IMPROVINGKNOWLEDGEOFFRACTIONS 1
Fazio, L. K., Kennedy, C., & Siegler, R. S. (2016). Improving children's knowledge of fraction magnitudes. PLOS ONE. doi: 10.1371/journal.pone.0165243
ImprovingChildren’sKnowledgeofFractionMagnitudes
LisaK.Fazio1#a*,CaseyA.Kennedy1,andRobertS.Siegler1,2
1DepartmentofPsychology,CarnegieMellonUniversity,Pittsburgh,PA,UnitedStatesofAmerica2TheSieglerCenterforInnovativeLearning,BeijingNormalUniversity,Beijing,China#aCurrentaddress:DepartmentofPsychologyandHumanDevelopment,VanderbiltUniversity,Nashville,TN,UnitedStatesofAmerica
Funding: This research was funded by the Institute of Education Sciences, U.S. Department of Education, ies.ed.gov, through Grants R324C10004, R305B100001, and R305A150262 to RSS and by support from Beijing Normal University to The Siegler Center for Innovative Learning, www.sieglercenter.net. The opinions expressed are those of the authors and do not represent views of the U.S. Department of Education or Beijing Normal University. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
*Correspondingauthor
E-mail:[email protected](LF)
IMPROVINGKNOWLEDGEOFFRACTIONS 2
Abstract
Weexaminedwhetherplayingacomputerizedfractiongame,basedonthe
integratedtheoryofnumericaldevelopmentandontheCommonCoreState
Standards’suggestionsforteachingfractions,wouldimprovechildren’sfraction
magnitudeunderstanding.Fourthandfifth-gradersweregivenbriefinstruction
aboutunitfractionsandplayedCatchtheMonsterwithFractions,agameinwhich
theyestimatedfractionlocationsonanumberlineandreceivedfeedbackonthe
accuracyoftheirestimates.Theinterventionlastedlessthan15minutes.Inour
initialstudy,childrenshowedlargegainsfrompretesttoposttestintheirfraction
numberlineestimates,magnitudecomparisons,andrecallaccuracy.Inamore
rigoroussecondstudy,theexperimentalgroupshowedsimilarlylarge
improvements,whereasacontrolgroupshowednoimprovementfrompracticing
fractionnumberlineestimateswithoutfeedback.Theresultsprovideevidencefor
theeffectivenessofinterventionsemphasizingfractionmagnitudesandindicate
howpsychologicaltheoriesandresearchcanbeusedtoevaluatespecific
recommendationsoftheCommonCoreStateStandards.
IMPROVINGKNOWLEDGEOFFRACTIONS 3
Introduction
Manychildrenandadultsstrugglewithfractions.OnoneNational
AssessmentofEducationalProgress(NAEP),anationwidetestgiventoaverylarge,
representativesampleofU.S.children,only49%ofeighthgraderscorrectlyordered
2∕7,1∕2and5∕9fromleasttogreatest.OnanotherNAEP,only55%of8thgraders
correctlysolvedasimplewordprobleminvolvingfractiondivision[1,2].Despite
fractioninstructionbeginninginelementaryschool,manypeoplefailtogainafirm
understandingoffractionsandharbormisconceptionsthroughhighschooland
college[3-6].
Thisisaseriousproblem,becauseunderstandingoffractionsisa
foundationalmathematicalskill.Earlyfractionknowledgestronglypredictslater
mathachievement[7-9],evenafterchildren’sIQ,readingcomprehension,working
memory,wholenumberarithmeticknowledge,race,ethnicity,andparental
educationandincomearestatisticallycontrolled[7].Moreover,asampleof1,000
U.S.Algebra1teachersidentifiedalackoffractionunderstandingasoneofthetwo
largestproblemshinderingtheirstudents’algebralearning[10].
Onemajorattempttoimprovechildren’smathematicsknowledgeingeneral,
andtheirfractionunderstandinginparticular,istheCommonCoreStateStandards
forMathematics(CCSS-M)[11].Developedbymathteachers,mathematicians,
principals,educationresearchers,andstatecontentexperts,thestandardshave
beenimplementedin43statesanddescribewhatchildrenshouldbeabletodoby
theendofeachgrade.Thestandardsaredesignedtobemorerigorousthanmost
existingstate-basedstandardsandtoemphasizedeeperunderstandingoffewer
IMPROVINGKNOWLEDGEOFFRACTIONS 4
topics.Forfractions,thestandardsemphasizethatchildrenshouldunderstand
fractionsasnumberswithmagnitudesthatcanbecomparedandordered,asshown
bytheclusterheadingsof“Developunderstandingoffractionsasnumbers”and
“Extendunderstandingoffractionequivalenceandordering”withinthesectionof
thedocumentonfractions.
AlimitationoftheCCSS-Mrecommendationsisthattheywerenotingeneral
basedonempiricalevidence,butratherontheprofessionaljudgmentsofpeople
withrelevantknowledge.Thiswasinevitable,giventheverylargenumberoftopics
thatarepartofthemathematicscurriculumandthelackofwell-controlled
experimentalstudiesregardingeffectiveteachingtechniquesformany,probably
most,ofthem.Nonetheless,itseemsessentialtotestkeyaspectsofthe
recommendationstodeterminetheirutility.
Whilethestandardsdonotdictatehowteachersshouldteachthecontentin
ordertoreachthestandards,theydosuggestthetypeoflearningactivitiesthat
shouldbeemphasized.Withinthedomainoffractions,onerecommendationisthat
thirdgradeinstructionshouldemphasizeaccurateplacementoffractionsona
numberline.ThisdiffersfromthetraditionalemphasisinU.S.mathematics
instructiononcircularandrectangulardiagrams,wherefractionsaregenerally
expressedasshadedpartsofawhole[12].Thestandards’focusonunderstanding
fractionsasnumberswithmagnitudedovetailswithrecentemphasiswithin
cognitivepsychologicaltheoriesonthecentralityofmagnitudeunderstandingto
mathematicalknowledge.
