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1. FLM Publishing Association
The Post-Piagetian Child: Early Mathematical Developments and a Role for Structured
Materials
Author(s): Tony Wing
Source: For the Learning of Mathematics, Vol. 29, No. 2 (Jul., 2009), pp. 8-13
Published by: FLM Publishing Association
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2. THE POST-PIAGETIANCHILD: EARLY
MATHEMATICALDEVELOPMENTSAND A
ROLE FOR STRUCTUREDMATERIALS
TONY WING
The firsttime I remembermeeting the term'mathematical
development' was in a UK government publication pre-
scribing a curriculumfor state-funded education of the
under-fives(DfEE/QCA,2000). It struckme as odd then,
andstill does, thatanythingso ill-defined,variedandplural
as whatis called 'mathematics'mightbe thoughtto have a
singular 'development', but there it was, officially sanc-
tionedandprescribedforteachersinaseriesof stagescalled
'SteppingStones'.
Thatofficialdocumentof 2000reflectedwidespreadagree-
ment within the communityof experts informingthe UK
governmentof thetimethatyoungchildren's'development'
of calculatingis foundeduponinitialmasteryof countingin
'everyday'situations,andthatadditionandsubtractionare
thereafterconstructedby childrenthroughtheinventionof a
predictable and necessary sequence of counting-in-ones
strategies(see, e.g., Fuson, 1992; Gray, 1991;Thompson,
1997).Interestingly,theview thatcounting-in-onesis thesin-
gularsufficientbasis of children'sinventionof addingand
subtractinghasoftenbeenreportedintheliteratureas partof
amuchdeeperparadigmshiftawayfromthedominationof a
Piagetianperspectiveduringthe 1960s,withinwhichthekind
of logicalunderstandingthatmightlie behindchildren's'pre-
conservation'countingactivitieshadnotablybeenquestioned.
Thisgeneralshift of thinkinghas also oftenbeen described
morepositivelyasinvolvingashifttowardsincreasingrecog-
nitionof theimportanceof Vygotsky'sdevelopmentaltheories
- sometimescharacterisedratherglobally as a shift froma
Piagetian 'individualistic' view of logico-mathematical
development toward the embracing of a thoroughgoing
'social' interpretationof thinkingandits origins.
Notable within this shifting interpretationto today's
'mathematicaldevelopment'has been the workof Walker-
dine,throughherapplicationof Foucault's ideasconcerning
'discourse'andthe historicallysituatednatureof knowing.
Her(1984, 1988) deconstructionof 'the Piagetianchild' of
the 1960sshowedhow thethenprevailingdiscourse,exem-
plified in the English Nuffield Mathematics Project,
'regulated'classroompracticesof the time, 'producingits
own truths'and a 'Piagetian child subject' as if 'natural'.
Herworkhas itself also inevitablyconstituteda key contri-
bution to the formationof a more recent 'post-Piagetian'
discourse.Sincethe 1980smanyotherinfluenceshavecon-
tributed to the regulating discourse of today's early
schooling (mostnotablythe influences of Vygotskyandof
socialanthropology),andthediscourseof todayis also 'pro-
ducingits own (historicallysituated)truths'in constructing
a socially natural'post-Piagetianchild subject'.The regu-
lating of classroom practices within the UK by today's
discoursehasbeeneffectedparticularlypowerfullythrough
the centralisedprescriptionsof 'informed'government,as
well as by the discursive practices of those contributing
experts by whom that government has considered itself
informed. (The implementation of a governmentbacked
NationalNumeracyStrategy(NNS) in Englandfrom 1998
onwardshasbeenoptimisticallycharacterisedby one of the
key civil servantsresponsible,MichaelBarber,as a period
of 'informedprescription'- see Fullan,2003, p. 4.)
The complexities andregulatingeffects of today's pre-
vailing discourse cannot be fully explored until history
advancesin ways thatwill permita yet newerretrospective
deconstruction(perhaps),froma yet to be constructedand
contrastingperspective.My concernhere andat this point
in history is simply to point to some instances where the
discourseof todayis producingits own truthsin anunhelp-
fully narrowing way, and to venture upon beginning an
examinationof the productionof whatmightreasonablybe
called the 'Vygotskianstudentsubject'.The 'mathematical
development' of a 'Vygotskian student' is what is impli-
cated most clearly within those expert views informing
recentandcurrentprescriptionsof UK government.Inpar-
ticular,I questionthe view thatchildren'naturally'develop
(orre-invent)additionandsubtractionstrategiesby passing
througha necessary sequence of counting-in-ones proce-
duresarisingwithineverydaycontexts,andthatnumerical
understandingmay only develop with the use of physical
objects and imagery somehow meaningfully 'situated'
within'everyday'pre-schoolcontexts.
