Lawrence Erlbaum Associates (Taylor & Francis Group)
Examining the Effects of Different Multiple Representational Systems in Learning Primary
Mathematics
Author(s): Shaaron Ainsworth, Peter Bibby, David Wood
Source: The Journal of the Learning Sciences, Vol. 11, No. 1 (2002), pp. 25-61
Published by: Lawrence Erlbaum Associates (Taylor & Francis Group)
Stable URL: http://www.jstor.org/stable/1466720
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THEJOURNAL
OFTHELEARNING
SCIENCES,
11(1),25-61
C 2002,Lawrence
Inc.
Erlbaum
Associates,
Copyright
Examiningthe Effectsof Different
MultipleRepresentational
Systemsin
LearningPrimaryMathematics
ShaaronAinsworth,PeterBibby,andDavidWood
EconomicandSocialResearchCouncilCentrefor Researchin
andTraining
Instruction,
Development,
University
of Nottingham
in schoolsand
arenowcommonplace
Multi-representational
learningenvironments
homes.Researchthathas evaluatedthe effectivenessof suchenvironments
shows
thatlearnerscanbenefitfrommultiplerepresentations
once theyhavemastereda
numberof complextasks.Oneof thekeytasksforlearningwithmultiplerepresentationsis successfultranslation
betweenrepresentations.
Inorderto explorethefactors
thatinfluencelearners'
translation
betweenrepresentations,
thisarticlepresents2 exwitha multi-representational
environment
wherethedifficultyof translatperiments
was manipulated.
Pairsof pictorial,mathematical,
or
ing betweenrepresentations
were
in
used
to
teach
children
of
3
mixedpictorialandmathematical
1
representations
conditionsaspectsof computational
estimation.In Experiment
experimental
1, all
childrenlearnedtobecomemoreaccurate
estimators.
Children
inthepictorialandthe
mathematical
conditionsimprovedintheirabilitytojudgetheaccuracyof theiresti2 exploredifthemixed
mates,butchildreninthemixedconditiondidnot.Experiment
condition'sdifficultieswithtranslation
weretemporary
additional
time
by requiring
to be spentonthesystem.Itwasfoundthatchildreninalltheexperimental
conditions
of estimation
Itis arguedthatthemixedcondiimprovedintheirjudgments
accuracy.
tion'sfailureto improveinExperiment
I wasdueto thedifficultiestheyexperienced
intranslating
information
betweendisparate
Theirsuccessin
typesof representation.
2
was
not
to
betweenrepresentations,
but
Experiment
explained by learning translate
thatcontained
allthenecessaryinforthroughtheadoptionof a singlerepresentation
mation.Thisstrategywasonlyeffectivebecauseof thewaythatinformation
wasdistributedacrossrepresentations.
andrequests
forreprints
should
besenttoShaaron
ESRC
Centre
forReCorrespondence
Ainsworth,
searchinDevelopment,
Instruction
andTraining,
Schoolof Psychology,
of Nottingham,
University
ParkNottingham,
UK NG72RDE-mail:
University
[email protected]
26
WOOD
AINSWORTH,
BIBBY,
The use of multipleexternalrepresentations
(MERs)to supportlearningis
environments.
in
and
in
classroom
traditional
computer-based
settings
widespread
andfractionssuchas 33%or 1/3areoftenpresentedto
Forinstance,percentages
childrenalongsidea drawingof a pie chartwith one thirdshaded.Learnersare
givenalgebrawordproblemsorearlyreadingbooksthatcontainpictures.Geometrysoftwarepackagessuchas GeometryInventor(LOGAL/ TangibleMath)allinkedto geometricalfigures.
low tablesandgraphsto be dynamically
to exenvironments
havedesignedcomputer-based
A numberof researchers
understandstudents'
to
MERs
that
can
contribute
the
developing
ploit advantages
ing. One areathathas seen particularactivityof this kind is the teachingof
mathematical
function.For example,FunctionProbe(Confrey,1991)provides
keystrokeactionsto helpstudentscometo
graphs,tables,algebra,andcalculator
theconceptof function.UsingMERs,it aimsto helplearnersdevelop
understand
by considering
aspectsof functionsuchas fieldof applicabildeepunderstanding
and
rate
of
change, patterns(e.g., Confrey& Smith,1994;Confrey,Smith,
ity,
is designedto supportspe&
Piliero, Rizzuti,1991).Eachof therepresentations
cific activities.Forexample,the graphwindowsupportsthe qualitativeexplorationof aspectssuchas shapeanddirection,whereasthetablewindowcanbe used
to functions.
to introducea moreexplicitexpressionof thecovariational
approach
these
of therelationbetween
FunctionProbesupportsanunderstanding
representationsby allowingstudentsto passfunctionsbetweenthedifferentwindowsand
themto grasptheconvergenceacrossrepresentations.
by encouraging
A similarapproach
totheteachingof functionscanbe seeninthe"VisualMathematics"curriculum
(e.g., Yerushalmy,1997).Again,emphasisis placedon the
This showedhow stuuse of MERs,andparticularly
graphicalrepresentations.
dents could come to understandfunctionsof two variablesby using a comreasonwith,andexplain
environment
thatallowedthemto construct,
puter-based
MERs.Brenneret al. (1997) showedthatstudentscouldbe successfullytaught
bothto representfunctionproblemsin MERsandto translate
betweenrepresentationssuchas tablesandgraphs.
educationperspectivesprovidean explaCognitivescienceandmathematics
nationof the benefitsof thesespecificapplications
of MERsto teachfunctions.
Forexample,LarkinandSimon(1987)contrasted
informationally
equivalentdiin termsof search,recognition,and
and sententialrepresentations
agrammatic
inference.They concludedthatdiagramsare searchedmoreefficiently;do not
have the high cost of perceptualenhancement
associatedwith sententialrepresentations;and exploit perceptualprocessesthus makingrecognitioneasier.
Given that representations
it is clearthat the use of
differso fundamentally,
with different
MERscan be beneficial.By combiningdifferentrepresentations
are
not
limited
the
learners
computational
by
properties,
strengthsand weaknesses of one particularrepresentation.
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
27
However,thisis onlyone of thereasonsto use MERs.Ainsworth(1999)protaxonomyof MERs.Sheidentifiedsevendifferentusesof MERs
poseda functional
classes--complement,constrain,
ineducational
softwarethatfallintothreeprimary
thatcontaincomplemenandconstruct.
Thefirstclassis theuseof representations
or
information
cognitiveprocesses.Forexample,the
tary
supportcomplementary
tothisuseofrepreabovecorresponds
LarkinandSimon(1987)exampledescribed
is usedtoconstrain
sentations.
Inthesecond,onerepresentation
possible(mis)interoften
pretationsin the use of another.For example,simulationenvironments
a lessfamiliaror
tohelplearners
interpret
providea familiarconcreterepresentation
abstractrepresentation.
Finally,MERscanbe usedto encouragelearnersto constructa deeperunderstanding
of a situation.For example,Kaput(1989, pp.
createsa whole
179-180)proposedthat"thecognitivelinkingof representations
thatis morethanthesumofitsparts...itenablesustoseecomplexideasinanewway
andapplythemmoreeffectively."
Dienes(1973)arguedthatperceptual
variability
learners
with
the
same
in
(the
opportuconceptsrepresentedvaryingways)provides
Incognitiveflexibilitytheory(e.g.,Spiro&Jehng,1990)
nityto buildabstractions.
theabilitytoconstruct
andswitchbetweenmultipleperspectives
of adomainis fundamentalto successfullearning.Mayer(e.g.,Mayer& Anderson,1992;Mayer&
atheoryofmulti-media
whichshowedthatstudents
Sims,1994)described
learning,
better
and
when
with
gain
problemsolving conceptual
theyarepresented
knowledge
bothtextandpictures.Inallthesecases,forlearners
to achievethemaximum
benefitsofMERs,theymustcometounderstand
notonlyhowindividual
representations
relatetoeachother.Thelatterconstitutes
a
operate,butalsohowtherepresentations
to learningthatMERscanmake.
uniquecontribution
of representations
benefitin
Althoughthecoordination
providesan additional
certainlearningsituations,
previousresearchhasshownthattheabilityto translate
between representationsdiffers markedlybetween experts and novices.
Tabachneck,Leonardo,and Simon(1994) reportedthatnoviceslearningwith
MERsin economicsdidnotattemptto translateinformation
betweenline graphs
andwritteninformation.
Thiscontrasted
withexpertperformance
wheregraphical
andverbalexplanations
wereintimatelyboundtogether.Apparently,
the deeper
of
the
facilitated
the
to
the
different
knowledge
experts
ability integrate
representationalformats.Kozma,Chin,Russel,andMarx(2000) discussedhow expert
chemistscanbe distinguished
fromnovicechemistsby theirintegrated
multi-representational
of
thatallowsexpertsto transunderstanding chemicalphenomena
late fromone representational
formatto another.Thebehaviorof theseexperts
contrastswiththeresearchof Schoenfeld,Smith,andArcavi(1993),whostudieda
studentlearningto understand
functionsusingtheGrapher
environment.
Theydescribedin detailthemappings
betweenthealgebraicandgraphical
representations
in thisdomain.Theyshowedhowa studentcouldappearto havemasteredfundamental componentsof a domainterms of either algebraor graphs.However, her
behavior with the representationswas often inappropriate,as she had not inte-
28
AINSWORTH,
BIBBY,WOOD
gratedherknowledgeacrossthem.Schoenfeldet al.'s microgeneticanalysisrevealedboththecomplexityof themappingsthatcanexistbetweenrepresentations
andtheproblemsthatcanensuewhenthosemappingsarenotmade.
tocoordinate
MERshasalsobeenfoundtobea farfromtrivial
Teachinglearners
of
(1991)examinedtheunderstanding
activity(deJonget al., 1998).Yerushalmy
functionsby35 fourteen-year-olds
afteranintensive3 monthcoursewithmulti-representational
software.In total,only 12%of studentsgaveanswersthatinvolved
considerations.
childrenwhousedtworepbothvisualandnumerical
Furthermore,
resentations
werejustas errorproneas thosewhouseda singlerepresentation.
If a learneris unableto translate,
orhasdifficultymappingtheirknowledgebetweenrepresentations,
thenthe uniquebenefitsof MERsmay neverarise.One
is to mabetweenrepresentations
of thetranslation
wayto examinetheimportance
the
extent
to
which
different
of
can
influence
the
nipulate
pairs representations
need
can
translation
To
do
we
to
understand
how
difthis,
process.
representations
fer.First,theycandifferin termsof theinformation
expressed.Second,theycan
differin thewaythattheinformation
is presented.
Theselevelsareoftenreferred
to as therepresented
andrepresenting
worlds(Palmer,1978).
Thisarticlereportstwo studiesthatexaminetheuse of MERs,varyingthedein
worldandholdingconstanttheinformation
greeof similarityintherepresenting
therepresented
world.Thisis achievedin thecontextof a computer-based
system
thatsupportschildrenlearningestimation.
