APOS:AConstructivistTheoryofLearning

inUndergraduateMathematicsEducationResearch

EdDubinsky,GeorgiaStateUniversity,USA

and

MichaelA.McDonald,OccidentalCollege,USA

Theworkreportedinthispaperisbasedontheprinciplethatresearchinmathematicseducation

isstrengthenedinseveralwayswhenbasedonatheoreticalperspective.Developmentofatheoryor

modelinmathematicseducationshouldbe,inourview,partofanattempttounderstandhow

mathematicscanbelearnedandwhataneducationalprogramcandotohelpinthislearning.Wedo

notthinkthatatheoryoflearningisastatementoftruthandalthoughitmayormaynotbean

approximationtowhatisreallyhappeningwhenanindividualtriestolearnoneoranotherconceptin

mathematics,thisisnotourfocus.Ratherweconcentrateonhowatheoryoflearningmathematics

canhelpusunderstandthelearningprocessbyprovidingexplanationsofphenomenathatwecan

observeinstudentswhoaretryingtoconstructtheirunderstandingsofmathematicalconceptsandby

suggestingdirectionsforpedagogythatcanhelpinthislearningprocess.

Modelsandtheoriesinmathematicseducationcan

?supportprediction,

?haveexplanatorypower,

?beapplicabletoabroadrangeofphenomena,

?helporganizeone’sthinkingaboutcomplex,interrelatedphenomena,

?serveasatoolforanalyzingdata,and

?providealanguageforcommunicationofideasaboutlearningthatgobeyondsuperficial

descriptions.

Wewouldliketoofferthesesixfeatures,thefirstthreeofwhicharegivenbyAlanSchoenfeldin

“Towardatheoryofteaching-in-context,”IssuesinEducation,bothaswaysinwhichatheorycan

contributetoresearchandascriteriaforevaluatingatheory.

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Inthispaper,wedescribeonesuchperspective,APOSTheory,inthecontextofundergraduate

mathematicseducation.Weexplaintheextenttowhichithastheabovecharacteristics,discussthe

rolethatthistheoryplaysinaresearchandcurriculumdevelopmentprogramandhowsuchaprogram

cancontributetothedevelopmentofthetheory,describebrieflyhowworkingwiththisparticular

theoryhasprovidedavehicleforbuildingacommunityofresearchersinundergraduatemathematics

education,andindicatetheuseofAPOSTheoryinspecificresearchstudies,bothbyresearcherswho

aredevelopingitaswellasothersnotconnectedwithitsdevelopment.Weprovide,inconnection

withthispaper,anannotatedbibliographyofresearchreportswhichinvolvethistheory.

APOSTheory

Thetheorywepresentbeginswiththehypothesisthatmathematicalknowledgeconsistsinan

individual’stendencytodealwithperceivedmathematicalproblemsituationsbyconstructingmental

actions,processes,andobjectsandorganizingtheminschemastomakesenseofthesituationsand

solvetheproblems.InreferencetothesementalconstructionswecallitAPOSTheory.Theideas

arisefromourattemptstoextendtothelevelofcollegiatemathematicslearningtheworkofJ.Piaget

onreflectiveabstractioninchildren’slearning.APOSTheoryisdiscussedindetailinAsiala,et.al.

(1996).Wewillarguethatthistheoreticalperspectivepossesses,atleasttosomeextent,the

characteristicslistedaboveand,moreover,hasbeenveryusefulinattemptingtounderstandstudents’

learningofabroadrangeoftopicsincalculus,abstractalgebra,statistics,discretemathematics,and

otherareasofundergraduatemathematics.Hereisabriefsummaryoftheessentialcomponentsofthe

theory.

Anactionisatransformationofobjectsperceivedbytheindividualasessentiallyexternaland

asrequiring,eitherexplicitlyorfrommemory,step-by-stepinstructionsonhowtoperformthe

operation.Forexample,anindividualwithanactionconceptionofleftcosetwouldberestrictedto

workingwithaconcretegroupsuchasZ20andheorshecouldconstructsubgroups,suchas

H={0,4,8,12,16}byformingthemultiplesof4.Thentheindividualcouldwritetheleftcosetof5as

theset5+H={1,5,9,13,17}consistingoftheelementsofZ20whichhaveremaindersof1whendivided

by4.