ImportanceofUnderstandingNumericalMagnitudes
IMPROVINGKNOWLEDGEOFFRACTIONS 5
Overthepastdecade,researchershaveidentifiedchildren’sunderstandingof
numericalmagnitudesasacentralcomponentoftheiroverallmathematical
knowledge.Children’sabilitytoapproximatenumericalmagnitudes,asmeasuredby
theiraccuracyatplacingnumbersonanumberline,estimatingtheanswersto
arithmeticproblemsand/orestimatingthenumberofobjectspresented,isstrongly
relatedtotheirmathachievementtestscoresbothconcurrently[13,14]and
longitudinally[15-17].Infact,children’saccuracyonanumberlinetaskin1stgrade
predictstheirgrowthinmathachievementthrough5thgrade,evenaftercontrolling
forintelligence,workingmemory,processingspeedandotherearlynumericalskills
[15].
Theserelationsarecausalaswellascorrelational.Childrenwhoparticipate
ininterventionsdesignedtoimprovetheirunderstandingofwholenumber
magnitudesshowimprovementsonuntrainedmagnitudetasks[18-20]andon
learningnovelarithmeticproblems[21,22].
Theintegratedtheoryofnumericaldevelopment[23,24]suggeststhatthis
relationbetweennumericalmagnitudeunderstandingandmathematics
achievementshouldholdnotjustforpositivewholenumbers,butforalltypesof
numbers.Amaintenetofthetheoryisthatacrucialpartofmathematical
developmentisunderstandingthatallnumbershavemagnitudesthatcanbe
orderedandcompared.Thus,understandingfractionmagnitudesisakeystepin
mathematicaldevelopment.
Aswithwholenumbers,understandingoffractionmagnitudesisrelatedto
overallmathachievementbothconcurrently[23,25]andlongitudinally[7-9].These
IMPROVINGKNOWLEDGEOFFRACTIONS 6
findingsandpreviousoneswithwholenumberssuggestthatitshouldbepossibleto
increasechildren’sfractionmagnitudeunderstandingandthattheimproved
understandingshouldtransfertotasksthatwerenottrained.Asdescribedbelow,
currentinterventionsdojustthat,butarelengthyandmultifacetedanddonotallow
forspecificationofthecomponentsthatproducethegains.Thepresentstudyseeks
toprovideevidencethatashort,simpleinterventioncanalsoimprovefraction
magnitudeunderstanding.
PriorInterventions
Previousstudieshaveshownincreasesinchildren’sfractionknowledge
followinginterventionsthatemphasizefractionmagnitudeunderstanding.Moss
andCase[26]createdarationalnumberscurriculumthatemphasizedconnections
amongpercentages,decimalsandfractions;comparingandorderingtheir
magnitudes;games,songs,andmonetarytransactions;andmanyotheractivities
involvingrationalnumbers.Incomparisontochildrenexposedtoatraditional
curriculum,childrenwhoreceivedthisexperimentalcurriculumwerebetterableto
solvenonstandardproblems(e.g.,“Whatis½of1∕8?”)andcompareandorder
rationalnumbers,thoughtheywereequivalentatsolvingstandardfraction
arithmeticproblems(e.g.,“Whatis3¼-2½?”)[26,27].SimilarlySaxe,Diakow,and
Gearhart[28]showedthatacurriculumunitthatemphasizedplacingintegersand
fractionsonnumberlineswasmoreeffectiveatimprovingstudents’understanding
offractionmagnitudesthanawell-regardedtraditionalcurriculum.Moreover,
Fuchsandcolleaguesfoundthatat-risklearnerswhoweretaughtusinga
curriculumthatfocusedoncomparingfractionsandplacingthemonanumberline
IMPROVINGKNOWLEDGEOFFRACTIONS 7
learnedmorethanchildrentaughtwithatraditionalcurriculumthatdescribed
fractionsaspartsofawhole[29,30].
Theseareimpressivedemonstrationsthatwell-conceivedinstructionover
manyweeks,whichincludesanemphasisonmagnitudeunderstanding,canproduce
largegainsinfractionknowledge.However,becausethestudieswereso
multifaceted,thesourceoftheireffectivenessremainsuncertain.Tobetterspecify
theprocessesthroughwhichinterventionsdesignedtoimprovefractionknowledge
exercisetheireffects,thepresentstudyexamineslearningfromabriefintervention
tightlyfocusedonfractionmagnitudes.
CurrentStudy
Priorresearchonchildren’sunderstandingofdecimalmagnitudesshowed
thatplayingabriefcomputergamewaseffectiveinimprovingunderstandingof
decimalmagnitudeswithbothAmerican[31]andGerman[32]children.Inthe
CatchtheMonstergame,childrenwerepresentedwitha0-1numberlineanda
decimalthatindicatedamonster’sposition.Childrenwereencouragedtousethe
decimaltoestimatethemonster’spositiononthelineandreceivedfeedback
regardingeachestimate’saccuracy.Iftheestimatewassufficientlyaccurate,the
monsterdiedadramaticdeath;ifnot,themonsterlaughedatandmockedthechild.
Weadaptedthispreviouslysuccessfulgametocreate,CatchtheMonsterwith
Fractions,whichhasaddedfeaturestodealwiththemorecomplexconceptof
fractionmagnitudes.CatchtheMonsterwithFractionshasthreemainfeatures.