SYSTEMDESCRIPTION
The computer-based
EstimationNotalearningenvironment(Computational
tion-BasedTeachingSystem;CENTS)usedintheseexperiments
children
supports
in learningto understand
estimation.
can
estimation
computational
Computational
be definedas theprocessof simplifyinganarithmetic
problemusingsomeset of
rulesor proceduresto producean approximate
but satisfactoryanswerthrough
mentalcalculation(Dowker,1992).Estimation
is notonlya usefulskillin its own
rightbuthasalsobeenimplicatedin developingnumbersense(Sowder,1992).
CENTSis designedto help9- to 12-year-old
childrenlearnsomeof thebasic
in thesuccessfulperformance
of computational
esknowledgeandskillsrequired
timation.It actsas an environment
forchildrento practiceandreflectupontheir
estimationskills.Thecentralpedagogicalgoalis to encouragelearnersto understandhowtransforming
numbersaffectstheaccuracyof answerswhenestimating.
Thisfocusstemsfromrecognition
of theimportance
of thisknowledgeindeveloping estimationskillsandnumbersense(e.g., Trafton,1986).It is a fundamental
of anestimate(LeFevre,Greenham,
&
componentforjudgingtheappropriateness
Waheed,1993;Sowder& Wheeler,1989)andis necessaryif post-compensation
(adjustingan estimateto makeit moreaccurate)is to be used. However,LeFevreet
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
29
al. (1993)foundthisknowledgewasunderdeveloped
in children'sviews of estiof
thedevelopment
is idealforsupporting
mation.A computer-based
environment
thisknowledgeas immediatefeedbackcanbe givenwhichis contingent
uponstudents'ownestimates.
CENTSsupportsa numberof estimationstrategiesidentifiedby previousresearch(e.g.,Reys,Rybolt,Bestgen,& Wyatt,1982).Onlytwoareusedin thecurto
Withrounding,numbersaretransformed
rentstudy:roundingandtruncation.
arithmetic
thenearestmultipleof 5, 10, 100,andso forth,andthentheappropriate
operationis applied.Forexample,19 x 69 couldbe roundedto 20 x 70 (seeupper
a newvaluefortheright-most
LHSof Figure1).Truncation
involvessubstituting
x
19
69
truncated
to 10 x 60 (see upper
the
can
be
same
digit(s).Using
example,
RHS of Figure1). Thesestrategieswere selectedin accordancewith teachers'
wishesandtheNationalCurriculum
(EnglandandWales).
CENTSpromotesthepredict-test-explain
cyclethathasbeenfoundto produce
betterunderstanding
in scienceeducation(e.g.,Howe,Rodgers,& Tolmie,1990).
estimate,performthe
UsingMERs,learnersmakepredictionsabouta particular
of theestimation
the
results
have
to
examine
and
then
the
estimation,
opportunity
children
After
each
in
the
of
their
logtheresults
problem,
process
light
predictions.
workbook.
in
an
on-line
of (at least)two differentestimationstrategies
Theydescribehow theytransformed
the numbers,whetherthe estimateis accurate,and
howdifficulttheyfoundeachestimationprocess.At theendof a session,children
areencouraged
to reviewthelogbookto investigatepatternsin theirestimates.
--S --7----il~uirit~ainq
trur~cation
E{tka
Zeros
Zeros
Answer
1400
aspo
Answer
600
w I
no
T
I
1410
500
19 X69
140
600
60 a
FIGURE1 An illustration
of a completedproblemusingpictorialrepresentations.
30
WOOD
BIBBY,
AINSWORTH,
Using CENTS
A typicalsequenceillustrated
forone strategy(rounding)
is as follows:
Giventheproblem-estimate19 x 69
1. Producetheintermediate
solution.
Roundto 20 x 70
solution.
2. Predicttheaccuracyof yourestimatebasedon theintermediate
Rep 1. Higherandcloseto theexactanswer
Rep2. Closeto theexactanswer
will be performed
usedthisprediction
Note.Dependingon therepresentations
valuesorby selectingpartof a picture(e.g.,placingthe
usingeithernumerical
crosson the splatwall).
3. Multiplythe "extracted"
digits.
2x 7=14
4. Adjustplacevalue.
(1)0 x (1)0 = (1)00
5. Respond.
1400
6. Receivefeedbackon the accuracyof the estimate.Thisallowsyou to also
evaluateyourjudgmentof the estimate'saccuracy.
Note.Dependingon therepresentations,
feedbackis providedusingeithernumericalvaluesorbyindicating
ofa
part picture(e.g.,thesplatonthesplatwall).
atstages1,3, and4 to
Helpis provideduponerrororbyrequestbythecomputer
ensurethatstudentsdonotfaildueto slipsornumberfacterrors(e.g.,a timestable
squareis availableto helpwithmultiplication).
INCENTS
REPRESENTATIONS
CENTSwasusedinthepresentstudiestoassesshowdifferentcombinations
of representations
mayaffecttheprocessandoutcomesof learning.Multiplerepresentationsareusedbothfordisplay(toillustrate
theaccuracyof accuratechildren'sestimates)andfor action(childrenpredicthow accuratetheyjudgetheirestimateto
arebasedonthepercentage
deviationof theestimatefrom
be).All representations
theexactanswer([estimate- exactanswer/ exactanswer]x 100),whichcaptures
bothdirectionandmagnitude
differences.Thisis a commonmeasureof theaccuracyof estimates(e.g., Dowker,1992).No matterhowthesurfacefeaturesof the
representationsdiffer, the deep structureis always based on this relationship.
MULTI-REPRESENTATIONAL
SYSTEMS
EXAMINING
31
candifferin two ways,eitherin the information
theyexpress
Representations
is presented,
thatis, therepresented
andrepreor in thewaythatthisinformation
In
this
we
will
consider
how
these
worlds
dimensection,
(Palmer,1978).
senting
sions may interactto produceeffectivelearning.We startwiththe represented
bothlevelsof explanation.
world,andfinallyintegrate
world,thentherepresenting
of representaParticular
attentionis paidto theeasewithwhicheachcombination
tionssupportstranslation.
RepresentedWorld
The informationrepresentedin CENTSvaries along two dimensions-the
amountof information
candisplaydirectionormagnitudesepa(representations
or
andthe resolutionof incan
both
dimensions
simultaneously),
rately
display
formation(eithercategoriesof 10%or continuousrepresentations
accurateto
in
of
content,combinations represen1%).Furthermore, termsof informational
tationscanbe eitherfullyredundant
(sameamountandresolutionof information
in bothrepresentations),
nonredundant
(no overlapbetweenamountandresolution of information),
or partiallyredundant
(someoverlapbetweenthe amount
and resolutionof informationin bothrepresentations).
In the experimentsdescribedin this article,partiallyredundant
systemswere used
representational
in CENTS,one
(Table1 andFigures2 to 4). In eachpairingof representations
representation
expressedthe magnitudeof estimationaccuracyin 10%bands
the articleas the
(archerytarget,histogram).Theseare referredto throughout
The
second
contains
direction
and
categoricalrepresentations.
magnitudeinformationwith continuousresolution(splatwall,numericaldisplay)andarecalled
continuousrepresentations.
World
Representing
CENTScandisplayestimation
of theinforaccuracyindifferentwaysindependent
mationexpressed.Representations
havetwo basiccomponents,
theirformatand
1
TABLE
inCENTS
Representations
RepresentingWorld
Representation
Splatwall
Numerical
Archerytarget
Histogram
Format
Pictorial
Mathematical
Pictorial
Mathematical
RepresentedWorld
Available Information
Resolution
Directionandmagnitude
Directionandmagnitude
Magnitude
Magnitude
Continuous
Continuous
(10%)
Categorical
Categorical
(10%)
Higher
Spot on
Lover
Rounmng
1400
Trumwcion
600
n
Trwmali6
600
Roundn
1400
FIGURE2 Pictorialrepresentations:
Splatwallandarcherytarget.
Percentage Away
409+
Predict
i+5%
Actual
+7%
-19%
30-40%I
20-30%I
-54%]
10-20%,
Rownig
Trwcamn
1400
600
0-10%
0%
1400600"IdMWaM
1400
600
FIGURE3 Mathematical
Numeralsandhistogram.
representations:
Percentage Away
Predict
-19
Actua +
-554.%
i
Roundmg Trmncaion
1400
[2tiatJ
FIGURE4
32
600
J
Rounkng
1400
Trntbnr
600
Mixedrepresentations:
Numeralsandarcherytarget.
MULTI-REPRESENTATIONAL
EXAMINING
SYSTEMS
33
Theformatof a representation
is themeansbywhichinformation
is preoperators.
sented.Theoperators
aretheprocessesby whichthatinformation
is manipulated.
Thesefactorstendto be integrated
in taxonomiesof representations
(e.g., Lesh,
Post,& Behr,1987;Lohse,Biolsi,Walker,&Rueler,1994).Onesimplebutuseful
distinctionproposedbyKaput(1987)is betweenambientsymbolsystems,suchas
picturesandnaturallanguage,andother,normallyschool-taught
representations
suchas graphs,tables,andschematicdiagrams
to as mathematical
(referred
representations).CENTStakesadvantageof this distinctionusingbothpictorialand
mathematical
representations.
Thepictorialrepresentations
in CENTSarebaseduponthemetaphor
of physical distance(Figure2). Thefirstrepresentation
is anarcherytargetthatrepresents
in bandsof 10%deviationsfromtheexactanswer.Theinmagnitudeinformation
nerband,forexample,represents
To indicatehowaccurate
00/o--10%.
theybelieve
theirestimateto be, usersselecta bandin thetarget,whichalsohighlightsa flag
withthatcolor.Thecomputershootsan arrowto showthe deviationof the estimatefromtheexactanswer.Thisallowsstudentsbothto see theaccuracyof their
estimateandto evaluatethesuccessof theirprediction
of estimation
The
accuracy.
secondpictorialrepresentation
is a "splatwall"
thatrepresents
bothmagnitude
and
directioninformation,
andis continuous.
Themetaphor
its
is
guiding design very
similarto the archerytargetbutincludesa directioncomponent.Childrenplace
crossesonthewallatsomedistanceeitheraboveorbelowthecenterof thewallto
theirpredictions.
arethenfiredto indicatetheaccuracyof their
represent
"Splats"
estimatesandtheirpredictions.
Themathematical
in CENTSaredesignedto parallelthepictorepresentations
rialrepresentations
in thata histogram
information
in bands
expressesmagnitude
of 10%,andthe continuousrepresentation
the
simplygives percentagedeviation
in numberswiththesignrepresenting
direction(Figure3).Judgment
of estimation
accuracyis indicatedon thehistogram
by drawinga lineacrossthebar.Feedback
is givenby the computeras it colorsin thebar.Boththepictorialrepresentations
aregraphicalbutthemathematical
canbe eithergraphical
representations
(thehistogram)ortextual(thenumericaldisplay).
