THE JOURNAL OF TEACHER ACTION RESEARCH 81€¦ · THE JOURNAL OF TEACHER ACTION RESEARCH 82 Journal...

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Transcript of THE JOURNAL OF TEACHER ACTION RESEARCH 81€¦ · THE JOURNAL OF TEACHER ACTION RESEARCH 82 Journal...

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DIFFERENTIATINGINSTRUCTIONTOINCREASECONCEPTUALUNDERSTANDINGANDENGAGEMENTINMATHEMATICSMelissaJ.MaininiandLeslieC.Banes

UniversityofCalifornia,Davis

AbstractThisaction-researchprojectaimedtoexploreseveralstrategiesthatteacherscanusetodevelopconceptualmathematicalunderstandingandincreasebehavioralengagementforstudentswithdifferinginstructionalneeds.Specifically,Iinvestigatedstrategiesfordifferentiatinginstruction,includingindividualizedinstructioninflexiblegroupsandtheuseofproblemswithmultipleentrypointsoverafive-monthperiodina6thgradeclassroom.Analysesfocusedonsixfocalstudents,allofwhomwereEnglishlearnersorformerEnglishlearners.Overall,findingssuggestthattheuseofflexibleinstructionalgroupsandmathproblemswithmultipleentrypointscanhelpteachersmeetthevariedneedsofstudents,thusallowingallstudentstobemoreengagedandsuccessfulintheclassroom.

Keywords:differentiatedinstruction,Englishlearners,flexiblesmallgroups,mathematics,conceptualunderstanding,studentengagement

Introduction

IcringeasIremembermyveryfirstdayasaclassroomteacher.ItwasarefreshingAugustmorningasIeagerlydeliveredmy6thgrademathlesson.IhadmeticulouslycreatedaPowerPointpresentationonplacevalue,completewithcolorfulfontsandengaginganimations.Ideliveredmycarefullycraftedlessonwithstereotypicalfirst-year-teacherenthusiasmandhandedoutaworksheetforindependentpractice.Mystudents,withtheirownfirst-day-of-schoolmotivation,diligentlybegancompletingtheproblems.Gradually,studentsbegantoraisetheirhandswithquestions.Iracedthroughtherowsofstudents,literallyjoggingfromdesktodesktryingtolendsupportandanswerquestions.Somestudentslookedatmewithembarrassment,nothavingbeenabletostartthefirstproblemontheirown.Othersreachedamomentofpanicassoonastheysawawordproblem.WhileIstruggledtoofferenoughsupporttothestudentswhoneededit,severalother

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studentsracedthroughtheproblemsandlookedatmewitheagereyes,asking,“Whatdowedonext?!”IrealizedIdidn’thaveananswerforthem.Forsomestudents,theproblemswerenotchallengingenoughtoextendtheirthinking,yetothershadbarelystarted.Iwondered,howcouldIbeaneffectiveteachertobothofthesesetsoflearners?ThisisthequestionIendeavoredtoanswerformyself,andforotherteacherswhofacesimilarchallenges.

Teachersoftenstrugglewithhowtodifferentiateinstructioninordertosimultaneouslymeettheneedsofallstudentsintheirclass.However,whetherornottogroupstudentsaccordingtotheircurrentmathematicsachievementhasbeenacontentiousissueineducationsincethe1980s(Boaler,2013).InCalifornia,theaveragesizeofapublicupperelementaryschoolclassroomisover25students,andaboutonethirdofthesearestillintheprocessoflearningthelanguageofinstruction(CalEdFacts,2014).Itisinevitablethatasingleclassroomwillrepresentawidevarietyofstudentabilitylevels,learningstyles,strengths,andneeds.Teachers,therefore,mustbewellequippedwithstrategiesthatallowthemtomaximizetheacademicandintellectualgrowthofalltypesoflearners,includingmeetingtheneedsofEnglishlearners(ELs)andformerELs.WeusethetermEnglishlearners(Els)herebecausewebelievethistermisfamiliartoourreaders.However,ourbeliefsaremoreinlinewithtermemergingbilingualinsteadofEnglishlearnerasawayofemphasizingthevalueofbilingualism(Garcia,2009).Inourownclassrooms,myco-authorandIhavenoticedstudentsoftenshowsignsofdisengagementwheninstructiondoesnotmatchtheircurrentlevelofunderstanding,beittooeasyortoodifficult.Whenitcomestomathematicsinstruction,onesizedoesnotfitall.Receivingindividualizedsupportandguidancecateredtotheirspecificneedsandstrengthscanallowallstudentstothriveacademicallyintheclassroom.

Thisaction-researchstudyaimedtouncoverhowtwodifferentstrategiesfordifferentiationcanbeusedina6thgradeclassroomwiththegoalofincreasingmathematicalunderstandingandbehavioralengagement.Althoughorchestratingseveralstudentgroupsatthesametimecanbechallenging,wefoundittobeaneffectivewayforonenewteachertomeetthedisparateneedsofstudents,allowingustousestrategiessuggestedbypastresearchtosupportELsinparticular.Thefollowingoverarchingquestionsguidedourinquiry:

1. Howcanflexible,smallgroupinstructionimpacttheconceptualunderstandingandbehavioralengagementofstudentswithdifferingneeds?

