Developing Effective Fractions Instruction for Kindergarten ...IES Practice Guide Developing Efective Fractions Instruction for Kindergarten Through 8th Grade September 2010 Panel

IESPRACTICEGUIDE WHATWORKSCLEARINGHOUSE

DevelopingEffectiveFractionsInstructionforKindergartenThrough8thGrade

NCEE20104039

DevelopingEffectiveFractionsInstructionDevelopingEffectiveFractionsInstructionforKindergartenThrforKindergartenThrough8thGradeough8thGrade

U.S.DEPARTMENTOFEDUCATION

ReviewofRecommendations

TheInstituteofEducationSciences(IES)publishespracticeguidesineducationtobringthebestavailableevidenceandexpertisetobearoncurrentchallengesineducation.Authorsofpracticeguidescombinetheirexpertisewiththefindingsofrigorousresearch,whenavailable,todevelopspecificrecommendationsforaddressingthesechallenges.Theauthorsratethestrengthoftheresearchevidencesupportingeachoftheirrecommendations.SeeAppendixAforafulldescriptionofpracticeguides.

Thegoalofthispracticeguideistooffereducatorsspecificevidence-basedrecommendationsthataddressthechallengeofimprovingstudents’understandingoffractionconceptsinkindergartenthrough8thgrade.Theguideprovidespractical,clearinformationoncriticaltopicsrelatedtotheteachingoffractionsandisbasedonthebestavailableevidenceasjudgedbytheauthors.

PracticeguidespublishedbyIESareofferedonourwebsiteatwhatworks.ed.gov/publications/practiceguides.Practiceguidespublishedtodateareshowninthefollowingtable.

Practice Guides Published

Relevant for All Grade

Levels

Relevant for Elementary

School

Relevant for Secondary

School

EncouragingGirlsinMathandScience(September 2007) OrganizingInstructionandStudytoImproveStudentLearning(September 2007) TurningAroundChronicallyLow-PerformingSchools(May 2008) UsingStudentAchievementDatatoSupportInstructionalDecisionMaking(September 2009) AssistingStudentsStrugglingwithReading:ResponsetoIntervention(RtI)andMulti-TierInterventioninthePrimaryGrades(February 2009)

EffectiveLiteracyandEnglishLanguageInstructionforEnglishLearnersintheElementaryGrades(December 2007)

ImprovingReadingComprehensioninKindergartenThrough3rdGrade(September 2010) ReducingBehaviorProblemsintheElementarySchoolClassroom(September 2008) AssistingStudentsStrugglingwithMathematics:ResponsetoIntervention(RtI)forElementaryandMiddleSchools(April 2009)

DevelopingEffectiveFractionsInstructionforKindergartenThrough8thGrade(September 2010) ImprovingAdolescentLiteracy:EffectiveClassroomandInterventionPractices(August 2008) StructuringOut-of-SchoolTimetoImproveAcademicAchievement(July 2009) DropoutPrevention(August 2008) HelpingStudentsNavigatethePathtoCollege:WhatHighSchoolsCanDo(September 2009)

http://ies.ed.gov/ncee/wwc/publications/practiceguides/


IESPracticeGuide

Developing Effective Fractions Instruction for Kindergarten Through 8th Grade

September 2010

Panel Robert Siegler (Chair) Carnegie Mellon University

Thomas Carpenter University of WisConsin–Madison

Francis (Skip) Fennell MCdaniel College, WestMinster, Md

David Geary University of MissoUri at ColUMbia

James Lewis University of nebraska–linColn

Yukari Okamoto University of California–santa barbara

Laurie Thompson eleMentary teaCher

Jonathan Wray hoWard CoUnty (Md) PUbliC sChools

Staff Jeffrey Max Moira McCullough Andrew Gothro Sarah Prenovitz MatheMatiCa PoliCy researCh

Project Officer Susan Sanchez institUte of edUCation sCienCes

NCEE20104039U.S.DEPARTMENTOFEDUCATION

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ThisreportwaspreparedfortheNationalCenterforEducationEvaluationandRegionalAssistance,InstituteofEducationSciencesunderContractED-07-CO-0062bytheWhatWorksClearinghouse,whichisoperatedbyMathematicaPolicyResearch.

Disclaimer

TheopinionsandpositionsexpressedinthispracticeguidearethoseoftheauthorsanddonotnecessarilyrepresenttheopinionsandpositionsoftheInstituteofEducationSciencesortheU.S.DepartmentofEducation.Thispracticeguideshouldbereviewedandappliedaccordingtothespecificneedsoftheeducatorsandeducationagencyusingit,andwithfullrealizationthatitrep-resentsthejudgmentsofthereviewpanelregardingwhatconstitutessensiblepractice,basedontheresearchthatwasavailableatthetimeofpublication.Thispracticeguideshouldbeusedasatooltoassistindecisionmakingratherthanasa“cookbook.”Anyreferenceswithinthedocumenttospecificeducationproductsareillustrativeanddonotimplyendorsementoftheseproductstotheexclusionofotherproductsthatarenotreferenced.

U.S.DepartmentofEducation

ArneDuncanSecretary

InstituteofEducationSciences

JohnQ.EastonDirector

NationalCenterforEducationEvaluationandRegionalAssistance

RebeccaMaynardCommissioner

September2010

Thisreportisinthepublicdomain.Althoughpermissiontoreprintthispublicationisnotnecessary,thecitationshouldbe:

Siegler,R.,Carpenter,T.,Fennell,F.,Geary,D.,Lewis,J.,Okamoto,Y.,Thompson,L.,&Wray,J.(2010).Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE#2010-4039).Washington,DC:NationalCenterforEducationEvaluationandRegionalAssistance,InstituteofEducationSciences,U.S.DepartmentofEducation.Retrievedfromwhatworks.ed.gov/publications/practiceguides.

WhatWorksClearinghousePracticeGuidecitationsbeginwiththepanelchair,followedbythenamesofthepanelistslistedinalphabeticalorder.

ThisreportisavailableontheIESWebsiteathttp://ies.ed.gov/nceeandwhatworks.ed.gov/publications/practiceguides.

AlternateFormats

Onrequest,thispublicationcanbemadeavailableinalternateformats,suchasBraille,largeprint,orcomputerdiskette.Formoreinformation,contacttheAlternateFormatCenterat202–260–0852or202-260-0818.

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TableofContents

DevelopingEffectiveFractionsInstructionforKindergartenThrough8thGrade

TableofContentsReviewofRecommendations . . . . . . . . . . . . . . . . . . . . . . . . .1

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

InstituteofEducationSciencesLevelsofEvidenceforPracticeGuides . . . . . . .3

IntroductiontotheDevelopingEffectiveFractionsInstructionforKindergartenThrough8thGradePracticeGuide . . . . . . . . . . . . .6

Recommendation1.Buildonstudents’informalunderstandingofsharingandproportionalitytodevelopinitialfractionconcepts . . . . . . . . . . . . 12

Recommendation2.Helpstudentsrecognizethatfractionsarenumbersandthattheyexpandthenumbersystembeyondwholenumbers.Usenumberlinesasacentralrepresentationaltoolinteachingthisandotherfractionconceptsfromtheearlygradesonward . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Recommendation3.Helpstudentsunderstandwhyproceduresforcomputationswithfractionsmakesense. . . . . . . . . . . . . . . . . . . . . . 26

Recommendation4.Developstudents’conceptualunderstandingofstrategiesforsolvingratio,rate,andproportionproblemsbeforeexposingthemtocross-multiplicationasaproceduretousetosolvesuchproblems . . . . . . . . 35

Recommendation5.Professionaldevelopmentprogramsshouldplaceahighpriorityonimprovingteachers’understandingoffractionsandofhowtoteachthem . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Glossary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

AppendixA.PostscriptfromtheInstituteofEducationSciences. . . . . . . . . . . . 49

AppendixB.AbouttheAuthors . . . . . . . . . . . . . . . . . . . . . . . . 51

AppendixC.DisclosureofPotentialConflictsofInterest. . . . . . . . . . . . . . . 54

AppendixD.RationaleforEvidenceRatings . . . . . . . . . . . . . . . . . . . 55

AppendixE.EvidenceHeuristic . . . . . . . . . . . . . . . . . . . . . . . . 68

Endnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

IndexofKeyMathematicalConcepts . . . . . . . . . . . . . . . . . . . . . 84

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TableofContentscontinued

ListofTablesTable1.InstituteofEducationScienceslevelsofevidenceforpracticeguides . . . . . . . .4

Table2.Recommendationsandcorrespondinglevelsofevidence . . . . . . . . . . . 11

TableD.1.StudiesofinterventionsthatusednumberlinestoimproveunderstandingofwholenumbermagnitudethatmetWWCstandards(withorwithoutreservations) . . . . . . . . . . . . . . . . . . . . . . . . . . 58

TableD.2.StudiesofinterventionsthatdevelopedconceptualunderstandingoffractioncomputationthatmetWWCstandards(withorwithoutreservations) . . . . . . . . . . . . . . . . . . . . . . . . . . 61

TableE.1.Evidence heuristic. . . . . . . . . . . . . . . . . . . . . . . . . . 68

ListofFiguresFigure1.Sharingasetofobjectsevenlyamongrecipients . . . . . . . . . . . . . . 14

Figure2.Partitioningbothmultipleandsingleobjects. . . . . . . . . . . . . . . . 15

Figure3.Studentworkforsharingfourpizzasamongeightchildren . . . . . . . . . . 16

Figure4.Findingequivalentfractionsonanumberline . . . . . . . . . . . . . . . 23

Figure5.Usingfractionstripstodemonstrateequivalentfractions. . . . . . . . . . . 24

Figure6.Fractioncirclesforadditionandsubtraction. . . . . . . . . . . . . . . . 28

Figure7.Redefiningtheunitwhenmultiplyingfractions. . . . . . . . . . . . . . . 29

Figure8.Usingribbonstomodeldivisionwithfractions. . . . . . . . . . . . . . . 30

Figure9.Ratiotableforaproportionproblem. . . . . . . . . . . . . . . . . . . 39

Figure10.Ratiotableforexploringproportionalrelations . . . . . . . . . . . . . . 40

ListofExamplesExample1.Measurementactivitieswithfractionstrips . . . . . . . . . . . . . . . 21

Example2.Introducingfractionsonanumberline. . . . . . . . . . . . . . . . . 22

Example3.Strategiesforestimatingwithfractions. . . . . . . . . . . . . . . . . 31

Example4.Problemsencouragingspecificstrategies . . . . . . . . . . . . . . . . 38

Example5.Whycross-multiplicationworks. . . . . . . . . . . . . . . . . . . . 39

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Recommendation1.Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. • Useequal-sharingactivitiestointroducetheconceptoffractions.Usesharingactivitiesthatinvolve

dividingsetsofobjectsaswellassinglewholeobjects.

• Extendequal-sharingactivitiestodevelopstudents’understandingoforderingandequivalenceoffractions.

• Buildonstudents’informalunderstandingtodevelopmoreadvancedunderstandingofproportionalreasoningconcepts.Beginwithactivitiesthatinvolvesimilarproportions,andprogresstoactivitiesthatinvolveorderingdifferentproportions.

Recommendation2.Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward. • Usemeasurementactivitiesandnumberlinestohelpstudentsunderstandthatfractionsarenumbers,

withallthepropertiesthatnumbersshare.

• Provideopportunitiesforstudentstolocateandcomparefractionsonnumberlines.

• Usenumberlinestoimprovestudents’understandingoffractionequivalence,fractiondensity(thecon-ceptthatthereareaninfinitenumberoffractionsbetweenanytwofractions),andnegativefractions.

• Helpstudentsunderstandthatfractionscanberepresentedascommonfractions,decimals,andper-centages,anddevelopstudents’abilitytotranslateamongtheseforms.

Recommendation3.Help students understand why procedures for computations with fractions make sense. • Useareamodels,numberlines,andothervisualrepresentationstoimprovestudents’understanding

offormalcomputationalprocedures.

• Provideopportunitiesforstudentstouseestimationtopredictorjudgethereasonablenessofanswerstoproblemsinvolvingcomputationwithfractions.

• Addresscommonmisconceptionsregardingcomputationalprocedureswithfractions.

• Presentreal-worldcontextswithplausiblenumbersforproblemsthatinvolvecomputingwithfractions.

Recommendation4.Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to crossmultiplication as a procedure to use to solve such problems. • Developstudents’understandingofproportionalrelationsbeforeteachingcomputationalprocedures

thatareconceptuallydifficulttounderstand(e.g.,cross-multiplication).Buildonstudents’developingstrategiesforsolvingratio,rate,andproportionproblems.

• Encouragestudentstousevisualrepresentationstosolveratio,rate,andproportionproblems.

• Provideopportunitiesforstudentstouseanddiscussalternativestrategiesforsolvingratio,rate,andproportionproblems.

Recommendation5.Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them. • Buildteachers’depthofunderstandingoffractionsandcomputationalproceduresinvolvingfractions.

• Prepareteacherstousevariedpictorialandconcreterepresentationsoffractionsandfractionoperations.

• Developteachers’abilitytoassessstudents’understandingsandmisunderstandingsoffractions.

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Acknowledgments

ThepanelgreatlyappreciatestheeffortsofJeffreyMax,MoiraMcCullough,AndrewGothro,andSarahPrenovitz,stafffromMathematicaPolicyResearchwhoparticipatedinthepanelmeet-

ings,summarizedtheresearchfindings,anddraftedtheguide.JeffreyMaxandMoiraMcCulloughhadprimaryresponsibilityfordraftingandrevisingtheguide.WealsothankShannonMonahan,CassandraPickens,ScottCody,NeilSeftor,KristinHallgren,andAlisonWellingtonforhelpfulfeed-backandreviewsofearlierversionsoftheguide, and Laura Watson-Sarnoski and Joyce Hofstetter for formatting and producing the guide.

RobertSiegler(Chair)ThomasCarpenter

Francis(Skip)FennellDavidGearyJamesLewis

YukariOkamotoLaurieThompson

JonathanWray

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LevelsofEvidenceforPracticeGuides

InstituteofEducationSciencesLevelsofEvidenceforPracticeGuides

ThissectionprovidesinformationabouttheroleofevidenceinInstituteofEducationSciences’(IES)WhatWorksClearinghouse(WWC)practiceguides.Itdescribeshowpracticeguidepanels

determinethelevelofevidenceforeachrecommendationandexplainsthecriteriaforeachofthethreelevelsofevidence(strongevidence,moderateevidence,andminimalevidence).

Thelevelofevidenceassignedtoeachrecom-mendationinthispracticeguiderepresentsthepanel’sjudgmentofthequalityoftheexistingresearchtosupportaclaimthatwhenthesepracticeswereimplementedinpastresearch,positiveeffectswereobservedonstudentoutcomes.Aftercarefulreviewofthestudiessupportingeachrecommendation,panelistsdeterminethelevelofevidenceforeachrecommendationusingthecriteriainTable1andtheevidenceheuristicdepictedinAppendixE.Thepanelfirstconsiderstherelevanceofindividualstudiestotherecom-mendation,andthendiscussestheentireevidencebase,takingintoconsideration:

• thenumberofstudies

• thequalityofthestudies

•whetherthestudiesrepresenttherangeofparticipantsandsettingsonwhichtherecommendationisfocused

•whetherfindingsfromthestudiescanbeattributedtotherecommendedpractice

•whetherfindingsinthestudiesareconsis-tentlypositive

Aratingofstrong evidencereferstoconsis-tentevidencethattherecommendedstrate-gies,programs,orpracticesimprovestudentoutcomesforawidepopulationofstudents.Inotherwords,thereisstrongcausalandgeneralizableevidence.

Aratingofmoderate evidence referseithertoevidencefromstudiesthatallowstrong

causalconclusionsbutcannotbegeneralizedwithassurancetothepopulationonwhicharecommendationisfocused(perhapsbecausethefindingshavenotbeenwidelyreplicated)ortoevidencefromstudiesthataregener-alizablebuthavesomecausalambiguity.Italsomightbethatthestudiesthatexistdonotspecificallyexaminetheoutcomesofinterestinthepracticeguidealthoughtheymayberelated.

Aratingofminimal evidence suggeststhatthepanelcannotpointtoabodyofresearchthatdemonstratesthepractice’spositiveeffectonstudentachievement.Insomecases,thissimplymeansthattherecommendedpracticeswouldbedifficulttostudyinarigor-ous,experimentalfashion;1inothercases,itmeansthatresearchershavenotyetstudiedthispractice,orthatthereisweakorcon-flictingevidenceofeffectiveness.Aminimalevidenceratingdoesnotindicatethattherecommendationisanylessimportantthanotherrecommendationswithastrongevi-denceormoderateevidencerating.

FollowingWWCguidelines,improvedoutcomesareindicatedbyeitherapositivestatisticallysignificanteffectorapositivesubstantivelyimportanteffectsize.2TheWWCdefinessubstantivelyimportant,orlarge,effectsonoutcomestobethosewitheffectsizesgreaterthan0.25standarddeviations.Inthisguide,thepaneldiscussessubstantivelyimportantfindingsasonesthatcontributetotheevidenceofpractices’effectiveness,evenwhenthoseeffectsarenotstatisticallysignificant.

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LevelsofEvidenceforPracticeGuidescontinued

Table1.InstituteofEducationScienceslevelsofevidenceforpracticeguides

StrongEvidence

Aratingofstrong evidencemeanshigh-qualitycausalresearchlinksthispracticewithpositiveresultsinschoolsandclassrooms.Theresearchrulesoutothercausesofthepositiveresults,andtheschoolsandclass-roomsaresimilartothosetargetedbythisguide.Strongevidenceisdemonstratedwhenanevidencebasehasthefollowingproperties:

• Highinternalvalidity:theevidencebaseconsistsofhigh-qualitycausaldesignsthatmeetWWCstandardswithorwithoutreservations.3

• Highexternalvalidity:theevidencebaseconsistsofavarietyofstudieswithhighinternalvaliditythatrepre-sentthepopulationonwhichtherecommendationisfocused.4

• Consistentpositiveeffectsonrelevantoutcomeswithoutcontradictoryevidence(i.e.,nostatisticallysignifi-cantnegativeeffects)instudieswithhighinternalvalidity.

• Directrelevancetoscope(i.e.,ecologicalvalidity),includingrelevantcontext(e.g.,classroomvs.laboratory),sample(e.g.,ageandcharacteristics),andoutcomesevaluated.

• Directtestoftherecommendationinthestudiesortherecommendationisamajorcomponentoftheinter-ventionsevaluatedinthestudies.

• Thepanelhasahighdegreeofconfidencethatthispracticeiseffective.

• Intheparticularcaseofrecommendationsonassessments,theevidencebasemeetsThe Standards for Educational and Psychological Testing (AmericanEducationalResearchAssociation,AmericanPsychologicalAssociation,andNationalCouncilonMeasurementinEducation,1999).

ModerateEvidence

Aratingofmoderate evidencemeanshigh-qualitycausalresearchlinksthispracticewithpositiveresultsinschoolsandclassrooms.However,theresearchmaynotadequatelyruleoutothercausesofthepositiveresults,ortheschoolsandclassroomsarenotsimilartothosetargetedbythisguide.Moderateevidenceisdemonstratedwhenanevidencebasehasthefollowingproperties:

• Highinternalvaliditybutmoderateexternalvalidity(i.e.,studiesthatsupportstrongcausalconclusions,butgeneralizationisuncertain)ORstudieswithhighexternalvaliditybutmoderateinternalvalidity(i.e.,studiesthatsupportthegeneralityofarelation,butthecausalityisuncertain).

•TheresearchmayincludestudiesmeetingWWCstandardswithorwithoutreservationswithsmallsamplesizesand/orotherconditionsofimplementationoranalysisthatlimitgeneralizability.

•TheresearchmayincludestudiesthatsupportthegeneralityofarelationbutdonotmeetWWCstan-dards;5however,theyhavenomajorflawsrelatedtointernalvalidityotherthanlackofdemonstratedequivalenceatpretestforquasi-experimentaldesignstudies(QEDs).QEDswithoutequivalencemustincludeapretestcovariateasastatisticalcontrolforselectionbias.ThesestudiesmustbeaccompaniedbyatleastonerelevantstudymeetingWWCstandardswithorwithoutreservations.

• Apreponderanceofpositiveeffectsonrelevantoutcomes.Contradictoryevidence(i.e.,statisticallysignifi-cantnegativeeffects)mustbediscussedbythepanelandconsideredwithregardtorelevancetothescopeoftheguideandintensityoftherecommendationasacomponentoftheinterventionevaluated.Ifoutcomesareoutofthescopeoftheguide,thisalsomustbediscussed.

• Thepaneldeterminedthattheresearchdoesnotrisetothelevelofstrongevidencebutismorecompellingthanaminimallevelofevidence.

• Intheparticularcaseofrecommendationsonassessments,theremustbeevidenceofreliabilitythatmeetsThe Standards for Educational and Psychological Testing,butevidenceofvaliditymaybefromsamplesnotadequatelyrepresentativeofthepopulationonwhichtherecommendationisfocused.

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LevelsofEvidenceforPracticeGuidescontinued

Table1.InstituteofEducationScienceslevelsofevidenceforpracticeguides(continued)

MinimalEvidence

Aratingofminimal evidencemeansthepanelconcludedtherecommendedpracticeshouldbeadopted;how-ever,thepanelcannotpointtoabodyofcausalresearchthatdemonstratestherecommendation’spositiveeffectandthatrisestothelevelofmoderateorstrongevidence.

IntermsofthelevelsofevidenceindicatedinTable1,thepanelreliedonWWCevidencestandardstoassessthequalityofevidencesupportingeducationalprogramsandpractices.WWCevaluatesevidenceforthecausalvalidityofinstructionalprogramsandpracticesaccordingtoWWCstandards.

Informationaboutthesestandardsisavailableathttp://ies.ed.gov/ncee/wwc/pdf/wwc_pro-cedures_v2_standards_handbook.pdf.EligiblestudiesthatmeetWWCevidencestandardsormeetevidencestandardswithreservationsareindicatedbyboldtextintheendnotesandreferencespages.

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http://ies.ed.gov/ncee/wwc/pdf/wwc_procedures_v2_standards_handbook.pdf


Introduction

IntroductiontotheDevelopingEffectiveFractionsInstructionforKindergartenThrough8thGradePracticeGuide

Thissectionprovidesanoverviewoftheimportanceofdevelopingeffectivefractionsinstruc-tionforkindergartenthrough8thgradeandexplainskeyparametersconsideredbythepanel

indevelopingthepracticeguide.Italsosummarizestherecommendationsforreadersandcon-cludeswithadiscussionoftheresearchsupportingthepracticeguide.

U.S.students’mathematicsskillshavefallenshortformanyyears,withtheramificationsofthisinadequateknowledgewidelyrecognized.

The1983reportA Nation at Risk relatedAmerica’ssafetyandprosperitytoitsmathematicalcompetenceandwarnedthatAmericanstudents’mathematicalknowledgewasinsufficienttomeetthechallengesofthemodernworld.Morethan25yearslater,U.S.students’mathematicalachievementcontinuestolagfarbehindthatofstudentsinEastAsiaandmuchofEurope.6OnlyasmallpercentageofU.S.studentspossessthemath-ematicsknowledgeneededtopursuecareersinscience,technology,engineering,ormath-ematics(STEM)fields.7Manyhighschoolgraduateslackthemathematicalcompetenceforawiderangeofwell-payingjobsintoday’seconomy.8Moreover,largegapsinmathemat-icsknowledgeexistamongstudentsfromdif-ferentsocioeconomicbackgroundsandracialandethnicgroupswithintheUnitedStates.9

ThesedisparitieshurtthenationaleconomyandalsolimittensofmillionsofAmericans’occupationalandfinancialopportunities.10

Poorunderstandingoffractionsisacriticalaspectofthisinadequatemathematicsknowl-edge.KnowledgeoffractionsdiffersevenmorebetweenstudentsintheUnitedStatesandstudentsinEastAsiathandoesknowl-edgeofwholenumbers.11Thislearninggapisespeciallyproblematicbecauseunderstandingfractionsisessentialforalgebraandothermoreadvancedareasofmathematics.12

Teachersareawareofstudents’difficultyinlearningaboutfractionsandoftenarefrustratedbyit.Inarecentnationalpoll,AlgebraIteachersratedtheirstudentsas

having“verypoorpreparationinrationalnumbersandoperationsinvolvingfractionsanddecimals.”13Thealgebrateachersrankedpoorunderstandingoffractionsasoneofthetwomostimportantweaknessesinstudents’preparationfortheircourse.

ManyexamplesillustrateAmericanstudents’weakunderstandingoffractions.Onthe2004NationalAssessmentofEducationalProgress(NAEP),50%of8th-graderscouldnotorderthreefractionsfromleasttogreatest.14 Theproblemisnotlimitedtorationalnumberswrittenincommonfractionnotation.Onthe2004NAEP,fewerthan30%of17-year-oldscorrectlytranslated0.029as29/1000.15 Thesamedifficultyisapparentinone-on-onetestingofstudentsincontrolledexperimentalsettings:whenaskedwhichoftwodecimals,0.274and0.83,isgreater,most5th-and6th-graderschoose0.274.16

Theseexamplesandothersledtheauthorsofthisguidetoconcludethefollowing:

AhighpercentageofU.S.studentslackconceptualunderstandingoffractions,evenafterstudyingfractionsforseveralyears;this,inturn,limitsstudents’abilitytosolveproblemswithfractionsandtolearnandapplycomputationalproceduresinvolvingfractions.

Thelackofconceptualunderstandinghasseveralfacets,including

• Notviewingfractionsasnumbersatall,butratherasmeaninglesssymbolsthatneedtobemanipulatedinarbitrarywaystoproduceanswersthatsatisfyateacher.

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Introductioncontinued

• Focusingonnumeratorsanddenominatorsasseparatenumbersratherthanthinkingofthefractionasasinglenumber.Errorssuchasbelievingthat3/8>3/5arisefromcompar-ingthetwodenominatorsandignoringtheessentialrelationbetweeneachfraction’snumeratoranditsdenominator.

• Confusingpropertiesoffractionswiththoseofwholenumbers.Thisisevidentinmanyhighschoolstudents’claimthatjustasthereisnowholenumberbetween5and6,thereisnonumberofanytypebetween5/7and6/7.17

Thispracticeguidepresentsfiverecommen-dationsintendedtohelpeducatorsimprovestudents’understandingof,andproblem-solvingsuccesswith,fractions.Recommen-dationsprogressfromproposalsforhowtobuildrudimentaryunderstandingoffractionsinyoungchildren;toideasforhelpingolderchildrenunderstandthemeaningoffractionsandcomputationsthatinvolvefractions;toproposalsintendedtohelpstudentsapplytheirunderstandingoffractionstosolveprob-lemsinvolvingratios,rates,andproportions.Improvingstudents’learningaboutfractionswillrequireteachers’masteryofthesubjectandtheirabilitytohelpstudentsmasterit;therefore,arecommendationregardingteachereducationalsoisincluded.

Recommendationsinthepracticeguideweredevelopedbyapanelofeightresearchersandpractitionerswhohaveexpertiseindifferentaspectsofthetopic.Panelistsincludeamath-ematicianactiveinissuesrelatedtomath-ematicsteachereducation;threemathematicseducators,oneofwhomhasbeenpresidentoftheNationalCouncilofTeachersofMath-ematics;twopsychologistswhoseresearchfocusesonhowchildrenlearnmathemat-ics;andtwopractitionerswhohavetaughtmathematicsinelementaryandmiddleschoolclassroomsandsupervisedotherelementaryandmiddleschoolmathematicsteachers.Panelmembersworkedcollaborativelytodeveloprecommendationsbasedonthebestavailableresearchevidenceandontheir

combinedexperienceandexpertiseregardingmathematicsteachingandlearning.

Scopeofthepracticeguide

Writingthisguiderequireddecisionsregard-ingtheintendedaudience,whichgradelevelstoexamine,whichskillsandknowledgetoconsider,andwhichtermstouseindescrib-ingtheresearchandrecommendations.Thepanelconsistentlychosetomaketheguideasinclusiveaspossible.

Audienceandgradelevel.Theintendedaudienceiselementaryandmiddleschoolteachers,mathematicssupervisors,teacherleaders,specialists,coaches,principals,par-ents,teachereducators,andothersinterestedinimprovingstudents’mathematicslearning.Gradelevelsemphasizedarekindergartenthrough8thgrade;almostallinstructioninfractionstakesplacewithinthisperiod,andthisisthepopulationstudiedinmostoftheavailableresearch.Theguidefocusesnotonlyoncomputationwithfractions,butalsoonskillsthatreflectunderstandingoffrac-tions,suchasestimatingfractions’positionsonnumberlinesandcomparingthesizesoffractions,becauselackofsuchunderstandingunderliesmanyoftheotherdifficultiesstu-dentshavewithfractions.

Content.Thisdocumentusesthetermfractionsratherthanrational numbers.Thetermfractionsreferstothefullrangeofwaysofexpressingrationalnumbers,includingdecimals,percentages,andnegativefractions.Thepanelmakesrecommendationsonthisfullrangeofrationalnumbersbecausestudents’understandingofthemiscriticaltotheiruseoffractionsincontext.

Theguide’sinclusivenessisfurtherevidentinitsemphasisontheneedforstudentstobeabletoperformcomputa-tionaloperationswithfractions;tounder-standthesecomputationaloperations;andtounderstand,morebroadly,whatfractionsrepresent.