First,inlinewiththeemphasisoftheCCSS-M(standard3.NF.A.1,Table1)andwith
children’slimitedunderstandingoffractionnotation,wepresentedaconceptual
IMPROVINGKNOWLEDGEOFFRACTIONS 8
frameworkforthinkingaboutthemeaningoffractionsthatemphasizedunit
fractions(fractionswith1asthenumerator,suchas1∕3and1∕8).Thisconceptual
frameworkwaspresentedattheoutsetoftheintervention.Itintroducedunit
fractions,showedchildrenhowtodividethenumberlineintothenumberof
segmentsindicatedbythedenominator,andpointedoutthatunitfractionswith
largerdenominatorsareclosertozeroonanumberlinethanunitfractionswith
smallerdenominators.Second,aftereachestimateofthemonster’sposition,the
childwasgivendetailedfeedback,asshowninFig1.Withthechild’sestimate
remainingvisible,thenumberlinewasdividedintothenumberofequal-size
segmentsindicatedbythedenominator,eachsegmentwaslabeledwiththefraction
thatexpressedthenumberofsegmentsbetweenitandtheorigin,andthe
magnitudeofthefractionwasemphasizedwithaboldedlinethatranfromzeroto
thepresentedfraction.Thisdetailedfeedbackallowedchildrentoseewhytheir
answerwascorrectorincorrectandprovidedadditionalopportunitiestolearn
aboutfractionmagnitudes.ItalsomatchedtherecommendationfromtheCCSS-M
(3.NF.A.2.B)thatchildrenshouldunderstandthatafractiona∕bonanumberline
consistsofasegmentsof1∕blength.Finally,practicewaspresentedinanengaging
gamesetting.Ratherthansimplylearningaboutfractions,childrenweretryingto
captureescapedmonsters.Thiswasdesignedtohelpchildrenremainedmotivated
evenwhentheywerestrugglingtoaccuratelyplacefractionsonthenumberline.
Fig1.Asamplecorrecttrialfromtheleastdemandinglevelof“Catchthe
MonsterwithFractions”(top)andasampleincorrecttrialfromthemost
demandinglevel(bottom).
IMPROVINGKNOWLEDGEOFFRACTIONS 9
Table1.RelevantCommonCoreStateStandardsforMathematics.Standard Text3.NF.A.1 “Understandafraction1∕basthequantityformedby1partwhenawhole
ispartitionedintobequalparts;understandafractiona∕basthequantityformedbyapartsofsize1∕b.”
3.NF.A.2 “Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.”
3.NF.A.2.A “Representafraction1∕bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1∕bandthattheendpointofthepartbasedat0locatedthenumber1∕bonthenumberline.”
3.NF.A.2.B “Representafractiona∕bonanumberlinediagrambymarkingoffalengths1∕bfrom0.Recognizethattheresultingintervalhassizea∕bandthatitsendpointlocatedthenumbera∕bonthenumberline.”
4.NF.A.2 “Comparetwofractionswithdifferentnumeratorsanddenominators,e.g.,bycreatingcommondenominatorsornumerators,orbycomparingtoabenchmarkfractionsuchas½.Recognizethatcomparisonsarevalidonlywhenthetwofractionsrefertothesamewhole.Recordtheresultsofcomparisonswithsymbols>,=,or<,andjustifytheconclusions,e.g.,byusingavisualfractionmodel.”
Onthepretestandposttest,childrencompletedthreetasksthatvariedin
theirdistancefromthepracticedtask.Inthemostsimilartask,childrenestimated
locationsoffractionsonnumberlines;thistaskdifferedfromthegameinthe
absenceoffeedback,thelackofagamesetting,andtheparticularfractions
presented.Next,afractionmagnitudecomparisontaskrequiredchildrentodecide
ifapresentedfractionwasgreaterthanorlessthan3/5.Thiswaslesssimilartothe
activitiesinthegame,inthatitrequiredrelativeratherthanabsolutejudgments.
MagnitudecomparisonassessedanotherCCSS-Mstandard,onethatstatesthat
childrenshouldbeabletocomparefractionswithunequalnumeratorsand
denominators(4.NF.A.2).Finally,inthefartransfertask,weexaminedchildren’s
memoryforfractions.Priorresearchhasshownthatwhenwholenumbersare
IMPROVINGKNOWLEDGEOFFRACTIONS 10
includedinvignettes,thenumberslaterrecalledbychildrenwithgoodmagnitude
knowledgeareclosertothenumbersthatwerepresentedthanarethenumbers
recalledbychildrenwithweakermagnitudeknowledge[33].Usingthesame
reasoning,weexpectedchildrenwithamorepreciseunderstandingoffractionsto
recallfractionmagnitudesmoreaccurately,evenwhentheydidnotrememberthe
exactfractionsthatwerepresented.Study1wasapreliminarystudywithasimple
pretest/posttestdesignconceivedtotestiftheinterventionwaseffective.Study2
builtontheresultsofStudy1totesttheinterventionagainstcontrolactivities.
Study1
Method
Participants
Theparticipantswere26fourthandfifthgradersattwourbancharter
schoolsandonesuburbanpublicschoolnearPittsburgh,PA(Mage=10.60years,
SD=0.49,58%4thgrade,58%female,69%Black,19%White,8%biracial,4%
Asian).Bothcharterschoolsservedaprimarilylow-incomepopulation,but
studentsatallthreeschoolsscoredatorabovethestateaverageonastandardized
testofmathachievement.
Materials
Fulldescriptionsofthematerials,instructions,andprocedureareprovided
intheonlinesupplement,S1Materials.Hereweprovidesummariesofeach.
