Reflective Discourse and Reflective Colective

8/3/2019 Reflective Discourse and Reflective Colective

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Reflective Discourse and Collective ReflectionAuthor(s): Paul Cobb, Ada Boufi, Kay McClain, Joy WhitenackSource: Journal for Research in Mathematics Education, Vol. 28, No. 3 (May, 1997), pp. 258-277Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749781 .

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Journal orResearch n MathematicsEducation1997, Vol. 28, No. 3, 258-277

Reflective Discourse andCollective ReflectionPaulCobb, VanderbiltUniversityAdaBoufi, National and CapodistrianUniversityofAthens, GreeceKayMcClain,VanderbiltUniversityJoyWhitenack,Universityof Missouri

The analysisin thispaper ocuses on therelationshipbetweenclassroomdiscourse andmath-ematicaldevelopment.Wegiveparticularttention o reflectivediscourse,nwhichmathematicalactivityis objectifiedand becomesanexplicittopicof conversation.We differentiatebetweenstudents'developmentof particularmathematical onceptsand theirdevelopmentof ageneralorientation o mathematical ctivity. Specific issues addressed nclude both the teacher'sroleand the role of symbolization n supporting eflective shifts in the discourse.We subsequentlycontrast uranalysisof reflectivediscoursewithVygotskianaccountsof learning hatalso stresstheimportancef social interactionndsemioticmediation.Wethenrelate hediscussion o char-acterizationsof classroomdiscoursederivedfromLakatos'philosophicalanalysis.

Thecurrenteformmovementnmathematicsducation lacesconsiderablempha-sis on the role that classroomdiscourse canplay in supporting tudents'concep-tualdevelopment.The consensuson thispoint transcends heoreticaldifferencesand includes researcherswho draw primarilyon mathematics as a discipline(Lampert, 990),on constructivistheory Cobb,Wood,&Yackel, 1993;Thompson,Philipp, Thompson,& Boyd, 1994), and on sociocultural heory(Forman,1996;vanOers,1996).Ourpurposenthisarticle s to suggestpossible elationshipsetweenclassroom discourse and the mathematicaldevelopmentof the studentswho par-ticipaten,andcontributeo it. To thisend,we focus on a particularypeof discoursethatwe callreflectivediscourse.Itis characterizedy repeated hifts such thatwhatthe students ndteacherdo in actionsubsequently ecomesanexplicitobjectof dis-cussion. Infact,we mighthave called it mathematizing iscourse because there sa parallelbetween ts structure ndpsychologicalaccountsof mathematical evel-opment n which actionsorprocessesare transformedntoconceptualmathemat-ical objects.In the course of the analysis,we also developthe relatedconstructofcollective reflection.This latternotion refersto thejoint or communalactivityofmakingwhat was previouslydone in action anobjectof reflection.

A previousdraftof this articlewas presentedat the Symposiumon Communication orUnderstanding n Classrooms,National Centerfor Research in Mathematical SciencesEducation,Madison,WI,October,1994.The researchreported n this paperwas supportedby the National Science FoundationundergrantNo. RED-9353587. Theopinionsexpresseddo notnecessarilyreflecttheviewsof the Foundation.The authorsaregratefulto ErnaYackel for herhelpfulcomments on apreviousdraft.


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Paul Cobb,AdaBoufi,KayMcClain,Joy Whitenack 259

In thefirstpartof thearticle,we discuss action-oriented ccountsof mathemat-ical development and then presentclassroom episodes to exemplify reflectivediscourse and collective reflection. In subsequentsections, we clarify the stu-dents' andteacher's contributionso the developmentof reflective discourse andconsiderpossiblerelationshipsbetweenindividualstudents'mathematicaldevel-opmentandthe classroomsocial processesin whichthey participate.Againstthisbackground,we relateouranalysisof classroomdiscourseto the characterizationofferedby Lampert 1990), andthen concludeby summarizinghepragmaticandtheoreticalsignificanceof the notion of reflective discourse.

MATHEMATICALACTIONSAND MATHEMATICALOBJECTSGrayandTall (1994) recentlyobservedthat "thenotion of actions orprocesses

becoming onceived s mental bjectshasfeaturedontinuallyn the iterature"p. 118).As anexample, heyrefer o Piaget's(1972)contentionhatactionson mathematicalentitiesat one level becomemathematicalbjects hemselvesat anotherevel.Piagetused the notionof reflectiveabstractiono account or suchdevelopmentswhereinthe resultof a mathematical ctioncan be anticipated ndtakenas a given, and theaction tself becomesanentitythatcan be conceptuallymanipulated.Mathematicseducatorsworking n thePiagetian radition ave called thisdevelopmental rocess"integration"Steffe, von Glasersfeld,Richards,& Cobb, 1983) and"encapsula-tion"(Dubinsky,1991). The influence of this Piagetianview can also be seen inVergnaud's 1982) discussionof theway inwhich studentsgradually xplicate he-orems-in-action hat areinitiallyoutsidetheirconscious awareness.Sfard's(1991) accountof mathematical evelopment s compatiblewiththatofPiaget n thatshepositsaprocessof reificationwherebyoperational rprocesscon-ceptionsevolve intoobject-like tructuralonceptions.Shemakesanimportanton-tributionygroundinghisview n detailed istoricalnalyses f avariety f mathematicalconceptsincludingnumberand function. In herview, the historicaldevelopmentof mathematics anbe seen as a "long sequenceof reifications,each one of themstartingwhere the formerends, each one of themaddinga new layerto the com-plex systemof abstractnotions" 1991, p. 16). Inaddition,she illustrateshow thedevelopmentof waysof symbolizinghassupportedhereificationprocessboth his-toricallyandin students'conceptualdevelopment.Thus,although he is careful oclarifythatthedevelopmentand use of symbolsis by itself insufficient,SfardandLinchevski 1994)contend hatmathematicalymbolsaremanipulablen a waythatwords arenot. Inherview, this contributes o the reificationof activityand to thedevelopmentof object-likestructural onceptions.Thegeneralnotionof processesbeing ransformedntoobjectsalso features romi-nently nFreudenthal'snalysis f mathematicalevelopment. orexample,hearguesthat"theactivityof the lowerlevel, that s theorganizingactivityby meansof thislevel, becomes anobjectof analysison thehigher evel; theoperationalmatterofthe lower evelbecomesa subjectmatter nthenext evel"(1973,p. 125).Theagree-menton thispointbetweenFreudenthalndPiaget(1972) is particularlyignificant