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Tohelpstudentsunderstandthefullrangeoffractions,thepanelsuggestseducatorseffectivelyconveythefollowing:

•Commonfractions,decimals,andpercentsareequivalentwaysofexpressingthesamenumber(42/100=0.42=42%).

•Wholenumbersareasubsetofrationalnumbers.

•Anyfractioncanbeexpressedinaninfinitenumberofequivalentways(3/4=6/8=9/12=0.75=75%,andsoon).

Boththestrengthsstudentsbringtothetaskoflearningaboutfractionsandthechal-lengesthatoftenmakelearningdifficultarecoveredinthisguide.Childrenenterschoolwitharudimentaryunderstandingofshar-ingandproportionality,conceptsonwhichteacherscanbuildtoproducemoreadvancedunderstandingsoffractions.18Thescopeoftheguideincludesdescribingtheseearlydevelopingconceptsandhowmoreadvancedunderstandingcanbebuiltonthem.Theguidealsodescribescommonmisconceptionsaboutfractionsthatinterferewithstudents’learning—forexample,themisconceptionthatmultiplyingtwonumbersmustresultinalargernumber—andhowsuchmisconcep-tionscanbeovercome.

Finally,theguideaddressesnotonlytheneedtoimprovestudents’understandingoffrac-tions,butalsotheneedtoimproveteachers’understandingofthem.FartoomanyU.S.teacherscanapplystandardcomputationalalgorithmstosolveproblemsinvolvingfrac-tionsbutdonotknowwhythosealgorithmsworkorhowtoevaluateandexplainwhyalternativeproceduresthattheirstudentsgeneratearecorrectorincorrect.19 Similarly,manyteacherscanexplainpart-wholeinter-pretationsoffractionsbutnototheressentialinterpretations,suchasconsideringfractionsasmeasuresofquantitiesthatofferprecisionbeyondthatofferedbywholenumbersorviewingfractionsasquotients.

U.S.teachers’understandingoffractionslagsfarbehindthatofteachersinnationsthatproducebetterstudentlearningoffractions,suchasJapanandChina.20Althoughsomeoftheinformationinthisguideisaimedatdeepeningteachers’understandingoffrac-tions,professionaldevelopmentactivitiesthatimproveteachers’understandingoffractionsandcomputationalproceduresthatinvolvefractionsalsoseemessential.

Summaryoftherecommendations

Thispracticeguideincludesfiverecommen-dationsforimprovingstudents’learningoffractions.Thefirstrecommendationisaimedatbuildingthefoundationalknowledgeofyoungstudents,thenextthreetargetolderstudentsastheyadvancethroughtheirelementaryandmiddleschoolyears,andthefinalrecommendationfocusesonincreasingteachers’abilitytohelpstudentsunderstandfractions.Althoughtherecommendationsvaryintheirparticulars,allfivereflecttheperspectivethatconceptualunderstandingoffractionsisessentialforstu-dentstolearnaboutthetopic,torememberwhattheylearned,andtoapplythisknowl-edgetosolveproblemsinvolvingfractions.Educatorsmayprofitablyadoptsomeoftherecommendationswithoutadoptingallofthem,butwebelievethatthegreatestbenefitwillcomefromadoptingalloftherecommen-dationsthatarerelevanttotheirclasses.

• Recommendation1istobuildonstu-dents’informalunderstandingofsharingandproportionalitytodevelopinitialfractionconcepts.Learningisoftenmosteffectivewhenitbuildsonexistingknowl-edge,andfractionsarenoexception.Bythetimechildrenbeginschool,mosthavedevelopedabasicunderstandingofshar-ingthatallowsthemtodividearegionorsetofobjectsequallyamongtwoormorepeople.Thesesharingactivitiescanbeusedtoillustrateconceptssuchashalves,thirds,andfourths,aswellasmoregen-eralconceptsrelevanttofractions,such

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asthatincreasingthenumberofpeopleamongwhomanobjectisdividedresultsinasmallerfractionoftheobjectforeachperson.Similarly,earlyunderstandingofproportionscanhelpkindergartnerscompare,forexample,howone-thirdoftheareasofasquare,rectangle,andcirclediffer.

•Recommendation2istoensurethatstudentsknowthatfractionsarenumbersthatexpandthenumbersystembeyondwholenumbers,andtousenumberlinesasakeyrepresentationaltooltoconveythisandotherfractionconceptsfromtheearlygradesonward.Althoughitseemsobvioustomostadultsthatfractionsarenumbers,manystudentsinmiddleschoolandbeyondcannotidentifywhichoftwofractionsisgreater,indicatingthattheyhavecursoryknowledgeatbest.Numberlinesareparticularlyadvantageousforassessingknowledgeoffractionsandforteachingstudentsaboutthem.Theypro-videacommontoolforrepresentingthesizesofcommonfractions,decimals,andpercents;positiveandnegativefractions;fractionsthatarelessthanoneandgreaterthanone;andequivalentandnonequiva-lentfractions.Numberlinesalsoareanaturalwayofintroducingstudentstotheideaoffractionsasmeasuresofquantity,animportantideathatneedstobegivengreateremphasisinmanyU.S.classrooms.

• Recommendation3istohelpstudentsunderstandwhyproceduresforcomputa-tionswithfractionsmakesense.ManyU.S.students,andeventeachers,cannotexplainwhycommondenominatorsarenecessarytoaddandsubtractfractionsbutnottomultiplyanddividethem.Fewcanexplainthe“invertandmultiplyrule,”orwhydividingbyafractioncanresultinaquotientlargerthanthenumberbeingdivided.Studentssometimeslearncom-putationalproceduresbyrote,buttheyalsooftenquicklyforgetorbecomecon-fusedbytheseroutines;thisiswhattends

tohappenwithfractionsalgorithms.For-gettingandconfusingalgorithmsoccurlessoftenwhenstudentsunderstandhowandwhycomputationalproceduresyieldcorrectanswers.

•Recommendation4involvesfocusingonproblemsinvolvingratios,rates,andproportions.Theseapplicationsoffractionconceptsoftenprovedifficultforstudents.Illustratinghowdiagramsandothervisualrepresentationscanbeusedtosolveratio,rate,andproportionproblemsandteach-ingstudentstousethemareimportantforlearningalgebra.Alsousefulisprovidinginstructiononhowtotranslatestate-mentsinwordproblemsintomathemati-calexpressionsinvolvingratio,rate,andproportion.Thesetopicsincludewaysinwhichstudentsarelikelytousefractionsthroughouttheirlives;itisimportantforthemtounderstandtheconnectionbetweentheseappliedusesoffractionsandtheconceptsandproceduresinvolvingfractionsthattheylearnintheclassroom.

• Recommendation5urgesteachereducationandprofessionaldevelopmentprogramstoemphasizehowtoimprovestudents’understandingoffractionsandtoensurethatteachershavesufficientunderstandingoffractionstoachievethisgoal.Fartoomanyteachershavedifficultyexplaininginterpretationsoffractionsotherthanthepart-wholeinterpreta-tion,whichisusefulinsomecontextsbutnotothers.Althoughmanyteacherscandescribeconventionalalgorithmsforsolvingfractionsproblems,fewcanjus-tifythem,explainwhytheyyieldcorrectanswers,orexplainwhysomenonstan-dardproceduresthatstudentsgenerateyieldcorrectanswersdespitenotlookinglikeaconventionalalgorithm.Greaterunderstandingoffractions,knowledgeofstudents’conceptionsandmisconceptionsaboutfractions,andeffectivepracticesforteachingfractionsarecriticallyimportantforimprovingclassroominstruction.

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Useofresearch

Therecommendationsinthispracticeguidearebasedonnumeroustypesofevidence,includingnationalandinternationalassess-mentsofstudents’mathematicalknowledge,asurveyofteachers’viewsofthegreatestproblemsintheirstudents’preparationforlearningalgebra,mathematicians’analysesofkeyconceptsforunderstandingfractions,descriptivestudiesofsuccessfulandunsuc-cessfulfractionslearners,andcontrolledexperimentalevaluationsofinterventionsdesignedtoimprovelearningoffractions.

Theresearchbasefortheguidewasidenti-fiedthroughacomprehensivesearchforstudiesoverthepast20yearsthatevalu-atedteachingandlearningaboutfractions.Thissearchwasdoneforalargenumberofkeywordsrelatedtofractionsteachingandlearningthatweresuggestedbythepanelmembers;theresultsweresupplementedbyspecificstudiesknowntopanelmembersthatwerenotidentifiedbythedatabasesearch,includingearlierworks.Theprocessyieldedmorethan3,000citations.Ofthese,132mettheWWCcriteriaforreview,and33metthecausalvaliditystandardsoftheWWC.

Insomecases,recommendationsarebasedonsuchrigorousresearch.ButwhenresearchwasrareordidnotmeetWWCstandards,the

recommendationsreflectwhatthisguide’spanelbelievesarebestpractices,basedoninstructionalapproacheshavingbeensuc-cessfullyimplementedincasestudiesorincurriculathathavenotbeenrigorouslyevalu-ated.ThepanelcouldnotfulfillitswishtobaseallrecommendationsonstudiesthatmetWWCstandards,inlargepartbecausefarlessresearchisavailableonfractionsthanondevelopmentofskillsandconceptsregardingwholenumbers.Forexample,the2nd Handbook of Research on Mathematics Teaching and Learning (NationalCouncilofTeachersofMathematics,2007)includes109citationsofresearchpublishedin2000orlateronwholenumbersbutonlyninecitationsofresearchonfractionspublishedoverthesameperiod.High-qualitystudiestestingtheeffective-nessofspecificinstructionaltechniqueswithfractionswereespeciallyscarce.Agreateramountofhigh-qualityresearchonfractionsisclearlyneeded,especiallystudiesthatcomparetheeffectivenessofalternativewaysofteachingchildrenaboutfractions.

Table2showseachrecommendationandthestrengthoftheevidencethatsupportsitasdeterminedbythepanel.Followingtherec-ommendationsandsuggestionsforcarryingouttherecommendations,AppendixDpres-entsmoreinformationontheresearchevi-dencethatsupportseachrecommendation.

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Table2.Recommendationsandcorrespondinglevelsofevidence

Levels of Evidence

Recommendation Minimal Evidence

Moderate Evidence

Strong Evidence

1. Buildonstudents’informalunderstandingofsharingandproportionalitytodevelopinitialfractionconcepts.

2. Helpstudentsrecognizethatfractionsarenumbersandthattheyexpandthenumbersystembeyondwholenumbers.Usenumberlinesasacentralrepresentationaltoolinteachingthisandotherfractionconceptsfromtheearlygradesonward.

3. Helpstudentsunderstandwhyproceduresforcomputationswithfractionsmakesense.

4. Developstudents’conceptualunderstandingofstrategiesforsolvingratio,rate,andproportionproblemsbeforeexposingthemtocross-multiplicationasaproceduretousetosolvesuchproblems.

5. Professionaldevelopmentprogramsshouldplaceahighpriorityonimprovingteachers’understandingoffractionsandofhowtoteachthem.

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Recommendation1

Buildonstudents’informalunderstandingofsharingandproportionalitytodevelopinitialfractionconcepts.Students come to kindergarten with a rudimentary understanding of basic fraction concepts. They can share a set of objects equally among a group of people (i.e., equal sharing)21 and identify equivalent proportions of common shapes (i.e., proportional reasoning).22

By using this early knowledge to introduce fractions, teachers allow students to build on what they already know. This facilitates connections between students’ intuitive knowledge and formal fraction concepts. The panel recommends using sharing activities to develop students’ understanding of ordering and equivalence relations among fractions.

Sharing activities can introduce children to several of the basic interpretations of fractions discussed in the introduction. Sharing can be presented in terms of division—such as by partitioning 12 candies into four equally numerous groups. Sharing also can be presented in terms of ratios; for example, if three cakes are shared by two children, the ratio of the number of cakes to the number of children is 3:2.

Although fractions are typically introduced by 1st or 2nd grade, both the sharing and the proportional reasoning activities described in this recommendation can begin as early as preschool or kindergarten.

Summaryofevidence:MinimalEvidence

Thisrecommendationisbasedonstudiesshowingthatstudentshaveanearlyunder-standingofsharingandproportionality,23

andonstudiesofinstructionthatusesharingscenariostoteachfractionconcepts.24How-ever,noneofthestudiesthatusedsharingscenariostoteachfractionconceptsmetWWCstandards.Despitethelimitedevidence,the

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Recommendation1continued

panelbelievesthatstudents’informalknowl-edgeofsharingandproportionalityprovidesafoundationforintroducingandteachingfractionconcepts.

Equalsharing.Childrenhaveanearlyunder-standingofhowtocreateequalshares.Byage4,childrencandistributeequalnumbersofequal-sizeobjectsamongasmallnumberofrecipients,andtheabilitytoequallyshareimproveswithage.25Sharingasetofdiscreteobjects(e.g.,12grapessharedamongthreechildren)tendstobeeasierforyoungchildrenthansharingasingleobject(e.g.,acandybar),butbyage5or6,childrenarereason-ablyskilledatboth.26

Casestudiesshowhowanearlyunderstand-ingofsharingcouldbeusedtoteachfrac-tionstoelementarystudents.27Intwostudies,teachersposedstoryproblemswithsharingscenariostoteachfractionconceptssuchasequivalenceandordering,aswellasfractioncomputation.Thestudiesreportedpositiveeffectsonfractionknowledge,buttheydonotproviderigorousevidenceontheimpactofinstructionbasedonsharingactivities.

Proportionalrelations.Thepanelbelievesthatinstructionalpracticescanbuildonyoungchildren’srudimentaryknowledgeofproportionalitytoteachfractionconcepts.Thisearlyunderstandingofproportionalityhasbeendemonstratedindifferentways.Byage6,childrencanmatchequivalentpropor-tionsrepresentedbydifferentgeometricfiguresandbyeverydayobjectsofdifferentshapes.28One-halfisanimportantlandmarkincomparingproportions;childrenmoreoftensucceedoncomparisonsinwhichoneproportionismorethanhalfandtheotherislessthanhalf,thanoncomparisonsinwhichbothproportionsaremorethanhalforbotharelessthanhalf(e.g.,comparing1/3to3/5 iseasierthancomparing2/3to4/5).29Inaddi-tion,childrencancompleteanalogiesbasedonproportionalrelations—forexample,halfcircleistohalfrectangleasquartercircleistoquarterrectangle.30

Althoughthereisevidencethatdescribesyoungchildren’sknowledgeofproportionality,norigorousstudiesthatmetWWCstandardshaveexaminedwhetherthisearly-developingknowledgecanbeusedtoimproveteachingoffractionconcepts.

Howtocarryouttherecommendation

1. Useequal-sharingactivitiestointroducetheconceptoffractions.Usesharingactivi-tiesthatinvolvedividingsetsofobjectsaswellassinglewholeobjects.

Thepanelrecommendsthatteachersofferaprogressionofsharingactivitiesthatbuildsonstudents’existingstrategiesfordividingobjects.Teachersshouldbeginwithactivitiesthatinvolveequallysharingasetofobjectsamongagroupofrecipientsandprogresstosharingscenariosthatrequirepartitioninganobjectorsetofobjectsintofractionalparts.Inaddition,earlyactivitiesshouldbuildonstudents’halvingstrategy(dividingsomethingintotwoequalsetsorparts)beforehavingstudentspartitionobjectsamonglargernum-bersofrecipients.Studentsshouldbeencour-agedtousecounters(e.g.,beans,tokens),createdrawings,orrelyonotherrepresenta-tionstosolvethesesharingproblems;then

teacherscanintroduceformalfractionnames(e.g.,onethird,onefourth,thirds,quarters)andhavechildrenlabeltheirdrawingstonamethesharedpartsofanobject(e.g.,1/3 or1/8ofapizza).Foroptimalsuccess,childrenshouldengageinavarietyofsuchlabelingactivities,notjustoneortwo.

Sharingasetofobjects.Teachersshouldinitiallyhavestudentssolveproblemsthatinvolvetwoormorepeoplesharingasetofobjects(seeFigure1).Theproblemsshouldincludesetsofobjectsthatcanbeevenlydividedamongsharers,sotherearenoremainingobjectsthatneedtobepartitionedintofractionalpieces.

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Recommendation 1 continuedRecommendation1continued

Intheseearlysharingproblems,teachersshoulddescribethenumberofitemsandthenumberofrecipientssharingthoseitems,andstudentsshoulddeterminehowmanyitemseachpersonreceives.31Teachersmightthenposethesameproblemwithincreasingnumbersofrecipients.32Itisimportanttoemphasizethattheseproblemsrequireshar-ingasetofobjectsequally,sothatstudentsfocusongivingeachpersonthesamenum-berofobjects.

Partitioningasingleobject.Next,teach-ersshouldposesharingproblemsthatresultinstudentsdividingoneormoreobjectsintoequalparts.Thefocusoftheseproblemsshiftsfromaskingstudentshow many thingseachpersonshouldgettoaskingstudentshow muchofanobjecteachpersonshouldget.Forexample,whenonecookieissharedbetweentwochildren,studentshavetothink

abouthowmuchofthecookieeachchildshouldreceive.

Teacherscanbeginwithproblemsthatinvolvemultiplepeoplesharingasingleobject(e.g.,fourpeoplesharinganapple)andprogresstoproblemswithmultiplepeoplesharingasetofobjectsthatmustbedividedintosmallerpartstoshareequally(e.g.,threepeoplesharingfourapples).Problemsthatinvolvesharingoneobjectresultinsharesthatareunit fractions(e.g.,1/3,1/4,1/9),whereasscenarioswithmultiplepeopleandobjectsoftenresultinnonunit fractions(e.g.,3/4).33

Thisdistinctionbetweenunitandnon-unitfractionsisimportant,becausewhenfrac-tionsarereducedtolowestterms,non-unitfractionsarecomposedofunitfractions(e.g.,3/4=1/4+1/4+1/4),buttheoppositeisnotthecase.Sharingsituationsthatresultinunitfractionsprovideausefulstartingpoint

Figure1.Sharingasetofobjectsevenlyamongrecipients

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Problem

Threechildrenwanttoshare12cookiessothateachchildreceivesthesamenumberofcookies.Howmanycookiesshouldeachchildget?

ExamplesofSolutionStrategies

Studentscansolvethisproblembydrawingthreefigurestorepresentthechildrenandthendrawingcook-iesbyeachfigure,givingonecookietothefirstchild,onetothesecond,andonetothethird,continu-inguntiltheyhavedistributed12cookiestothethreechildren,andthencountingthenumberofcookiesdistributedtoeachchild.Otherstudentsmaysolvetheproblembysimplydealingthecookiesintothreepiles,asiftheyweredealingcards.

Recommendation 1 continuedRecommendation 1Recommendation 1Recommendation 1Recommendation 1 continuedcontinuedRecommendation1continued

Figure2.Partitioningbothmultipleandsingleobjects

Problem

Twochildrenwanttosharefiveapplesthatarethesamesizesothatbothhavethesameamounttoeat.Drawapicturetoshowwhateachchildshouldreceive.

ExamplesofSolutionStrategies

Studentsmightsolvethisproblembydrawingfivecirclestorepresentthefiveapplesandtwofigurestorepresentthetwochildren.Studentsthenmightdrawlinesconnectingeachchildtotwoapples.Finally,theymightdrawalinepartitioningthefinalappleintotwoapproximatelyequalpartsanddrawalinefromeachparttothetwochildren.Alterna-tively,asinthepicturetotheright,childrenmightdrawalargecirclerepresentingeachchild,twoappleswithineachcircle,andafifthapplestrad-dlingthecirclesrepresentingthetwochildren.Inyetanotherpossibility,childrenmightdivideeachappleintotwopartsandthenconnectfivehalfapplestotherepresentationofeachfigure.

forintroducingfractionnames,especiallybecausesomechildrenthinkthatallfractionalpartsarecalledone-half.34

Thepanelalsosuggestsstartingwithprob-lemsthatinvolvesharingamongtwo,four,oreightpeople(i.e.,powersoftwo).35Thisallowsstudentstocreateequalpartsbyusingahalvingstrategy—dividinganobjectinhalf,dividingtheresultinghalvesinhalf,andsoon,untilthereareenoughpiecestoshare(seeFigure2).36Eventually,studentsshouldsolvesharingproblemsforwhichthey

cannotuseahalvingstrategy.Partitioningabrownieintothirds,forexample,requiresthatstudentsanticipatehowtoslicethebrowniesothatitresultsinthreeequalparts.Studentsmaybetemptedtouserepeatedhalvingforallsharingproblems,butteachersshouldhelpstudentsdevelopotherstrate-giesforpartitioninganobject.Oneapproachistohavestudentsplacewoodensticksonconcreteshapes,withthesticksrepresentingtheslicesorcutsthatastudentwouldmaketopartitiontheobject.37

2. Extendequal-sharingactivitiestodevelopstudents’understandingoforderingandequivalenceoffractions.

Teacherscanextendthetypesofsharingactivitiesdescribedintheprevioussteptodevelopstudents’understandingoforderingandidentifyingequivalentfractions.Theover-allapproachremainsthesame:teachersposestoryproblemsthatinvolveagroupofpeople

sharingobjects,andstudentscreatedrawingsorotherrepresentationstosolvetheprob-lems.However,teachersusescenariosthatrequirefractioncomparisons oridentificationofequivalentfractionsandfocusondifferentaspectsofstudents’solutions.

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Sharingactivitiescanbeusedtohelpstudentsunderstandtherelativesizeoffractions.Teacherscanpresentsharingscenarioswithanincreasingnumberofrecipientsandhavestu-dentscomparetherelativesizeofeachresult-ingshare.Forexample,studentscancomparethesizeofpiecesthatresultwhensharingacandybarequallyamongthree,four,five,orsixchildren.38Teachersshouldencouragestu-dentstonoticethatasthenumberofpeoplesharingtheobjectsincreases,thesizeofeachperson’ssharedecreases;theyshouldthenlinkthisideatoformalfractionnamesandencour-agestudentstocomparethefractionalpiecesusingfractionnames(e.g.,1/3ofanobjectisgreaterthan1/4ofit).

Whenusingsharingscenariostodiscussequivalentfractions,teachersshouldconsidertwoapproaches,bothofwhichshouldbeusedwithscenariosinwhichthenumberofsharersandthenumberofpiecestobesharedhaveoneormorecommonfactors(e.g.,fourpizzassharedamongeightchildren):

•Partitionobjectsintolargerorsmallerpieces.Onewaytounderstandequivalentsharesistodiscussalternativewaystoparti-tionandreceivethesameshares.39Studentscanthinkabouthowtosolveasharingsce-nariousingdifferentpartitionstoproduceequalshares.Suchpartitioningmayrequiretrialanderroronthepartofstudentsto

identifywhichgroupingsresultinequalshares.Studentsmightcombinesmallerpiecestomakebiggeronesorpartitionbig-geronesintosmallerpieces.Forexample,tosolvetheproblemofeightchildrensharingfourpizzas,studentsmightpartitionallfourpizzasintoeighthsandthengiveeachchildfourpiecesofsize1/8.Alternatively,studentscoulddivideeachpizzaintofourthsandgiveeachperson2/4,ordivideeachpizzaintohalvesanddistribute1/2toeachchild.Studentsshouldunderstandthatalthoughtherearedifferentwaystopartitionthepizza,eachpartitioningmethodresultsinequivalentshares.

•Partitionthenumberofsharersandthenumberofitems.Anotherwaytohelpstudentsunderstandequivalenceistoparti-tionthenumberofsharersandobjects.40

Forexample,ifstudentsarriveat4/8fortheprobleminthepreviousparagraph,theteachercouldaskhowtheproblemwouldchangeifthegroupsplitintotwotablesandateachtablefourchildrensharedtwopiz-zas.Studentscancomparethenewsolutionof2/4totheiroriginalsolutionof4/8 toshowthatthetwoamountsareequivalent(seeFigure3).Todrivehomethepoint,theeightchildrencouldthensitatfourtables,withtwochildrenateachtablesharingasinglepizza—andreachingthemorefamiliarconceptof1/2.

Figure3.Studentworkforsharingfourpizzasamongeightchildren

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Recommendation 1 continuedRecommendation 1Recommendation 1Recommendation 1Recommendation 1 continuedcontinuedRecommendation1continued

Anotherwaytoteachequivalentfractionswithsharingscenariosistoposeamissing-valueprobleminwhichchildrendeterminethenum-berofobjectsneededtocreateanequivalentshare.Forexample,ifsixchildrenshareeightorangesatonetable,howmanyorangesareneededatatableofthreechildrentoensureeachchildreceivesthesameamount?41Theproblemcouldbeextendedtotableswith12children,24children,or9children.Tosolvetheseproblems,studentsmightidentifyhowmuchonechildreceivesinthefirstscenarioandapplythattothesecondscenario.Alter-natively,theycouldusethestrategydescribedaboveandpartitionthesixchildrenandeightorangesattheoriginaltableintotwotables,sothatthenumberofchildrenandorangesatthe

firstnewtableequalthenumberofchildrenandorangesatthesecondnewtable.

Hereisanotherexamplethatallowsstudentstoexploretheconceptofequalpartitioning:if24childrenaregoingoutforsandwiches,and16sandwicheshavebeenordered,whatarethedifferentwaysthechildrencouldsitattablesanddividethesandwichessotheywouldallreceivethesameamount?Optionsmightincludehavingonebigtableof24chil-drenand16sandwiches,havingfourtablesofsixchildrenandfoursandwichesateach,eighttablesofthreechildrenandtwosand-wichesateach,andsoon.

3.Buildonstudents’informalunderstandingtodevelopmoreadvancedunderstandingofproportional-reasoningconcepts.Beginwithactivitiesthatinvolvesimilarpropor-tions,andprogresstoactivitiesthatinvolveorderingdifferentproportions.

Earlyinstructioncanbuildonstudents’infor-malunderstandingtodevelopbasicconceptsrelatedtoproportionalreasoning.Teachersshouldinitiallyposeproblemsthatencour-agestudentstothinkabouttheproportionalrelationsbetweenpairsofobjects,withoutnecessarilyspecifyingexactquantities.Forexample,teacherscouldusethestoryofGoldilocks and the Three Bears todiscusshowthebigbearneedsabigchair,themedium-sizedbearneedsamedium-sizedchair,andthesmallbearneedsasmallchair.42

Thefollowinglistprovidesexamplesofdif-ferentrelationsrelevanttoearlyproportionalreasoningthatcanbeexploredwithstudents:

• Proportionalrelations.Teacherscandis-cussstoriesorscenariosthatpresentbasicproportionalrelationsthatarenotquanti-fied.Forexample,aclasscoulddiscussthenumberofstudentsitwouldtaketobalanceaseesawwithone,two,orthreeadultsononeend.Creatingmoreandlesssaturatedliquidmixtureswithlemonademixorfood

coloringcanfacilitatediscussionscompar-ingthestrengthorconcentrationofdiffer-entmixtures.

• Covariation.Teachersshoulddiscussproblemsthatinvolveonequantityincreasingasanotherquantityincreases.Examplescouldincludetherelationbetweenheightandclothingsizeorbetweenfootlengthandshoesize.43

• Patterns.Simplerepeatingpatternscanbeusefulfordiscussingtheconceptofratio.Forexample,studentscouldcom-pleteapatternsuchasbluestar,bluestar,redsquare,bluestar,bluestar,redsquare,bluestar,bluestar,redsquare,andsoon.44Teacherscanthendiscusshowmanybluestarsthereareforeveryredsquare,havestudentsarrangethestarsandsquarestoshowwhatgetsrepeated,havestudentschangethepat-terntoadifferentratio(e.g.,threebluestarstooneredsquare),orhavestudentsextendthepattern.45

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Potentialroadblocksandsolutions

Roadblock1.1. Students are unable to draw equalsize parts.

SuggestedApproach.Letstudentsknowthatitisacceptabletodrawpartsthatarenotexactlyequal,aslongastheyrememberthatthepartsshouldbeconsideredequal.

Roadblock1.2.Students do not share all of the items (nonexhaustive sharing) or do not create equal shares.

SuggestedApproach.Althoughchildrenhaveanintuitiveunderstandingofsharingsituations,theysometimesmakemistakesintheirattemptstosolvesharingproblems.Studentsmaynotsharealloftheitems,espe-ciallyifasharingscenariorequirespartition-inganobject.Teachersshouldhelpstudentsunderstandthatsharingscenariosrequiresharingalloftheobjects—possiblyevennotingthateachchildwantstoreceiveasmuchasheorshepossiblycan,sonoobjectsshouldremainunaccountedfor.

Studentsalsomightnotcreateequalsharesbecausetheydonotunderstandthatdeal-ingoutequal-sizeobjectsresultsinanequalamountforeachperson.46Inthiscase,teacherscandiscusshowdealingoutobjectsensuresthateachpersonreceivesanequalamountandcanencouragestudentstoverifythattheydividedtheitemsequally.

Equalsharingisimportantbecauseitlaysafoundationforlaterunderstandingofequiv-alentfractionsandequivalentmagnitudedifferences(e.g.,understandingthatthedif-ferencebetween0and1/2isthesameasthedifferencebetween1and11/2 orbetween73and731/2).

Roadblock1.3. When creating equal shares, students do not distinguish between the number of things shared and the quantity shared.

SuggestedApproach.Youngerstudentsinparticularmayconfuseequalnumbersofshareswithequalamountsshared.47Forexample,ifstudentsareaskedtoprovideequalamountsoffoodfromaplatewithbothbigandsmallpieces,achildmightgiveoutequalnumbersofpiecesoffoodratherthanequalamounts.Thismisunderstandingmaystemfromlimitedexperiencewithsituationsinwhichentitiesofdifferentsizesaredealtoutorshared.

