The Role of Reflection in Math Learning

7/30/2019 The Role of Reflection in Math Learning

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The Role of Reflection in Mathematics LearningAuthor(s): Grayson H. WheatleySource: Educational Studies in Mathematics, Vol. 23, No. 5, Constructivist Teaching: Methodsand Results (Oct., 1992), pp. 529-541Published by: SpringerStable URL: http://www.jstor.org/stable/3482851 .

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GRAYSON H. WHEATLEY

THEROLEOFREFLECTIONNMATHEMATICSEARNING

ABSTRACr.Reflective abstractions central o the theoryof constructivism s putforthby vonGlasersfeld. n"coming oknow",personsmakemajor ognitiveadvancesby taking heiractionsasobjects of thought Leamersmove beyondbeing"in the action"whenthey engagein reflection.Thereare serious limitations n the 'explain-practice"methodof instruction ndactive learing.Performing ven self-generatedmathematical perationsdoes not have the powerwhichresultsfromreflectingon theactivity.Problem-centeredearning, n instructionaltrategywhich hasbeenshown to providerichopporunities or reflection,s exanmined.he natureof reflection n mathe-maticalactivity s alsoconsidered.

The thesis of this paperis thatreflectionplays a criticallyimportant ole inmathematicsearningandthat ustcompleting asks s insufficient.An instruc-tionalstrategywhichencourageseflection s described, ndthe role of commu-nication and assessment in problem-centered learning and its effect onreflection is considered.Evidence is presented that encouragingreflectionresults n greatermathematics chievement, ven on standardizedests whichstressprocedures ndparticularonventions.

PROBLEM-CENTERED LEARNINGProblem-centeredearning s an instructionaltrategybased on constructivism(von Glasersfeld,1987). In particular,t is assumedthatstudentswill givemeaning o theirexperiences n idiosyncraticwaysandthatattempts o imposemathematical rocedures e.g., subtractingwhole numbersusing the standardregroupingmethod)areineffectiveand,in fact,detrimental.t is also assumedthat earnings facilitated y opportunitieso communicatendnegotiatemath-ematicalmeaning.Mathematicss not "outthere n the realworld"but is thelearner'sorganizingactivity.Self-constructedmathematicalelationships reviableif they standthe test of subsequent xperience,whichincludes nterac-tionswithothers.Teachersn theMathematics earningProjectat theFloridaStateUniversityLaboratorychooluseproblem-centeredearng as theirprimarynstructionalstrategy(Wheatley,1991). In problem-centeredearning, he teacherselectstasks fromresourcematerials not a textbook)which she believes could beproblematic or the students,or she may devise tasksfor a specialpurpose.Thesetasksarepresentedwith a minimum f teacher alkandthestudentshenwork on the tasksin pairs.The students n pairsare at a similarstage in theirEducational udies inMathematics 3:529-541, 1992.0 1992KluwerAcademicPublishers.Prined in theNetherlands.

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530 GRAYSONH. WHEATLEY

mathematicaldevelopment.The pairs attemptto reach consensus on theiranswers hroughnegotiation.In the smallgroupsetting, herearerichopportu-nities for students o constructmathematicalelationships.Afterabout30 min-utes, the teacherorganizes he class for a presentationf solutionmethodsbythe students.The presentations re made to the class, not to the teacher.Theclass has theobligationof tryingto understandheexplanations, nd thismayinvolve requestsfor clarificationwith ensuing negotiationsbetween students.Theteacher's olein this whole-classdiscussions to facilitate henegotiations.At timesshemay uxtapose womethodsor framean issuefordiscussion.Sincethe teachercan attachrelative mportanceo the various ssues, she canguidethe discussion n whatshe believes to be morefruitfuldirections.Throughoutthe small-group nd whole-classdiscourse he staunchlymaintains non-judg-mentalstance.In thiswaystudents andevelop ntellectual utonomy ndneednot try to "read" heteacher o learn he"right"way to do thetasks.Extensiveuse is madeof non-routineasksandevery effort s madeto encourage tudentsto reflect on theirsolutionactivity.A majorgoalof problem-centeredearningis tocreatea culture orinquiry Richards, 990).