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Whenanactionisrepeatedandtheindividualreflectsuponit,heorshecanmakeaninternal

mentalconstructioncalledaprocesswhichtheindividualcanthinkofasperformingthesamekindof

action,butnolongerwiththeneedofexternalstimuli.Anindividualcanthinkofperforminga

processwithoutactuallydoingit,andthereforecanthinkaboutreversingitandcomposingitwith

otherprocesses.Anindividualcannotusetheactionconceptionofleftcosetdescribedabovevery

effectivelyforgroupssuchasS4,thegroupofpermutationsoffourobjectsandthesubgroupH

correspondingtothe8rigidmotionsofasquare,andnotatallforgroupsSnforlargevaluesofn.In

suchcases,theindividualmustthinkoftheleftcosetofapermutationpasthesetofallproductsph,

wherehisanelementofH.Thinkingaboutformingthissetisaprocessconceptionofcoset.

Anobjectisconstructedfromaprocesswhentheindividualbecomesawareoftheprocessasa

totalityandrealizesthattransformationscanactonit.Forexample,anindividualunderstandscosets

asobjectswhenheorshecanthinkaboutthenumberofcosetsofaparticularsubgroup,canimagine

comparingtwocosetsforequalityorfortheircardinalities,orcanapplyabinaryoperationtothesetof

allcosetsofasubgroup.

Finally,aschemaforacertainmathematicalconceptisanindividual’scollectionofactions,

processes,objects,andotherschemaswhicharelinkedbysomegeneralprinciplestoforma

frameworkintheindividual’smindthatmaybebroughttobearuponaproblemsituationinvolving

thatconcept.Thisframeworkmustbecoherentinthesensethatitgives,explicitlyorimplicitly,

meansofdeterminingwhichphenomenaareinthescopeoftheschemaandwhicharenot.Because

thistheoryconsidersthatallmathematicalentitiescanberepresentedintermsofactions,processes,

objects,andschemas,theideaofschemaisverysimilartotheconceptimagewhichTallandVinner

introducein“Conceptimageandconceptdefinitioninmathematicswithparticularreferencetolimits

andcontinuity,”EducationalStudiesinMathematics,12,151-169(1981).Ourrequirementof

coherence,however,distinguishesthetwonotions.

Thefourcomponents,action,process,object,andschemahavebeenpresentedhereina

hierarchical,orderedlist.Thisisausefulwayoftalkingabouttheseconstructionsand,insomesense,

eachconceptioninthelistmustbeconstructedbeforethenextstepispossible.Inreality,however,

whenanindividualisdevelopingherorhisunderstandingofaconcept,theconstructionsarenot

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actuallymadeinsuchalinearmanner.Withanactionconceptionoffunction,forexample,an

individualmaybelimitedtothinkingaboutformulasinvolvingletterswhichcanbemanipulatedor

replacedbynumbersandwithwhichcalculationscanbedone.Wethinkofthisnotionasprecedinga

processconception,inwhichafunctionisthoughtofasaninput-outputmachine.Whatactually

happens,however,isthatanindividualwillbeginbybeingrestrictedtocertainspecifickindsof

formulas,reflectoncalculationsandstartthinkingaboutaprocess,gobacktoanactioninterpretation,

perhapswithmoresophisticatedformulas,furtherdevelopaprocessconceptionandsoon.Inother

words,theconstructionofthesevariousconceptionsofaparticularmathematicalideaismoreofa

dialecticthanalinearsequence.

