Implications of Competing Interpretations of Practice

Implications of Competing Interpretations of Practice for Research and Theory in Mathematics Education Author(s): Ruhama Even and Baruch B. Schwarz Source: Educational Studies in Mathematics, Vol. 54, No. 2/3, Connecting Research, Practise and Theory in the Development and Study of Mathematics Education (2003), pp. 283-313 Published by: Springer Stable URL: http://www.jstor.org/stable/3483199 . Accessed: 18/01/2011 10:08 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=springer. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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RUHAMAEVEN1 and BARUCHB. SCHWARZ2

IMPLICATIONS OF COMPETING INTERPRETATIONS OF PRACTICE FOR RESEARCH AND THEORY IN MATHEMATICS EDUCATION *

ABSTRACT. In this paperwe explorethe issue of interdependencyof theoryandresearch findings in the context of researchon the practice of mathematicsteaching and learning at school. We exemplify how analyses of a lesson by using two differenttheoretical perspectiveslead to differentinterpretationsand understandingsof the same lesson, and discuss the implicationsof competinginterpretationsof practicefor researchandtheoryin mathematicseducation. KEY WORDS:activitytheory,researchmethodology,theoryand practice,verbalanalysis

INTRODUCTION

In their chapteron 'Competingparadigmsin qualitativeresearch',Guba and Lincoln (1994) point to an interesting,sometimes unforeseen, connection between the theoreticalframeworkused by the researcherand the findingsof the research.They claim that "theoriesand facts are quite interdependent- that is, that facts are facts only within some theoretical framework"(p. 107). Moreover,they assert,"Notonly arefacts determined by the theorywindow throughwhich one looks for them,but differenttheory windows might be equally well supportedby the same set of 'facts.' " (p. 107). In this article,we explorethis issue of interdependencyof theory and researchfindings in the context of researchon the practiceof mathematics teaching and learningat school. We exemplify how analyses of a piece of practice- a lesson - by using two differenttheoreticalperspectives lead to differentinterpretationsandunderstandingsof the same lesson, and we discuss the implicationsof competing interpretationsof practice for researchand theoryin mathematicseducation. Several decades ago, researchin mathematicseducation was mainly whatwas called 'cognitiveresearch'.The individualstudent'smathematics conceptions and ways of learningwere the focus of attention.In the last decade the focus of researchin mathematicseducationstartedto extend *

Note: Both authorscontributedequallyto this paper. EducationalStudiesin Mathematics 54: 283-313, 2003.

W9(? 2003 KluwerAcademicPublishers. Printedin the Netherlands.

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from the individualstudent'scognition and knowledge to include socioculturalaspectsof mathematicslearning.Mathematicseducationresearchers beganto focus on studentandteacherparticipationin learning/teaching activities and on different kinds of interaction- between teaching and learningand between knowledge and practice.Currently,both cognitive andsocio-culturaltheoreticalorientationsarecommonin researchin mathematics education. Consequently,we decided to use them as lenses for the two analyses of the lesson. To be able to highlightthe contrastin the interpretationsderivedwhen using differenttheoreticalperspectivesin the limited space of a paper,we deliberatelychose to use extremevariations of these two theoreticalorientations. The articleconsists of five parts.The firstpartprovidesrelevantbackgroundinformationabout the lesson. In the second part, we analyse the lesson from a cognitive science perspective. In the third part, we analyse the lesson from a socio-culturalperspective.The fourthpart of the manuscriptconfronts the two analyses and asks whether these analyses are compatible,or rather,whether,one of the analyses can or should be ruledout. Finally,the fifthpartdiscussesmethodologicalissues concerning the theory of and researchregardingeducationalpractice. In particular, we consider to what extent research findings can practicallybe called into questionby theory and vice-versa. We conclude by claiming that in mathematicseducation,pluralisticas well as absolutistviews of research are both sterile and that there is a need for adopting a critical dialectic approach.

BACKGROUND

The lesson to be analyzedwas partof an innovativeyearlongintroductory course on functionsfor B-level grade-ninestudents.In Israel,ninthgrade students are usually assigned to A-, B-, or C-level mathematicsclasses (A being the highest). A-level classes include about 60% of thi?national studentcohortandB-level classes about40%of the cohort(C-levelclasses comprisea small percentage).The functioncourse was firstdesigned,created,implemented,andinvestigatedwithinA-level classes. The curriculum developersaimedfor the A-level studentsto investigateproblemsituations with computerizedtools, raise hypotheses, collaborateon solving problems, explain and discuss their solutions, and reflect on their learningin individualor collective writtenreports(Hershkowitzet al., 2002). After two years of development,researchand implementation,the projectteam decidedto try the functioncoursein B-level classes, attemptingto keep the main characteristicsof the tasks similarto those in the A-level course.

COMPETINGINTERPRETATIONS

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Oranim school received a 30m long wire to fence a rectangularvegetable garden lot. The lot is adjacentto the school wall, so thatthe fence has three sides only.

a. Find four possible dimensionsfor the lot and theircorrespondingareas. b. For which dimensionsdoes the lot have the largestarea? c. If one of the dimensions is 1Im. what is the area of the lot'?Can you find anotherlot with the same area?If you can, find its dimensions;if not. explain. d. How many lots with the following areasare there:80mr?150m2?

Figure 1. The Fence Problem.

The teacherof this trialhas been a centralmemberof the curriculumdevelopmentteam. She is an experiencedhigh-school mathematicsteacher. In her dual role as curriculumdesigner and teacher she regularlysought innovationsin content and ways of teaching, and systematicallyreflected on her own teachingandthe learningprocesses of her students.Duringthe yearprecedingthe presentexperiment,she successfullytaughtthe function course to an A-level class in the high school where she regularlyteaches mathematics- a religious all-girl academicorientedschool (Schwarzand Hershkowitz,1999). The experimentalB-level class belonged to the same school. The lesson we analyse in this articlewas partof a multi-lessonactivity thattook place ratherearly in the school year.The activitywas builton the 'Fence Problem'(see Figure 1). The 'Fence Problem'was designedto createthe need to move between symbolic and graphicrepresentations.Moreover,as this activity led students naturallyto produce various representativesof the same function, the teacher,togetherwith the otherdesign team members,assumedthat it would enhance studentlearningof the concept of function1.And indeed, when working on the 'Fence Problem' with the A-level students in the previousyear, the teachercapitalizedon the computerartefactsproduced by studentsand co-constructedsignificantaspects of the functionconcept with these students(Schwarzand Hershkowitz,2001). Before workingon the 'Fence ProblemSituation',the B-level studentsin the presentexper-

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iment had already learnedhow to use graphiccalculators(TI-81). They learnedto drawgraphswhen the formulais known,to 'walk' on the graph and observethe changesin the displayedcoordinatesof the pointer,andto 'distort'a graphandcreatedifferentgraphicalrepresentativesby choosing different 'viewing windows' on the graphiccalculators.The work on the Fence Problemhas spreadover severallessons. One of themis the focus of this manuscript.This lesson is designed for studentsto learnaboutdifferent representationsof functions,and how to use the graphiccalculatorto solve problemsthatrequirethe passagefrom one representationto another (mainly,between algebraicand graphicrepresentations),paying attention to manipulating'viewing windows' on the graphiccalculatoraccordingly. The following two sections present analyses of the lesson from two different theoreticalperspectives.The analyses are targetedtoward the generalaim of evaluatingwhetherthe intentionof the designerof the Fence Problem (creatingthe need and developing the ability to move between symbolic and graphicrepresentations)was attained.We first analyse the lesson froma classical cognitiveperspectiveandthen froma socio-cultural perspective.

VERBAL ANALYSIS: A CLASSICAL COGNITIVE APPROACH

Generaldescription Cognitive Science focuses on processes and representations.While many researchersovertlyclaim thatthey adopta cognitiveapproach,the specific terms 'Cognitive Science' and 'Cognitive Psychology' generally imply that complex psychological activitycan be reducedto primitivefunctions (reducibility),andthatpsychologicalentitiessuch as representations,skills or memory can be characterizedindependentlyfrom context. We adopt herethis hard-lineCognitiveScience stance.The most well-knownmethod for analysingverbaldataused to be protocolanalysis(EricssonandSimon, 1984), which focuses on processes of problem solving aiming to create computer-modelsimulation.Accordingto protocolanalysis,the subjectis an individualwho undertakesa sequenceof problemstatesas (s)he applies permissibleoperators.Computer-modelsimulationis the ultimategoal of protocol analysis, indicatingthathumanbehaviourscan be representedas step-by-steppredictableprocesses. VerbalAnalysis (Chi, 1997) reflects a theoreticalshift in CognitiveScience froma searchfor a cyberneticview of mentalprocessesto a questfor mentalrepresentations2. The aim is thennot to predictbehaviouron problemsolving but to investigatementalmodels and representationsthatexplain humanbehaviour.For the analysis in this