Catchthemonsterwithfractions.Theexperimentalinterventionconsisted
ofbriefconceptualinstruction,practiceinplacingfractionsonanumberline,and
IMPROVINGKNOWLEDGEOFFRACTIONS 11
feedbackimmediatelyaftereachestimate.First,thechildrenreceived
approximately3minutesofconceptualinstructionaboutunitfractions.The
instructiondefinedunitfractionsasfractionsinwhichthenumeratoris1,andit
thenprovidedseveralillustrations.Next,childrenwereshownhowtolocateunit
fractionsona0-1numberline,andshownthatunitfractionswithlarger
denominatorscorrespondtosmallersegmentsofthenumberline.Finally,children
weretoldthatthenumeratorinthefractionindicatedthenumberofsegmentsof
theunitfraction(e.g.,3∕4is3ofthe1∕4segments).Duringtheinstruction,visualswere
presentedonthecomputerscreenwhiletheexperimenterreadthescriptand
pointedattherelevantpartofthedisplay.
Next,childrenplayedCatchtheMonsterwithFractions.Theyweretoldthat
themonstershadescapedandthattheirknowledgeoffractionswasneededto
recapturethem.Oneachtrial,childrensawanumberlinewith“0”ontheleftend,
“1”ontherightend,andafractionabovethelinethatindicatedthemonster’shiding
place.Childrenwereinstructedtocatchthemonsterbyclickingthecomputer’s
mouseatthenumberlinelocationwherethefractionbelongs.
Afterchildrenclickedonthenumberline,theysimultaneouslysawthe
locationthattheyclicked,thecorrectlocationofthefraction,andwhethertheir
estimatewascloseenoughtocatchthemonster(Fig1).Inaddition,thenumberline
wassegmentedintothenumberoflabeledpartsindicatedbythedenominator,and
thelinewasboldedfrom0tothelocationonthenumberlineofthepresented
fraction.Afterviewingthefeedbackfortwoseconds,childrencouldmoveontothe
nexttrial.Aschildrenimprovedatlocatingthefractions,themonstergotsmaller
IMPROVINGKNOWLEDGEOFFRACTIONS 12
andrequiredmorepreciseestimatesofthefractions’locations.Atthestartofthe
game,estimateshadtobewithin20%ofthemonster’smidpointtocatchit.Thus,as
showninthetoppanelofFig1,whenthefractionpresentedwas2∕5,themonster
wouldbecaughtbyestimatesrangingfrom1∕5to3∕5.Ifthechildcorrectlyanswered
fiveconsecutivetrialsatthattolerance,asmallermonsterwaspresented,andthe
estimateshadtobewithin15%ofitsmidpoint.Whenchildrenansweredfive
consecutivetrialscorrectly,theymovedtothefinallevel,whereestimateshadtobe
within10%ofthemidpoint(bottompanel,Fig1).Childrenplayedthegameforten
minutesoruntiltheyansweredfiveconsecutiveproblemscorrectlyatthemost
demandinglevel.
To-be-estimatedfractionsweredrawnfromthesetof45fractionswith
integernumeratorsanddenominatorslessthanorequaltotenandmagnitudes
between0and1,exclusive.Oneachtrial,afractionwasrandomlydrawnfromthe
setwithoutreplacement.Becausefractionsthatmetthesespecificationstendedto
havelargedenominators(1fractionwithadenominatorof2but8witha
denominatorof9),onceadenominatorwaspresentedthreetimes,nomore
fractionswiththatdenominatorwereshownuntiltheentiresetwasexhausted.
Then,thesetwasreshuffled,andthedenominatorlimitswerereset.Thishelped
ensurethatchildrenreceivedpracticewithsmalleraswellaslargerdenominators.
Pre-andposttestmeasuresoffractionknowledge.
Numberlineestimation.Childrenwerepresented120–1numberlines.
Aboveeachlinewasato-be-estimatedfraction.Childrenmadetheirestimatesby
movingthecursortothedesiredpositionandclickingthemousebutton.Twosets
IMPROVINGKNOWLEDGEOFFRACTIONS 13
offractionsservedasthepretestandposttest.Eachchildsawonesetonthepretest
andtheotherontheposttest,counterbalancedacrossparticipants.Eachset
includedthreefractionsfromeachquarterofthenumberline;nodenominator
occurredmorethanthreetimesinaset.
Magnitudecomparison.Oneachtrial,childrenwereshowna0-1numberline
withthelocationof3∕5markedandaskedwhethereachof15fractionswaslessthan
orgreaterthan3∕5.Onesetoffractionswaspresentedatpretestandanotherat
posttest,counterbalancedacrossparticipants.Ineachset,eightfractionswere
smallerthan3∕5andsevenwerelarger.Denominatorsrangedfromthreetoten;no
denominatorappearedmorethanthreetimesinaset.
Recall.Ontherecalltask,childrenwerereadshortvignettesandaskedto
rememberinformationfromthestory.Childrenwereinstructedtopayattentionto
thespecificsofthestory,becausetheywouldbequestionedaboutwhattheyheard.
Eachvignettecontainedtwofractionsandsomefillerinformation.Samplevignettes
andquestionsareshowninTable2.
Table2.SampleItemsfromtheFractionRecallTask.
Story Question1 Question2 Question3
Jamieorderedapepperonipizzaforherandherfriends.Jamieate4∕7ofthepizzaandherfriendSamate2∕9ofthepizza.
WhatkindofpizzadidJamieorder?
WhatfractionofthepizzadidJamieeat?
WhatfractionofthepizzadidSameat?
Sarahwantedtofindsomethingtoreadatherschool’slibrary.Sarah’sfavoritebooksareabouthorses.Ononeshelf,shesaw
Whatfractionofthebookswerefiction?
Whatfractionofthebookswerenonfiction?
WhatareSarah’sfavoritebooksabout?