APOSTheorycanbeuseddirectlyintheanalysisofdatabyaresearcher.Inveryfinegrained

analyses,theresearchercancomparethesuccessorfailureofstudentsonamathematicaltaskwiththe

specificmentalconstructionstheymayormaynothavemade.Ifthereappeartwostudentswhoagree

intheirperformanceuptoaveryspecificmathematicalpointandthenonestudentcantakeafurther

stepwhiletheothercannot,theresearchertriestoexplainthedifferencebypointingtomental

constructionsofactions,processes,objectsand/orschemasthattheformerstudentappearstohave

madebuttheotherhasnot.Thetheorythenmakestestablepredictionsthatifaparticularcollectionof

actions,processes,objectsandschemasareconstructedinacertainmannerbyastudent,thenthis

individualwilllikelybesuccessfulusingcertainmathematicalconceptsandincertainproblem

situations.Detaileddescriptions,referredtoasgeneticdecompositions,ofschemasintermsofthese

mentalconstructionsareawayoforganizinghypothesesabouthowlearningmathematicalconcepts

cantakeplace.Thesedescriptionsalsoprovidealanguagefortalkingaboutsuchhypotheses.

DevelopmentofAPOSTheory

APOSTheoryaroseoutofanattempttounderstandthemechanismofreflectiveabstraction,

introducedbyPiagettodescribethedevelopmentoflogicalthinkinginchildren,andextendthisidea

tomoreadvancedmathematicalconcepts(Dubinsky,1991a).Thisworkhasbeencarriedonbya

smallgroupofresearcherscalledaResearchinUndergraduateMathematicsEducationCommunity

(RUMEC)whohavebeencollaboratingonspecificresearchprojectsusingAPOSTheorywithina

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broaderresearchandcurriculumdevelopmentframework.Theframeworkconsistsofessentiallythree

components:atheoreticalanalysisofacertainmathematicalconcept,thedevelopmentand

implementationofinstructionaltreatments(usingseveralnon-standardpedagogicalstrategiessuchas

cooperativelearningandconstructingmathematicalconceptsonacomputer)basedonthistheoretical

analysis,andthecollectionandanalysisofdatatotestandrefineboththeinitialtheoreticalanalysis

andtheinstruction.Thiscycleisrepeatedasoftenasnecessarytounderstandtheepistemologyofthe

conceptandtoobtaineffectivepedagogicalstrategiesforhelpingstudentslearnit.

ThetheoreticalanalysisisbasedinitiallyonthegeneralAPOStheoryandtheresearcher’s

understandingofthemathematicalconceptinquestion.Afteroneormorerepetitionsofthecycleand

revisions,itisalsobasedonthefine-grainedanalysesdescribedaboveofdataobtainedfromstudents

whoaretryingtolearnorwhohavelearnedtheconcept.Thetheoreticalanalysisproposes,intheform

ofageneticdecomposition,asetofmentalconstructionsthatastudentmightmakeinorderto

understandthemathematicalconceptbeingstudied.Thus,inthecaseoftheconceptofcosetsas

describedabove,theanalysisproposesthatthestudentshouldworkwithveryexplicitexamplesto

constructanactionconceptionofcoset;thenheorshecaninteriorizetheseactionstoformprocesses

inwhicha(left)cosetgHofanelementgofagroupGisimaginedasbeingformedbytheprocessof

iteratingthroughtheelementshofH,formingtheproductsgh,andcollectingtheminasetcalledgH;

andfinally,asaresultofapplyingactionsandprocessestoexamplesofcosets,thestudent

encapsulatestheprocessofcosetformationtothinkofcosetsasobjects.Foramoredetailed

descriptionoftheapplicationofthisapproachtocosetsandrelatedconcepts,seeAsiala,Dubinsky,et.

al.(1997).

Pedagogyisthendesignedtohelpthestudentsmakethesementalconstructionsandrelatethem

tothemathematicalconceptofcoset.Inourwork,wehaveusedcooperativelearningand

implementingmathematicalconceptsonthecomputerinaprogramminglanguagewhichsupports

manymathematicalconstructsinasyntaxverysimilartostandardmathematicalnotation.Thus

students,workingingroups,willexpresssimpleexamplesofcosetsonthecomputerasfollows.