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paper,we chose to use VerbalAnalysis as a representativeof the cognitive science approachas we aim to explainthe lesson participants'(teacherand students)behavioursby investigatingtheir mentalmodels and representations. We describefirstthe VerbalAnalysis methodologysuggestedby Chi (1997) in generalterms,and then use it to analysethe lesson. The first steps suggested by Chi are to reduce or sample the protocols, identify an appropriateunit of analysis and segment the reduced protocol.The researcherneeds to developor choose a coding scheme, and to operationaliseevidence for coding, that is, to decide which utterances constituteevidencethatcan be translatedinto a specificcode. The last steps of VerbalAnalysis are depictingthe mappedformalism,seeking patterns in the depicteddata,and interpretingthe patterns. Adoptingthe VerbalAnalysis methodfor analysingprotocolsof classroom lessons to uncover mental representationsis quite problematicas classroom protocols include participationof multiple agents. It is then difficult,if not impossible,to isolate an individualto study his/hermental representationsor learningprocesses. One way to cope with this problem is to identify the set of studentsas a coherententity interactingwith the teacher.Such identificationis possible when the group of studentsseem to work withoutconflicts or clashes amongthemselves,as was the case in the lesson we analyse: studentsdid not object to each other,did not even speak with each other,and generallywaited for the teacher'sinvitationto intervene.3This identificationenablesthe analysisof a bulk of verbaldata from classroomactivities(Leinhardt,1987; 1989; Leinhardtand Schwarz, 1997). Verbalanalysis of the lesson Researchquestion. AdoptingVerbalAnalysis as a methodologicalmethod impinges on the questionsfor inquiry.We alreadymentionedthe general aim for the analysis of the lesson, evaluatingwhetherthe intentionof the designerof the Fence Problem(creatingthe need and developingthe ability to move between symbolic and graphicrepresentations)was attained. Adoptinga cognitive approachmeans focusing on mentalrepresentations, abilities,or capacities.A specificanda naturalresearchquestionaccording to the Cognitive Science perspectiveis the following: To which extent do studentsconceive the passage to a new graphicalrepresentationof a functionas a problemsolving strategyduringthe lesson? Reducingthe protocols. The 50-minutelesson was videotapedand then transcribed.Observationof the videotapeindicatedthatthe teacheralternated between two basic types of instruction:whole-groupteacher-centred

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discussions and small-group(pairs) work. By means of types of instruction, the lesson was comprisedof six parts:a whole-groupdiscussion occurredduringthe first,thirdand fifthpartsof the lesson; small groupwork duringthe second, fourthand sixth parts.We decided to restrictour analysis to the threewhole-groupdiscussionpartsof the lesson. This, because these parts comprised a large enough portion of the lesson (about onehalf of the class time), providedrich and coherentverbaldata,enablingus to consider the whole group of studentsas an entity interactingwith the teacher. Segmentingthe reducedprotocol. The lesson centredon developingstudents' understandingof differentrepresentationsof functions,mainlytheir ability to choose an appropriateand suitablerepresentation,flexibility in moving from one representationto another and in connecting different representationsof the same function.Consequently,we chose the passage from one representationto anotheras a naturalboundaryfor segments. Moreover,there were times when there was not a change of representations, but rathera change in the way the representationwas used: as a display representationor as an action representation.A display representation means that known informationwas translatedor recapitulated; an action representationmeans that objects were transformedthroughactions such as computing,solving, stretching,etc. (Kaput,1992). A passage within the same representationbetween referringto it as a display or an actionrepresentationconstitutedanotherboundaryfor segmentation.Such segmentationwas feasible as talk in the lesson generally referredto one representationat a time, and as the teacher explicitly providedscaffolding to reasoning.The differentrepresentationsof function in the lesson were algebraic,tabular,graphic,and verbal.We identified19 segmentsin the reducedprotocol.The two authorssegmentedthe protocol separately. Two other researchersin mathematicseducationparticipatedin the segmentation,leadingto 90% interraterreliabilityon segmentingand coding. Disagreementswere resolvedby discussion. Developing a coding scheme. We developed a coding scheme that correspondsto our focus on students'understandingof functionsin different representations.We firstdistinguishedbetween segmentswhere the fence context was the focus and those where the context was sparse.For each case, we labelled each segment accordingto the representationthat was the focus. In the context of the fence we identified the following representations:situation(Sit), graph (Gr), table (Tab), algebra(Al). In the 'context-free'segments we identifiedthe latterthree. For each segment,


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we also identifiedwho initiatedthe passage to a new representation(the teacher or the students),who was the trigger for this passage, and what was the nature of the interlocutor(s)'response (if any). We define five kinds of responsiveness:Accompanyingtalk refers to talk in which the interlocutorattends to the other's talk without elaboration,typically acknowledgingthatthe interlocutorfollows the other'stalk.Elaboratingtalk refersto talk in which the interlocutorelaboratesutterancesand expresses deepercognitive involvement.Oppositionrefersto talk in which the interlocutorexplicitlyexpressesdisagreementandobjection.Puzzlementpoints to talk expressing confusion, perplexity or bewilderment.Finally, nonresponsivenessrefers to the absence of talk subsequentto the initiator's talk4.We also distinguishbetween a passage to a differentrepresentation that is embeddedin the context of the Fence problemsituationand a passage to a differentrepresentationthat is disconnectedfrom a situational context. Moreover,as we assume that the natureof the class discourse is relatedto studentunderstanding,we characterisethe types of utterances at each segment accordingto the following categories:Presentation(Pr), Short Questions (SQ), Extended Questions (EQ), Short Answers (SA), ExtendedAnswers (EA), Rephrasing/re-voicing(R), Objection(Ob). Operationalizingevidencefor coding. As explainedearlier,the transcript of the lesson was treated as if it were a protocol of two persons (the teacherandthe 'students')solving a problemtogether.We constructedthe in a form of four columns.The firstcolumn designates transcript-protocol the segment and line-numbers,the second - the teacher'stalk, the thirdthe students'talk, and the last column designatesthe teacher'sbehaviour. This form of protocol is asymmetricin that it presentsthe teacher'sbehaviourbut not the students'.The reason for that was that the group of studentswas conceived as one entity and thereforeit was impossible to specify individualstudent'sbehaviour.Such a form of transcriptis appropriate for the parts of the lesson we chose to analyse because they are relativelyteacher-centred. Figure 2 shows the beginningof the lesson. The teacherrecalls first a previouslesson in whichthe studentsproducedgraphsusing theirgraphing calculators (lines 1-4 and 6-8). She then asks students short questions (lines 10, 12) aimingat showingthatmanycorrectgraphsmay be obtained when using different 'frames' (on the graphic calculator)for the same curve. The teacherwaves her hands in the air and uses gestures to show that the frame may presentlargeror smallerportionsof the graph.Then the teacherindicatesthat graphscan serve to find informationand recalls that the graphthey had worked on in the previous lesson helped to infer

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A I 2 3 4 5 6 7 8 9 10 ll B12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 C27 28 29 30 31 32 33

Students

Teacher

Segm.

Whenwe work-edon thecomputerlastweek, som-eof the girLsdid notknowat the beginningin whatrangeto draw the graphso that it would be possible to see the whole picture.Andsomeof the girlsgot thispicture.Ri,ht?

Teacher'snon-verbal actions T drawsa graph on theboard.

Yes. No. T addsa frame to previousgraph.

1 am askingif somethinglike this reallyhappened.Why did it happen? Becausewe actuallycut. Thiswas. niore or less, our screenand we cut the graph.Had we added ftiorenumbersto the numberline. it would have added to thispicture.Is thisgraphcorrect? moreinformationi

T pushes'theframeof the araphwithherhands. Yes

fromit? Yes.Canlwe rindall theinformation No... Bit we found whatwe wanted. T explainswhatshe meant by recallinga previous homeworkquestionfor whichtheneeded information is notvisibleon the givengraph

It mightbe thatwe foundwhatwe wantedto. butit might have been thathad I askedadditionalquestionsthenwe would have not found. OK, that means there was a questioti:Whatis the side so that the lot would be the largest.is it possibleto see it fromthis?Yes, it is possible to see it fromthis. Therewas a question:Are theretswo lots so that bothhave the same area?No, therewas not sucha questionTherewas a question:If one side [theone parallelto the wall is 11,whatis the areaandis therean additionallot with the same area'?Right? Now. is it possiblethatI won'tbe ableto answersucha question?

T noticesthatshe chosea questionthatwasnot partof thehomework. Yes

Why? Becauseherewe haveanotherplace, then.. T pointsto a potentialplace for I lIy=t 11on the visible partof the graphon the

Thatmeansthatif II werein thisplace,forexamplethen what happened? We can't see what happensin what follows,right'?

calculator, then to its

pointwhich -symmetrical" is outsidethe visiblepartof the gra. D34 So todaywe wantto practice.Umm,firstof all we want 35 to say thatwe hada formulain the fenceproblem.