IMPROVINGKNOWLEDGEOFFRACTIONS 14
that3∕7ofthebookswerefictionandonanothershelf2∕5ofthebookswerenonfiction. Mr.Smithaskedthechildreninhisclasshowtheylikedtotravelbest.5∕8ofthechildreninhisclasslikedairplanesbestand3∕10ofthechildreninhisclasslikedcarsbest.Mr.Smithlikestotravelbytrain.
HowdoesMr.Smithliketotravel?
Whatfractionofthechildrenlikedairplanesbest?
Whatfractionofthechildrenlikedcarsbest?
Aftertheexperimenterreadthestoryaloud,thechildwasaskedtocount
backwardsbythreesfromatwo-digitnumberfor15seconds.Then,threequestions
aboutthestorywerepresented.Childrenwereaskedtorecallbothfractionsand
oneotherdetailaboutthestory.Sixvignetteswerepresentedasthepretestand
anothersixastheposttest,counterbalancedacrossparticipants.
Procedure
Allchildrenparticipatedindividuallyinaquietroomattheirschoolduring
theschoolday.Pretest,intervention,andposttesttogethertookapproximately30
minutesandoccurredduringthesamesession.Childrenfirstcompletedthefraction
recall,numberlineestimation,andmagnitudecomparisonpretests,inthatorder.
TheythenreceivedtheconceptualinstructionandplayedCatchtheMonsterwith
Fractions.Finally,theycompletedtheposttesttasksinthesameorderasonthe
pretest.ApprovalforthestudywasprovidedbytheCarnegieMellonUniversity’s
InstitutionalReviewBoard,andwrittenconsentwasobtainedfromboththeparent
andchild.
Results
IMPROVINGKNOWLEDGEOFFRACTIONS 15
PerformanceonCatchtheMonsterwithFractions
Childrencompletedamedianof30trialswhileplayingthecomputergame,
andtheirestimatesweremarkedasincorrectonanaverageof25%(SD=16)of
trials.Mostparticipants(16of26,61%)successfullycompletedallthreelevelsafter
anaverageof5.5minutes(SD=1.5)and27trials(SD=9).Forthepurposeof
comparison,theminimumnumberoftrialsrequiredtocompletethegamewas15.
Amongtheothertenchildren,fivereached,butdidnotcomplete,levelthree,three
reachedleveltwo,andtwodidnotpasslevelone.
Fractionnumberlineestimation
Estimationaccuracywasmeasuredusingpercentabsoluteerror(PAE),
definedas:PAE=(|Participant’sAnswer–CorrectAnswer|)/NumericalRangeX
100.Forexample,ifaparticipantwasaskedtolocate1∕5ona0–1numberline,and
markedthelocationcorrespondingto1∕4;PAEwouldbe5%((|0.25-0.20|)/1X100).
PAEvariesinverselywithaccuracy;thehigherthePAE,thelessaccuratethe
estimate.
Estimatesimprovedgreatlyfrompretesttoposttest,t(25)=-4.73,p<.001,d
=-1.10.PretestPAEaveraged19%(SD=11),posttestPAEaveraged10%(SD=6).
Magnitudecomparison
Accuracyonthemagnitudecomparisontaskalsoimproved,from62%
correct(SD=19)onthepretestto70%(SD=19)ontheposttest,t(25)=2.37,p=
.026,d=0.48.
Recall
IMPROVINGKNOWLEDGEOFFRACTIONS 16
Iftheinterventionimprovedfractionmagnitudeunderstandingandthat
knowledgeinfluencedencodingandretrieval,recalloffractionsinthevignettes
shouldbeclosertotheoriginalvaluesontheposttestthanonthepretest.Weused
PAEasourmeasureofrecallaccuracy.Whileallto-be-rememberedfractionswere
lessthanone,childrenoccasionallyrecalledfractionsaboveone,whichcouldleadto
aPAE>100%.PAEsonthose2%oftrialsweretrimmedto100%.
Asexpected,pretestaccuracyonthememorytaskwasrelatedtopretest
numberlineestimationPAE,r(24)=.54,p=.004,andmagnitudecomparison
accuracy,r(24)=-.47,p=.014,demonstratingthatfractionknowledgewasrelated
torecallaccuracy(thenegativecorrelationwasduetohighermagnitude
comparisonaccuracyandlowerPAEreflectingbetterknowledge).Thisreplicates
thefindingsofThompsonandSiegler[33]andextendstheirfindingsfromwhole
numberstofractions.
Especiallystriking,recallaccuracyofthefractionsinthevignettesimproved
afterplayingCatchtheMonsterwithFractions.PAEdecreasedfrom20%(SD=12)
onthepretestto16%(SD=9)ontheposttest,t(25)=-2.47,p=.021,d=-0.52.
Recalloftheexactfractionsinthevignettesdidnotincreasefrompretestto
posttest,3.12vs.3.15,t<1,nordidcorrectrecalloffillerinformation,3.77vs.3.65,
t<1.Thesefindingsprovideddiscriminantvalidityfortheinterpretationthat
improvedrecallaccuracyreflectedimprovedknowledgeoffractionmagnitudes,
ratherthanimprovedmemoryforexactnumericalinformationorimprovedfacility
withthetypesofquestionsasked.