Onewaytoaddressthismisconceptionistousecolorcuestohelpstudentsdistinguishbetweenthequantitybeingsharedandthenumberofitemsbeingshared.48Forexample,inascenarioinwhichbothoftwoidenticaltoydogsaresaidtobehungry,childrencouldbeaskedwhetherthedogswouldhavethesameamounttoeatifonedogreceivedfivelargeredpiecesofpretendfoodandtheotherdogfivesmallgreenpiecesofpretendfood.

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Recommendation2

Helpstudentsrecognizethatfractionsarenumbersandthattheyexpandthenumbersystembeyondwholenumbers.Usenumberlinesasacentralrepresentationaltoolinteachingthisandotherfractionconceptsfromtheearlygradesonward.Early fractions instruction generally focuses on the idea that fractions represent parts of a whole (e.g., onethird as the relation of one part to a whole that has three equal parts). Although the partwhole interpretation of fractions is important, too often instruction does not convey another simple but critical idea: fractions are numbers with magnitudes (values) that can be either ordered or considered equivalent.

Many common misconceptions—such as that two fractions should be added by adding the numerators and then adding the denominators—stem from not understanding that fractions are numbers with magnitudes. Not understanding this can even lead to confusion regarding whether fractions are numbers. For example, many students believe that fourthirds is not a number, advancing explanations such as, “You cannot have four parts of an object that is divided into three parts.”49 Further, many students do not understand that fractions provide a unit of measure that allows more precise measurement than whole numbers; these students fail to realize that an infinite range of numbers exists between successive whole numbers or between any two fractions.50 Reliance on partwhole instruction alone also leaves unclear how fractions are related to whole numbers.

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An effective way to develop students’ understanding of fractions as numbers with magnitudes is to use number lines. Number lines can clearly illustrate the magnitude of fractions; the relation between whole numbers and fractions; and the relations among fractions, decimals, and percents. They also provide a starting point for building students’ number sense with fractions and provide a way to represent negative fractions visually, which can otherwise be a challenging task. All of these types of understanding are crucial for learning algebra and other more advanced areas of mathematics.

Summaryofevidence:ModerateEvidence

Evidenceforthisrecommendationprimarilycomesfromstudiesdemonstratingtheuseful-nessofnumberlinesfordevelopingnumbersensewithwholenumbers.Thesestudiesusednumberlinerepresentationstoteachpreschoolandearlyelementarystudentsaboutthemagnitudesofwholenumbers.51Anadditionalstudyshowedhownumberlinescanbeusedtoteachdecimalssuccessfully.52AllofthesestudiesmetWWCevidencestandards.Moreover,accuracyinlocatingwholenumbersonnumberlinesisrelatedtomathematicalachievementamongstudentsinkindergartenthrough4thgrade,andaccuracyinlocatingdecimalsonnumberlinesisrelatedtoclass-roommathematicsgradesamong5th-and6th-graders.53Thepanelbelievesthatgiventheapplicabilityofnumberlinestofractionsaswellaswholenumbers,thesefindingsindicatethatnumberlinescanimprovelearningoffrac-tionsinelementaryandmiddleschool.

Numberlineswithwholenumbers.Playingalinearboardgamewithwholenumbersforaboutonehour(four15-minutesessionsoveratwo-weekperiod)improvedunderstandingofnumericalmagnitudesbypreschoolersfromlow-incomebackgrounds.54

Thegameinvolvedmovingamarkeroneortwospacesatatimeacrossahorizontalboardthathadthenumbers1to10listedinorderfromlefttorightinconsecutivesquares.Twoadditionalstudiesshowedthevalueofothernumberlineproceduresforimprovingknowledgeofwholenumbermagnitudes.Estimatingthelocationsof10numbersona0-to-100numberlineimproved1st-graders’abilitytolocatewholenumbersonthenumberline;55andshowing1st-grade

studentstheaddendsandsumsofadditionproblemsonanumberlineincreasedthelikelihoodthatstudentscorrectlyansweredtheproblemslater.

Numberlineswithdecimals.Inanotherstudy,numberlineswereusedtoteachdecimalconceptsto5th-and6th-gradestudents.56Theteachingtechniqueinvolvedprovidingstudentswithpracticelocatingdecimalsonanumberlinedividedintotenthsandwithaprompttonoticethetenthsdigitforeachnumber.Thesestudentswerelatermoreaccurateinlocatingdecimalsonanumberlinethanstudentswhosenumberlineswerenotdividedintotenthsanddidnotreceiveprompts.Forallstudentsinthestudy,abefore-and-aftercomparisonshowedthatconceptualunderstandingoffractionsimprovedafterlocatingdecimalsonanumberline.Thislastfindingissuggestiveevidence,becausethereisnocomparisongroupofstudentswhodidnotuseanumberline.

AnotherstudyexaminedaDutchcurriculumthatusednumberlinesandmeasurementcon-textstoteachfractions.57Studentsinthetreat-mentgrouplocatedandcomparedfractionsonanumberlineandmeasuredobjectsintheclassroomusingastripthatcouldbefoldedtomeasurefractionalparts.AlthoughthisstudydidnotmeetWWCevidencestandards,theauthorsreportedpositiveeffectsonmiddleschoolstudents’numbersensewithfrac-tions.58Twoadditionalstudiesthatwerenoteligibleforreviewfoundmixedresultsofusinganumberlinetoteachfractionconcepts.Bothstudiesnotedchallengesthatstudentsfaceinunderstandingfractionsonnumberlines.59

Forexample,onestudyreportedthatstudentshaddifficultyfindingequivalentfractionson

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anumberlinepartitionedintosmallerunits(e.g.,finding1/3onanumberlinedividedintosixths).60

Otherevidencethatisconsistentwiththerec-ommendationincludesastudyshowingtherelationbetweenskillatestimatinglocations

ofdecimalsonanumberlineandmathgradesfor5th-and6th-gradestudents,61

andamathematician’sanalysisindicatingthatlearningtorepresentthefullrangeofnumbersonnumberlinesisfundamentaltounderstandingnumbers.62


1. Usemeasurementactivitiesandnumberlinestohelpstudentsunderstandthatfrac-tionsarenumbers,withallthepropertiesthatnumbersshare.

Whenstudentsviewfractionsasnumbers,theyunderstandthatfractions,likewholenumbers,canbeusedtomeasurequantities.Measurementactivitiesprovideanaturalcon-textinthisregard.63Throughsuchactivities,teacherscandeveloptheideathatfractionsallowformoreprecisemeasurementofquan-titiesthandowholenumbers.

Teacherscanpresentsituationsinwhichfrac-tionsareusedtosolveproblemsthatcannotbesolvedwithwholenumbers.Forexample,theycanaskstudentshowtodescribetheamountofsugarinacookierecipethatneedsmorethan1cupbutlessthan2cups.

Teacherscanthenshowstudentsthevariousmeasurementlinesonameasuringcupandconveytheimportanceoffractionsindescrib-ingquantities.Teachersshouldemphasizethatfractionsprovideamorepreciseunitofmeasurethanwholenumbersandallowstudentstodescribequantitiesthatwholenumberscannotrepresent.Fractionstrips(alsoknownasfractionstripdrawings,stripdiagrams,barstripdiagrams,andtapedia-grams)arelengthmodelsthatallowstudentstomeasureobjectsusingfractionalpartsandreinforcetheideathatfractionscanbeusedtorepresentquantities(seeExample1).

Example1.Measurementactivitieswithfractionstrips

Teacherscanusefractionstripsasthebasisformea-surementactivitiestoreinforcetheconceptthatfrac-tionsarenumbersthatrepresentquantities.64

Tostart,studentscantakeastripofcardstockorconstructionpaperthatrepresentstheinitialunitofmeasure(i.e.,awhole)andusethatstriptomeasureobjectsintheclassroom(desk,chalkboard,book,etc.).Whenthelengthofanobjectisnotequaltoawholenumberofstrips,teacherscanprovidestudentswithstripsthatrepresentfractionalamountsoftheoriginalstrip.Forexample,astudentmightusethreewholestripsandahalfstriptomeasureadesk.

Teachersshouldemphasizethatfractionstripsrepre-sentdifferentunitsofmeasureandshouldhavestu-dentsmeasurethesameobjectfirstusingonlywholestripsandthenusingafractionalstrip.Teachersshoulddiscusshowthelengthoftheobjectremainsthesamebuthowdifferentunitsofmeasureallowforbetterprecisionindescribingit.Studentsshouldrealizethatthesizeofthesubsequentlypresentedfractionstripsisdefinedbythesizeoftheoriginalstrip(i.e.,ahalfstripisequaltoone-halfthelengthoftheoriginalstrip).

1/2 1/4

Usingfractionstripstomeasureanobject

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2.Provideopportunitiesforstudentstolocateandcomparefractionsonnumberlines.

Teachersshouldprovideopportunitiesforstudentstolocateandcomparefractionsonnumberlines.Theseactivitiesshouldincludefractionsinavarietyofforms,includingproperfractions(2/3),improperfractions(5/3),mixednumbers(12/3),wholenumbers(4/2),decimals(0.40),andpercents(70%).

Teacherscaninitiallyhavestudentslocateandcomparefractionsonnumberlineswiththefractionsalreadymarked(e.g.,anum-berlinewithmarksindicatingtenths).Pre-segmentednumberlinesavoidthedifficultystudentshaveinaccuratelypartitioningthenumberline.Thesenumberlinesalsoareusefulforlocatingandcomparingfractionswhoselocationsareindicated(e.g.,3/8and5/8 onanumberlinewitheighthsmarked)andfractionswhosedenominatorisafactoroftheunitfractionsshownonthenumberline(e.g.,1/4and3/4onalinewitheighthsmarked),aswellasfractionswithotherdenominators(e.g.,1/7,3/5).Forexample,studentsmightcomparethelocationsof7/8and3/4onanum-berlinemarkedwitheighths.Theseactivitiesshouldincludeopportunitiesforstudentstolocatewholenumbersonthenumberlineandcomparetheirlocationstothoseoffractions,includingonesequivalenttowholenumbers(e.g.,locating1and8/8).

Numberlinesalsocanbeusedtocomparefractionsofvaryingsizestowholenumbersgreaterthanone(locating10/3 onanumberlinewith0attheleftend,5attherightend,and1,2,3,and4markedinbetween).Exam-ple2providesastrategythatcanbeusedtointroducestudentstotheideaoflocatingfractionsonanumberline.

Comparingfractionswithdifferentdenomina-torsonapre-segmentednumberlinecanbecomplicatedforyoungstudents—forexample,comparing3/8and1/3onanumberlinedividedintoeighths.Tohelpstudentsunderstandsuchproblems,teacherscanlabelnumberlineswithonefractional-unitsequenceabovethenumberlineandadifferentfractional-unitsequence

Example2.Introducingfractionsonanumberline

Thefollowingexampledescribesonewaytointroducetheideaoflocatingfractionsonanumberline,emphasizingthatfractionsarenumberswithquantities.

Toillustratethelocationof3/5ona0-to-5numberline,theteachermightfirstmarkandlabelthelocationof1andthendividethespacebetweeneachwholenumberintofiveequal-sizeparts.Afterthis,theteachermightaddthelabels0/5,1/5,2/5,3/5,4/5,and5/5inthe0–1partofthenumberlineandhighlightthelocationof3/5.65Displayingwholenumbersasfractions(e.g.,5/5)allowsteacherstodiscusswhatitmeanstodescribewholenumbersintermsoffractionsandtoclarifythatwholenumbersarefractionstoo.

belowthenumberline.Forexample,whenaskingstudentstocompare1/3and3/8,teachersmightlabeleighthsabovethenumberlineandthirdsbelowit.Suchnumberlinesallowstu-dentswhoarerelativelyearlyintheprocessoflearningaboutfractionstolocateandcomparefractionswithdifferentdenominatorsandtothinkabouttherelativesizeofthefractions.66

Teachersalsoshouldprovidestudentswithopportunitiestolocateandcomparefractionsonnumberlinesthatareminimallylabeled—forexample,oneswiththelabels0,1/2,1,11/2,and2.Thisapproachisalmostaneces-sityforfractionswithlargedenominators(e.g.,dividinganumberlineinto28thsisdif-ficult)andencouragesstudentstothinkaboutthelocationoffractionsrelativetothelabeledlandmarks.67Forexample,teacherscanhavestudentslocate6/7onanumberlinemarkedwith0,1/2,and1.

Forawhole-classactivity,teacherscandrawanumberlineontheboardandhavestudentsmarkestimatesofwheredifferentfractions

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The following example describes one way to introduce the idea of locating fractions on a number line, emphasizing that fractions are numbers with quantities.

To illustrate the location of 3/5 on a 0-to-5 number line, the teacher might first mark and label the location of 1 and then divide the space between each whole number into five equal-size parts. After this, the teacher might add the labels 0/5, 1/5, 2/5, 3/5, 4/5, and 5/5 in the 0–1 part of the number line and highlight the location of 3/5.65 Displaying whole numbers as fractions (e.g., 5/5) allows teachers to discuss what it means to describe whole numbers in terms of fractions and to clarify that whole numbers are fractions too.

Recommendation 2 continued


fall.Asthenumberlinefillsup,teacherscanguideadiscussionaboutfractionsyettobeplaced,highlightingtheneedtopreservethecorrectorder.Insertingdecimalsandpercent-agesonthatsamenumberlinecanteachadditionalvaluablelessons.

Finally,teachersshouldencouragestudentstothinkaboutthedistancebetweentwofractions:Forexample,studentscouldcompare1/12and1/4andconsiderwhether1/12iscloserto1/4 or0.Similarly,0.3or0.45couldbecomparedtolocationsmarked0,1/2,and1,or0,0.5,and1.

3.Usenumberlinestoimprovestudents’understandingoffractionequivalence,frac-tiondensity(theconceptthatthereareaninfinitenumberoffractionsbetweenanytwofractions),andnegativefractions.

Inadditiontobeingusefulforcomparingposi-tivefractionmagnitudes,numberlinesalsocanbevaluableforteachingequivalentfractions,negativefractions,andfractiondensity.Numberlinesare,ofcourse,nottheonlywaytoteachtheseconcepts,butthepanelbelievestheyarehelpfulforimprovingstudents’understanding.

Numberlinescanbeusedtoillustratethatequivalentfractionsdescribethesamemagni-tude.Forexample,askingstudentstolocate2/5 and4/10 onasinglenumberlinecanhelpthemunderstandtheequivalenceofthesenumbers.Teacherscanmarkfifthsabovethelineandtenthsbelowit(orviceversa)tohelpstudentswiththistask.Althoughviewingequivalentfractionsasthesamepointonanumberlinecanbechallengingforstudents,68thepanelbelievesthattheabilitytodosoiscriticalforthoroughunderstandingoffractions.

AdiscussionofequivalentfractionsshouldbuildonpointsmadeinStep1aboutfractionsonthenumberline.Forexample,teacherscandividea0-to-1numberlineintohalvesandquartersandshowthat1/2and2/4occupythesame,orequivalent,pointonthenumberline(seeFigure4).Studentscanusearulertoiden-tifyequivalentfractionsonthestackednumberlinesshowninFigure4,identifyingfractionsthatoccupythesamelocationoneachnum-berline.Fractionstripsalsocanbeusedtoreinforcetheconceptofequivalentfractionsbyallowingstudentstomeasurethedistancebetweentwopointsusingdifferent-sizedfrac-tionstrips(seeFigure5).

Numberlinesalsocanbeusedtohelpstu-dentsunderstandthataninfinitenumberoffractionsexistbetweenanytwootherfrac-tions.Thisisonewayinwhichfractionsdiffer

Figure4.Findingequivalentfractionsonanumberline

UseofnumberlinestoteachequivalenceoffractionsinaJapanesecurriculum

Source:AdaptedfromShoseki(2010).

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Figure5.Usingfractionstripstodemonstrateequivalentfractions

0 2

1/2 1/2 1/2 1/2

1/2 1/2+ 1/2+ 1/2+ 4/2=

1/4 1/4 1/4 1/4 8/4+ + +1/4 1/4 1/4 1/4+ + + + =

8/44/2= =2

1/4 1/4 1/4 1/4 1/4 1/4 1/4 1/4

fromwholenumbersandcanbeadifficultconceptforstudentstograsp.69Teacherscanhelpstudentsunderstandthisconceptbyaskingthemtomakesuccessivepartitionsonthenumberline,creatingsmallerandsmallerunitfractions.70Forexample,studentscandividewholenumbersegmentsinhalftocreatehalves,andthendivideeachhalfintohalvestocreatefourths,thendivideeachfourthintohalvestocreateeighths,andsoon(thisactivityalsocanbedonewiththirds,ninths,twenty-sevenths,etc.).Suchdivisionsshowstudentsthattheyalwayscanpartitionanumberlineusingsmallerunitfractions.71

Thesamecanbedonewithdecimalsandpercents—suchasbyshowingthat0.13,0.15,and0.17areamongtheinfinitenumbersthatfallbetween0.1and0.2,andthat2%fallsbetween0%and10%.

Thepanelfurtherrecommendsthatteachersusenumberlineswhenintroducingnegativefractions.Teachingnegativefractionsinapart-wholecontextcanbedifficult,becausetheideaofanegativepartofawholeisnon-intuitive.Butthenumberlineprovidesastraightforwardvisualrepresentationoffractionslessthanzero,aswellasfractionsgreaterthanzero.

Byprovidingnumberlinesthatincludemarksandlabelsforzero,forseveralpositivefrac-tions,andforseveralnegativefractionswiththesameabsolutevaluesasthepositivefrac-tions,teacherscanhelpconveythesymmetryaboutzeroofpositiveandnegativefractions.Andbyplacingpositiveandnegativefractionsintostories—possiblyaboutlocationsaboveandbelowsealeveloraboutmoneygainedorlost—teacherscanillustrateadditionandsubtractionofbothtypesoffractions.

4.Helpstudentsunderstandthatfractionscanberepresentedascommonfractions,deci-mals,andpercentages,anddevelopstudents’abilitytotranslateamongtheseforms.

Studentsneedabroadviewoffractionsasnumbers.Thatincludesunderstandingthatfractionscanberepresentedasdecimalsandpercentsaswellascommonfractions.Teachersshouldclearlyconveythatcommonfractions,decimals,andpercentsarejustdif-ferentwaysofrepresentingthesamenumber.

Numberlinesprovideausefultoolforhelpingstudentsunderstandthatfractions,decimals,andpercentsaredifferentwaysofdescrib-ingthesamenumber.Byusinganumberline

withcommonfractionslistedaboveitanddecimalsorpercentagesbelowit,teacherscanhelpstudentslocateandcomparefrac-tions,decimals,andpercentsonthesamenumberline.Forexample,teacherscanpro-videstudentswithanumberlinemarkedwith0and1,andstudentscanbeaskedtolocate3/4,0.75,and75%onit.Inaddition,whenstu-dentsusedivisiontotranslateafractionintoadecimal,theycanplotboththefractionandthedecimalonthesamenumberline.

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Roadblock2.1.Students try to partition the number line into fourths by drawing four hash marks rather than three, or they treat the whole number line as the unit.72

SuggestedApproach.Whenusinganumberlinewithfractions,studentsmustbetaughttorepresentfourthsasfourequal-sizesegmentsbetweentwowholenumbers.Teachersshoulddemonstratethatinsertingthreeequallyspacedhashmarksbetween,say,0and1dividesthespaceintofourequalsegments,orfourths.Thisrulecanbegeneralizedsothatstudentsknowthatdividingthenumberlineinto1/nunitsrequiresdrawingn –1hashmarksbetweentwowholenumbers.

Roadblock2.2. When students locate fractions on the number line, they treat the numbers in the fraction as whole numbers (e.g.,placing 3/4between 3 and 4).

SuggestedApproach.Thismistakereflectsacommonmisconceptioninwhichstudentsapplytheirwholenumberknowledgetofractions—viewingthenumbersthatmakeupafractionasseparatewholenumbers.Themisconceptioncanbeaddressedbypresentingstudentswithcontrastingcases:forexample,havingthemlocate3and4ona0-to-4numberline,thenidentifying3/4asafractionbetween0and1,andfinallydiscuss-ingwhyeachfractiongoeswhereitisplaced.

Roadblock2.3. Students have difficulty understanding that two equivalent fractions are the same point on a number line.

SuggestedApproach.Studentsoftenhavetroubleinternalizinghowpartitionsthatlocateonefraction(e.g.,eighthspartitionsforlocat-ing4/8)alsocanhelplocateanequivalent

fraction(e.g.,1/2).Onewaytoaddressthislackofunderstandingistoshowstudentsonesetofnumericallabelsabovethenumberlineandanothersetoflabelsbelowit.Thus,halvescouldbemarkedjustabovethelineandeighthsjustbelowit,andteacherscouldpointouttheequivalentpositionsof1/2and4/8,of1and8/8,of11/2and12/8,andsoon.Anotherapproachisforstudentstocreateanumberlineshowing1/2andanothernumberlineshowing4/8andthencomparethetwo.Teach-erscanlineupthetwonumberlinesandleadadiscussionaboutequivalentfractions.

Roadblock2.4. The curriculum materials used by my school district focus on partwhole representations and do not use the number line as a key representational tool for fraction concepts and operations.

SuggestedApproach.Althoughitisimpor-tantforstudentstounderstandthatfractionsrepresentpartsofawhole,thepanelnotesthatthisisonlyoneuseoffractionsandthereforerecommendstheuseofnumberlinesandmeasurementcontextstodevelopacomprehensiveunderstandingoffractions.Manipulativesthatoftenareusedtorepresentpart-wholeinterpretations,suchasfractioncirclesandfractionstrips,alsocanbeusedtoconveymeasurementinterpretations,butconsiderablecareneedstobetakentoavoidstudentssimplycountingpartsofthefractionstriporcirclethatcorrespondtothenumera-torandtothedenominatorwithoutunder-standinghowthenumeratoranddenominatortogetherindicateasinglequantity.Usingnumberlinesthatareunmarkedbetweentheendpointscanavoidsuchcountingwithoutunderstanding.Sometextbooksusenumberlinesextensivelyforteachingfractions;teach-ersshouldexaminethosebooksforideasabouthowtousenumberlinestoconveytheideathatfractionsaremeasuresofquantity.

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Recommendation3

Helpstudentsunderstandwhyproceduresforcomputationswithfractionsmakesense.Students are most proficient at applying computational procedures when they understand why those procedures make sense. Although conceptual understanding is foundational for the correct use of procedures, students often are taught computational procedures with fractions without an adequate explanation of how or why the procedures work.

Teachers should take the time to provide such explanations and to emphasize how fraction computation procedures transform the fractions in meaningful ways. In other words, they should focus on both conceptual understanding and procedural fluency and should emphasize the connections between them. The panel recommends several practices for developing understanding of computational procedures, including use of visual representations and estimation to reinforce conceptual understanding. Addressing students’ misconceptions and setting problems in realworld contexts also can contribute to improved understanding.

Summaryofevidence:ModerateEvidence

Thepanelbasedthisrecommendationinlargepartonthreewell-designedstudiesthatdemonstratedtheeffectivenessofteachingconceptualunderstandingwhendevelopingstudents’computationalskillwithfractions.73

Thesestudiesfocusedondecimalsandwererelativelysmallinscale;however,thepanelbelievesthattheirresults,togetherwithextensiveevidenceshowingthatmeaningful

informationisrememberedmuchbetterthanmeaninglessinformation,providepersuasiveevidenceforthisrecommendation.74 Addi-tionalsupportfortherecommendationcomesfromfourstudiesthatshowedapositiverela-tionbetweenconceptualandcomputationalknowledgeoffractions.75

Thestudiesthatcontributedtotheevidencebaseforthisrecommendationusedcomputer-basedinterventionstoexaminethelink

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betweenconceptualknowledgeandcomputa-tionalskillwithdecimals.Sixth-gradestudentscompletedthreelessonsondecimalplacevalue(i.e.,conceptualknowledge)andthreelessonsonadditionandsubtractionofdecimals(i.e.,proceduralknowledge).76Iteratingbetweenthetwotypesoflessonsimprovedstudents’proceduralknowledge,comparedwithteach-ingalloftheconceptuallessonsbeforeanyoftheproceduralones.Inanotherstudy,5th-and6th-gradestudentspracticedlocatingdecimalsonanumberlineusingacomputer-basedgame.Dividingthenumberlineintotenthsandencouragingstudentstonoticethetenthsdigitimproved5th-and6th-gradestudents’abilitytolocatedecimalsonanumberline(comparedtonotprovidingtheprompts).77

Researchalsoshowsapositiverelationshipbetweenstudents’conceptualandproceduralknowledgeoffractions.Thatis,childrenwhohaveabove-averageconceptualknowledgealsotendtohaveabove-averageknowledgeofcomputationalprocedures.Studiesof4th-and5th-gradersandof7th-and8th-gradersindi-catedthatconceptualknowledgewaspositivelyrelatedtocomputationalproficiencyaftercon-trollingforpriormathachievement,arithmeticfluency,workingmemory,andreadingability.78

Inaddition,conceptualknowledgeofdecimalspredictedstudents’abilitytolocatedecimalsonanumberline.79Whilethesestudiesshowacorrelationbetweenconceptualandproceduralknowledge,theydidnotexaminetheeffective-nessofinterventionsthatdevelopconceptualknowledgetoimproveproceduralknowledge.

Thepanelalsoidentifiedevidencethatspe-cificallyaddressedtwoofthefourstepsforimplementingthisrecommendation.

Useofrepresentations.Evidenceidenti-fiedbythepanelsupportstherecommended

practiceofusingvisualrepresentationsandmanipulativesduringinstructiononfractioncomputation(Step1).Twowell-designedstud-iesfoundthattheuseofmanipulativesandpictorialrepresentationshadapositiveeffectoncomputationalskillwithfractions.80Oneofthesestudiesfocusedonfractioncircles(setsofcircles,inwhichthefirstisawholecircle,thesecondisdividedinhalf,thethirdisdividedinthirds,etc.).81Theotherstudyhadstudentsuseavarietyofmanipulativesforlearningcomputationalprocedureswithfrac-tions,includingfractionsquaresandfractionstrips.82AthirdstudyexaminedtheRationalNumberProjectcurriculum,whichempha-sizestheuseofmanipulativesasoneofmanycomponents.83Theauthorsofthestudyreportedthatthecurriculumhadapositiveeffectonfractioncomputationabilities.How-ever,manipulativeswereonlyonecomponentofthismultifacetedcurriculum,andthestudyprovidedinsufficientinformationfortheWWCtocompleteareview,sotheconclusionsthatcanbedrawnfromthestudyregardingtheroleofmanipulativesarelimited.

Realworldcontexts.Thepanelidentifiedevidencerelatedtotheuseofreal-worldcon-textsforimprovingskillatexecutingcompu-tationalprocedureswithfractions(Step4).84

Inoneofthestudies,personalizingproblemsfor5th-and6th-gradestudentsimprovedtheirabilitytosolvedivisionproblemswithfractions.85Theotherstudyfoundthatpos-ingproblemsineverydaycontextsimproved11- and12-year-oldstudents’abilitytoorderandcomparedecimals.86Additionalstudiesarguedfortheuseofreal-worldcontextsforteachingproceduresforcomputingwithfrac-tionsbutdidnotproviderigorousevidencethatsuchinstructioncausesimprovementinfractioncomputation.87

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1. Useareamodels,numberlines,andothervisualrepresentationstoimprovestu-dents’understandingofformalcomputationalprocedures.

Teachersshouldusevisualrepresentationsandmanipulatives,includingnumberlinesandareamodels,thathelpstudentsgaininsightintobasicconceptsunderlyingcom-putationalproceduresandthereasonswhytheseprocedureswork.Forexample,whenteachingadditionorsubtractionoffractionswithunlikedenominators,teachersshouldusearepresentationthathelpsstudentsseetheneedforcommondenominators.

Thereareseveralwaysteacherscanuserepresentationstoilluminatekeyunderlyingconcepts:

•Findacommondenominatorwhenaddingandsubtractingfractions.Acommonmistakestudentsmakewhenfacedwithfractionsthathaveunlikedenominatorsistoaddbothnumeratorsanddenominators.88Certainrepresenta-tionscanprovidevisualcuestohelpstudentsseetheneedforcommon

denominators.Forexample,teacherscandemonstratethatwhenaddingpiecescorrespondingtofractionsofobjects(e.g.,adding1/2ofacircleand1/3ofacircle),convertingboth1/2and1/3tosixthspro-videsacommondenominatorthatappliestobothfractionsandallowsthemtobeadded(Figure6).Discusswithstudentswhymultiplyingdenominatorsalwaysindicatesacommondenominatorthatcanbeusedtoexpressbothoriginalfractions.

•Redefinetheunitwhenmultiplyingfractions.Multiplyingtwofractionsrequiresfindingafractionofafraction.Forexample,whenmultiplying1/4by2/3,studentscouldstartwith2/3oftheoriginal(usuallyunmentioned)unitandfind1/4 ofthisfractionalamount.Pictorialorconcreterepresentationscanhelpstudentsvisual-izethisprocesstoimprovetheirunder-standingofthemultiplicationprocedure.Forexample,studentscanshadeinwith

Figure6.Fractioncirclesforadditionandsubtraction

1/6

1/6

1/6

1/2 1/2

1/3

1/6

1/6

5/6

+

+

=

=

1/3

1/6

1/61/6

1/6

1/6

Adding1/2+1/3usingfractioncircles

Source:AdaptedfromCramerandWyberg(2009).

