Studies in Reflecting Abstraction

336 Pages·2001·103.19 MB·English

by  Jean Piaget

Studies in ReflectingAbstraction
Jean Piaget
edited and translated by
Robert L.Campbell
Firstpublished2001byPsychologyPress
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Recherchessurl’abstractionréfléchissante.English
Studiesinreflectingabstraction/JeanPiaget[andhiscollaborators];editedand
translatedbyRobertL.Campbell.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN1—84169-157-7
1.Abstractioninchildren.2.Schemas(Psychology)inchildren. I.Piaget,Jean,
1896- 11.Campbell,RobertL.III.Title.
BF723.A26R43132000
155.4'1343--dc21
00-044553
ISBN1-84169-157-7
Contents
REFLECTINGABSTRACTIONINCONTEXT(RobertL.Campbell) 1
I. Structures,equilibration,andreflectingabstraction 1
II. Originsofthepresentvolume 7
III. Piaget’sfirstthoughtsonreflectingabstraction 8
IV. ReflectingabstractioninthefullcontextofPiaget’s
late-periodthinking 14
V. Doweengageinreflectingabstractionwheneverweare
notrelyingonperception? 21
VI. Thetranslation 25
VII. Acknowledgments 26
PARTONE:LOGICO-ARITHMETICALORALGEBRAIC
ABSTRACTION 29
Acknowledgments 32
1. Abstraction,differentiation,andintegrationintheuseof
elementaryarithmeticoperations(withAlina Szeminska) 33
2. Theconstructionofcommonmultiples
(withJean-LouisKaufmannandJ.-F.Bourquin) 55
3. Theinversionofarithmeticoperations(withAlbertMoreau) 69
4. Abstractionandgeneralizationduringtransfersofunits
(withPierreMaessinger) 87
vi CONTENTS
5. Problemsofclassinclusionandlogicalimplication(withDaphné
Vaelin-Liambey andIoannaBerthoud-Papandropoulou) 105
6. Theformationofanalogies(withJacquesMontangeroand
J.-B.Billeter) 139
7. FromconcreteformsoftheKleingrouptothe[NRCgroup
(withAlbertoMunari) 153
I. Rotationsof180° 156
11. Systemsoffourobjectsorclasses 162
PARTTWO: THEABSTRACTIONOFORDER 169
8. Additiveandexponentialseries(withThaliaVergopoulo) 171
9. Conditionsonreadingoffcomplexadditiveseries 179
I. Readingoffandcontinuingalreadygivenseries
(withJeanineCuazandJacquelineCambon) 179
II. Abstractioninthecourseofimitatinganotherperson’sactions
(withJean-JacquesDucret) 188
10. Orderingpracticalactivity 193
I. Directordering(withSilviaDayanandEdithDekkers) 193
11. Directandinverseorderingsofactions
(withMadeleineSpycherand-Claudelein) 199
11. Changesinorderingornecessarybacktracking
(withAlex Blanchet) 207
ConclusionsofPartTwo 215
PARTTHREE:THEABSTRACTIONOFSPATIAL
RELATIONSHIPS 217
12. Relationsbetweenthesurfaceareaandtheperimeterofrectangles
(withJean-PaulBronckartandEdouardRappeduCher) 219
13. Themovementsofasuspendedprojectile
(withMichelangeloA.FliickigerandIsabelleFliickiger) 233
14. Diagonals(withMargotLavalléeandMariaSoléSugranes) 249
15. Thedisplacementofareferencepointinasystemofcyclicmovements
(withEdithAckermannandNicoleCox) 257
16. Abstractionfromdisplacementsandfromtheircoordinations
(withJacquelineCambonandJeanineCuaz) 265
17. Rotationsandtranslations(withJacquesdeLannoy) 281
CONTENTS vii
18. Therotationofabararoundapivotduringthesensorimotorperiod(with
ClaudeMonnier) 289
ConclusionsofPartThree 297
GENERALCONCLUSIONS 303
I Projection 304
II. Thecreationofnoveltiesspecifictoreflectingabstraction 307
III. Equilibration,thesourceofnovelties,andrelationships
betweentheintensionsandextensionsofstructures 312
IV. Empiricalandreflectingabstraction 317
Authorindex 323
Subjectindex 327
Reflectingabstraction in context
RobertL.Campbell
|.STRUCTURES, EQUILIBRATION,
AND REFLECTINGABSTRACTION
AnintroductiontoJeanPiaget’sconceptofreflectingabstractionwillimprove
theuser-friendlinessofwhatyouareabouttoread.Butmakingsuchapresenta-
tionislikedesigningusablesoftware.Itcannotbedoneinasinglepass;ittakes
iterations.