2. Howcanproblemswithmultipleentrypoints(Low-Floor-High-Ceilingproblems)serveasawaytodifferentiateinstructionforstudentswithdifferingneeds?

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Teacher-ResearcherPositionality.Thefirstauthorwastheclassroomteacher.Atthetimeofdatacollection,Iwasinmyfirstyearofteaching.ThisprojectwasconductedaspartofmyMastersofEducationdegreeatalarge,research-orienteduniversity.Ilaterextendedtheanalysesofmyprojectforpublication.Unlessotherwisenoted,useofthepronoun“I”throughoutthetextreferstome,and“we”referstobothauthors.

Thesecondauthorisaformerbilingualteacherandwastheinstructorofthetwo-quarterresearchmethodscourseIattendedaspartofmyMaster’sprogram.Ascourseinstructor,sheguidedthedesignofthisinquiryproject,encouragedmetosharewhatIlearnedwithothereducators,andcollaboratedwithmeinwritingthisarticle.

FlexibleAbilityGrouping.Abilitylevelgroupingremainsacontroversialtopicineducation,andresearchonitsimpactremainsmixed.Whilesomeresearchhasdemonstratedpositiveeffects,otherresearchsuggestsabilitygroupingcanhaveanegativesocialimpactonstudents(Hallam,Ireson,Mortimore,&Davies,2000).Incontrast,seminalworkbySlavin(1987)demonstratesthatabilitylevelgroupingcanbeaneffectiveinstructionalstrategy,aslongastheabilitygroupsareconfinedtoaspecificsubject(e.g.mathorreading).Slavinfurthersuggeststhatabilitygroupingallowshigherachievingstudentstobeexposedtoanappropriatelyacceleratedpaceofinstruction,whileallowinglowerachievingstudentstoreceivemoreattentionandpractice.

Similarly,Gibbons(1991)discussesthenotionofskillsgrouping:thearrangementofstudentsintogroupsbasedontheirabilitylevelsandneeds.Skillsgrouping,whendonefluidlyandforashortperiodoftime,allowsstudentstobeexposedtoinstructionandcontentthatmatchtheircurrentneedsandlevelofunderstanding.Small-groupinstructionisdefinedassituationsinwhichthreeormorestudentsworkonacommonmathematicaltask(Jansen,2012).

ConceptualUnderstanding.Forthepurposesofthispaper,Idefineconceptualunderstandingasastudent’sabilityto“[recognize]and[understand]coreunderlying[mathematical]ideas”(Burnsetal.,2015)andtorecognizehowsuchideasareinterrelated(NationalResearchCouncil,2001).Thisstandsincontrasttoproceduralskill,whichisastudent'sabilitytoexecutethestepsneededtosolveaproblem(Rittle-Johnson,Siegler,&Alibali,2001).Ialsoexploredhowastudent’sabilitytoapplymathematicalconceptstoreal-worldsituationsisconnectedtoconceptualunderstanding.Studentswithconceptualunderstandingarebetterabletoapplymathematicalknowledgelearnedatschooltosituationsineverydaylifethanstudentswhoonlyhaveproceduralskills(Kilpatrick,Swafford,&Findell,2001).Further,theCommonCoreStateStandardsstatethat

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“mathematicallyproficientstudentscanapplythemathematicstheyknowtosolveproblemsarisingineverydaylife,society,andtheworkplace,”thusemphasizingtheimportanceofconceptualunderstanding(CCSS.Math.Practice.MP4,2017).

BehavioralEngagement.Ialsosoughttoexplorehowsmall-groupinstructionmightimpactthebehavioralengagementofstudents.Behavioralengagementisdefinedasstudents’activeparticipationinlearningactivities(Wang,Berlin,&Berlin,2014).Askingquestions,sharinganswers,andmakingrelatedcommentsallmaybeindicatorsofbehavioralengagement.Studentengagementisessential,asithasbeenshownbyresearchtobeanindicatorofacademicachievement(Dotterer&Lowe,2011).

LiteratureReview

Small-groupInstructioninMathematics.Pastresearchhasdemonstratedthatsmall-groupinstructioncanbeusedtoenhancestudentlearning.Forexample,KazemiandStipek(2001)describehowsmall-groupdiscussioninwhichteacherspressstudentsforjustificationoftheirmathematicalideascanhelpstudentsmovebeyondtheircurrentlevelofunderstanding.Buildingonthisresearch,Webbandcolleagues(2009)comparedthenatureofelementarymathteachers’interactionswithstudentsinsmallgroupswiththeirinteractionsduringwhole-classinstruction.Resultssuggestthatteacherprobingofstudents’ideasinsmallgroupsmaybemoreeffectivethanprobingduringwhole-classinstruction,leadingtohigherinstancesofcorrectandcompletemathematicalexplanations.Giventhis,small-groupinstructionservedasthefoundationformydatacollection.

StrategiesforSupportingEnglishLearnersandFormerEnglishLearners.ResearchhasdemonstratedthatsomestrategiesaremoreeffectivethanothersatsupportingELs.Specifically,Walqui(2006)identifiedthefollowingrelevantstrategies:modeling,bridging(connectingnewconceptstopriorknowledge),andcontextualizing(relatingconceptstoeverydaysituationsandlanguage).ThesestrategiescanbeappliedtofurthersupportELsduringinstructioninflexibleabilitygroups.InadditiontosupportingstudentscurrentlyclassifiedasELs,researchsuggestsmanystudentswhohavebeenreclassifiedEnglishproficient(formerELs)stillrequirelanguagesupportinordertosucceedwiththedemandsofcontentarealiteracy(Haas,Huang,&Tran,2014).