The role of counting and the 'everyday' in
today's mathematical development
The Vygotskianstudentmoves throughthreekey develop-
mentalstages,eachcharacterisedintermsof socialinstitutions:
pre-school, school, andwork (adolescence). These Vygot-
skian social divisions have proved durable over time,
maintaining a set of constructed oppositions in today's
discourse between an 'everyday' world and a world of
'schooling', andbetween'schooling'andanopposedworld
of 'work'. Developing and strengthening these discur-
sive divisions,contributionsfromsocial anthropologyhave
8 For the Learning of Mathematics 29, 2 (July, 2009)
FLM Publishing Association, Edmonton, Alberta, Canada
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3. characterisedpracticesof a numberof childrenin Brazilian
streets,of dairyworkers,andof Liberiantailorsas likewise
opposedto those of traditional(orformal)'schooling', and
concludedthata 'situated'cognitionexplainshow children's
likewiseconstructed'transitions'betweensuchsocialworlds
areinevitablyproblematic.Withinthisdiscourse,a preferred
'mathematicaldevelopment' is producedthatnecessarily
beginswithin'everyday'social(andsituatedly'meaningful')
practicesof homes (oranywhereelse 'not-school'), andis
mosteffectively builtuponby enlightenedpracticeswithin
schools thatprogressto the abstractobjectsof mathematics
directlyfrom 'everyday'situationsandmaterialsfound in
'reallife', thus obviatingthe constructedandproblematic
'transitions'(seeAubrey,1997).Theseconstructedandoppos-
ing social worlds,andthedifficultiesof transitionsbetween
them,areevidencedwithinthediscoursebyopposingdescrip-
tionsof selectedexamplesandsocial interactionsofferedas
representativeof the'everyday',of 'schooling',andof 'work'
generally (see Lave, 1988 and Nunes & Bryant, 1996 for
influentialexamples).Suchdiscursiveobjectification(Sfard,
2008, chap.2) of somethingcalled 'theeveryday',although
in some ways inevitablein generaldiscussion, is neverthe-
less notreal,andeasilyobscuresimportantfeaturesof actual
('lived')everyday(andschool,andwork)situationsin which
bothlearningandteachingoccur.
Duringone sequenceof a UK NNS (2000) trainingvideo
called EveryChild Counts,a young child is seen 'making
his own arithmeticdiscoveries'withinthe 'everyday'situa-
tion of cooking with his mother,at home. The voice-over
explainshow, "Theeverydayenvironmentprovidesoppor-
tunitiesfor all childrento formmathematicalconcepts for
themselves, in theirown way"suggesting thatthe insights
apparentlyarisingonthisoccasionaresufficientlyaccounted
for by the 'everydayness' of the 'opportunities' and the
child's own mentalactivity in discussion with his mother
withinthemeaningful'situation'.Whatis notthoughtwor-
thy of mention however is the fact that this particular
'everyday' environmentplaced before the child a 4 X 3
structuredarrayof spaces in the formof a bakingtray,and
thatthe child uses this structure(by repeatedlypointingas
he speaks and works) to explain his latest whole number
calculatingwithin'12'to his mother.A reasonableinterpre-
tationis thatitis thestructurehe findswithinthe 'everyday'
situationthatis supportinghis new insights,notthe 'every-
day' natureof the baking tray and the 'situated' cooking
activity.Similarly,it is the structureimplicatedwithinthe
'workpractices'of dairyworkersin anotherexamplethatis
exploitedin theircalculating(see Scribner,1997).
To objectify 'the everyday'within a Vygotskian(or any
other)discourseeffectivelyconcealsaninfinitevarietycalled
up by the question "Whose'everyday'?' (and similarly,
"Whose'schooling'?', 'Whose'work'?'), andcan thusalso
lead to key assumptionsaboutwhat is to count as 'every-
day' in research studies. The whole corpus of studies
purportingto showthatyoungchildren'sinventionof calcu-
lating'naturallydevelops'througha predictablesequenceof
counting-in-onesstrategies(Fuson,1992;Thompson,1997)
involvesyoungchildrenrespondingto wordproblemsin sit-
uationsandcircumstancesrestrictedto 'everyday'material,
thatis, wordproblemsframedwithin 'everyday'contexts,
and only discrete objects andfingers available to support
children'sproblemsolving.Thus,the verychildrenwho are
claimedtobe demonstratinga 'naturaldevelopment'towards
the inventionof additionandsubtractionthroughcounting-
in-onesstrategieshaveactuallybeenplacedwithina history
andconditionsthatruleout otherpossibilities. The discur-
sive 'truth'thatchildren'slearningin school is built upon
pre-schoolunderstandingsthatare'situated'withinan'every-
day' world has determinedthe situated conditions of the
studies themselves. In Walkerdine's(1988) terms, these
'empirical'researchcontributionsallowthediscourseto
[...] producethe possibility of certainbehavioursand
then readthemback as 'true', creatinga normalizing
vision of the 'natural'child, (p. 5)
Childrenuse counting-in-onesstrategiesto solve addition
andsubtractionproblemsin circumstanceswhereotherpos-
sibilities have not arisen.Suchconditionsdo not disclose a
'natural development', no matter how many times the
researchis replicated,merelyone possible pathto progress
withinwhich representationalpossibilities have been arbi-
trarilyorconsciouslyrestrictedto 'theeveryday'.