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260 ReflectiveDiscourseand CollectiveReflection

given thatFreudenthal ascriticizedotheraspectsof Piaget's theory.LikeSfard,Freudenthaland his collaborators root much of their work in the history ofmathematics.Perhapsbecause of thiscommonality, hey also emphasizethe rolethatdevelopingandusingbothinformaland standard ymbolicschemes canplayin supporting he transition rom action to object.Itshouldbe notedthat hisbriefdiscussionof action-orientedevelopmentalhe-orieshasnecessarilyeenselective ndhasomitted number f importantontributionsincludinghoseof HarelandKaput 1991),Mason 1989),andPirieandKieren1994).Forourpurposes,t sufficesto note two centralassumptionshatcut across he workof the theoristswe have referenced.The firstis that students'sensory-motor ndconceptualactivity s viewed as the sourceof theirmathematicalways of knowing.The second assumption s thatmeaningfulmathematical ctivityis characterizedby thecreation ndconceptualmanipulationf experientiallyealmathematicalbjects.In the remainder f thisarticle,we focuson atypeof classroomdiscourse hatappearsto support tudents'reificationof their mathematical ctivity.

BACKGROUND:THEFIRST-GRADECLASSROOMThe two sample episodesthat we will present or illustrativepurposesare bothtaken romafirst-gradelassroom n whichwe conducted year-longeaching xper-iment.Theepisodeswere selected to clarifythe notionsof reflectivediscourseandcollective reflectionand are not intended o demonstrate xemplarypractice.Thisclassroom s of interest n that he mathematical evelopment f all 18childrenwassignificant.Their earningwas documentedn a seriesof video-recordednterviewsconducted n September,December,January,andMay. Our focus in these inter-viewswasonthe children's volvingarithmeticalonceptions ndstrategies.A vari-

etyof taskswerepresentedneachset of interviews,ncluding othhorizontal umbersentencesandwordproblems.The tasks n theSeptember ndDecember nterviewsinvolved numbers o 20, whereas hosein theJanuary ndMayinterviews nvolvednumbers o 100. In theSeptembernterviews,all 18 childrenusedcountingstrate-gies thatranged romcountingall on their ingers ocountingon andcountingdown.However,only2 childrendevelopedderived-fact rthinking-strategyolutions pon-taneously,and each did so on onlyone occasion.Incontrast,11childrenused non-countingstrategiesroutinely n theDecember nterviews o solve all or almost allthe taskspresented.Another5 childrenusedthinking trategieso solve at least halfthe taskspresented.A comparison f theremaining children'sperformancen theSeptemberand December interviews indicates that they had developed moresophisticatedcountingmethods.In theJanuaryandMayinterviews, the tasks included wordproblemssuchasthe following:Joe and Bob each graba handfulof candiesout of ajar.Joe gets 53 andBobgets28. Howmanydoes Joeneed toputbackso thathe andBobhave the samenumberof candies?The mostsophisticatedolutions bservedntheJanuarynterviews ccurredwhen


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7 of the children solved addition sentences such as 16 + 9 = and 28 + 13 =by developingefficientnoncounting trategies.However,none of the childrenwereableto solveanyof thewordproblems resented.ncontrast, 2 of thechildren olvedall or almostall thetaskspresented n theMayinterviewsby using relativelyeffi-cient noncounting trategies hat nvolvedthe conceptualcoordinationof unitsoftenandone. Theremaining6 childrenhadall madesignificantprogresswhencom-paredwith theirperformancen theDecember andJanuarynterviews.Mostnowusedthinking trategies outinely,andall but1childattemptedo developsolutionsthat involved unitsof ten and one.Detailedanalysisof the interviewscan be foundin Cobb,Gravemeijer,Yackel,McClain,and Whitenack in press)and Whitenack 1995). Ourpurpose n givingthisverybriefsummaryof theinterviewresults s merelyto indicate hat,by tradi-tionalstandards,hefirst-gradeeacherwasreasonably uccessful n supporting erstudents'mathematical evelopment.Explanations f her successnecessarilymakereference o a number f issuesincluding heinquirymicroculturestablishednherclassroomandthesequencesof instructional ctivitiesshe used.Anadditionalssuethatappearedoplayanimportantoleconcerns he natureof classroomdiscourse.

REFLECTIVEDISCOURSE:AN INITIALEXAMPLEAs we have noted, the children used a range of counting methods in theSeptember nterviews.The analysisof theseinterviews also indicated hat five ofthe childrencould not use their ingersas substitutes orother tems. Forexample,thetaskspresentedn Septemberncludedelementary ddition toryproblems uchas, "Canyou pretend hatyou have threeapples" childnodsaffirmatively),"Canyou pretend hat I have two apples" child nods), "Howmanyapplesdo we havetogether?You have threeandI have two." Forthese five children, hepossibilityof puttingup fingersas perceptual ubstitutes or the applesdid not ariseand,as

a consequence, they were not able to enter the situation described in the taskstatement.Thesefindingswere consistentwith ourinitial classroomobservationsof thechildrenand influenced he firstsequenceof instructional ctivities thatwedeveloped ncollaborationwith the teacher.Thissequence nvolvedfingerpatterns,spatialpatterns,and theconceptualpartitioning ndrecomposingof collections ofup to 10 items.The firstepisodewe will presentoccurreda weekafter heSeptembernterviewswerecompletedand nvolvedaninstructionalctivitydesigned o supporthe devel-opmentof flexiblepartitioninge.g., a collectionof five itemsconceptualizedntheimaginationas four and one or three andtwo as the needarises).Theteacherusedanoverheadprojectorandbeganby showingapictureof twotrees,one larger hantheother,andfive monkeys.Sheasked he children everalquestionsabout hepic-ture andexplainedthatthemonkeyscouldplayin thetrees and could swingfromone tree to theother.Shethenasked,"Iftheyall wanttoplayin thetrees, ustthinkof ways thatwe can see all five monkeysin the two trees-all five monkeysto bein two trees."