Readerswho are moderately acquainted with Piaget’s ideas will have a
questionrightaway.Almostsurelyitis“reflectiveabstraction?thattheyhave
heardor readabout, not “reflectingabstraction.” Inthe past,most English-
speaking Piagetscholars usedthe genericphrase“reflective abstraction.” In
fact,IinitiallyplannedtotranslatethetitleofthisbookasStudiesinReflective
Abstraction.ButIchangedmymindaftermakingaroughtranslationofthefirst
twochapters.
In this book, Piaget proposes a basic developmental process called
abstractionre'fléchissante(literally,reflectingabstraction).At the sametime
he regularly calls attention to a higher-order refinement of this process:
abstraction réfléchie (literally, reflected abstraction). In turn, abstraction
réfléchiehasitsownhigher-orderrefinement,whichPiagetcallsmétaréflexion
(metareflection). Throughout the book, Piaget’s interpretations ofdevelop-
mental trends usually require him to distinguish reflecting from reflected
abstraction,andofteninducehimtobringinmetareflectionaswell.Itisthis
hierarchy oftypes ofreflectingabstractionthat haspromptedmeto render
2 STUDIESINREFLECTINGABSTRACTION
abstractionréfléchissanteliterally;nothingelsewouldallowthetranslationto
payfullheedtoPiaget’snuances,andStudiesinReflectingAbstractionisvery
muchabookofnuances.l
Havingexplainedmychoiceofterminology,Iwilltrytodescribereflecting
abstractionasoneofthebareessentialsofPiaget’sgeneticepistemology.This
taskisnotsoeasyasitmightappear,becausereflectingabstractionbecameone
oftheessentialsinPiaget’sframework;itdidnotstartoutthatway.Piagetand
his collaborators didnot get aroundto a focused investigationofreflecting
abstractionuntil he hadbeenelaborating andtesting his theories ofhuman
developmentforover50years.Reflectingabstractiondidnotevenappearasa
theoreticalconceptuntil30years intothisprogramofresearch.
Consequently Iwill begin with a back-of-the-envelope presentation,
intendedtohelpthereadergetsomesenseofreflectingabstractionbeforebeing
confrontedwiththecomplexwebofissuesthatPiagetraisesinthisbook.Iwill
takeupthe entanglementsbetweenreflectingabstractionandotherdevelop-
mental processes later on in this introduction, and only afterIhave said
something about the developmental history of this idea in Piaget’s own
thinking.
Jean Piaget (1896—1980) memorably labeled his framework not as
psychology,butasgeneticepistemology:anintrinsicallydevelopmentaltheory
of human knowledge. Genetic epistemology is about two things: what
knowledgeconsistsof,andthewaysinwhichthisknowledgedevelops.
ForPiaget,knowledgeisnotamatterofpicturesorsentencesorsymbolic
data structures in the mind that correspond to structures in the world;
knowledgeisbasicallypragmaticoraction-oriented.Knowledgeisfundamen-
tally operative; it isknowledgeofwhat to dowith somethingundercertain
possible conditions. Or it is knowledge of what that thing will do under
different conditions. (Piagetdidadmit aninferiorspecies ofknowledgethat
merely pictured static configurations; he called the lesser kindfigurative
knowledge.ForPiaget,perceptionand—usually—languagearefigurative.But
figurativeknowledgeisnotwheretheactionis;Piagetregularlyinsiststhatit
hasnothingimportanttocontributetohumandevelopment.)