Vygotsky'swritinghas not been the only influencecon-
tributing to objectification of the 'everyday' (anywhere
'not-school' in children's lives), which is why I preferto
call the constructedsubjectof today'sprevailingdiscourse
morebroadlythe 'post-Piagetianchild' [1] to signify both
manifold contributions and a place in history [2]. Hans
Freudenthal(1973) arguedthathis preferredapproachto
Elementaryarithmeticinstruction is integrated [...]
withthevariouslifeactivitiesof thechild,(pp. 139-40,
emphasisadded)
Freudenthal'sinfluence is particularlysignificant because
developersof the RealisticMathematicsEducation(RME)
approach he inspired frequently oppose a constructed
'everyday' ('realistic') worldto an objectified 'traditional
schooling' not only in the sense of noting difference (as
Vygotskyhaddone)butinafurtherevaluativesenseaswell.
WithinRMEthe favoured 'realistic' approachesarecom-
monly contrastedto a pejoratively termed 'transmission
teaching' apparentlycharacteristicof 'traditionalschool-
ing' - which is of coursealso to be seen by opposition,and
also pejoratively,as 'unrealistic'.
Evaluativediscursive objectification of the 'everyday',
the 'realistic'(whose'realistic'?),andof 'traditionalschool-
ing' enables what Goldin (2008) has called 'dismissive
theorizing'.Once it is discursivelyestablishedthat'natural
development'builds uponearly 'everyday' (or 'realistic')
experiences,thenorganised'schooling'mustbe adaptedto
build uponthose very 'everyday' experiences if it is to be
successful.Materialsdesignedspecificallyfor aneducative
function (such as Cuisenaire rods and Dienes blocks)
become anathemathrough being confined to a world of
'schooling'- inthistheystandopposedto 'everyday'objects
and 'realistic'scenarios,andbecome therefore(againpejo-
ratively) artificial, being outside everyday and realistic
experience,andtherebyeffectivelymeaninglessto children.
More'empirical'evidencecanbe accumulatedshowingthat
such artificialdevices 'don't work' [3], andtheorizingcan
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4. be calleduponto explainwhy this is so. Gravemeijeret al
(2000) for instance,claim thatthe 'realistic'and 'situated'
approachof RMEmaybe
[...] contrastedwith the traditionaltransmissionview
of instruction in which mathematical symbols are
treatedas referringunambiguouslyto fixed, given ref-
erents.The teacher'srole in this traditionalscheme is
typicallycastas thatof explainingwhatsymbolsmean
andhow they areto be used by linking themto refer-
ents. Frequently this involves the use of concrete
materialsor visualmodels designedto ensurethatstu-
dentslearnmathematicsmeaningfully.Implicitin this
approachis the assumptionthatsuch models embody
the mathematical concepts and relationships to be
learned.However,fromourperspective,the explana-
torypowerof such didacticmodels can be seen "only
in theeye of thebeholder",(p. 226)
And in Gravemeijer, Lehrer, van Oers, and Verschaffel
(2002),
As expert adults, we are able to see these [number]
relationshipsin the materialbecause we have already
constructedthese relationships, but for the students
who have not alreadyconstructedthese relationships,
the Dienes blocks arejust pieces of wood. Thisdoes
notleave theteachermanyoptions,otherthanto spell
out the correspondences between the blocks and the
algorithmin detail The consequences of thatpolicy,
however,will soonerbe rotealgorithmization[sic]than
understanding,(pp. 11-12, emphasisadded)
Icallsucharguments'dismissivetheorizing'becauseacare-
fully limitedandflawedtheoreticalargumentis constructed
andpositioned as the alternativeto the writers' own pre-
ferredrationale. In Gravemeijeret al (2002, pp. 11-14),
forinstance,it is arguedthattheuse of 'manipulatives'such
as Dienes blocks involves presuming that the materials
somehow 'embody' numberideas, and thatthis entails a
commitmentto questionablydualist'correspondence'theo-
riesof truth.Itis furthersuggestedthatin practiceit is only
possibleto interpretDienes blocks as 'embodying'number
ideas once one has alreadydeveloped those same number
ideasby some other[socio-cultural]means. Dienes blocks
(in this case) aredismissed as the concomitantof a flawed
teachingtheory,whichtheoryitself maybe used to explain
the 'empiricalconfirmation'of the inevitablypoorteaching
thatresults from using them. Thus these artificial (purely
pedagogic)materialsmaybe safelydismissedfromourcon-
siderationon boththeoreticalandempiricalgroundsanda
morenatural'development'preferredsolely from 'every-
day' and 'realistic'situations.(Note thata failed (embodi-
ment)theoryis putforwardto explainwhy such pedagogic
materialsmustfail in use; RMEitself also featurespurely
pedagogic materials (such as bead strings and arithmetic
racks) but their use is explained with the support of an
entirelydifferent'learningtrajectory'theory.)