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The collection of monkeysremainedvisible throughout he ensuingdiscussionso thatall the childrenmightbe able to participateby proposingvariousways inwhich the monkeyscould be in thetrees.The teacherdrewa vertical inebetweenthe trees and then recordedthe children's suggestions, creating a table in theprocess. For example, the following exchange occurredafter the teacher hadrecorded wo children'scontributionsshowninFigure1).Thecontributions f thefive childrenwho could not initiallyuse theirfingersas perceptual ubstitutesaremarkedwith asterisks.

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Figure 1. Graphic or the "monkeys n the trees" ask

Anna*: I think hat hree ouldbe inthe ittle reeand wocouldbeinthebigtree.Teacher: OK, hreeouldbe nthe ittleree,wocould e inthebigtree writes12 etweenthetrees].So,still3 and2 but heyare ndifferentrees histime; hree nthelittleone and wo inthebigone.Linda, ouhaveanotherway?Linda*: Fivecouldbeinthebigone.Teacher: OK, ivecouldbe inthebigone[writes ]and henhowmanywouldbe in thelittleone?Linda*: Zero.Teacher: [Writes ].Anotherway?Anotherway,Jan?Jan: Four ouldbe inthe ittle ree,one inthebigtree.

We note in passingthatas partof herrole, the teachergave acommentary romtheperspective f onewho could udgewhichaspectsof thechildren'sactivitymightbe mathematically ignificant.Thus, she relatedAnna's proposalto the previoussuggestionby saying,"Stillthreeandtwo buttheyare in adifferent ree thistime."


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of theepisodethat eads us to classify it as anexampleof reflectivediscourse.Thesignificanceof thistypeof discourse ies in itsrelationshipo theaccountsof math-ematical development given by Piaget (1972), Sfard (1991), and Freudenthal(1973).Weconjecturehatopportunitiesrose orthechildren o reflecton andobjec-tifytheirprioractivityas theyparticipatednthe discourse. notherwords, he chil-drendid nothappen o spontaneously egin reflectingat thesamemoment.Instead,reflectionwas supportedand enabledby participationn the discourse.To accountfor this featureof reflectivediscourse, t mightatfirstglanceseemreasonableo suggest hat hechildren ngaged nacollectiveactof reflective bstrac-tion. However,Piaget's (1972) notion of reflectiveabstractions a psychologicalconstructhatrefers o theprocessbywhich ndividual hildren eorganizeheirmath-ematicalactivity.The contention hat he sampleepisodeinvolvedacollective actof reflectiveabstractionhereforempliesthateach of the children eorganizedheirmathematicalctivity, herebymakinga significant onceptual dvance. n ourview,thisassumeddirect ink between he socialprocessof reflectivediscourseand ndi-vidualpsychologicalprocessof reflectiveabstractions too strong.We insteadcon-jecturethatchildren'sparticipationn thistypeof discourseconstitutesconditionsfor thepossibilityofmathematicalearning,butthat t does notinevitablyresult neachchildreorganizingisorhermathematicalctivitycf.Cobb,Jaworski,&Presmeg,1996).In thisformulation,he link between discourseandpsychologicalprocesseslike reflective abstractions indirect.Thisperspectiveacknowledges hatboththeprocess of mathematical earning and its products, increasingly sophisticatedmathematicalways of knowing,are social throughandthrough.However,it alsoemphasizesthatchildrenactivelyconstruct heir mathematical nderstandings stheyparticipaten classroom ocialprdcesses.Toemphasizehis ndirectinkbetweenthe individualandcollectiveaspectsof mathematical evelopment,we distinguishbetween hepsychologicalprocessof reflectiveabstraction ndthe communalactiv-ity of collective reflectionthat occurs as childrenparticipate n reflective dis-course. Thus, in the case of the sample episode, we infer that the childrencollectivelyreflectedon theirprioractivityof generatinghepossiblewaysthe mon-keyscouldbe in thetwotrees.However, nferencesabout he reflectiveabstractionsandconceptual eorganizationshatparticularhildrenmighthave madewhilepar-ticipating n the sampleepisode requirea detailedpsychologicalanalysis.This discussionof collective reflectionquestions heview that he childrenweremerelycarried longby a discoursehatdeterminedheir ndividualhinking.Wefur-therhighlight he children'sagencyby notingthat heycontributedo the shiftsthatoccurrednthe discourse.Consider,orexample, heoccasionwhentheteacher skedthe childrenf therewasa waythat heycouldbe sure heyhadgenerated ll thepos-sibilities. twasat this uncturehatJordan airedup possibilitiesnvolving he samenumber ombinations.ngivingthisexplanation, e reflectedon andreorganizedheclass's prioractivityas recordedn thetable.Had none of thechildrenbeenable torespond o the teacher'squestion n thisway, the shiftin discoursewouldnothaveoccurred. t is therefore easonableo saythat ndividual hildren ontributedo thedevelopmentof the discourse hatsupported nd sustainedcollective reflection.


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REFLECTIVEDISCOURSEAND MATHEMATICALLEARNINGWehave takencaretoclarify hat heproposed elationshipetweenreflectivedis-course andconceptualdevelopment n mathematicss speculative.Inconsideringwhatchildrenmight earnwhen heyengage n suchdiscourse,we distinguishetween(a)theirconstruction f specificmathematical onceptionsand(b)thegeneralori-entation o mathematical ctivitythatparticipationn thediscoursemightfoster.

ConceptualDevelopment nDiscourseWithregard o the first of thesetwo issues, there s some indication hatthe dis-

cussions of partitioningwereproductive orsome of the children.Thepartitioningactivitywasrepeatednthemonkeys-and-treesettinghedayafter hesample pisode,and henagain3 weekslaterusingthesettingof adouble-decker us(vandenBrink,1989).The childrenwere toldthat herewere,say,tenpassengers n a doubledeckerbus andwere askedto decidehow manycould be on the topdeck and how manycould be on the bottomdeck.On bothdays,theteacher ecorded hechildren's ug-gestions,and hepossibility f organizingheirproposalsntopairsbecameanexplicittopicof conversation.On the secondday,thechildrenwere also askedtocompleteanactivitysheet that nvolved generatingpartitions orthe thedouble-deckerbusproblem.Of the 16 childrenpresent,8 consistentlygeneratedcommutativepairsand3 producedmore sophisticatedorganizations.Fora taskinvolving7 passen-gers, thepossibilitiesincluded0 7 6 5 4 3 2 1

7 0 1 2 3 4 5 6and

7 0 6 1 5 2 4 3

0 7 1 6 2 5 3 4where

70

signified7 peopleon thetopdeck of thebus andnone on thebottomdeck. No dis-cerniblepatterns ouldbe detected n thewrittenwork of theremaining5 children.


