THEJOURNALOFTEACHERACTIONRESEARCH 24
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
MORETHANWORDS:STRUGGLINGREADERS’COMPREHENSIONOFWORDPROBLEMSLeannaR.MacDonaldandLeslieC.Banes
UniversityofCalifornia-Davis
AbstractBeforetheyareabletosolvemathematicalwordproblems,studentsmustbeabletoreadandcomprehendtheproblems.Althoughchallengingforallstudents,strugglingreadersfaceadditionalcognitivedemandswhensolvingwordproblemsthatproficientreadersdonot.Inthisactionresearchstudy,fourfocalstudents,includingtwoEnglishlearnersandtwonativeEnglishspeakers,weregivenamultiplicativecomparisonproblemandpromptedtoretellitintheirownwords,solvetheproblemusingaselectedstrategy,andthenretelltheproblemagain.Aretellrubricwasusedtoanalyzestudents’comprehensionbymeasuringthecompletenessoftheretell,whileadrawingandwritingrubricmeasuredstudents’visualrepresentationsandmetacognitivestrategiesinvolvedincomprehension.ResultssuggestdiscussionprovidedopportunitiesforstudentstolearnfromoneanotheranddrawingsespeciallyhelpedEnglishLearners’whocommunicatedtheirunderstandingthroughsymbolsandvisuals.Writingsupportedstudents’metacognitiveskillsleadingtogreatercomprehension,butmaybeproblematicforstudentsatthebeginningstagesofEnglishacquisition.Itisimperativeforteacherstoprovideopportunitiesforstrugglingreaderstodiscuss,draw,andwriteaboutwordproblemstosupporttheircomprehensionandtoextendtheseskillstomathintherealworld.
Keywords:elementarymath,wordproblems,readingcomprehension,Englishlearners
Introduction
Thisarticledescribesanactionresearchprojectconductedbyafirst-yearteacherwiththegoalofincreasingherunderstandingofhowstrugglingreaderscomprehendmathematicalwordproblems.Suggestionsandimplicationsforinstructionarediscussed.
LiteratureReview
Substantialresearchinthefieldofmathematicshasbeenconductedontheuseofdiscussionsandthinks-aloudstocomprehendmathematics.Researchershavefoundthatwhenstudentsareprovidedwithopportunitiestoengageinmeaningfulmathematicaldialogue,students’comprehensionofamathematicalsituationincreasesassodoestheirmathematicaljustifications(Bargh&Schul,1980;Chi,2000;Frankeet
THEJOURNALOFTEACHERACTIONRESEARCH 25
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
al.,2009;King,1992;Rogoff,1991).Therefore,discussioncanprovideanotherwayforstudentstointeractwithtext,hopefullyleadingtomoreunderstandingofapresentedmathematicalsituationinagivenwordproblem.Otherresearchershavefoundthattheuseofpicturesanddrawingssupportsstudents’comprehensionandconceptualunderstandingofmathematics(Marinoetal.,2010;O’Connelletal.,2005).Drawingapicturemighthelpstudentsconnectwhattheythinkorsayinaretelltoatangiblerepresentationintheirsolutionprocess.Italsoallowsstudentstocommunicatetheircomprehensionofawordprobleminmultipleways.Givingstudentsopportunitiestoprovideamentalmodeltoamathproblemallowsthemtointernalizeandcontemplateamathematicalsituation(Dexter&Hughes,2011).EdensandPotter(2008)foundthatdrawingscanreducethelinguisticdemandcommonlyfoundinwordproblems—animportantfindingwhencontemplatingstrategiesforELswhoarestrugglingreaderstouse.Therefore,usingdrawingsasastrategytosupportstudents’comprehensionofwordproblemsisimportanttoexplore.Writinghasalsobeenfoundtobeausefulstrategyasitsupportsstudents’metacognition(Artz&Armour-Thomas,1992;Carr&Biddlecomb,1998;Powell,1997;Pugalee,2001),animportantthoughtprocessforstudents’comprehension.Althoughthetestsubjectsfromtheseresearchprojectsaremucholderinage(whichseemsquestionabletousewritingforfourthgraders)JulietBaxterandhercolleagues(2005)foundthat7thgraderswithwritingandreadingdisabilities,operatingatleasttwoyearsbelowgradelevel,showedmultipleinstancesofstudents’comprehensionandconceptualunderstandingofproblemswhengivingopportunitiestojournaltheirmathematicalthinking.Accordingly,studentswhofacedadditionalacademicchallenges,whichaffectedtheirabilitytowrite,stillbenefitedfromwritingabouttheirmathematicalthinking.Forthisreason,providingopportunitiesforstudentstowriteabouttheirthinkingisimportanttoinvestigateaswritingcouldfostertheircomprehension.