Consilience
As an alternativeto such 'dismissive theorizing' Goldin
(2008, p. 196) in his discussion of representationsuggests
thata spirit of consilience may be more helpful to us all,
withinwhich 'a coherenceandcompatibilityof knowledge
in differentdomains'cancontributeto a 'theoreticalframe-
workfor mathematicseducationthatcan unify useful and
valid ideas'. Ina similarspirit,Bruner(2006) invitedus all
to 'celebratethe divergence' of Piaget andVygotsky,not-
ing, "Justas depthperceptionrequiresa disparitybetween
two views of a scene, so in the humansciences the same
maybe true:depthdemandsdisparity"(p. 195).Withinsuch
anopennessof spirit,I suggest thatmanydifferent'mathe-
maticaldevelopments'arebothpossibleandsuccessful,that
as Dienes himselfproposed(see Dienes & Holt, 1973)chil-
dren may build mathematical understandingwith both
'everyday' and with pedagogic materials, and that key
choices to be made between representationalpossibilities
(and hence 'developments') when teaching mathematics
will be influenced significantly by underlyingviews (not
always either conscious or expressed) upon what is to be
called 'mathematical'.Importantfurtherconsiderationsin
practiceconcerncommunicationandeconomyin teaching.
Withregardto whatis called 'mathematical',I begin by
notingthatinmy experiencethewordis alwaysusedincon-
nection with actions, objects, and/or situations at least
involving structureof some kindor kinds(apartfromtriv-
ial nominal uses such as, "Whathomeworkhave you got
today?" "Oh, mathematics."). Structure may be being
sought,found,formed,recognised,described,developedor
utilised - it doesn'tmatterwhich. Secondly,althoughthere
may be many who still regard'mathematics'as a body of
knowledgeI note thatall experiencesthatarecalled 'math-
ematical' also involve action of some kind(s) at some
level(s); peopledo 'mathematics'.Structureandactionmay
thus be seen as togethernecessary to what is deemed dis-
tinctively'mathematical'.Also, for Gattegno,
[...] the essence of the mentalattitudewhich charac-
terises the mathematician is the substitution of the
virtualfor the actual,the performingof virtualactions
which areenvisaged as being real or realisable,once
there is no reason to suppose that they might not
become actual.(Cuisenaire& Gattegno,1957,p. 57)
Whilefor Freudenthal,
[...] whateveryou thinkaboutit, the exclamation"so
it goes on" is mathematics,it is the firstmathematics
mankindproducedandindividualsareproducing.Itis
great and importantmathematics, it is first and last
mathematics;it is the loftiest andthe most profound
mathematics.(1973, p. 173)
It is difficult to imagine how any 'virtual action' in Gat-
tegno's sense, or any observation'so it goes on' would be
possiblewithoutsome form(s)of structurehavingbeendis-
cerned.Already,withtheseviews, therearisesthepossibility
of distinguishingwithinany situationwhenactionmay be
deemed 'mathematical'. This is not a trivial observation
since it raises significant questions aboutwhetheractions
involving'theuseof numbersaslabels'(partof theUKgov-
ernment prescribed 'mathematical development' -
DfES/QCA,2003) canreasonablybe called 'mathematical'
at all [4], and invites furtheranalysis of many 'situated'
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5. socialpracticesto determinepreciselywhichactionsamong
all those occurringin the situationareto be picked out as
'mathematical'. Indeed, in some situations the 'situated-
ness' of the understandinginvolved might be the very
qualitythatprecludesit fromqualifyingas 'mathematical'.
Note I am not claiming that Gattegno and Freudenthal
havesomehowtogetherauthoritativelycapturedwhatsome
ideal 'thing' called 'mathematics'is; I am observing that
thedistinguishingof anychilddevelopment,oranyinstance
of a 'practice within a community' as 'mathematical', is
contingentuponanimplicitorexplicitview of whatis to be
called 'mathematical',andthatin theirview 'mathemati-
cal' activity essentially involves some form or forms of
virtualaction.Ina spiritof consilienceI amsuggestingthat
all experiencesdeemed'mathematical'atleastinvolve both
structureandaction,andI invitereadersfurtherto consider
an additionaloperationalcriterionthatit could be virtual
action in particularthat is distinctively 'mathematical'.
Muchhasbeensaidandwrittenabout'theobjectsof math-
ematics' over millennia that resonates with movement
betweenwhatis 'real'andwhatis 'virtual'(see Sfard,2008,
chap.6); when the use of wordssuch as 'two', 'three'etc.
switches fromadjectivalto nominalin children'sdevelop-
ment(andfrom'real'actionto 'virtual'object)we areinclined
tojudgethattheyhave(re-)inventedthevirtualmathemati-
cal objectswe call 'numbers'.Thislastobservationreminds
us thatessentiallyinvolvedwithin'mathematical'activityis
whatwe maycall 'mathematical'communication- certainly
self-communicationandpossiblyalso interpersonal;in thisI
follow Sfard's(2008) operationaldefinitionof thinkingas
'individualizedinterpersonalcommunication'(p.81).