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Significantly, 4 of these latter childrenwere amongthe 5 who had been unableto use their fingers as perceptual substitutes in the Septemberinterviews. Ittherefore seems importantto consider how these children might have inter-preted hepriorwhole-classdiscussions.Possibly, heydid notreorganizehe resultsof prioractivity when they attempted o understandwhatotherchildren and theteacherweresayinganddoing(cf. Steffe, Firth,& Cobb,1981).Forthem,the dis-coursemighthave been aboutfindingways themonkeyscould be in the trees orthe passengerscould be on the bus, per se. It should be noted that this possibil-ity would constitute an advance when compared with their activity in theSeptember nterviews.Theanalysis f the children'swrittenworkserves oemphasizeurtherhatalthoughparticipationn reflectivediscoursesupportsand enables ndividualreflectionon,andreorganization f, prioractivity,it does notcause it, determine t, orgenerateit. Thus,in the view we areadvancing, t is the individualchildwho has to do thereflecting ndreorganizing hileparticipatingnandcontributingothedevelopmentof the discourse.Thisimpliesthat he discourseandtheassociated ommunalactiv-

ityof collectivereflectionbothsupport ndareconstituted ythe constructive ctiv-ities of individualchildren.

TheDevelopmentof a MathematizingOrientationWe can address he second aspectof the students' earning-their generalori-entation o mathematical ctivity-by consideringanepisodethatoccurred n thesameclassroomalmost5 monthsafter he first. The instructional ctivitiesused atthe time of the secondsampleepisodeweredesigned o supporthechildren's truc-turingof numbersupto 100intocompositeunits,particularly f ten and one. Oneof the anchoring ituations hat the teacherhadpreviously ntroducedwas that ofMrs.Wright's candy shop,where loose candieswerepacked nto rolls of ten. Tointroduce this particular nstructionalactivity, the teacherexplained that Mrs.Wrightwas interruptedwhen she was countingoutcandies andputting hem intorolls. She thenposed thefollowing question:Teacher: What f Mrs.Wright as43piecesofcandy nd he s packinghemntorolls.What redifferent ays hat hemight ave43pieces fcandy-howmany ollsandhowmanypiecesmight hehave?Sarah,what'sonewayshemight indthem?The discussionproceededsmoothly,in thatthechildren,as a group, generatedthe variouspossibilities with little apparentdifficulty. Their contributionswereas follows:

Sarah*: Fourrollsand threepieces.Elizabeth*:Forty-threepieces.Kendra: She mighthave two rolls and 23 pieces.Darren: She could have threerolls, 12 pieces, I mean 13pieces.Linda*: Onerolland33pieces.Theteacher or herpartrecordedeach of theirsuggestionson a boardas shownin Figure3.


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4r 3p 43p 2r 23p00000o 0000 OO0000 f 00l0000 O08 8 8 8 8 8000 O

3r 13p 1r 33p00 00000 OO0

00 0

8 888

Figure3. The teacher'srecordof the students'suggestions

Karen hen indicated hatshe hadsomethingshe wantedto say andwent to theboardat the front of the room.Karen: Well, see, we've done all theways. We had 43 pieces....Teacher: OK.Karen: And, ee,wehad43pieces pointso43p]and ightherewe havenone olls,andrightherewe haveone roll[pointso Ir33p]Teacher: OK,I'mgoing to number hese,there'sone way ... no rolls [writes"0"next to

43p].Karen: Andthere'sone roll, there's2 rolls, thenthere's3, andthere's4.Teacher: [Numbers hecorrespondingpictures1, 2, 3, 4].

Thisexchangeis reminiscentof the firstepisodein thatboth concern he possi-ble ways of partitioninga specified collection. However, a crucial differencebetween he twoepisodesconcerns hejustifications ivenforthe claim thatallpos-sibilitieshad been found.Inthe firstepisode,Elizabeth xplained hatshe could notthink of any moreways. It was not until the teacherasked the childrenhow theycouldknow for sure hat heyhadfoundall thewaysthat he discoursemovedbeyondwhatmightbe termedempiricalarguments. ncontrast,Karen ustifiedherclaimbyorderinghepossibilitieshathadbeengeneratedwithoutprompting. hus,almostseamlessly, hediscourse hifted romgeneratinghepossiblewaysthe candiesmightbe packedto operatingon the results of thatgenerativeactivity.As theexchangecontinued,t becameapparenthatmanyotherchildren ook theneedtoproduce nordering s self-evident. an oinedKaren t theboard ndproposedan alternativeway of numbering hepictures:


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268 ReflectiveDiscourse and CollectiveReflection

Jan: I thinkyoushouldnumberhem, ikeputthem norder.Likethatonefirst[pointsto 43p], thenthatone [ir 33p].Teacher: [Erases he0 next to 43p andwrites 1]Call this number wo [ir 33p]?Jan: 'Causethat'sthe firstway [43p], 'cause it's no rolls.Teacher: OK. [Writes2 and3 next to Ir 33p and 2r23p, respectively.]

However,at thispoint,Janbeganto focus on the numberof rolls in eachconfigurationrather than on the order that each configuration would be produced when packingcandy and eventually said that she was confused.Another child, Casey, said that he thought he knew what Jan meant and joinedher and Karen at the board.Casey: She means like there'snonerighthere[43p]andthat's[number]one, and thenthere's one [roll]righthere[ir 33p], thatmakes[number]2; there's two [rolls]right here [2r 23p], that makes [number]3; there's three [rolls] right here[3r 13p], that makes [number] our;and there's four[rolls]rightthere[4r3p];thatmakes[number] ive.Teacher: [Labelstheconfigurationsas Caseyspeaks.]Casey: Because that'sthefirstone [43p],that'sthesecond[ir 33p]one, that'sthe third[2r23p]....Jan: No, I thinkthat should be the third[3r 13p].Casey: I'm notcountingrolls.

