Multisensory Algebra through concrete to representational to abstract instruction for middle school students with learning difficulties PDF
Preview Multisensory Algebra through concrete to representational to abstract instruction for middle school students with learning difficulties
MULTISENSORYALGEBRATHROUGHCONCRETETOREPRESENTATIONAL TOABSTRACTINSTRUCTIONFORMIDDLESCHOOLSTUDENTSWITH LEARNINGDIFFICULTIES BRADLEYSTEVENWITZEL ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHE UNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHE REQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2001 ACKNOWLEDGMENTS ThesuccessoftheCRAprogramwouldnothavebeenfehbythestudentsor teachershadthecontributionsofsomanybeennolessthanextraordinary. Toeach personinvolvedinthemanyaspectsofthisproject,Iamverythankful. Additionally,1 alsomustthankeachpersonwhocontributedtomymakingittothedissertationstage~a processIoftenwonderedifIwouldeverreachorwanttoreach. AsIhavethankedso manypeoplealongthisjourney,Ireservethissectionforspecialacknowledgmentsto thosewhocontributedspecificallytotheculminationofmydoctoraldegree. IwishtothankPinellasCountyadministratorsandteachersforpushingthis projecttocompletion. DoctoralcandidateSuziDavis,wholeadstheSecondaryMath ProgramsinPinellasCounty,wasverysupportiveinplacingthisprograminasmany middleandhighschoolsaspossible. Dr.Stevenlachinimayhavepresentedredtapeto me,buthealsoshowedmehowtountiemyhandsfromit. Activeadministratorssuchas Mrs.WhiteandMrs.Bucklespressedtheirteacherstoparticipate,whileMs.Crosby helpedadministermessagesandmaterialstothoseinvolved. Tothestudentsandparentswhoagreedtoparticipateinthisproject,Igivespecial thanks. Theiropen-mindednesstoinnovativeresearchenablesnewadvancesinthemany fieldsofeducationthathelptopusheducation'sstateoftheart. Withouttheir willingnesstolearnmoreandwanttheirchildrentosucceed,noneofthisworkcould succeed. ii TothededicatedteachersIowemuchmorethancertificatesandpositive recommendations. Thetimeinvolvedinthedevelopmentofeachstepoftheprogram andthepreparationforeachday'slessonexceededallexpectations. IwanttothankMs. JaneAlsten,Ms.KimAndrusko,Mrs.PauletteBaretto,Mrs.BillieBomstein,Mr.Brad Daniel,Mrs.JackieJurek,Mrs.BarbaraKries,andMrs.MarshaPagnottafortheir efforts. Theirsuggestionswerepositiveandthecommentsinspiring. Ionlyhopeall teachersremainasdedicatedastheywere. Iwanttothankmycommittee--Dr.CecilMercer,Dr.MaureenConroy,Dr. NancyCorbett,andDr.DavidMiller-fortheirsupportthroughoutmydoctoralprogram. FromquestionstocommentsmycommitteeassistedmeinwhatevercapacityIneeded withouthesitation.Althoughdemanding,theirdirectionswerehelpful,theirrequests werereasonable,andtheirconfidenceinmewasnoticeable.Fromthem1learnedabout pilotprograms,programdevelopment,teachersustainability,statisticalreasoning,andthe concernsinmathematicseducation,especiallywithstudentswithdisabilities. Imustthankseparatelyahandfulofpeoplewhopushedmetosucceedthroughout mygraduateworkbeyondthatwhichIknewIcould. Dr.CecilMercerpulledmeaside forextrawork(thanksalot)threeyearsagoinmyfirstsemesterasamaster'sstudent. Eversincemyfirstclasswithhim,heledmetostartmyownresearchinareasthat intriguedme. Insteadofaskingmetocompletehisprojects,hetooktimeoutofhis scheduletohelpmeworkonmine. Hesetupsituationsformetoexploremycapabilities withinthefieldofspecialeducation. Idonotknowhowtorepayhimforhisassistance