Somestudents T interrupt

Figure 2. The beginningof the lesson.

dimensionsof differentlots thathave the same area(lines 15-23). She also statesthatsome graphsmay not allow the inferenceof neededinformation. Next the teacherturnsto be more specificaboutthe previouslesson, recalls that it was about the Fence ProblemSituation,and reconstituteswith the studentsthe formula(S(x)) of the areaof the lot (segmentD, only the first two lines 34 and 35 are displayed). The excerptpresentedin Figure2 includesfour segments:A, B, C, and the beginningof D. These segments are delineatedby the representation on which the public talk centers.In lines 1-11 (segmentA) the talk refers solely to graphs.In lines 12-26 (segmentB) the talkcenterson the problem situation(intertwinedwith the graph in the background).In lines 27-33 (segment C) the teacher and the students returnto exclusively refer to the graphicrepresentation.In segment D (lines 34-48) they refer to the algebraicrepresentation.All four segmentsare in the context of the fence problem.Examinationof the firstline in each segment- the transitionlines

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12, 27 and 34 - indicatesthatthe initiativeto pass to a new representation is thatof the teacher.Finally,examinationof the types of utterancesresults in the following coding. Forteacher'sutterances:Presentationin lines 1-4, 15-17, ExtendedQuestionsin lines 6-10, 17-25, 31-33, ShortQuestions in lines 4 (end), 10 (end), and 27. As for the students'utterances:Short Answers in lines 5, 11, 13-14, 26, and 28-30.

formalism. Depictingthe coded dataof this study Depictingthemapped was conductedin two ways. One way is displayedin Figure 3. The segments of the lesson are markedin the first row (segments I, S and V

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indicatethe small-groupworkpartsof the lesson which are not partof the reducedprotocol we analyse). Each of the following four rows designate one of the four representationsto which the class talk refers when the context concerns the Fence situation. Sit designates segments in which participantsfocus specificallyon objects and events of the Fence task;Gr, Tab, and Al designate segments in which the talk refers to graph, table and algebra/formula(respectively).The last threerows similarlydesignate the representationalsystems to which the talk refers, but here, with no connection to the Fence situationalcontext but ratherrelatedto a formal task. Circles representthe teacherand squares,the students.For each segment we mark who initiates the passage to a new representationor the change in the way the representationwas used (display or action) by an arrowtargetedto the initiator.The trigger for the passage to a different representationis markedat the beginningof the arrow.The natureof responsivenessis markedas following: gray representsaccompanyingtalk, black - elaborating talk, inscribed question mark - puzzlement, crossed -

opposition,and white - non-responsiveness. For example, Figure 3 displays an arrow ending in segment B with a black circle and a gray squarebesides it. This means that the teacher was the one to initiatethe passage from the graphicalrepresentationof the Fence situationin segmentA to an interpretationconcerningthe situation itself in segmentB, andthatstudents'responsesaccompaniedthe teacher's move. SegmentF presentsa teacher'sinitiativeto move from an algebraic representationto work that focuses solely on the situation(see Figure4). She asks students,who are actively discussing with her how to compute S(5), where S(x) = (30-2x)x is the formulafor the area of the lot: "What is the meaning in the story of S(5)?".The studentsdo not understandher questionandrespond:"Whatdo you meanby 'What'sthe meaning'?"and experience difficultiesin following the teacher'sguiding questions (e.g., "If the length of the side is 5 then the area of -"). This episode is rep-

resentedin Figure 3 by an arrowfrom the algebraicrepresentationto the situationending with a black circle and a question-markedsquarebesides it; the latterrefers to students'puzzlement.Segment H in Figure 3 illustrates students' opposition, when the teacher tries to detach the formula S(x) = (30-2x)x from the story, change the S to L and asks studentsto computeL(-3). Studentsoppose this requestby questioningthe legitimacy of substitutingnegativenumbers:"Howcan therebe a -3 length?" In segmentK, these arethe 'students'(in fact, one student)who initiate the passageto a tabularrepresentationas a responseto the teacher'srequest to find how to solve the equationL(x) = -152. The teacherelaborateson

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TABLEI

Teacher

Pr

SQ

EQ

SA

EA

R

27%

42%

754

4%

2%

19%

9%

6%

76%

6%

Ob

(68%) Students

3%

(32%)

this suggestionandinvites,in segmentL, a studentto suggest anotheridea; an idea relatedto the formulathat the teacherhas seen this studentworking on duringthe small group work. This studentpasses to the algebraic representationin segmentL. Consequentlythe arrowin L originatesfrom the teacherandis targetedto the studentwho initiatesthe passageto a new representation. Horizontalarrowsin Figure 3 (e.g., from D to E) representa passage within the same representationbetweendisplay and actionrepresentation. The two arrowsoriginatingfrom the crossed squarein segment 0 indicate that in segment P the teacher initiatedtalk about both the Fence situation and its graphicalrepresentationsubsequentlyto the opposing expressed by students.The question-markedsquarein segment P indicates that studentswere puzzled aboutthis teacher'sinitiative.The crossed circle in segment P indicates that the teacherreferredto the Fence situation. This was a teacher response to students' reference to the Fence situationto indicate that the Fence situationis irrelevantto the problem under consideration.Figure 3 displays in the beginning of each wholeclass partof the lesson (segments A, J, and T) a dashedarrowto indicate thatthe interlocutor(in fact always the teacher)initiatedthe referenceto a representation. Figure 3 presents a comprehensivepicture about initiatives and the nature of responsivenessregardingthe passing from one representation to another.This picturecontributesto our understandingof the students' mentalrepresentations- the focus of our cognitive analysis. But this way of depictingthe coded data does not presentinformationon the natureof the utterancesthat comprisedthe talk. Anothermethod we use to depict the coded dataof this studyfocuses on the categoriesof utterances.TableI presents the percentagesof the following kinds of utterance:Presenta-

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tion (Pr), Shortquestions(SQ), ExtendedQuestions(EQ), ShortAnswers (SA), ExtendedAnswers (EA), Rephrasingor re-voicing (R), and Objection (Ob). Patternsdetectedin the depicteddatain Figure3 andTableI arepresented in the next step of the VerbalAnalysis. Seekingpatternand coherencein the depicteddata. Figure3 shows that the Fence problem situation served as context for the first whole class discussion(segmentsA-H), except for segmentH. The second whole class discussion focused mainly on work in a formal context detachedalmost completelyfrom the Fence situation(segmentJ - 0). It is only in segment P and partly in segments 0 and Q that there is a referenceto the Fence situation. In the short last whole class discussion, the class focused on graphicalmanipulationswithoutreferenceto the Fence situation(segments T and U). The aim of the lesson was to develop students'understandingof different representationsof functions,theirability to choose an appropriateand suitable representation,their flexibility to move from one representation to anotherand to connect differentrepresentationsof the same function. To interpretthe lesson accordingto this perspectivewe seek patternand coherence in initiative,responsiveness,disagreementor puzzlementthat characterizethe passages from one representationto another.Figure 3 shows that it was the teacher who made most of the initiatives to pass to a new representation(teacher- 16, students- 3). Studentsusually accompaniedthe teacher or elaboratedher initiatives.On seven occasions (segments A, B, C, G, H, J and T) students'talk is of an accompanying nature.On eight occasions studentsrespondwith elaboration(segmentsD, E, K, L, M, N, R andU), andon fouroccasionsthey arepuzzled (segments F, 0, P and Q). On two occasions (segmentsH and0), studentsoppose the teacher'sinitiative5. Whatemergesfromthe depicteddatain Figure3 is thatthe teacherhad a majorrole in initiatingthe passage to a new representationthroughout the lesson while the studentsgenerallyonly respondedto the teacher'sinitiative.In aboutone thirdof the passages studentsreactedto the teacher's initiativeswith elaboratingutterances,and in one thirdby accompanying talk. An examinationof the protocolindicatesthatthe elaboratedtalk was often done by a few specific studentsonly (e.g., in K, L, and M). TableI thatshows the natureof all the utterancesduringclass talk (and not only those relatedto the passagebetweenrepresentations)confirmsthe centralrole played by the teacher.About two-thirdsof the utteranceswere the teacher'sand she was the only one to make Presentations(aboutone-


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fourthof her utterances).When studentsdid participatein the class talk, they typicallyrespondedwith ShortAnswers(76%of students'utterances) to teacher's Short Questions (42% of the teacher'sutterances).Students almost never asked questions. Extended Questions as well as Extended Answers throughoutthe lesson were rare,both from the students'and the teacher's sides. Therefore,it appearsthat the elaborationrepresentedin Figure 3 (one thirdof students' responses to teacher'sinitiativesto pass to a new representation)reflectthe activityof special students,and that in generalstudents'talk is laconic. Interpretingthe data. The lesson centredon work with differentrepresentations of functions, on choosing appropriateand suitable representations, andon being flexible in connectingrepresentationsandpassingfrom one to another.However,throughoutthe lesson studentsalmost neverinitiated the passage to a new representation.They passively accompanied the teacher'smoves, and displayedlow level thinkingonly. Possible interpretationsfor this behaviourmight be thatfor the studentsto use multiple representationsto solve this problemis a cognitiveobstacle,or thatexpectation of the teacher'scontrolof the strategyon behalf of the class poses a barrier. Let us examine the adequacyof these interpretations.In orderto check the adequacyof the latterinterpretationwe examine students'behaviour duringsmall groupwork(which was not partof the VerbalAnalysis). This examinationindicates that students' behaviourduring small group work was similarto their behaviourduringwhole class instructionin that they almost never initiatedthe passage to a new representationand generally displayedlow level thinking.Consequently,it is not likely thatexpectation of the teacher'scontrolof the strategyon behalf of the class is the source of students'behaviour. To support and verify the interpretationthat for the students to use multiplerepresentationsto solve this problemis a cognitive obstacle, we analyze in detail part of the lesson. The teacherwrote on the boardL(x) = (30-2x)x and asked the studentsto solve the following tasks: L(-3) =, L(-10) = , L(20) =, L( ) = -200 and L( ) = -152. The students worked

in pairs on the tasks. For the first three tasks they chose to use the calculator unit of their graphiccalculators.However, for the last two tasks substitutingnumbersin a formulawas not possible and the studentsdid not know how to find the missing numbers.Some found the numbersby trialanderror.Still, no one has attemptedto use theirgraphiccalculatorsto constructthe graphof L(x) = (30-2x)x and to find the needed information fromthe graph,althoughthis methodhas alreadybeen used in this class in