As theexchangecontinued, t became ncreasinglyevident thatJan,Karen,anda third hild,Anna*,on one side,andCaseyon the othersideweretalkingpasteachother.Jan, Karen,and Anna*repeated hattheirnumbering cheme was based onthe numberof rolls,andCaseycontinued o protest hathe was notcountingrolls.At this juncture, the teacher intervened and initiated a final shift in the discourse.She first explained Jan, Karen, and Anna's approach, stressing that "the way thatthey were thinking about numbering them and naming them was by how many rollsthat they used to make 43 candies."Casey: I wasn'tcounting herolls,I wascountinghow theywentinorder.Like thatonewas the firstone, and thatone was the secondone.

[Severalchildren ndicated hey disagreedor did notunderstand.]Casey: I wasn'tcountingthe rolls. I wasn'tgoing like "oneroll, two rolls, threerolls,fourrolls."Teacher: OK,Casey,now let me say whatyou weresayingandyou listen to me andseeif I saywhatyou said.[Addresses heclass.]Now what t means s that herearetwo differentways of namingthem.

She henwenton to contrastherationalesor hetwoapproaches,tressinghat"Caseywastalking boutustnaming hema differentway."Jan nterruptedheteacher osay,"Now I understand,"and then spontaneously explained Casey's approachherself.Two major shifts in the discourse can be discerned in this second episode. In thefirst, the various partitionings of 43 candies that the children generated were objec-tified and treated as entities that could be ordered. In the second shift, the activity ofordering he configurations tself became an explicit topic of conversation.To the extentthat the children participatedin these shifts, they could be said to be engaging in theactivityof mathematizing.The first of these shifts in which thepartitioningswere treated


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asentitieswasrelatively outine ormanyof the children.Thissuggests hat hegen-eralorientationhe childrenweredevelopingwhentheyparticipatedn reflectivedis-coursewasthatof mathematizing.n thisregard,we note with Bauersfeld 1995):

Participatingn theprocesses f amathematicslassrooms participatingn a cultureofmathematizing.hemany kills,whichanobserveran dentify ndwilltakeasthemainperformancef theculture,orm heproceduralurfacenly.Theseare hebricksof thebuilding, ut hedesignof thehouseof mathematizings processed tanotherlevel.Asit is withculture,hecoreofwhat s learnedhrougharticipations when odo what and how to do it .... The corepartof schoolmathematics nculturationomesintoeffecton themeta-level nd s "learned"ndirectly.p.460)Ourconjecture s that a crucialaspectof what is currently alled a mathemati-cal disposition(NationalCouncil of Teachersof Mathematics,1991)is developedin this indirectmanneras childrenparticipaten reflective classroomdiscourse.

SYMBOLIZINGAND THETEACHER'SROLEThusfar,we haveconsideredhowparticipationnreflectivediscoursemightsup-port students' mathematical earning. In doing so, we have conjecturedaboutbothspecific conceptualdevelopments hat nvolve the transition romprocessto

objectandthemoregeneraldevelopment f amathematizingrientation. henotionof reflectivediscoursealsohelpsclarifycertainaspectsof the teacher'srole.Inourview, one of theprimaryways in whichteacherscanproactivelysupport tudents'mathematicaldevelopment s to guide and,as necessary, nitiate shifts in the dis-course suchthatwhat was previouslydone in action can become anexplicit topicof conversation.This was exemplifiedin the firstepisodewhen the teacher niti-ateda shiftbeyondwhatwe termed mpirical erification y asking,"Istherea waythatwe could be sure andknow thatwe've gottenall theways [that ive monkeyscould be in thetwotrees]?"Theensuingshiftin thediscourse hatoccurred anbeviewed as an interactional ccomplishmentn that t also dependedon the contri-butionmadebyone of thechildren, ordan. herole that heteacher's uestionplayedin this exchangewas, in effect, thatof an invitation,or anoffer, to stepbackandreorganizewhathadbeen done thusfar.'It is importanto clarifythat nitiatingandguidingthedevelopmentof reflectivediscourserequiresconsiderablewisdom andjudgmenton the teacher' part.Onecan, for example, imaginea scenario n which a teacherpersistsin attempting oinitiate shift nthediscoursewhennoneof the students ivesaresponsehat nvolvesreflectiononprioractivity.Theveryrealdangers, of course, hatanintended cca-sion forreflectivediscoursewilldegeneratento a socialguessinggame nwhichstu-dents ry oinferwhat heteacherwants hem osayanddo(cf.Bauersfeld, 980;Voigt,1985).Inlightof thispossibility, heteacher's ole ininitiating hifts nthe discourse'It can be argued hat he instructional ctivities houldbe modifiedso that he need to "know or sure"arises in an apparentlyspontaneousway. However, as a practicalmatter,this ideal is not alwaysobtainable. t is therefore ssentialthat he teacherbepreparedo take the initiative n facilitating hiftsin thediscourse.


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mightbethought f asprobingo assesswhether hildren anparticipatentheobjec-tificationfwhat heyarecurrentlyoing.Suchaformulationcknowledgeshe eacher'proactive ole nguiding hedevelopment f reflectivediscoursewhilesimultaneouslystressing oth hat uchdiscourse s aninteractionalccomplishmentnd hat tudentsnecessarilyhaveto make an active contributiono its development.A secondaspectof the teacher's oleapparentnbothepisodes s thewayinwhichshedevelopedsymbolicrecordsof the children'scontributions.Ofcourse,one canimaginea scenario n whichways of notatingcould themselves have been a topicfor explicit negotiation.Forourpurpose, he crucialpointis not who initiated hedevelopment f the notational chemes,but thefactthat he recordsgrewoutof stu-dents'activityn abottom-upmannercf.Gravemeijer,npress),and hat heyappearedto play animportant ole in facilitatingcollective reflectionon thatprioractivity.We canclarifyboth the influence of therecordsand theirsupportive oleby con-sideringthe first of the two episodes.There,the influence of the tablethe teachermadewhilerecording hechildren'scontributionsirstbecameapparent uring hefollowing exchange:Linda*: Five [monkeys]couldbe in thebig one.Teacher: OK, ivecouldbe in thebigone[writes ]and henhowmanywouldbe in thelittleone?Linda*: Zero.Teacher: [Writes0.]