OfferingChoiceasDifferentiatedInstruction.Similarly,researchhasalsorevealedstrategiesforenhancingbothbehavioralengagementandconceptualunderstandinginstudentswhoare“advanced,”oraheadofthemajorityoftheclass.Tomlinson(2005)arguesteachersshouldacceleratethepaceofinstructionformoreadvancedlearnersandofferopportunitiestomakechoices.Whenstudentsareallowedtomakedecisionsaboutthematerialstheyuse,theproblemstheysolve,ortheassignmentstheycomplete,they

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generallymakechoicesthataremoreappropriatefortheirneedsthanwhatcanbeofferedtotheclassasawhole;this,inturn,improvesmotivationandhelpspreventdisruptivebehavior(Bluestein,2008).

MathproblemswithMultipleEntryPoints.SeveralmatheducatorshaveadvocatedforLow-Floor-High-Ceiling(LFHC)tasks,alsocalledLow-Threshold-High-Ceilingtasks,asawayofprovidingmeaningfulactivitiestodifferenttypesoflearners.LFHCtaskscanbeaccessibletoallstudentsbecausetheyhavemultipleentrypoints;studentscanbegintheproblematdifferentlevels.However,theseproblemsalsocanbeextendedtohigherlevelsdependingonstudents’abilitylevels.ThefollowingproblemisanexampleofaLFHCtask,adaptedfromYouCubed(2016):

Foreachpartoftheproblem,startwithasquaresheetofpaperandmakefoldstoconstructanewshape.Then,explainhowyouknowtheshapeyouconstructedhasthespecifiedarea.

1. Constructasquarewithexactly¼theareaoftheoriginalsquare.Explainhowyouknowthatthisnewsquarehas¼ofthearea.

2. Constructatrianglewithexactly¼theareaoftheoriginalsquare.Explainhowyouknowthatthisnewtrianglehas¼ofthearea.

3. Constructasquare(i.e.notarectangle)withexactly½theareaoftheoriginalsquare.Explainhowyouknowthatthisnewsquarehas½ofthearea.

Suchtasksallowstudentstoworkattheirownpace,whilealsoprovidingopportunitiesforchallengeandcriticalthinking(Bernander&Metke,n.d.).LFHCtasksaredesignedtoallowstudentsto“showwhattheycando,notwhattheycan’t”(NRICH,2011).Thus,theseproblemsofferthepotentialfordifferentiatinginstructionwhileallowingallstudentstoaccessgrade-levelconcepts.

Methodology

ContextandParticipants.Allresearchwasconductedwhilethefirstauthorwastheteacherofrecordinaself-contained6thgradeclassroom.Of28students,17wereELs,andsixhadbeenreclassifiedfluentEnglishproficient(R-FEP).AllELswerenativeSpanishspeakers.School-wide,77.5%ofstudentsqualifiedforFree-and-ReducedLunch,anindicatoroflowsocio-economicbackground.AllclassroominstructionandstudentdiscussionstookplaceinEnglish.Basedonthepreviousyear’sstandardizedmathscores,71%oftheclassclassifiedas“StandardNotMet,”29%as“StandardNearlyMet,”andnostudentswereclassifiedas“StandardMet.”

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Table1displaysbackgroundcharacteristicsofthesixfocalstudents.Comparedtotheirpeers,studentsassignedtoInterventionGroup1werethemostinneedofsupportinregardstodivisionandrelatedwordproblems.Incontrast,studentsinInterventionGroup2haddemonstratedanabilitytoworkatafasterrateofinstructionthanthemajorityoftheirpeersandhadproventhattheycouldsuccessfullyperformrelevantskillsindependently.

Studentsweregroupedfluidlybasedontheirabilitylevelofrelatedconcepts.Whenforminggroups,Ilargelyreliedonstudentdatacollectedduringthelesson.Forexample,studentsansweredquestionsonwhiteboardsduringthelessons,participatedinpair-sharesandclassdiscussions,andwereencouragedtoaskquestions.Listeningandrecordingstudents’responsesoffereddatasourcesthathelpedmetrackhowwellcertainstudentscomprehendedtheconceptathand.Additionally,myownknowledgeofstudentstrengthsandneedsfurtherhelpedmeformulategroups.

Becauseofthefluidnatureofthegroupingprocess,thecompositionandsizeofthesmallgroupsweredifferentforeachlesson.Forthepurposesofthisstudy,however,thesixfocalstudentsremainedinthesamegroupsthroughoutallsixroundsofdatacollection.ItisimportanttonotethatsmallgroupingisastrategythatIusedfrequentlyinmyclassroomacrosssubjects,notjustduringdatacollection.Asaresult,thesixfocalstudentswereassignedtodifferentgroupsduringotherlessons.Whilesomeresearchhasarguedthatabilitylevelgroupingcannegativelyimpactstudentssocially,Ifoundthatmanystudentsinmyclasswantedtobeinthesmallgroupthatreceivedinterventioninstruction,andthus,moreteacherattention.ItshouldbenotedthatstandardizedtestscoresareincludedinTable1asasourceofbackgroundinformation,notasameansofhowstudentswereassignedtogroups.