Problem-centeredearningcontrastssharplywith the "explain-practice"methodof instructionwhich s standardn traditionalchoolmathematics.Thelimitationsof such directinstructionhave been well documentedby Confrey(1990). Furthermore, roblem-centeredearning is not to be confusedwith"active earning" r what s sometimes alled"hands-onmath"n whichmanip-ulativesare usedto helpstudents earnplace valueandalgorithmic rocedures.A salientfeatureof problem-centeredearning s listening to students.Inorderto select tasks,we musthavesome idea of students'mathematicalon-structions. teffe (1990) insists thatmathematicseachersmustattempt o con-struct the mathematicalknowledgeof theirstudents;we must adapt to thestudentsrather hanexpectthe students o do the adapting.Taskswhichhavelearningpotentialarebest formulated y basingdecisionson ourconstructionsof thestudents' urrentmathematicalnowledge atherhanon a fixedcurricu-lum.Forexample,we wantGrade2 students o be able to finddifferencesofwholenumbers, ut to expectthemto do so whenthey havenotconstructedenas anabstract nitis anexercise n futilityandfrustrationorbothstudents ndteachers.Many of the mathematical ifficulties tudentshave in schoolresultfromtheimposition f methodswhichcannotbe understoody them.Oncestu-dentsconstruct en as anabstractterableunit,addition ndsubtraction f two-digitnumbers reeasilyaccomplished.



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THE ROLE OF REFLECTION IN MATHEMATICS LEARNING 531ASSESSMENT IN PROBLEM-CENTERED LEARNING

Informed rofessional udgmentIn theMathematics earning roject, eachers ssesspupilsusing nformed ro-fessional udgmenLGradesarenotgivenfordailyworkand testsarenotadmin-isteredexceptfortherequired tateandnationalassessments.nfact,at notimedoes the teachercommunicate judgmentof the students'mathematics. hedoes notcollectworksheets,mark he ones thatarewrong,or have students or-rect their "mistakes".nstead, he teacherkeeps notes of students'activityinwhich she considerstheir persistence,confidence, co-operation, ommunica-tion,andthequalityof theirmathematicalonstructions.In problem-centeredearning, he teacherhas many opportunities o learnabout hequalityof students'work.She observesstudents' nteractionn pairsas they engage n thedailytasksand as they present heirsolutions o the classfordiscussion.Shealsoarrangesimeto talkwithstudentsndividuallyndthusgain nsights nto theirmathematicalonstructions.Onone occasion,as I was observingGrade2 studentspresentingheirsolu-tions, Mrs.Jones,the teacher,hearda student hinking ut loud. Tracywas notpresentingherideas to another, he was just taLkingo herself abouthow shedid the problem.Mrs.Jonessaid to me, "Didyou hearwhatTracy ust said?That s the fist timeTracyhas used tenin sucha powerfulway".This teacherwas so in tunewith thereasoningof her 27 pupilsthatshe noticed hatone ofthemhad madea significant onstruction. he noted his advancen herlog forlateruse. Rather han relying on test grades andaveragesof daily scores onworksheets, his teacherassessed pupils by using her informedprofessionaljudgment.And she was inforrned ecauseshe arranged pportunitieso learnabout the quality of her students' work. Her assessment was professionalbecauseshe wasknowledgeable boutelementary choolchildren'smathemat-ics, in this case the importance f constructing en as an abstractunit. Mrs.Jones asserted hatshe had muchgreater onfidence n her assessmentof stu-dents'mathematicshanever beforebecause hecouldnow describe o a parent,or to anyoneelse, what herstudentscouldandcould not do. In problem-cen-tered earning,eachers lsohavestudentskeepa portfolioof theirmathematicsworkwhich ncludes amplesof dailyactivitiesas well as specialprojects.Standardized estsThe data show that elementary tudentsengaged n problem-centeredearningdevelop greatermathematicalompetence hanstudents aughtusing the con-ventionalexplain-practicemethod even on standardized ests which feature