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the last series of lessons. Morethanthat,the very same graph- its formula was denotedthen as S(x) = (30-2x)x - was examinedanddiscussedin this lesson. Whenthe whole class discusses the tasks, one studentsuggestedto constructthe equation(30-2x)x = -152 and to solve it (note thatthe class did not learn yet how to solve quadraticequations).The teachertries to help the studentsrealize that passing from the algebraicrepresentationto the graphicis a desirablepossibilityhere.This is illustratedin the protocol of segmentsM to R as displayedin Figure5. In segment M the teachertries to lead the studentsto the conclusion that it is impossible to use algebrato attaina solution. She flags that the attemptto solve an equationis good (lines 236-237) andexplainswhy this leads to an impasse (lines 240-246). As shown in line 246, the reaction of the studentsto the impasse is to ask whether guessing is legitimate. The teacher wants studentsto reuse the strategythey have alreadyused in the Fence situation:to draw a graphand to read the result. Therefore she asks for anotherway to solve the problem and mentions the Fence situation.Studentsstill remainwith the same representationandrecallthat it is possible to use anotherformulaby using anothervariable,the second side of the lot (lines 253-255). Attemptingto remindstudentsabout the graphicway, the teacherwaves her handsin a parabolalike manner(lines 260-263), accompaniesher handmoving with the questionwhetherthere is anothermethod, and repeats it until students answer in a telegraphic style 'Graph!'(line 264). The use of the graphto solve the problemthen becomes easy for the students.They are also able to find 'windows' on their graphic calculatorsresembling given graphs, and to 'solve' other algebraicproblemsgraphically.This supportsour interpretationsthat it is the connectionsbetweenrepresentationsandthe initiativeto pass to a new representationthatis difficult. This conclusion fits with findings of other cognitive studies where it was found that translatingfrom one representationto anotheris cognitively difficult(e.g., Even, 1998;Janvier,1978, 1987; SchwarzandDreyfus, 1995). While it is frequentlyarguedthatthe multiplicityof externalrepresentationsfacilitatesthe learningof concepts as it helps in the integration of perspectives (e.g., Goldman, 1976; Spiro, 1989), the multiplicity of externalrepresentationsis also a cognitive load thathinderslearningprocesses (Sweller and Cooper,1985; Sweller and Chandler,1994). Learners prefer then to compartmentalizewhen solving problemsinvolving functions: they tend to undertakemanipulationsin the same representation unless they areinvitedto do so. Linkingthe findingsof the VerbalAnalysis with the cognitive load theoryis one possibility.Anotherpossibility is to link the findingsto the rich literatureabouttransferandclaim thatthe diffi-

COMPETING INTERPRETATIONS

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culties studentsencounteredare relatedto problemsof transfer.However, transferinvolves passing to a new problemwhile in our case the problem remainsthe same. Consequently,we conclude thatstudents'difficultiesin the lesson can be explainedby the cognitive load theoryin thatthe use of multipleexternalrepresentationsis a cognitive obstacle.

ACTIVITY THEORY APPROACH

ActivityTheorynotions In this section we approachthe analysis of the lesson from a different theoreticalperspective- from a socio-culturalapproach.A socio-cultural approachto mind begins with the assumptionthat action is mediatedand that it cannotbe separatedfrom the milieu in which it is carriedout. It is assumedthatin orderto understandthe individualit is necessaryto understand the social relationsin which the individualexists (Wertsch,1991). For analysing the lesson we use the socio-culturalframeworkof Activity Theory (Leont'ev, 1981). Out of the range of socio-culturaltheories (Vygotskian, situated, etc.) we chose Activity Theory because its characteristics,as describedbelow, allowed us to illustrateour claim clearly since Activity Theory focuses on motives and we claim that the key for comprehendingthe lesson accordingto a socio-culturalperspectiveis to identify the differentmotives of the participants.We will not providehere a full descriptionof ActivityTheory.Ratherwe will highlightcentralcharacteristicsthatwill be useful for our analysisof the lesson. ActivityTheory views the structureof social interactionas sourceof the structureof human thinking.The unitof analysisaccordingto ActivityTheoryapproachis the humanactivity.The basic characteristicof activityis its objectorientation. An activity's object is its real motive. Need is always an essential partof an activity.Activities are chains of actions relatedby the same object and motive. Actions can be understoodonly within the activityin which they are embedded.Actions arethe basic 'components'of humanactivitiesthat translatethem into reality.The actions thatconstitutean activityare energised by its motive, andaredirectedtowardconsciousgoals. Apartfromits intentionalaspect(whatmustbe done), an actionhas an operationalaspect (how it can be done), definedby the objectivecircumstances.Operations are the means by which an action is carriedout. They dependdirectlyon the conditionsunderwhich a concretegoal is attained.

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Researchquestionand data source Similarly to the case of the Cognitive Science perspective,the Activity Theory perspective impinges on the general aim of evaluatingwhether the intentionof the designerof the Fence Problem(creatingthe need and developing the ability to move between symbolic and graphicrepresentations) was attained.This is translatedinto the following researchquestion: What is the natureof passing from one representationto anotherin the activity in which the teacherand the studentsparticipatedduringthe lesson? The analysis is based on severalsources of data:the video-tapeof the lesson, classroomobservation,the protocolused for the cognitiveanalysis, its verbalanalysis and the two ways of depictingthe coded data (Figure3 and Table I). While the two last sources were productsof the Cognitive Science perspective,they are used here as artefactswithout necessarily adoptingthe interpretationsof this perspective. Theteacher Teacher'smotive. When viewing a lesson as an activity,attentionshould be directedto identifyingthe motive of the participantsin the activity(i.e., the teacherand the students),and the goals of the actions undertakenby these participantsduringthe activity.The teacher'soverall motive when teaching mathematics,as expressed by her in numerousformal and informaldiscussions,is that her studentsunderstandand learnmathematics in meaningfulways. More specifically in this lesson, as expressedby the curriculumdevelopmentteam of which she is a member,her motive is thather studentslearnaboutdifferentrepresentationsof functionsand understandhow to use the graphiccalculatorto solve problemsthat require the passage from one representationto another.This teacher'smotive is relatedto the need to choose the Rangeoptionin a cleverway when solving problemsusing the graphiccalculator.Otherwise,some solutionsmay not be shown on the screenand would be missed. Teacher'sactions and goals. The teacher'sactions duringthe lesson are derivedfrom her overalland specific motives.For example,she begins the lesson (i.e., the activity) with the action of presentingthe studentswith a problematicsituationof having a 'correct'graphthat does not provide all the relevantinformation.Her goal is to convey thata uniquegraphical representativecannotdisplay all characteristicsof a function.Also, thatin orderto solve a specific problem,such as, "Whatis the other side of the lot if one side is 11?", the problemsolver needs to choose an appropriate display on the graphiccalculator.Otherwise,for example,only one of the

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INTERPRETATIONS COMPETING Segm. F67

Teacher'snonverbalactions

Students

Teacher

T points to 100

What Ls the meaning in the story of what is

68 on the board 69 70 71 72 No. of 1(00also. What is the meaningof 73 everythingwrittenhere:S(5) = 100? Whatis 74 the meaningin the story?We havea story. 75 76 77 78 If the lengthof theside is 5 thenthe areaof 79 80 81 82 The areaof what? 83 84 Of the fence? Of what? 85 86 Of whereyou putthe fence? 87 88 The areainside 89 90 The areaof 91 92 If we substitutehere5 forx, we saidwe'll get 93 100. And in the story it means that if the 94 lengthof the side is 5 metersthenthe areaof 95 what? 96 97 Thenthe areaof the lot is 100squaredmeters 98 99 Youdon't haveto. G 100 .. .Story.I havca formula.Writeit downas a 101 title whereveryou want 1)2 103 Yes. To writeon a new page H104 And I forgot for a moment the fence 105 problem. I got this kind of formulaI give 106 this formulaa differentname. so you know 107 that1 forgotthe fence problem. We'llcall it. 108 for exantple 109 11(1 L. OK, then L(x) = (30-2x)x. I wantyou to IlI makethe followingcalculations. 112 1.13 No. with... You may use a calculator. 114 115 1 forgotthe fence. I just havea formula. 116 1J17 118 Why is it L andnot S? Becausenow it is not 119 the fenceproblem.Andwe will soon explain 120 why this is not the fence problem.But, it is a

andS(5) Whatdoes it mean'whatis the meaning'? What is the question? [Is it] what is the meaningof 100'?