Discussion
IMPROVINGKNOWLEDGEOFFRACTIONS 17
Performanceonallthreeoutcomemeasuresimprovedfrompretestto
posttest.Thus,theinterventionshowedpromiseforimprovingstudents’fraction
understanding.However,Study1utilizedasimplepretest/posttestdesign,sothe
increasescouldhavebeenduesimplytoincreasedfamiliaritywiththeoutcome
measures,experimentalsituation,orsomeotherthirdvariable.Therefore,to
replicatetheinitialfindingsandtesttheeffectivenessoftheinterventionmore
rigorously,inStudy2werandomlyassignedchildrentoexperimentalandcontrol
groups.Participantsinthecontrolgrouppracticedplacingfractionsonanumber
lineforthesamenumberoftrialsasparticipantsinStudy1,butdidnotreceive
feedbackorinstruction,whereastheexperimentalgroupreceivedthesame
instructionasinStudy1.Thiscontrolgroupwasdesignedspecificallytoexamineif
thegainsseeninthefirststudywereduetoadditionalexposuretofractionsand
fractionnumberlineestimationorifthegainswereduetosomecombinationofthe
instruction,gameandfeedback.
Study2
Method
Participants
Participantswere51fifthgraders(Mage=10.73,SD=0.39,61%female,
69%White,20%Black,8%biracial,2%Asian,2%Hispanic),whocamefromthree
schoolsnearPittsburgh,PA:anurbancharterschool,aprivateCatholicschool,anda
suburbanpublicschool.Studentsatallthreeschoolsscoredatorabovethestate
averageonastandardizedtestofmathachievement.Ofthe51children,25
IMPROVINGKNOWLEDGEOFFRACTIONS 18
practicedplacingfractionsonnumberlinesand26playedCatchtheMonsterwith
Fractions.Duetoacomputermalfunction,oneparticipantwasexcludedfromthe
controlconditiononthefractionmagnitudecomparisontask.
Controlactivities
ChildreninthecontrolconditionofStudy2placedfractionsona0–1
numberlinefor30trialswithoutfeedback.Thecomputerprogramthatpresented
problemsandrecordedresponseswasthesameasinStudy1andinthe
experimentalgroupofStudy2.The30trialsmatchedthemediannumberoftrials
duringCatchtheMonsterwithFractionsinStudy1;itwassomewhathigherthanthe
medianof24trialsfortheexperimentalgroupinStudy2.Thefractions’
denominatorsrangedfrom2-10,asintheexperimentalcondition.
Procedure
Participantswererandomlyassignedtoeitherthecontrolorexperimental
condition.TheexperimentalgroupreceivedthesameinterventionasinStudy1
featuringboththeconceptualinstructionandplayingCatchtheMonsterwith
Fractions.Participantswereagaintestedindividuallyinaquietspaceintheirschool.
ThethreepretestandposttestmeasureswereunchangedfromStudy1.
Results
PerformanceonCatchtheMonsterwithFractions
Ofthe26childrenwhoreceivedtheintervention,18(69%)metthecriterion
atthehighestlevelafteranaverageof5.1minutes(SD=1.5)and24trials(SD=10).
Oftheeightchildrenwhotimedoutwithoutmeetingthatcriterion,fourreachedbut
IMPROVINGKNOWLEDGEOFFRACTIONS 19
didnotcompletelevelthree,threereachedbutdidnotcompleteleveltwo,andone
didnotcompletelevelone.Acrossallchildrenwhoreceivedtheintervention,the
mediannumberoftrialsreceivedwas24;amongtheestimates,21%(SD=17)were
markedasincorrect(toofarfromthecorrectlocationofthefraction).
Fractionnumberlineestimation
A2(Condition:experimentalorcontrol)X2(Timeoftest:pretestor
posttest)ANOVArevealedaninteractionbetweenconditionandtimeoftest,F(1,
49)=10.42,p=.002,η2p=.18.AsshownintheleftmostpanelofFig2,childrenwho
receivedtheexperimentalinterventionweremuchmoreaccurateatplacing
fractionsonthenumberlineontheposttest(PAE=10%,SD=9)ascomparedtothe
pretest(PAE=18%,SD=13),t(25)=-4.14,p<.001,d=-0.86.Incontrast,
estimationaccuracyofchildreninthecontrolgroupdidnotchangefrompretest
(PAE=17%SD=13)toposttest(PAE=16%,SD=13),t(24)=-1.54,p=.136,d=-
0.31.
Fig2.Meanpretestandposttestperformanceonfractionnumberline
estimation,magnitudecomparison,andrecallforchildreninthecontroland
experimentalconditions(Study2).
Fractionmagnitudecomparison
AparallelANOVAindicatedthatconditionandtimeoftestalsointeractedon
thefractionmagnitudecomparisontask,F(1,48)=4.32,p=.043,η2p=.08.Asshown
inthemiddlepanelofFig2,playingthegametendedtoimprovechildren’s
IMPROVINGKNOWLEDGEOFFRACTIONS 20
magnitudecomparisonaccuracyfrompretest(75%correct,SD=18)toposttest
(80%correct,SD=15),t(25)=1.87,p=.074,d=0.35.Childreninthecontrolgroup
againshowednopretest-posttestimprovement(74%correct,SD=20;71%correct,
SD=22),t(24)=1.13,p=.272,d=0.23.
Fractionrecall
AsinStudy1,PAEs>100%weretrimmedto100%(4%oftrials).Pretest
fractionrecallPAEwasagaincorrelatedwithbothnumberlineestimationPAE,
r(49)=.29,p=.038,andmagnitudecomparisonaccuracy,r(49)=-.32,p=.023.
Onthismeasure,timeoftestandconditiondidnotinteract(F<1;rightmost
panelofFig2).AsinStudy1,recallofchildrenwhoreceivedtheintervention
improvedfrompretest(PAE=20%,SD=14)toposttest(PAE=14%,SD=8),t(25)
=-2.28,p=.031,d=-0.49.Unexpectedly,however,childrenwhopracticedfraction
numberlineestimationwithoutfeedbackalsoimproved,thoughtheimprovement
wassmallerandmarginallysignificant(pretestPAE=21%,SD=19,posttestPAE=
18%,SD=18),t(24)=-1.80,p=.085,d=-0.36.