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Figure7.Redefiningtheunitwhenmultiplyingfractions

Loriisicingacake.Sheknowsthat1cupoficingwillcover2/3ofacake.Howmuchcakecanshecoverwith1/4cupoficing?

verticallines2/3 ofasquarecakedrawnonpaperandthenshadeinwithhorizontallines1/4 ofthecake’sshadedarea,resultinginaproductrepresentedbythecross-hatchedarea(Figure7).89Thisapproachillustrateshowtoredefinetheunit—initiallytreatingthefullcakeasthewhole,andthentreatingtheverticallyshadedportionofthecakeasthewhole.

• Divideanumberintofractionalparts.Dividingfractionsisconceptuallysimilartodividingwholenumbers,inthatstudentscanthinkabouthowmanytimesthedivi-sorgoesintothedividend.Forexample,1/2÷1/4canberepresentedintermsof“Howmany1/4sarein1/2?”

Teacherscanuserepresentationssuchasribbonsoranumberlinetohelpstudentsmodelthedivisionprocessforfractions.Studentsusingribbonscancuttworib-bonsofequalsizeandthenseparateoneintofourthsandoneintohalves.Toshowthedivisionproblem1/2÷1/4,studentscan

findouthowmanyfourthsofaribbonfitontoone-halfofaribbon,whenthewholeribbonwasthesamelengthinbothcases(seeFigure8).90Similarly,ateachercandrawanumberlinewithbothfourthsandhalveslabeledtoshowstudentsthattherearetwo1/4segmentsin1/2.Teacherscanhelpstudentsdeepentheirunderstand-ingofthedivisionprocessbypresentingproblemsinwhichthedivisor,dividend,orbotharegreaterthanone,andprob-lemsinwhichthequotientisnotaninte-ger,suchas13/4dividedby1/2.

Teachersshouldconsidertheadvantagesanddisadvantagesofdifferentrepresentationsforteachingproceduresforcomputingwithfractions.Akeyissueiswhethertherepre-sentationadequatelyreflectsthecomputationprocessbeingtaught,allowingstudentstomakelinksbetweenthetwo.

Teachersalsoshouldthinkaboutwhetherarepresentationcanbeusedwithdifferenttypesoffractions—properfractions(5/8),

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Figure8.Usingribbonstomodeldivisionwithfractions

¼ ¼ ¼ ¼

½ ½

¼ ¼

½

Step 1. Divide a ribbon into fourths.

Step 2. Divide a ribbon of the same length into halves.

Step 3. Find out how many fourths of a ribbon can fit into one-half of the ribbon.

Two fourths fit into one-half of the ribbon.

So, ½÷ ¼= 2.

Students use ribbons to solve ½÷ ¼

mixednumbers(13/8),improperfractions(11/8),andnegativenumbers(–1/2).Forexam-ple,areamodelsmayreadilyillustrateaddi-tionoffractionswithpositivenumbersbutdonotaseasilylendthemselvestoexplainingadditionoffractionswithnegativenumbers.Incontrast,numberlinescanbeusedtoexplainboth.

Representationsthatstudentshaveusedtolearnothermathematicalconcepts,especially

otherfractionconcepts,maybeparticularlyuseful.Forexample,manystudentslearntorepresentdecimalsusingbase-10blocksor100grids(10by10squares,witheachsquarerepresenting1/100andthewholesquarerepre-senting1).Familiaritywiththisrepresentationalsomighthelpstudentsunderstandaddingandsubtractingdecimalandcommonfrac-tions.Forexample,100gridscanbeusedtoillustratethatadding2.34+1.69isthesameasadding234/100+169/100.

2. Provideopportunitiesforstudentstouseestimationtopredictorjudgethereason-ablenessofanswerstoproblemsinvolvingcomputationwithfractions.

Whenteachingproceduresforcomputingwithfractions,teachersshouldprovideopportunitiesforstudentstoestimatethesolutionstoproblems.Estimationrequiresstudentstousereasoningskillsandthusleadsthemtofocusonthemeaningofproce-duresforcomputingwithfractions.91 Teach-erscanaskstudentstoprovideaninitialestimateandtoexplaintheirthinkingbefore

havingthemcomputetheanswer.92Students,inturn,canusetheestimatestojudgethereasonablenessoftheiranswers.

Toimproveestimationskills,teacherscandiscusswhetherandwhystudents’solutionstospecificproblemsarereasonable;theyalsocanaskstudentstoexplainthestrate-giestheyusedtoarriveattheirestimatesand

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comparetheirinitialestimatestothesolutionstheyreachedbyapplyingacomputationalalgorithm.Consideranexample:astudentmightestimatethatthesolutionof1/2+1/5 ismorethan1/2butlessthan3/4,since1/5 issmallerthan1/4.Ifthestudentthenincorrectlyaddsthenumeratorsanddenominatorstoproducethesum2/7,theteachercannotethatthisanswercannotberightbecause2/7islessthan1/2.93Fromthere,theteachercanguidethestudenttoidentify,understand,andcor-recttheproceduralerror.

Estimationislikelytobemostusefulwithprob-lemsinwhichasolutioncannotbecomputedquicklyoreasily.Thereisnopointaskingstu-dentstoestimatetheanswertoaproblemthatcanbesolvedquicklyandaccuratelybymentalcomputation,suchas7/9–5/9.

Teachingstudentseffectiveestimationstrate-gies(Example3)canmaximizethevalueofestimationfordeepeningunderstandingofcomputationsinvolvingfractions.

Example3.Strategiesforestimatingwithfractions

Strengtheningestimationskillscandevelopstudents’understandingofcomputationalprocedures.

Benchmarks.Onewaytoestimateisthroughbenchmarks—numbersthatserveasreferencepointsforesti-matingthevalueofafraction.94Thenumbers0,1/2,and1areusefulbenchmarksbecausestudentsgenerallyfeelcomfortablewiththem.Studentscanconsiderwhetherafractionisclosestto0,1/2,or1.Forexample,whenadding7/8 and3/7,studentsmayreasonthat7/8 iscloseto1,and3/7 iscloseto1/2,sotheanswerwillbecloseto11/2.95Further,ifdividing5by5/6,studentsmightreasonthat5/6iscloseto1,and5dividedby1is5,sothesolutionmustbealittlemorethan5.96

RelativeSizeofUnitFractions.Ausefulapproachtoestimatingisforstudentstoconsiderthesizeofunitfractions.Todothis,studentsmustfirstunderstandthatthesizeofafractionalpartdecreasesasthedenominatorincreases.97Forexample,toestimatetheanswerto9/10+1/8,beginningstudentscanbeencour-agedtoreasonthat9/10 isalmost1,that1/8iscloseto1/10,andthatthereforetheanswerwillbeabout1.Moreadvancedstudentscanbeencouragedtoreasonthat9/10 isonly1/10 awayfrom1,that1/8 isslightlylargerthan1/10,andthereforethesolutionwillbeslightlymorethan1.Theprinciplecanandshouldbegeneral-izedbeyondunitfractionsonceitisunderstoodinthatcontext.Keydimensionsforgeneralizationincludeestimatingresultsofoperationsinvolvingnon-unitfractions(e.g.,3/4÷2/3),improperfractions(7/3÷3/4),anddecimals(0.8÷0.33).

PlacementofDecimalPoint.Acommonerrorwhenmultiplyingdecimals,suchas0.8×0.9or2.3×8.7,istomisplacethedecimal.Encouragingstudentstoestimatetheanswerfirstcanreducesuchconfusion.Forexample,realizingthat0.8and0.9arebothlessthan1butfairlyclosetoitcanhelpstudentsrealizethatanswerssuchas0.072and7.2mustbeincorrect.

3. Addresscommonmisconceptionsregardingcomputationalprocedureswithfractions.

Misconceptionsaboutfractionsofteninterferewithunderstandingcomputationalprocedures.Thepanelbelievesthatitiscriticaltoidentifystudentswhoareoperatingwithsuchmiscon-ceptions,todiscussthemisconceptionswiththem,andtomakecleartothestudentswhythemisconceptionsleadtoincorrectanswersandwhycorrectproceduresleadtocorrectanswers.

Teacherscanpresentthesemisconceptionsindiscussionsabouthowandwhysomestu-dents’computationproceduresyieldcorrectanswers,whereasothers’donot.Thegroupwilllikelyfindthatmanycomputationalerrorsresultfromstudentsmisapplyingrulesthatareappropriatewithwholenumbersorwithothercomputationaloperationswithfractions.

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Somecommonmisconceptionsaredescribednext,togetherwithrecommendationsforaddressingthem.

•Believingthatfractions’numeratorsanddenominatorscanbetreatedasseparatewholenumbers.Acommonmistakethatstudentsmakeistoaddorsubtractthenumeratorsanddenomina-torsoftwofractions(e.g.,2/4+5/4=7/8 or3/5–1/2=2/3).98Studentswhoerrinthiswayaremisapplyingtheirknowledgeofwholenumberadditionandsubtractiontofrac-tionproblemsandfailingtorecognizethatdenominatorsdefinethesizeofthefrac-tionalpartandthatnumeratorsrepresentthenumberofthispart.Thefactthatthisapproachisappropriateformultiplicationoffractionsisanothersourceofsupportforthemisconception.

Presentingmeaningfulproblemscanbeusefulforovercomingthismisconception.Forexample,ateachermightpresenttheproblem,“Ifyouhave3/4ofanorangeandgive1/3ofittoafriend,whatfractionoftheoriginalorangedoyouhaveleft?”Subtract-ingthenumeratorsanddenominatorsseparatelywouldresultinananswerof2/1 or2.Studentsshouldimmediatelyrecog-nizetheimpossibilityofstartingwith3/4 ofanorange,givingsomeofitaway,andend-ingupwith2oranges.Suchexamplescanmotivatestudentstothinkdeeplyaboutwhytreatingnumeratorsanddenominatorsasseparatewholenumbersisinappropriateandcanleadthemtobemorereceptivetodiscussionsofappropriateprocedures.

•Failingtofindacommondenominatorwhenaddingorsubtractingfractionswithunlikedenominators.Studentsoftenfailtoconvertfractionstoequivalentformswithacommondenominatorbeforeaddingorsubtractingthem,andinsteadjustinsertthelargerdenominatorinthefractionsintheproblemasthedenomina-torintheanswer(e.g.,4/5+4/10=8/10).99

Thiserroroccurswhenstudentsdonotunderstandthatdifferentdenominatorsreflectdifferent-sizedunitfractionsandthat

addingandsubtractingfractionsrequiresacommonunitfraction(i.e.,denominator).Thesameunderlyingmisconceptioncanleadstudentstomakethecloselyrelatederrorofchangingthedenominatorofafractionwithoutmakingthecorrespondingchangetothenumerator—forexample,byconvertingtheproblem2/3+2/6into2/6+2/6.Visualrepresentationsthatshowequivalentfractions—suchasanumberlineorfractionstrip—againcanillustratetheneedforbothcommondenominatorsandappropriatechangesinnumerators.

•Believingthatonlywholenumbersneedtobemanipulatedincomputationswithfractionsgreaterthanone.Whenaddingorsubtractingmixednumbers,studentsmayignorethefractionalpartsandworkonlywiththewholenumbers(e.g.,53/5–21/7=3).100 Thesestudentsareeitherignoringthepartoftheproblemtheydonotunderstand,misunderstandingthemeaningofmixednumbers,orassumingthatsuchproblemssimplyhavenosolution.101

Arelatedmisconceptionisthinkingthatwholenumbershavethesamedenomina-torasafractionintheproblem.102 Thismisconceptionmightleadstudentstotranslatetheproblem4–3/8into4/8–3/8 andfindananswerof1/8.Whenpresentedwithamixednumber,studentswithsuchamisconceptionmightaddthewholenumbertothenumerator,asin31/3 ×6/7 =(3/3+1/3)×6/7=4/3 ×6/7=24/21.Helpingstudentsunderstandtherelationbetweenmixednumbersandimproperfractions,andhowtotranslateeachintotheother,iscrucialforworkingwithfractions.

•Treatingthedenominatorthesameinfractionadditionandmultiplicationproblems.Studentsoftenleavethedenominatorunchangedonfractionmultiplicationproblemsthathaveequaldenominators(e.g.,2/3 ×1/3=2/3).103Thismayoccurbecausestudentsusuallyencountermorefractionadditionproblemsthanfractionmultiplicationproblems;thismightleadthemtogeneralizeincorrectly

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tomultiplicationthecorrectprocedurefordealingwithequaldenominatorsonaddi-tionproblems.Teacherscanaddressthismisconceptionbyexplainingtheconceptualbasisoffractionmultiplicationusingunitfractions(e.g.,1/2 ×1/2=halfofahalf=1/4).Inparticular,teacherscanshowthattheproblem1/2 ×1/2isactuallyaskingwhat1/2of1/2 is,whichimpliesthattheproductmustbesmallerthaneitherfractionbeingmultiplied.

•Failingtounderstandtheinvertandmultiplyprocedureforsolvingfractiondivisionproblems.Studentsoftenmisapplytheinvert-and-multiplyproce-durefordividingbyafractionbecausetheylackconceptualunderstandingoftheprocedure.Onecommonerrorisnotinvertingeitherfraction;forexample,astudentmaysolvetheproblem2/3÷4/5 bymultiplyingthefractionswithoutinverting4/5(e.g.,writingthat2/3÷4/5=8/15).104 Othercommonmisapplicationsoftheinvert-and-multiplyruleareinvertingthewrongfraction(e.g.,2/3÷4/5=3/2 ×4/5)orinvert-ingbothfractions(2/3÷4/5=3/2 ×5/4).Sucherrorsgenerallyreflectalackofconcep-tualunderstandingofwhytheinvert-and-multiplyprocedureproducesthecorrect

quotient.Theinvert-and-multiplyproce-duretranslatesamulti-stepcalculationintoamoreefficientprocedure.

Thepanelsuggeststhatteachershelpstudentsunderstandthemulti-stepcalcu-lationthatisthebasisfortheinvert-and-multiplyprocedure.Teacherscanbeginbynotingthatmultiplyinganynumberbyitsreciprocalproducesaproductof1,andthatdividinganynumberby1leavesthenumberunchanged.Thenteacherscanshowstudentsthatmultiplyingbothfractionsbythereciprocalofthedivisorisequivalenttousingtheinvert-and-multiplyprocedure.Fortheproblem2/3÷4/5=(notethatwereferto2/3asthedividendand4/5 asthedivisor):

•multiplyingboththedividend(2/3)anddivisor(4/5)bythereciprocalofthedivisoryields(2/3 ×5/4)÷(4/5 ×5/4).

•multiplyingtheoriginaldivisor(4/5)byitsreciprocal(5/4)producesadivisorof1,whichresultsin2/3 ×5/4 ÷1,whichyields2/3 ×5/4.

• thus,theinvertandmultiplyprocedure,multiplying2/3 ×5/4,providesthesolution.

4. Presentreal-worldcontextswithplausiblenumbersforproblemsthatinvolvecom-putingwithfractions.

Presentingproblemswithplausiblenumberssetinreal-worldcontextscanawakenstu-dents’intuitiveproblem-solvingabilitiesforcomputingwithfractions.105Thecontextsshouldprovidemeaningtothefractionquan-titiesinvolvedinaproblemandthecomputa-tionalprocedureusedtosolveit.Real-worldmeasuringcontexts,suchasrulers,ribbons,andmeasuringtapes,canbeuseful,ascanfood—bothdiscreteitems(e.g.,cartonsofeggs,boxesofchocolates)andcontinuousones(e.g.,pizzas,candybars).106 Studentsthemselvescanbeahelpfulsourceofideasforrelevantcontexts,allowingteacherstotailorproblemsarounddetailsthatarefamil-iarandmeaningfultothestudents.107 School

events,suchasfieldtripsorclassparties,trackandfielddays,andongoingactivitiesinothersubjects,alsocanserveasengagingcontextsforproblems.

Teacherscanhelpstudentsmakeconnec-tionsbetweenareal-worldproblemandthefractionnotationusedtorepresentit.Insomecases,studentsmaysolveaproblemframedinaneverydaycontextbutbeunabletosolvethesameproblemusingformalnotation.108 Forinstance,theymightknowthattwohalvesequalawholebutanswerthewrittenproblem1/2+1/2 with2/4.Teach-ersshouldhelpstudentsseetheconnectionbetweenthestoryproblemandthefraction

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notationandencouragethemtoapplytheirintuitiveknowledgeinbothsituations.Whiletryingtomakeconnections,teacherscan

directstudentsbacktothereal-worldstoryproblemiftheirstudentsneedtoeaseintounderstandingtheformalnotation.109


Roadblock3.1.Students make computational errors (e.g., adding fractions without finding a common denominator) when using certain pictorial and concrete object representations to solve problems that involve computation with fractions.

SuggestedApproach.Teachersshouldcare-fullychooserepresentationsthatmapstraight-forwardlytothefractioncomputationbeingtaught.Forexample,whenteachingfractionaddition,arepresentationshoulddemonstratetheneedforaddingsimilarunitsandthusleadstudentstofindacommondenominator.Useofsomerepresentationscanactuallyreinforcemisconceptions.Inonestudy,theuseofdotpaperforaddingfractionsledstudentstomoreoftenusetheincorrectstrategyofadd-ingnumeratorswithoutfindingacommondenominator.110 Representationsthathold

unitsconstant,suchasameasuringtapewithmarkedunits,canhelpstudentsseetheneedforcommonunitfractions.

Roadblock3.2. When encouraged to estimate a solution, students still focus on solving the problem via a computational algorithm rather than estimating it.

SuggestedApproach.Estimationshouldbepresentedasatoolforanticipatingthesizeandassessingthereasonablenessofananswer.Teachersshouldfocusonthereason-ingneededtoestimateasolutionandshouldemphasizethatestimationisapreliminarysteptosolvingaproblem,notashortcuttoobtaininganexactanswer.Teacherswhoposeproblemsthatcannotbesolvedquicklywithmentalcomputation(e.g.,problemssuchas5/9+3/7ratherthan5/8+3/8)willlikelyavoidthisroadblock.

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Recommendation4

Developstudents’conceptualunderstandingofstrategiesforsolvingratio,rate,andproportionproblemsbeforeexposingthemtocrossmultiplicationasaproceduretousetosolvesuchproblems.Proportional reasoning is a critical skill for students to develop in preparation for more advanced topics in mathematics.111

When students “think proportionally,” they understand the multiplicative relation between two quantities.112 For example, understanding the multiplicative relation in the equation Y = 2Xmeans understanding that Yis twice as large as X (and not that X is twice as large as Y, which is what many students think). Contexts that require understanding of multiplicative relations include problems that involve ratios (i.e., the relation between two quantities, such as the ratio of boys to girls in a classroom), rates (i.e., the relation between two quantities measured in different units, such as distance per unit of time), and proportions (i.e., two equivalent ratios). Proportional reasoning often is needed in everyday contexts, such as adjusting recipes to the number of diners or buying material for home improvement projects; thus proportional reasoning problems provide opportunities to illustrate the value of learning about fractions.

The panel recommends that teachers develop students’ proportional reasoning prior to teaching the crossmultiplication algorithm, using a progression of problems that builds on their informal reasoning strategies. Visual representations are particularly useful for teaching these concepts and for helping students solve problems. After teaching the crossmultiplication algorithm, teachers should return to the informal reasoning strategies, demonstrate that they and the algorithm lead to the same answers on problems for which the informal reasoning strategies are applicable, discuss why they do so, and also discuss problems that can be solved by the crossmultiplication algorithm that cannot easily be solved by the informal strategies.

A caution for teachers: Evidence from many types of problemsolving studies, including ones involving ratio, rate, and proportion, indicates that students often learn a strategy to solve a problem in one context but cannot apply the same strategy in other contexts.113 Stated another way, students often do not recognize that problems with different cover stories are the same problem mathematically.114 To address this issue, teachers should point to connections among problems with different cover stories and illustrate how the same strategies can solve them.

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Evidencefortherecommendationcomesfromconsensusdocumentsthatemphasizetheimportanceofproportionalreasoningformathematicslearning,aswellasthepanel’sexpertopinion.115 Additionally,thepanelseparatelyreviewedevidencerelevanttoparticularactionstepswithintherecom-mendation.Theseactionstepsaresupportedbycasestudiesdemonstratingthevarietyofstrategiesstudentsusetosolveratio,rate,andproportionproblems;arigorousstudyofmanipulatives;andtwowell-designedstudiesthattaughtstrategiesforsolvingwordproblems.

Buildingondevelopingstrategies.Threesmallcasestudiesprovidedevidencethatstudentsuseavarietyofstrategiestosolveproportionalreasoningproblems(Step1).116

Somestudentsinitiallyappliedabuildupstrategy(e.g.,tosolve2:3=x:12,theyadded2:3fourtimesuntiltheyreached8:12,andthensaidx=8),whereasothersappliedastrategythatfocusedonthemultiplicativerelationbetweentworatios(e.g.,tosolve2:3=x:12,theyidentifiedtherelationbetweenthedenominators[3×4=12]andappliedthisrelationtodeterminethemiss-ingnumerator[2×4=8],thensaidx=8).However,thesestudiesdidnotexaminewhetherbasinginstructiononthesestrategiesimprovedstudents’proportionalreasoning.Thepanelbelievesthatstudents’proportionalreasoningcanbestrengthenedthroughpresentingaprogressionofproblemsthatencouragesuseofthesestrategiesandthatprovidesabasisforrealizingthatthecross-multiplicationprocedurecansolvesome,butnotall,typesofproblemsmoreefficientlythanotherstrategies.

Usingrepresentations.Theevidencesup-portingtheuseofmanipulativesandpicto-rialrepresentationstoteachproportionality

conceptsislimited(Step2).However,onestudythatmetWWCstandardsfoundthattheuseofamanipulativeimproved4th-graders’abilitytovisualizeandcomparetworatios,whichimprovedtheirabilitytosolvemixtureproblems,comparedtostudentswhohadnoexposuretotheseproblemsorthemanipula-tive.117InanotherstudythatmetWWCstan-dards,studentsimprovedtheirabilitytosolvemissingvalueproportionproblemsbyrepre-sentinginformationfromtheseproblemsinadatatablethathighlightedthemultiplicativerelationshipsbetweenquantities.118Athirdwell-designedstudyfoundapositiveimpactonstudentlearningofcollaborativelycon-structingpictorialrepresentationsrelativetousingteacher-generatedrepresentations.119

Thesestudiesindicatethatmanipulativesandpictorialrepresentationscanbeeffectiveteachingtools;however,theprinciplesthatdeterminewhentheyareandarenothelpfulremainpoorlyunderstood.

Teachingproblemsolvingstrategies.Thepanelalsoidentifiedlimitedevidencesupportingtherecommendationtoteachstrategiesforsolvingwordproblemsinvolv-ingratiosandproportions(Step3).Theinterventionsexaminedinthesestudiestaughtmiddleschoolstudentsafour-stepstrategyforsolvingratioandproportionwordproblems.120Thisstrategydevelopedstudents’understandingofcommonproblemstructures,directedstudentstouseadia-gramtoidentifykeyinformationneededtosolveaproblem,andencouragedstudentstocomparedifferentsolutionstrategies.Oneofthesestudiesfocusedonstudentswithlearningdisabilities,whiletheothersampledstudentswithadiversemixofabilitylev-els.121 Bothstudiesfoundapositiveeffectontheaccuracyofstudents’solutionstoratioandproportionproblems.

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1.Developstudents’understandingofproportionalrelationsbeforeteachingcomputa-tionalproceduresthatareconceptuallydifficulttounderstand(e.g.,cross-multiplication).Buildonstudents’developingstrategiesforsolvingratio,rate,andproportionproblems.

Opportunitiesforstudentstosolveratio,rate,andproportionproblemsshouldbeprovidedpriortoteachingthecross-multiplicationalgorithm.122 Teacherscanuseaprogressionofproblemsthatbuildsonstudents’develop-ingstrategiesforproportionalreasoning.123 Inparticular,teacherscaninitiallyposeproblemsthatallowsolutionsviathebuildupandunitratiostrategiesandprogresstoproblemsthatareeasiertosolvethroughcrossmultiplication.Encouragingstudentstoapplytheirownstrate-gies,discussingwithstudentsvariedstrate-gies’strengthsandweaknesses,andhelpingstudentsunderstandwhyaproblem’ssolutioniscorrectareadvisable.124 Ifstudentsdonotgeneratethesestrategiesontheirown,teach-ersshouldintroducethestrategiesaswaysofsolvingratio,rate,andproportionproblems.

Teacherscaninitiallyposestoryproblemsthatallowstudentstouseabuildupstrategy,inwhichtheyrepeatedlyaddthenumberswithinoneratiotosolvetheproblem(seeExample4).125Problemsthatfacilitatetheuseofthebuildupstrategyshouldhaveaninte-gralrelationbetweenthecomponentnum-bersinthetworatios—arelationinwhichthenumbersinoneratiocanbegeneratedbyrepeatedlyaddingnumbersintheotherratio,allowingstudentstobuilduptotheunknownnumber.Forexample,theratios2:3and10:15haveanintegralrelation,becauserepeatedlyadding2sand3stothefirstratioleadsto10:15.Thus,initialproblemsshouldinvolveratiosforwhichstudentscaneasilyapplyabuildupstrategy,suchas,“Johnisbakingbreadforsomefriends.Heuses2cupsofflourforevery3friends.Ifhewantstomakebreadfor15friends,howmanycupsofflourshouldheuse?”

Next,teacherscanpresentsimilarproblems,butwithlargernumbers,thatdemonstrate

tostudentshowtime-consumingitcanbetoadduprepeatedlytotheunknownvalue.Studentswillseetheadvantageofmultiply-inganddividingratherthandependinguponrepeatedaddition.Forexample,inthebakingbreadproblem,Johncouldbebakingbreadforall54studentsinthe5thgrade.

Teacherscanthenpresentproblemsthatcannotbesolvedimmediatelyeitherthroughrepeatedadditionorthroughmultiplyingordividingagivennumberbyasingleinte-ger(seeExample4).Theseareproblemsthatinvolveratioswithoutanintegralrela-tion,suchasx/6=3/9.Suchproblemscanbesolvedbytheunitratiostrategy,whichinvolvesreducingtheknownratio(3/9)toaformwithanumeratorof1andthendeter-miningthemultiplicativerelationbetweenthenewunitratioandtheratiowiththeunknownelement(x/6).Themultiplicativerelationbetweenthedenominatorsintheunitratioandtheunknownratiocanthenbeusedtosolveforthemissingelement.Forexample,x/6=3/9couldbesolvedbyexpressing3/9as1/3,identifying2/2asthenumberthatcouldbeusedtomultiply1/3 andobtainadenominatorof6withoutchangingthevalueof1/3,multiplying1/3 by2/2toobtain2/6,andanswering“x=2.”126

Thesametypeofreasoningcanbeusedtosolveproblemsforwhichtheanswerisnotawholenumber;forexample,“Susanismakingdinnerfor6peopleandwantstousearecipethatserves8people.Therecipefor8callsfor2cupsofcream.Howmuchcreamwillsheneedtoserve6?”Thiscontextpresentstheproblemas2:8asx:6.Studentscouldsolvethisproblembyreasoningthatsince2cupsofcreamserve8people,1cupofcreamwouldserve4people,and11/2 cupsofcreamwouldserve6.

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Problemssuchasthoseinthelastparagraphcanbeusedtohelpstudentsrecognizetheadvantagesofastrategythatcansolveproblemsregardlessoftheparticularnum-bers.Cross-multiplicationcanbeintroducedassuchanapproach.Problemsthatdonotinvolveintegralrelationsandcannoteasilybereducedtounitfractionswillhelpstudentsseetheadvantagesofcross-multiplication,whichisessentiallyaproceduretocreateequivalentratios.Itsusecanbeillustratedwithproblemssuchasthosepresentedinthepreviousparagraphthatweresolvedwithaunitstrategy.Forexample,studentscouldbeencouragedtosolvethelastproblem

withthecross-multiplicationstrategy:writingxtheequation2/8= /6andcross-multiplyingto

findthemissingvalue.Afterstudentsarriveatthesameanswerof11/2,teacherscanleadstudentsinadiscussionofwhytheunitratioandcross-multiplicationproceduresyieldthesameanswer(seeExample5).Studentsshouldpracticebothwithproblemsthataresolvedeasilythroughinformalreasoningandmentalmathematicsandwithproblemsthataresolvedeasilyusingcross-multiplicationbutnotthroughthebuilduporunitratiostrategies.Teacherscanencouragestudentstodiscusshowtoanticipatewhichapproachwillbeeasiest.

Example4.Problemsencouragingspecificstrategies

Ratio,rate,andproportionproblemscanbesolvedusingmanystrategies,withsomeproblemsencouraginguseofparticularstrategies.Illustratedbelowarethreecommonlyusedstrategiesandtypesofproblemsonwhicheachstrategyisparticularlyadvantageous.

BuildupStrategy

Sampleproblem.IfStevecanpurchase3baseballcardsfor$2,howmanybaseballcardscanhepurchasewith$10?

Solutionapproach.Studentscanbuilduptotheunknownquantitybystartingwith3cardsfor$2,andrepeat-edlyadding3morecardsand$2,thusobtaining6cardsfor$4,9cardsfor$6,12cardsfor$8,andfinally15cardsfor$10.