attheconceptualandimplementationstagesofeachofmyadventures. WhileCecil MerceristhereasonIenrolledattheUniversityofFlorida,myfamilypushedmeto iii completewhatIstarted. Ithankmygrandmother,Mrs.GerdaKohnke,whoseinsightand supportthroughoutmygraduateworkallowedmetoworkwithfewerdistractions. I thankmybrotherMarkforhishumorandmoralsupportwhileIbalancedfulltime employmentwhilecompletingthisdissertation. Tomyparents.Dr.RonaldWitzeland ElisabethWitzel,Icouldneversayenough. Theirhighexpectationscontinuedtopush metosucceedafterIhadclaimedfailure. Theirknowledgeofmyroleasadoctoral studenthelpedmedealwiththemanypressuresandrealizethereisamuchbiggerlightat theendofthetunnelthancanbeseenfromwithinthewallsofaninstitution'swalls. Anaudiblesighofreliefgoestomywife,Isabelle. Ithankherforputtingupwith methroughoutthislengthyprocess. Goingfromateacherwithcontinualpraiseanda lowbutsteadysalarytothefinancialuncertaintyofafulltimestudentbeingtoldhestill hasmuchtolearnisachallengingandhumblingchange. Toendurethistransition requiredmetopartakeinintrospection. Thismoratoriumcamewithapricetothose aroundme. Isabelleenduredmyalteringmoodstheseyearsandurgedmenottochange. WithouthersanityImayhaveconfusedmyinitialpurposeforachievingmyterminal degree--tohelpasmanystudentswithdisabilitiessucceedinacademicswhomotherwise mayhavestruggled. iv 1 TABLEOFCONTENTS Page ACKNOWLEDGMENTS ii ABSTRACT viii CHAPTERS 1 INTRODUCTION 1 Rationale 6 DefinitionofTerms 7 2 REVIEWOFRELATEDLITERATURE 10 ArithmetictoAlgebraGap 1 AlgebraasaContinuationofArithmetic 13 BridgingtheGap 18 PreparingPriorKnowledge 19 PotentialforArithmeticDifficultiesforStudentswithLearning Disabilities 21 EffectiveMathInstructionalStrategiesforStudentswithLearning Disabilities 23 ExplicitInstruction 23 CueingPriorKnowledgethroughScaffolding 25 DevelopingMotivationthroughRelevancy 26 Constructivism 27 PracticalConstructivism 28 MisconceptionsaboutConstructivism 32 ImplicitInstruction 33 ConstructivistMath 34 ExplicitInstructionandCueingPriorKnowledge 34 RelevancyandMotivation 35 UseofCRAwithBasicMathFacts 36 CRAPrinciplesUsedwithAlgebra 38 AlgebrathroughConcreteManipulatives 38 AlgebraUsingRepresentations 40 ProblemswithCRAUse 41 V AnEmpirically-basedAlgebraCRAModel 44 ElementaryversusSecondarySchool 45 Summary 46 3 METHODSANDPROCEDURES 48 Introduction 48 Hypotheses 48 Methods 49 SettingsandParticipants 49 PretestMeasures 57 PosttestMeasures 59 Follow-upMeasures 60 Fidelity 60 ResearchDesign 61 Procedures 61 LargeNDesignstudy 62 TeacherTraining 63 PretestConditions 63 TreatmentGroup 64 ComparisonGroup 66 Materials 67 ExperimentalDesignandAnalysis 69 4 RESULTS 71 Introduction 71 OverallFindings 72 FindingsbyRepeatedMeasures 72 InstructionalDifferences 75 TestDifferenceperGroup 76 ErrorPatternAnalysisforEachGroup 77 TreatmentIntegrity 79 Summary 80 5 DISCUSSION 81 Introduction 81 SummaryofHypothesesandResults 81 TheoreticalImplicationsoftheResearchFindings 82 PracticalImplicationsoftheResearchFindings 83 LimitationstothePresentStudy 85 ImplicationsforFutureResearch 88 Summary 9q vi APPENDICES A PRETESTANDPOSTTESTASSESSMENT 91 B FOLLOW-UPASSESSMENT 93 C TREATMENTFIDELITYCHECKLIST 95 D CASESTUDYPILOTFORLARGENSTUDY 96 E SCRIPTEDLESSONEXAMPLES 99 F LEARNINGWORKSHEETS 106 REFERENCES 125 BIOGRAPHICALSKETCH 133 vii AbstractofDissertationPresentedtotheGraduateSchool