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532 GRAYSON H. WHEATLEY

computation nd standardypesof problems Cobb,Wood,Yackel,Nicholls,Wheatley,Trigatti, ndPerlwitz,1991;KamiiandLewis, 1991;Nicholls,Cobb,Yackel,Wood,andWheatley,1990).In theMathematics earningProject, heCaliforniaTestof Basic Skills wasadministeredo two Grade3 classesof 27 studentswhohadbeenusingproblem-centeredeaming or twoyears.Thegradeplacementwas 3.7 on computations,6.0 on concepts,and4.4 on the section labeledproblemsolving.Theirtotalmathscore of 4A was 1.4 grade evels above the normof 3.7. Prior o usingconstructivism sa basisforprogram evelopment,hescoresat thisschoolhad

beenbelowgrade evel on all subsections f thetest Thesefindingsare similarto theresultsof thestudiescited above.Elementarychoolchildren ngaged nproblem-centeredearningcompare avorablyon standardizedests with stu-dentseducatedna moreconventionalmanner.Because of the importanceattached to such test scores, they cannot beignoredand are thusreportedn thispaper.However, t is not at all clearthatthe available standardized ests are an accurateindicatorof mathematical

knowledge.Kamiiand Lewis (1991)have shownthatsuch resultscanbe quitemisleading.Theyinterviewedtudentswhohadperformedwell onstardizedtests andreported hatfrequentlyhesestudentsassociated ittle meaningwiththeircomputational rocedures; hey were unableto respondmeaningfullyonon-routine roblemswhichrequiredmore thanthe use of a set procedure.ncontrast,studentswho had not scoredparticularlyhigh on the tests but hadexperienced meaning-basedmathematics rogramwereableto do quite wellwithnon-routineroblems.

PersistenceOne factor to consider n assessinga pupil's mathematicss persistence.Towhatextentdoes a student taywith a problem?norder odo mathematicst isnecessary o engagein a task on a sustainedbasis.Mathematicss not a set ofdiscrete acts andprocedures utaninterconnectedetof relationships. hus, tis necessary o investconsiderablenergyandattentionnmaking ense of a sit-uation,workingout a solution,andassessingthereasonableness f theconclu-sion. Doing mathematicsrequirespersistence, and any assessment systemshould ake ntoconsiderationhedegree o whichstudents repersistent.Math-ematicalpowerresults romanunusualnvestment f attention vera consider-able periodof time. In the Mathematics eaning Project,we havefoundthatstudentsbecomemuchmorepersistent,oftenworkingon a single taskfor anentireperiod.



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THE ROLE OF REFLECTION IN MATHEMATICS LEARNING 533EFFECTS OF DIRECT INSTRUCTION

The widespreaduse of the term "application"uggests that mathematics sviewed by many as a formal system to be learnedin the abstractand then"kappliedo the realworld".Studentsdo not easily give meaning o abstractor-mulationswhichare not in their experience.As learners,we give meaning onew experiencesn termsof ourpreviousexperiences.When mathematicalor-mulations low frommeaningful xperiences, tudentsare able to constructamathematicalworld whichis coherentanduseful. Rather hanbeginningwithabstract ormulationso real worldproblems,mathematicsnstruction ouldbegin withsituations romwhichgeneralizationsan be made.Oftenthe use of manipulativess supposed o makethe abstractornulationsof mathematicsomprehensibleo students.Using concreteobjects to "show"studentsa mathematicalonceptor relationships still based on the "abstract-first"conceptionof learning.When,as Gravemeijer1990) suggests,emphasisis placed on mathematizing rom potentiallymeaningfulsituations,studentshave the opportunityo construct xperience-based nowledge.Childrenwhohave the opportunityo experience uchleaming environmentsreunlikely oact as Talithadid in the examplebelow. In mathematicsearning, he intentionto makesense is essential Erlwanger, 973). Neither heabstract-firstorpro-cedures-first pproach o learning ostersthe intention o makesense. As anexample of the effects of direct instruction,consider Talitha's mathematicsworldas inferredroma clinical nterview.

Talitha, Grade5 studentwho hadexperiencedonventional,extbook-basedmathematicsnstruction,was interviewed o determinehe natureof her math-ematics ShawandJakubowski, 991).Talithawas considered y herteacher obe amongthe top 15percentof the class and was indeedproficientn carryingoutalgorithmic rocedureswith wholenumbers.Forexample,during he inter-view she confidently omputed,using the standard borrowing"lgorithm, hedifferencebetween2,005 and 1,237. Butheractionon a subsequent askindi-cated thatshe was not makingsense of her computational ctivity.The task"235 minus341" waspresented erbally.She wrote t down in a vertical ormatandbegan"applying"he method orperformingmulti-digit ubtractionhehadbeen trained o use in subtractingwhole numbers see Figure 1). She did notattemptoconsider hemeaning f thequestion.Shedid not say that t wouldbeimpossible assuming he set of wholenumbers), rsay that he resultwouldbea negativenumber;heblindlybeganusinga procedure he hadbeen trained ouse- a procedurewhichapparently ad ittle if anymeaning or her.