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Table1:FocalStudentBackgroundCharacteristics

FocalStudents

(*Nameshavebeenchanged)

Group CommonCoreStandardizedMathAssessment

CaliforniaEnglishLanguageDevelopmentTest(CELDT)

Alex* Intervention1 StandardNotMet EarlyAdvanced(4)

Brandon Intervention1 StandardNotMet R-FEP

Diana Intervention1 StandardNotMet Intermediate(3)

David Intervention2 StandardNearlyMet R-FEP

Megan Intervention2 StandardNearlyMet R-FEP

Sammy Intervention2 StandardNearlyMet R-FEP

Themathcontentemphasizedininterventionswasguidedbythedistrict-adoptedcurriculum,GoMath!(HoughtonMifflinHarcourt).

DataCollectionandAnalysis.Ipositionthisstudyas“teacheractionresearch”drawingonCochran-Smith’sandLytle’s(1993)definition,“systematicandintentionalinquirycarriedoutbyteachers”intheirownclassroomsforthepurposeoftakingactionthathasthepotentialtoimprovelearning(p.3).Inthisstudy,Ianalyzedtheperformanceandgrowthofsixfocalstudents,conductingresearchintwophases.SeeTable2foradescriptionofdatacollectionacrossphases.Phaseoneconsistedoffourroundsofdatacollectionfocusingontheuseofsmall-groupdifferentiatedinstructiontomeettheuniqueneedsofeachgroup.Phasetwoconsistedoftworoundsofdatacollectionexploringtheuseofsmall-groupLFHCtasksasanotherstrategyfordifferentiatinginstruction.Thebasicformatofinstructionwasthesameforallrounds.BeforeIfacilitatedsmall-groupinstruction,Iconductedawhole-grouplessoninwhichImodeledamathematicalconceptandcorrespondingskills.Isubsequentlydividedstudentsintosmallgroupsof5-7studentsbasedoncurrentabilitylevelsforthespecificmathconcept.Behavioralengagementdatawascollectedduringbothwhole-groupandsmall-groupinstruction.

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Table2:SummaryofDataCollection

PhaseOne PhaseTwo

Round

1

Round2

Round3 Round4 Round5 Round6

ConceptualUnderstandingData

Pre-

Interview,

Quiz1

Quiz2 Quiz3 Post-

Interviews,

Quiz4

Pre-

Interviews,Writtenfree-responsequestion,Quiz5

Writtenfree-responsequestion,Quiz6

BehavioralEngagementData

Observ-ationalfieldnotes,Videotapedsmall-

groupinstruction

Observ-ationalfieldnotes,Videotapedsmall-groupinstruction

Observ-ationalfieldnotes,Videotapedsmall-groupinstruction

Observ-ationalfieldnotes,Videotapedsmall-groupinstruction

PhaseOne:DifferentiatedSmall-GroupInstruction

InterventionGroup1.BecausethisgroupconsistedofELsandformerELswhostruggledtounderstandwordproblems,Iworkeddirectlywiththegroupusingthestrategyofcontextualizing(Walqui,2006).Iledstudentsindiscussingandvisuallyrepresentingproblemsthatrequiredstudentstoapplythemathematicalconcepttaughtduringwhole-groupinstruction.

Forexample,thefocusofthelessonsandactivitiesinRound4wasOrderofOperations.Duringsmall-groupinstruction,studentswereaskedtosolvethefollowingproblem,(½+¾)÷2.StudentsandIcollaborativelycontextualizedthisproblembycreatinga‘real-world’scenariothatdescribedtheproblemandcreatedapicturetorepresentthescenario.Thestudentscontextualizedthefractionssayingtheyrepresented“½ofachocolatecakeand¾ofacheesecake.”Theythencreatedpicturestorepresenteachmathematicaloperation.

InterventionGroup2.Thisgroupreceivedanadaptedassignmentthatrequiredthemtoapplythesamemathematicalconcept,butwithlargernumbersandmultiplesteps.Ialsoencouragedstudentstoshowmultiplewaysofsolvingeachproblemtoofferagreaterchallengeandopportunitiesforstudentstomakeconnectionsbetweensolutionstrategies

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(NationalResearchCouncil,2001;Tomlinson,2005).Studentswereallowedtochoosehowtheyvisuallyrepresentedeachproblem(Tomlinson,2005).Studentslargelyworkedindependently,however,whennecessary,Iprovidedstudentswith‘hints’andguidanceinwhichIreferencednotesandanchorchartsandemphasizedkeywordstohelpstudentsunderstandthesituationdescribedinwordproblemsastheyworked.

PhaseTwo:Low-Floor-High-CeilingTasks

MysecondphaseofdatacollectionexploredtheuseoftwoLFHCtaskswithstudentsworkinginflexibleabilitylevelgroupsfortworoundsofdatacollection.AllgroupsworkedonthesameLFHCtask,however,aswasthecaseinPhaseOne,IfocusedmydatacollectionandanalysisonthesixfocalstudentsinInterventionGroups1and2.IdesignedtheLFHCtasksbasedonactivitiesandinformationpublishedbyCambridgeUniversity'sNRICH(2011)andStanfordUniversity’sYou-Cubed(2016).BothLFHCtasksweredesignedwiththeintentionofbeingaccessibletoallstudents,yetopen-endedenoughthatstudentscouldexplorethematmoreadvancedlevelsifappropriate(Cohen,1999).Studentswereencouragedtofocusontheexplorationaspectofthetasksandtoconsidermultiplesolutionsandapproachestothetasks.