Whatdoes it mean'whatis the meaning'? Thatthe area...That the surfacearea...That the lengthof the side is 5... The areaof The length of the side... Of the surface of... Not surfaceandnot... Of the fence... Of whereyou putthe fence... The lot. Well,whatdid I say? ...thelot T writeson the blackboard whilespeaking T continues writing

Of the lot Thiswas homework.To writeit down?

T "frames" S(x)=(30-2x)x To writeon a new page T writes(30-2x)xon the boardagainanderases everythingbesidesthe formulaandthegraph. L T writeson the board: L(-3)=?,L(-l0)=?. L(20)=? Witha calculator? How can therebe a -3 length? Whyis it L? It is somethingelse.

Signalswithherhandsthat "thisis over"

formula that we got from the fence problem.

Figure4. Difficultiesstudentshave to follow the teacher.

two possible solutions to the above problem is seen. This introductory action is designed to motivatethe students;to help them understandand adopt the teacher's motive. Then the class discusses the meaning in the 'story' (i.e., the Fence situation) of substitutinga numberin the algebraic representationof the area of the lot (see Figure4, segment F). Being awareof her students'difficultiesin performingalgebraictechniques,the

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teachercapitalizeson the lesson as an opportunityto practicesubstitution of numbersin algebraicexpressionsin a contextthatgives meaningto this procedure.In anotheraction laterin the lesson (Figure4, segmentsG and H), the teacherdetaches the algebraicexpressionthat representsthe area of the lot from the situationthatgave it rise. In fact, she literallydetaches it by erasingthe drawingof the lot from the board,changingthe name of the function from S to L, and declaringexplicitly that "this is a different problem".The teacherperformsthis detachmentin orderto allow the use of numbersnot suitable for the Fence situation(e.g., negative numbers). This way the teacher aims to create a need in her studentsto choose a Range that would enable them to see the relevantpart of the graph.The teacherkeeps workingwith the same algebraicexpressionthroughoutthe entire activity, as a means to connect the differentactivity components. Workwith the same algebraicexpressionalso highlightsthe differentways one can approachthe same algebraic expression:as a representativeof a situation and as a mathematicalobject that can be examined with no connectionto a specific situation. To help the readerdevelopa morecompletepictureof the whole lesson, below is a schematicdescriptionof the chainof the teacher'sactionsin the lesson. The teacher proposes a motivationalquestion:Does the given calculator'sgraphicrepresentationprovideall the relevantinformation? * The teacherinitiatesa whole class examinationof a test case (one side of the figureis 11, the othercannotbe found from the visible partof the given graphon the graphiccalculator). * The teacherinitiatesa whole class discussionon the meaningof substitutingnumbersin the algebraicrepresentationof the functionthat representsthe areaof the lot. * The teacherliterallydetachesthe algebraicrepresentationof the function from the Fence situationthatgave it rise. * The teacher asks the students to work in small groups on finding images and pre-imagesof the functionL(x) = (30-2x)x. The teacher chooses (an on-the-spotdecision) images so that the corresponding pre-imagescould be found by the studentsonly if they use an appropriateframein the graphicrepresentationor by trial-and-error. * The teacherinitiatesa whole groupdiscussion on the criticalrole of choosing an appropriateRange thatwould enable one to find needed informationfromthe graph. Teacher'soperations. Throughoutthe lesson, the teacher'soperations(in an Activity Theory sense) reflect her desire to engage her studentsin the


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activity while being attentiveto their understandings.She never lectures. Rather,she (a) asks studentsquestions; (b) listens and responds to their questions,remarksand suggestions;(c) assigns tasks for them to work on in small groups;and (d) tries to conduct whole group discussions. From an Activity Theory perspective,these are operationsas they are undertaken automaticallyand are part of the teacher's daily work in school (Leinhardtand Greeno, 1986). They are aimed to help the studentsbecome involved, understand,and develop a sense of sharedownershipof the activity'smotive as well as the actions' goals. The actionslisted above and these operationsalso suit the teacher'sgeneral approachto learning and her belief that studentslearnby constructingtheirknowledgethrough active participationin mathematicalinvestigations. Thestudents Let us now turn to examine the motives and goals of the students.We contendthatin the lesson manystudentsdid not sharethe teacher'smotive. Consequently,many studentsparticipatedin the same lesson as the teacher but in a differentactivity. To better understandthe nature of students' participation,we take a closer look below at a sampleof lesson's episodes. Thesearchfor illustrativeexamples. Let us examine,for a start,the first two minutesof the lesson (see Figure2). The teacherbegins the lesson by remindingthe class of a difficultythey encounteredin a previous lesson when they did not know what Range to choose in orderto see the 'whole picture'(segmentA). She then tries to find a specific example to illustrate this problem.She firstchooses the example:"Whatis the side so thatthe lot would be the largest?"(lines 18-19). However,as she quicklyrealizes,the visible partof the graphthey have on the graphiccalculatordoes show the answerto this question:"Yes,it is possible to see it from this" (lines 1920). She then turnsto look for an example from the homeworkquestions (probablyin orderto workwith somethingalreadyfamiliarto the students) and suggests: "Therewas a question:Are theretwo lots so thatboth have the same area?"(lines 20-21). The teacherquickly notices that she chose a question that was not part of the students' homework:"No, there was not such a question"(line 21-22). Only on her thirdattemptshe finds an appropriateexample:"Therewas a question:If one side is 11, what is the area and is there an additionallot with the same area?Right? Now, is it possible thatI won't be able to answersuch a question?" Throughoutthese two minutes, it is the teacherwho does most of the talking. Many studentsappearin the video-tape uninvolvedand uninterested. The teacher searches for an appropriateillustration,a search that

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could be illuminatingfor attentive,quick and involvedstudents.However, such a search seems to have the opposite impact on less attentive and disengaged students, such as many of those in this B-level class. These studentsseem to lose the aim for this searchand the goals of the teacher's actions in general. How meaningfulis meaning? Let us examine anotherlesson excerpt in the same manner.This excerptstarts6.5 minutesinto the lesson and lasts for 2.5 minutes. Between this excerpt and the previous one the teacher spent some time on partlyremindingthe studentsof, partlyreconstructing, the formulafor the area of the rectangularlot. In the previouslesson the class workedwith two formulas:the formulafor the other side of the lot: F(x) = 30-2x, wherex is one of the two equalsides, andthe formulafor the area:S(x) = (30-2x)x. Because of the similaritybetweenthe two formulas many students mix them up now and the teacher decides to clarify the differencebetween the two. Consequentlythe teacherasks the studentsto substitute5 in the area formula. She writes on the board:S(5) = . After overcoming the problem that 5 should be substitutedtwice: (30-2.5).5, insteadof only once: (30-2x).5, the teacherwrites on the boardstudents' suggestion:S(5) = (30-2 x 5).5 = 100. The teachercircles the two 'ends': S(5) and 100, and asks about the meaning in the story (i.e., in the Fence problemsituation)of what is circled on the board(segmentF in Figure4). As we can learnfrom students'responsesthey did not understandwhat the teachermeant when she asked: "Whatis the meaning in the story?" The few attentivestudentskept asking the teacher what the question is (or means) (lines, 69-71, 75) or suggestedwhat seemed like wild guesses (lines 76-77, 80, 83, 85). Researchon algebralearningindicatesthatmany studentssee no connectionsbetweenthe algebraicmanipulationsthey perform and the story or context from which these operationsemerge (e.g., Kieran, 1992). For such students,the meaningof S(5) and 100 is usually thatif one writes 5 insteadof x and performsthe operationsappropriately then one gets 100. Consequently,it was almostimpossiblefor the students to understandwhatthe teachermeant.As manyof the studentsin this class appearedinattentiveto the discoursethat took place between the teacher and some of the other students,it is also questionablewhetherthey even cared to understand.As understandingthis question does not have any implicationsfor participatingin the next partof the lesson, the discussion of the 'meaningin the story' remainedmeaninglessfor many students. "A rose by any other name would smell as sweet" (Romeo and Juliet). After trying, with much difficulty, to establish meaningful connections