Methodology
Context.ThisinquiryprojectwasconductedinafourthgradeclassroomatapublicTitle1schoollocatedinasuburbanneighborhoodknownforgunandgangviolence.About86%ofstudentsatApplegateElementary(pseudonym)receivefreeandreducedlunch,anindicatorofhighlevelsofpoverty.Forty–twopercentofstudentsareELs(EnglishLearners)andApplegateisaprogramimprovementschoolthathasastrongfocusonliteracy.
Theclassroomconsistedof29students,ofwhich14wereELs.AccordingtotheCaliforniaEnglishLanguageDevelopmentTest(CELDT)thatmeasuresstudents’Englishproficiencyonascaleof1-5,theaverageELlevelintheclasswas3,representinganintermediatelevelEnglishproficiency.OftheELstudentsintheclassroom,13students’nativelanguagewasSpanishandonestudents’nativelanguagewasHmong.Fifteen
THEJOURNALOFTEACHERACTIONRESEARCH 26
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
studentswereHispanic,sixwhite,fiveAfricanAmerican,andtheremainingthreewerePacificIslander,AlaskaNative,orHmong.Twenty-twostudentswerereadingbelowgradelevel,with13studentsreadingoneormoreyearsbehindgrade-levelnorms.
Forthisinquiryproject,fourfocalstudentswereselectedforindepthdataanalysis.Thesefocalstudentswereeitherreadingatafirstorsecondgradelevel,andwerechosenbecausetheyrepresentthereadinglevelsofalargeportionoftheclass.TwoofthefocalstudentswereELswithCELDTlevels1and2(beginningandearlyintermediate)andtwostudentswereEOs(studentswhospeakEnglishonly).AcombinationofELandEOstudentswerechoseninordertoexplorehowparticularinstructionalstrategiesmaysupportstudentswithdifferentlinguisticneeds.Thefourfocalstudentsareusuallypassiveandtheirvoicesgounheardingroupdiscussion.Theyareusuallyhesitantwhensolvingwordproblemsindependentlyandwaitforotherstoprovidesuggestionsoranswers.Aprimarygoalofthisprojectistoprovidestudentswithmoreopportunitiestosharetheirideasandgaintheconfidenceneededtoattemptwordproblems.Priortothisstudy,theteacherprimarilyhadstudentsattemptwordproblemsasawholeclasswithguidingquestionstoscaffoldstudentsthinking.However,theteacherdesiredamorestudent-centeredapproachbyprovidingherstudentswithstrategiestheycouldusetoengageinmathematicalthinking.
Purpose.Inthefieldofmathematics,comprehensioniscrucialforstudents’successinwordproblemsandinrealworldapplications.Notonlymuststudentsunderstandwhatawordproblemisaskinginreal-lifecontexts,theymustalsobeabletointeractwiththetextofwordproblemstosolvethem.Inamathematicaltext,readingbecomesevenmoredifficultastextisnotalwaysreadfromlefttoright(dependingifstudentsneedtoalsointerpretandreferencegraphs/tables)anditisusuallyvisuallycomplexastherearecallouts,sidebarsofgraphs,historicalfacts,and/orpracticeproblems(Bartonetal.2002).Thesetasksbecomeespeciallydifficultforstrugglingreadersastheyfaceadditionalcognitivedemandsthatproficientreadersdonot.Whenreadingwordproblems,strugglingreadersareaskedtosimultaneouslydecodetext,alreadyanareaofdifficulty,whilecomprehendingandrelatingthesewordstomathematics.Readingcomprehensionisstronglycorrelatedwithstudents’successonmathematicalwordproblems(Vilenius-Tuohimaa,Aunola,&Nurmi,2008).Therefore,findingstrategiesthatstudentscanusetohelplessenthecognitivedemandofwordproblemsisimportanttoprovideeducationalequityforourstrugglingreadersinorderforthemtoreachstandardsandusetheseskillsintherealworld.