MeasurementofConceptualUnderstanding.Studentscompletedathree-questionquizattheendofeachroundofdatainPhaseOneandPhaseTwo.Eachquizincludedthreetypesofquestions(asymbolicrepresentation,apictorialrepresentation,andawordproblem)becauseanabilitytoapplythesameconcepttodifferentrepresentationsisanindicatorofconceptualunderstanding(Panasuk,2010).IscoredeachquizusingamodifiedversionofthepubliclyavailablerubricforextendedmathematicalresponseitemscreatedbytheSmarterBalancedAssessmentConsortium(seeAppendixA).Ichosetousethisrubricbecauseitisusedtomeasurestudents’mathematicalunderstandingonthestandardizedassessmentsalignedwiththeCommonCoreStandards.

Iconductedstudentinterviewsforeachphaseoftheproject.Pre-Interviewsservedasbaselinedataforconceptualunderstanding.Interviewsweresemi-structuredandmeanttoassessstudents’conceptualunderstanding.PhaseOneinterviewsfocusedondivision,asdivisionwasarecurringconceptthatstudentsdealtwithovertheentirecourseofPhaseOne.Specifically,theinterviewquestionsweremeanttoassesswhetherornotstudentswereabletoidentifyaconnectionbetweentheconceptofdivisionandothermathematicalconceptsandskills.Forexample,onequestionasked,“Whenyouaredividing,whatskillsdoyouusetohelpyou?”Anotherquestionasked,“Whenyouthinkaboutdivision,whatotherconceptsinmathmightdivisionberelatedorconnectedto?”Iaudiotaped,transcribed,andthencodedtheinterviewsbasedonfourcodingcategoriescreatedthroughadeductiveprocess(seeAppendixB).

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DuringPhaseTwo,Icollectedpre-interviewdataonstudents’abilitytoapplyagivenmathematicalconcepttoreal-worldsituations,anotherindicatorofconceptualunderstanding(Kilpatricketal.,2001).Again,interviewquestionsemphasizedconceptualunderstanding,withquestionssuchas,“Inyourownwords,whatisarea?”and“Whenmightweuseareainoureverydaylives?”SimilartothePhaseOneinterviews,Iaudiotapedandtranscribedtheinterviews,andthensubsequentlycodedthembasedonthreecategories.

InPhaseTwo,Ialsoaskedstudentstocreateandsolvetheirown“real-worldwordproblem”viaawrittenresponsequestionattheendofeachround.IcodedstudentresponsesusingthesamecodingcategoriescreateddeductivelyforthePhaseTwoPre-Interviewdata(seeAppendixC).

MeasurementofBehavioralEngagement.Icapturedfeaturesofstudents’behavioralengagementbytallyingtheinstancesinwhichthesixfocalstudentsshowedoneofthefollowingindicatorsofbehavioralengagementduringwhole-classinstruction:answeringaquestion(voluntaryorinvoluntary),askingamathrelatedquestion,orsharingamathrelatedcomment/answer.Ithenanalyzedthevideorecordingsofsmall-groupinstructionforInterventionGroups1and2,countingthenumberoftimeseachofthesixfocalstudentsshowedoneoftheaforementionedindicatorsofbehavioralengagement.

Iusedthebehavioralengagementdatacollectedduringwhole-groupinstructionasbaselinedata.Iusedthisdatatothenprojectthenumberofbehavioralengagementindicatorsthatstudentswoulddemonstrateduringsmall-groupinstructionforeachround.Tocreatetheprojections,Ifirstdeterminedhowmanytimesmorestudentsthewhole-classinstructionhadascomparedtoeachsmallgroup.Ithenmultipliedthisnumberbythenumberofbehavioralindicatorsshownduringwhole-classinstructionforeachfocalstudenttocreateaprojection.Finally,IcomparedmyprojecteddataforeachfocalstudenttothedataIgatheredduringsmall-groupinstruction.

ResultsandDiscussion

DevelopingConnectionsBetweenConcepts.Theculminationofdatasuggestsstudents’conceptualunderstandingandbehavioralengagementincreasedoverthecourseofthestudy.DuringthePhaseOnepre-interviews,onlythreestudentsidentifiedaconnectionbetweendivisionandanothermathematicalconceptorskill(seeAppendixB).Thesedatasuggeststudentswerestilldevelopinganabilitytoidentifyhowdivisionisinterrelatedtoothermathconcepts(NationalResearchCouncil,2001).

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Post-interviewdatasuggestallsixfocalstudentsdeepenedtheirconceptualunderstandingofdivisionineachofthefourcodingcategories.Infact,allsixfocalstudentswereabletoidentifythatmultiplicationisanecessaryskillforperformingdivision.Forexample,Alexstated,“Todivideyouneedtopracticeyourmultiplication.”

Additionally,twostudentsrecognizedthatmultiplicationistheinverseoperationofdivision,whilefourstudentsarticulatedthatmultiplicationisrelatedtodivision.InSammy’swords,“Becauselikeit’slike[multiplicationanddivision]arebasicallytheoppositeofeachother.”Whilenostudentswereabletoidentifythatdivisioncaninvolveothermathematicaloperationsinthepre-interview,threestudentswereabletodosointhepost-interview.