Theserepresentations
arepairedin CENTSin orderto test predictionsconcerningthe relativeeffectivenessof differentMERsin supportinglearning.A
fully pictorialsystemwas producedfromthe archerytargetandsplatwallanda
fully mathematicalsystem fromthe histogramand the numericaldisplay.A
mixed systemwas createdby combiningone mathematical
and one pictorial
representation
(Figure4). To maximizethe differencesin the representing
worldsfor the mixedsystem,it was decidedto use the archerytargetas thepictorialrepresentation
andthenumericdisplayas themathematical
one.Thiscombines a graphicalwith a textual representation,whereas the alternative
combinationof histogramandsplatwallarebothgraphicalrepresentations.
The
mixed representationsin CENTS may be an ideal combination,in that pictorial
34
WOOD
AINSWORTH,
BIBBY,
canbe usedto bridgeunderstanding
to the moresymbolicones.
representations
In addition,Dienes(1973) arguedfor the linkingof imageryandsymbolismin
mathematics
education.Themixedsystemcamecloserto achievingthisthaneitherthe pictorialor the mathematical
systems.
the Representing
and RepresentedWorlds
Integrating
To predicthow specificcombinationsof representations
used in CENTSmay
influencelearning,we needto considerthe represented
world,the representing
world, and the interactionbetween them. First is the representedworld.
Childrenin all experimentalconditionsinteractwith the same represented
worldthroughone representation
containingdirectionandmagnitudeinformation andanotherrepresentation
thatprovidesonly magnitudeinformation.
Table 2 shows the space of learningoutcomes given a causal relationship
betweenrepresentation
use duringthe intervention
andjudgmentof estimation
at
Three
lead
to
and
threeto failureat judgsuccess
accuracy posttest.
paths
ment of estimationaccuracy.
Learners
canimproveinjudgingtheaccuracyof estimatesduringtheinterventionphasein threeways.First,theycouldlearnto correctly
accujudgeestimation
on
both
and
translate
between
these
racy
representations
successfully
We
would
these
students
to
at
well
representations.
expect
perform
posttest(Path
without
A). Second,childrencouldlearnto acteffectivelyon eachrepresentation
between
This
should
lead
also
to
successful
translating
representations.
learning
outcomes(PathB). We returnto thequestionof howto determine
thedifference
TABLE
2
PossiblePathsto Learning
OutcomesforChildren
External
UsingMultiple
inCENTS
Representations
Path
PathA
PathB
PathC
PathD
PathE
PathF
Rep.1
ContinuousDirection
andMagnitude
Rep.2
Categorical
Magnitude
Only
+
+
+
-
+
+
+
-
-
Translation
+
+
-
Learning
Outcomes
+ve
+ve
+ve
-ve
-ve
-ve
Note. Column2 to 4, the sign indicatesif learnershavemasteredthe cognitivetasksassociatedwith
using eithera particularrepresentationortranslatingbetweenrepresentations(a "+"indicatesmastery;a
"-" incompletemastery).Column5 representsthepredictedpositive ornegativelearningoutcomes.
MULTI-REPRESENTATIONAL
EXAMINING
SYSTEMS
35
betweenPathA andPathB in thefinalsectionof thisarticle.Third,learnerscould
mastercontinuous
thecategorical
withoutmastering
representation
representation
ortranslating
betweenrepresentations.
Thisagainshouldleadto successfullearncontainsalltheinformation
ingoutcomes(PathC) asthisrepresentation
necessary
to completethetask.
Similarly,therearethreewaysin whichlearnerscanfailto improveatjudging
theaccuracyof estimates.Wheninteracting
withCENTS,studentscouldprovide
the samewrongansweron bothrepresentations.
Inthiscase,theyhavemastered
translation
withoutmastering
judgmentof estimation
accuracyoneitherrepresentation.Wewouldnotexpectthesestudentstoperformwellatposttest,astheyhave
not demonstrated
the necessaryskillsduringthe intervention
(PathD). Alternabutnotthecontintively,childrencouldlearntousethecategorical
representation
uous representation.
When this happens,translationbetweenrepresentations
cannotoccur.Thisdoesnotprovidethemwithalltheinformation
theyneedtoperformwell atposttest(PathE).Finally,theydo notlearntojudgeestimationaccuand do not translatebetweenthem,
racy with either of the representations,
atposttest(PathF).
resultingin poorperformance
The natureof the representing
worldis expectedto influencethe particular
that
a
child
is
to
learningpath
likely followin CENTS.In linewithexistingliteratureon the propertiesof individualmathematical
or pictorialrepresentations,
a seriesof hypotheseswas generatedconcerningthe ease of masteryof the differentrepresentations
in CENTS.Bothof thepictorialrepresentations
shouldbe
to
understand
in
and
use
that
little
mathematical
relativelyeasy
they require
knowledge,can be consideredas ambientsymbolsystems(Kaput,1987),and
makeuse of perceptual
processesto supportinferences(Larkin& Simon,1987).
In addition,childrenwithlowermathematical
aptitudemaybe ableto use these
representationsmore successfullythan the other types of representations
& Snow, 1977).Onthe otherhand,the mathematical
(Cronbach
representations
in CENTSareless easyto understand
thantheirpictorialequivalentsas theyrequiremore specialistknowledgeand make less use of perceptualprocesses.
childrenshouldtake longerto learn
Comparedwith pictorialrepresentations,
CENTS'mathematical
representations.
Theeaseof translation
betweendifferentMERsis alsolikelyto differdependent on the natureof therepresenting
worlds.Inthe case of the representations
in
to consider
CENTS,childreninteractwith estimationaccuracyrepresentations
twoaspectsof theirestimates;whethertheestimateis higherorlowerthantheexactcalculation,
andhowmuchit deviatesfromtheexactcalculation.
Thefirstrepresentation
withwhichthechildreninteractrequiresthemto considerbothof these
andthesecondrepresentation
aspectsof estimation,
requiresthemonlyto consider
the latterdimension.Forexample,if a childhasalreadymadea judgmentabout
howfarawayanestimateis fromtheexactcalculation,
translation
enablesthemto
use this in the second representationto informtheirjudgmentaboutthe direction
36
AINSWORTH,BIBBY, WOOD
of thisdifference.Ontheotherhand,if a childhasalreadydecidedthemagnitude
thentheyhaveto carrythe
anddirectionof theestimatefromtheexactcalculation,
and
information
to
the
second
magnitude
representation usethisto constraintheir
with
occurswhenchildren
this
Thus,translation
problemsolving
representation.
the
of
from
to theotherand
their
one
carry product
representation
problemsolving
makeuse of thisinformation.
It is predictedthatthepictorialsystemof representations
in CENTSwill faciliin theserepresentations
tatetranslation
as theformatandoperators
areof a similar
in
kind.In the sameway,themathematical
CENTS
shareformat
representations
andoperators,andthusin combination
it shouldbe relativelystraightforward
to
translatebetweenthem.However,themixedsystemof representations
in CENTS
combinesrepresentations
thatapplyto
thatvaryboththeformatandtheoperators
thoserepresentations.
In thiscase,translation
betweenrepresentations
maywell
provemoredifficultthaneitherthepictorialormathematical
sysrepresentational
temsin CENTS.
theseanalyses,we arein apositionto formspecifichypothesesasto
Combining
whichlearningpathschildreninteracting
withdifferentMERswillfollow.Wehave
thattranslation
will playa beneficialrolein
betweenrepresentations
hypothesized
learningtojudgeestimation
accuracy.However,if thelearnermastersthecontinuousdirectionandmagnitude
thengiventhenatureoftherepresented
representation,
it
is
to
in
of
withouttranslatworld, possible improve judgment estimation
accuracy
to
This
sets
in
leads
two
outcomes
this
ing.
ofpossible
experiment-thosewherethe
learnerstranslate
andthosewheretheydonot.Theaboveanalysissuggeststhatthe
will facilitatelearningof eachindividual
pictorialsystemof representations
representationand also the translation
betweenrepresentations.
Learningoutcomes
shouldtherefore
bepositive.Themostlikelypathis therefore
PathA (Table2).With
to
the
mathematical
each
is
difficult
to
the
learn,particularly
respect
representations,
however
if
translation
is
translation
numerals,
relativelyeasy.Consequently,
helps
thelearnerto masterthemathematical
we wouldexpectto observe
representation,
PathA. However,if translation
playsnobeneficialroleinthecontext,wewouldexD.
Path
in
the
mixedsystem,thepictorialarcherytargetshouldberelapect
Finally
numerals
tivelyeasyto learnwhereasthemathematical
mayprovemoredifficult.
translation
will be difficult.Themostlikelypathis PathE.
Furthermore,
EXPERIMENT
1
Design
A two-factormixeddesignwasused.Thefirstfactorvariedthe systemsof representations.
Therewerefourgroupsof participants:
threeexperimental
groupsanda
controlgroup.Oneexperimental
groupreceiveda "picts"systemof representations
(targetand splatwall),anothera "math"system (histogramandnumerical),andthe
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
37
finalgroupa "mixed"system(targetandnumerical).
Thefinalgroupwasa no-intervention
controlthatparticipated
inthepenandpapertests,buttooknootherpart
in theexperiment.
Thesecondfactor,time,waswithinsubjects.Childrenwereasto
the
different
conditionsbasedon theirscoreson a mentalmathtestsuch
signed
thateachgrouphadapproximately
the samemeanandstandard
deviation.Each
conditionhadsimilarnumbersof boysandgirls,andthemeanageof theparticipantsdidnotdiffersignificantly.
DependentMeasures
To evaluatethe way thatrepresentation
use interactedwith learning,a number
of dependentvariableswere requiredto assess (a) learningoutcomes,and (b)
processmeasuresof systemusage.Learningoutcomesweremeasuredbasedon
a paperandpentestthatassessedlearners'estimationknowledgeandskillsprior
to and after interactionwith CENTS.An examplequestionfrom the test is
shownin Figure5. Eachitemon the paperandpentest requiredthe childrento
estimateanswersto either2 x 2 or 3 x 3 digitmultiplication
problems.They
werealso askedto statehowmuchtheirestimatedifferedfromtheexactanswer.
Theaccuracyof the estimateswas providedby the percentagedeviationof each
estimatefromthe exactanswer.Judgment
of estimationaccuracywas calculated
as the absolutedifferencebetweenthe predictedaccuracyandthe actualaccuracy of each estimate.If differentrepresentational
systemsin CENTSlead to
differentlearningoutcomesthesearemostlikelyto be observedin thechildren's
judgmentsof the accuracyof theirestimates.