Hegartyandcolleagues(1995)definecomprehensioninmathematicsonatieredscalewhichinvolves(1)understandingtheproblem,(2)formingaplantosolvetheproblemand(3)carryingouttheplanbysolvingit.Forthepurposesofthisproject,comprehensionisdefinedasunderstandingthemathematicalsituationdescribedinawordproblemandbeingabletoformaplantosolveit.
MultiplicativeComparisonProblemsaredefinedas“involvingacomparisonoftwoquantitiesinwhichoneisdescribedasamultipleoftheother”(Carpenter,Fennema,
THEJOURNALOFTEACHERACTIONRESEARCH 27
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
Franke,Levi,&Empson,2015,p.66).Theseproblemswereemphasizedinthisinquiryprojectbecausetheyincludelanguagethatcanbeparticularlydifficultforstudents(Stern,1993),suchasunderstandingthemeaningof“twiceasmany,”andtheyareprevalentin4thgrademathcurriculaandassessments.
Asdocumentedbystateanddistrictreadingassessments,thecurrentfourthgradeclassatApplegatestruggleswithreadingandcomprehendingtext.Moreover,atthebeginningoftheyear,studentscompletedathirdgrademathtestcontainingwordproblems,manyofthemmultiplicativecomparisonproblems.Theclassaverageonthistestwasonly51%,indicatinganeedforadditionalsupportwiththiskindofproblem.Multiplicativewordproblemshavetheaddedadvantagesofofferingawindowintostudents’understandingoftheproblem,becauseitisdifficulttogetacorrectanswersimplyby“numbergrabbing”(Littlefield&Rieser,1993)—wherestudentspickthenumbersseeninawordproblemandrandomlychoseanoperationwithoutfullyunderstandingthemathematicalsituationdescribed.
Thefollowingresearchquestionsguidedthedesignofthisinquiryproject:
1.)Whatstrategiescanstrugglingreadersusetobettercomprehendmultiplicativecomparisonwordproblems?
2.)DoELsandEOscomprehendwordproblemsdifferently,andifso,how?
3.)Whatpartsofwordproblemsarestudentsstrugglingwith?
DataCollectionandAnalysis.Theeffectivenessofretellstomonitorandaidcomprehensioniswellknownthroughouttheliteracyresearchcommunity(Brown&Cambourne,1987;Hoyt,1999;Mowbray,2010).Therefore,foreachofthethreeroundsofdatacollection,studentswerepromptedto(1)retellapresentedwordproblemintheirownwords,(2)solvetheproblemusingoneofthethreestrategies,and(3)retelltheproblemagain.Theteacherconductedallthreerounds.Students’retellswereevaluatedwithaRetellRubric(Ambrose&Molina,2014),whichunpackswordproblemsintotheircomponentelements,indicatingthepartsoftheproblemstudentsunderstoodorattendedto,andthepartstheydidnot.Theelementsofeachwordproblemthatwereanalyzedarethenumbers,theunits,themathematicalrelationship,andthequestion.Foreachelement,studentswhocorrectlyretoldthatpartoftheproblemreceivedascoreof2,studentswhoretoldanelementdifferentlyfromhowitwasstatedintheproblemreceivedascoreof1,andstudentswhoomittedanelementcompletelyreceivedascoreof0(AppendixA).
ForRound1,studentsweregivenahandoutofthefollowingproblem:Thomasbuiltafencethatwas12timesaslongasTerry’s.Terrybuiltafencethatwas4feetlong.HowlongwasThomas’sfence?Theywerepromptedtoretelltheproblemintheirownwordsafterrereadingtheproblemasmanytimesastheyneeded.Foreachretell,studentsworkedwithmeone-on-onesothattheirpeersdidnotinfluencetheirresponses.Then,
THEJOURNALOFTEACHERACTIONRESEARCH 28
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
inpartners(ELsandEOswerepartneredtogether),studentswerepromptedtodiscussthewordproblemtooneanother,notingsimilaritiesordifferencesinthinkingabouttheproblem(aclassroomnorm).Here,students’conversationswereaudio-recordedandtranscribed.Independently,studentssolvedtheproblemontheirhandout.Theteachertookfieldnotesofstudents’explanationstotheirsolutions.Lastly,studentswereagainpromptedtoretellthewordproblemintheirownwords,rereadingtheproblemasmanytimesastheyneeded.Theteacherrereadthetranscribeddiscussionandcomparedwhatstudentssaidintheirexplanationstothewrittenworkonthehandout.Thisallowedthemestoemergeaboutthematchandmismatchbetweenstudents’oraldiscourseandtheirwrittensolutionstrategies.