Here,the teacherpresumably skedLindahow many monkeyswerein the smalltreeonlybecauseshe wasrecordingLinda'sproposaln thetable.In otherwords, hetableinfluencedwhat countedas a completecontribution. ater n theepisode,thetablegreatly acilitated heprocessof empirically heckingwhetherparticular ar-titioningsof the five monkeyshadalready eenproposed.This wasthe firstoccasionwhen entries n the tablewerepointed o, andspokenof, as signsthatsignifiedvar-iouspartitionings.hus, herewas a reversal f thesignifier-signifiedelationcf.Kaput,1991).Later,whenJordanxplained isapproachfpairinghepartitionings,epointedto the table entriesas he spokeaboutmonkeys n trees.Indoingso, he appearedolook hroughhe able ntrieso see thepartitioningsboutwhichhespoke.Thus,althoughwe asobservers andistinguishbetweenthesignifiedandsignifier, he tableentriesand he variouspartitionings,hetwo seemed obeinseparableorhim nthat hetableentriesmeantparticular artitionings f monkeys.We could give a similar accountof the role that the picturesof candiesplayedin the secondepisode.Thereagain,thedistinctionbetweensignifiedandsignifierappearedo be reversed nthecourseof theepisode.Further, s in the firstepisode,shifts n the discoursewereaccompanied y changes n the functionof thesymbolicrecordsuntil,eventually,therecordsbecameexplicitobjectsof discourse hatthechildren ookedthrough o see theprioractivitythattherecordssymbolized.Bothhere and in the first episode, the importantrole attributed o symbolizationishighlycompatiblewiththetheoryof realisticmathematics ducationdevelopedbyFreudenthal nd his collaborators cf. Treffers, 1987;Streefland,1991). It is alsoconsistentwiththeresultsof Sfard' (1991)historicalanalyses,which indicate hat


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Paul Cobb,AdaBoufi, KayMcClain,Joy Whitenack 271

thedevelopment f waysof notatinghasrepeatedly layedacentral ole in themath-ematician's transition romprocesslike operativeconceptionsto objectlikestruc-turalconceptions. This is not to say, however, that either the developmentofwaysof symbolizingorchanges n discoursenevitablyeadtochanges nindividualstudents' hinking.We further larifyourpositionon this issue in the next section.

THE INDIVIDUALIN DISCOURSEThere re trong arallelsetween urdiscussion f reflective iscourse ndVygotsky'(1978) sociohistorical nalysisof development.The readermightthereforeassume

thattheanalysiswe havepresented s a straightforwardlaboration f Vygotsky'sposition.To guardagainstthis interpretation,we takeVygotsky's workas a pointof contrast.Althoughwe will emphasizedifferences n perspectives, he intellec-tualdebt we owe Vygotsky should be readily apparent.We have, in fact, refinedourpositionby engagingin anargumentwith thevoice of Vygotsky as expressedin his writings.Vygotsky emphasized two primaryinfluences on conceptual development,social interaction ndsemioticmediation cf. van derVeer&Valsiner,1991).With

regard o the firstof these nfluences,heposited hat nterpersonalelations re nter-nalized from the interpsychologicalplaneand reconstituted o formthe intrapsy-chologicalplaneof psychologicalfunctions.Ourspeculation hatthe firstgradersdevelopeda mathematizing rientation s they participatednreflectivediscoursemightappearoexemplify hisprocessof internalizationrom he nterpersonalomainto thepsychologicaldomain.The reader ould therefore onclude hatwe havefol-lowed Vygotskyin contending hatthe children nternalizeddistinctiveaspectsofthis collective activity.In our discussionof reflectivediscourse,we also addressed he secondinfluenceon conceptualdevelopmentconsideredby Vygotsky, thatof semioticmediation.Hereagain,Vygotsky gave social andculturalprocessespriorityover individualpsychologicalprocesses.He argued hat n thecourseof development, ultural oolssuch as oral and written anguageare nternalized ndbecomepsychologicaltoolsforthinkingcf.Rogoff,1990).In the twoepisodes, hemeansof notatingheteacherused to record he children'scontributions ouldbe characterized s cultural ools,andthechanges hatoccurred n the roleplayedby thesymbolscouldbe interpretedassteps nthis nternalizationrocess.Onemight,nfact,betemptedofollowLeont'ev(1981)and alkof thestudents ppropriatingheculturalools ntroducedytheteacher.In such an account,the ways of notatingwould be treatedas objectivemediatorsthatservedto carrymathematicalmeaningfromone generation o the next.Beforedifferentiatingurposition romthatof Vygotsky,we muststress hatwedo agreewithmanyof thecentral enets of his theory.Forexample,we agreethatchildren'smathematicaldevelopment s profoundly nfluencedbothby the face-to-face interactionsand by the culturalpractices in which they participate.Inaddition,we believe thatVygotsky was rightwhen he contended hatthinking sinextricablybound up with the cultural tools that are used (cf. Dirfler, 1996;


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Kaput,1991;Thompson,1992).Thus,as LeshandLamon(1992) note, it is diffi-cult for us to imaginehow the worldmighthave beenexperiencedbefore the con-ceptualmodelsandassociatednotation chemes thatwe now take forgrantedweredeveloped.The issue at hand s not, therefore, hatof determiningwhethersocialandculturalprocesses nfluence ndividualpsychologicalprocesses.Instead,t con-cernsthe specific natureof therelationshipbetweenthese two domains,andit isherethat we depart romVygotskiantheory.