To determinehow alternative
MERsmayleadto differencesin learningoutmeasures
were
derived
from the children'sinteractionswith
comes, process
CENTS.Foreachof thesix stagesof interaction
withCENTS,thechildren'skey
aboveinthesyspressesandmouseclickswererecorded
bythesystem(described
tem descriptionof the article).ForStages1, 3, 4, and5 of the interactions
with
1.Estimate:
19x 69
is
myestimate
1400
much verymuch
verymuch muchless
less
justless
exactly justmore more
less I
more
more
I
I
I the same
30%or
30%to
20%to
0%
10%to
0%to
10%to
20%to
30%or
below 1 20%less 10%less I 0%less
10%more120%more130%more above
+
FIGURE5 Anexamplequestion(withanswer)fromthepenandpapertest.
38
AINSWORTH,
BIBBY,WOOD
CENTS,therewere no differencesin the children'sexperiences.Furthermore,
theirinteractions
withCENTSduringthesestagesof strategicsupportwereconstrainedby thesystem.Forexample,at Stage3 whenthechildrenmultiplytheextracteddigits,helpwouldbe providedif theymadean error.Hence,no directly
with the systemcouldbe obrelevantdifferencesin the patternsof interaction
conditions.
servedforthesestagesbetweenthedifferentexperimental
a
AtStage2, childreninteracted
MERsaccording
totheirassigned
with different
on
AtStage6,thenatureofthefeedbackprovideddepended
condition.
experimental
withCENTSthatdiffered
theMERs.Giventhatthisistheonlypartoftheinteraction
betweenthe experimental
conditions,thisis wherethebehavioral
systematically
analysisfocused.
Wheninteracting
witheachof therepresentations
(e.g.,thesplatwall)children
weregiventhe taskof judginghowfartheirestimatewas fromtheexactanswer.
Boththesplatwallandthenumerical
abouttheextent
displayprovidedinformation
towhichchildren's
withrespecttobothmagnitude
anddijudgmentswereaccurate
rection.Thearcherytargetandhistogram
aboutmagnionlyprovidedinformation
tude. For the continuousrepresentations,
the absolutedifferencebetweenthe
children'sjudgmentof estimationaccuracyandtheactualvaluethatshouldhave
beenselected,giventhe particular
This
problem,was calculatedas a percentage.
on-linemeasuremapsdirectlyontothescoreof judgmentof estimationaccuracy
calculatedforthepaperandpentests.Similarly,forthecategorical
representations,
theabsolutedifferencebetweentheselectedandcorrectcategorieswascalculated.
If differencesinlearningoutcomesareobservedforthejudgments
acof estimation
on
the
these
either
and
be
related
to
should
the
curacy
paper pentests,
judgmentsof
estimationaccuracyon the continuousrepresentation,
thecategoricalrepresentation,orboth.
Theextenttowhichchildrencoordinated
theirinteractions
withthetworepresentationsis measurable
thecorrelation
betweentheirjudgments
of estibyexamining
mationaccuracyonthecontinuous
andcategorical
Thismeasureis
representations.
similarin kindto thatuseby SchwartzandDreyfus(1993)to measureintegration
acrossrepresentations
intheirmultirepresentational
software.Giventhattherepresentational
withonerepresentation
both
redundant,
systemsarepartially
containing
anddirectioninformation
andthesecondrepresentation
magnitude
only
containing
itis onlypossibleto deriveameasureofcoordination
forthe
information,
magnitude
Ifchildrenlearntocoordinate
theiruseofrepresentations
on
magnitude
component.
thisdimension,thereshouldbe a highpositivecorrelation
betweentheirjudgments
onthetworepresentations.
ifchildrencantranslate
ofestiTherefore,
theirjudgment
mationaccuracyfromthe firstrepresentation
withwhichtheyinteracted
ontothe
secondrepresentation,
thentheyshouldshowapproximately
thesamepercentage
deviationfromtheexactansweronbothrepresentations.
Ifthereis nocorrelation
betweenthetwomeasuresofjudgmentof estimation
accuracy,thenthechildrencannot be coordinatingtheiruse of the two representations.If the differentMERshave
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
39
differentimpactsonthelearningoutcomes,thenthedegreetowhichtheinteractions
arecoordinated
maypredictthosedifferences.
Participants
Forty-eight
mixed-ability
year-5pupilsfroma statejuniorschooltookpartin the
in
experiment.
Theyranged agefrom9:9to 10:8years.All thechildrenwereexperiencedwithmousedrivencomputers.
Materials
A generaltestofmentalmathematics
wasconstructed
bycombiningexercisesfrom
bookstwo andthreeof "ThinkandSolve MentalMaths"(Clarke& Shepherd,
1984).Itwaspilotedwithaparallelclassthatwasnottakingpartintheexperiment.
Thepenandpapertestrequired
thechildrento estimateananswerto a multiplicationproblem.Therewere20 questions,eight3-digitby 3-digitproblems(e.g.,
213 x 789)andtwelve2-digitby 2-digitproblems(e.g.,21 x 78).Toprobetheunthatchildrenhadintothe accuracyof theirestimates,theywererederstanding
to
state
howmuchtheirestimatedifferedfromtheexactanswer(judgment
quired
of estimationaccuracy,see Figure5).
Thenatural
Categorieswerelabeledin bothnaturallanguageandpercentages.
in
labels
used
the
and
test
were
the
same
as
those
used
language
pen paper
by childrento describetheirestimatesin theironlinelogbooks.Theresolutionusedfor
thepretestsandposttestswasthe sameas thatusedin the categoricalrepresentationsandthe logbookwas presentin all threeconditions.
Procedure
weregiventhe mentalmathtests in theirclassroom.
Pretest. Participants
Theclassteacherreadtheitemsto thechildrenandallowedthemto querymisunderstooditems.Childrenwerealloweda shortbreakaftereachblockoften items.
Intotal,thetesttookabout30 minto complete.
The estimationtests weregiventhe followingday.The instructions
stressed
thatexactanswerswerenotrequired
andencouraged
guessingratherthanleaving
an answerblank.The judgmentof estimationaccuracymeasurewas also explained.Thechildrenwereallowedto proceedat theirownpacethroughthetest
andgenerallytook between20 and40 minto completeit. Oneparticipant
was
stoppedafteranhour.Threeparallelversionsof eachtestwerecreatedand,to prevent copying, childrenseated close togetherwere given differentversions.
40
AINSWORTH,
BIBBY,WOOD
Computerintervention. The computerinterventionbeganthe following
week.Participants
usedthecomputer
ina quietspace.Toensuresuffiindividually
cientpracticewiththesystem,eachchildusedCENTStwice,separated
byapproxiwasbetween80 and100
matelytwo weeks.Thetotaltimespenton thecomputer
demonstrated
howtouseCENTSandthenstayedtoprovide
min.Theexperimenter
if
children
became
confused
about
howto operatethesystem,butnodirect
support
was
teaching given.
Thechildrenwereseteightquestionsthattheyhadto answerbybothtruncation
androunding.All questionspresentedweregenerateddynamically.
Eachchild
startedwitha two-by-twoproblemandwas graduallyintroduced
to largerproblems(two-by-three
andthree-by-two).
Thefinaltwowerethree-by-three
multiplication problems.After each problem,childrenfilled in the on-line logbook
recordingdetailsof theiractivities.
Posttest. Children
receiveda parallelversionof theestimation
testwithin10
daysof theirsecondcomputersession.
RESULTS
Thedesignfortheanalysesof thepenandpapertestswas4 (control,math,mixed,
wasbetweengroups,and
picts)by 2 (pretest,posttest).Thefirstfactor,Condition,
thesecondfactor,Time,wasa withinsubjectsmeasure.Thenumberof participants
in eachcell is 12 forall analysesunlessotherwisestated.
LeamingOutcomes
Children'sperformance
on pretestsandposttestswas analyzedto determinehow
CENTS
the acquisitionof computational
estimationskills.
effectively
supported
Themostcommonlyusedmeasureof estimation
is thepercentage
deperformance
viationof theestimatefromtheexactanswer.Forexample,anestimateof 2500for
thesum"53x 52"is 9.3%awayfromtheexactanswer.Thesupportforlearningto
estimatewasheldconstantoverall threeexperimental
conditions,so theonlyexandcontrolconpecteddifferencesonthismeasurewerebetweentheexperimental
ditions.In contrast,performance
on thejudgmentof the accuracyof an estimate
wasexpectedto differbetweentheexperimental
asthiswasthefocusof
conditions,
the differentMERs.
Estimation
accuracy. The results fromone participantwere removed.
Her results were 10 standard deviations above the mean at pretest. Table 3
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
41
shows thatchildren'spretestscores were very inaccurate.The averagepercentagedeviationof the estimatefromthe correctanswerwas 96%.This createdtwo problems.First,the datawere sufficientlynonhomogenousthatno
transformcould be used. Second, this measurehas traditionallyonly been
used on deviationsof up to 40%. Consequently,other measuresof performancewere designed.
One difficultywith using a percentagedeviationmeasureis that a large
numberof childrenperformedappropriate
correctfront-end
transformations,
but failed at place value correction.To distinextraction,and multiplication,
guish betweenthose childrenwho only failedat the finalstep fromthosewho
used incorrectstrategiesorjust guessedanswers,the estimateswerecorrected
for orderof magnitude.A childanswering12,000to "221 x 610"(i.e., failing
to correctby one orderof magnitude)wouldbe correctedfrom-91% to -11%
inaccurateby this measure.However,a guess of 25,000 wouldremain-81%
inaccurate.This measurewas designedto identifythe childrenwho weregeneratingplausibleestimates,only failingat the orderof magnitudecorrection
(Table 3).
Analysisusinga 4 x 2 ANOVAon the correctdatafounda significantmain
effectof time,F(1, 44) = 10.84,MSE= 295,p < 0.002, anda significantinteractionbetweenconditionandtime,F(3, 44) = 3.01, MSE= 159,p < 0.04. Simple
maineffectsanalysisfoundno significantdifferencesbetweenthe conditionsat
pretest,butthereweredifferencesat posttest,F(3, 88) = 4.57, MSE= 227,p <
0.02. The controlcondition'sperformance
did not change,but all threeexperimentalconditionsimprovedsignificantly:
mixed,F(1, 44) = 4.58,MSE= 159,p
< 0.04; math,F(1, 44) = 7.42, MSE = 159,p < 0.01; and picts,
=
F(l, 44) 7.025,
MSE= 159,p < 0.02. It seemsthatchildrencan learnto estimatewithCENTS
andthatthe observedimprovements
in performance
werenot due solely to the
effects of repeatedtesting.