ForRound2,studentsweregivenahandoutofthefollowingproblemandpromptedtoretellitintheirownwordsafterrereading:Thegiraffeinthezoois3timesastallasthekangaroo.Thekangaroois6feettall.Howtallisthegiraffe?Theteacherthenaskedstudentstosolvetheproblembydrawingapictureofit.Oncetheyreachedasolution,theteachertookfieldnotesofstudents’explanationsfortheiranswers.Studentswerethenpromptedtoretelltheproblemintheirwordsagain.Toanalyzestudents’drawings,aniterativeprocessofcreatingcodeswasusedbyresearchingthenecessarycomponentsinadrawingneededtocomprehendawordproblem(Dexter&Hughes,2011;Edens&Potter,2008)(AppendixB).VandeWalle’s(2012)four-pointrubricwasmodifiedbyaddinganothercategorytitled“Outstanding,”indicatingnoerrorsinthedrawing.Thisrubric,(AppendixC),allowedstudents’worktobeanalyzedagainstasetofdesiredlearnerresponsesandtoplacetheirunderstandingsonadevelopmentalcontinuum.Studentswerenotexpectedtoreceiveaperfectscoreonthedrawingrubricasstudents’drawingswereintendedforthemtomakesenseoftheproblem—nottopresenttoanaudience.
Forround3,studentsretoldthefollowingwordproblemintheirownwords:Jilllived5timesasmanymilesasLeodidfromtheocean.Leolived20milesfromtheocean.HowmanymilesdidJilllivefromtheocean?Theteacherthenaskedstudentstosolvetheproblembyjournaling,orwritingtheirthoughts,aboutit.Theteacheremphasizedthattheirgrammarandspellingwasnotimportant.Afterstudentsjournaledontheirhandoutandarrivedatasolution,studentsretoldtheproblemagain.Students’writingwasanalyzedintwodifferentways.First,usingAmbroseandMolina’s(2014)retellrubricwasusedtoassessstudents’writing.Thiswasanalyzedtodetermineifstudents’retellsweredifferentwhentheywerewrittenfromwhentheywerespoken.
Second,students’writingwasanalyzedbygoingthroughadeductiveprocessofcreatingaMetacognitiveStrategiesRubric(AppendixD).Metacognitivestrategieswereanalyzedbecauseresearchindicatesastrongcorrelationbetweenmetacognitiveskillsandstudents’comprehension(Lippmann&Linder,2007).DrawingontheworkofTanner(2012),whosuggeststeacherssupportthedevelopmentofmetacognitionbyaskingself-reflectivequestionsforplanning,monitoring,andevaluating,theteachercreatedarubrictoassessstudents’metacognitionwithintheirmathematicalwriting.Studentsreceivedofscoreof1for“yes”andascoreof0for“no.”Atotalpossiblescore
THEJOURNALOFTEACHERACTIONRESEARCH 29
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
was3,althoughitwasnotexpectedstudentswouldreceiveamaximumscoreastheywereaskedtojournalabouttheproblemonlyforthemselvesandwerenottoldtheirwritingwouldbereadbyadistantaudience.
ResultsandDiscussion
FindingsfromRound1suggestthefocalstudents’comprehensionofwordproblemsimprovedwhentheyweregivenopportunitiestodiscussaproblemwiththeirpeers.Firstgradelevelreadersimprovedthemostsignificantly,asseeninthefiguresbelow.Forexample,Sharron’sabilitytoincludeelementsoftheoriginalwordproblemduringthesecondretellingincreasedby50%,andDominick’ssecondretellingincreasedby38%.YusufandSharronwereabletoidentifytherelationshipoftheproblemaftertheirpeerdescribeditinthestudentdiscussion.Thesestudentswhoinitiallystruggledwiththeconceptlearnedfromtheirpeeroncegivenopportunitiestodiscuss.Thisfindingissupportedbyotherresearchstudiesasstudentslearnfromoneanotherwhenengaginginstudent-talk(Frankeetal.,2009).