Basedonthesedata,itappearsthatstudentsinbothInterventionGroups1and2deepenedtheirconceptualunderstandingofdivisioninPhaseOne.WhileitisdifficulttodrawcomparativeconclusionsaboutstudentsinGroups1and2duetothesmallsamplesize,thedatasuggeststudentsinInterventionGroup2hadasomewhatstrongerconceptualunderstandingofdivisionthanstudentsinInterventionGroup1.Evenso,post-interviewdataencouraginglysuggeststudentswereincreasinglyidentifyingconnectionsacrossmathematicalconceptsand,thus,hadexpandedtheirconceptualmathematicalunderstanding(NationalResearchCouncil,2001).

ConnectingConceptstoReal-worldApplications.InPhaseTwo,Iassessedconceptualunderstandingbasedonstudents’abilitytoapplymathematicalconceptstoreal-worldsituations(Kilpatricketal.,2001).Pre-interviewdatasuggestedstudentswerenotfullyabletoapplyagivenmathematicalconcept,inthiscasearea,toareal-worldsituation(seeAppendixC).Whenaskedtodescribeasituationinwhichareawouldbeusedduringpre-interviews,onlyonestudentwasabletodescribeaspecificsituation,andonlythreestudentswereabletoidentifyanddescribetherelevantmathematicaloperationneededtosolveanareaproblem.Forexample,Sammydescribed,“Likeisn’titbaseandheight?Andthey’rekindoflikeputtogetherandyouknow…youmultiplythemtobeabletofindthearea.”

AfterengaginginLFHCtasks,studentswereaskedtowriteandsolvetheirown“real-worldwordproblem”aboutareaorvolume.DataacrossthetworoundswithdifferentLFHCtasksrevealedthatmorestudentswereabletoapplymathematicalconceptswhencompletingthewrittenresponsequestionsthantheywereduringthepre-interview.Allsixstudentswereabletodescribeaspecificreal-worldexampleinvolvingthegivenmathematicalconcept(area)inRound5,andfourstudentswereabletodosoinRound6,whichfocusedontheconceptofvolume.Thetwostudentswhowerenotabletocreateareal-worldexampleofvolumeinRound6insteadwroteawordproblemaboutarea,suggestingthatthesestudentsneedmoreopportunitiestoexplorethedifferencesandconnectionsbetweenareaandvolume.

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Overall,datasuggeststudentsdiddevelopagreaterabilitytoapplymathematicalconceptstoreal-worldsituationsafterengaginginLFHCtasks.Forexample,onestudentstated:“IhaveatissueboxandIwanttoseehowmuchtissuesfitintoit.Whatisthevolumeiftheheightis8in,thewidthis6in,andthelengthis10in?”StudentsweremorefrequentlyabletodescribespecificsituationsinvolvingagivenmathconceptinRounds5and6thantheywereduringthepre-interview.ThisindicatestheLFHCmayhavehelpedstudentsdeepentheirconceptualunderstanding.

ApplyingConceptstoDifferentTypesofQuestions.Quizdatarevealedseveralinterestingtrends.Incomparingaveragescoresforeachtypeofquestionacrossquizzes,wefoundnomeaningfuldifferences;thetypeofquestionthatstudentsstruggledwithmostoneachofthesixquizzesvariedbyround.Whilestudentsdidnotdemonstratecompleteconceptualunderstandingoftheconceptsthatwereassessedbyeachquiz,thefactthattheywereabletoatleastpartiallyanswerthreedifferentrepresentationsofthesameconceptsuggestsstudentswere,indeed,beginningtodevelopconceptualunderstanding(Panasuk,2010).

Figure1.Graphofquizscoresforfocalstudents.Thisgraphillustratesthatstudentswereabletoapplymathematicalconceptstodifferenttypesofquestions.

Theoverallquizscoresforeachfocalstudent(seeFigure1)showvariedresults.TheresultsforSammy(InterventionGroup2)showsignsoftheceilingeffect;despitethefactthat

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instructionandquizzesbecomeprogressivelymorechallengingoverthecourseofthedatacollection,thequizzeswerelikelynotchallengingenoughforSammy.Evenso,thecombinationofSammy’squizscores,interviewdata,andwrittenresponsesrevealthatSammyhadlikelydevelopedconceptualmathematicalunderstanding.ThisappearstoalsobeatleastpartiallytruefortheothertwostudentsinInterventionGroup2,MeganandDavid;whileMeganandDavid’sscoresfluctuatedoverthecourseofthefourrounds,theyultimatelydemonstratedsignsofconceptualunderstandingbasedoninterview,written-response,andquizdata.

StudentsinIntervention1alsoshowedsignsofimprovementinregardstoconceptualunderstanding.AllthreestudentsinInterventionGroup1showedupward-trendingquizscoresoverthecourseofthesixrounds.ThiswasespeciallytrueforBrandonandDiana.Basedsolelyonquizdata,itappearsthatstudentshadnotyetfullydevelopedconceptualmathematicalunderstanding.Consideringinterviewandwritten-responsedatainconjunctionwithquizscores,however,suggeststhatstudentswerecertainlyintheprocessofdevelopingconceptualunderstanding.EachstudentinInterventionGroup1wasabletorecognizeatleastoneconnectionbetweendivisionandanothermathematicalconcept,andeachwasabletoidentifyatleastonespecificreal-worldexampleofaconcept.Inshort,myquiz,interview,andwritten-responsedatasuggestthatsmall-groupdifferentiatedinstructionandLFHCtasksdoappeartohelpstudentsdevelopconceptualunderstanding(Jansen,2012).