After she had successfully performed he subtractionn the ones and tenscolumns,she said,"Youcan't takethree rom one so you have to borrow romhere [13 tens]". Since there was nothing to the left of the one in the hundreds



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placeto borrow rom and sheknewshe hadto borrow, he borrowed romtheright! Her answerwas 884. When she was laterasked about this she said,"Maybe shouldnot haveborrowedromthe 13tensbut from he5 ones".It isclear thatTalitha s in a foreignland whenaskedto thinkaboutmathematics.She is not makingsense of heractivity; n fact, there s no intention o makesense of mathematics. orher,that s.nothow thegameis played.As a resultofschoolinstruction,he sees her taskas followingdirections ivenby theteach-er.Direct nstructionwas not effective n helpingTalithado mathematicsmean-ingfully.

Fig. 1. Talitha's computation.

ACTIVE LEARNINGRecentlyhere asbeenanattemptomoveaway romdirectnstruction.omeeducatorsavecome o seeteachers'emonstrationsf proceduress ineffec-tiveandhaveadvocatedctiveearningNational ouncil fTeachersfMath-ematics,989).Oftenwhenmanipulativesreusednteachingmathematics,heteacheremonstratesheway heyare obeusedand tudentsre eft ittle ree-dom ogivemeaningo theexpenencen waysthatmake ense o them;heway thematerials reto be used is prescribed.There s the mistakenbeliefonthe part of the teacherthat the mathematics s apparent n the materials, orexample,"base en"blocks(Cobb,Yackel,Wood,1992).This is basedon thebeliefthatmathematicss "out here" ndthatmodels"show"heconcepts.Thedemonstrationithconcretematerialss quiteappealingecauseheconceptsareso vivid orthosewhohavealreadymade heconstruction.hus, heresthemistakeneliefthatsincewe, as adults, ansee themathematicsn theblocks,hestudents illtoo.But he"seeing"equiresheveryconstructionheactivitys intendedoteach.



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THE ROLE OF REFLECTION IN MATHEMATICS LEARNING 535

The use of manipulatives oes not always promoteunderstandingnd theconstructionf meaning.Forexample, f students reaskedto calculate34 takeaway 18,they mightusemanipulativeso obtain heiranswer.Theymight1) countout 34 blocks;2) remove18 blocks;3) countthe number emaining; nd4) writetheanswer16.Whilesuchactivitymaybe meaningful ndresult n ananswer, t doesnot nec-essarily lead to die development f thinking trategiesandefficientcomputa-tionalprocedures.Whilestudents nvolved n thistypeof activitymaybe lesslikely to be justmanipulatingmeaningless ymbols, herearefew opportunitiesfor learning.They may very well be routinizinga primitiveand inefficientmethod oradding wo wholenumbers for example,countingout 14objects,countingout 17 objects,andthencounting hemall. Incontrast,f thelearingenvironmentncourageshestudentso devise theirownmethodsandtaketheiractivityas anobjectof thought,hentheir nitialprimitivemethodsmaybecomevaluable as a basis for the construction f othermathematical elationships.Withoutthe intention o give meaningto theiractivity,studentsmay just begoing through he motions and not doing mathematics.Only when studentsform he intention f making enseof theirexperiences o they ointhe commu-nity of inquirers ndeffectively akechargeof theirownlearning.

ASPECTS OF PROBLEM-CENTERED LEARNINGReflectionIn mathematicsearning, eflection s characterized y distancingoneself fromthe actionof doingmathematicsSigel, 1981).It is onethingto solvea problemand it is quiteanother o takeone's own actionas anobjectof reflection. n theprocess of reflection,schemes of schemesare constructed a second-orderconstruction. ersonswho reflecthave greater ontrolover theirthinkingandcandecidewhichof severalpaths otake,rather han implybeing n the action.