AlthoughMarywasabletocorrectlyrestatetheprobleminherownwordswith100%accuracyduringthepreandpost-discussionretellings,therewasamisalignmentbetweenwhatshesaidandhowshesolvedtheproblem.Theotherthreestudents’retellmatchedhowtheysolvedtheproblem.Thisfindingmayindicatethelimitationsofusingretellsasawaytogaugewhetherstudentsunderstandwordproblems.Perhapsasacompensationstrategyforlowreadingability,somestudentsareabletomemorizeandrestateaproblemwithoutreallyunderstandingthepresentedmathematicalsituation.
Figure1:GraphofPre-andPost-DiscussionRetellScores.
AsimilarphenomenonoccurredinRound2whenstudents’retellsappeartoworsenaftertheuseofthedrawingstrategy,seeninFigure2.However,threestudentssolvedtheproblemcorrectlyandwereabletoexplaintheirsolutions,demonstratingtheircomprehension.TheaveragescoreontheDrawingRubricwas3.5outof5.Nostudentincludedarepresentationoftheunknown,whichismirroredbytheirretells.However,thedrawingwasintendedtosupportstudents’comprehension,nottopresentafullpicturetoanaudience.Drawingsprovidedfocalstudentswithanalternativemeanstocommunicatetheircomprehensionofthewordproblem.Thiswasespeciallybeneficial
02468
PrePost PrePost PrePost PrePost
Mary Dominick Yusuf Sharron
Rubricscores
DiscussionRetells
Question
Relationship
Units
Numbers
THEJOURNALOFTEACHERACTIONRESEARCH 30
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
fortheELstudents(DominickandYusuf),perhapsduetoareductioninlinguisticdemandthroughdrawing.Dominickrepresentedtherelationshipbetweentheanimals’heightswithanarrowandYusufdrewabarindicatingdifferencesinheight.ThesesamplesarefoundinAppendixE.Studentsoftenpointedtotheirdrawingstocontextualizewhattheywereexplainingtotheteacherintheirplanningprocesstosolvetheproblem,afindingwhichisaffirmedbypriorstudiesofcommunicationviadrawinginmathematics(Dexter&Hughes,2011;Edens&Potter,2008).Althoughtheirretellswereincomplete,students’comprehensionofthewordproblemwasrepresentedthroughtheirdrawingsandexplanations.
Figure2:GraphofPre-andPost-DrawingRetellScores.
FindingsfromRound3suggeststudentsbenefitedfromwritingaboutthemathproblem,displayedinFigure3.Comparingthepre-andpost-retells,weseethatYusufandSharronhadsimilardifficultiesretellingtherelationshipinvolvedinthisproblem.ThissuggeststhatinterpretingrelationshipsincomparisonwordproblemsareequallydifficultforEOandELstudents.BothELstudents,YusufandDominick,alsosharesimilarimprovementsintheirpost-retellasbothstudentswereabletocorrectlyidentifyingthenumbersintheproblem.Thissuggeststhattheirwritingmighthavehelpedtheminternalizetheproblemandassociatednumbersatdeeperlevelthanverballyspeaking.ThisissupportedbyBaxterandcolleagues’(2005)whofoundthatstudents’comprehensionofwordproblemswasmoreevidentintheirwritingthanintheiroraldiscourse.Sharron’sandYusuf’swritingincludedmoreelementsofthewordproblemthantheirpreviousretell,suggestingthatwritinggavestudentsanopportunitytothinkabouttheproblemmoredeeply.However,thispatternisreversedforDominick,whoincludedonlytherelationshipintheprobleminhiswritingandnootherelementsoftheproblem,asseeninFigure4.ThissuggeststhatforCELDTlevel1students,writingmaynotbeaseffectiveastrategytosupportstudents’thinkingandcomprehensionofaproblem.ThisfindingisaffirmedbyresearchersEdens&Potter(2008)whofoundthatlinguisticdemandsaredecreasedforELstudentswhenpicturesareutilizedbutareincreasedwhenwritingtasksarerequired.Itappearsthatwritingmayhaveencouragedstudentstoformulateaplantosolve(AppendixF).However,whenverballyprompted,studentsillustratedevenmoremetacognitiveskills.For
0
2
4
6
8
PrePost PrePost PrePost PrePost
Mary Dominick Yusuf Sharron
Rubricscores
DrawingRetells
Question
Relationship
Units
Numbers
THEJOURNALOFTEACHERACTIONRESEARCH 31
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
example,Yusufwrote,“youcandoaddup”butwhenaskedbytheteacherwhywouldweaddhestated,“Uh!No!You’resupposetotimes!Causeitsays5times.Somultiply!”Thissuggeststhatespeciallywhenpairedwithteacherquestioning,writingcanhelpstudentsthinkabouttheirjustificationforsolvingaprobleminaparticularway.