Vygotsky 1978)arguedhat hequalities f mentaldevelopment rederivedrom,andgeneratedby,thedistinctiveproperties f the sociocultural rganization f theactivities n which theindividualparticipatesvanOers,1996).Thelinkagehepro-posedbetween he wo domainsstherefore direct ne(cf.Cobb, aworski,&Presmeg,1996). This, it will be recalled,is the relationthatwe questionedwhen we intro-duced henotionof collectivereflection.Theacceptance f this relationmpliesthatchildren'sdevelopmentof a mathematizing ttitudecanbe accounted ordirectlyin termsof theirparticipationn reflectivediscourse,without the need to refer totheir ndividual onstructive ctivities.Elsewhere,we havenoted hat hisapproachemphasizesthe social and culturalbasis of personalexperience(Cobb, 1995). Inourview, it is anappropriate pproach o takewhenaddressinga varietyof issuesincluding those that pertain to diversity and the restructuringof the school.Therefore, ur ntent s not toreject hisapproach utof hand.We doquestion,how-ever,theexplanatorypowerit provides n addressing he issues underdiscussionin this article. What is requiredis an analytical approachthat is fine-grainedenough to account for qualitativedifferences in individualchildren's thinkingeven as they participaten the same collective activities(cf. Confrey,1995).Therelevance of this criterion became apparentduringthe discussion of the firstepisodewhenwe observedhat5 of the 16children idnotappearohavereorganizedpriorpartitioning ctivity.Thisindicates hatthethinkingof somechildrenbutnotothersreflectedthe organizationof the social activityin whichthey participated.Our ationaleorpositing n ndirectinkagebetween ocialandpsychologicalrocessesis thereforepragmaticandderivesfrom our desireto accountfor such differencesin individualchildren'sactivity.As we have noted,thisview implies thatpartic-ipation n anactivitysuch as reflectivediscourseconstitutes he conditions or thepossibility f learning, ut t is the studentswhoactuallydothe earning.Participationin reflectivediscourse,therefore,canbe seen both to enable andconstrainmath-ematicaldevelopment,butnot to determine t.Given the current nterest in the philosophy and pedagogy of John Dewey(1926), we close this discussion of the relation between individual and socialprocessesby notingthatthepositionwe have outlined s closer to that of Deweythanto Vygotsky (1978). ForDewey, as forVygotsky, learningwas a processofenculturationnto growingandchangingtraditionsof practice.However,Deweyalso stressed the contributionsof actively interpretingstudents.Similarly, wehaveattemptedo illustrate hatstudents ontributeo thedevelopmentof thecom-munalactivities n whichtheyparticipate. hefollowingsummaryhatDewey gaveof his positionhas a remarkably ontemporary ingto it:


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Thecustoms,methods ndworkingtandardsf thecalling onstitute"tradition,"ndinitiationnto hetraditions themeansbywhich hepowers f learners rereleasedanddirected. utwe should lsohave osaythat heurgeorneedof an ndividualojoin nanundertakings anecessaryrerequisitefthetradition'seinga factorn hispersonalrowthnpowerand reedom;ndalsothathe hasto seeon his ownbehalfandnhisownway he elationsetween eans ndmethodsmployednd esultschieved.Nobody lse canseeforhimandhe can'tseebyjustbeing"told," lthoughherightkindoftellingmayguidehisseeingand hushelphimseewhathe needs osee.(1926,p.57;italics noriginal.Quoted yWestbrook,991,p.505)

DISCOURSEON DISCOURSEThusfar,we haveplacedouranalysisof reflective discourse n a broader heo-retical context by contrastingit with Vygotskian theory. We now relate it to

Lampert's 1990)influentialdiscussionof discoursenreformclassrooms.Lampertderiveshervision of the idealform of classroomdiscourse roma consideration fmathematicss adiscipline.DrawingparticularlynLakatos1976)andP6lya 1954),she argues hatthediscourseof mathematicianss characterized y a zig-zag fromconjectureso an examination f premises hroughhe useof counter-examples. neof herprimary oals is to investigatewhether t is possiblefor students oengageinmathematical ctivitycongruentwiththisportrayal f disciplinarydiscourse.As aconsequence,her focus is, forthemostpart,on how theteacherand students nter-act as they talk aboutand do mathematics.We will suggestthatour discussionofreflectivediscourse omplementshis workbyfocusingon what he teacher nd stu-dentscreatendividuallyndcollectivelyn thecourseof such nteractions.irst,how-ever,we tease outfurtheraspectsof Lampert'sanalysis.We note with Billig (1987) that a zig-zag between the generalandtheparticu-lar,or betweenconjecturesandrefutations, s not specific to mathematics,but ischaracteristicf argumentationngeneral.A further spectof Lampert's1990)analy-sis differentiatesmathematical ndscientificdiscourse romotherkindsof discourse.Shegives particular mphasis o threemaxims hatP61ya 1954)believed areessen-tial when making "a ready ascent from observations to generalizations,and areadydescent from thehighest generalizationso the mostconcreteobservations"(p.7). Thesemaximsconcern he intellectualourageand ntellectual onestyneededtorevisea belief whenthere s a goodreason o change t,andthewise restrainthatshouldbe exercisedso thatbeliefs are notchangedwantonlyand withoutgoodrea-son.Significantly,hesemaximsand herelated isionof classroomdiscourseappearto correspond losely to four commitments hatBereiter 1992) contendsarecen-tralto scientific discourse andmakescientificprogresspossible:

1. A commitment o work towardcommonunderstandingatisfactory o all;2. A commitment to frame questions and propositionsin ways that enableevidence to be brought o bearon them;3. A commitment o expandthebodyof collectively validpropositions;and


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4. A commitmentoallowanybelief to be subjectedo criticismf it willadvancethe discourse.(p. 8)Bereitercontends thatthese fourcommitmentsdistinguishmathematicalandscientific discourse from other forms of discourse, including philosophical,legal, and political discourse. His analysis therefore substantiatesLampert'sclaim that she is teachingher students o "acton thebasisof whatP61lyaalls 'themoral qualities of the scientist"' (1990, p. 58). In this respect, her analysismakes an importantcontribution.Inconsideringhesubstance f classroomdiscourse,we observewithGravemeijer(inpress) hatLampert1990)takespuremathematics s hermodelwhenshedrawson Lakatos(1976) andP61lya1954). Gravemeijer rguesthatappliedmathemat-ics can alsobe animportantourceof analogieswhen attention enterson students'mathematicaldevelopment,particularly t the elementary chool level. In devel-opingthis analogy,he suggeststhatpart fa[classroom]iscussionsabouthe nterpretationf thesituationketchednthe[applied]ontextualroblem.notherart f thediscussionocuses ntheadequacyndtheefficiencyfvariousolutionrocedures.his an mplicateshift fattentionowardsareflectionn he olutionrocedurerom mathematicaloint f view. Emphasisdded.]This sketch is highly compatiblewith our account of reflective discourse.Thetwo sample episodes both focus on what Gravemeijer would call contextproblems andinvolve shifts that lead to the development of collective reflec-tion. Consequently, whereas Lampert's primaryconcern is with the commit-ments that make progress possible as argumentszig-zag from conjectures torefutations,we are more interested n theprocessof mathematization s it occursin the course of such discussions. It is in this sense that we suggest the two