If childrenareto 'developmathematically'then,it seems
to me theremust be communicating,structureand action
involved. WithFreudenthalandGattegno,one could also
expect thereto be virtualaction, indeed given the distinc-
tive ways in which mathematicalobjects are formed one
would also expect movement between real and virtual
action.Keyfeaturesof situationswithinwhichchildrenwill
'developmathematically'thenconcernwhichobjects,which
actions,andwhichstructuresareinvolved,andimportantly
whenandhow these elements introducethemselves or are
introduced.Crucialdecisionsforteachers(as Freudenthal's
notionof 'mathematizing'suggests) arewhen,how, andby
whomstructureis to be introduced;importantly,how much
structureis tobe 'offered',andhowmuch(andwhich)struc-
turingis to be expectedof children?Pedagogiceconomy is
to bejudgedin thisas well.
Variouskindsof physicallystructuredpedagogicmateri-
als have been invented to support children's number
development, and are well documented at least since the
time of Froebel (1782-1852). To these may be addedfur-
therstructuredmaterials such as notches on sticks, lines
drawnin sand,andvariousforms of abacusthatwere per-
hapsneverdesignedwith a pedagogicpurposein mindbut
nevertheless may be most useful in teaching. All of these
physicalmaterialsmay affordrepresentationalpossibilities
with the potentialto supportchildren's active encounters
with structure;different materialsmay thus be seen with
different'representationalaffordances'.I suggest thatsuch
affordancesmay be seen more or less to emphasise either
numbernotationconventions(suchas beadstrings,abacus
andDienes blocks)ornumberrelations(suchas Cuisenaire
andNumicon);unstructuredmaterials(suchas countersand
cubes)maybe activelystructuredinuseto emphasiseeither.
Emphasisuponnumbernotation in teaching (for instance
'place value') may reflect a primaryconcernwith 'mathe-
matics' as symbolic system (cf. Vygotsky?),andemphasis
uponnumberrelationsmay reflecta primaryconcernwith
'mathematics'as studyof structure(cf. Piaget?).Calculat-
ing of courseinvolves both,butnot alwayssimultaneously;
comparemultiplyingby
'10'witha relatingof 'factors'.
Unifying 'useful and valid ideas'?
Vygotskyused his pre-school 'everyday', 'school', and
'work'distinctionsto markout andcharacterisebroaddif-
ferencesthathe claimedlaybetweenapproachesto 'concept
formation'. In his view, development is not simply built
upon 'everyday concepts' but is the productof two-way
links between 'the concreteandempirical'and 'conscious
awareness and volition'; his link is the zone of proximal
andactualdevelopment(see Hedegaard,2007). His charac-
terisationof 'proximaldevelopment'bringsto the fore one
key manifestationof the importanceof social interactionin
development,andof coursethekeyroleforhimof language
in thedevelopmentof thinking.It is importantto notehow-
ever that nowhere in this vision does he suggest that the
onlyphysicalobjectschildrencanlearnwithbefore'school'
are'everyday'objects;his pre-schoolworldis simplychar-
acterisedby a distinctive andsocial approachto 'concept
formation'- a mannerof learningthatonly happenscontin-
gently to involve 'everyday' materials.And Gravemeijer
(2002) is correct in suggesting that when children meet
Dienes blocks (or Cuisenairerods) for the first time they
are'justpieces of wood'. I suggestthatPiaget's attentionto
the logico-mathematical developmental significance of
action andstructuremaybe alignedwith Vygotsky's atten-
tionto the importanceof social interactionandlanguageto
outlineperfectlyfeasible 'mathematicaldevelopments'that
arerestrictedneitherto 'everyday'objectsnorto 'realistic'
scenarios. I furthersuggest that there are some potential
economies andadvantagesto be gainedthroughthe affor-
dances of particularstructuredmaterials that emphasise
numberrelationswithinsuchdevelopments.
Contraryto views expressedby criticsof Dienes blocks,
it is possible to outlineproductive'mathematicaldevelop-
ments'involvingpedagogicstructuredphysicalmaterialsin
either 'dualist'or 'non-dualist'terms;it is not necessaryto
theorise about the existence of 'meanings' of words and
symbols andpieces of wood for such perceptuallyaccessi-
ble materialto prove effective in use. Also pragmatically,
fora teacherit is notnecessaryto choose betweenontologi-
cal and/orepistemologicalpositionsunless to do so results
in improvementsto practice.I referto a particularnon-dual-
ist accountnext because it offers the importantpossibility
of novel empiricalinvestigationwith a view thenceto ped-
agogicalimprovement.
In Sfard(2008), thereis offered a carefully non-dualist
account of mathematical development that involves no
objectified'everyday'or 'realistic'worlds,andconsequently
no dismissal of particularphysical materialsas 'artificial'.