UnitRatioStrategy

Sampleproblem.Yukaribought6balloonsfor$24.Howmuchwillitcosttobuy5balloons?

Solutionapproach.Studentsmightfigureoutthatif6balloonscosts$24,then1ballooncosts$4.Thisstrat-egycanlaterbegeneralizedtooneinwhicheliminatingallcommonfactorsfromthenumeratoranddenomina-toroftheknownfractiondoesnotresultinaunitfraction(e.g.,aproblemsuchas6/15=x/10,inwhichreducing6/15resultsin2/5).

CrossMultiplication

Sampleproblem.Luisusuallywalksthe1.5milestohisschoolin25minutes.However,todayoneofthestreetsonhisusualpathisbeingrepaired,soheneedstotakea1.7-mileroute.Ifhewalksathisusualspeed,howmuchtimewillittakehimtogettohisschool?

Solutionapproach.Thisproblemcanbesolvedintwostages.First,becauseLuisiswalkingathis“usualspeed,”studentsknowthat1.5/25=1.7/x.Then,theequationmaybemosteasilysolvedusingcross-multiplication.Multiplying25and1.7anddividingtheproductby1.5yieldstheanswerof281/3minutes,or28minutesand20seconds.ItwouldtakeLuis28minutesand20secondstoreachschoolusingtheroutehetooktoday.

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Example5.Whycrossmultiplicationworks

Teacherscanexplainwhythecross-multiplicationprocedureworksbystartingwithtwoequalfractions,suchas4/6 =6/9.Thegoalistoshowthatwhentwoequalfractionsareconvertedintofractionswiththesamedenominator,theirnumeratorsalsoareequivalent.Thefollowingstepshelpdemonstratewhytheprocedureworks.

Step1. Startwithtwoequalfractions,forexample:4/6=6/9.Step2. Findacommondenominatorusingeachofthetwodenominators.

a. First,multiply4/6by9/9,whichisthesameasmultiplying4/6by1.

b. Next,multiply6/9by6/6,whichisthesameasmultiplying6/9by1.

Step3. Calculatetheresult:(4×9)=

(6×6)(6×9) (9×6)

Step4. Checkthatthedenominatorsareequal.Iftwoequalfractionshavethesamedenominator,thenthenumeratorsofthetwoequalfractionsmustbeequalaswell,so4×9=6×6.

Notethatinthisproblem,4×9=6×6isaninstanceof(a×d=b×c).

Asaresult,studentscanseethattheoriginalproportion,4/6=6/9,canbesolvedusingcross-multiplication,4×9=6×6,asaproceduretocreateequivalentratiosefficiently.

2. Encouragestudentstousevisualrepresentationstosolveratio,rate,andproportionproblems.

Thepanelrecommendsthatteachersencour-agetheuseofvisualrepresentationsforratio,rate,andproportionproblems.Teach-ersshouldcarefullyselectrepresentationsthatarelikelytoelicitinsightintoaparticularaspectofratio,rate,andproportionconcepts.Forexample,aratiotablecanbeusedtorep-resenttherelationsinaproportionproblem(seeFigure9).Toidentifytheamountofflourneededfor32peoplewhenarecipecallsfor1cupofflourtoserve8,studentscanusearatiotabletorepeatedlyadd1cupofflourper8peopletofindthecorrectamountfor32people(i.e.,theycanusethebuildupstrategy).Alternatively,studentscanusetheratiotabletoseethatmultiplyingtheratioby4/4(i.e.,4timestherecipe)providestheamountofflourneededfor32people.Thisvisualrepresentationprovidesaspecificrefer-entthatteacherscanpointtoastheydiscusswithstudentswhymultiplicationleadstothesamesolutionasthebuildupstrategy.

Figure9.Ratiotableforaproportionproblem

CupsofFlour

1 2 3 4

NumberofPeopleServed

8 16 24 32

Inadditiontousingtheratiotableasatoolforsolvingproblems,teacherscanuseittoexploredifferentaspectsofproportionalrelations,suchasthemultiplicativerelationswithinandbetweenratios.IntheratiotableinFigure10,thenumberofcupsofflourneededisalways2.5timesthenumberofpeople;thus,theratiobetweenthemisalways2.5:1.

AsdiscussedinRecommendation3,teachersshouldnotalwaysproviderepresentationstostudents;theysometimesshouldencourage

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themtocreatetheirownrepresentations—inthiscase,representationsofratios,rates,andproportions.127Priortoformalinstructioninratios,studentstendtousetabularorothersystematicformsofrecordkeeping,128 whichcanhelpthemunderstandthefunctionalrelationbetweenrowsorcolumnsorthenumbersinaratio.129 Teachersshouldhelpstudentsextendtheseandotherrepresen-tationstoabroadrangeofratio,rate,andproportionproblems.

Figure10.Ratiotableforexploringproportionalrelations

CupsofFlour

5 7.5 10 12.5

NumberofPeopleServed

2 3 4 5

3. Provideopportunitiesforstudentstouseanddiscussalternativestrategiesforsolvingratio,rate,andproportionproblems.

Thegoalistodevelopstudents’abilitytoidentifyproblemswithacommonunderlyingstructureandtosolveproblemsthataresetinavarietyofcontexts.130 Instructionmightfocusonthemeaningfulfeaturesofdifferentproblemtypes,includingratioandpropor-tionproblems,sothatstudentscantransfertheirlearningtonewsituations.Forexample,studentsmightfirstlearntosolverecipeproblems,suchas,“Arecipecallsfor3eggstomake20cupcakes.Ifyouwanttomake80cupcakes,howmanyeggsdoyouneed?”Hav-inglearnedtosolvesuchproblems,studentsmightthenbeaskedtosolvesimilarproblemswithdifferentcontexts,suchas:“Building3dog-housesrequires42boards;howmanyboardsareneededtobuild9doghouses?”131

Teachersalsoshouldhelpstudentsidentifykeyinformationneededtosolveaproblem.Oncestudentscanidentifythekeyinforma-tioninaproblem,theycanbetaughttousediagramstorepresentthatinformation.132

Suchdiagramsshouldnotsimplyrepresentthestoryproblemindiagramform;theyalsoshouldidentifytheinformationneededtosolvetheproblemandtherelationbetweendifferentquantitiesintheproblem.Teachersshouldencouragestudentstousedifferentdiagramsandstrategiestoarriveatsolutionsandshouldprovideopportunitiesforstudentstocompareanddiscusstheirdiagramsandstrategies.133

Thepanelsuggestsusingreal-lifecontextsbasedonstudents’experiences.Afewexam-plesareprovidedhere:134

•Unitprice.Teacherscanposeproblemsbasedontheunitpriceofanobject,suchascomparingthevalueoftwoitems(e.g.,a16-ouncecanofsodafor$0.89anda12-ouncecanofsodafor$0.62)anddeter-mininghowmuchacertainamountofanitemcostsgiventhecostperunitandthenumberofunitspurchased.Thecontextofunit-priceproblemscanbebuyingorsellingproduceatagrocerystore,cansofpaintatahardwarestore,oranyotherpurchasingsituation.

•Scaling.Studentscansolveproblemsrelatedtotheenlargementorreductionofaphoto,drawing,orgeometricshape(e.g.,doublethewidthanddoublethelengthofaphototocreateanewphotowhoseareaisfourtimesthatoftheoriginal).Anotherexampleofscalingisusingamaplegendtofindtheactualdistancebetweentwocities,basedontheirdistanceonthemap.

• Recipes.Recipesandcookingprovideusefulsettingsforratioandproportionproblems,forexample,“Ifarecipecallsfor1eggand3cupsofmilk,andthecookwantstomakeasmuchaspossibleusingall8eggsshehas,howmuchmilkisneeded,assumingthattheratioofeggsto

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milkintheoriginalrecipeismaintained?”Studentsalsocanrevisearecipetomakemoreorlessofthefinalamount,insitua-tionsthatcallforchangingthenumberofservingsoramountsofingredientsusingequivalentratios.

•Mixture.Problemsrelatedtothemixtureoftwoormoreliquidsprovideanothercontextforposingratioandproportionproblems.Studentscancomparetheconcentrationofamixture(e.g.,comparetherelativeamountofoneliquidtotheamountofanotherliquidinamixture)ordeterminehowtomaintaintheoriginalratiobetweenliquidsinamixtureiftheamountofoneoftheliquidschanges.

• Time/speed/distance.Studentscanbetoldthetime,speed,anddistancethatonecartraveledandthevaluesofanytwoofthesevariablesforasecondcarandthenbeaskedforthevalueofthethirdvariableforthesecondcar.Forexample,theycouldbetoldthatcarAtraveledfor2hoursatarateof45milesperhour,soittraveled90miles.ThentheycouldbetoldthatcarBtraveledatthesamespeedbuttraveledonly60milesandbeaskedtodeterminetheamountoftimethatcarBtraveled.


Roadblock4.1.Many students misapply the crossmultiplication strategy.

SuggestedApproach.CarefullypresentingseveralexamplesofthetypeshowninExam-ple5canhelpstudentsunderstandthelogicbehindthecross-multiplicationprocedureandwhytheratioswithintheproblemneedtobeinthecorrectformfortheproceduretowork.Makingsurethatstudentsunderstandthelogicofeachstepinthedemonstrationtakestime,butitcanpreventmanyfutureerrorsandmisunderstandings.

Roadblock4.2.Some students rely nearly exclusively on the crossmultiplication strategy for solving ratio, rate, and proportion problems, failing to recognize that there often are more efficient ways to solve these problems.

SuggestedApproach.Teachersshouldprovidestudentsopportunitiestouseavarietyofstrategiesforsolvingratio,rate,andpropor-tionproblemsandinitiallypresentproblemsthatareeasiesttosolvewithstrategiesotherthancross-multiplication.Forexample,teach-erscanpresentproblemsinwhichtherelation

withinthegivenratioisintegral(e.g.,5/15)andtherelationbetweenthecorrespondingnum-bersacrossthetworatiosisnot(e.g.,5/15=6/x).Thesetypesofproblemsmayencouragestu-dentstousepriorknowledgeofmultiplicativerelationsbetweennumeratoranddenominatorwithintheratiowherebothareknown.Requir-ingstudentstosolveproblemsmentally(with-outpencilandpaper)alsocanincreasetheuseofstrategiesotherthancross-multiplicationandbuildnumbersensewithfractions.

Roadblock4.3. Students do not generalize strategies across different ratio, rate, and proportion contexts.

SuggestedApproach.Inadditiontoprovid-ingstudentswithproblemsacrossavarietyofcontextsandteachingavarietyofrate,ratio,andproportionproblem-solvingstrategies,teachersshouldstrivetolinknewproblemswithprevi-ouslysolvedones.Teacherscanregularlyhavestudentsjudgewhenthesamesolutionstrategycouldbeusedfordifferenttypesofproblems.Forexample,teacherscandemonstratehowinforma-tionintwotypesofproblems,suchasrecipesandmixtureproblems,canbeorganizedinthesamewayandthencomparesolutionproceduresforthetwotypesofproblemssidebyside.

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Recommendation5

Professionaldevelopmentprogramsshouldplaceahighpriorityonimprovingteachers’understandingoffractionsandofhowtoteachthem.Teachers play a critical role in helping students understand fraction concepts. Teaching for understanding requires that teachers themselves have a thorough understanding of fraction concepts and operations—including deep knowledge of why computation procedures work. Appropriate use of representations for teaching fractions, a key aspect of the panel’s recommendations, requires that teachers understand a range of representations and how to use them to illustrate particular points.

An awareness of common misconceptions and of inappropriate strategies students use to solve fractions problems also is crucial for effective instruction in this area. The panel believes that preservice teacher education and professional development programs must develop teachers’ abilities in each of these areas, especially given considerable evidence that many U.S. teachers lack deep understanding of fraction concepts.135


Despitethelimitedevidencerelatedtothisrecommendation,thepanelbelievesteach-ersmustdeveloptheirknowledgeoffrac-tionsandofhowtoteachthem.Researchershaveconsistentlyfoundthatteacherslackadeepconceptualunderstandingof

fractions,136andthatteachers’mathematicalcontentknowledgeispositivelycorrelatedwithstudents’mathematicsachievement.137

Takentogether,thesefindingssuggestagreatneedforprofessionaldevelopmentinfractionconcepts.Regardless,theevidenceratingassignedbythepanelrecognizesthelimitedamountofrigorousevidenceonthe

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effectsofprofessionaldevelopmentactivi-tiesrelatedtofractions.

Inonewell-designedstudy,teacherswhoreceivedtrainingonfractionconcepts,onstudents’understandingoffractions,onstu-dents’motivationforlearningmath,andonhowtoassessstudents’knowledgeoffrac-tionsimprovedstudents’conceptualunder-standingoffractionsandtheirabilitytocomputewithfractions.138 However,anotherwell-designedstudyfoundnoimpactonstudentachievementinfractions,decimals,percentages,andproportions,despiteoffer-ing7th-gradeteachersupto68hoursofprofessionaldevelopmentonrationalnum-bersthroughasummerinstituteandone-dayseminars.139 TwootherstudiesthatmetWWCstandardsprovidedtrainingonhowstudentsdevelopknowledgeandskillsrelatedtospecificmathconcepts.Oneofthesestud-iesfocusedonwholenumberadditionandsubtractionandfoundimprovementsin

students’wholenumbercomputationandsolutionstowordproblems.Thesecondstudyprovidedteachertrainingonstudents’algebraicreasoningandreportedapositiveimpactonstudentlearning.

Researchindicatesthatmanyelementaryschoolteachershavelimitedknowledgeoffractionconceptsandprocedures.140 Inter-viewswithU.S.elementaryschoolteachersshowedthatahighpercentageofthemwereunabletoexplaincomputationalproceduresforfractions.141 Anotherstudyfoundthatsomeelementaryschoolteachershaddiffi-cultyorderingfractions,addingfractions,andsolvingratioproblems.142Manyoftheteach-erswhosolvedproblemscorrectlycouldnotexplaintheirownproblem-solvingprocess.Thepanelviewsthislimitedknowledgeoffractionsasproblematic,givenevidencethatteachers’mathematicalcontentknowledgeisrelatedtostudents’learning.143


1. Buildteachers’depthofunderstandingoffractionsandcomputationalproceduresinvolvingfractions.

Toprovideeffectivefractionsinstruction,teachersneedadeepunderstandingoffrac-tionconceptsandoperations.Inparticular,teachersneedtounderstandthereasoningbehindcomputationsthatinvolvefractionssotheycanclearlyandcoherentlyexplaintostu-dentswhytheprocedureswork,notjustthesequenceofstepstotake.Withoutaconcep-tualunderstandingoffractioncomputation,teachersarenotlikelytohelpstudentsmakesenseoffractionoperations.144 Therefore,teacherpreparationandprofessionaldevelop-mentactivitiesmustsupportadeeperlevelofunderstandingoffractions.145

Teachersshouldhaveopportunitiestogainbetterunderstandingoffractionsalgorithmsbysolvingproblemsandexploringthemeaningofalgorithms.146 Oneapproachistoposeproblemsthatprovokedeep

discussionsofthealgorithms,possiblyusingadvancedversionsofexamplesfromteachers’lessons.147Forexample,teachersmightsolveaprobleminwhichtheyhavetoequallydistributefractionalpartsofcakeamonganumberofpeople(e.g.,3cakesdis-tributedamong8people),whereasstudentsmightbeaskedtodistributeawholenumberofcookies(e.g.,18cookiesamong6people).Particularlyusefulareproblemsoractivitiesthatleadteacherstoquestionwhyanalgo-rithmworksortoexaminewhattheydoanddonotunderstandaboutanalgorithm.148

Althoughteacherscanaddresstheseprob-lemsontheirownorinsmallgroups,mak-ingtimefordiscussioniscrucial.

Havingteachersestimateanswerstofractionsproblemsanddiscussthereasoningthatledtotheestimatesalsocanbeuseful.All

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activitiesshouldeventuallylinkbacktotheclassroom,withopportunitiesforteacherstodiscusshowtheywouldrespondtostudents’questionsaboutwhyestimationisvaluableandthelogicthatseparateseffectiveandlesseffectiveestimationprocedures.

Professionaldevelopmentshouldnotfocusexclusivelyonfractiontopicscoveredattheteacher’sgradelevel.Teachersmustunder-standfractionconceptscoveredintheentireelementaryandmiddleschoolcurriculaandshouldknowhowtheseconceptsfitwithin

thebroadermathcurriculum.Awarenessoffractionconceptstaughtinearliergradesensuresthatteacherscanbuildonwhatstu-dentsalreadyknow;italsocanhelpteachersidentifyandaddresscommonmisconceptionsthatstudentsmighthavedeveloped.Under-standingfractionconceptsandothermoreadvancedmathematicsthatwillbecoveredinlatergradeshelpsteacherssetgoalsandthinkabouthowtheirteachingcanprovidefoundationsforideasthatstudentswillencounterinthefuture.

2. Prepareteacherstousevariedpictorialandconcreterepresentationsoffractionsandfractionoperations.

Touseconcreteandpictorialrepresentationseffectively,teachersmustunderstandhowtheserepresentationslinktofractionconceptsandhowtheycanbeusedtoimprovestudentlearning.Teachereducationandprofessionaldevelopmentactivitiesshouldprepareteach-erstousesuchrepresentationsforteachingfractionsandshouldhelpteachersunder-standhowtherepresentationsrelatetotheconceptsbeingtaught.

Teachersmightlearn,forexample,thatdiagramsofsharingscenarioscanhelphigh-lightthelinkbetweenfractionsanddivision(i.e.,thequotientinterpretationoffractions)byallowingstudentstorepresentfractionswithequalshares(e.g.,2largebrowniessharedamong5children).Numberlinescanfocusstudentsonmeasurementinterpreta-tionsoffractions,withfractionsrepresent-ingadistancebetweentwonumbers.Area

models—particularlyrectangularones,butmodelsusingothershapesaswell—canbeusedtodepictpart-wholerepresentationsoffractions.

Developmentactivitiesshouldprovideoppor-tunitiesforteacherstointegraterepresenta-tionsintofractionslessons.149Inaddition,teachersneedtounderstanddifficultiesthatmightarisewhentheyuseapictorialorconcreterepresentationtoteachfractions.Forexample,studentsmayviewtheentirenum-berline,ratherthanthedistancebetweentwonumbers,astheunitwhenlocatingfractions(e.g.,theymightinterpretthetaskoflocat-ing3/4ona0-to-5numberlineaslocatingthepoint75%ofthewayacrossthenumberline).Professionaldevelopmentactivitiesneedtohelpteachersanticipatemisconceptionsandlearningproblemsthatarelikelytoarise,andidentifywaysofaddressingthem.

3. Developteachers’abilitytoassessstudents’understandingsandmisunderstandingsoffractions.

Professionaldevelopmentactivitieswithteachersshouldemphasizehowstudentsdevelopanunderstandingoffractionsandtheobstaclesstudentsfaceinlearningaboutthem.150Informationfromresearchonthedevelopmentoffractionlearningshouldbeprovidedinthesediscussions.151

Onemethodthatisusefulformeetingthisgoalistoprovideteacherswithopportuni-tiestoanalyzeandcritiquestudentthinkingaboutfractions.Thiscanbedonebyexam-iningstudents’writtenworkorwatchingvideoclipsofstudentssolvingproblemsthataredesignedtoprovideinsightinto

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students’thinking.152Forexample,teacherscanbeaskedtoanalyzesourcesofstudents’difficultyonproblemssuchas,“Paigehad3boxesofcereal.Eachboxwas2/3full.Ifthecerealinthe3boxeswerepouredintoemptyboxes,howmanyboxeswoulditfill?Userectangulardrawingsoranumberlinetodisplayyourreasoning.”Teacherscanbeaskedtovideo-recordstudents’performanceonsuchproblemsbeforeaprofessionaldevelopmentsession;thenteacherscanbringstudents’workorvideoclipstothesessionandusethemasabasisfordiscussion.

Teachersshouldknowthetypesofmistakesstudentsmostoftenmakewhenworkingwithfractionsandalsoshouldunderstandtheunderlyingmisconceptionsthatcausethem.Analyzingstudents’workisausefulwaytoidentifyproblemareasandtogaininsightintostudents’thoughtprocesses.Tobemosteffective,teachersmustknowhowtodesignproblemsthatdiagnosethesourceoferrors.Forexample,teachersmightstructureadecimal-orderingproblemtoassesswhetherstudentsunderstandplacevalue(e.g.,order-ingthefollowingdecimalsfromsmallesttolargest:0.2,0.12,0.056).

Preserviceandin-serviceactivitiesshouldhelpteachersunderstandresearchonchil-dren’sknowledgeoffractions;theresearchpresentedshouldbechosentoinformteach-ers’assessmentactivitiesandinstruction.Forexample,researchhasshownthatstudentsoftenhavedifficultywithfractionnamesandwithunderstandingthevalueoffractions.153

Whetherstudentsinagivenclassroomarehavingsuchdifficultycanbeassessedbyaskingthemtostatefractionsthatlabelthelocationsofhatchmarksonanumberlinewithendpointsof0and1.Suchanassess-mentmightindicatethatstudentsrefertoavarietyoflocationsas1/2,orthattheyviewfractionswithlargerdenominatorsaslargerthanfractionswithsmallerdenominators(e.g.,theymightthinkthat1/8>1/3).Suchapatternmightleadtoanengagingandproductivediscussionofhowthesystemfornamingfractionsworksandwhythatnamingproceduremakessense.Moregener-ally,developmentactivitiesshouldprovideopportunitiesforteacherstopracticewritingorselectingproblemsthataccuratelyassessstudents’understandingandtouseassess-mentresultstodesignusefullessons.


Roadblock5.1. Administrators or professional development personnel might argue that the topic of fractions is just one of many that elementary and middle school teachers must be prepared to teach and that their district, program, or school cannot devote more time or resources to it.

SuggestedApproach.Thepanelrecog-nizesthattimeandresourcesforprovidingprofessionaldevelopmentarelimited.How-ever,aconvincingargumentcanbemadefordevotingsometimeandresourcestothistopic:(1)fractionsareacriticalfoundationformoreadvancedmathematics,(2)manyteacherslacksufficientunderstandingoffractiontoteachthetopiceffectively,and

(3)U.S.studentslagfurtherbehindthoseinothercountriesinsolvingproblemswithfractionsthaninsolvingproblemswithwholenumbers.154Thepanelbelievestheneediscriticalforelementaryandmiddleschoolteacherstoreceiveprofessionaldevel-opmentrelatedtotheircontentknowledgeoffractionsandtotheteachingoffractions,includingdecimals,percentages,ratios,rates,andproportions.Thepanelsuggeststhatschoolanddistrictleadersconsiderfractionsahighpriorityforprofessionaldevelopment.

Roadblock5.2. Some teachers have difficulty with whole number topics, such as multiplication and division, that are related to the teaching of fractions.

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SuggestedApproach.Adeepunderstand-ingofwholenumbermultiplicationanddivision,includingwhyandhowcommoncomputationalalgorithmswork,isessen-tialforteachingfractionseffectively.Whenselectingordesigningprofessionaldevelop-mentactivitiesrelatedtofractions,educationleadersshouldconsiderwhetherreviewingthesekeywholenumbertopicsisaneces-saryprerequisiteforteachersintheparticu-larschoolordistrict.

Roadblock5.3.Some teachers do not think additional professional development involving fractions is necessary.

SuggestedApproach.Althoughmostteach-ersareabletocomputewithfractions,manydonothaveastrongconceptualbackgroundregardingfractionsoranunderstandingofthelogicunderlyingcomputationalalgo-rithmsusedforsolvingfractionproblems.Byfirstdeterminingifteachersknowwhyandhowcommoncomputationalalgorithms(e.g.,invertandmultiply)workandwhycertainstepswithinalgorithmsarenecessary(e.g.,establishingcommondenominatorsforaddi-tionandsubtraction),educationleaderscandecidewhetherprofessionaldevelopmentinvolvingfractionsisanimportantneedintheirschoolsordistricts.

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GlossaryGlossaryGlossary

CCommonfraction–Afractionwrittenintheforma /b,wherebotha andb areintegersandbdoesnotequalzero(e.g.,3/4,6/5,–(1/8)).Covariation–Ameasureofhowmuchtwoquantitieschangetogether.Forexample,theextenttowhichonequantityincreasesasanotherquantityincreases.

DDenominator–Foranyfractiona /b,thedenominatoristhenumberbelowthehashline.Thedenominatorrepresentsthedivisorofadivisionproblem,orthenumberofpartsintowhichawholeamountisdivided(e.g.,forthefraction2/3,thedenominator3referstoawholedividedintothreeparts).

EEqualsharing–Theactivityofcompletelydistributinganobjectorsetofobjectsequallyamongagroupofpeople.

Equivalentfractions–Fractionsthatrepresentthesamenumericalvalue;equalfractions.Forexample,2/4and4/8arebothequalto1/2;therefore2/4,4/8,and1/2areequivalentfractions.

FFractiondensity–Theconceptthatbetweenanytwofractionsthereisanotherfraction.Forexample,thefraction1/4 isbetween0and1/2;thefraction1/8 isbetween0and1/4;andthefraction1/16isbetween0and1/8.Oneconsequenceofthisfactisthatbetweenanytwofractionsthereareaninfinitenumberoffractions.

IImproperfraction–Afractionwithanumeratorthatisgreaterthanorequaltothedenominator.Examplesofimproperfractionsinclude5/5,9/8,and14/9.

MMixednumber–Afractionwrittenasawholenumberandafractionlessthanone.Examplesofmixednumbersinclude12/3,43/8,and–25/6.Multiplicativerelation–Arelationbetweentwoquantitiesinwhichonequantitycanbemultipliedbyafactortoobtainasecondquantity.

NNumerator–Foranycommonfractiona /b,thenumeratoristhenumberabovethehashline.Thenumeratorrepresentsthedividendofadivisionproblemorthenumberoffractionalpartsrepresentedbyafraction(e.g.,forthefraction2/3,thenumerator2representsthenumberofthirds).

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Glossarycontinued GlossaryGlossarycontinued continued

PPercent–Anynumberexpressedasafractionorratioof100(i.e.,withadenominatorof100).Forexample,75%isequivalentto0.75or75/100.Proportion–Anexpressionoftwoequivalentratiosorfractions.Aproportionisanequationwrittenintheforma /b=c /d,thusindicatingthatthetworatiosareequivalent.

Proportionalreasoning–Theliteratureconsistsofseveraldifferentdefinitionsofproportionalreasoning.Onabasiclevel,thetermmeansunderstandingandworkingwiththeunderlyingrelationsinproportions.155

Othersdescribeproportionalreasoningastheabilitytocompareonerelativeamounttoanother,156ortheabilitytounderstandmultiplicativerelationsorreasonaboutmultiplicativesituations.157

QQuotient–Thesolutiontoadivisionproblem.Forexample,3isthequotientforthefollowingdivisionproblem:12÷4=3.

RRationalnumber–Anynumberthatcanbeexpressedintheforma /b wherea andbarebothintegersandb doesnotequalzero.Rationalnumberscantakemanydifferentforms,includingcommonfractions,ratios,decimals,andpercents.

Rate–Therelationbetweentwoquantitiesmeasuredindifferentunits.Forexample,distanceperunitoftime.

Ratio–Therelationbetweentwoquantities.Forexample,theratio2:3mightrepresenttherelation-shipofthenumberofboystogirlsinaclassroom,ortwoboysforeverythreegirlsintheclass.

UUnitfraction–Afractionwithanumeratorofone(e.g.,1/3,1/11).Unitratio–Aratiowithadenominatorofone(e.g.,5:1,9:1).

WWholenumbers–Thesetofnumbersstartingwithzeroandincreasingbyone(i.e.,0,1,2,3…).

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AppendixA

PostscriptfromtheInstituteofEducationSciences

Whatisapracticeguide?TheInstituteofEducationSciences(IES)publishespracticeguidestosharerigorousevidenceandexpertguidanceonaddressingeducation-relatedchallengesnotsolvedwithasingleprogram,policy,orpractice.Eachpracticeguide’spanelofexpertsdevelopsrecommendationsforacoherentapproachtoamultifacetedproblem.Eachrecommendationisexplicitlyconnectedtosupportingevidence.Usingstandardsforrigorousresearch,thesupportingevidenceisratedtoreflecthowwelltheresearchdemonstratesthattherecommendedpracticesareeffective.Strongevidencemeanspositivefindingsaredemonstratedinmultiplewell-designed,well-executedstudies,leavinglittleornodoubtthatthepositiveeffectsarecausedbytherecommendedpractice.Moderateevi-dencemeansthatwell-designedstudiesshowpositiveimpacts,butsomequestionsremainaboutwhetherthefindingscanbegeneralizedorwhetherthestudiesdefinitivelyshowthatthepracticeiseffective.Minimalevidencemeansdatamaysuggestarelationshipbetweentherecommendedpracticeandpositiveoutcomes,butresearchhasnotdemonstratedthatthepracticeisthecauseofpositiveoutcomes.(SeeTable1formoredetailsonlevelsofevidence.)

Howarepracticeguidesdeveloped?

Toproduceapracticeguide,IESfirstselectsatopic.TopicselectionisinformedbyinquiresandrequeststotheWhatWorksClearinghouseHelpDesk,formalsurveysofpractitioners,andalimitedliteraturesearchofthetopic’sresearchbase.Next,IESrecruitsapanelchairwhohasanationalreputationandexpertiseinthetopic.Thechair,workingwithIES,thenselectspaneliststoco-authortheguide.Panelistsareselectedbasedontheirexpertiseinthetopicareaandthebeliefthattheycanworktogethertodeveloprelevant,evidence-basedrecommendations.IESrec-ommendsthatthepanelincludeatleastonepractitionerwithrelevantexperience.