1*Istillreferto“reflectiveabstraction”inmyowntheorizing;seeRobertL.Campbelland
MarkH.Bickhard,Knowinglevelsanddevelopmentalstages(Basel:S.Karger,1986)andRobert
L.Campbell and Mark H.Bickhard,Types ofconstraints on development: An interactivist
approach,DevelopmentalReview,12,311—338(1992).Theinteractivistconceptionofknowl-
edgeanddevelopmentthatIwork inincorporatesasharpdifferentiationbetweenlearningand
reflectiveabstraction,whereas Piaget(aswe will see)didnotdifferentiatenearly so sharply
betweenequilibrationandreflectingabstraction.Moreover,ininteractivisttheorythereisnoneed
todistinguishbetweenvarietiesofreflectiveabstraction:Adifferenttheoreticalprinciple—the
hierarchyofknowinglevels(KnowingLevel1,KnowingLevel2,KnowingLevel3,etc.)——does
muchofthework that Piagetaccomplishesbydistinguishingbetweenreflectingabstraction,
reflectedabstraction,andmetareflection.
REFLECTINGABSTRACTIONINCONTEXT 3
ForPiaget,operativeknowledgeconsistsofcognitivestructures.Themost
basictypeofstructure,alreadyavailabletoinfants,isthesensorimotoraction
scheme.2Atypicalearlyschemeisonethatspecifieshowtogetthemobilethat
hangsoverthe baby’scribmoving(bykickingit,for instance).3Duringthe
course ofdevelopment,morepowerfulandsophisticatedkindsofstructures
become possible, such as the structures ofconcrete operations that, during
middlechildhood,permitchildrentoengageinelementarykindsofreasoning
aboutclassification,number,andputtingthingsinorder.WhilePiagetinvested
agooddealofeffortindescribingstructuresmathematically,usingsystemsof
abstractalgebraorthe symbolismofmathematicallogic,the specifics ofhis
mathematicaltreatment areusuallynotatissueinthisvolume.
Ifknowledge consists of cognitive structures, then development comes
downtowhatstructuresdo.AccordingtoPiaget,operativestructuresneverjust
sit there; they demand to be applied. When the knowing subject applies a
cognitivestructuretothe environment,orassimilatesthe environmenttothe
structure, the structure may work without modification; it may fillfill the
knower’s goals as expected. But insome cases assimilationwill be unsuc-
cessful.Inthosecases,applyingtheschemewillnotleadtoattainingthegoalas
expected.Forinstance,ifayoung childhasacquiredafly-swatting scheme,
applyingthatschemetoanotherhouseflyisroutineassimilation,andnormally
thegoalwillbereached(theworstthatwouldusuallyhappenismissingthefly
andhavingto try again). Butifthe childattempts to apply the scheme.toa
differentkindofflyinginsect,sayahornet,theresultwillprobablybeunsuc-
cessfulassimilation:swattingahornetresultsingettingstung;
Whenunsuccessfulassimilationhastakenplace,thechildneedstomodify
the scheme,inordertoaccommodateittotheenvironment.Forinstance,the
childmaynotswatbeforecheckingvisuallywhethertheinsectlookslikeafly
(inwhich case swatting is truly indicated) or like a hornet (inwhich case
somethingdifferentisindicated,likemovingawayquietly).Accommodating
theschemetotheenvironmentmaymeanputtingrestrictionsonit(e.g.,usethis
fly-swatting scheme only with non-stinginginsects)ordifferentiatingitinto
oneormoresubschemes.Onoccasion,entirelynewschemeswillneedtobe
constructedifsuccessfulaccommodationistotakeplace.