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between the manipulationsin the algebraicrepresentationand the story of the problemsituation,the teacherdetachesthe formulafrom the Fence story.This occursnine and a half minutesinto the lesson. The teacherdoes not explain the rationalefor this move to the class. Rather,she focuses on helping the studentsunderstandthat now they are changingthe 'rules of the game'. She does it by putting a frame aroundthe formulaS(x) = (30-2x)x on the boardand telling the studentsto copy it into a new page in their notebooks (Segment G in Figure 4). She then tells the students that she forgot the fence problem.The teachererases everythingfrom the boardbut the expression (30-2x)x and the graphand gives the algebraic expressiona new name, L, to emphasisethat it is not relatedto the Fence story anymore(lines 104-1 10, segmentH in Figure4). Afterthe formulais detachedfromthe Fence story,it becomes possible to work with negativenumbersand the teacherasks the studentsto make the following calculations:L(-3) = , L(-10) = , L(20) = (lines 110-111, segment H). Laterthe teacheradds two more exercises: L( ) = -200 and L( ) = -152 and writes 'A challenge'. The studentsappearconfused about the requestto work with negativenumbersand the teachertries to explain that L is differentfrom S and that this is not the Fence problemanymore (lines 114-121). To betterunderstandthis students'confusion one should note that by this point the studentshave encounteredthree similarformulas: F(x) = 30-2x (for the other side of the lot where x is one of the two equalsides), S(x) = (30-2x)x (for the areaof the lot) andL(x) = (30-2x)x (a formuladetachedfrom the Fence story).The firsttwo were constructedin a previouslesson and, as alreadymentioned,earlierin this lesson students kept mixing them up. Also, S was changed to L without an explanation and this change happenedonly a shorttime afterthe teachermade a great deal of the meaningin the Fence storyof substituting5 in S(x) = (30-2x)x, emphasizingthe connection to the Fence story: that if the length of the side is 5 then the areaof the lot is 100. One should also keep in mind that this is a class of low-achievingstudentsthat often appearinattentive.For these studentsa 'sudden' appearanceof - 3, for example, does not make much sense nor does the teacher'sclaim that "it is not the fence problem" (lines 118-9). For them changing the name of the formula from S to L does not seem enough to signal that the formulanow is not connectedto the fence story. For them, having a negative numberfor x, which just a minute ago meant a side of a rectangularlot, could be anotherindicator thatmathematicsdoes not make sense. The nature of the need to pass to another representation. One of the teacher'sgoals when designing the lesson was for the studentsto feel the

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R. EVEN AND B.B. SCHWARZ Students

Teacher

Segm

M234 235 236 237 238 239 240 241 242 243 244

Is it rightto say thatx(30-2x) mustbe equalto -152?

N245 246 247 248 0249 250 251 252 P253 254 255 Q256 257 258 259 260 261 262 263 264 265

andwe will be able to do...

Teacher'snon-verbal actions

Yes Yes. this is a very good idea.Whatis the problemwith this idea? Difficultto solve. Thereare two... Correct.We don't know how to solve such an equation because the x appearstwice. If we try to develop the parentheses,we obtain an equationwe don't know to solve.. So we'll learnto solve such an equationperhaps at the end of next year, We'll have to guess? 1 didn't meanto guess. althoughsoeiTtines it's worth to know how to guess Are there other ways you can describe this creature. other ways to describe the story. This not a story becausethelot alreadydisappeared.but...

T pointsat the formula. T flags her handsin a sign of "over" Whatwe did...

Whatdid you do? We did 301did not nean whetherthereis anotherformula Something totally different. What did we do with surface of the lot. when we wanted to know, for example.if for a given surfacethereis more... To divide.! To divide, I don't thinkwe did [pauscl Whatdid we do with the conmputer? Ipausel

[pausel Ipausel Graph!

T drawsin the aira convex arc T continuesflagging her hanids

Forwhichpurposethe graph? To check

R266 Is it possiblethatif we have a graph.we can get 267 information? 268 269 Let's tryto do a graph

Of course.we can!

Figure 5. Incompatibilityof meanings:The teacherfails at conveyingthe role of graphs.

need to pass to another representation when solving problems. Also, when passing from algebraic to graphic representation she wanted students to realise that there is a need to carefully choose the appropriate Range in the graphing calculator in order to find solutions to various problems. To attain these goals she, for example, asked students to find L( ) = -200 and L( ) = -152. Knowing that the students do not know yet how to solve quadratic equations algebraically, the teacher anticipated that they would use their graphic calculators to draw the appropriate graph and then would realise that they need to choose the Range cleverly. However, students 'solved' the first item immediately, as -200 was the answer to L(20) = , a previous problem presented by the teacher (unfortunately the teacher did not check the numbers ahead of time). The second item L( ) = -152 was either not solved at all or solved by trial and error. The whole class discussion of this task (Figure 5) illustrates the teacher's struggle to lead the students to a realisation that passing to a graphical rep-


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resentationis needed. The teacherstartsthe discussion by acknowledging one student'sidea to solve the equationx(30-2x) = -152 algebraically.The teacherevaluatesthis idea as good (Lines 236-237) and with the participation of some otherstudentsconcludesthatthis methodis not appropriate here because they do not know yet how to solve such equations (i.e., quadratic)(Lines 238-244). A studentthen asks:"Willwe have to guess?" (Line 246). The teacherdoes not encouragethe choice of this methodfor this case (Lines 247-248), as she wants studentsto switch representations, drawan appropriategraphand readthe solutionsfrom it, i.e., to reuse the strategythey have used to solve problemsconnectedto the Fence situation. She tries to provide studentswith 'hints' that would help them recall the use of graphs.She keeps askingfor anothermethod(Lines 249-250, 256), mentions the Fence situationeven though she instructedstudentsbefore to forget about it (lines 250-251, 256-258, 262), and waves her hands in a parabolalike manner(Lines 260-263). Finally, after the teacher asks: "Whatdid we do with the computer?"(Line 262), one student 'catches' the hint and replies: "Graph!"(Line 264). Fromthis point on studentsuse graphsto solve this and similarproblemseasily. However,such a use of the graphicrepresentationis rootedin what may be describedas students' need to guess what the teacherhas in mind and their need to follow the teacher'sinstructionsratherthanin a genuine mathematicsneed to dovthis in orderto solve the problem. General nature of studentparticipation. Analysis of the lesson observations and of the video-recordedlesson indicates that many studentsin the class appearuninvolvedand uninterestedin the intellectualchallenges presentedby the teacherto the whole class. Only a small numberof students participatedin the whole class discussions thatthe teacherstroveto conduct.Even then, as TableI and Figure 3 also show, their participation was rathershallow and of low-level, characterizedby short answers to teacher's short questions. The majorityof the studentsbecame involved in the mathematicstasks only during the small group work parts. But also then, most of the studentswere not involved in exploration,problem solving, debates among themselves, and the like, as was common to the A-level studentsthe teacherhas taughtin a previousyear.Rather,many of the currentstudentstendedto call for teacherhelp afterminimalattempts to solve the tasks, asking her to tell them what the correct answers are. In general, studentsseemed satisfiedwhen they reachedthe correctfinal answersand frustratedwhen they did not.

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Students'motive. Manyof the students'actionsduringsmall groupwork seem to be guided by the goal of reaching correct answers quickly and painlessly.Duringwhole class discussionsmanystudentsseem uninvolved and uninterested,doing the minimumpossible. Takinginto account students' ways of participationin the lesson and that these studentsexperienced many failures and difficultiesin their previous mathematicsstudies, it is reasonableto determinethat the motive of many students was survivingthe lesson. The many failures and difficultiesin their previous mathematicsstudies taught them that active involvementdoes not bring with it good results.They learnedin previousyears thatthey are not good in mathematics,and therefore,would not probablybe able to solve the problemsposed by the teacher.They also knew thattheirgradeswould not be affectedby the natureof theirparticipationin plenaryactivities. So, a good strategywould be to stay uninvolvedratherthan expose their misunderstandings.Still, they learnedin previousyears thatit is importantto reach correctfinal answersin mathematics.Consequently,when assigned individualtasks, they were focused on finding correctfinal answers.Not expecting to be able to reason by themselves, they gave up after short attempts,expecting the teacherto fulfil her role in helping them have the final answers,not aiming at understandingthe mathematics. Same lesson butdifferentactivities The analysis of the lesson from an Activity Theory perspectivesuggests that the teacher and the studentsin the lesson participatedindeed in the same lesson but in differentactivities. Their motives were differentand consequentlythe differentactions that constitutethe activity and are energized by its motive were different.In general,the teacher'smotive was thather studentslearnmathematicsmeaningfully.More specificallyin the lesson, her motive was that her studentslearn about differentrepresentations of functions and understandhow to use the graphic calculatorto solve problems that requirethe passage from one representationto another,mainly,between graphicand algebraicrepresentations.We showed how the teacher'sactions duringthe lesson were derivedfrom her overall and specific motives. Her actions were well connected,one leading to another,creatinga complete whole. However,beauty is in the eye of the beholder- whatthe teachersees is not necessarilyseen by the otherlesson participants- the students. In the excerptsanalysedabovewe exemplifiedhow variouslesson components were experienceddifferentlyby the teacherand by the students. Many studentsdid not makeconnectionsbetweenthe differentpartsof the lesson. Furthermore,these studentsseem to conceive the lesson as com-