Thepanelreceivesageneraltemplatefordevelopingapracticeguide,aswellasexamplesofpublishedpracticeguides.Panel-istsidentifythemostimportantresearchwithrespecttotheirrecommendationsandaugmentthisliteraturewithasearchofrecentpublicationstoensurethatsupportingevidenceiscurrent.Thesearchisdesignedtofindallstudiesassessingtheeffectivenessofaparticularprogramorpractice.ThesestudiesarethenreviewedagainsttheWhatWorksClearinghouse(WWC)standardsbycertifiedreviewerswhorateeacheffective-nessstudy.WWCstaffassistthepanelistsin

compilingandsummarizingtheresearchandinproducingthepracticeguide.

IESpracticeguidesarethensubjectedtorigorousexternalpeerreview.ThisreviewisdoneindependentlyoftheIESstaffthatsupportedthedevelopmentoftheguide.Acriticaltaskofthepeerreviewersofapracticeguideistodeterminewhethertheevidencecitedinsupportofparticularrecommenda-tionsisup-to-dateandthatstudiesofsimilarorbetterqualitythatpointinadifferentdirec-tionhavenotbeenoverlooked.Peerreviewersalsoevaluatewhetherthelevelofevidencecategoryassignedtoeachrecommendationisappropriate.Afterthereview,apracticeguideisrevisedtomeetanyconcernsofthereview-ersandtogaintheapprovalofthestandardsandreviewstaffatIES.

AfinalnoteaboutIESpracticeguides

Inpolicyandotherarenas,expertpanelstypicallytrytobuildaconsensus,forgingstatementsthatallitsmembersendorse.Butpracticeguidesdomorethanfindcom-monground;theycreatealistofactionablerecommendations.Whenresearchclearlyshowswhichpracticesareeffective,thepanelistsusethisevidencetoguidetheirrecommendations.However,insomecases,researchdoesnotprovideaclearindication

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AppendixAcontinued

ofwhatworks,andpanelists’interpretationoftheexisting(butincomplete)evidenceplaysanimportantroleinguidingtherecom-mendations.Asaresult,itispossiblethattwoteamsofrecognizedexpertsworkingindependentlytoproduceapracticeguideonthesametopicwouldcometoverydiffer-entconclusions.Thosewhousetheguidesshouldrecognizethattherecommendationsrepresent,ineffect,theadviceofconsultants.However,theadvicemightbebetterthan

whataschoolordistrictcouldobtainonitsown.Practiceguideauthorsarenationallyrecognizedexpertswhocollectivelyendorsetherecommendations,justifytheirchoiceswithsupportingevidence,andfacerigorousindependentpeerreviewoftheirconclusions.Schoolsanddistrictswouldlikelynotfindsuchacomprehensiveapproachwhenseek-ingtheadviceofindividualconsultants.

InstituteofEducationSciences

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AppendixB

AbouttheAuthors

Panel

RobertSiegler,Ph.D.,istheTeresaHeinzProfessorofCognitivePsychologyatCarn-egieMellonUniversity.Hiscurrentresearchfocusesonthedevelopmentofestimationskillsandhowchildren’sbasicunderstandingofnumbersinfluencestheirestimationandoverallmathachievement.Thegeneralover-lappingwavestheoryofcognitivedevelop-ment,describedbySieglerinhis1996bookEmerging Minds,hasprovenusefulforunder-standingtheacquisitionofavarietyofmathskillsandconcepts.HisotherbooksincludeHow Children Discover New Strategies,How Children Develop,andChildren’s Thinking.SieglerwastheTischDistinguishedVisit-ingProfessoratTeachersCollege,ColumbiaUniversity,for2009/10andwasamemberofthePresident’sNationalMathematicsAdvisoryPanelfrom2006to2008.HereceivedtheBrothertonFellowshipfromtheUniversityofMelbournein2006andtheAmericanPsycho-logicalAssociation’sDistinguishedScientificContributionAwardin2005.

ThomasCarpenter,Ph.D.,isemeritusprofessorofcurriculumandinstructionattheUniversityofWisconsin–MadisonanddirectoroftheDiversityinMathematicsEducationCenterforLearningandTeach-ing.Hisresearchinvestigateshowchildren’smathematicalthinkingdevelops,howteach-ersusespecificknowledgeaboutchildren’smathematicalthinkingininstruction,andhowchildren’sthinkingcanbeusedasabasisforprofessionaldevelopment.Dr.CarpenterisaformereditoroftheJournal for Research in Mathematics Education.AlongwithEliza-bethFennema,MeganFranke,andothers,hedevelopedtheCognitivelyGuidedInstruc-tionresearchandprofessionaldevelopmentproject.Heiscurrentlyfocusedonissuesofequityandsocialjusticeinmathematicsteachingandlearning.

Francis(Skip)Fennell,Ph.D.,isaprofessorofeducationatMcDanielCollegeandproject

directoroftheElementaryMathematicsSpe-cialistsandTeacherLeadersProject.Dr.Fennelliswidelypublishedinprofessionaljournalsandtextbooksrelatedtoelementaryandmiddlegrademathematicseducation,andheservedontheNationalMathematicsAdvisoryPanel,chairingtheConceptualKnowledgeandSkillsTaskGroup.In2008,hecompletedatwo-yearpresidencyoftheNationalCouncilofTeachersofMathematics.HeservedasoneofthewritersofthePrinciples and Standards for School MathematicsandtheCurriculum Focal Points,bothfortheNationalCouncilofTeach-ersofMathematics.Dr.Fennelliscurrentlyprincipalinvestigatoronaprojectaimedatdesigningagraduatecurriculumforelemen-tarymathematicsteacherleaders,creatingaclearinghouseofexistingmaterials,anddevelopingsupportmaterialsforelementarymathematicsspecialistsandteacherleaders.

DavidGeary,Ph.D.,isacognitive-develop-mentalpsychologist,aswellasaCurators’ProfessorandThomasJeffersonProfessorinthedepartmentofpsychologicalsciencesattheUniversityofMissouri.Hehasbeenstudy-ingdevelopmentalandindividualdifferencesinbasicmathematicalcompetenciesformorethan20yearsandiscurrentlydirectingalongitudinalstudyofchildren’smathematicaldevelopmentandlearningdisorders.Gearyistheauthorofthreebooks,includingChildren’s Mathematical Development,andtheco-authorofafourth.Inaddition,heservedontheNationalMathematicsAdvisoryPanelandwasoneoftheprimarycontributorstothe1999MathematicsFrameworkforCaliforniaPublicSchoolsforkindergartenthroughgrade12.DistinctionsreceivedincludetheChancellor’sAwardforOutstandingResearchandCreativeActivityintheSocialandBehavioralSciencesandaMERITawardfromtheNationalInsti-tutesofHealth.

W.James(Jim)Lewis,Ph.D.,istheAaronDouglasProfessorofMathematicsattheUniversityofNebraska–Lincoln(UNL),aswellasdirectoroftheschool’sCenterforSci-ence,Mathematics,andComputerEducation.Dr.Lewisisprincipalinvestigatorfortwo

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AppendixBcontinued

NationalScienceFoundationMathSciencePartnerships,NebraskaMATHandtheMathintheMiddleInstitutePartnership.HewaschairoftheConferenceBoardoftheMathematicalSciencescommitteethatproducedThe Mathematical Education of Teachersandco-chairoftheNationalResearchCouncilcommitteethatproducedthereportEducating Teachers of Science, Mathematics, and Technology: New Practices for the New Millennium.Dr.Lewiswasalsoco-principalinvestigatorforMathMatters,aNationalScienceFoundationgranttorevisethemathematicseducationoffutureelementaryschoolteachersatUNL.

YukariOkamoto,Ph.D.,isaprofessorinthedepartmentofeducationattheUniversityofCalifornia–SantaBarbara.Herworkfocusesoncognitivedevelopment,theteachingandlearningofmathematicsandscience,andcross-culturalstudies.Sheisparticularlyinterestedinchildren’sacquisitionofmath-ematical,scientific,andspatialconceptsandparticipatedinthevideostudiesofmathemat-icsandscienceteachingaspartoftheThirdInternationalMathematicsandScienceStudy(TIMSS).Dr.Okamoto’srecentpublicationsincludeFourthGraders’ Linking of Rational Number Representation: A Mixed Method Approach and Comparing U.S. and Japanese Elementary School Teachers’ Facility for Linking Rational Number Representations.

LaurieThompson,M.A.,has10yearsofexperienceasanelementaryteacherandmathresourceteacher.Herexperience

includesteaching1st,3rd,4th,and5thgradesinCarrollCounty,Maryland;LoudonCountyPublicSchools,Virginia;andKatyIndependentSchoolDistrict,Texas.AsanelementarymathresourceteacherinLoudonCounty,Ms.Thompsonworkedwithelemen-tarymathteacherstoteam-teachlessons,organizeguidedinstructionalcenters,andconductsmall-groupinstruction.Inthisrole,shealsodevelopedandevaluatedmathematicslessonsandmaterialsforkindergartenthrough5th-gradeclassrooms.Shehasservedasamentorandteamleaderfornewteachersandparticipatedinprofessionallearningcommunities.

JonathanWray,M.A.,istheinstructionalfacilitatorforsecondarymathematicscur-ricularprogramsintheHowardCounty(Maryland)PublicSchoolSystem.Herecentlycompletedatwo-yeartermaspresidentoftheMarylandCouncilofTeachersofMath-ematics(MCTM).Mr.WraywasselectedastheMCTMOutstandingTeacherMentorin2002andashisdistrict’sOutstandingTech-nologyLeaderinEducationbytheMarylandSocietyforEducationalTechnologyin2004.HeservesontheeditorialpanelofTeaching Children Mathematics,apeer-reviewedjournalproducedbytheNationalCouncilofTeachersofMathematics.Mr.Wrayalsohasservedasaclassroomteacherforprimaryandintermedi-ategrades,agifted/talentedresourceteacher,anelementarymathematicsresourceteacher,acurriculumandassessmentdeveloper,andaneducationalconsultant.

StaffJeffreyMaxisaresearcheratMathematicaPolicyResearchwithexperienceconductingevaluationsintheeducationarea.Hiscur-rentworkfocusesonteacherqualityissues,includingmeasuresofteachereffective-ness,thedistributionofteacherquality,andteacher-compensationreform.Mr.MaxalsocontributestotheWhatWorksClearinghouse,previouslyworkingonthepracticeguidethataddressesaccesstohighereducation.His

priorexperienceincludesteachingkindergar-teninaNewOrleanspublicschool.

MoiraMcCulloughisaresearchanalystatMathematicaPolicyResearchandhasexperi-encewitheducationevaluationsandresearch.Ms.McCulloughhasworkedextensivelyfortheWhatWorksClearinghouse.Sheisacerti-fiedreviewerofstudiesacrossseveralareasandcoordinatedtheelementaryschoolmathtopicarea.Shecontributedtothepractice

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AppendixBcontinued

guideaddressingaccesstohighereducationandtoresearchperspectivessynthesizingexpertrecommendationstostatesandschooldistrictsforuseoffundsfromtheAmericanRecoveryandReinvestmentAct.Shealsohasexperiencewithmeasuresofteachereffec-tivenessinmathematics.

AndrewGothroisaresearchanalystatMath-ematicaPolicyResearch.Hehasexperienceprovidingresearchsupportandconductingquantitativedataanalysisontopicsrangingfromchilddevelopmenttoantipovertypro-grams.Mr.Gothrosupportedthepanelonthisprojectbyidentifyingandorganizingrelevant

research,synthesizingfindingsfromreviewedstudies,andcraftinglanguageforanaudienceofeducationpractitioners.

SarahPrenovitzisaresearchassistant/programmeratMathematicaPolicyResearch.Shehasexperienceprovidingresearchsupportandconductingdataanalysisforstudiesofteacherincentiveprogramsandprofessionaldevelopmentprograms,aswellasprogramstosupportandencourageemploymentforpersonswithdisabilities.Shealsohasdevel-opedcompanionmaterialstoaccompanyacurriculumforHeadStartstaffonusingcontin-uousassessmenttoshapeclassroompractice.

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AppendixC

DisclosureofPotentialConflictsofInterest

Practiceguidepanelsarecomposedofindividualswhoarenationallyrecognizedexpertsonthetopicsaboutwhichtheyaremakingrecommendations.IESexpectstheexpertstobeinvolvedpro-fessionallyinavarietyofmattersthatrelatetotheirworkasapanel.Panelmembersareaskedtodisclosetheseprofessionalactivitiesandinstitutedeliberativeprocessesthatencouragecriticalexam-inationoftheirviewsastheyrelatetothecontentofthepracticeguide.Thepotentialinfluenceofthepanelmembers’professionalactivitiesisfurthermutedbytherequirementthattheygroundtheirrecommendationsinevidencethatisdocumentedinthepracticeguide.Inaddition,beforeallpracticeguidesarepublished,theyundergoanindependentexternalpeerreviewfocusingonwhethertheevidencerelatedtotherecommendationsintheguidehasbeenpresentedappropriately.

Theprofessionalactivitiesreportedbyeachpanelmemberthatappeartobemostcloselyassoci-atedwiththepanelrecommendationsarenotedbelow.

JimLewisreceivesroyaltiesasanauthorofMath Vantage,amathematicscurriculumformiddleschoolstudents.Thisprogramisnotmentionedintheguide.

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AppendixD

RationaleforEvidenceRatingsa

Thepanelconductedaninitialsearchforresearchfrom1989to2009onpracticesforimprovingstu-dents’learningoffractions.Thesearchfocusedonstudiesofinterventionsforteachingfractionstostudentsinkindergartenthrough8thgradethatdidnotexclusivelyfocusonstudentswithdiagnosedlearningdisabilities;studiesexaminedstudentsinboththeUnitedStatesandothercountries.

Panelistsidentifiedmorethan3,000studiesthroughthisinitialsearch,including125withcausaldesignsreviewedaccordingtoWhatWorksClearinghouse(WWC)standards.Twenty-sixofthestud-iesmetevidencestandardswithorwithoutreservations.Giventhelimitedresearchonpracticesforimprovingstudents’fractionknowledge,thepanelexpandeditssearchbeyondfractionstoidentifystudiesrelevantfornumberlines(Recommendation2)andprofessionaldevelopment(Recommendation5).Thisledpanelmemberstoanadditionalsevenstudiesthatmetstandardswithorwithoutreserva-tions.ThepanelalsoexaminedstudiesthatdidnothavedesignseligibleforaWWCreviewbutwererel-evanttotherecommendations,includingcorrelationalstudies,casestudies,andteachingexperiments.

Recommendation1.Buildonstudents’informalunderstandingofsharingandproportionalitytodevelopinitialfractionconcepts.

Levelofevidence:MinimalEvidence

Thepanelassignedaratingofminimal evidencetothisrecommendation.Therecom-mendationisbasedonsevenstudiesshowingthatstudentshaveanearlyunderstandingofsharingandproportionality158andtwostud-iesofinstructionthatusedsharingscenariostoteachfractionconcepts.159 However,noneofthestudiesinthislattergroupmetWWCstandards.Despitethislimitedevidence,thepanelbelievesthatstudents’informalknowl-edgeofsharingandproportionalityprovidesafoundationforteachingfractionconcepts.

Thepanelseparatelyexaminedtheresearchonsharingactivitiesandproportionalrela-tionsforthisrecommendation.

Sharingactivities.Childrenhavetheabilitytocreateequalsharesatanearlyage.Childrenasyoungasage5cancompletebasicsharingtasksthatinvolveevenlydistributingasetof12or24objectsamongtwotofourrecipi-ents.160Inonestudy,most5-year-oldchildrencoulddothisevenwhenusingdifferent-size

units(i.e.,equallydistributingsingle,double,andtripleblocks).Theabilitytocreateequalsharesimproveswithage,with6-year-oldchildrenperformingbetterthan4-and5-year-olds.161Sharingcontinuousobjectsismoredifficultforyoungchildrenthansharingasetofobjects:childreninonestudyhadmoredif-ficultysharingaropeamongthreerecipientsthansharingasetofcrackers.162

Children’sunderstandingofhowtosharedoesnotnecessarilyextendtounderlyingfractionconcepts.Manystudentsdonotunderstandthatsharingthesamesetofobjectswithmorepeopleresultsinsmallersharesforeachperson.163Onestudythatpotentiallymetstandardsshowedanimprovedunderstandingofthisconceptamongkindergartenstudentswhoweregivenresultsfromsharingscenarioswithdifferentnumbersofsharers(i.e.,differ-entdenominators).164Thisstudydemonstratedthepotentialforusingsharingactivitiesasthebasisforteachingearlyfractionconcepts.However,areviewofthestudycouldnotbecompletedbecauseinsufficientinforma-tionwasprovidedonthenumberofschoolsassignedtoeachcondition.

Twocasestudiesshowedhowanearlyunderstandingofsharingcouldbeusedtoteachfractionstoelementarystudents.165

aEligiblestudiesthatmeetWWCevidencestandardsormeetevidencestandardswithreservationsareindicatedbyboldtextintheendnotesandreferencespages.

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AppendixDcontinued

Inbothstudies,ateacherposedvariousstoryproblemsbasedonsharingscenariostoteachfractionconceptssuchasequivalenceandordering,aswellasfractioncomputa-tion.Forexample,studentssolvedproblemsaboutpeoplesharingvaryingamountsoffoodortheuseofseatingarrangementstoshareasetofobjectsindifferentways.Theinstructioninbothstudiesincludedstoryproblemsbasedonrealisticsituations,opportunitiesforstudentstousetheirowndrawingsandstrategiestoobtainsolutions,andwhole-classdiscussions.

OneofthesestudiesexaminedaDutchcur-riculumfor4th-gradestudentsbutdidnotmeetstandardsbecauseonlyoneclassroomwasassignedtothetreatment.166 Theotherstudypresentedafive-weekinstructionalunitto171st-gradestudentsbutlackedacontrolgroup,soitdidnothaveareviewabledesign.However,bothstudiesreportedpositiveresultswithusingsharingscenariostoteachfractionconcepts.167

Proportionalrelations.Youngchildrenhaveanearlyunderstandingofproportionalrela-tions.Threestudiespresentedproportionswithgeometricfiguresoreverydayobjectsandhadstudentsidentifyorcreateamatch-ingproportion.168Forexample,inonestudy,theexperimenterpretendedtoeataportionofapizzaandhadchildrenpretendtoeatthesameproportionfromaboxofchocolates.169

Inanotherstudy,studentsmatchedpropor-tionsrepresentedbyboxesfilledwithblueandwhitebricks.170Byage6,childrenmatchedequivalentproportionsinallofthesestudies.

One-halfplaysanimportantroleinchildren’searlyproportionalreasoningabilities.Childrenperformedbetterwhenchoosingbetweenoptionsthatcrossedthehalfpoint(i.e.,onefigurewasmorethanone-halffilled,andtheotherwaslessthanone-halffilled)thanbetweentwoproportionsthatwerebothmorethanorlessthanone-halffilled.171 Chil-drentendedtohavemoredifficultymatchingproportionsrepresentedbydiscreteobjectsthanbycontinuousobjects.172

Children’searlyunderstandingofproportionalrelationsalsoisreflectedintheirabilitytosolvebasicanalogies.Analogiesaresimilartopropor-tionsinthatstudentsmustidentifyarelationinthefirstsetofitemsandthenapplythisrelationtoasecondsetofitems.Onestudyfoundthatchildrenages6and7performedabovechanceonanalogiesbasedonsimplepatternsorpro-portionalequivalence.173Forexample,studentscouldcompletetheanalogy,“Halfcircleistohalfrectangleasquartercircleistoquarterrectangle.”Onestudyfoundthatchildrencouldmaptherelativesizesofitemswithinathree-itemsettotherelativesizesofitemswithinanothersetofthreeobjects.174Forexample,whentheexperimenterselectedthelargestofthreedifferent-sizecups,childrencouldpickthecorrespondingcupfromtheirsetofthreecups.

Thepaneldidnotidentifystudiesmeetingstandardsthatexaminedtheeffectofusingthisearlyknowledgetoteachfractioncon-cepts.However,onestudythatpotentiallymetstandardsexaminedawaytoimprovestudents’abilitytomatchequivalentpropor-tions.175Theauthorprovided6-to8-year-oldchildrenwithfeedbackandexplanationsabouthowtousethehalfboundarytoidentifyequivalentproportions.Thisstrategyfocusedchildrenonthepart-partrelationbetweenshadedandunshadedareasusedtorepresentproportions;theauthorreportedpositiveeffectsonchildren’sabilitytoidentifywhichoftwoglasseswasmorefull—and,therefore,onwhetherstudentscoulddifferentiatebetweenabsoluteandrelativeamountsofwater.

Recommendation2.Helpstudentsrecognizethatfractionsarenumbersandthattheyexpandthenumbersystembeyondwholenumbers.Usenumberlinesasacentralrepresentationaltoolinteachingthisandotherfractionconceptsfromtheearlygradesonward.

Levelofevidence:ModerateEvidence

Thepanelratesthisrecommendationasbeingsupportedbymoderate evidence,basedon

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AppendixDcontinued

threestudiesthatmetWWCstandardsandusednumberlinestoteachstudentsaboutthemagnitudesofwholenumbers;176onestudythatmetWWCstandardsandshowedthatinstructionwithnumberlinesimprovedstu-dents’understandingofdecimalfractions;177

andfourstudiesthatshowedstrongcorrela-tionsbetweennumberlineestimateswithwholenumbersandperformanceonarith-meticandmathematicalachievementtests.178

Anotherstudydemonstratedthatapropertyofnumberlineestimatesthathasbeendocumentedextensivelywithwholenumbersalsoispresentwithfractions(specifically,logarithmictolineartransitionsinpatterns).Thissuggeststhatrepresentationsofnumeri-calmagnitudesinfluenceunderstandingoffractionsaswellasofwholenumbers.179 Thepanelbelievesthatgiventheclearapplicabilityofnumberlinestofractionsaswellaswholenumbers,thesefindingsindicatethatnumberlinescanimprovefractionlearningforelemen-taryandmiddleschoolstudents.

Theevidencetosupportthisrecommenda-tionincludesstudiesthatexaminedtheuseofnumberlinesandotherlinearrepresentationstoteachwholenumberandfractionconcepts.

Numberlinesforwholenumberconcepts.(seeTableD.1.)Threestudiesthatmetstan-dardsfoundthatbrieflyplayingalinearboardgamewithnumbersimprovedpreschoolstudents’understandingofwholenumbermagnitude.180 Inthestudies,studentsfromlow-incomebackgroundsplayedanumericboardgame20to30timesoverthecourseoffourtofivesessionslasting15to20minuteseach.Thegameinvolvedmovingamarkeroneortwospacesatatimeacrossahorizontalboardthathadthenumbers1to10listedinorderfromlefttorightinconsecutivesquares.Studentsusedaspinnertodeterminewhethertomakeoneortwomovesandthensaidoutloudthenumbertheyhadspunandthenumbersonthesquaresastheymoved.Theexperimenterplayedthegamewitheachchildandhelpedeachcorrectlynamenum-bers.Controlstudentsintwoofthestudiesplayedthesamegamebutwithcolorsrather

thannumbers,181andcontrolstudentsintheotherstudycompletedcountingandnumber-identificationtasks.182

Thelinearboardgame,whichthepanelviewsasaproxyfornumberlines,hadapositiveeffectonstudents’abilitytocomparethesizeofwholenumbers.Authorsofthethreestudiesreportedsignificanteffectsizesof0.75,0.99,and1.17onaccuracyincompar-ingwholenumbers(from0to10).183 Thelinearboardgamealsoimprovedparticipatingstudents’abilitytolocatewholenumbersonanumberlineaccurately.Thesestudiesmeasuretheaccuracyofstudents’numberlineesti-matesusingameasurecalledpercentabso-luteerror,whichisthedifferencebetweenastudent’sestimateandtheactualnumberdividedbythescaleofthenumberline.Twoofthestudiesfoundeffectsizesforpercentabsoluteerrorof0.63(authorreported)and0.86(WWCcalculated).184Oneofthestudiesalsoreportedthatplayingthegamesignifi-cantlyimprovedstudents’abilitytolearntheanswerstoadditionproblemsonwhichtheyreceivedfeedback.185

ResearchsupportingtheuseofnumberlineswithwholenumbersincludestwoadditionalstudiesthatmetWWCstandards.Oneofthestudieshadstudentsplace10evenlyspacednumbersonanumberlinebeforelocatingnumbersona0-to-100numberline.186 Theauthorsreportthatthisapproachledtoasubstantivelyimportantbutnotsignificantincreaseintheaccuracyofstudents’numberlineestimates,whereasstudentsinthecontrolgroup,wholocatedonenumberatatime,didnotimprove.187(TheWWCdefinessubstan-tivelyimportant,orlarge,effectsonoutcomestobethosewitheffectsizesgreaterthan0.25standarddeviations.188)

Thesecondsupportingstudyusedanumberlinetoimprove1st-gradestudents’perfor-manceonadditionproblemsforwhichtheyhadbeentrained.189 Treatmentgroupstudentsviewedtheaddendsandsumsoffouraddi-tionproblemsonanumberline;controlgroupstudentsreceivedfeedbackontheproblems

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AppendixDcontinued

TableD.1.StudiesofinterventionsthatusednumberlinestoimproveunderstandingofwholenumbermagnitudethatmetWWCstandards(withorwithoutreservations)

Citation GradeLevel AnalysisSampleSize Intervention Comparison

SieglerandRamani Preschool 36students Studentsplayalinear Studentsplayalinear(2008) boardgamewith boardgamewithcolors.

numbers.

RamaniandSiegler Preschool 112students Studentsplayalinear Studentsplayalinear(2008) boardgamewith boardgamewithcolors.

numbers.

SieglerandRamani Preschool 88students Studentsplayalinear Studentsparticipatein(2009) boardgamewith countingandnumber-

numbers.190 identificationtasks.

SieglerandBooth 1stand2nd191 55students Studentsplace10evenly Studentsusenumber(2004) spacednumbersona lineswithoutlocating

numberline. evenlyspacednumbersfirst.

BoothandSiegler(2008)

1st 52students Studentsreceiveanumberlineshowingaddendsandsums

Studentssolvetrainedadditionproblemswith-outanumberline.

fortrainedadditionproblems.192

butdidnotuseanumberline.Theauthorsreportedthattreatmentgroupstudentsweremorelikelythancontrolgroupstudentstoanswerthesameadditionproblemscorrectlylater.Inaddition,thestudynotedthatthenumberlineexperienceledtoimprovedqual-ityoferrorsontheadditionproblems(errorsthatwereclosertothecorrectanswer).

Thepanelalsoidentifiedevidenceshowingarelationbetweenstudents’accuracyinlocat-ingwholenumbersonanumberlineandgeneralmathachievement.193Thesestudiesshowapositivesignificantrelationbetweenthelinearityofnumberlineestimatesandgeneralmathachievementforstudentsinkindergartenthrough4thgrade,withcorrela-tionsrangingfrom0.39to0.69.Theaccuracyofnumberlineestimates(i.e.,howcloseanumberistoitsactualposition)waspositivelyrelatedtogeneralmathachievement,withonestudyfindingasignificantrelationrang-ingfrom0.37to0.66);194 anadditionalstudyfoundpositivebutnon-significantrelationsfor1st-and2nd-gradersinoneexperimentandsignificantpositiverelationsfor2nd-and4th-gradersinanotherexperiment.195

Numberlinesforteachingfractionconcepts.OnestudythatmetWWCstandards

examinedtheuseofnumberlinesforcom-paringthemagnitudeofdecimals.196 Sixty-onestudentsin5thand6thgradesplayedacomputergameinwhichtheylocatedadecimal’spositionona0-to-1numberline.Studentsinthetreatmentandcontrolgroupscompleted15problemsduringsessionslast-ingabout40minutes.Thestudyinvolvedthreetreatmentgroupsthatreceivedinter-ventionsdesignedtohelpstudentscorrectlyrepresenttheproblem:thefirsttreatmentgroupreceivedapromptforstudentstonoticethetenthsdigitofeachdecimal,thesecondgroupusedanumberlinewiththetenthsplacemarked,andthethirdgroupreceivedboththepromptsandmarkedtenthsonthenumberline.Studentsinthecontrolgroupalsosolvedcomputer-basednumberlineproblems,butwithouttheassistanceoftheseinterventions.

Sincestudentsinboththetreatmentandcon-trolgroupsusednumberlines,thestudydoesnotprovidecausalevidenceforwhetherusingnumberlinesimprovesstudents’understand-ingofdecimals.However,theresultsindicatethatfocusingoncertainaspectsofthenum-berline—specifically,noticingandmarkingthetenthsplace—ledtosignificantimprove-mentsinstudents’abilitytolocatedecimals

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AppendixDcontinued

onanumberline.Thecombinationofplac-ingtenthsmarkingsonthenumberlinesandpromptingstudentstonoticeandthinkaboutthem(treatmentgroup3)significantlyimprovedstudents’abilitytolocatedecimalfractionsonanumberlinerelativetowhenneitherwaspresent(effectsizeof0.57).Whenstudentsonlyreceivedthetenthsmarkings(treatmentgroup1)oronlyheardtheprompts(treatmentgroup2),theinterventionsdidnothaveasignificanteffect.Thepanelbelievesthisoutcomeindicatesthatthecombinationofthepromptsandmarkings,togetherwithuseofthenumberline,leadstoincreasedunder-standingofdecimals’magnitude.