Z20:={0..19};

op:=|(x,y)->x+y(mod20)|;

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H:={0,4,8,12,16};

5H:={1,5,9,13,17};

Tointeriorizetheactionsrepresentedbythiscomputercode,thestudentswillconstructmore

complicatedexamplesofcosets,suchasthoseappearingingroupsofsymmetries.

Sn:={[a,b,c,d]:a,b,c,din{1,2,3,4}|#{a,b,c,d}=4};

op:=|(p,q)->[p(q(i)):iin[1..4]]|;

H:={[1,2,3,4],[2,1,3,4],[3,4,1,2],[4,3,2,1]};

p:=[4,3,2,1];

pH:={p.opq:qinH};

Thelaststep,toencapsulatethisprocessconceptionofcosetstothinkofthemasobjects,canbevery

difficultformanystudents.Computeractivitiestohelpthemmayincludeformingthesetofallcosets

ofasubgroup,countingthem,andpickingtwocosetstocomparetheircardinalitiesandfindtheir

intersections.Theseactionsaredonewithcodesuchasthefollowing.

SnModH:={{p.opq:qinH}:pinSn};

#SnModH;

L:=arb(SnModH);K:=arb(SnModH);#L=#K;LinterK;

Finally,thestudentswriteacomputerprogramthatconvertsthebinaryoperationopfromanoperation

onelementsofthegrouptosubsetsofthegroup.Thisstructureallowsthemtoconstructabinary

operation(cosetproduct)onthesetofallcosetsofasubgroupandbegintoinvestigatequotient

groups.

Itisimportanttonotethatinthispedagogicalapproach,almostalloftheprogramsarewritten

bythestudents.Onehypothesisthattheresearchinvestigatesisthat,whethercompletelysuccessfulor

not,thetaskofwritingappropriatecodeleadsstudentstomakethementalconstructionsofactions,

processes,objects,andschemasproposedbythetheory.Thecomputerworkisaccompaniedby

classroomdiscussionsthatgivethestudentsanopportunitytoreflectonwhattheyhavedoneinthe

computerlabandrelatethemtomathematicalconceptsandtheirpropertiesandrelationships.Once

theconceptsareinplaceintheirminds,thestudentsareassigned(inclass,homeworkand

examinations)manystandardexercisesandproblemsrelatedtocosets.

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Afterthestudentshavebeenthroughsuchaninstructionaltreatment,quantitativeand

qualitativeinstrumentsaredesignedtodeterminethementalconceptstheymayhaveconstructedand

themathematicstheymayhavelearned.Thetheoreticalanalysispointstoquestionsresearchersmay

askintheprocessofdataanalysisandtheresultsofthisdataanalysisindicatesboththeextentto

whichtheinstructionhasbeeneffectiveandpossiblerevisionsinthegeneticdecomposition.

Thiswayofdoingresearchandcurriculumdevelopmentsimultaneouslyemphasizesboth

theoryandapplicationstoteachingpractice.

Refiningthetheory

Asnotedabove,thetheoryhelpsusanalyzedataandourattempttousethetheorytoexplain

thedatacanleadtochangesinthetheory.Thesechangescanbeoftwokinds.Usually,thegenetic

decompositionintheoriginaltheoreticalanalysisisrevisedandrefinedasaresultofthedata.Inrare

cases,itmaybenecessarytoenhancetheoveralltheory.Animportantexampleofsucharevisionis

theincorporationofthetriadconceptofPiagetandGarcia(1989)whichisleadingtoabetter

understandingoftheconstructionofschemas.Thisenhancementtothetheorywasintroducedin

Clark,et.al.(1997)wheretheyreportonstudents’understandingofthechainrule,andisbeingfurther

elaborateduponinthreecurrentstudies:sequencesofnumbers(Mathews,et.al.,inpreparation);the

chainruleanditsrelationtocompositionoffunctions(Cottrill,1999);andtherelationsbetweenthe

graphofafunctionandpropertiesofitsfirstandsecondderivatives(Baker,et.al.,submitted).Ineach

ofthesestudies,theunderstandingofschemasasdescribedabovewasnotadequatetoprovidea

satisfactoryexplanationofthedataandtheintroductionofthetriadhelpedtoelaborateadeeper

understandingofschemasandprovidebetterexplanationsofthedata.