It is notenough orstudents o complete asks;we mustencouragetudents oreflect on theiractivity.For example,being askedto justify a methodof solu-tion will oftenpromotereflection.This may occur in the small-group ettingwhena learningpartner sks, "Willthatwork?" r it may occur in the whole-class discussionwhen thepresenter s asked to clarifyan explanation. inally,carefully elected askscancauseperturbation hichresults nreflection.Whensolversreflectonsolutionactivity hey"distance"hemselves romthe



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activityand"hold heactivity n thought"Sigel, 1981).In thisway, theymaketheir activityan objectwhich can be examined. As Cifarelliobserved n hisstudyof problem olvers,Reflectionsmadepossiblesolvers'anticipationsnd n manycasessubsequentolutionactivity edto higher evel reflections .. anticipations f potentialproblematic ituationsgave rise to novelsolutionactivityandeventuallyed solvers ohighlevel reflections.Cifarelli,1988,p. 195)It is possible thatstudentsmaybe so active thattheyfail to reflectand thusdonot learn.We can keep studentsso busy thatthey rarelyhave time to thinkaboutwhattheyaredoing,andtheymayfail to becomeawareof theirmethodsandoptions.In fact,there s an implicitmessagethattheyarenotsupposed othinkaboutwhattheyaredoing.Rarelyare American tudentsencouragedoreflect on their mathematicalactivity. As Stigler andPerry (1990) report,"Japanesetudents,n particular,penda fargreater mountof timethaneitherChineseorAmerican hildren ngaging n reflectiveverbalization boutmathe-matics" p. 52).

Once studentshaverepeatedlydetermined ifferencesof whole numbers ycountingout blocks andarefaced with a taskfor which theirprimitivemethodis inefficient for example,92 take away37 - opportunities nd encourage-mentto reflecton theiractivitymay leadto the construction f an efficientself-generated algorithm. By becoming conscious of their organizing activity,students an moreeffectivelymodifytheiractions.Theeffects of doingwithoutreflection re seenin the examplebelow.

The followingproblemwas presented o a Grade3 student:"You have 24.How manythreeswill you need to have36?".Sue countedby threes rom24 to36, keeping rackof thenumber f threeson herfingers.She reportedouras ananswer.She was able to solve the problemusinga meaningfulmethod.Whenasked to explainwhatshe did,herresponsewas to repeat heaction.She wasnot able to takeher activityas an objectof thought;hat s, she was not able tothinkaboutheractivity.She was "inthe action"rather han"reflecting n heraction"SchOn, 983).

As an exampleof reflectionin mathematics earning,consideran episodewithJim,a beginningGrade3 student.My goal in the interviewwas to engageJimin mathematical ctivityso thatI could inferhis mathematicalchemesfordealingwith whole numbers. beganwitha mentalarithmetic ask and pro-ceeded withthe intention f findinga task whichhe would take as a problem.Theprotocols shownbelow.Interviewer: What s 21 takeaway19?

Jim: One ... no, TWO!I: What s 31 takeaway28?



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THE ROLE OF REFLECTIONIN MATHEMATICSLEARNING 537

J: 12.I: What s 31 takeaway 29?J: 11.I: What s 31 takeaway30?J: [Longpause]One?

WhenJim was asked,"Whats 31 takeaway30",he paused orquitea while.Since he hadpreviouslyanswered11, to 31 takeaway 29, he was confrontedwith a perturbationn his activity.For some timebeforehe spoke,he wasobvi-ously deepin thought.I inferredhathe wasanticipatinghe resultof potentialactivities. He was reflecting, that is, modifying, his cognitive structures(schemes) o account or the conflict n hisresponseswhichhe nowrecognized.When he finally responded,he said "one"in a quizzical way, as if to say,"Basedon whatI said before,it should be 11 but I know it is one". He wasrestructuring is schemes,performinga majorreorganization.When he wasasked again, "What s 31 takeaway 29?",he responded, Two ... I've got itnow".He hadre-establishedquilibrium, nd thetensioncreatedby thecogni-tive conflict was reduced.He hadbecomeawareof his two ways of thinkingabout he task.

Later n the interview,I placedsix cubes in frontof him andasked,"Howmany?"Hecounted hemandresponded six".Next,I placeda tenrod with thesix cubes and asked "How manynow?". He had previouslyestablishedbycounting hatthe rodrepresenteden. Therewasa long pauseduringwhichhethought ntentlyabout hetask.We couldsay thathis reflectionwas causedbyhis anticipation.He couldhave easily determined he numberby countingbyones,buthe wasnot satisfied o obtain heanswerby thatmethod. believe hedid not immediatelycountby ones becausehe was in the processof givingmeaning o tenas more than ust a namein a sequence.He was attempting oanticipate he resultof one or more potentialactionsbut was unableto antici-pate the resultof any particularction.Jim couldnot think,"sixand ten more ssixteen",as an adultmightdo, becauseten didnot exist as an iterableunit forhim.After helong pausehe counted he six ones,countedup six on the ten rod,said thatsix and six are 12, andcounted 13, 14, 15, 16. Manyfactorsplayedinto his decisionto determine ow manymadeup thisparficularollection.Hecould havefoundoutby countingby oneorperhaps wo. Hecould have startedwithten andcountedup six to arriveat sixteen.Becauseof thelong pauseandtheobviousconcentration, inferthathe was formulating plan.He may alsohavebeen consideringwhatwould be appropriaten the situation how hisactionswouldbe perceivedby me. Inany event,he was clearlythinkingabouthis thinkingandanticipatingheresultsof potential ctions.Thisis an example