Figure3:GraphofPre-andPost-WritingRetellScores.
Figure4:GraphofElementsIncludedinStudents’Writing.
ConclusionandImplications
Thisactionresearchprojectinvestigatedstrugglingreaders’comprehensionofwordproblemsutilizingthreedifferentstrategies:discussingwithapartner,drawing,andwriting.Studentswiththelowestreadingscoresshowedthemostimprovementafterdiscussingthewordproblemwithapartner.Thissuggeststeachersshouldpartnerstudentsofdifferentreadinglevelstogetherbecausestrugglingreadersseemtobenefitwhenpartneredwithhigher-leveledreadingpartners.Drawingandwritingaboutwordproblemscanalsobeeffectivestrategiesforstrugglingreaders,especiallythosewhoareEnglishlearners.DrawingwasespeciallyhelpfulforELstudentsperhapsbecauseit
02468
PrePost PrePost PrePost PrePost
Mary Dominick Yusuf Sharron
Rubricscores
RetellScoresBeforeandAfterWriting
Question
Relationship
Units
Numbers
0
2
4
6
8
Mary Dominick Yusuf Sharron
ElementsinStudents'Writing
Writing-RetellQuestion
Writing-RetellRelationship
Writing-RetellUnits
Writing-RetellNumbers
THEJOURNALOFTEACHERACTIONRESEARCH 32
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
offersanopportunitytocommunicatemathematicalunderstandingthroughsymbolsandvisualrepresentations.Writingalsoseemedtosupportstudents’comprehensionofwordproblems,althoughitcreatedadditionalchallengesforDominick,anELwithbeginninglevelEnglishproficiency.Thus,teachersmaywanttobecautiouswhenusingwriting-to-learnstrategieswithstudentswhoareatthebeginningstagesofEnglishdevelopment.BeginningELsmaybebetterabletocommunicatetheirunderstandingthroughothermodes,suchasdrawing,discussion,orusingmanipulatives.
Multipleandvariedmeasuresofassessmentmayhelpteachersgainadeeperunderstandingoftheirstudents’comprehensionofwordproblems.Althoughtheyprovidevaluableinformationaboutstudents’understandingofaproblem,usingretellsalonemaybemisleading.Forexample,Maryretoldtheproblemwith100%accuracybuttherewasamisalignmentbetweenwhatshesaidandhowshesolvedtheproblem.Furthermore,acrossallthreeroundsMary’spreandpostretellscoresremainedunchanged,althoughshedidnotalwayssolvetheproblemscorrectly.Thissuggestsretellsalonearenotenoughtodeterminewhetherastudentunderstoodtheproblem.Multipleassessments,suchausingacombinationofretells,drawings,writing,andoraldiscussion,canofferabroader,perhapsmoreaccurateviewofstudents’comprehensionofmathematicalwordproblems.Futureresearchcanexplorehowdifferentkindsofpromptsforwritingcanelicitdifferencesinstudents’mathematicalwriting,andwhetheritismorebeneficialforstudentstodrawapictureofthemathematicalsituationbeforeorafterdiscussingtheproblemwithpeers.