Acomparisonofstudents’conceptualunder-standingofdecimalsbeforeandaftertheinterventionprovidesadditionalevidenceontheusefulnessofnumberlines.Playingthecomputer-basednumberlinegameledtoimprovementsintreatmentandcontrolstu-dents’conceptualunderstandingofdecimals,includingtheirabilitytocomparerelativemagnitudesoffractions,identifyequivalentfractions,andunderstandplacevalue.Thisissuggestiveevidence,becausethereisnocomparisongroupofstudentswhodidnotuseanumberline.

Anotherstudyexaminedtheuseofnumberlinesinfractionsinstructionbutdidnotmeetstandards.197ThestudycomparedtwoDutchcurriculaoverthecourseofaschoolyear.Onecurriculumfocusedontheuseofnumberlinesandmeasurementcontextstoteachfractions;theothercurriculumusedcirclesandpart-wholerepresentationsoffractions.Studentsinthetreatmentgroupmeasuredobjectsusingdifferent-sizebarsandcomparedfractionsonanumberline.Theauthorsreportedpositiveeffectson9-to10-year-olds’understandingoffractions.However,thestudydidnotmeetstandards,becauseonlyoneclassroomofstudentswasassignedtothetreatment.Anotherproblemininterpretingthestudywasthattheexperi-mentalgroupencouragedstudentinteraction,whereasthecontrolgroupstudentsprimarily

workedalone.Asaresult,distinguishingtheeffectoftheseinstructionalapproachesfromtheeffectofthecurriculumwasnotpossible.

Twoadditionalstudiesthatwerenoteligibleforreviewfoundmixedresultsofusinganumberlinetoteachfractionconcepts.198

Onestudyexaminedusinganumberlinetoteachfractionadditiontoaclassof6th-gradestudents.Basedonclassroomobservationsandinterviewswiththeteacherandtwostudents,theauthorsfoundthatstudentshaddifficultyviewingpartitionsonanumberlineasfixedunits,aswellasdifficultyassociatingequivalentfractionswithasinglepointonthenumberline.Minordifferencesinhowtheteacherpresentedthenumberlineaffectedwhetherstudentsviewedthepartitionsasfixedunits.

Thesecondstudydescribedthreesmallcasestudiesoffractioninstructionthatusednumberlinesforrepresentingandorderingfractions.199Inthisstudy,4th-and5th-gradestudentshadtroublelocatingfractionsonanumberlinewhenfractionswereinreducedformandthenumberlinewasorganizedbyasmallerunitfraction(e.g.,theyhaddifficultylocating1/3onanumberlinedividedintosixths).However,theauthorsalsoreportedthatnumberlineinstructionimprovedstu-dents’abilitytoworkwithfractions.

Additionalevidence.Othertypesofevi-dencealsosupportedtheimportanceofdevelopingstudents’abilitytounderstandfractionsonanumberline.Students’abilitytolocatedecimalsonanumberlineisrelatedtogeneralmathachievement.Astudyof5th-and6th-gradeGermanstudentsfoundasignificantpositivecorrelationbetweenstudents’skillinestimatingthelocationofdecimalsonanumberlineandtheirself-reportedmathematicsgradesinschool.200Inaddition,amathematician’sanalysisindicatedthatlearningtorepresentthefullrangeofnumbersonnumberlinesisfundamentaltounderstandingnumbers.201

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AppendixDcontinued

Recommendation3.Helpstudentsunderstandwhyproceduresforcomputationswithfractionsmakesense.

Levelofevidence:ModerateEvidence

Thepanelratedthisrecommendationasbeingsupportedbymoderate evidence,basedonstudiesspecificallyrelatedtoconceptualandproceduralknowledgeoffractions.ThisevidenceratingisbasedonthreerandomizedcontrolledtrialsthatmetWWCstandardsanddemonstratedtheeffectivenessofteachingconceptualunderstandingwhendevelopingstudents’computationalskillwithdecimals.202

Interventionsthatiteratedbetweeninstruc-tiononconceptualknowledgeandproceduralknowledgehadapositiveeffectondecimalcomputation.203Althoughthestudiesfocusedondecimalsandwererelativelysmall-scale,thepanelbelievesthatthethree,togetherwiththeextensiveevidencethatmeaning-fulinformationisrememberedmuchbetterthanmeaninglessinformation,204providepersuasiveevidencefortherecommendation.Additionalsupportfortherecommendationcomesfromfourcorrelationalstudiesof4th-,5th-and6th-gradestudentsthatshowedsignificantrelationsbetweenconceptualandproceduralknowledgeoffractions.205Consen-susdocuments,suchasAdding It UpandtheNationalMathematicsAdvisoryPanelreport,alsosuggesttheimportanceofcombininginstructiononconceptualunderstandingwithproceduralfluency.206

Panelmembersfocusedtheirreviewonstudiesthatspecificallyexaminedinterven-tionstodevelopstudents’understandingoffractioncomputation.ThreerandomizedcontrolledtrialsthatmetWWCstandardssupporttherecommendation.207Twoofthestudiesusedcomputer-basedinterventionstocomparedifferentwaysoforderingconcep-tualandproceduralinstructionfor6th-gradestudents.208Thestudies’treatmentgroupsalternatedbetweenconceptuallessonsondecimalplacevalueandprocedurallessonsonadditionandsubtractionofdecimals;the

controlgroupscompletedalloftheconcep-tuallessonsbeforereceivinganyofthepro-cedurallessons.Theinterventionconsistedofsixlessons,duringwhichstudentssolvedwordproblemswhilereceivingfeedbackfromthecomputerprogramasneeded.Bothoftherelativelysmall-scalestudiesfoundpositiveeffectsofiteratingbetweenconceptualandprocedurallessons.Onerandomlyassigned26studentsandfoundalarge,significanteffectoncomputationalproficiencywithdecimals(effectsize=2.38);theotherstudyrandomlyassignedfourclassroomsandfoundasubstantivelyimportant,butnotsignificant,effect(effectsize=0.63).

Thethirdstudyexaminedaninterventiondesignedtoimprovestudents’conceptualunderstandingofhowtolocatedecimalsonanumberline.209Init,5th-and6th-gradestudentspracticedlocatingfractionsonanum-berlineusingacomputer-basedgamecalledCatchtheMonster.Studentsinthetreatmentgroupsreceivedeitheraprompttonoticethetenthsdigitoranumberlinedividedintotenths—twointerventionsthatthepanelviewsasbuildingstudents’conceptualknowl-edge.Controlstudentsdidnotreceivethepromptsanduseda0-to-1numberlinewith-outthetenthsmarked.Bothtreatmentshadasignificant,positiveeffectonstudents’abilitytolocatedecimalsonanumberlinewithoutthepromptsorthetenthsmarked.Receivingboththepromptsandthenumberlinewiththetenthsmarkedhadagreaterimpactthanreceivingthetwointerventionsseparately.

Thepanel’srecommendationalsoissup-portedbycorrelationalevidencethatshowsasignificantrelationbetweenstudents’concep-tualandproceduralknowledgeoffractions.Hechtetal.(2003)administeredavarietyofassessmentsto1055th-graders,andHecht(1998)assessed1037th-and8th-graderstoexaminehowconceptualunderstandingandproceduralskillarerelated.HechtandVagi(inpress)includedasampleof1814th-and5th-graderstomeasuretherelationbetweenconceptualandproceduralknowledge.Thestudiesmeasuredbothconceptualand

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AppendixDcontinued

TableD.2.StudiesofinterventionsthatdevelopedconceptualunderstandingoffractioncomputationthatmetWWCstandards(withorwithoutreservations)

CitationGradeLevel

AnalysisSampleSize Intervention Comparison Outcome EffectSize210

Rittle-Johnsonand

6th 4classrooms Studentscompletesixcomputer-based

Studentscompletesixcomputer-based

Computationalproficiency

0.63,ns

Koedinger(2002)

lessonsoncomputa-tionwithdecimals,alternatingbetween

lessonsoncomputa-tionwithdecimals,completingallofthe

withdecimals

conceptualandproce-durallessons.

conceptuallessonsbeforetheprocedurallessons.

Rittle-Johnsonand

6th 26students Studentscompletesixcomputer-based

Studentscompletesixcomputer-based

Decimalarithmetic

2.83,sig

Koedinger(2009)

lessonsoncomputa-tionwithdecimals,alternatingbetween

lessonsoncomputa-tionwithdecimals,completingallofthe

conceptualandproce-durallessons.

conceptuallessonsbeforetheprocedurallessons.

Rittle-Johnson,Siegler,andAlibali(2001)

5thand6th

61students Whenlocatingdeci-malsonanumberline,studentsreceiveaprompttonoticethetenthsdigitandusea0-to-1number

Whenlocatingdecimalsonanumberline,studentsuseda0-to-1numberlinewithoutthetenthsmarked.

Locatingdecimalsonanumberline

0.57,sig

linewiththetenthsmarked.

ns=notsignificantsig=statisticallysignificant

proceduralknowledgeoffractionsandfrac-tioncomputation.Allthreestudiesfoundthataftercontrollingforotherfactors,conceptualknowledgeoffractionssignificantlypredictedstudents’abilitytosucceedatfractioncom-putationandestimation.Whilethesestudiesshowacorrelationbetweenconceptualandproceduralknowledge,theydonotestablishwhetherinterventionstodevelopconceptualknowledgeimproveproceduralknowledge.

Inanotherexperiment,Rittle-Johnson,Siegler,andAlibali(2001)211foundthat5th-gradestudents’understandingofdecimals(i.e.,relativemagnitudeandequivalence)wassignificantlyrelatedtotheirabilitytolocatedecimalsonanumberline.212Con-trollingforinitialproceduralknowledge,conceptualknowledgewasfoundtoaccountfor20%ofperformancevarianceonatestofproceduralknowledge.

Manipulativesandrepresentations.Thepanelidentifiedevidencethatsupportsthefirstactionstep,whichrecommendsusingmanipu-lativesandvisualrepresentationstoteachfrac-tioncomputation.Tworandomizedcontrolledtrials,bothunpublisheddissertations,thatmetWWCstandardsfoundthatusingmanipulativeshadapositiveeffectonfractioncomputa-tion.213Nishida(2008)214conductedarelativelysmall-scalestudyontheuseoffractioncirclestoteachnumerator-denominatorrelationsandotherfractionconcepts.Thestudyfoundthathavingstudentsusefractioncircles,ratherthanobservingteachers’useofthem,significantlyimprovedstudents’understandingoffractionconceptsrelevanttocomputation(effectsize=0.73).Theuseofmanipulativefractioncirclesalsohadasubstantivelyimportant,butnotstatisticallysignificant,effectonfractionunder-standing,comparedwiththeuseofpicturesoffractionscircles.

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AppendixDcontinued

Thesecondstudyfoundthatusingavarietyofmanipulativestosupplementa3rd-gradefractionscurriculumimprovedstudents’understandingoffractionsandfractioncom-putation.215Thestudy’sunitincludedlessonsonfractionmagnitude,equivalence,addition,andsubtraction.Teachersinthestudyusedmanyofthesamematerials,butteachersinthetreatmentgroupalsoemployedvariousmanipulativesandmodels,includingfrac-tionsquares,fractiongames,fractionstrips,pizzas,fractionspinners,cubes,gridcards,paperstrips,virtualmanipulatives,cutouts,andshapes.Theuseofthesemanipulativeshadasubstantivelyimportant,butnotstatis-ticallysignificant,effectonatextbookassess-mentoffractionknowledgeandcomputation(effectsize=0.60).

Arandomizedcontrolledtrialthatpoten-tiallymeetsstandardsexaminedtheuseofmanipulativesandreal-worldcontextsforteachingfractions.216ThestudyexaminedacurriculumdevelopedbytheRationalNumberProject(RNP)thatemploysamulti-prongedapproachincorporatingmanipula-tives,real-worldcontexts,andestimationandfocusesonbuildingstudents’quantita-tivesenseoffractions.Teachersof5th-and6th-gradestudentswererandomlyassignedtouseeithertheRNPcurriculumoroneoftwocommercialcurriculathatincludedminimaluseofmanipulatives(AddisonWesley MathematicsorMathematics Plus).TheRNPcurriculumhadasignificantpositiveeffectonfractioncomputationandestimation(effectsize=0.27and0.65,respectively).However,thestudyprovidedinsufficientinformationtoassesssampleattrition,andamountofuseofmanipulativeswasonlyoneofmanydifferencesbetweenthecurricula,makingitdifficulttodistinguishwhichaspectsoftheinterventionledtothepositiveoutcomes.

Realworldcontextsandintuitiveunderstanding.Useofreal-worldconceptsalsocanimprovefractioncomputationprofi-ciency(Step4).ArandomizedcontrolledtrialthatmetWWCstandardsindicatedthatusinginformationfromstudentstopersonalize

lessonsonfractiondivisionsignificantlyimprovedtheirabilitytosolvefractiondivisionwordproblems.217Studentsinthetreatmentconditionreceivedinstructionviacomputer-assistedlessonsbasedoncontextssuggestedbythestudents;controlstudentsweretaughtusingabstractlessonswith-outsuchcontexts.Thetreatmenttargeted5th-and6th-gradestudentsduringasingle-lessonunitonfractiondivision.

Aquasi-experimentaldesignstudythatpotentiallymeetsstandardsevaluatedtheimpactofpracticingfractioncomputationwithproblemssetineverydaycontexts.218

Overthecourseofthreedays,studentsinthetreatmentgroupsolvedcontextualizedprob-lemsinvolvingcomputationwithdecimals.Problemsincludedreferencestosoft-drinkbottles,monetaryexchanges,andmeasure-ment.Thecontrolgroupsolvedsimilarprob-lemsbutwithoutanycontextualreferences.Basedontheauthor’scalculations,instructionusingcontextualizedproblemssignificantlyimprovedthestudents’abilitytoorderandcomparedecimals.Thestudyhadasmallsampleof1611- and12-year-oldsfromNewZealand;itpotentiallymetstandardsbecauseinsufficientinformationexistedtodemon-stratethatthetreatmentandcontrolgroupswereequivalentatbaseline.

Recommendation4.Developstudents’conceptualunderstandingofstrategiesforsolvingratio,rate,andproportionproblemsbeforeexposingthemtocrossmultiplicationasaproceduretousetosolvesuchproblems.


Thepanelassignedaratingofminimal evidencetothisrecommendation.Evidencefortheoverallrecommendationcomesfromconsensusdocumentsthatemphasizetheimportanceofproportionalreasoningformathematicslearning.219Thepanelsepa-ratelyreviewedevidenceforthethreeactionstepsthatcomprisethisrecommendation.

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AppendixDcontinued

Theseactionstepsaresupportedbycasestudiesdemonstratingthevarietyofstrate-giesstudentsusetosolveratio,rate,andproportionproblems;astudyofmanipula-tivesthatmetWWCstandards;andtwostud-iesthatmetstandardsandtaughtstrategiesforsolvingwordproblems.

Buildingonearlydevelopingstrategiesforsolvingproportionalityproblems.Evidenceforthefirstactionstepisbasedoncasestudiesthatexaminestudents’strate-giesforsolvingproportionalityproblems.Nostudiesbothmetstandardsandexam-inedtheeffectofusingstudents’developingstrategiestoimprovetheirunderstandingofproportionality.However,thepanelbelievesthatthefindingsofthesecasestudiesprovideabasisforusingaprogressionofproblemsthatbuildsonthesestrategiestodevelopstudents’proportionalreasoning.

Aliteraturereviewofearlyproportionalreasoningfoundthatstudentsinitiallytendtorelyonstrategiesthatbuildupadditivelyfromoneratiotoanother.220Studentswhousethisapproachmaynotunderstandthemultiplicativerelationsbetweenratios.221

Toillustratethispoint,acasestudyof214th- and5th-gradestudentsdescribedfourdevelopmentallevelsforsolvingproportion-alityproblems.222Oneimportantdifferenceamongtheselevelswaswhetherthedevelop-mentallevelsonlyinvolvedbuildingupfromsmallertolargerratiosorwhethertheyalsoincludedtheknowledgethatratios,likefrac-tions,canbereduced.

Carpenteretal.(1999)andLamon(1994)suggestedthattreatingratiosassingleunitsisanimportantdevelopmentalstepforproportionalreasoning.InCramer,Post,andCurrier(1993),8th-gradestudentsweremorelikelythan7th-gradestudentstosolvepro-portionalityproblemsbytreatingtheratioasaunitandbyfindinganequivalentfraction.Bothstudiesconfirmthatstudentshavemoredifficultywithproportionalityproblemsthatinvolvenon-integerrelations.223

Usingvisualrepresentationsandmanipulatives.Visualrepresentationsandconcretemanipulativescanincreasestudents’proficiencyinsolvingrate,ratio,andproportionproblems.Inarandomizedcontrolledtrialthatmetstan-dards,Fujimura(2001)evaluatedtheimpactofprovidingstudentswithconcretemanipulativestosolvemixtureproblems.Japanesestudentsin4thgradereceivedamanipulativetoassisttheminsolvingaproportionprobleminvolv-ingthemixtureoftwoliquids.Studentsusedthemanipulativetovisuallyrepresenttheunitrate,ortheamountoforangeconcentrateforeachunitofwater.Completingaproblemusingthemanipulativeimprovedstudents’abilitytolatersolvethesametypeofmixtureproblemswithoutthemanipulative.Studentsinthetreat-mentgroupperformedsignificantlybetterthanstudentswithnoexposuretomixtureproblemsduringtheintervention(effectsize=0.74).Thetreatmenthadasubstantivelyimportant,butnotstatisticallysignificant,effectrelativetoacontrolgroupinwhichstudentsreceivedaworksheettocalculatetheunitratetosolvemixtureproblems(effectsize=0.34).

Aninstructionalstrategythattaughtstudentstouseadatatableforrepresentinginforma-tioninamissingvalueproportionproblemhadasignificantpositiveeffectonthestu-dents’abilitytosolvetheseproblems.InastudythatmetWWCstandards,7th-gradersweretaughtaproblem-solvingstrategyinwhichtheyidentifiedtheproblemtype,rep-resentedtheprobleminatable,determinedthemultiplicativerelationbetweentheknownquantities,andthenappliedthatrelationtocalculatetheunknownquantity.224Research-ersrandomlyassignedfiveclassroomstoreceiveinstructionineithertheabovestrategyorasubstituteapproachinwhichstudentslearnedtorecognizetheproblemstructure,solvetheproblembysubstitut-ingintegersforanycomplexnumbers,andthenresolvetheproblemwiththecomplexnumbers.After10lessons,studentsinthetreatmentgroupperformedbetterthanthoseinthecontrolgrouponmissingvaluepropor-tionproblems.

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AppendixDcontinued

Inanotherrandomizedcontrolledtrialthatmetstandards,Terweletal.(2009)investigatedtheeffectivenessofinstructing5th-gradestudentstosolvepercentageproblemsbyconstructingrepresentationscollaborativelyinsteadofusingteacher-maderepresentationsandgraphs.Thisinterventionhadasubstantivelyimportant,butnotstatisticallysignificant,impactonstu-dentperformanceonaresearcher-constructedposttestofproblemsolvingwithpercentages(effectsize=0.41).

Strategiesforsolvingwordproblems.Theliteratureonteachingstrategiesforwordproblemsincludesmanystudiesout-sidethescopeofthisguide—studiesthatfocusonstudentsin9thgradeorabove,low-performingstudents,andstudentswithlearningdisabilitiesorontopicsotherthanratio,rate,orproportion.225Initsreviewofavailableresearch,theNationalMathematicsAdvisoryPanelusedthesestudiestosup-porttheteachingofexplicitstrategiesforsolvingwordproblemswithlow-performingstudentsandstudentswithlearningdis-abilities.226However,forthisactionstep,thepanelsoughtevidencespecificallyrelatedtostudentswithoutdiagnosedlearningdisabili-tiesupto8thgradeandtoratio,rate,andproportionwordproblems.

Tworandomizedcontrolledtrialsthatmetstandardsexaminedafour-stepstrategyforteachingstudentstosolveratioandproportionwordproblems.227Thestrategyinvolvedaschema-basedapproachinwhichstudentsidentifytheproblemtype,repre-sentcriticalinformationfromtheprobleminadiagram,translateinformationintoamathematicalequation,andsolvetheprob-lem.Keyaspectsoftheapproach,whichwasdesignedtoaddressconcernsaboutthelimi-tationsofdirectinstruction,includeteachingstudentstoidentifyunderlyingproblemstructures,suchasthroughschematicdia-grams,andcomparingandcontrastingdif-ferentsolutionstrategiesandproblemtypes.Oneofthestudiesfocusedonstudentswithlearningdisabilities(i.e.,16ofthe19stu-dentshadadiagnosedlearningdisability),228

andtheotherincludedstudentswithamorediverseabilityrange.229Xin,Jitendra,andDeatline-Buckman(2005)foundasignificantpositiveeffect(albeitwithstudentswithlearningproblems)ofanapproachthattaughtstudentstoidentifytheproblemtypeandrepresenttheproblemusingadiagram.Studentsinthecomparisongroupalsolearnedstrategiesforsolvingwordproblemsbutfocusedmoreondrawingpicturestorepresenttheproblems.Jitendraetal.(2009)foundasubstantivelyimportant,butnotstatisticallysignificant,effectofteachingthefour-stepstrategyonresearcher-developedtestsofratioandproportionwordproblems,relativetoteachingwordproblemswithadistrict-adoptedmathematicstextbook(effectsize=0.33and0.38,immediateanddelayedposttests,respectively).230

Athirdrandomizedcontrolledtrial,MooreandCarnine(1989),alsoexaminedanexplicitstrategyforteachingstudentstosolveratioandproportionwordproblems.Thisstudymetstandardsbutisoutsidethereviewprotocolbecauseitincludedstudentsin9ththrough11thgradesandfocusedonspecialeducationandlow-performingstudents.Thepanelviewsthestudyasprovidingsupplementalevidencetosupporttherecom-mendation.TheWWCdidnothavesufficientinformationtocalculateeffectsizes,butthestudy’sauthorsreportthatteachingstudentsexplicitrulesandproblem-solvingstrategiessignificantlyimprovedtheirproficiencyinsolvingratiowordproblemsrelativetostu-dentstaughtusingabasalcurriculum.

Recommendation5.Professionaldevelopmentprogramsshouldplaceahighpriorityonimprovingteachers’understandingoffractionsandofhowtoteachthem.


Thepanelassignedaminimal evidence ratingtothisrecommendationbecauseoflimitedrigorousevidenceontheeffectsoffractions-relatedprofessionaldevelopmentactivities.

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AppendixDcontinued

Toevaluatethisrecommendation,thepanelsoughtevidencethatprofessionaldevelopmentthatfocusesspecificallyonfractionsimprovesstudentoutcomes.231Twostudiesthatfocusedondevelopingteachers'knowledgeoffractionsmetstandards.Theprofessionaldevelopmentinthefirststudyaddressedthefirsttwoactionstepsfortherecommendationandfoundposi-tiveeffectsonstudentlearning;thesecondstudyaddressedallthreeactionstepsbutdidnotfindasignificanteffectonstudents'under-standingoffractions.232Twootherstudiesmetstandardsandprovidedevidencefortherecom-mendation’sthirdstep—developingteachers’understandingofstudents’mathematicalthink-ing—butfocusedonwholenumberadditionandalgebraicreasoningratherthanonfrac-tions.233Ahandfulofotherstudiespotentiallymetstandardsbutdidnotexaminefractionsordidnotprovideprofessionaldevelopmentdirectlyrelevanttotherecommendation.234

Despitethelimitedevidenceontheeffectsofprofessionaldevelopmentactivitiesonteach-ers’understandingoffractionconceptsandskills,thepanelbelievestheneedtodevelopteachers’knowledgeoffractionsandofhowtoteachthemiscritical.Teachers’mathemati-calcontentknowledgeispositivelycorrelatedwithstudents’mathematicsachievement,235

andresearchershaveconsistentlyfoundthatteachersintheUnitedStateslackadeepcon-ceptualunderstandingoffractions.236Takentogether,thesefindingssuggestthatprovid-ingprofessionaldevelopmentonfractionconceptsisimportant.

Professionaldevelopmentrelatedtofractions.OnerandomassignmentstudymetstandardsandexaminedaprofessionaldevelopmentprogramcalledIntegratedMathematicsAssessment(IMA).Thispro-gramaddressedteachers’understandingof(1)fractionconcepts,(2)howstudentslearnfractions,(3)students’motivationformathachievement,and(4)assessment.237

Teacherslearnedaboutfractionconceptsthroughactivitiesandexercisesthatweremorecomplexversionsofthoseforstudents.

Tounderstandstudents’thinking,teachersexaminedstudentworkandvideotapesofstudentssolvingproblemsandexploredstudents’difficultiesinlearningfractions.TheIMAtrainingconsistedofafive-daysummerinstituteand13follow-upsessionsforupperelementaryteachers.TeachersassignedtotheIMAprofessionaldevelopmentprogramachievedasignificantimprovementintheirstudents’conceptualunderstandingoffrac-tions,comparedwithteachersintheteachersupportgroup,whometninetimestoreflectontheirinstructionalpractices.TheIMAtraininghadasubstantivelyimportant,butnotstatisticallysignificant,effectonstudents’abilitytocomputewithfractions.

Amorerecentstudyofprofessionaldevelop-mentrelatedtofractionsalsometWWCstan-dardsbutdidnotfindasignificanteffectonstudents’learningoffractionconcepts.238Thestudyexaminedtwoprofessionaldevelopmentprogramsfor7th-gradeteachersin12districtsacrossthecountry.Teachersinthetreatmentschoolswereeligibleforabout68hoursoftrainingthroughathree-daysummerinstituteandfive1-dayseminarspairedwithtwo-dayin-schoolcoachingvisits.Theprofessionaldevelopmentfocusedonconceptualandpro-ceduralskillinrationalnumbertopics,aswellasmathematicsknowledgeforteaching.Thisincludedidentifyingthekeyaspectsofmath-ematicalunderstanding,recognizingcommonerrorsmadebystudents,andselectingrep-resentationsforteachingfractions.Activitiesincludedsolvingmathproblemsandreceivingfeedbackontheirsolutions,discussingcom-monstudentmisconceptions,andplanningles-sons.Teachersinthecontrolschoolsreceivedtheexistingprofessionaldevelopmentpro-videdbythedistrict.However,theprofessionaldevelopmentdidnothaveasignificantimpactonstudents’understandingoffractions,deci-mals,percentages,orproportions.

Professionaldevelopmentrelatedtoothermathematicstopics.Findinglittleevidencerelatedspecificallytofractions,thepanelexpandeditsreviewtoincludeprofes-sionaldevelopmentthatfocusedonother

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AppendixDcontinued

mathtopics.Twoadditionalstudiesmetstandardsandimplementedtrainingrelevanttothethirdactionstepoftherecommenda-tion—developingteachers’understandingofstudents’mathematicalthinking.239

Onestudyexaminedafour-weeksummerworkshop(20hours)aimedatdevelopingteachers’knowledgeofhowchildrenlearnwholenumberadditionandsubtractionconcepts.240Teachersparticipatinginthepro-gram,calledCognitivelyGuidedInstruction(CGI),learnedaboutchildren’ssolutionstrate-giesandhowtoclassifyproblemtypes,dis-cussedhowtoincorporateinformationfromtheCGIworkshopintotheclassroom,andplannedinstructionaccordingly.Comparedwithacontrolgroupofteacherswhoreceivedfourhoursofworkshopsonproblemsolvingandtheuseofnonroutineproblems,theCGIprogramhadasubstantivelyimportant,butnotstatisticallysignificant,effectoncomputa-tionproblemsandadditionandsubtractionwordproblems.Thedifferingamountsoftimethatteachersspentinthetwoconditionsalsolimitedinterpretationofthefindings.

Thesecondstudy,arandomizedcontrolledtrialstudyconductedinalarge,urbandistrictwith1st-through5th-gradeteachers,alsoexaminedaprofessionaldevelopmentprogramfocusedondevelopingteachers’understandingofstudents’mathematicalthinking.241Emphasizingunderstandingoftheequalsignandusingnumberrelationstosimplifycalculations,thetraininginthisstudywasdesignedtoimproveteachers’abilitytoincorporatealgebraicreasoningintoelementarymathematics.Teacherslearnedtomakesenseofstudents’strategiesforsolvingproblems,tolinkstudents’think-ingtokeymathematicalideas,andtoleadmathematicalconversationswithstudents.Theprogramincludedaninitialmeetingandeightmonthlyafter-schoolwork-groupmeet-ings(atotalofabout16.5hours),aswellasatrainerwhospentahalf-dayaweekateachschooltoprovideadditionalsupport.Resultsfromthestudyshowedthatthisprofes-sionaldevelopmentsignificantlyimproved

students’understandingoftheequalsignandstudents’useofrelational-thinkingstrat-egiesforsolvingcomputationsbutnotforsolvingequations(i.e.,withlettersrepresent-ingunknownquantities).