307

posed of severalunrelatedparts,where the teacher'sgoals for performing differenttasks were unclear(and, likely, not of interest).Students' goals were different from the teacher's - theirs usually centred on obtaining correctfinal answersto the problemsassignedby the teacher,not on making sense of the mathematicalsituationnor on using connectionsbetween differentrepresentationsof functions as a tool to solve problems.While the teacherdesignedtasksthatcentredon a cleveruse of the graphicalrepresentationof functions,manystudentswere stuckin performingalgebraic manipulations,not makingany connectionsto the overallactivity.Student then appearedto do all the neededwork,but actuallyparticipatedin a very differentactivitythanthe teacher.As such, the natureof the students'participationin the activitywas very differentfromwhatthe teacherhadwished for and did not contributeto the developmentof the activity as designed by the teacher.While this may seem odd when the teacher'sdesign of the activity is considered,it makes perfect sense when 'surviving'is taken as the student'smotive. The B-level class students,who have a history of learningdifficulties and low achievementin years of traditionalmathematicsteaching, aimed to survivethe mathematicslesson. Consequently,they behavedaccording to common beliefs about school mathematicsadoptingsocial and sociomathematicalnorms prevalentin traditionalmathematicsclasses (Even and Lappan, 1994; Lampert,1990; Schoenfeld, 1988; Yackel and Cobb, 1996). These studentsconsideredthe reachingof correctfinal answersas central,attributingalmost no importanceto the solution processes. They gave up after short and minimal attemptsto solve teacherassigned problems, assuming that if they cannot solve the problem immediatelythey would not be able to solve it at all, based on a commonbelief and also on theirown experiencesas low achievers.They also expected the teacherto be thejudge of the correctnessof theiranswers,attributingno credibilityto their and theirpeers' own reasoning.They felt thatit is the teacher'sduty to 'help' studentswhen they encounterdifficulties and when they could not react to her queries, and that it is her role to ease the achievementof the lesson ritual.

THE COMPATIBILITY OF THE TWO APPROACHES

Both the CognitiveVerbalAnalysis andthe Activity Theorybased analysis of the lesson indicatethatthingsarenot going smoothlyin this lesson. Both analysesshow thatstudentsdo not behavemathematicallyas desiredby the teacher.However,the two approachessuggest differentinterpretationsof the situationand of the sourcesto the problemsobserved.The VerbalAna-

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lysis points to students' cognitive difficulties in integratinginformation from differentrepresentationsin order to constructmathematicalknowledge and to students'cognitive difficultiesin initiatingthe passage from one representationof functionto anotherin orderto solve a problem.These cognitive difficultiesare takenas independentof context. The plausibility of this cognitive interpretationis supportedby othercognitive studiesthat show that readinginformationfrom one representationto anotheris cognitively difficult,and thatthe multiplicityof representationsis a cognitive load. The ActivityTheorybasedanalysisleads to an alternativeinterpretation of the lesson. This perspectivesuggests that the teacherand the students participatedin the same lesson but in differentactivities, where different motives,goals, beliefs andnormsregardingschool mathematicsdroveand guided them. As a result, students' ways of participatingin the lesson were differentfrom what the teacherhad wished for. The plausibilityof this socio-culturalinterpretationis again supportedby other studies that show how socio-culturalaspects of school mathematics,context, classroom culture,practicesand norms are central to students' mathematical behavior. An immediateresponseto the discrepancybetween the two interpretations might be: How could differentmethodologiesyield differentexplanations of the same phenomenon?Which one is correct?When addressing these disturbingquestionsit is importantto note thatthe two analyseswere guided by two different research questions. The research question that guided the cognitive analysis was: To what extent do studentsconceive the passage to a new graphicalrepresentationof a function as a problem solving strategyduringthe lesson? However,the Activity Theory based analysis was conductedin relationto a differentresearchquestion:What is the natureof the activity in which the teacher and the studentsparticipated duringthe lesson? Naturally,answers to differentquestions may not coincide. Still, anotherquestion then emerges: Are the two answers providedby the two perspectivescompatible? As we have alreadymentionedearlier,for severaldecadesmathematics educationresearchused to focus on cognitive developmentof mathematical concepts.Recently,the focus of researchin mathematicseducationhas extendedfromthe individualstudent'scognitionandknowledgeto contextual,socio-culturalandsituatedaspectsof mathematicslearningandknowing. The practicesandcultureof the classroomcommunity(e.g., the nature of social engagements and norms) have become an importantfactor in studyinglearningprocesses, and mathematicseducationresearchersstarted to incorporatethe two perspectives- cognitiveandsocio-cultural- into


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a complex view of mathematicslearning (e.g., Cobb, Stephan,McClain and Gravenmeijer,2001). This new focus signals a shift from examining humanmental functioningin isolation to consideringcultural,social, institutionaland historicalfactorsas well. The mathematicseducationcommunityincreasinglyembracesthe view that,like cognitiveaspects,cultural and social processes are integralcomponentsof mathematicslearningand knowing. In line with the currenttrend in the field of researchin mathematics educationwe respondby proposingthat the lesson we analyzed,which is partof the practiceof teaching and learningmathematics,is too complex to be understoodusing only one perspective.The cognitive analysisof the lesson presentedabove did not take account of socio-culturalaspects of students'participationin the lesson. Rather,it focused only on the complexity of cognitive processes, such as, the initiatingof a passage to a new representation.Similarly,the Activity Theorybased analysis did not refer to cognitive difficultiesthat studentsmay have encountered.Rather,it focused only on socio-culturalaspects, such as, the relevanceof passingto a graphicalrepresentationto students'activity.Moreextensively,contrasting irrelevancewith cognitive obstacles stressesthe extentto which these two factorsare interdependent.In otherwords,the juxtapositionof irrelevance and cognitive obstacles raises questions about the natureof the link that binds them. Is the link causal (because passage to new representationsis difficult, studentsare not willing to engage in activities involving such a passage)? Does this link representa tendency among weak students?To what extent can properguided participationof the teacher diminish the cognitive obstacles and does this participationnecessarilyleads to a better engagementof the students?These questions, as well as many others (concerningthe possible mediationaltools thatcan be used), are raisedas interestingoutcomes of the confrontationbetween the two methodologies we adopted.In brief, the confrontationof the two methodologiesraises new generalquestionsconcerningresearchandpedagogy. Is INTEGRATION THE ANSWER? We may conclude from the discussion above that a more complete understandingof the complicatedpracticeof teachingandlearningmathematics requiresthe use of both cognitive and socio-culturalperspectives.Indeed, other researchersin mathematicseducationhave reachedthis conclusion. For example, Cobb, Yackel and their colleagues, who have been engaged for more than a decade in a researchand developmentproject,began the projectintendingto focus on learningprimarilyfrom a cognitiveperspect-

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ive. However,it becameapparentto themthatthey neededto broadentheir interpretativestanceby developinga sociological perspective(Yackeland Cobb, 1996). The recommendationto use bothcognitiveandsocio-cultural perspectivesin orderto developmore completeunderstandingof the complicatedpracticeof teachingand learningmathematicssettles what at first seemed to be two conflictinginterpretationsof the same phenomenonand is graduallybeing embracedby the mathematicseducationcommunity. However,such a resolutionraises severalissues. One issue is whether the use of differenttheoreticalperspectivesalways leads to compatibility. The socio-culturalanalysiswe conductedfocused on the generalnatureof students' engagementin the lesson and less on content issues. Does the complementarynatureof the analyses stem from the fact that the cognitive analysis was concernedwith contentissues whereasthe socio-cultural analysis focused primarilyon the natureof general participation?Would the use of cognitive analysis and socio-culturalanalysis always lead to compatibility?The following example(suggestedto us by Steve Lerman), thatlooks at differentexplanationsfor success and failure,shows thatthis is not necessarily the case. Cognitive explanationsto success and failure that attributethem to low naturalability would be incompatiblewith socio-culturalexplanationsthat attributesuccess and failure to the social structureof the pedagogicrelationship. This question of compatibilityis not necessarily related to the case of cognitive and socio-culturalperspectives.Rather,it is a more general one, and is relevantto any case where multiple theoreticalperspectives are applied, such as in the case of using the lenses of varioustheoretical socio-culturalperspectives.Theoreticalperspectivesframe the posing of researchquestions and provide meaning to the researchquestionsposed. They guide the choice of the methods for data collection and analysis, assist with the interpretationsof the findings,and supportthe development of new ideas. Therefore,bringingtheoreticalperspectivestogethermay not always be possible. But, should we even aim at, and embrace, the idea of harmonising differenttheoreticalapproaches?An epistemologicalissue emerges then thatis crucialto the advancementof researchin generaland to the field of mathematicseducationin particular.The two interpretationsof the lesson we presentedabove fit well with Guba and Lincoln's (1994) reflections on the relationsbetween theory and researchfindings. Not only theoretical perspectivesframethe posing of researchquestionsand the choice of the methodology,but, as mentionedat the beginning of this paper,Guba and Lincoln claim that "theoriesand facts are quite interdependent- that is, that facts are facts only within some theoreticalframework"(p. 107).


