Multisensory Algebra through concrete to representational to abstract instruction for middle school students with learning difficulties

145 Pages·2001·5.3 MB·English

by  Witzel, Bradley Steven

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.
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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
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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.
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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.
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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