Additionalevidence.Thereisfurtherevidencethatstudents’achievementispositivelyrelatedtoteachers’mathematicsknowledgeforteaching—forexample,theirskillatexplainingmathconcepts,under-standingstudentstrategies,andprovidingrepresentations.Astudyof6991st-and3rd-grademathteachersfoundapositiverelationbetweenteachers’mathknowledgeforteachingandstudents’learninggainsinmathaftercontrollingforstudentandteachercharacteristics.242 Althoughthisstudydidnotspecificallyfocusonfractions,itdemonstratedtheimportanceofteachers’mathcontentknowledgeforteaching.

ProfessionaldevelopmentwithfractionsisneededbecausemanyU.S.teacherslackadeepconceptualunderstandingoffractions.243

AstudycomparingChineseandAmericanteachersfoundthatonly9ofthe21U.S.teacherswhotriedtocalculate13/4÷1/2 didsocorrectly,whereasall72Chineseteacherscorrectlycompletedtheproblem.244U.S.teacherscouldnotrepresentorexplaindivisionwithfractions,andmanyconfusedthealgorithmfordividingfractionswiththealgorithmsforadding,subtracting,andmulti-plyingfractions.

Otherstudieshavereportedsimilarfindings.Astudyof218elementaryschoolteachersinMinnesotaandIllinoisfoundthatmanyteacherscouldnotsolvecomputationprob-lemsinvolvingfractionsandthatmostofthosewhocorrectlysolvedproblemscouldnotprovideacorrectexplanationoftheirsolutions.245Forexample,almosthalfoftheteachersintheMinnesotastudyincorrectlysolvedasubtractionprobleminvolvingfrac-tions(1/3–3/7).Further,astudyof46preser-vicemiddleschoolteachersatauniversityinTexasfoundthatmostteachersknewtheprocedurefordividingwithfractionsbutdid

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AppendixDcontinued

notunderstandwhytheprocedureworkedandcouldnotjudgewhetheranalternativeprocedureforsolvingadivisionproblemwithfractionswascorrect.246

Thesestudiesclearlyindicatethatteach-ers’understandingoffractionsneedstobe

upgraded.However,therecommendationregardingprofessionaldevelopmentislargelybasedonthepanel’sexpertise,becauseofthelimitedevidenceregardingtheeffectsofprofessionaldevelopmentactivitiesfocusedonfractions.

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AppendixE

EvidenceHeuristic

Thisappendixcontainsaheuristicforcategorizingtheevidencebaseforpracticeguiderecommenda-tionsasstrong evidence,moderate evidence,orminimal evidence.Thisheuristicisintendedtoserveasaframeworktoensurethatthelevelsofevidenceareconsistentlyappliedacrosspracticeguideswhileatthesametimeclarifyingthelevelsforpanelistsandeducators.Thecoredocumenttoaccom-panythisheuristicisthe“InstituteofEducationScienceslevelsofevidenceforpracticeguides”(Table1inthispracticeguide).

TableE.1.Evidenceheuristic

CriteriaforaStrongEvidenceBaseThiscriterionisnecessaryforastronglevelofevidence.

Highinternalvalidity(high-qualitycausaldesigns).StudiesmustmeetWWCstandardswithorwithoutreservations.247

Highexternalvalidity(requiresaquantityofstudieswithhigh-qualitycasualdesigns).StudiesmustmeetWWCstandardswithorwithoutreservations.248

Effectsonrelevantoutcomes—consistentpositiveeffectswithoutcontradictoryevidence(i.e.,nostatisticallysignificantnegativeeffects)instudieswithhighinternalvalidity.249

Directrelevancetoscope(i.e.,ecologicalvalidity)—relevantcontext(e.g.,classroomvs.laboratory),sample(e.g.,ageandcharacteristics),andoutcomesevaluated.

Directtestoftherecommendationinthestudies,ortherecommendationisamajorcomponentoftheinterventiontestedinthestudies.

Forassessments,meetsThe Standards for Educational and Psychological Testing. Panelhasahighdegreeofconfidencethatthispracticeiseffective.

CriteriaforaModerateEvidenceBaseThiscriterionisnecessaryforamoderatelevelofevidence.

Highinternalvaliditybutmoderateexternalvalidity(i.e.,studiesthatsupportstrongcausalconclu-sions,butgeneralizationisuncertain)ORstudieswithhighexternalvaliditybutmoderateinternalvalidity(i.e.,studiesthatsupportthegeneralityofarelation,butthecausalityisuncertain).

•TheresearchmayincludestudiesgenerallymeetingWWCstandardsandsupportingtheeffec-tivenessofaprogram,practice,orapproachwithsmallsamplesizesand/orotherconditionsofimplementationoranalysisthatlimitgeneralizability.

•TheresearchmayincludestudiesthatsupportthegeneralityofarelationbutdonotmeetWWCstandards;250however,theyhavenomajorflawsrelatedtointernalvalidityotherthanlackofdemonstratedequivalenceatpretestforquasi-experimentaldesignstudies(QEDs).QEDswith-outequivalencemustincludeapretestcovariateasastatisticalcontrolforselectionbias.ThesestudiesmustbeaccompaniedbyatleastonerelevantstudymeetingWWCstandards.

Effectsonrelevantoutcomes—apreponderanceofevidenceofpositiveeffects.Contradictoryevidence(i.e.,statisticallysignificantnegativeeffects)mustbediscussedbythepanelandcon-sideredwithregardtorelevancetothescopeoftheguideandintensityoftherecommendationasacomponentoftheinterventionevaluated.

Relevancetoscope(i.e.,ecologicalvalidity)mayvary,includingrelevantcontext(e.g.,classroomvs.laboratory),sample(e.g.,ageandcharacteristics),andoutcomesevaluated.

Intensityoftherecommendationasacomponentoftheinterventionsevaluatedinthestudiesmayvary.

Forassessments,evidenceofreliabilitythatmeetsThe Standards for Educational and Psychological Testingbutwithevidenceofvalidityfromsamplesnotadequatelyrepresentativeofthepopulationonwhichtherecommendationisfocused.

Thepanelisnotconclusiveaboutwhethertheresearchhaseffectivelycontrolledforotherexplanationsorwhetherthepracticewouldbeeffectiveinmostorallcontexts.

Thepaneldeterminesthattheresearchdoesnotrisetothelevelofstrongevidencebutismorecompellingthanaminimallevelofevidence.

(continued)

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CriteriaforaMinimalEvidenceBaseThiscriterionisnecessaryforaminimallevelofevidence.

Expertopinionbasedondefensibleinterpretationsoftheory(ortheories).Insomecases,thissimplymeansthattherecommendedpracticeswouldbedifficulttostudyinarigorous,experimentalfashion;inothercases,itmeansthatresearchershavenotyetstudiedthispractice.

Expertopinionbasedonreasonableextrapolationsfromresearch:

•Theresearchmayincludeevidencefromstudiesthatdonotmeettstrongevidence(e.g.,casestudies,qualitativeresearch).

•Theresearchmaybeoutofthescopeofthepracticeguide.

•Theresearchmayincludestudiesforwhichtheintensityofthereconentoftheinterventionsevaluatedinthestudiesislow.

hecrit

mme

eriafor

ndation

moder

asaco

ateor

mpo

Theremaybeweakorcontradictoryevidence.

Inthepanel’sopinion,therecommendationmustbeaddressedaspartofthepracticeguide;however,thepanelcannotpointtoabodyofresearchthatrisestothelevelofmoderateorstrong.

AppendixEcontinued

TableE.1.Evidenceheuristic(continued)

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-

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Endnotesa

1. Formoreinformation,seetheWWCFre-quentlyAskedQuestionspageforpracticeguides,http://ies.ed.gov/ncee/wwc/refer-ences/idocviewer/doc.aspx?docid=15.

2. SeetheWWCguidelinesathttp://ies.ed.gov/ncee/wwc/pdf/wwc_procedures_v2_stan-dards_handbook.pdf.

3.Thisincludesrandomizedcontroltrials(RCTs),quasi-experimentaldesigns(QEDs),regressiondiscontinuitydesigns(RDDs),andsingle-casedesigns(SCDs)evaluatedwithWWCstandards.

4. Iftheonlyevidencemeetingstandards(withorwithoutreservations)isSCDs,theguide-linessetbytheSCDstandardspanelwillapply.Forexternalvalidity,therequirementsareaminimumoffiveSCDresearchpapersexaminingtheinterventionthatmeetevi-dencestandardsormeetevidencestandardswithreservations,thestudiesmustbecon-ductedbyatleastthreedifferentresearchteamsatthreedifferentgeographicalloca-tions,andthecombinednumberofexperi-mentsacrossstudiestotalsatleast20.

5. Incertaincircumstances(e.g.,acomparisongroupcannotbeformed),thepanelmaybaseamoderateratingonmultiplecorrela-tionaldesignswithstrongstatisticalcontrolsforselectionbiasthatdemonstrateconsis-tentpositiveeffectswithoutcontradictoryevidence.

6. Baldietal.(2007);Gonzalesetal.(2009).

7. NationalAcademyofSciences(2007).8.McCloskey(2007);NationalAcademyof

Sciences(2007);Rivera-Batiz(1992).9. Starkey,Klein,andWakeley(2004).

10.Rivera-Batiz(1992).11. Mullisetal.(1997).12.NationalMathematicsAdvisory Panel

(2008).13.Hofferetal.(2007).14.NationalCouncilofTeachersofMathematics

(2007).15.Kloosterman(2010).16. RittleJohnson,Siegler,andAlibali

(2001).17. VamvakoussiandVosniadou(2004).

18.DavisandPepper(1992);FrydmanandBry-ant(1988);HuntingandSharpley(1988);Pepper(1991);Singer-FreemanandGoswami(2001);SpinilloandBryant(1991,1999).

19.Ma(1999).20.Ma(1999);Moseley,Okamoto,andIshida

(2007).21.DavisandPepper(1992);FrydmanandBry-

ant(1988);HuntingandSharpley(1988);Pepper(1991).

22.Singer-FreemanandGoswami(2001);SpinilloandBryant(1991,1999).

23.DavisandPepper(1992);FrydmanandBry-ant(1988);HuntingandSharpley(1988);Pepper(1991);Singer-FreemanandGoswami(2001);SpinilloandBryant(1991,1999).

24.Empson(1999);Streefland(1991).25.DavisandPepper(1992);FrydmanandBry-

ant(1988);HuntingandSharpley(1988);Pepper(1991).

26.Chen(1999);HuntingandSharpley(1988).27. Empson(1999);Streefland(1991).28.Singer-FreemanandGoswami(2001);Spinillo

andBryant(1991,1999).29.SpinilloandBryant(1991,1999).30.Goswami(1989,1995).31. Empson(1999);Streefland(1991).32.Empson(1999).33.Teachersshouldalsonotethatsharingsitu-

ationswithmorepeoplethanobjectsresultinproperfractions,whereassharingsitua-tionswithmoreobjectsthanpeopleresultinimproperfractionsormixednumbers.

34.Empson(1999);Streefland(1991).35.Empson(1999).36.PothierandSawada(1990).37. Ibid.38.Empson(1999);Streefland(1991).39. Ibid.40.Streefland(1991).41. Ibid.42.ResnickandSinger(1993).43. Ibid.44.WarrenandCooper(2007).45. Ibid.

aEligiblestudiesthatmeetWWCevidencestandardsormeetevidencestandardswithreservationsareindicatedbyboldtextintheendnotesandreferencespages.Formoreinformationaboutthesestudies,pleaseseeAppendixD.

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http://ies.ed.gov/ncee/wwc/references/idocviewer/doc.aspx?docid=15

http://ies.ed.gov/ncee/wwc/references/idocviewer/doc.aspx?docid=15




Endnotescontinued

46.DavisandPitkethly(1990).47. FrydmanandBryant(1988).48.Ibid.49.Mack(1993).50.Sophian(2007).51. BoothandSiegler(2008);Ramaniand

Siegler(2008);SieglerandBooth(2004);SieglerandRamani(2008,2009).

52.RittleJohnson,Siegler,andAlibali(2001).

53.BoothandSiegler(2006,2008);Schneider,Grabner,andPaetsch(2009);Siegler&Booth(2004).

54.RamaniandSiegler(2008);SieglerandRamani(2008,2009).

55. SieglerandBooth(2004).56.RittleJohnson,Siegler,andAlibali

(2001).57.KeijzerandTerwel(2003).58.Ibid.59.Brightetal.(1988);Izsak,Tillema,andTunc-

Pekkan(2008).60.Brightetal.(1988).61.Schneider,Grabner,andPaetsch(2009).62.Wu(2002).63.Sophian(2007).64.Izsak,Tillema,andTunc-Pekkan(2008);

Yanik,Helding,andFlores(2008).65.Adapted from Beckmann (2008); Wu

(2002).66.Beckmann(2008);Lamon(2005).67.VandeWalle, Karp, andBay-Williams

(2010).68.Brightetal.(1988);Izsak(2008).69.NiandZhou(2005).70.Lamon(2005);Niemi(1996).71.NiandZhou(2005).72.Brightetal.(1988).73.RittleJohnsonandKoedinger(2002,

2009);RittleJohnson,Siegler,andAlibali(2001).

74.Anderson(2004);SchneiderandPressley(1997).

75.Hecht(1998);Hecht,Close,andSantisi(2003);HechtandVagi(inpress);RittleJohnson,Siegler,andAlibali(2001).

76.RittleJohnsonandKoedinger(2002,2009).

77. RittleJohnson,Siegler,andAlibali(2001),Experiment2.

78.Hecht(1998);Hecht,Close,andSantisi(2003);HechtandVagi(inpress).

79.Rittle-Johnson,Siegler,andAlibali(2001),Experiment1.

80.Hudson Hawkins (2008); Nishida(2008).

81. Nishida(2008),Experiment2.82.HudsonHawkins(2008).83.Cramer,Post,anddelMas(2002).84.AnandandRoss(1987);Irwin(2001).85.AnandandRoss(1987).86.Irwin(2001).87.Mack(2000);Mack(2001);Rittle-Johnsonand

Koedinger(2005).88.CramerandWyberg(2009).89.Sowderetal.(1998).90.Bulgar(2009).91.VandeWalle, Karp, andBay-Williams

(2010).92.Smith(2002).93.Hecht(1998).94.Johanning(2006).95.Smith(2002).96.GerverandSgroi(1989).97. CramerandWyberg(2009);Cramer,Wyberg,

andLeavitt(2008);Hecht(2003).98.Kerslake (1986); Mack (1995); Painter

(1989).99.Sharp(2004).

100.TatsuokaandTatsuoka(1983).101. Hackenberg(2007).102.Mack(1995).103.Painter(1989).104.Ashlock(2009);BarashandKlein(1996).105.KamiiandWarrington(1995);Mack(2001);

WarringtonandKamii(1998).106.Mack(1990,1993).107. AnandandRoss(1987);Irwin(2001).108.Mack(1990).109. Ibid.110. Cramer,Wyberg,andLeavitt(2008).

((7171))

Endnotescontinued

111. Lesh,Post,andBehr(1988);NationalCouncilofTeachersofMathematics(2000);NationalMathematicsAdvisoryPanel(2008).

112. Smith(2002);Stemn(2008).113.NationalMathematicsAdvisory Panel

(2008).114. Kloosterman(2010);Lamon(2007);Thomp-

sonandSaldanha(2003).115. Lesh,Post,andBehr(1988);NationalCouncil

ofTeachersofMathematics(2000);NationalMathematicsAdvisoryPanel(2008).

116. Carpenteretal.(1999);Cramer,Post,anddelMas(2002);Lamon(1994).

117. Fujimura(2001).118. Sellke,Behr,andVoelker(1991).119. Terweletal.(2009).120. Jitendraetal.(2009);Xin,Jitendra,and

DeatlineBuchman(2005).121. Thestudywithmostlylearningdisabled

studentsfallswithinthepanel’sprotocolbecausesomeofthestudentsarelowper-formingbutnotlearningdisabled.Thepro-tocolincludesstudiesthatdidnotfocussolelyonlearningdisabledstudents.

122.Thisstepfocusesprimarilyonstrategiesforsolvingtwotypesofproportionproblems(Lamon,2005):(1)missing-valueproblemsinwhichstudentsaregivenonecompleteratioandanotherratiowithamissingvaluethatstudentsmustidentify,and(2)compari-sonproblemsinwhichstudentsdeterminewhethertworatiosareequivalent.

123.Carpenteretal.(1999);Cramer,Post,andCurrier(1993);Lesh,Post,andBehr(1988).

124. Carpenteretal.(1999).125. Ibid.126.Carpenteretal.(1999);Lamon(2007).127. Kent,Arnosky,andMcMonagle(2002);Ter-

weletal.(2009).128.ResnickandSinger(1993).129.Lamon(1993).130. Jitendraetal.(2009).131. Xin,Jitendra,andDeatlineBuchman

(2005).132.Hembree(1992);Jitendraetal.(2009).133. Jitendraetal.(2009);Xin,Jitendra,and

DeatlineBuchman(2005).134. Bottge(1999);Helleretal.(1989).

135.LiandKulm(2008);Ma(1999);Newton(2008);Postetal.(1988).

136.Hill,Rowan,andBall(2005).137. LiandKulm(2008);Ma(1999);Newton

(2008);Postetal.(1988).138. Saxe,Gearhart,andNasir(2001).139. Garetetal.(2010).140. Garetetal.(2010);Ma(1999);Postetal.

(1988).141. Ma(1999).142.Postetal.(1988).143.Hilletal.(2005).144.Ma(1999).145.LiandKulm(2008).146.Althoughafulldiscussionofhowtostruc-

tureprofessionaldevelopmentisbeyondthisguide’sscope,thepanelprovidesbasicsuggestionsthataddressthegoalofthisstep.

147. Marchiondra(2006);Saxe,Gearhart,andNasur(2001).

148.Tirosh(2000).149. Saxe,Gearhart,andNasir(2001);Wu

(2004).150. Saxe,Gearhart,andNasir(2001).151. Carpenteretal.(1989).152. Jacobsetal.(2007);Saxe,Gearhart,and

Nasir(2001).153.Kloosterman(2010);NationalCouncilof

TeachersofMathematics(2007).154.Mullisetal.(1997).155. NationalResearchCouncil(2001).156.SophianandWood(2007).157. CramerandPost(1993);VandeWalle,Karp,

andBay-Williams(2010).158.DavisandPepper(1992);FrydmanandBry-

ant(1988);HuntingandSharpley(1988);Pepper(1991);Singer-FreemanandGoswami(2001);SpinilloandBryant(1991,1999).

159. Empson(1999);Streefland(1991).160.Chen(1999);DavisandPepper (1992);

FrydmanandBryant(1988);HuntingandSharpley(1988);Pepper(1991).

161. Chen(1999).162.HuntingandSharpley(1988).163.Sophian,Garyantes,andChang(1997).164.Ibid.

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Endnotescontinued

165.Empson(1999);Streefland(1991).166.Streefland(1991).167. Empson(1999).168.Singer-FreemanandGoswami(2001);Spinillo

andBryant(1991,1999).169.Singer-FreemanandGoswami(2001).170.SpinilloandBryant(1991).171. SpinilloandBryant(1991,1999).172.Boyer,Levine,andHuttenlocher(2008);

Singer-FreemanandGoswami(2001);SpinilloandBryant(1999).

173.Goswami(1989).174. Goswami(1995).175. Spinillo(1995).176. RamaniandSiegler(2008);Sieglerand

Ramani(2008,2009).177. RittleJohnson,Siegler,andAlibali

(2001).178. BoothandSiegler(2006,2008);Sieglerand

Booth(2004);SieglerandRamani(2009).179.OpferandDeVries(2008).180. RamaniandSiegler(2008);Sieglerand

Ramani(2008,2009).181. RamaniandSiegler(2008);Sieglerand

Ramani(2008).182. SieglerandRamani(2009).183. RamaniandSiegler(2008);Sieglerand

Ramani(2008,2009).TheWWCcalcu-latedtheeffectsizeforSieglerandRamani(2008);theeffectsizesreportedforRamaniandSiegler(2008)andSieglerandRamani(2009)werereportedbytheauthors(theauthorsdidnotprovidesufficientinformationfortheWWCtocalculatetheeffectsize).

184. SieglerandRamani(2008);RamaniandSiegler(2008).SieglerandRamani(2009)foundasignificanteffectofthenum-berboardgameonpercentabsoluteerrorbutdidnotreportaneffectsizeorprovidesufficientinformationtocalculateit.

185. SieglerandRamani(2008,2009).186. SieglerandBooth(2004).187.Theauthorsdidnotreportwhetherthe

differencebetweentreatmentandcontrolgroupswassignificant.

188.FollowingWWCguidelines,improvedout-comesareindicatedbyeitherapositivestatisticallysignificanteffectorapositive

substantivelyimportanteffectsize.Inthisguide,thepaneldiscussessubstantivelyimportantfindingsasonesthatcontributetotheevidenceofpractices’effectiveness,evenwhenthoseeffectsarenotstatisti-callysignificant.SeetheWWCguidelinesathttp://ies.ed.gov/ncee/wwc/pdf/wwc_pro-cedures_v2_standards_handbook.pdf.

189. BoothandSiegler(2008).190.Thisstudyalsoincludedacircularnumeric

boardgametreatmentthatwasnotcon-sideredaspartoftheevidenceforthisrecommendation.

191. Thisstudyalsoincludedkindergartenstu-dents,buttheauthorsfocusedtheiranalysisresultson1st- and2nd-graderssincethetreatmentdidnothavetheexpectedeffectforkindergarteners.

192. Thisstudyalsoincludedatreatmentgroupthathadstudentsgenerateanumberlinewiththeaddendsandsums(insteadofpro-vidingthemwithacomputer-generatedline).Thesetwotreatmentsdidnotsignificantlydifferinthepercentageofcorrectanswers.

193.BoothandSiegler(2006);SieglerandBooth(2004).

194. SieglerandBooth(2004).195.BoothandSiegler(2006).196. RittleJohnson,Siegler,andAlibali

(2001).197. KeijzerandTerwel(2003).198.Brightetal.(1988);Izsak,Tillema,andTunc-

Pekkan(2008).199.Brightetal.(1988).200.Schneider,Grabner,andPaetsch(2009).201.Wu(2002).202.RittleJohnsonandKoedinger(2002,

2009);RittleJohnson,Siegler,andAlibali(2001).


204.Anderson(2004);SchneiderandPressley(1997).

205.Hecht(1998);Hecht,Close,andSantisi(2003);HechtandVagi(inpress);RittleJohnson,Siegler,andAlibali(2001).

206. NationalMathematicsAdvisoryPanel(2008);NationalResearchCouncil(2001).

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Endnotescontinued

207. RittleJohnsonandKoedinger(2002,2009);RittleJohnson,Siegler,andAlibali(2001).


209.RittleJohnson,Siegler,andAlibali(2001),Experiment2.

210.Forap-value<0.05,theeffectsizeissig-nificant(sig);forap-value≥0.05,theeffectsizeisnotsignificant(ns).

211. RittleJohnson,Siegler,andAlibali(2001),Experiment1.

212. Ibid.213. Hudson Hawkins (2008); Nishida

(2008).214. Nishida(2008),Experiment2.215. HudsonHawkins(2008).216.Cramer,Post,anddelMas(2002).217. AnandandRoss(1987).218. Irwin(2001).219. NationalCouncilofTeachersofMathematics

(2000);NationalMathematicsAdvisoryPanel(2008).

220.ResnickandSinger(1993).221. Lamon(1994).222.Carpenteretal.(1999).223.Carpenteretal.(1999);Cramer,Post,and

Currier(1993).224. Sellke,Behr,andVoelker(1991).225.Bassok(1990);Carroll(1994);Cooperand

Sweller(1987);Lewis(1989);LewisandMayer(1987);ReedandBolstad(1991);SwellerandCooper(1985).

226.NationalMathAdvisoryPanel(2008).227. Jitendraetal.(2009);Xin,Jitendra,and

DeatlineBuchman(2005).228.Xin,Jitendra,andDeatlineBuchman

(2005).229.Jitendraetal.(2009).230.Jitendraetal.(2009)reportedasignifi-

cantpositiveeffectofthetreatmentontheproblemsolvingposttest.However,whentheWWCappliedaclusteringcorrection,sincestudentsinthestudywereclusteredinclassrooms,theresultswerenotsig-nificant.Foranexplanation,seetheWWCTutorialonMismatch.FortheformulastheWWCusedtocalculatethestatistical

significance,seetheWWCProceduresandStandardsHandbook.

231. Thepaneldidnotreviewstudiesthatmea-suredtheeffectofprofessionaldevelopmentonteacherknowledge,althoughareviewbytheNationalMathAdvisoryPaneldidnotidentifyanystudieswithacomparisongroupdesign.

232. Garetetal.(2010);Saxe,Gearhart,andNasir(2001).

233. Carpenteretal.(1989);Jacobsetal.(2007).

234.Cole(1992);MeyerandSutton(2006);Niess(2005);Ross,Hogaboam-Gray,andBruce(2006);Sloan(1993).

235.Hill,Rowan,andBall(2005).236.LiandKulm(2008);Ma(1999);Newton

(2008);Postetal.(1988).237. Saxe,Gearhart,andNasir(2001).Although

thestudyalsousedaquasi-experimentaldesigntocomparetwomathcurricula,onlytheprofessionaldevelopmentportionofthisstudy,whichusedarandomassignmentdesign,isrelevantforRecommendation5.

238.Garetetal.(2010).239.Carpenteretal.(1989);Jacobsetal.

(2007).240. Carpenteretal.(1989).241. Jacobsetal.(2007).242.Hill,Rowan,andBall(2005).243.LiandKulm(2008);Ma(1999);Postetal.

(1988).244.Ma(1999).245.Postetal.(1988).246.LiandKulm(2008).247. Thisincludesrandomizedcontroltri-

als(RCTs),quasi-experimentaldesigns(QEDs),regressiondiscontinuitydesigns(RDDs),andsingle-casedesigns(SCDs)evaluatedwithWWCstandards.

248. Iftheonlyevidencemeetingstandards(withorwithoutreservations)isSCDs,theguidelinessetbytheSCDstandardspanelwillapply.Forexternalvalidity,therequirementsareaminimumoffiveSCDresearchpapersexaminingtheinterven-tionthatmeetevidencestandardsormeetevidencestandardswithreserva-tions,thestudiesmustbeconductedby

((7474))

Endnotescontinued

atleastthreedifferentresearchteamsat andfloor,theassessment’sitemgradients,threedifferentgeographicallocations,andthecombinednumberofexperimentsacrossstudiestotalsatleast20.

whethertheassessmentwasoveralignedwiththeintervention,andtheappropriate-nessoftheassessmentforthesampleto

249. Whenevaluatingwhethereffectsarecon- whichitwasapplied.sistentorcontradictory,considerthepsy-chometricpropertiesoftheassessments.Forexample,effectsarelesslikelytobe

250. Incertaincircumstances(e.g.,acomparisongroupcannotbeformed),thepanelmaybaseamoderateratingonmultiplecorrelational

detectedifanassessmentisunreliable.Psychometricpropertiestoconsiderincludereliability,thepresenceoflimitedorcon-

designswithstrongstatisticalcontrolsforselectionbiasthatdemonstrateconsistentpos-itiveeffectswithoutcontradictoryevidence.

strainedvariance,theassessment’sceiling

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IndexofKeyMathematicalConcepts

addition,9,19,20,24,27,28,30-34,36-39,43,46,57-59,60,62,63,65,66

commonfraction,6,8,9,24,30,47,48

compare,7,9,13,15,16,17,20,22-23,24,25,27,36,40,41,48,57,58,59,62

conceptualunderstanding,6,8,11,20,26-34,35,42,43,59,60,61,62,65,66

crossmultiplication,11,35, 37-39,41

decimals,6,7,8,9,20,21,22,23,24,26,27,30,31,43,45,48,57,58,59,60,61,62,65

density,23-24,47

division,8,9,12-17,18,19,22,24,25,27,29,30,31,33,37,38,44,45,46,47,48,62,66,67

equalsharing,8,11,12-17,18,44,47,55,56

equivalence,8,9,12-13,15-18,19,20,22-25,32,35,38-39,41,47,48,56,59,61-63

estimation,7,20,21,22,26,30,31,34,43,44,57,58,59,61,62

improperfractions,22,30,31,32,47

invertandmultiply,9,33,46

magnitude,7,9,16,18,19-23,31,32,56-57

measurement,8,9,19-21,23,25,33-34,35,44,59,62,

mixednumbers,22,30,32,47

multiplication,8,9,28-29,31,32-33,35,36,38,39,41,45-46,63,66

multiplicativerelations,35-41,47,48,62-63

negativefractions,7,9,20,23-24,30

numberline, 7,9,11,19-25,27,28-30,32,44,45,55,56-59,60-61

ordering,6,12-17,19,20,23,27,43,45,56-60,62

partitioning,12-18,21,22,24,25,59

percents,7-8,9,20,22,23,24,43,45,48,64,65,

proceduralfluency,6,8,9,11,26-34,35-36,37-39,41,42-45,60-62,65-67

proportions,8,9,11,12,13,17,35,36,37-41,43,45,48,55,56,62-64,65

rate,9,11,35,36,37-41,45,48,62-64

ratio,9,11,12,17,35,36,37-41,43,45,48,62-64

ratiotable,39-40

rationalnumbers,6,7-8,43,48,65

reciprocal,33

representations,9,11,13,15,19-25,26,27,28-30,32,34,35,36,39-40,42,44,56-59,61,63-64,65,66

scaling,40

subtraction,9,24,27,28,30,32,43,45,60,62,66

unitfraction,14,22,24,31,32,33,34,38,48,59

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