AbouttheAuthors
LeannaR.MacDonaldteachesfourthgradeinaTitleIpublicschool.ShereceivedhercredentialsandmasterdegreeineducationfromUC,Davisandisinterestedinresearchandinstructionthathelpshergainadeeperunderstandingofherstudents'mathematicalthinking.Email:[email protected]
LeslieC.Banesisaformerbilingualelementaryschoolteacher.SheisaninstructorfortheMAEducationprogramatUC,DavisinCaliforniaandcollaborateswithteacherstoexploretherelationshipbetweenmathematicsandlanguagelearning. Email:[email protected]
THEJOURNALOFTEACHERACTIONRESEARCH 33
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
ReferenceListAmbrose,R.,&Molina,M.(2014).Spanish/Englishbilingualstudents’comprehensionof arithmeticstoryproblemstexts.InternationalJournalofScienceandMathematicsEducation,12 (6),1469-1496.Artz,A.F.andArmour-Thomas,E.(1992).Developmentofacognitive-metacognitive frameworkforprotocolanalysisofmathematicalproblemsolvinginsmallgroups. CognitionandInstruction9(2),137–175.Bargh,J.A.,&Schul,Y.(1980).Onthecognitivebenefitofteaching.JournalofEducational Psychology,72,593-604.Barton,M.L.,Heidema,C.,&Jordan,D.(2002).Teachingreadinginmathematicsandscience. EducationalLeadership,60(3),24–28.Baxter,J.A.,Woodward,J.,&Olson,D.(2005).Writinginmathematics:Analternativeformof communicationforacademicallylow-achievingstudents.LearningDisabilitiesResearch& Practice,20,119–135.Brown,H.&Cambourne,B.(1987).Readandretell:Astrategyforthewhole language/naturallearningclassroom.Portsmouth:Heinemann.Carpenter,T.P.,Fennema,E.,Franke,M.L.,Levi,L.,Empson.(2015).Children’sMathematics: CognitivelyGuidedInstruction(2nded.).Portsmouth,NH:Heinemann.Carr,M.&Biddlecomb,B.(1998).Metacognitioninmathematics:Fromaconstructivist perspective’,inD.J.Hacker,J.DunloskyandA.C.Graesser(eds.),Metacognitionin EducationalTheoryandPractice,LawrenceErlbaumAssociates,Mahweh,NJ.Chi,M.T.H.(2000).Self-explainingexpositorytexts:Thedualprocessesofgenerating inferencesandrepairingmentalmodels.InR.Glaser(Ed.),Advancesininstructionalpsychology: Educationaldesignandcognitivescience(pp.161-238).Hillsdale,NJ:LawrenceErlbaum.Dexter,D.D.,&Hughes,C.A.(2011).Graphicorganizersandstudentswithlearning disabilities:Ameta-analysis.LearningDisabilityQuarterly,34(1),51–72.Edens,K.,&Potter,E.(2008).Howstudents“unpack”thestructureofwordproblem:graphic representationsandproblemsolving.SchoolScience&Mathematics,180(5),184-196.Franke,M.,Webb,N.,Chan,A.,Ing,M.,Freund,D.,Battey,D.(2009).Teacherquestioningto elicitstudents’mathematicalthinkinginelementaryschoolclassrooms.Journalof TeacherEducation,60(4),380-392.Hegarty,M.,Mayer,R.E.,Monk,C.E.(1995).Comprehensionofwordproblems:A comparisonofsuccessfulandunsuccessfulproblemsolvers.JournalofEducationalPyschology, 87(1),18-32.Hoyt,L.(1999).Revisit,Reflect,Retell:StrategiesforImprovingReadingComprehension. USA:Heinemann(Portsmouth).King,A.(1992).Facilitatingelaborativelearningthroughguidedstudentgeneratedquestioning. EducationalPsychologist,27,111-126.