Thetriadmechanismconsistsinthreestages,referredtoasIntra,Inter,andTrans,inthe

developmentoftheconnectionsanindividualcanmakebetweenparticularconstructswithinthe

schema,aswellasthecoherenceoftheseconnections.TheIntrastageofschemadevelopmentis

characterizedbyafocusonindividualactions,processes,andobjectsinisolationfromothercognitive

itemsofasimilarnature.Forexample,inthefunctionconcept,anindividualattheIntralevel,would

tendtofocusonasinglefunctionandthevariousactivitiesthatheorshecouldperformwithit.The

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Interstageischaracterizedbytheconstructionofrelationshipsandtransformationsamongthese

cognitiveentities.Atthisstage,anindividualmaybegintogroupitemstogetherandevencallthemby

thesamename.Inthecaseoffunctions,theindividualmightthinkaboutaddingfunctions,composing

them,etc.andevenbegintothinkofalloftheseindividualoperationsasinstancesofthesamesortof

activity:transformationoffunctions.Finally,attheTransstagetheindividualconstructsanimplicit

orexplicitunderlyingstructurethroughwhichtherelationshipsdevelopedintheInterstageare

understoodandwhichgivestheschemaacoherencebywhichtheindividualcandecidewhatisinthe

scopeoftheschemaandwhatisnot.Forexample,anindividualattheTransstageforthefunction

conceptcouldconstructvarioussystemsoftransformationsoffunctionssuchasringsoffunctions,

infinitedimensionalvectorspacesoffunctions,togetherwiththeoperationsincludedinsuch

mathematicalstructures.

ApplyingtheAPOSTheory

IncludedwiththispaperisanannotatedbibliographyofresearchrelatedtoAPOSTheory,its

ongoingdevelopmentanditsuseinspecificresearchstudies.Thisresearchconcernsmathematical

conceptssuchas:functions;varioustopicsinabstractalgebraincludingbinaryoperations,groups,

subgroups,cosets,normalityandquotientgroups;topicsindiscretemathematicssuchasmathematical

induction,permutations,symmetries,existentialanduniversalquantifiers;topicsincalculusincluding

limits,thechainrule,graphicalunderstandingofthederivativeandinfinitesequencesofnumbers;

topicsinstatisticssuchasmean,standarddeviationandthecentrallimittheorem;elementarynumber

theorytopicssuchasplacevalueinbasennumbers,divisibility,multiplesandconversionofnumbers

fromonebasetoanother;andfractions.Inmostofthiswork,thecontextforthestudiesarecollegiate

levelmathematicstopicsandundergraduatestudents.Inthecaseofthenumbertheorystudies,the

researchersexaminetheunderstandingofpre-collegemathematicsconceptsbycollegestudents

preparingtobeteachers.Finally,somestudiessuchasthatoffractions,showthattheAPOSTheory,

developedfor“advanced”mathematicalthinking,isalsoausefultoolinstudyingstudents’

understandingofmorebasicmathematicalconcepts.

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Thetotalityofthisbodyofwork,muchofitdonebyRUMECmembersinvolvedindeveloping

thetheory,butanincreasingamountdonebyindividualresearchershavingnoconnectionwith

RUMECortheconstructionofthetheory,suggeststhatAPOSTheoryisatoolthatcanbeused

objectivelytoexplainstudentdifficultieswithabroadrangeofmathematicalconceptsandtosuggest

waysthatstudentscanlearntheseconcepts.APOSTheorycanpointustowardspedagogicalstrategies

thatleadtomarkedimprovementinstudentlearningofcomplexorabstractmathematicalconceptsand

students’useoftheseconceptstoprovetheorems,provideexamples,andsolveproblems.Data

supportingthisassertioncanbefoundinthepaperslistedinthebibliography.