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6. Insteadthereis a carefullydevelopedaccountof how action
withany physicalobject(s)maybecome implicatedwithin
aVygotskian(usedina broadsense here;Sfard,importantly
formy argument,writesof thinkingas 'individualizedinter-
personalcommunication'ratherthanas 'internalisedspeech
or language') discursive 'mathematicaldevelopment'. On
such a view structuredpedagogic materialssuch as Cuise-
naire rods and Dienes blocks may become equivalent
'realizations'of numbersignifiersalongsideanyotherforms
- concrete,iconic, or symbolic - linked withinwhat Sfard
calls 'realizationtrees'. Differing affordancesof different
physicalobjectsthusoffera rangeof 'Vygotskian'develop-
mental possibilities, some perhapsmore economical and
effectivewithsome childrenthanothers.
Forbriefandillustrativedetail,I addressheretwo crucial
aspectsof numberdevelopment:thereificationof counting
processesintomathematicalobjectssignifiedwithwordssuch
as 'two', three',etc., andchildren'sperception/construction
of relationsbetweensuchobjects.Itis therelatingof number
objectsto each otherandbetween the nameswe give them
thatcharacteriseseffectivecalculatingandits developments;
inthisrelatingwe cannoteacrucialroleplayedbybothstruc-
tureandstructuring.Inprevailingaccountsof development
(see Fuson,1992;Sfard,2008) we findtransitionsdescribed
fromcounting-in-onesproceduresto calculating(relating)
with 'whole' numberobjects, withperceptual structuring
apparentlyplayingno mediatingrole. It is worthexploring
thepotentialeconomiesandeffectivenessinvolvedin enac-
tiveandvisualstructuringhoweveraschildren'develop'their
thinkingbeyondcounting-in-onestoward'numbers',forit is
noted often enough thatmany children remainlimited to
counting-in-onesproceduresinsteadof calculatingby relat-
ingwholenumbers(see, e.g., Gray,1991).
Reifying a counting procedureinto a virtual object (a
number)is both a mysterious process and not inevitable;
some childrenappearto achievethisonly partially.Thereis
a kind of 'compression' involved, and as Sfard observes
(2008, p. 171), there is '[...] replacement of talk about
processes with talk about objects'. Mysteriously though,
the virtualnumberobjectsomehowjust 'arises'duringthis
time.Inthesameway thatSfardacknowledgescrucialroles
for visual mediation in interpersonalcommunication and
forrealizationsof mathematicalsignifiersindifferentmedia,
I suggest that there is opportunityfor a helpful visually
structuringrole withinthe developmentalprocess of reifi-
cation itself - a structuringin two senses: the real objects
involved may themselves be structured{e.g., fingers), but
discreterealobjectsmay also be structuredin a systematic
wayso as to invite a relatingstructurebetweenthe various
virtual objects (i.e., numbers) that are produced. Dienes
blocks, arithmeticracks,andbeadstringsinvolve structur-
ings that reflect notation conventions; Numicon pieces
involve structuringsthatreflect initially numberrelations
andsubsequentlynotation.Cuisenairerods invite a devel-
opment that will proceed in parallel with reification of
counting-in-onesinsteadof buildinguponit, withdiscourse
developing aroundordinalrelationsbetween ungraduated
lengthsusuallybeforediscourseconcerningcardinalvalues
is subsumed.In short,the variousanddifferentstructuring
actionswith,andthe physicalstructuresof, the realobjects
implicatedwithinthetalkin a reifyingprocessmaysupport
thatprocess significantly; the lack of structurein random
arrangementsof discretephysicalobjectsoffersonly mini-
mal supportto childrenwhose mathematicaldevelopment
dependsuponthembothreifyingseparatedcounting-in-ones
procedures and seeing connections between the virtual
objects thatresult. Gestaltpsychology offers manyexam-
ples, and thoughtful discussion, of human perceptual
structuringthatarerelevanthere(Wertheimer,1945).
Some concluding remarks
Justas thereis no reasonto assumethatchildrenmayonly
'develop mathematically'with 'everyday'objects,thereis
no reasonto assumethatchildrenmayonly (oreven princi-
pally)form'numbers'byreifyingcounting-in-onesprocedures.
Cardinalitymaybe determinedbygroupingdiscreteobjects
into systematic visual patterns,as well as by counting-in-
ones, andsuchactivestructuringmayformarichperceptual
basis for exploring of both numbernotation and number
relations[5]. Nor shouldwe forgetthatGattegnoproposed
a perfectlyfeasible mathematicaldevelopmentbasedupon
relationsof magnitude(bothordinalandalgebraic)before
cardinalityis studiedatall.
Dismissive theorizing acts to supportideological posi-
tions that risk precluding significant possibilities until
ideology has, eventually,to be abandoned(Goldin,2008).