TABLE
3
Estimation
andTimeUsingPercentageDeviation
Scores(Experiment
Accuracy
byCondition
1)
Control
OutcomeMeasure
Deviation
Deviationcorrected
for
placevaluemagnitude
Time
M
Mixed
SD
M
Math
SD
M
Picts
SD
M
SD
Pretest 89.8% 16.9 88.9% 9.5 101% 62.5 102% 55.7
Posttest 82.7% 13.9 60.7% 24.6 55.3% 45.1 57.6% 34.1
Pretest 38.6% 10.6 37.3% 18.1 38.1% 15.6 40.7% 12.16
Posttest
42.1%
10.4
27.1%
14.5
24.0%
17.2
27.6%
19.1
42
BIBBY,WOOD
AINSWORTH,
of estimationaccuracywas
Judgmentof estimationaccuracy. Judgment
measuredto explorethe insightsthe childrenhaveintotheprocessof estimation
andhowanestimatedifferedfromtheexactanswer.Thiswasassessedona 9-point
to indicatehowfarawaytheirestimatewasfromtheexscaleusedby participants
actanswer.Theresponseswerecodedas thedifferencebetweenthecategorythat
theyshouldhaveselectedgiventheirestimateandthosethattheyactuallyselected
and8 (e.g.,a childselects
(Table4). Thisprovidesa scorebetween0 (agreement)
was "verymuchless").
vey muchmorewhen theirestimate
Theanalysisrevealeda significantmaineffectof time,F(1, 44) = 8.25,MSE
betweenconditionandtime,F(3,
= 0.64,p < 0.01, anda significantinteraction
44) = 3.28, MSE = 0.64, p < 0.03. The only significant differencesbetween the
conditions were at posttest, F(3, 88) = 4.14, MSE = 0.98, p < 0.01. The perfor-
manceof boththe controlconditionandmixedconditiondidnotchangesignificantly. However, scores for the math condition,F(1, 44) = 5.73, MSE = 0.98, p
< 0.02, and the picts condition,F(1, 44) = 4.67, MSE = 0.98, p < 0.003, did im-
prove significantly.
ProcessMeasures
The behavioralprotocolswere analyzedto determinehow the use of different
MERsresultedin these differentiallearningoutcomes.The firstmeasurewas
judgmentof estimationaccuracyandthe secondmeasurewas representational
coordination.
Judgmentof estimationaccuracywas assessedseparatelyfor the
two representations.
Continuousjudgmentof estimationaccuracy. Thecontinuous
representationsusedin theexperiments
werethenumerical
displayin themixedandmath
conditionsandthe "splatwall"
in thepictscondition.Thedatafromthesplatwall
wererecodedaspercentage
modelthatdrives
deviationscoresusingtheunderlying
therepresentation.
AnANOVAwasconducted
ontheon-linedatafromthetwotri4
TABLE
of Estimation
andTime(Experiment
1)
Judgment
byCondition
Accuracy
Control
Time
Pretest
Posttest
Math
Mixed
Picts
M
SD
M
SD
M
SD
M
SD
3.26
3.58
0.91
1.18
3.06
2.67
1.74
0.97
3.06
2.28
0.91
0.81
3.46
2.44
0.62
1.32
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
43
alswithCENTS.Thedesignwas3 (math,mixed,picts)x 2 (Time1,Time2). The
firstfactorwasbetweengroups,andthesecondwaswithinsubjects[Table5]).
Thesedatadidnotpasshomogeneity
of variancetests,andtherefore
weretransformedusinga naturallog function.Therewas a significantmaineffectof time,
betweentime
F(1, 33) = 8.03,MSE= 0.27,p < 0.01, anda significantinteraction
and condition,F(2, 33) = 4.08, MSE = 0.12, p < 0.03. Simple main effects only
identifiedsignificantdifferencesbetweenthe conditionsat Time2, F(2, 66) =
4.44,MSE=0.19,p < 0.02,withboththepictsandthemathconditionsperforming
betteratTime2 thanthemixedconditions(Figure6). Children
in themathconditionwho usedthe numericalrepresentation
a significantimprovedemonstrated
ment in performance,F(1, 33) = 14.67, MSE = 0.12, p < 0.001.
TABLE
5
of Estimation
andTime(Expteriment
Judgment
Accuracy
Representation,
1)
byCondition,
Mixed
ProcessMeasure
Time
M
SD
Math
M
SD
Picts
M
SD
Continuous
deviation Time1 19.4% 11.45 21.0% 12.75 13.8% 4.34
reps.percentage
Time2 18.0% 5.66 11.8% 6.00 11.7% 3.94
Categorical
reps.categorydifferences Time1 1.17 0.26 1.24 0.55 1.04 0.29
Time2
1.14 0.38 0.86 0.30 0.93 0.38
c 20%
o
Mixed(Numerical)
Maths(Numerical)
Picts (Splatwall)
C
10%
TimeI
Time2
of estimation
oncontinuous
FIGURE6 Judgment
accuracy
representations
by conditionand
time(Experiment
1).
44
AINSWORTH,
BIBBY,WOOD
Categorical
judgmentof estimationaccuracy. Theaboveanalysiswas
ofthecategorical
(thetarget
repeatedforthemagnitude
component
representations
in thecaseof mixedandpictsconditions,andthehistogramin the
representation
mathcondition).
Therewasa significant
effectoftime,F(1,33)= 4.59,MSE=0.12,
betweenconditionandtimewasnotsignificant(Figure7);
p < 0.04.Theinteraction
however,theoverallpatternof resultswassimilartothoseforthecontinuous
representations.
of thejudgment
of estimacoordination.Themeasures
Representational
tionaccuracyprovidedataonhowstudentscometo understand
therepresentation
andthetask,butdonotsaywhetherchildrenrecognizetheconnections
betweenthe
As
children's
of
the
representations.
understanding
system
multirepresentational
improved,theirbehaviorshouldhavebecomesimilaracrossbothrepresentations.
To obtaintherepresentational
coordination
measure,thechildren'sjudgmentsof
estimationaccuracyonthetworepresentations
werecorrelated
(Table6, Figure8).
Wepredictedthatdepending
ontheexperimental
condition,childrenwoulddiffercoordination.
entiallyimprovein representational
Thereis a trendforthecorrelations
tobehigherontheseconduseof thesystem,
F(1, 33)= 3.629,MSE=0.06,p < 0.065.Therewereno significantdifferencesbetweentheconditionsat Time1,buttherewereatTime2, F(2, 66) = 3.60,MSE=
forthemathcondi0.11,p < 0.04. Simplemaineffectsshowedan improvement
tion, F(1, 33) = 3.73, MSE = 0.06, p < 0.06, and the picts condition,F(1, 33) =
1.4
0
1.2
o
Mixed(Archery
Target)
Maths(Histogram)
Picts(Archery
Target)
0S1.0
0
0.8
Time1
Time2
FIGURE7 Judgment
of estimation
oncategorical
accuracy
representations
byconditionand
time(Experiment
1).
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
45
3.824,MSE= 0.06,p < 0.06.However,themixedconditionshowednoevidenceof
improvedcoordination.
DISCUSSION
twodifferentaspectsof theestimation
CENTSsupports
process-producinganeshowthatestimaterelatesto theexactanswer.Giventhat
timateandunderstanding
anestimatewasheldconstantoverthethreedifferentverthesupportforproducing
sions of CENTS,we expectedno strikingdifferencesbetweenthe experimental
conditionsin termsof theirestimationaccuracy.Thiswaswhatwe observed.Bedeviationfromtheexactanswerwas
foreexposuretoCENTS,themeanpercentage
thatthe childrendid notknowhow to applyestimation
96%.Thisdemonstrates
strategiescorrectly.Inorderto assesschildren'sestimatesmorecloselygiventhis
TABLE6
Correlations
VetweenJudgment
of Estimation
on TwoRepresentations
by
Accuracy
andTime(Experiment
Condition
1)
Mixed
Math
Picts
Time
M
SD
M
SD
M
SD
Time1
Time2
0.37
0.31
0.36
0.31
0.47
0.67
0.42
0.37
0.37
0.57
0.33
0.23
0.8
Mixed
0.6
Maths
o
Picts
0.4
0.2
Time1
Time2
FIGURE8 Correlations
betweenjudgments
of estimation
accuracyon tworepresentations
by conditionandtime(Experiment
1).
46
WOOD
AINSWORTH,
BIBBY,
low performance,
answerswerecorrectedfororderof magnitudeandthe results
re-examined.
Themodifiedpercentage
deviationscoresof thethreeexperimental
aftertheintervention,
butthecontrol
conditionsshowedsignificantimprovement
condition'sscoresdidnotimprove.Thechildrenin thethreeexperimental
conditionsbecameequallycompetentestimators
CENTS.
by using
childrenneedto understand
In orderto becomeflexible,accurateestimators,
how transforming
solutionaffectsthe accunumbersto producean intermediate
the
of thisskillby
of
the
estimate.
We
encouraged development
racy
subsequent
two
to
the
of
representarequiringchildren judge accuracy theirestimatesusing
tions.Thisskillis theonemostdirectlysupported
by theMERsandthereforeany
effectof conditionwouldbe expectedto manifestitselfinjudgmentof estimation
thatgiventhevaryingnatureof thedifferentrepresenaccuracy.Wehypothesized
tationalsystems,notallcombinations
wouldleadto successfuloutcomes.Penand
papermeasuresof thejudgmentof estimation
accuracyshowedthatchildreninthe
mathandpictsconditionsmadesignificantimprovements
in thisskill.However,
childreninthemixedconditionbecamesignificantly
moreaccurateestimators
(estimationaccuracy)withoutbecomingbetterat knowinghow accuratetheiranswerswere(judgment
of estimationaccuracy).
Toexaminehowtheproperties
of thedifferent
representational
systemsresulted
in thisoutcome,thebehaviorof thechildrenusingCENTSwasexamined.Theresultsshowedthatbothcontinuous
andmagnitude)
andcategorical
(direction
(maghave
a
similar
nitude) representations
strikingly
pattern,althoughthe only
statisticallysignificantinteractionswere for the continuousrepresentations.
Childrenin thepictsconditiondisplayeda trendforbetterjudgmentof estimation
thisgoodperformance
atTime2. ThemathconaccuracyatTime1andmaintained
ditionbecamebetteratjudgingtheaccuracyof theirestimatesovertime.Theirlow
initialperformance
seemstoindicatethattherewasasignificant
costassociated
with
Onceunderstood,
theserepresenlearninghowtousemathematical
representations.
tationswereusedsuccessfully.Judgments
of estimationaccuracywiththemixed
didnotimproveoverthesessions.Relativeto theotherconditions,
representations
in themixedconditionwereworseatjudgingestimationaccuracy.
participants
Bothrepresentations
thatwereusedby themixedconditionwerealsopresent
in oneof theotherconditions-thetargetwasusedby thepictsconditionandthe
numericalrepresentation
were
by the mathcondition.Whentherepresentations
wereableto use themsuccessfully.It
employedin theseconditions,participants
was onlywhentheserepresentations
werecombinedin themixedconditionthat
this poorperformance
was observed.Hence,the explanationof this difference
lies in thecombination
of therepresentations
ratherthanin thenatureof theindividualrepresentations.