OftheUniversityofFloridainPartialFulfillmentofthe RequirementsfortheDegreeof DoctorofPhilosophy MULTISENSORYALGEBRATHROUGHCONCRETETOREPRESENTATIONAL TOABSTRACT INSTRUCTIONFORMIDDLESCHOOLSTUDENTSWITH LEARNINGDIFFICULTIES By BradleyStevenWitzel December2001 Chair: CecilD.Mercer Department: SpecialEducation Learningalgebraisacomplextask,especiallyforstudentswithmathlearning disabilitiesanddifficulties. Afewpublishersandresearchershaveproducedcurriculum programstohelplow-performingstudentswithinitialalgebrainstruction,butnoneof theseprogramshavebeenshowntohelpstudentspastsolvinginverseoperations. This researchexaminedanewlydevelopedconcrete-to-representational-to-abstract(CRA) algebraprogramthatdoesgeneralizetomorecomplexequations. Thirty-fourmatchesof studentswithdisabilitiesandstudentsatriskfordisabilitieswereobtainedfromthe358 studentswhoparticipatedinthisprogramacrossfourseparatemiddleschoolsinanurban countyinFlorida. Studentswerematchedaccordingtopretestscore,standardizedmath testscores,thesamecurrentmathteacher,similarageandsamegradelevel,andsemester averages. All358studentsparticipatedinthe19-lessoncurriculumthattookstudents fromreducingalgebraicexpressionstosolvingmultiplestepsinglevariable viii transformationalalgebraequations. Teacherstaughtone-halfofthematchesusing traditionalinstructionwhiletheotherhalfweretaughtusingtheCRAprogram.Students' scoreswerecomparedacrosspretests,posttests,andfollow-uptests. Studentswho receivedCRAinstructionshowedstatisticallysignificanthighergrowthin transformationalequationsthantraditionallytaughtpeersfrompretesttoposttestand follow-uptestscores,F(2,66)-13.888,p=0.000). PosthocWestsrevealedthatthe studentswholearnedthroughthisnewlydevelopedCRAalgebraprogramoutperformed theirtraditionallytaughtpeersonboththeposttestandthe3-weekfollow-uptest. The positiveresultsfromthisresearchraisethepossibilityofteachingalgebratostudents withlearningproblemsthroughwell-constructed,hands-on,andpictorialinstruction first. ix 5 CHAPTER 1 INTRODUCTION Forthepasttwodecades,America'spublicschoolshavecomeunderattackfor students'poorperformanceinmathematics. StartingwithANationatRisk(National CommissiononExcellenceinEducation[NCEE],1983),theU.S.wasdepictedasa decliningnationformathematicseducation. FindingsfromtheThirdInternational MathematicsandScienceStudy(TIMSS)(NationalCenterforEducationalStatistics [NCES],1999)indicatedthatU.S.studentscomparedpoorlyonmeasuresofmath achievementtostudentsfrom40othercountries,andthisperformancebecamemore evidentasstudentsenteredsecondaryschool. WhileU.S.fourth-gradestudentswere abovetheinternationalaverageinmathandscienceachievement,eighth-gradestudents werenot. Infact,bythetimeU.S.studentsreachedeighthgrade,theywerebelow average. Bythefinalyearofsecondaryschool,U.S.studentswereamongthelowestin theworldinbothmathandscienceachievement.Interestingly,resultsfromTIMSS (NCES,1999)indicatedthattheseventh-grademathcurriculumformostnationswas equivalenttotheU.S.ninth-gradecurriculum. PriortoTIMSS,U.S.educatorswereconcernedwithpoormathematics achievementnationally,specificallywiththedropinmathematicsachievementfrom fourthtotwelfthgrades(NCES,1999). Infact,heightenedconcernoverthepasttwo decadesresultedinavarietyofreformmovementsinmathematicseducation. These reformmovementswerespurredbyANationatRisk(NCEE,1983)andledtoan 1
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