Piagetbelievedthatschemestendtogetapplied,soassimilationandaccom-
modationhappennaturally.Heregardeddevelopmentasdrivenbyprocesses
that tend toward a balance, or equilibrium, between assimilation and
2*FrequentlymistranslatedschemainEnglish.PiagetusedtheFrenchwordschemeforthe
basicoperativestructureswearediscussinghere.HereservedtheFrenchwordschémaforsome-
thingdifferentandfarlessbasic.
3* SeeJeanPiaget,Lanaissancede1’intelligencechezl’enfant(Neuchatel:Delachauxet
Niestlé,1936;translatedbyMargaretCookasTheoriginsofintelligenceinchildren,NewYork:
InternationalUniversitiesPress,1952).
4 STUDIESINREFLECTINGABSTRACTION
accommodation. Hence, his most general account of development was an
accountofequilibration.Theequilibrationofcognitivestructuresmaytakemore
complicatedformsdependingonthetypeofstructure.Equilibrationalsovaries
indetail,dependingonwhetherstructuresareassimilatingandaccommodating
thephysicalenvironmentorotherstructureswithintheknowingsubject.4
Evena fully elaboratedtreatment ofequilibration leaves something out,
however. Basic equilibration pertains to interactions with the physical
environment; the more refined forms pertain to internal adjustments and
integrationswithin the knowingsubject. Butnone ofthem explainhow we
cometoknowaboutourownprocessesofknowing,oraboutthecoordinations
ofourownactions. How,for instance,dowe becomeableto give a correct
descriptionofhowwe navigatethroughawire maze?(Thisisatask thatwe
mostlikelymasteredat apracticallevellongbefore—seeChapter 11 inthis
book.)Howdowebecomeabletorecognizethenumberoftimesthatweadded
some numberofobjects (Chapter2 here)? How dowe buildnew cognitive
structuresthatareactuallyaboutoldstructures (aswhenwe constructformal
operations,whichPiagetregardsasoperationsonconcreteoperations)?
Thisiswhereabstractionre'fléchissante,reflectingabstraction,comesinto
Piaget’stheory.Itsspecialfunctionisabstractingpropertiesofourprocessesof
knowing,orouractioncoordinations.Itdoessointwophases.Thefirstphase,
réfléchissement,projectsastructureatalowerdevelopmentallevel(suchasthe
actioncoordinationofinterest)ontoahigherlevel(wherethecoordinationmay
nowbeunderstoodconsciouslyandexplicitly).The secondphase,réflexion,
reorganizes the structure at the higher level; an explicit understanding of
something about our knowledge or our actions is not a mere copy ofthe
previouscognitivestructure,anditneedstobeabletobeintegratedwithother
new structures at the higherlevel.Piaget’splay onwords, emphasizingthe
opticalmeaningofreflectioninthefirstcase,andthecognitivemeaninginthe
second case, does not work in English, so we will call these two phases
projectionandreflectioninthistranslation.
Consideranexample.InChapter2ofthisbook,Piagetexamineschildren’s
thinking about multiplication. Multiplication looks like “just” repeated
addition,yetchildrenfinditmuchmoredifficultthanaddition.Thereasonfor
this difference, accordingto Piaget,'isthat inorder to understandmultipli-
cation, children have to do two things. Ofcourse, they have to be able to
recognizehowmuchtheyareaddingeachtime.Thisisempiricalabstraction—
4* See Jean Piaget, L’équilibration des structures cognitives: Probléme central du
développement(Paris:PressesUniversitairesdeFrance,1975;translatedbyTerranceBrownand
KishoreJulianThampyasTheequilibrationofcognitivestructures:Thecentralproblemofintel-
lectualdevelopment,Chicago:UniversityofChicagoPress,1985).Anearliertranslationofthis
book,issuedin1977underadifferenttitle,isemphaticallynotrecommended.