Thetranslation
betweenindividualrepresentations
maybe a crucialaspectof
learningto use MERs.We hadhypothesizedthatthisprocessof translationmaywell
be problematicfor the childrenin the mixed condition,but relativelyeasy for chil-
MULTI-REPRESENTATIONAL
EXAMINING
SYSTEMS
47
Toexplorethis,a measureof representational
dreninthepictsandmathconditions.
thatasexperience
withthesystemincoordination
wasdeveloped.Itwaspredicted
be
a
trend
toward
should
as
there
creased,
by the
increasing
convergence measured
two
on
the
different
ofestimation
correlation
betweenthejudgments
repreaccuracy
wasfoundinboththemathandpictsconditions,
butnotin
sentations.
Convergence
werelessableto
themixedcondition.Thisfailuretoconvergesuggeststhatchildren
translatebetweenthemixedrepresentations.
These combinationsof processandlearningoutcomemeasuresallow us to
identifywhichof the six learningpathsforjudgmentof estimationaccuracywe
foreachcondition.Childrenin the
proposedin Table2 bestexplainperformance
accuracywitheachreprepictsconditionlearnedto masterjudgmentof estimation
coordination
sentation,to show increasingconvergenceon the representational
measure,andto be successfulat posttest.We propose,thatas predicted,learning
PathA bestdescribesthisbehavior.Childrenin themathconditionalso showed
successfuloutcomesanddemonstrated
coordinahigh level of representational
tion.In comparison
withthepictscondition,theytooklongerto mastertherepresentations.However,by the end of their secondCENTSsession, they were
judgingestimationaccuracyeffectivelywithbothmathematical
representations.
We hypothesizedthatif learnersin the mathconditionwere ableto benefitby
theirproblemsolvingbetweenrepresentations
translating
theywouldbe successful.Weconcludethereforethatthebehaviorof childreninthisconditionalsocorrespondedto learningPathA. Finally,forthemixedcondition,we hadpredicted
that,dueto the difficultyin masteringthe complexmathematical
representation
andthedifficultyof translating
betweenrepresentations,
themostlikelyPathwas
E. Thiswasnotwhatwe observed.Instead,giventhelowrepresentational
coordinationscoresandthefailureto improveatjudgingestimation
with
accuracy either
the
best
of
is
Path
F.
Webelievethatthe
representation,
explanation performance
reasonwhy childrenperformed
less well thanwe expectedin this conditionwas
thattheirattemptsto translatewhentranslation
was difficultinterfered
withthe
successfullearningof theindividualrepresentations.
Thesefindingsareconsistentwiththeideathatcombinations
of differentrepresentations
donotalwaysproducetheoptimum
benefits.Rather,similarrepresenting
worldsmayleadtohigherperformance
thanmixingrepresenting
worlds.However,
thechildrenusedCENTSforlessthantwohours.Itcouldbe thecasethatourconcernsaboutthelackof observable
benefitsofmixedrepresentations
donotreflecta
worlds
long-term
difficulty.Rather,it couldsimplybethatwithmixedrepresenting
childrentakelongerto learnto translate
betweenrepresentations.
EXPERIMENT
2
Boththerepresented
andrepresenting
worldsin CENTSwerenewto theparticipantsin Experiment1. This placedespecially heavy learningandworkingmemory
48
AINSWORTH,
BIBBY,WOOD
demandsuponthechildren.Theobservation
thatthelearnerswiththemixedrepresentational
is
consistent
with
systemstruggled
cognitiveloadanalysesof learning
Chandler
&
Sweller,1992;Sweller,1988).Cognitiveloadaccountssuggest
(e.g.,
thatthetaskdemandsareinitiallyveryhighwhenlearnersareintroduced
to aproblem.However,withpractice,components
ofthetaskbecomeautomated,
freeingup
resourcesforotheraspectsofthetask.Ifthisis thecase,thedifficultiesthatthechildrenexperiencecoordinating
mixedrepresentational
systemsarelikelyto be only
in the short-term.
withthe
Whenchildrenbecomemoreexperienced
problematic
andwithestimation
of mixedreprelearningenvironment
problems,coordination
sentationsshouldimprove.
Thishypothesiswas testedby addingtwo furtherintervention
sessionsto the
experiment,
producinga totalof fourCENTStrialsin all.1Giventhe successful
of childrenin themathandpictsconditionin Experiment
1,we conperformance
tinuedto predictthattheywoulddemonstrate
positiveoutcomesandfollowPath
A. However,if thechildrenwiththemixedMERshadnotimprovedatrepresentationalcoordination
orto learn
by thefourthsession,thenthefailureto coordinate
therepresentations
wasunlikelyto be dueto the lackof familiarity
withthetask.
Weneededto determine
whichlearningpathchildreninthemixedconditionwere
If
demonstrated
successfullearningoutcomes,theymayhavefolfollowing. they
lowedpathsA, B, orC (Table2). If theydidnotimproveatjudgingtheaccuracyof
theirestimates,theycouldhavetakenoneof pathsD, E, orF.
Design
Thisexperimentemployedthe samerepresentations
andbasicdesignas Experiment1.A two-factormixeddesignwasused.Thefirstfactorvariedthesystemsof
Thisresultedin threeexperimental
conditionsof 12participants
representations.
consistingof thosewhoreceived"picts"(targetandsplatwall),"math"
(histogram
andnumerical),
and"mixed"(targetandnumerical)
Thefinalconrepresentations.
ditionwasano-intervention
controlwhotookthepenandpapertests.A secondfacwereassignedto a conditionbasedontheir
tor,time,waswithinsubjects.Children
scoresona mentalmathtest.Eachconditionhadsimilarnumbers
of boysandgirls
andthemeanage of theparticipants
didnotdiffersignificantly.
Participants
Forty-eight
year-5andyear-6pupilsfroma statejuniorschooltookpartin theexNonehadtakenpartinExperiment
1.Thechildrenrangedinagefrom9:5
periment.
to 11:2years.All thechildrenwereexperienced
withmousedrivencomputers.
'This numberof sessions was also thoughtto constitutethe maximumamountof time that a child
could be expected to use such a focused learningenvironmentin a UK classroom.
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
49
Materials
Thesewereidenticalto thoseusedin Experiment
1.
Procedure
2 followedthesameprocedure
asExperiment
1,exceptthatchildrenin
Experiment
the experimental
conditionusedCENTSa totalof fourtimes.Thetotaltimethey
spenton thecomputerwasbetween150and220 min.
RESULTS
LeamingOutcomes
Penandpapermeasuresweretakento examinewhetherthecomputer
intervention
children
to
become
accurate
estimators.
As
boththeacbefore,
taught
successfully
andcorrected
forplacevalue)andjudgment
curacyof theirestimates(uncorrected
of estimationaccuracyscoreswereexamined.
Estimationaccuracy. As seen in Table7, the pretestperformance
of the
childrenwaslow.Theestimateswereonaverage87%awayfromtheexactanswer.
Atposttest,theexperimental
conditions
weremuchcloserwithanaverage28%deviation.No analysiswasperformed,
as thedatawereextremelyheterogeneous.
A secondmeasureof accuracyis thecorrectedpercentagedeviation.Thisadjusted children'sanswersto the correctorderof magnitudeand hence distinbutfailed
guishedbetweenchildrenwhoperformed
transformations,
appropriate
at finalplacevaluecorrection,fromthosewho usedinappropriate
strategiesor
simplyguessedananswer(Table7).
Analysisby a 4 x 2 ANOVAyieldedsignificantmaineffectsof condition,F(l1,
42) = 6.28, MSE = 105.1,p < 0.002, and time, F(1, 42) = 147.33, MSE= 60.9, p <
0.001.Therewas a significantinteraction
betweenconditionandtime,F(3, 42) =
conditionsimprovedover
21.38, MSE= 60.9,p < 0.001. All the experimental
time: mixed, F(1, 42) = 111.65, MSE = 60.9, p < 0.001; math,F(l, 42) = 68.44,
MSE = 60.9, p < 0.001; and picts, F(l, 42) = 31.81, MSE = 60.9, p < 0.001. The
controlconditiondidnotimprove.
Theimprovements
inchildren'sestimation
skillsafteranintervention
phaseusing CENTSwerethereforereplicatedconvincinglyby thisexperiment.
werecodedas thedifJudgmentof estimationaccuracy. Theresponses
ferencebetween theirjudgmentof the accuracyof theirestimatesandthe category
50
AINSWORTH,
BIBBY,WOOD
thatshouldhavebeenselected,giventheactualestimate(Table8).Thiswasexaminedusinga 4 x 2 ANOVA.
Thereweremaineffectsof condition,F(l, 42) = 6.81, MSE= 1.27,p < 0.001,
andtime,F(1, 42) = 110.38,MSE= 0.71,p < 0.0001,anda significantinteraction
between time and condition,F(3, 42) = 6.04, MSE= 1.27,p<0.002. All conditions
control,F(l, 42)= 4.87,MSE= 0.71,p < 0.04;mixed,F(1,
improvedsignificantly:
42) = 57.83, MSE = 0.71, p < 0.0001; math, F(1, 42) = 47.04, MSE = 0.71, p <
0.0001;andpicts,F(1, 42) = 18.65,MSE= 0.71,p < 0.0001.Simplemaineffects
showedno differencesbetweenthe conditionsat pretest,butthereweredifferencesatposttest,F(3, 84) = 11.37,MSE= 0.99,p < 0.001.Theexperimental
conditionsweresignificantly
betteratjudgingtheaccuracyof theirestimatesthanthe
controlconditionatposttest:mixedversuscontrol(q= 6.17,p < 0.001),mathversus control (q = 5.93, p < 0.001), and picts versus control(q = 5.33, p < 0.01).
Theresultsof the analysisof judgmentof estimationaccuracythereforediffer fromExperiment1. Here,childrenin all the experimental
conditionsimtheir
over
time.
a
result
is
with
theproposal
Such
consistent
proved
performance
thatmixedrepresentations
are only problematicinitially.In orderto examine
more closely how the differentrepresentational
systemsmay have affected
the
behavioral
learning,
protocolsgeneratedduringthe interventionsession
were examined.
7
TABLE
Estimation
andTimeUsingPercentageDeviation
Scores(Experiment
Accuracy
byCondition
2)
Control
Outcome
Measure
Time
%Deviation
%Deviationcorrected
forplacevalue
magnitude
M
SD
Mixed
M
Math
SD
M
Picts
SD
M
SD
Pretest 89.2% 14.83 83.6% 6.81 92.3% 24.22 84.3% 7.33
Posttest 85.8% 11.87 27.8% 18.98 20.1% 15.42 36.7% 45.11
Pretest 42.7% 4.48 50.8% 7.11 46.2% 8.85 40.7% 11.69
Posttest 43.6% 11.07 17.2% 12.13 18.7% 7.73 22.0% 6.89
TABLE
8
of Estimation
andTime(Experiment
Judgment
Accuracy
byCondition
2)
Control
Time
Pretest
Posttest
Mixed
Math
Picts
M
SD
M
SD
M
SD
M
SD
4.81
4.04
1.09
0.92
4.60
1.99
0.87
0.79
4.53
2.07
0.68
1.4
3.82
2.27
0.94
1.1
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
51
ProcessMeasures
To examinehowthechildren'sperformance
changedwithexperienceon CENTS
andtheeffectsofthedifferent
anumber
wereperformed.
ofanalyses
As
conditions,
forExperiment
1,twotypesof measureswereexamined:thosethatanalyzedhow
thechildren'sunderstanding
of thedomainwasreflectedintheiruseofrepresentatheirappreciation
of howtherepresentations
relate
tions,andthosethatmeasured
to eachother.