AsinStudy1,therewasnochangefrompretesttoposttestinrecallofthe
exactfractions,3.55vs.3.55,t<1,orofthefillerinformation,3.94vs.4.29,t(50)=
1.48,p=.146,d=0.21,suggestingthattheincreasedaccuracyofrecallwasspecific
tothemagnitudeinformation.
ExistingKnowledgeandAcquisitionofNewKnowledge
Wealsoexaminedtheamountlearnedfromthegameamongchildrenwho
hadaboveorbelowaverageinitialknowledgeoffractions.Forthisanalysisweused
acombinedsampleofallthechildrenwhoplayedCatchtheMonsterwithFractions
IMPROVINGKNOWLEDGEOFFRACTIONS 21
inStudies1and2.Wefirstcheckedwhetherthe4thand5thgradersshowedequal
gainsfromplayingthegame.Childrenatbothgradelevelsshowedsimilargains
frompretesttoposttestonnumberlineestimation(4th6.33,5th9.38,t(50)=1.02,p
=.313),magnitudecomparison(4th7.56,5th6.13,t<1)andfractionmemory(4th
4.15,5th5.25,t<1).
Wethenstandardizedchildren’sperformanceonthethreepretestmeasures
andaddedthethreez-scorestogethertocreateacompositemeasureofinitial
fractionknowledge(numberlineestimationPAEandfractionmemoryPAEwere
reversescoredsothathigherscoresindicatedmoreknowledgeforallthreetasks).
Thiscompositemeasurewasnotcorrelatedwithgradelevel,r(50)=.09,p=.515.
Studentswithlessinitialknowledgeshowedgreaterimprovementsthan
peerswithmoreinitialknowledge.Oneachtask,therewasasignificantnegative
correlationbetweeninitialfractionknowledgeandgainsfrompretesttoposttest;
numberlineestimationr(50)=-.48,p<.001;magnitudecomparisonr(50)=-.30,p
=.033;fractionmemoryr(50)=-.39,p=.004.Atminimum,thisshowsthatthe
gamewaseffectivewithstudentswhostartedwithlessknowledge.Atmaximum,it
suggeststhatthegameisespeciallyeffectivewithsuchstudents.Relevant
descriptivestatisticsareshowninTable3.
Table3.DescriptiveStatisticsofPretestFractionKnowledgeandGainsfrom
PretesttoPosttestforChildrenwhoplayedtheCatchtheMonsterGame.
M SD Range
Pretestfractionknowledge(z-score) 0 2.31 -5.57–3.61
IMPROVINGKNOWLEDGEOFFRACTIONS 22
NLEGain(PAE) 8.50 9.78 -9.80–39.35
MagCompGain(%correct) 6.54 15.54 -27.67–53.33
FractionMemoryGain(PAE) 4.93 10.78 -11.80–37.36
Although18childrentimedoutofthegamewithoutcompletingLevel3(the
highestlevel),thesechildren’sfractionknowledgeimprovedfrompretestto
posttestonbothnumberlineestimationandfractionrecall:numberlineestimation
pretestPAE28%(SD=12),posttest17%(SD=9),t(17)=-3.78,p=.001,d=-0.91;
fractionrecallpretestPAE26%(SD=18),posttest18%(SD=10),t(17)=-2.74,p=
.014,d=-0.83.Magnitudecomparisonaccuracyofthesechildrendidnotchange,
pretest59%correct(SD=16),posttest63%correct(SD=15),t(17)=1.00,p=.331,
d=0.26.
GeneralDiscussion
Acrosstwostudies,childrenwhoreceivedconceptualinstructionabout
fractionsthatwasinaccordwiththeCommonCoreStateStandardsandthenplayed
CatchtheMonsterwithFractionsshowedlargeimprovementsinfractionmagnitude
understanding.Theymoreaccuratelyplacedfractionsonanumberline,compared
fractionmagnitudes,andrememberedthemagnitudesoffractionsinstoriesonthe
posttestthanthepretest.Thesegainswereconsistentlyfoundacrossbothstudies.
Incontrast,childrenwhopracticedplacingfractionsonnumberlineswithout
feedbackdidnotshowsignificantimprovementsonanyofthethreetasks.
AmountandBreadthofLearning
IMPROVINGKNOWLEDGEOFFRACTIONS 23
Theknowledgegainsofchildrenwhoreceivedtheinterventionwerequite
large.The4thand5thgradersinthesestudiesfinishedtheinterventionwithaverage
PAEsonthefractionestimationtaskthatweresimilartothoseof8thgradersfrom
Siegler,ThompsonandSchneider[23]andmoreaccuratethanthoseof8thgraders
inSieglerandPyke[25].Yet,childreninthepresentstudybeganwithPAEsof18%
and19%,slightlyworsethanthe15%observedfor6thgradersinSiegler,Thompson
andSchneider[23]andsimilartothe18%inSieglerandPyke[25].Thus,afterthe
present15-minuteintervention,childrenwereasaccurateaspeerswithseveral
yearsofadditionalschooling.
Moreover,theinterventionwasatleastaseffectiveforchildrenwhostarted
withlessknowledgeasforoneswhostartedwithmore.Ratherthanfindingthe
typical“Mattheweffect”wherehigh-knowledgechildrenlearnmorethanlow-
knowledgechildren[34,35],childrenwhostartedwithlowerknowledgeshowed
greaterimprovement.EvenchildrenwhowereunabletofinishCatchtheMonster
withFractionsintheallottedtimeshowedimprovementinfractionestimation
accuracyandmemoryforfractions.