BehavioralEngagement.StudentsinbothInterventionGroups1and2clearlyshowedanincreaseinbehavioralengagementduringsmall-groupinstructionascomparedtowhole-groupinstruction(seeFigure2).ThiswastrueinbothPhaseOneandPhaseTwo.IfoundnomeaningfuldifferencesbetweenthebaselinedataforInterventionGroups1and2;thefocalstudentsinbothgroupsshowedanaverageof1.1behavioralengagementindicatorsduringwhole-classinstruction.

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Figure2.Graphofbehavioralengagementindicatorsforfocalstudents.Thisgraphillustratesthatstudentsinbothgroupsweremorebehaviorallyengagedduringsmall-groupinstructionthantheywereduringwhole-groupinstruction.

Allsixfocalstudentsshowedmoreindicatorsofbehavioralengagementduringsmall-groupinstructionascomparedtowhole-classinstruction,aswellasthesmall-groupprojecteddata.Iprojectedthatstudentswould,onaverage,show5.2indicatorsofbehavioralengagementduringsmall-groupinstruction.Inreality,studentsinInterventionGroup2exhibitedanaverageof41.7behavioralengagementindicatorsduringPhaseOne,and42indicatorsduringPhaseTwo.StudentsinInterventionGroup1demonstratedanaverageof15indicatorsofbehavioralengagementduringPhaseOne,and37indicatorsinPhaseTwo.

Thesedata,therefore,suggestthatstudentsinbothsmallgroupsweremorebehaviorallyengagedduringsmall-groupinstructionthantheywereduringwhole-classinstruction(Jansen,2012).Evenwhenthesizedifferentialbetweenthewhole-classandsmall-groupinstructionwascontrolledfor,studentsstillshowednotablymoresignsofbehavioralengagementduringsmall-groupinstruction.StudentsinInterventionGroup2exhibitedalmostanidenticalnumberofbehavioralengagementindicatorsduringPhaseOne,whenstudentsparticipatedinsmall-groupdifferentiatedinstruction,andduringPhaseTwo,whenstudentscompletedLFHCtasks.Thissuggeststhatbothinstructionalstrategieswereaneffectivewaytobehaviorallyengagehigherperformingstudents.

StudentsinInterventionGroup1,however,showedsignificantlymoreindicatorsofbehavioralengagementduringPhaseTwo(37indicators)ascomparedtoPhaseOne(15

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indicators).ThissuggeststhattheuseofLFHCtasksisawaytomoreequallyengagestudentsatbothendsoftheachievementspectrum(Cohenetal.,1999).

ImplicationsandConclusion

Theculminationofdatasuggeststhatflexiblesmall-groupinstructioncan,infact,beusedtoenhancebothconceptualunderstandingandbehavioralengagementforstudentsatoppositeendsoftheachievementspectrum.Futureresearchisneededtofurtherinvestigatetheuseofflexiblegroups,howeverthisresearchsuggeststhatdifferentiatedinstructionandlow-floor-high-ceilingtasksarebothpromisingtoolsfortargetingtheuniqueneedsofadvancedandinterventionsmallgroups.StudentsinGroups1and2allshowedsignsofgrowthinconceptualunderstandingandbehavioralengagement.

Webelievethattheseinstructionalpracticesandfindingsmaybegeneralizabletootherclassroomswithsimilarpopulationsofstudents.Manyteachersfacethechallengeofsimultaneouslycateringinstructiontostudentswithdifferentlearningstyles,needs,andstrengths.ThisresearchsuggeststhatflexibleabilitylevelgroupingandLFHCtaskshavethepotentialtodifferentiateinstructioneffectively.

MayaAngelouoncesaid,“….indiversitythereisbeautyandthereisstrength.”Whilemosteducatorswouldagreewiththewisdombehindthisquote,webelievemanywouldalsoaddthateffectivelymeetingstudents’diverselearningneedsisoneofthemostchallengingandworthyclassroomgoals.Thisinquiryprojectexploredstrategiesofafirst-yearteacherthatallowedhertomeetherdiversestudents’needsandsupportallstudentsinsucceedingintheclassroom,offeringaglimpseintothepotentialofsmall-groupinstructionthatencouragesstudentstobeinvolvedinthelearningprocess.

AbouttheAuthors

MelissaJ.Maininiisasixthgradeelementaryteacher.ThisresearchisanextensionofherMAEducationprogramattheUniversityofCaliforniaDavis.Email:[email protected]

LeslieC.Banesisaformerbilingualelementaryteacher.SheiscurrentlyaninstructorfortheMAEducationprogramattheUniversityofCaliforniaDavisandcollaborateswithteacherstoexploretherelationshipbetweenmathematicsandlanguagelearning.Email:[email protected]

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ReferencesBernander,S.,&Metke,J.(n.d.).LowFloorHighCeilingTasks.RetrievedNovember11,2016from http://www.wismath.org/Resources/Documents/Annual%20Conference/210JMetke- Low%20Floor%20High%20Ceiling%20PP.pdf.