Freudenthal(1983), in his passionateadvocacyof number
linesdismissedCuisenairerodsasopposedtoworkwithsuch
imagery (p. 10). In his false implicationof a necessity for
choicebetweennumberlinesandCuisenairerodshe ignored
therichpossibilitiesof involvingbothwithin'mathematical
developments'.The 'part-whole'relationshipsso fundamen-
tal to effective calculating and rational number can be
exploredmost effectively with such visual realizationsand
in their'universality'of possibilitytherodspresentno nec-
essary obstacle to experience of continuitywith a number
line. Consilience,I suggest,offersa richerfuture;following
Bruner,we limitthe depthof ourunderstandingin advance
by askingourselvesto choosebetweenPiagetandVygotsky,
similarlyin givingourselvesmanyotherfalse choices.
Vygotsky's characterisationof developmentalstages in
termsof socialinstitutionsinvolvedthegeneralisingof social
practicesof histimeandplace.Discursivelyobjectifying'the
everyday'and'schooling'and'work'subsequentlyasif these
opposingconstructionswereasrealandpresentto thesenses
asmaterialsubstancesrisksdiscoursethatproducesonlyarbi-
trarilynarrowtruths,andmayblindustodetailsof significance
inactuallivedexperience.Thereisnomagicalpropertyinher-
ingin 'everyday'oranyotherperceptuallyaccessibleobjects;
anyphysicalobjectmaybecomea 'communicationmediator'
withinthepractisingof ahumandiscourse.
Workingwith operationaldefinitions of what is called
'mathematical',of 'mathematicalcommunication',andof
'thinking'as individualised interpersonalcommunication
opens up explorationof discursive 'mathematicaldevelop-
ments' thatboth recognises Vygotsky's insights regarding
the social dimension of thinkingandmakes possible new
avenuesof empiricalenquiry.AddingGestaltobservations
concerningperceptualstructuring(withothers)to suchdia-
logic discoursewidensenquirystill further.
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7. Thereis no space within a single article such as this to
lay outthe detailof varying'mathematicaldevelopments';
my aim in this paperis simply to pointto theirpossibility.
Much work remains to be done investigating perceptual
activity within discursive developments, andin particular
the possibilities of visual structuring- how much, when,
how, andby whom?And within this, investigation of not
just notationstructures,butnumberrelationsas well. Simi-
larly,empiricalinvestigationof 'routines','rituals','deeds'
and'explorations'(Sfard,2008, chaps.7 & 8) withindevel-
opments involving variously structured(andstructuring)
visual 'communicationmediators'could revealmuchto us
thatremainsclosed whenwe restrictourselvesunnecessar-
ily to just 'everyday' objects and 'realistic' scenarios.
Childrenaremoreinventivethanthatandas Gattegnooften
pointed out, the evidence we have daily from children's
learning of their mother tongues shows us just what an
incredibleeye (andear,andfeeling) they have for pattern
andstructure.
Acknowledgements
I amverygreatlyindebtedto AnnaSfard,JohnMason,Kev
Delaney,andWacekZawadowskifor theirmost thoughtful
commentsuponearlierwritingtowardthis article,andfor
theirencouragement.
Endnotes
[1]AdistinctionbetweenPiagetian'child'developmentandVygotskian'stu-
dent'developmentis important,andto discuss the constructedsubjectof
today'sprevailingpost-Piagetiandiscourseas either'child'or 'student'risks
oversimplifyinga complex andshiftingdiscursivesituation.Since I later
arguethatPiagetianchilddevelopmentisnotsimplytobereplacedbyaVygot-
skianperspective,andbecauseIamhereconcernedtodiscussthemathematical
developmentof 'students'whoarealso 'children',I havepreferredto usethe
term'thepost-Piagetianchildsubject'.Readersshouldbe awarehoweverthat
fromthispointI amreferringto a moretechnicallycorrect,butstylistically
moreawkward,constructed'post-Piagetianchild/student'subject.
[2]EvenPiagethimselfhadobserved,"Itis a greatmistaketo supposethata
childacquiresthe notionof numberandothermathematicalconceptsjust
fromteaching.Onthe contrary,to a remarkabledegreehe develops them
himself,independentlyandspontaneously"(Piaget,1953,p. 2).
[3] Walkerdine(1988) for example, forensically savages an episode of
poorteachingwithanalysisof severalpagesof transcript(pp. 159-182) in
orderto demonstratethatthe strugglingteacher's 'reading'of herclass-
room use of Dienes's materialswas wildly differentfrom the children's
own experiences(see also Hart,1989).
[4] Theuse of numeralsto designatebusroutes,ortelephoneaddresses,is
distinguished within the French language (for instance) by the use of
'numéro'as opposedto the cardinal'nombre';the English languagecur-
rentlyinvolves no such lexical distinction,bothuses being coveredby the
use of thesameword'number'.Describing'the use of numbersas labels'
as 'mathematical'maybe one instancein whicha socially anthropological
view of 'mathematics'as 'social practiceswithin communities'is impli-
catedwithoutacknowledgementorexplanation.