Theseresultssuggestthatclassroomactivitiesfocusedoninculcating
understandingofunitfractionsandlocationsoffractionsonnumberlines(activities
emphasizedintheCCSS-M)arelikelytoimprovestudents’understandingoffraction
magnitudes.However,suchactivitiesmustalsoincludewell-designedfeedback.
Practiceinplacingfractionsonnumberlinesisinsufficienttoimprovestudent
knowledge.Futureresearchshouldexaminetheeffectivenessoftheintervention
whendoneinwhole-classsituationsandhowtoexpandstudents’fraction
IMPROVINGKNOWLEDGEOFFRACTIONS 24
understandingtoincludeallfractions,ratherthansimplytheproperfractions
(thosefrom0-1)usedinthepresentstudies.
Inadditiontodemonstratingthevalueofnumberlinebasedfraction
activities,theseresultsprovidenewtheoreticalevidencefortheconnectionamong
differenttypesoffractionmagnitudeknowledge.Theintervention,whichdealt
specificallywithplacingfractionsonanumberline,producedgainsnotonlyon
numberlineestimationbutalsoonfractionmagnitudecomparisonandmemoryfor
fractions.Thus,thestudentswerenotsolelygainingproceduralknowledgeofhow
toplacefractionsonnumberlines,butratheremergedwithabetterunderstanding
offractionsmoregenerally.Theresultsfromthememorytaskareparticularly
strikinginhowdifferentthetaskwasfromthetasktrainedduringtheintervention.
Thisfartransferprovidesfurtherevidencefortheassumptionoftheintegrated
theoryofnumericaldevelopment[23,24]thatunderstandingnumerical
magnitudesiscrucialforawiderangeofmathematicalknowledge.
LimitationsandFutureDirections
Severallimitationsofthepresentinvestigationshouldbenoted.Inboth
studies,theposttestwaspresentedimmediatelyaftertheintervention.Thus,one
clearavenueforfutureresearchistoinvestigatethepersistenceofthebenefits.
Doestheinterventionchangethewaythatchildrenunderstandfractionsinthe
long-term,orareboostersessionsnecessarytoconsolidatethechange?Inaddition,
wepurposefullydevelopedtheCatchtheMonsterwithFractionsgametoinclude
threeelementsthatwehypothesizedtobebeneficialforlearning:unitfractions
instruction,elaborativefeedback,andanengaginggamecontext.Nowthatithas
IMPROVINGKNOWLEDGEOFFRACTIONS 25
beenestablishedthattheinterventionhasrelativelybroad,positiveeffects,future
researchisneededtodeterminewhichofthesefeaturesarecrucialforlearning.
Finally,oursamplewasethnicallydiverse,butthestudentsattendedschoolswith
averageoraboveaveragemathachievement.Establishingtheeffectivenessofthe
interventionwithstudentsatschoolswithlowermathematicsachievementisthus
anotherimportantgoal.Thegreaterlearningofchildrenwithlesspretest
knowledgesuggeststhatthepresentapproachwillbeeffectiveatsuchschools,but
thatremainstobedemonstrated.Finally,theidealtimingoftheintervention
remainstobeestablished.Wehypothesizethattheinterventionisparticularly
effectivewhenstudentshavesomebutnotgreatknowledgeoffractions(aswas
trueformoststudentsinthepresentstudies),butagain,theidealtimingofthe
interventioniscurrentlyunknown.
TestingandImprovingtheCommonCoreStateStandards
Moregenerally,thepresentstudiesillustrateamethodthatseemsapplicable
totestingthecommoncorestatestandardsintheircurrentformanddeveloping
research-basedmodificationswherenecessary.Thepresentresultssupportthe
standards’suggestionthatunderstandingunitfractionsandlearningtoplace
fractionsonnumberlinesarekeycapabilitiesthathelpdeepenchildren’s
understandingoffractions.However,thepresentresultsdonotsuggestthatallof
thestandardsareequallyeffective;whetherthatisthecaseremainstobeseen.
Experimentallytestingthestandards’recommendationscanhelpmakethem
increasinglyevidence-based.Itisalsoimportanttonotethatourfindingsdonot
meanthatothertypesoffractioninstructionarenoteffective.Wefoundpositive
IMPROVINGKNOWLEDGEOFFRACTIONS 26
resultsafteremphasizingunitfractionsandplacingfractionsonnumberlines,but
thatdoesnotmeanthatotherinstructionaltechniqueswouldnotshowsimilar
gains.However,givenstudents’currentstruggleswithfractions,asdocumentedin
theintroduction,wewouldsuggestthatthetypicalmodesofinstructionarenot
servingthecurrentneedsofstudents.
Thisapproachoftestingtheeffectsofimplementingrecommendationsfrom
thestandardsisentirelyinkeepingwiththespiritwithwhichthestandardswere
proposed.Theauthorsofthestandardsexplicitlystated,“Onepromiseofcommon
statestandardsisthatovertimetheywillallowresearchonlearningprogressions
toinformandimprovethedesignofstandardstoamuchgreaterextentthanis
possibletoday”[11].Weencourageotherinvestigatorstotestadditionalspecific
recommendationsofthestandards,sothattheresearchcommunitycanproducethe
scientificevidencenecessaryforinformeddebateandimprovementsinthismajor
educationalinitiative.
IMPROVINGKNOWLEDGEOFFRACTIONS 27
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SupportingInformation
S1Materials.Fulldescriptionsofthematerials,instructions,andprocedure
forboththeinterventionandthepretest/posttestmeasure.
S1Data.DatafromStudy1.
S2Data.DatafromStudy2.
IMPROVINGKNOWLEDGEOFFRACTIONS 33
Figure1
IMPROVINGKNOWLEDGEOFFRACTIONS 34
Figure2
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