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SmarterBalancedAssessmentConsortium.(n.d.).SmarterBalancedAssessments.RetrievedNovember11, 2015fromhttps://www.smarterbalanced.org/wp-content/uploads/2015/08/Smarter-Balanced- Mathematics-General-Rubrics.docx.StandardsforMathematicalPractice.(2016).CommonCoreStateStandardsInitiative.RetrievedNovember11, 2016fromhttp://www.corestandards.org/Math/Practice.Tomlinson,C.(2005).QualityCurriculumandInstructionforHighlyAbleStudents.TheoryIntoPractice,160- 166.Walqui,A.(2006).ScaffoldinginstructionforEnglishLanguageLearners:Aconceptualframework. InternationalJournalofBilingualEducationandBilingualism,9(2),159-180.Wang,Z.,Bergin,C.,&Bergin,D.A.(2014).Measuringengagementinfourthtotwelfthgradeclassrooms:The ClassroomEngagementInventory.SchoolPsychologyQuarterly,29(4),517-535.Webb,N.M.,Franke,M.L.,De,T.,Chan,A.G.,Freund,D.,Shein,P.,&Melkonian,D.K.(2009).‘Explaintoyour partner’:teachers'instructionalpracticesandstudents'dialogueinsmallgroups.CambridgeJournal ofEducation,39(1),49-70.YouCubed.(2016).LowFloorHighCeilingArchives-YoucubedatStanfordUniversity.RetrievedMay4,2016, fromhttps://www.youcubed.org/grade/low-floor-high-ceiling/.

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AppendixA:QuizRubric

AdaptedfromSmarterBalancedMathematicsGeneralRubrics

RetrievedNovember11,2015fromhttps://www.smarterbalanced.org/wp-content/uploads/2015/08/Smarter-Balanced-Mathematics-General-Rubrics.docx.

0 1:PartialUnderstanding

2:ReasonableUnderstanding

3:FullandCompleteUnderstanding

Studentdidnotattemptproblem/answerisnotinterpretable

Student’sresponsecontainssomeoftheattributesofanappropriateresponse.However,theresponseshowsevidenceofinsufficientmathematicalknowledge,errorsinfundamentalmathematicalprocedures,and/orotheromissionsorirregularities.

Studentaddressesmostofthetaskinamathematicallysoundmanner.Theresponsecontainssufficientevidenceofthestudent’scompetenceinproblemsolving,reasoning,and/ormodeling,butnotenoughevidencetodemonstrateafullunderstandingoftheprocessesheorsheappliestothespecifiedtask.

Studentaddressesthetaskinamathematicallysoundmanner.Theresponsecontainsevidenceofthestudent’scompetenceinproblemsolving,reasoning,and/ormodeling,andcontainsthecorrectfinalanswer.

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AppendixB:PhaseOne:PreandPostInterviewData

ConceptualUnderstandingofDivision

A=Advanced,I=Intervention

Code StudentExample(s) FocalStudentsRound1Pre-Interview

Round4Post-Interview

Studentrecognizesthatmultiplicationistheinverseoperationofdivision.

“Becausedividingisliketheoppositeofmultiplication…”

Sammy(A) Sammy(A),Megan(A)

Studentidentifiesthatmultiplicationisaconceptrelatedtodivision.

“It’srelatedtomultiplying.”

David(A) Sammy(A),David(A),Alex(I),Diana(I)

Studentidentifiesthatmultiplicationisanecessaryskillforperformingdivisionsuccessfully.

“Likeknowingyourtimestables.”“Todivideyouneedtopracticeyourmultiplication.”

Sammy(A),David(A),Brandon(I)

Sammy(A),Megan(A),David(A),Alex(I),Brandon(I),Diana(I)

Studentidentifiesthatdivisioncaninvolveothermathematicaloperations.

“Weusesubtractionindivision.Andwehavetousemultiplication.”

Megan(A),David(A),Alex(I)

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AppendixC:PhaseTwo:Interview&WrittenResponseData

ConceptualUnderstandingofAreaandVolume

A=Advanced,I=Intervention

Code StudentExample(s)

FocalStudentsRound5Pre-Interview

Round5WrittenResponse

Round6WrittenResponse

Studentgivesavagueexampleofascenarioinwhichareaorvolumeapplies.

“Sayyou’rebuildingahouse.Youneedtofindtheareatobeabletomakeitperfect.”

Alex(I),Brandon(I),Diana(I),Sammy(A),Melany(A)

David(A)

Studentgivesanexampleofaspecificscenarioinwhichareaorvolumeapplies.

“Whenyou’regoingtopaintaroom,youuseareatoknowhowmuchpainttouse.”

David(A) Sammy(A),Megan(A),David(A),Alex(I),Brandon(I),Diana(I)

Sammy(A),Megan(A),Alex(I),Brandon(I)

Studentcorrectlyidentifiesanddescribestheoperationusedtosolvethesituationdescribedinthescenario.ORStudentcorrectlysolvesthesituationdescribedinthescenario.

“Youwouldmultiplythebasetimestheheight.”“YouwouldusetheA=bhformula.”

Sammy(A),David(A),Brandon(I)

Sammy(A),Megan(A),David(A),Diana(I)

Sammy(A),Megan(A)