Continuous
judgmentof estimationaccuracy. Thismeasureexamined
of
estimation
Estimation
judgment
accuracy
usingthecontinuous
representations.
wasrepresented
as numerals
formixedandmathconditions
andas a
accuracy
forthepictscondition.A 3 x 2 ANOVAwasconducted
withtheon-line
"splatwall"
datafromthe participants'
firstandlast trialswith CENTS.The designwas 3
(mixed,math,pictures)x 2 (Time1, Time4 [Table9]).
Analysisrevealeda maineffectof time,F(1, 31) = 44.11,MSE= 24.1,p <
0.0001. Therewas also a trendtowardsa maineffect of condition,F(2, 31) =
betweentimeandcon2.99, MSE= 19.16,p < 0.065.A trendforan interaction
dition,F(2, 31) = 2.87, MSE= 24.1,p < 0.07, was also observed(Figure9).
Thereweresignificantdifferencesbetweenthe conditionsafterthe firstsession
on the computer,F(2, 62) = 4.80, MSE= 24.6,p < 0.012, butnot afterall four
sessions.At Time 1, childrenin the picts conditionwere performingsignificantlybetterthanthe otherconditions:
pictsversusmath(q = 4.10,p < 0.05)and
=
<
versus
mixed
picts
(q 4.02,p 0.05). Howeverby Time4, the otherexperimentalconditionshad improvedsignificantly,but the picts conditionhad not
improvedfurther;mixed,F(1, 31) = 17.35,MSE= 24.1,p < 0.001;andmath,
F(1, 31) = 28.5, MSE = 24.1, p < 0.001.
Categoricaljudgmentof estimationaccuracy. As with the continuous
thereweremaineffectsof time,F(1, 31) = 7.29,MSE= 0.11,p <
representations,
TABLE
9
of Estimation
andTime(Experiment
Judgment
Accuracy
byCondition,
Representation,
2)
Mixed
Process Measure
Time
Continuous
deviation Time1
reps.percentage
Time4
Categorical
reps.categorydifferences Time1
Time4
M
19.2%
10.9%
1.15
1.10
SD
5.69
4.50
0.28
0.27
Math
M
19.3%
8.2%
1.30
0.78
Picts
SD
6.31
2.96
0.41
0.30
M
14.0%
9.8%
0.91
0.84
SD
4.27
3.06
0.24
0.43
52
AINSWORTH,
BIBBY,WOOD
0.012,andcondition,F(2, 31)= 3.29,MSE=0.12,p< 0.05(seeFigure10)withthe
childrenin thepictscondition(archerytarget)predictingsignificantly
moreaccu=
<
wasa sigthan
the
There
mixed
condition
rately
(archery
target;q 3.55,p 0.05).
nificantinteraction
betweenconditionandtime,F(2, 31) = 3.65,MSE= 0.11,p <
0.05. Simplemaineffectsanalysisshowedthereweresignificantdifferencesbetweenthe conditionsatTime1,F(2, 62) = 3.94,MSE= 0.11,p < 0.025,withthe
betterperformance
thanthemathconpictsconditiondemonstrating
significantly
=
<
dition(histogram;
At
betweenthecondiq 4.01,p 0.05). Time4, thedifferences
tionsapproached
F(2,62)= 2.95,MSE= 0.11,p < 0.06.Theonlyconsignificance,
ditionto changesignificantly
overtimewasthemathcondition,F(l, 31) = 13.78,
MSE= 0.11,p < 0.001.
Unlike Experiment1, differenceswere foundbetweenthe categoricaland
continuousrepresentations.
In this experiment,
bothof the mathrepresentations
were associatedwith poorerperformance
initially,but improvedsignificantly
over time.Usingthe pictsrepresentations,
children'sjudgmentsof the accuracy
of theirestimateswereinitiallybetterandby Time4, theyhadverysimilarperformanceto the mathcondition.However,therewas a dissociationforthemixed
condition.Judgmentof estimationaccuracywiththecontinuousmixedrepresentationimprovedovertime,whereaschildren'sperformance
withthe categorical
mixed representation
did not.
coordination.Thisanalysis
wasdesigned
toexamine
the
Representational
behavioracrossthetworepresentations.
If children'sunsimilarityof participants'
of
the
their
behavior
shouldhave
derstanding
representational
systemimproved,
20%
Mixed(Numerical)
S
Maths(Numerical)
Picts(Splatwall)
C10%
10%
Time1
Time4
FIGURE9 Judgments
of estimation
oncontinuous
and
accuracy
representations
bycondition
time(Experiment
2).
SYSTEMS
EXAMINIGMULTI-REPRESENTATIONAL
53
becomesimilaracrossbothrepresentations.
Thiswasexaminedby correlating
the
on
judgmentsof estimationaccuracy the two differentrepresentations.
Analysis
was by a 3 x 2 ANOVA,(Table10andFigure11).
Therewere main effects of condition,F(2, 31) = 9.84, MSE= 0.11, p <
0.001,andtime,F(2, 31) = 5.78,MSE= 0.07,p < 0.002.Therewas a significant
interaction
betweenconditionandtime,F(2, 31) = 4.80,p < 0.02. Simplemain
effects showedsignificantdifferencesbetweenthe conditionsat Time 4, F(2,
62) = 14.68, MSE = 0.11, p < 0.0001. For both the math and picts conditionthe
correlationbetweenthejudgmentsof estimationaccuracyincreasedovertime,
F(1, 31) = 8.79, MSE= 0.07,p < 0.01, andF(l, 31) = 11.84,MSE= 0.07,p <
0.002. Themixedconditionshowedno evidenceof increasedcoordination
even
afterfour trialson the computer.
DISCUSSION
Thisexperiment
wasdesignedto determine
whetherthedifferentpatternsof convergenceofjudgmentsof estimation
accuracyfoundin Experiment
1 wereattenuatedoreliminatedafterlongerperiodsof taskexperience.A secondgoalwassimthebeneficialeffectsof CENTSonaspects
plyto replicatethefindingsconcerning
of estimationperformance.
Withrespectto the secondgoal, the effectsconcerningestimationaccuracy
werereplicated.Childrenin all threeconditionsimprovedon the penandpaper
tests. As may be expected,given the extendedexperienceof using CENTSin
was greaterthanin Experiment1. ForjudgExperiment2, this improvement
1.4
1.2
Mixed(ArcheryTarget)
o
Maths(Histogram)
1.0
Picts (ArcheryTarget)
o
0
0.8
0.6
Time1
Time4
FIGURE10 Judgment
of estimationaccuracyon categorical
representations
by condition
andtime(Experiment
2).
54
AINSWORTH,
BIBBY,WOOD
TABLE
10
Correlations
BetweenJudgment
of Estimation
onTtwoRepresentations
Accuracy
by
Condition
andTime(Experiment
2)
Mixed
Math
Picts
Time
M
SD
M
SD
M
SD
Time1
Time4
0.16
0.10
0.22
0.36
0.38
0.72
0.33
0.25
0.27
0.66
0.34
0.26
0.8
0.6
Mixed
Maths
0.4
Picts
0.2
Time1
Time4
FIGURE11 Correlations
betweenjudgments
of estimation
ontworepresentations
accuracy
andtime(Experiment
bycondition
2).
mentsof estimationaccuracy,childrenin all experimental
conditionsimproved.
Thisresultcontrastswiththe resultsof Experiment
where
themixedcondition
1
didnot improve,butis consistentwiththehypothesisthatmixedrepresentations
for a shortperiodof timewhenthe initialtaskdemands
only proveproblematic
are great.
Processmeasureswereexaminedto determine
if thishypothesiswassupported
the
use
children's
of
of estimation
by
representations.
Judgment
accuracywiththe
a
showed
similar
of
categoricalrepresentations
strikingly
pattern resultsto experimentone.At Time1, childrenin thepictsconditionweremoreaccuratethanchildrenin the otherconditions.By Time4, the mathconditionhad significantly
improvedtheirjudgmentsof estimationaccuracyandhad very similarperformanceto thepictscondition.Themixedconditionshowedno improvement
with
the categoricalrepresentation.
the
the
use
of
the
However, findingsconcerning
continuousrepresentations
didnotmatchtheresultsof Experiment
I so exactly.
andthemathcondiAgain,the pictsconditionshowedbetterinitialperformance
EXAMINING
MULTI-REPRESENTATIONAL
SYSTEMS
55
tionsignificantly
overtime.However,in contrast
to Experimprovedperformance
iment 1, the mixed representations
conditionalso improvedsignificantly.In
thenumerical
summary,
(presentinthemathandmixedconditions)
representation
wasusedsuccessfullyby childrenin bothconditions.However,thearcherytarget
in mixedandpictscases)was used successfully
(the categoricalrepresentation
whenit was combinedwithanotherpictorialrepresentation
butnot whencombinedwith a mathematical
This
the
representation. replicates findingin Experiment1 thattheway a representation
is usedcanbe influencedby thepresenceof
otherrepresentations
withwhichit is paired.
The majorconcernaddressedby this experiment
was whetherchildren'suse
of mixedrepresentations
wouldbecomeincreasingly
coordinated
withextended
wereonlyproblematic
due
practice.It was arguedthat,if mixedrepresentations
to initialtaskdemands,foursessionsshouldhaveprovidedsufficientexperience
for evidenceof coordination
to becomeapparent.
Boththemathandpictsconditionsbecamesignificantlymorecoordinated
overtime.However,evenafterfour
sessionson the computer,the mixedcondition'sbehaviordid not. This result
is not solely dueto the initial
suggeststhatfailureto coordinaterepresentations
demands.
learning
Unlikechildrenin themathandpictsconditions,childrenin themixedcondition didnot learnto translateacrosstherepresentations.
If theyhad,theiruse of
bothrepresentations
wouldhavebeenequallyeffective.We suggestthatthisled
themto abandontheirattemptto learnabouttheproperties
of oneof therepresentations(categorical)andto concentrate
on the otherrepresentation
(continuous).
This followsfromthe observation
thatthe mixedconditionshoweddissociation
betweenjudgmentsof estimationaccuracyusingthe categorical(archerytarget)
andcontinuous(numerical)
Onthecontinuous
but
representations.
representation,
noton thecategoricalone,theirperformance
improved.Onereasonwhylearners
is thatit containsboththedimayhavefocusedon thecontinuousrepresentation
rectionandthemagnitude
information
necessaryforlearning
judgmentof estimation accuracy,makingit possibleto meetthe taskdemands.If this inferenceis
valid,thenthechildrenin thisconditionmayhavemadea strategicdecisionabout
howto approach
thetask.
