improving children’s knowledge of fraction magnitudes

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)


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.


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


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


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


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


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


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).


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


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


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


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


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?


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


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


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


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


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


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


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


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


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


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


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


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


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.


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SupportingInformation

S1Materials.Fulldescriptionsofthematerials,instructions,andprocedure

forboththeinterventionandthepretest/posttestmeasure.

S1Data.DatafromStudy1.

S2Data.DatafromStudy2.


Figure1


Figure2

improving children’s knowledge of fraction magnitudes

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