UsingtheAPOSTheorytodevelopacommunityofresearchers

Atthisstageinthedevelopmentofresearchinundergraduatemathematicseducation,thereis

neitherasufficientlylargenumberofresearchersnorenoughgraduateschoolprogramstotrainnew

researchers.Otherapproaches,suchasexperiencedandnoviceresearchersworkingtogetherinteams

onspecificresearchproblems,needtobeemployedatleastonatemporarybasis.RUMECisone

exampleofaresearchcommunitythathasutilizedthisapproachintrainingnewresearchers.

Inaddition,aspecifictheorycanbeusedtounifyandfocustheworkofsuchgroups.The

initialgroupofresearchersinRUMEC,about30total,madeadecisiontofocustheirresearchwork

aroundtheAPOSTheory.Thiswasnotforthepurposeofestablishingdogmaorcreatingaclosed

researchcommunity,butratheritwasadecisionbasedoncurrentinterestsandneedsofthegroupof

researchers.

RUMECwasformedbyacombinationofestablishedandbeginningresearchersin

mathematicseducation.ThusoneimportantroleofRUMECwasthementoringofthesenew

researchers.HavingasingletheoreticalperspectiveinwhichtheworkofRUMECwasinitially

groundedwasbeneficialforthosejustbeginninginthisarea.AtthemeetingsofRUMEC,discussions

couldfocusnotonlyonthedetailsoftheindividualprojectsastheydeveloped,butalsoonthegeneral

theoryunderlyingallofthework.Inaddition,thegroup’sgeneralinterestinthistheoryandfrequent

discussionsaboutitinthecontextofactiveresearchprojectshasledtogrowthinthetheoryitself.

Thiswasthecase,forexample,inthedevelopmentofthetriadasatoolforunderstandingschemas.

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Astheworkofthisgroupmatures,individualsarebeginningtouseothertheoreticalperspectivesand

othermodesofdoingresearch.

Summary

Inthispaper,wehavementionedsixwaysinwhichatheorycancontributetoresearchandwe

suggestthatthislistcanbeusedascriteriaforevaluatingatheory.Wehavedescribedhowonesuch

perspective,APOSTheoryisbeingused,inanorganizedway,bymembersofRUMECandothersto

conductresearchanddevelopcurriculum.Wehaveshownhowobservingstudents’successinmaking

ornotmakingmentalconstructionsproposedbythetheoryandusingsuchobservationstoanalyzedata

canorganizeourthinkingaboutlearningmathematicalconcepts,provideexplanationsofstudent

difficultiesandpredictsuccessorfailureinunderstandingamathematicalconcept.Thereisawide

rangeofmathematicalconceptstowhichAPOSTheorycanandhasbeenappliedandthistheoryis

usedasalanguageforcommunicationofideasaboutlearning.Wehavealsoseenhowthetheoryis

groundedindata,andhasbeenusedasavehicleforbuildingacommunityofresearchers.Yetitsuse

isnotrestrictedtomembersofthatcommunity.Finally,weprovideanannotatedbibliographywhich

presentsfurtherdetailsaboutthistheoryanditsuseinresearchinundergraduatemathematics

education.

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AnAnnotatedBibliographyofworks

whichdeveloporutilizeAPOSTheory

I.Arnon.Teachingfractionsinelementaryschoolusingthesoftware“FractionsasEquivalence

Classes”oftheCentreforEducationalTechnology,TheNinthAnnualConferenceforComputersin

Education,TheIsraeliOrganizationforComputersinEducation,BookofAbstracts,Tel-Aviv,Israel,

p.48,1992.(InHebrew).

I.Arnon,R.NirenburgandM.Sukenik.Teachingdecimalnumbersusingconcreteobjects,The

SecondConferenceoftheAssociationfortheAdvancementoftheMathematicalEducationinIsrael,

BookofAbstracts,Jerusalem,Israel,p.19,1995.(InHebrew).