THEJOURNALOFTEACHERACTIONRESEARCH 34
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
Lippmann,R.K.,&Linder,C.(2007).Metacognitiveactivityinthephysicsstudentlaboratory: Isincreasedmetacognitionnecessarilybetter?MetacognitionandLearning,2(1),41-56.Littlefield,J.,&Rieser,J.J.(1993).Semanticfeaturesofsimilarityandchildren'sstrategiesfor identificationofrelevantinformationinmathematicalstoryproblems.Cognition&Instruction,11, 133-188.Marino,M.,Black,A.,Hayes,M.,Beecher,C.(2010).Ananalysisoffactorsthataffect strugglingreaders’achievementduringatechnology-enhancedSTEMastronomycurriculum. JournalofSpecialEducationTechnology,25(3),35-47.Mowbray,T.(2010).Thepowerofreadandretell.PracticallyPrimary15(2),10-12.O'Connell,S.R.,Beamon,C,Beyea,J.M.,Denvir,S.S.,Dowdall,L.A.,Friedland,N.G.,etal. (2005).Aimingforunderstanding:Lessonslearnedaboutwritinginmathematics.Teaching ChildrenMathematics,12,192-199.Osman,M.E.,&Hannafin,M.J.(1992).Metacognitionresearchandtheory:Analysisand implicationsforinstructionaldesign.EducationalTechnologyResearchandDevelopment,40(2), 83-99.Powell,A.B.(1997)Capturing,examiningandrespondingtomathematicalthinking throughwriting.TheClearingHouse71(1),21–25.Pugalee,D.K.(2001).Writing,mathematicsandmetacognition:Lookingforconnections throughstudents’workinmathematicalproblemsolving.SchoolScienceand Mathematics101(5),236–245.Rogoff,B.(1991).Guidanceandparticipationinspatialplanning.InL.Resnick,J.Levine,&S. Teasley(Eds.),Perspectivesonsociallysharedcognition(pp.349-383).Washington, DC:AmericanPsychologicalAssociation.Stern,E.(1993).Whatmakescertainarithmeticwordproblemsinvolvingthecomparisonofsets sodifficultforchildren?JournalofEducationalPsychology,85(1),7-23.Tanner,K.B.(2012).PromotingStudentMetacognition.CBE-LifeSciencesEducation,11(2), 113-120.VandeWalle,J.A.;Karp.K.S.;Bay-Williams,J.M.(2012).ElementaryandMiddleSchool Mathematics:TeachingDevelopmentally(8thed.).USA:Pearson.Vilenius-Tuohimaa,PM;Aunola,K.;Nurmi,JE.(2008).Theassociationbetweenmathematical wordproblemsandreadingcomprehension.EducationalPsychology,28(4),40.
THEJOURNALOFTEACHERACTIONRESEARCH 35
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
AppendixA:RetellRubric
Element Same
(2)
Different
(1)
Missing
(0)
Numbers Containsbothnumbersfromoriginalproblem.
Containstwonumbers.Atleastonenumberisdifferentthanoriginal.
Atleastonenumberintheproblemismissing.
Units Unitisidentifiedinproblem. Unitisdifferentfromoriginalproblem.
Unitismissing.
Relationship Relationshipisconsistentwithwordproblem.
Relationshipdiffersfromwhatwasdescribedinwordproblem.
Relationshipismissing.
Question Containsaquestionconsistentwithwordproblems.
Containsaquestionwithmeaningdifferentfromoriginal.
Questionismissing.
THEJOURNALOFTEACHERACTIONRESEARCH 36
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
AppendixB:ElementsWithinDrawing
Student Kangaroo Giraffe Kangaroo’s height
Giraffe’s relationship
Unknown
Rubric Score
X X X X X X X
THEJOURNALOFTEACHERACTIONRESEARCH 37
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
AppendixC:DrawingRubric
NoAttempt
0
Unsatisfactory
1
Marginal
2
Proficient
3
Excellent
4
Outstanding
5
Thetaskisnotcompleted.Noelementsindrawingarecorrectlyindicated.
Fragmentsofaccomplishment,butlittletonosuccess.Onlyoneelementindrawingiscorrectlyindicated.
Partoftaskisaccomplished,butlackofevidenceofunderstanding.Twoelementsindrawingarecorrectlyindicated.
Couldworktofullaccomplishmentwithminimalfeedback.Threeelementsarecorrectlyindicated.
Drawingmeetsdemandsoftask.Mayhaveminorerrors.Fourelementsarecorrectlyindicated.
Drawingmeetsdemandsoftaskwithnoerrorsandallfiveelementsarecorrectlyindicated.
THEJOURNALOFTEACHERACTIONRESEARCH 38
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
AppendixD:MetacognitiveRubric
Student Planning
Doesthestudentwriteaboutakindofplantosolvetheproblem?
Monitoring
Doesthestudentwriteaboutanycontemplationtheyarehavingabouttheproblem?
Evaluating
Doesthestudentchecktomakesureananswermakessense,ordoublechecktheirthinking?
X X X X
THEJOURNALOFTEACHERACTIONRESEARCH 39
JournalofTeacherActionResearch- Volume3,Issue3,2017,<practicalteacherresearch.com>,ISSN#2332-2233©JTAR.AllRights
AppendixE:Drawings
Yusuf’sDrawing Dominick’sDrawing