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of a learner ngaged nreflection.As anotherexampleof an instructionalettingwhichpromotesreflection,consider heQuickDrawactivityusedin the Mathematics earningProject. nthe QuickDrawactivity,a complexgeometric igure (see Figure2) is brieflyshown to the students,and they are asked to drawwhat they see. Thentheteacherasks, Vhat didyousee andhowdidyoudraw t?".In thiswaythe stu-dents' mathematicsare acknowledged.It is quite a different matterfor the

teacher o ask, "DidyourdrawingmatchwhatI showedyou?Did you get itright?".Thelanguage animplythat here s justonewayof thinkingabout hefigure (the teacher'sway) and thatthe students' ask is to see it the teacher'sway. Thequestion"Whatdid you see?"encourages tudents o give meaningotheirexperiencen waysthatmakesenseto them.As a varietyof interpretationsarepresentedby the students, he teachercan express delight at the diversityandcreativity n thechildren'smathematics.

Forexample, heshape n Figure2 was shownto Grade4 students ndtheyreported eeing (1) a cube, (2) a hexagonwitha Y in it, and(3) two diamondsandtwo triangles.As studentspresented lternativenterpretations,heelementof surpriseat anothers'constructioned to reflection, and studentsbegantoreport till otherwaysof seeingthefigure.Therewas anexpressionof amaze-ment andjoy in realizing he figurecould be viewed in so many ways.Whentheteacherresponded on-judgmentally,ot"pushing" ne interpretationveranother, he studentswerefree to constructheirown mathematics ather hantryto determinewhattheywere supposed o do. In thisactivity, he discoursepromotedeflection.

Fig. 2. Quick Draw shape.

JUSTIFICATIONOptimalconditions or learning xist when the individualmustdefenda posi-



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THE ROLEOF REFLECTIONIN MATHEMATICSLEARNING 539

tionhe or she has taken.However,social normsmustbe negotiatedbeforethepotential an berealized.1) The groupmembersmustassumetheobligationof trying o makesense of

theexplanation.2) Personspresentinga solutionor explanationmustpresenta self-generatedsolution.

3) Solutionsmust be presentedn a "proofsand refutations"etting(Lakatos,1976). That is, group membersrecognize the obligation to constructaresponse to any challenge to theirexplanation an explanationwhichincorporates constructionf thequestioner.

4) Thepurpose f thedialogue s notto be rightbut to makesense.5) The purposeof any questionraised by a memberof the groupis to givemeaning o theexplanation;t is a sincereandgenuinequestion.While it may requireseveral weeks or even months o negotiate hese socialnorms,onceaccomplished,earnings greatly acilitated. n theprocessof justi-fying a solutionmethod, eflectiononactivity s natural.

SUMMARYThe limitationsof the conventional xplain-practicemethodof instruction rebeingwidely recognized.Confrey 1990) has described he effectson learnersof such direct nstruction. hawandJakubowski1991)describedhow Talithaoperated n numerals singruleswhichapparently ad ittlemeaningorher.Inattempting o overcome the limitationsof directinstruction, ducatorshaveoften movedto active earning.Yet simplyhavingstudents ngage n meaning-ful activitiesdoes notalwaysresult n a desirable evelof mathematicalompe-tence.Mathematicsearningdoes notalwaysresult romdoing,no matterhowwell theactivitiesaredesigned.However, t hasbeen argued n this paper hatestablishinga learningenvironmentn whichreflectingon actions s encour-agedcanbe quiteeffective.Establishing nenvironmentwhichencourages eflection s a complexenter-prise.Inaddition o selectingtasks,the teachermust negotiate ocial norms nwhichstudents anbecomea community f inquirers. othstudents nd eachermust learnto talk mathematicsand learn to listen. Effective mathematicsinstruction1) promotes utonomy ndcommitment;2) is basedontheoreticalmodelsof children'smathematicsearning;3) assumes hatmeaningmustbe negotiated;