I.Arnon.Refiningtheuseofconcreteobjectsforteachingmathematicstochildrenattheageof

concreteoperations,TheThirdConferenceoftheAssociationfortheAdvancementoftheMathematical

EducationinIsrael,BookofAbstracts,Jerusalem,Israel,p.69,1996.(InHebrew).

I.Arnon.Inthemind’seye:Howchildrendevelopmathematicalconcepts–extendingPiaget's

theory.Doctoraldissertation,SchoolofEducation,HaifaUniversity,1998a.

I.Arnon.Similarstagesinthedevelopmentsoftheconceptofrationalnumberandtheconceptof

decimalnumber,andpossiblerelationsbetweentheirdevelopments,TheFifthConferenceofthe

AssociationfortheAdvancementoftheMathematicalEducationinIsrael,BookofAbstracts.Be’er-

Tuvia,Israel,p.42,1998b.(InHebrew).

ThestudiesbyArnonandhercolleagueslistedabovedealwiththedevelopmentof

mathematicalconceptsbyelementaryschoolchildren.Havingcreatedaframeworkthat

combinesAPOStheory,Nesher’stheoryonLearningSystems,andYerushalmy’sideasof

multi-representation,sheinvestigatestheintroductionofmathematicalconceptsasconcrete

actionsversustheirintroductionasconcreteobjects.Sheestablishesdevelopmentalpathsfor

certainfraction-concepts.Shefindsthatstudentstowhomthefractionswereintroducedas

concreteactionsprogressedbetteralongthesepathsthanstudentstowhomthefractionswere

introducedasconcreteobjects.Inaddition,thefindingsestablishthefollowingstageinthe

developmentofconcreteactionsintoabstractobjects:afterabandoningtheconcretematerials,

andbeforeachievingabstractlevels,childrenperformtheconcreteactionsintheirimagination.

ThiscorrespondstotheinteriorizationofAPOStheory.

M.Artigue,Ense?anzayaprendizajedelanálisiselemental:?quésepuedeaprenderdelas

investigacionesdidácticasyloscambioscurriculares?RevistaLatinoamericanadeInvestigaciónen

MatiemáticaEducativa,1,1,40-55,1998.

Inthefirstpartofthispaper,theauthordiscussesanumberofstudentdifficultiesandtriesto

explainthemusingvarioustheoriesoflearningincludingAPOSTheory.Students’

unwillingnesstoacceptthat0.999…isequalto1isexplained,forexample,byinterpretingthe

formerasaprocess,thelatterasanobjectsothatthetwocannotbeseenasequaluntilthe

studentisabletoencapsulatetheprocesswhichisageneraldifficulty.Inthesecondpartofthe

paper,theauthordiscussesthemeasuresthathavebeentakeninFranceduringthe20th

Centurytoovercomethesedifficulties.

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M.Asiala,A.Brown,D.DeVries,E.Dubinsky,D.MathewsandK.Thomas.Aframeworkfor

researchandcurriculumdevelopmentinundergraduatemathematicseducation,ResearchinCollegiate

MathematicsEducationII,CBMSIssuesinMathematicsEducation,6,1-32,1996.

Theauthorsdetailaresearchframeworkwiththreecomponentsandgiveexamplesofits

application.Theframeworkutilizesqualitativemethodsforresearchandisbasedonavery

specifictheoreticalperspectivethatwasdevelopedthroughattemptstounderstandtheideasof

Piagetconcerningreflectiveabstractionandreconstructtheminthecontextofcollegelevel

mathematics.Forthefirstcomponent,thetheoreticalanalysis,theauthorspresenttheAPOS

theory.Forthesecondcomponent,theauthorsdescribespecificinstructionaltreatments,

includingtheACEteachingcycle(activities,classdiscussion,andexercises),cooperative

learning,andtheuseoftheprogramminglanguageISETL.Thefinalcomponentconsistsof

datacollection