Childrenin the pictsconditionlearnedto masterjudgmentof estimationacshowedincreasingconvergenceon the reprecuracywith each representation,
sentationalcoordination
measure,andwere successfulat posttest.We propose,
thatas predicted,PathA best explainsthis behavior(Table2). Childrenin the
mathconditionalso showedsuccessfuloutcomesanddemonstrated
highlevel of
coordination.
In comparison
withthe pictscondition,theytook
representational
longerto masterthe representations.
However,by the endof theirfinalCENTS
session,theywerejudgingestimationaccuracyeffectivelywithbothmathematical representations.
We hypothesized
thatif learnersin the mathconditionwere
able to benefit by translatingtheir problemsolving between representationsthey
56
AINSWORTH,
BIBBY,WOOD
wouldbe successful.We concludethereforethatchildrenin this conditionalso
showedPathA.
Forthe mixedcondition,we arguedin Experiment
I thatthesechildrenwere
Path
F.
neither
mastered
the
northecategorical
continuous
following
They
representationandshowedno evidenceof translating
betweenthoserepresentations.
We expectedthatin Experiment
2, giventheadditional
time,thosechildrencould
mastersomeof thesetaskdemands.We foundthatthey successfullylearnedto
withoutmastering
judgeestimationaccuracywiththe continuousrepresentation
thecategorical
ortranslation.
thepaththatbestdescribes
Therefore,
representation
theirbehavioris PathC.
GENERALDISCUSSION
The resultsof theseexperiments
canbe explainedby consideringthe properties
of the representational
systemsandthenatureof therelationsbetweentherepresentedandrepresenting
worlds.Withrespectto the information
by
represented
each
of
was partiallyredundant.
The continuous
CENTS,
pair representations
whereasthe
representation
presentedbothmagnitudeanddirectioninformation,
As
information.
we notedin
categoricalrepresentation
onlypresentedmagnitude
the introduction,
muchresearchhas suggestedthat,if learnerstranslatebetween
thatdisplaydifferentaspectsof the represented
representations
world,they can
be expectedto gainmorerobustandflexibleknowledge.However,one of the
in CENTSpresentsall of the information
aboutthe represented
representations
worldneededto completethe task successfully.Thus,failingto translatebetween representations
shouldnot provedisastrous,providedthatthe learneris
ableto use andmasterthe representation
thatcontainsall of the relevantinformationaboutthe represented
world.
Thenatureoftherepresented
worldprovidesanexplanation
fortheapparent
dissociationinlearningoutcomesforthemixedcondition.InExperiment
in
1,learners
thisconditiondidnotcoordinate
therepresentations.
Theyalsodidnotlearnto use
eitheroftherepresentations
tojudgeestimation
InExperiaccuracyindependently.
ment2, althoughlearnersagainfailedtocoordinate
their
representations, improved
withthecontinuous
enabledthemtodiscoverandlearn
performance
representation
tojudgetheaccuracyof theirestimates.Theseresultssuggestthat,if translation
betweenrepresentations
is notrequired
forsuccessfultaskperformance,
thenattemptinfluenceon learningoutcomes.
ing to translatemaywell havea detrimental
In this finalsectionof the article,we turnto considertwo important
further
in
questionsconcerningthe processesof translatingbetweenrepresentations
CENTS-how we candetermine
if learnersweretranslating
theirproblemsolving
behaviorfromonerepresentation
to anotherandwhatrepresenting
worldfactors
influencedthis process.
MULTI-REPRESENTATIONAL
SYSTEMS
EXAMINING
57
In this article,we haveproposedthatchildrenin the mathematical
andpictorial conditionslearnedto use each representation
and, in addition,learnedto
translatebetweenthe two representations
(PathA). Yet, it couldbe the casethat
childrenin these conditionsbecamesuccessfulwith bothrepresentations
indeand
never
carried
the
of
their
results
from
one
pendently
problemsolving
representationto another(PathB). We believethatthis explanation
is less likelyfor
the followingreasons.First,if eachrepresentation
was learnedin isolationthen
at
of
the
task
estimation
proficiency
judging
accuracyshouldalwaysprecede
at
translation.
for
In
the
fact,
proficiency
pairsof mathematical
representations
the reverseseemsto be the case.Childrenin this conditionarethe mostcoordinatedat Time 1 butleastproficientatjudgmentsof estimationaccuracywiththe
is learnedin isolation,behavioron
Second,if a representation
representations.
thatrepresentation
shouldnot changedependingon otherrepresentations
with
whichit is paired.Yet, in bothexperiments
behavioron a representation
was influencedby its pairing.For example,the archerytargetrepresentation
was always used successfullyto makejudgmentsof estimationaccuracywhenpaired
withthe splatwallrepresentation,
butnotwhenpairedwiththe numericalrepresentation.Finally,we can appealto cognitiveeconomy.If childrenhad comto arriveat
pleteda significantamountof problemsolvingwitha representation
an answerto a tasktheyfindcomplex,it wouldseemlikelythattheywouldrememberthe outcomeof thisprocessandcarryit to a newrepresentation.
We artherefore
that
it
is
more
that
children
in
the
gue
likely
pictsandmathconditions
wereableto learnhowthe presentedrepresentations
relatedto eachotherrather
thanworkingindependently
on each new representation.
However,to identify
how learnersdevelopan understanding
of the relationbetweenrepresentations,
we stillneedanaccountthatdescribesthecognitiveprocessesandstrategiesthat
learnersuse.
We now turnto the questionof whatrepresentational
factorsinfluencedthe
was apparentlyeasy for the math
processof translationandwhy coordination
andpicts condition,but difficultfor the mixedcondition.The pictorialrepresentationsin CENTSwere each baseduponthe same metaphor.Eachrepresents proximityas physicaldistancefrom a goal. Both the formatand the
arealmostidentical.A secondsimilaritybeoperatorsfortheserepresentations
tween the pictorialrepresentations
relies on the methodof interactionwith
those representations.
A directmanipulationinterfacewas used to act upon
both representations.
In addition,pictures(andnaturallanguage)are ambient
symbolsystems(Kaput,1987).Childrenof this age shouldhavehadconsiderable opportunity
to interpretlanguageandpictures,butrelativelylittle experience with otherrepresentations.
Hence,it mightbe expectedthattranslation
betweentwo familiartypes of representations
wouldbe moreeasily achieved.
Obviously, these similarities need not necessarily apply to all combinations of
pictorial representations.
58
AINSWORTH,
BIBBY,WOOD
also occurred
Translation
betweenthe differentmathematical
representations
successfully.This mightseem moresurprisingas thereis less similarityin the
formatandoperatorsassociatedwiththeserepresentations.
Thehistogramrepresentationwas graphicalandexploitedperceptual
processes.By contrast,the numericaldisplaywastextualandtheinterfaceto therepresentations
wasdifferent.
Thehistogramwas acteduponby directmanipulation
andthe numericaldisplay
via the keyboard.Theserepresentations
wererelativelyunfamiliar
to childrenof
this age. However,it is proposedthatmappingbetweenthe representations
was
facilitatedas bothrepresentations
usednumbers.Dufour-Janvier,
Bednarz,and
that
children
that
believed
two
Belanger(1987) suggested
only
representations
wereequivalentif theybothusedthe samenumbers.It is possiblethatthe numberscouldbe usedby learnersto helpthemtranslate
betweenthetwo mathematical representations.
Thefailureto coordinate
themixedrepresentations
mayalsobe dueto a numberof factorsthathavebeenidentifiedinpreviousresearch.
Themixedrepresentations differin termsof modality-the archerytargetrepresentation
is graphical
andthenumericaldisplayis textual.Theinterfaceto therepresentations
involved
both directmanipulation
and the keyboard.This multirepresentational
system
combinedmathematical
andnonmathematical
representations.
Amongstothers,
betweenthesetypesof representation.
Kaput(1987)hasmadea strongdistinction
Researchonmultimodal
whenchildrenareacquiring
newmathematifunctioning
cal concepts(e.g.,Watson,Campbell,& Collis,1993)andresearchonwordalgebra problems(e.g., Tabachneck,Koedinger,& Nathan, 1994) suggest that
differenttypesof representation
mayalsoleadto completelydifferentstrategies.
research
on
Finally,
novice-expertdifferences(e.g., Chi, Feltovich,& Glaser,
1981)wouldpredictthatlearnerswouldfindit moredifficultto recognizethesimwhentheirsurfacefeaturesdiffer.Consequently,
ilaritybetweenrepresentations
we cansee thatforthemixedrepresentations
usedin CENTS,failureof overlap
occurredat a numberof levels.
A definitivestatement
of thefactorsthataffecttranslation
betweenrepresentationsrestsuponan integrativetaxonomyof representations
andtheiruse. Given
theresultsof theexperiments
thehere,it is likelythatsuchanintegrative
reported
of theproperties
of boththerepresented
andreporywillrequireanunderstanding
resentingworlds.Withintherepresenting
world,a widevarietyof factors-such
as themodalityof therepresentations
level of abstraction,
(textualvs. graphical),
type of representations
(staticvs. dynamic),typeof strategies,andinterfacesto
representations-couldinfluencethe learner'sabilityto coordinaterepresentations.Intherepresented
theresolution
world,theamountof availableinformation,
of information,
andinformation
canalso contribute
to the ease with
redundancy
whichmulti-representational
of the contribusystemsmaybe used.Irrespective
tionseachof thesefactorsmakesto thetranslation
process:Thegenerallessonthat
we can learnfromthis researchis thatthe more the formatand operatorsofrepre-
MULTI-REPRESENTATIONAL
EXAMINING
SYSTEMS
59
sentationsdifferthemoredifficultlearnerswill findtranslating
andintegrating
informationacrossrepresentations.
Thisresearchemphasizes
of explicitlyconsidering
theimportance
thedesignof
in termsof the represented
environments
world,the repremultirepresentational
betweenthesetwo levels.It demonstrates
that
sentingworld,andtheinteractions
withsimilarrepresenting
to translate
worlds,whereit is relativelystraightforward
betweenrepresentations,
therearesubstantial
gainsin learning.Whenthe reprebetween
sentingworldsaredissimilar,learnerscan finddifficultyin translating
However,if learnersfocustheirattentionon singleappropriate
representations.
thiscanresultin successfullearningoutcomes.Thisis onlypossirepresentation,
ble if the designof therepresented
worldensuresthatthisonerepresentation
enall
the
information.
capsulates
necessary
ACKNOWLEDGMENTS
Thisresearchwas supported
by theEconomicandSocialResearchCouncilatthe
EconomicandResearchCouncilCentreforResearchin Development
Instruction
andTraining.
Theauthorswouldliketo thankBenWilliams,Dr.CarolineWindrum,
andtwo
thisarticle.
anonymousrefereesfortheirhelpin preparing
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