approachesare complementary.CONCLUSIONS

Throughouthediscussion,we have suggestedthatthe notion of reflectivedis-course is of interest or bothpragmaticand theoreticalreasons.Pragmatically, nanalysisof reflectivediscourse larifieshowteachersmightproactively upportheirstudents'mathematicaldevelopment n ways compatiblewith recent reformrec-ommendations. t thereforehas implications or in- andpreservice eacherdevel-opment.Inthisregard,we have used the notionof reflectivediscourse o guidetheeditingof classroomvideorecordingswhenpreparing ypermedia ases for teacherdevelopment Goldmanet al., 1994).This notion also proveduseful when devel-oping instructionalactivities both in collaborationwith the first-gradeteacherandat otherclassroom esearch ites.Inparticular,nstructionalequenceswerefre-quentlydesignedso that t mightbe possiblefor the teacher o initiate shiftsin thediscourseby capitalizingon the students'mathematical ontributions.Theoretically,we haveargued hatreflectivediscourses a useful constructn thatit suggests possible relationshipsbetween classroom discourseand mathematical


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(Eds.),Assessmentof authenticperformance n school mathematicspp. 17-62). Washington,DC:AmericanAssociation for the Advancementof Science.Mason, J. (1989). Mathematicalabstractionas the result of a delicate shift of attention. For theLearningof Mathematics,9(2), 2-8.NationalCouncilof Teachers f Mathematics.1991).Professional tandardsor teachingmathematics.Reston,VA: Author.Piaget,J. (1972). Theprinciples of geneticepistemology.London:Routledge& KeganPaul.Pirie, S., & Kieren,T. (1994). Growth n mathematical nderstanding:How can we characterizet andhow can we representt? EducationalStudies nMathematics,26, 61-86.P61ya,G. (1954). Inductionandanalogy in mathematics.Princeton,NJ: PrincetonUniversityPress.Rogoff, B. (1990). Apprenticeship n thinking:Cognitive development n social context. Oxford:OxfordUniversityPress.Sfard, A. (1991). On the dual nature of mathematicalconceptions:Reflections on processes andobjectsas differentsides of the samecoin. EducationalStudies nMathematics,22, 1-36.Sfard,A., &Linchevski,L. (1994).Thegainsandpitfallsof reification-the case of algebra.EducationalStudiesinMathematics,26, 87-124.Steffe,L. P.,vonGlasersfeld,E.,Richards, .,&Cobb,P. (1983).Children's ounting ypes:Philosophy,theory,andapplication.New York:PraegerScientific.Steffe,L. P.,Firth,D., & Cobb,P. (1981).On the natureof countingactivity:Perceptual nit tems.FortheLearningof Mathematics,1(2), 13-21.Streefland, . (1991).Fractionsn realisticmathematicsducation.Aparadigm fdevelopmentalesearch.Dordrecht,The Netherlands:Kluwer.Thompson,P. W. (1992). Notations,principles,and constraints:Contributions o the effective use ofconcretemanipulativesnelementarymathematics. ournalor Research n MathematicsEducation,23, 123-147.Thompson,A. G., Philipp,R. A., Thompson,P. W., & Boyd, B. A. (1994). Calculational ndconcep-tualorientationsnteachingmathematics.n D.B. Aichele Ed.),ProfessionalDevelopmentor TeachersofMathematics, he 1994 Yearbook f theNationalCouncilof Teachers f Mathematicspp.79-92).Reston,VA: NationalCouncilof Teachersof Mathematics.Treffers,A. (1987). Threedimensions:A modelofgoal andtheorydescription nmathematicsnstruc-tion-the WiskobasProject.Dordrecht,The Netherlands:Reidel.vandenBrink,F. J. (1989). Realistischrekenonderwijsanjonge kinderer.Utrecht,The Netherlands:VakgroepOnderzoekWiskundeonderwijs Onderwijscomputerentrum,Rijksuniversiteit trecht.van derVeer, R., & Valsiner,J. (1991). UnderstandingVygotsky:A quest or synthesis. Cambridge,MA: Blackwell.vanOers,B. (1996).Learningmathematics s meaningfulactivity.In P. Nesher,L. Steffe,P.Cobb,G.Goldin,& B. Greer Eds.),Theoriesofmathematicalearning pp.91-114). Hillsdale,NJ: Erlbaum.Vergnaud,G. (1982).Cognitiveanddevelopmental sychologyand researchnmathematics ducation:Some theoreticalandmethodological ssues. For theLearningof Mathematics,3(2), 31-41.Voigt, J. (1985). Patterns and routines in classroom interaction.Recherches en Didactique desMathematiques, , 69-118.Vygotsky, L. S. (1978). Mind and society: The developmentof higher psychological processes.Cambridge,MA:HarvardUniversityPress.


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AuthorsPaul Cobb, Professorof MathematicsEducation,PeabodyCollege, Box 330, VanderbiltUniversity,Nashville,TN 37203; [email protected] Boufi,Assistant rofessor f Mathematics ducation,National ndCapodistrianniversity f Athens,Pedagogics Department f ElementaryEducation,33, IppokratusStreet,Athens,GreeceKayMcClain,SeniorLecturernMathematicsducation,eabodyCollege,Box330,Vanderbiltniversity,Nashville,TN 37203; [email protected] Whitenack, AssistantProfessorof Education,Universityof MissouriatColumbia,212 TownsendHall, Columbia,MO 65211; CIJW@ HOWME.MISSOURI.EDU

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Reflective Discourse and Reflective Colective