The pressure of the curriculum and the preparations for thematriculation examinations in the upper grades created an emphasis on thetechnical procedures. Thus the teachers liked solutions that they thought


were easier when solving, explaining, and understanding. Note also thatteachers preferred to teach a problem in a particular way if they consideredit more convincing.

All the factors that were found to influence teachers’ problem-solvingpreferences may be seen as grounded in the rules of mathematicalclassroom discourse and its socio-mathematical norms and practices estab-lished in the classroom community (Cobb, 2000; Sfard, 2000). As Sfardpoints out, roles of the participants in a mathematical discourse affect whatthey see as a purpose of the activity, what they count as a convincing argu-ment, and what they see as a required form of argument. On the one handteachers’ teaching and learning experiences were reflected in the findingsof this study. On the other hand, norms that were founded in the mathema-tical activities of the study influenced problem-solving preferences of mostof the teachers. In terms of Cobb (2000), the teachers when participating inthe study were involved as learners in contributing to the establishment ofthe social norms. Consequently they “reorganized their individual beliefsabout their own role, others’ roles, and the general nature of mathematicalactivity” (Cobb, 2000, p. 322).

The activity described in this paper was found to furnish an effectiveprofessional-development environment. The supplementary data in thisstudy show that the teachers acquired a shared conviction that they hadto create a variety of challenging learning conditions for their students.The teachers found both symmetry and solving problems in different waysto be attractive not only for themselves but for their students as well, andtherefore some teachers started implementing this in their own lessons.

The activity presented in this paper served as a model for profes-sional development of mathematics teachers and may be recommended forimplementation in different professional development programs focusingon other big ideas or different learning environments. Implementation ofthis activity in a professional development program developed teachers’awareness of the importance of such problem-solving characteristics asthe elegance of a solution and the mathematical challenge and inspirationof a solution in teaching mathematics.

One of the limitations of this study is its limited application toclassroom practice. Even though the teachers’ self-reports allowed usto see that some teachers implemented their learning experience in theclassroom, no systematic investigation of this issue took place. It mightbe interesting to examine systematically how often and in which way theteachers who took part in this study use symmetry in their classroom. Dothese teachers indeed teach their students to solve problems in different

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ways in other mathematical topics? If they do, how do their students learnin this way? How do teachers’ preferences for using different problem-solving strategies influence and depend on their students’ mathematicalperformance and problem-solving preferences?

EPILOGUE

I would like to think that if Ron asked me now “Why don’t they teach usto solve problems using symmetry?” I could answer him: “Some teachersdo”. The teachers who took part in the second stage of the study continuedtheir participation in the professional-development project during twoyears following the study. The teachers voluntarily informed me abouttheir different experiences related to using symmetry in the classroomand in other situations. They told me their stories when we met at theworkshops, they sent me e-mails and called me to share their experiencesof using symmetry. From this supplementary data it appeared that someteachers started to use symmetry on their own and to search for the casesin which symmetry might be applied. The teachers often were surprisedthemselves that they used symmetry even when they did not plan it.Their problem-solving behavior had changed. Most of the teachers beganto believe in symmetry as a powerful mathematical tool. The followingexcerpt from the interview with the teachers who participated in the projecttwo years after the end of the intervention exemplifies such a change:

Activities related to symmetry were very important. These activities left me with manyopen questions that I could not answer quickly. Several times it took me more than a weekto find a solution for these questions. These activities influenced my approach to teachingproblem solving and my definition of a good student. Before these activities I thought thatany good student could solve any problem immediately. Now when my students fail tosolve a problem and feel unsatisfied because of this failure, I ask students to talk aboutthings they do not understand and to solve the problem again. I think that ability to thinkabut difficulties, to talk about mathematics and to try solving a problem several times candefine a good student. This is the main change in my approach to problem solving.

ACKNOWLEDGMENTS

The paper is based on the D.Sc. thesis of the author submitted to theTechnion – Israel Institute of Technology, Haifa, under the supervisionof Abraham Berman and Orit Zaslavsky and was partly supported bythe Miriam and Aaron Gutwirth Memorial Fellowship. The study was


conducted within the framework of “Tomorrow 98” project in the UpperGalilee – A Model for Improving Mathematics High School Education,Orit Zaslavsky – Project Director.

I would like to thank Sara Hershkowiths, Barbara Jaworski, BebaShternberg, Anna Sfard and Michal Yerushalmy for their helpfulcomments on the earlier versions of the paper.

NOTES

1 Note that for all the reported relationships between the ways in which teachers char-acterize different problem-solving strategies and teachers’ preferences regarding the usethese strategies significant corrections were found. However these correlations were ofdifferent strengths. Space does not allow details to be included in this paper.2 For reasons of length, it has not been possible to include details of all the problems. Ifreaders are interested in these problems they can contact the author directly.

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APPENDIX 1


APPENDIX 2

Response questionnaire

Please, complete the following table: list advantages and disadvantages of the

different solution strategies that were presented in your card.

Way 1: Regular Solution Way 2: Based on Symmetry

Advantages

Disadvantages

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APPENDIX 3

Characterization questionnaire

Please compare the two ways of solving the problem presented in Part I of your original card.Complete the table.

Way 1 Way 2 Is this an advantage or a(mark X if (mark X if disadvantage of the solution?

appropriate) appropriate) Advantage Disadvantage Difficult todetermine

Mark X where appropriate

1. The way is easier when solvingthe problem.

2. The way is more interesting.

3. The way is more challenging(needs more creative thinking).

4. The way is easier to understand.

5. The way is more conventional.

6. The way is more convincing.

7. The way develops thinking more.

8. The way is easier to explain.

9. The way is more beautiful.

10. The way is easier to teach.


APPENDIX 4

Preference Questionnaire

Please complete the table regarding different ways of solving of the problempresented in your card.

Way 1 Way 2 Another way Why?(Mark X if (Mark X if (What is it?)

appropriate) appropriate)

In which way did you solvethe problem from Part II of

the first card?

In which way did you explainthe problem from Part 1 of the

first card to your peer?

Which way do you like better?

Which way would you use inyour classroom if you were to

teach problems of this kind?

Faculty of EducationUniversity of HaifaMount CarmelHaifa, 31905IsraelE-mail: [email protected]

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Problem-Solving Preferences of Mathematics Teachers: Focusing on Symmetry