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4) has sense makingas a goal;5) involvesthe negotiation f social norms; nd6) encourages ndfacilitates eflection.We have solid evidencethat nstructionwhich has reflectionas a primary om-ponentenablesstudents o construct obustmathematicalelationships. tudentswho haveexperienced roblem-centeredearning, n which reflection s central,areable to solve non-routineroblems ndto construct ew knowledge.

REFERENCESCifarelli,V. V.: 1988, TheRole of Abstraction s a LearningProcess in Mathematical roblemSolving.Unpublished octoraldissertation, urdueUniversity,WestLafayette,N.Cobb, P., Wood,T., Yackel,E., Nicholls,J., Wheatley,G., Trigatti,B., andPerlwitz,M.: 1991,"Assessmentof a problem-centeredearningsecond-grademathematics roject",Journal orResearch nMathematics ucation 22, 3-29.Cobb,P., Yackel,E.,andWood,T.: 1992,"Aconstructivistltemativeo therepresentationaliewof mindin mathematics ducation", ournal or Researchin MathenaticsEducation,23(1),2-33.Confrey,J.: 1990, 'Whatconstructivismmpliesforteaching", ournal or Research nMathemat-icsMonographNo. 4, 107-122.Erlwanger, .: 1973, "Benny'sperceptionof rulesandanswers n IPImathematics",ournalofMathematical ehavior1(2), 1-26.Gravemeijer,K.: 1990,"Context roblemsandrealisticmathematicsnstruction."n K. Gravemei-jer, M. van denHeuvel,andL Streefland,eds.), ContextFree ProductionTestsand Geometryin RealisticMathematicsEducation,ResearchGroup or MathematicalEducation ndEduca-tionalComputerCentre,StateUniversity f Utrecht,TheNetherlands.Kamii,C. andLewis,B.: 1991, "Achievementests in primarymathematics: erpetuatingower-order hinking", rithmetic eacher38(9),4-9.Lakatos, .: 1976,ProofsandReftaions, Cambridge niversityPress,Cambridge.NationalCouncilof Teachers f Mathematics, ommission n StandardsorSchoolMathematics.:1989,CurriculumndEvaluation tandards,Author,Reston,VA.Nicholls,J., Cobb,P., Yackel,E.,Wood,T., andWheatley,G.: 1990,"Assessingyoungchildren'smathematicalearning". n G. Kulh (ed.), AssessingHigher OrderThinkingn Mathematics,AmericanAssociationortheAdvancementf Science,Washington,DC,pp. 137-154.Richards, .: 1990, "Mathematicaliscussions".n E. vonGlasersfelded.),RadicalConstructivisminMathematics ducation,KluwerAcademicPublishers,Dordrecht.Schon,D. A.: 1983,TheReflectivePractitioner,BasicBooks,New York.Shaw,K. L. andJakubowski, .:1991, "Teachershangingorchanging imes",FocusonLearningProblemsnMathematics, 3(4), 13-20.Sigel, L.A.: 1981,"Socialexperience n the development f representationalhought:Distancingtheory."nI. E. Sigel, D. M. Brodzinksy, ndR. M. Golinkoff eds.),NewDirections nPiage-tianTheory ndPractice,LawrenceErlbaun,Hillsdale,NJ.Steffe, L.: 1990,"Adaptivemathematicseaching", n T. CooneyandC. Hirsch(eds.),TeachingandLearningMathematicsn the 1990s.NationalCouncilof Teachers f Mathematics, eston,VA, pp.41-51.Stigler,J. W. and Perry,M.: 1990, "Mathematicseaming in Japanese,Chinese,andAmericanclassroons"InG. B. Saxe andM. Gearharteds.),Children'sMathematics. ewDirectionsorChildDevelopment, o.41. Jossey-Bass,SanFrancisco, p.27-54.



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THE ROLE OF REFLECTION IN MATHEMATICS LEARNING 541von Glasersfeld,E.: 1987, TheConstruction f Knowledge, ntersystems ublications, easide,CA.Wheatley,G.: 1991, "Constructivist erspectiveson scienceand mathematicsearning",Science

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