[5] I am gratefulto WacekZawadowskifor pointingout the significance
of Pythagoreanreasoninghere;since linguistic deductive logic had not
been inventedin theirtime, the demonstratingof numericalrelationships
by earlyPythagoreansfrequentlyinvolvedthedynamicimageryof 'show-
ing'. I am loathto believe thathumanshave somehow lost this capacity
since thesubsequentinventionof formaldeductivereasoning.
References
Aubrey,C. (1997) 'Children'searlylearningof numberin school andout',
in Thompson,I. (ed.), Teachingand learning early number,Bucking-
ham,UK, OpenUniversityPress,pp.20-27.
Bruner,J. (2006) Insearch ofpedagogy volumeII: the selected worksof
JeromeS. Bruner,London,UK, Routledge.
Cuisenaire,G. andGattegno,C. (1957)Numbersincolour:a newmethodof
teachingarithmeticinprimaryschools(3rdedn.),London,UK,Heinemann.
DfEE/QCA (2000) Curriculumguidancefor thefoundation stage, Lon-
don,UK, QCA.
DfES/QCA(2003) Foundationstageprofile,London,UK, QCA.
Dienes, Z. andHolt,M. (1973) Let'splay maths,London,UK, Penguin.
Freudenthal,H. (1973) Mathematicsas an educational task, Dordrecht,
NL, Reidel.
Freudenthal,H. (1983) Thedidactical phenomenology of mathematical
structures,Dordrecht,NL, Reidel.
Fullan, M. (2003) Changeforces witha vengeance, London, UK, Rout-
ledgeFalmer.
Fuson,K. (1992) 'Researchon learningandteachingadditionandsubtrac-
tionof whole numbers',in Leinhardt,G., Putnam,R. andHattrup,R.A.
(eds.), Analysis of arithmeticfor mathematicsteaching, Hillsdale,NJ,
Erlbaum,pp. 53-187.
Goldin,G, (2008) Perspectiveson representationin mathematicallearning
and problem solving', in English, L. D. (éd.), Handbookof interna-
tional research in mathematics education (2nd edn.), Oxford, UK,
Routledge,pp. 176-201.
Gravemeijer,K., Cobb,P.,Bowers, J. andWhitenack,J. (2000) 'Symbol-
izing, modeling, andinstructionaldesign', in Cobb,P.,Yackel,E. and
McClain, K. (eds.), Symbolizingand communicatingin mathematics
classrooms:perspectiveson discourse,tools, and instructionaldesign,
Mahwah,NJ,Erlbaum,225-273.
Gravemeijer,K., Lehrer,R., vanOers,B. andVerschaffel,L. (eds.) (2002)
Symbolizing,modeling and tool use in mathematicseducation, Dor-
drecht,NL, Kluwer.
Gray,E. M. (1991) 'An analysis of divergingapproachesto simple arith-
metic: preference and its consequences', Educational Studies in
Mathematics22, 551-574.
Hart,K. (1989) 'Place value: subtraction', in Hart,K., Johnson, D. C,
Brown,M., Dickson, L. andClarkson,R (eds.), Children'smathemati-
cal frameworks 8-13: a study of classroom teaching, Windsor,UK,
NFER-NELSON,pp. 8-45.
Hedegaard,M. (2007) 'Thedevelopmentof children'sconceptualrelation
to the world, with focus on concept formationin preschoolchildren's
activity', in Daniels, H., Cole, M. andWertsch,J. V. (eds.), TheCam-
bridgecompanionto Vygotsky,New York,NY, CambridgeUniversity
Press,pp.246-275.
Lave, J. (1988) Cognition inpractice, Cambridge,UK, CambridgeUni-
versityPress.
Nunes,T.andBryant,P.(1996) Childrendoingmathematics,Oxford,UK,
Blackwell.
Piaget, J. (1953) 'How childrenformmathematicalconcepts', Scientific
American189(5), 74-79.
Scribner,S. (1997) 'Mindin action:a functionalapproachto thinking',in
Cole, M, Engstrom,Y.andVasquez,O. (eds.),Mind,culture,andactiv-
ity: seminal papers from the Laboratory of Comparative Human
Cognition,Cambridge,UK, CambridgeUniversityPress,pp. 354-368.
Sfard,A. (2008) Thinkingas communicating: humandevelopment, the
growthof discourses, and mathematizing,New York,NY, Cambridge
UniversityPress.
Thompson,I. (ed.) (1997) Teachingand learningearly number,Bucking-
ham,UK, OpenUniversityPress.
Walkerdine,V. (1984) 'Developmentalpsychology andthe child-centred
pedagogy:the insertionof Piaget into early education', in Henriques,
J., Hollway,W.,Urwin,C, Venn,C. andWalkerdine,V.(eds.), Chang-
ing thesubject,London,UK, Methuen,pp. 153-202.
Walkerdine,V. (1988) Themasteryof reason:cognitivedevelopmentand
theproductionof rationality,London,UK, Routledge.
Wertheimer,M. (1945) Productive thinking, New York,NY, Harper&
Brothers.
13
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