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Theoryandresearchare then trappedin a vicious circle. As was illustrated in this article, researchfindings are interpretedin ways that confirmthe theories that serve as research lenses, and correspondinglysupportthe theories.Ourpropositionto harmonisethe two approachesis in line with the directioncurrentlyembracedby leading mathematicseducators.But perhapsthis is the pointwherewe need to ask ourselves,as a researchcommunity,whethersuch an approachindeed opens possibilitiesfor breaking the vicious circle. Would integrationof differenttheoreticalperspectives allow us to develop new ways of thinkingthat breakthe bordersof, and give rise to, theoriesthatare radicallydifferentfrom currentones? Should we at all aim at the developmentof radicallynew theories?Is this even possible? NOTES 1. The term representativedesignates any sign standing for a mathematicalobject (a function, a geometricalfigure, etc.) As mathematicalobjects are non-material,these signs appearin mathematicalrepresentations(what Kaput [1992] calls notationsystems). Representativesof the same mathematicalobject generallycannotdirectlydisplay all the propertiesof this object:for example,graphicalandtabularrepresentatives of infinite functions are partial.Different studies have shown that the manipulation of various representativesfrom different representationsvery often fosters concept learning(Schwarzand Dreyfus, 1995; Schwarzand Hershkowitz,1999). 2. There are other substantialdifferencesbetween protocol analysis and verbalanalysis. First, the techniquefor collecting verbaldata in protocol analysis is thinkaloud protocols. Experimentalstudies have shown thatthis techniqueenhanceslearning.It was thus abandonedin verbalanalysisin which collection of datais done in interviewsor in problem-solvingsessions. Secondly,protocolanalysistends to model humanbehavior througha computationalmodel whose elementsarepre-definedin contrastwith verbal analysis in which elements stem from the analysis of data. For other differences,see Chi (1997). 3. We are awarethat the identificationis generallyproblematic.It was possible here because the discoursein the class was generallynot argumentative.The groupfunctioned as if there was complete agreement among participants(although it was probably partly a way to socialize). Had the discourse been argumentative,we should have distributedthe analysis among participants.In such a case, the analysis would have been much more complicated. 4. There is a contrastbetween protocol analysis for which the coding scheme is predefined and verbal analysis for which, like here, the coding scheme stems from the researchquestion.It is clear,though,thatcoding schemescreatedfor specific research purposesareapplicablein otherresearchstudieswith the same purposes.Forexample, the coding schemes initially created to observe the effects of self-explanationson learning (Chi, Bassok, Reimann and Glaser, 1989) were used and refined in other studiesto broadenthis observation(Chi, de Leeuw et al., 1994;Neumanand Schwarz, 1998, 2000). 5. The identificationof all the studentsto one single entity may seem here problematic. When we claim 'studentsoppose' or 'studentsare puzzled', we are aware that only

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some of them opposed and thatsome expressedpuzzlement.However,when students neither oppose nor show puzzlement, yet do not react to the interventionsof their peers, they take a (passive) decision to accept to be representedby these peers. In this sense, the term 'students'is justified.

REFERENCES Chi, M.T.H.: 1997, 'Analyzing the content of verbal data to representknowledge: A practicalguide, TheJournalof the LearningSciences 6, 271-315. Chi, M.T.H.,Bassok, M., Lewis, M., Reimann,P. and Glaser,R.: 1989, 'Self-explanations: How studentsstudy and use examples in learningto solve problems',CognitiveScience 13, 145-182. Chi, M.T.H.,de Leeuw, N., Chiu, M.H. and LaVancher:1994, 'Elicitingself-explanations improvesunderstanding',CognitiveScience 18, 439-477. Cobb, P., Stephan,M., McClain,K. andGravemeijer,K.: 2001, 'Participatingin classroom mathematicalpractices',TheJournalof the LearningSciences 10(l&2), 113-163. Eriksson and Simon, H.: 1984, Protocol Analysis: VerbalReports as data, MIT Press, Cambridge,MA. Even, R.: 1998, 'Factors involved in linking representationsof functions', Journal of MathematicalBehavior 17(1), 105-121. Even, R. and Lappan,G.: 1994, 'Constructingmeaningful understandingof mathematics content', in D.B. Aichele and A.F. Coxford (eds.), Professional Developmentfor Teachersof Mathematics,1994 Yearbook,NCTM, Reston, VA, pp. 128-143. Goldman,A.I.: 1976, 'Discriminationand perceptualknowledge', Journal of Philosophy LXXIII(20),771-791. Guba,E.G. andLincoln,Y.S.: 1994, 'Competingparadigmsin qualitativeresearch,in N.K. Denzin and Y.S. Lincoln (eds.), Handbookof QualitativeResearch,Sage, CA, pp. 105117. Hershkowitz,R., Dreyfus,T., Ben-Zvi, D., Friedlander,A., Hadas,N., Resnick,T., Tabach, M. and Schwarz, B.B.: 2002, 'Mathematicscurriculumdevelopment for computeractivity,'in L.D. English (ed.), ized environments:A designer-researcher-teacher-learner Handbookof InternationalResearchin MathematicsEducation:Directionsfor the 21th century,KluwerAcademic Publishers,pp. 657-694. Janvier C.: 1978, The Interpretationof Complex Cartesian Graphs RepresentingSituations: Studiesand TeachingExperiments,Doctoraldissertation,Universityof Nottingham. Janvier C. (ed.): 1987, Problems of Representationin the Teachingand Learning of Mathematics,LawrenceErlbaumAssociates, Hillsdale, New Jersey. Kaput,J.J.: 1992, 'Technologyand mathematicseducation',in D.A. Grouws(ed.), Handbook of Researchon MathematicsTeachingand Learning,NationalCouncil of Teachers of Mathematics,Reston, VA, pp. 515-556. Kieran,C.: 1992, 'The learning and teaching of school algebra', in D.A. Grouws (ed.), Handbookof Researchon MathematicsTeachingand Learning,McMillan,New York, pp. 390-419. Lampert,M.: 1990, 'When the problem is not the question and the solution is not the answer:Mathematicalknowing and teaching',AmericanEducationalResearchJournal 27(1), 225-282. Leinhardt,G.: 1987, 'Developmentof an expertexplanation:An analysis of a sequenceof subtractionlessons', Cognitionand Instruction4(4), 225-282.

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Leinhardt,G.: 1989, 'Mathlessons: A contrastof novice and expertcompetence',Journal for Researchin MathematicsEducation20(1), 52-75. Leinhardt, G. and Greeno, J.G.: 1986, 'The cognitive skill of teaching', Journal of EducationalPsychology78(2), 75-95. Leinhardt,G. and Schwarz,B.B.: 1997, 'Seeing the problem:An explanationfrom Polya', Cognitionand Instruction15, 395-434. Leont'ev, A.N.: 1981, 'The problemof activity in psychology', in J.V. Wertsch(ed. and trans.),The Conceptof Activityin Soviet Psychology,Sharpe,Armonk,NY, pp. 37-7 1. Neuman,Y. and Schwarz,B.B.: 1998, 'Is self-explanationwhile solving problemshelpful? The case of analogicalproblemsolving', BritishJournalof EducationalPsychology68, 15-24. Neuman, Y. and Schwarz, B.B.: 2000, 'Substitutingone mystery for another:The role of self-explanationsin solving algebra word-problems',Learning and Instruction10, 203-220. Schoenfeld, A.: 1988, 'When good teaching leads to bad results:The disastersof "well taught"mathematicsclasses', EducationalPsychologist23, 145-166. Schwarz,B.B. and Dreyfus, T.: 1995, 'New actions upon old objects: A new ontological perspectiveof functions',EducationalStudiesin Mathematics29, 259-291. Schwarz, B.B. and Hershkowitz,R.: 1999, 'Prototypes:brakes or levers in learning the function concept? The role of computertools', Journalfor Research in Mathematics Education30, 362-389. Schwarz, B.B. and Hershkowitz,R.: 2001, 'Productionand transformationof computer artifactstowardsthe constructionof mathematicalmeaning',Mind, Cultureand Activity 8(3), 250-267. Spiro, R.J., Feltovitch, P.J., Coulson, R.L. and Anderson, D.K.: 1989, 'Multiple analogies for complex concepts: antidotesfor analogy-inducedmisconceptionsin advanced knowledgeacquisition',in S. VosniadouandA. Ortony(eds.), Similarityand Analogical Reasoning,CambridgeUniversityPress, New York,NY, US, pp. 498-531. Sweller, J. and Cooper, G.A.: 1985, 'The use of worked examples as a substitutefor problemsolving in learningalgebra',Cognitionand Instruction2, 59-89. Sweller,J. andChandler,P.: 1994, 'Why some materialis difficultto learn', Cognitionand Instruction12(3), 185-233. Wertsch,J.V.: 1991, Voicesof the Mind,HarvesterWheatsheaf,London. Yackel, E. and Cobb, P.: 1996, 'Sociomathematicalnorms, argumentation,and autonomy in mathematics',Journalfor Researchin MathematicsEducation27, 458-477.

1Departmentof Science Teaching, WeizmannInstituteof Science, Rehovot76100, Israel E-mail:[email protected] 'School of Education, TheHebrewUniversity, Jerusalem91904, Israel E-mail: [email protected]

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