Copyright by HirokoKawaguchiWarshauer 2011
TheDissertationCommitteeforHirokoKawaguchiWarshauerCertifiesthat thisistheapprovedversionofthefollowingdissertation: TheRoleofProductiveStruggleinTeachingandLearning MiddleSchoolMathematics Committee: SusanEmpson,Supervisor JamesBarufaldi EdmundT.Emmer AnthonyPetrosino PhilipUriTreisman
TheRoleofProductiveStruggleinTeachingandLearning MiddleSchoolMathematics by HirokoKawaguchiWarshauer,B.A.;M.S. Dissertation PresentedtotheFacultyoftheGraduateSchoolof TheUniversityofTexasatAustin inpartialfulfillment oftherequirements forthedegreeof DoctorofPhilosophy TheUniversityofTexasatAustin December2011
Dedication TomyhusbandMaxandchildrenAmy,Nathan,Lisa,andJeremy
Acknowledgements Iamdeeplygratefultomydissertationadvisor,SusanEmpson,who suggestedthetopicofmydissertationandwhoprovidedgentleguidance,keen insights,andinfinitepatienceoverthecourseofmyresearchandwriting.ithank herforsupportingmylearningasiexperiencedtheverytopicichosetostudy: productivestruggle. Iamalsogratefultomycommitteemembers,UriTreisman,Anthony Petrosino,EdmundEmmer,andJamesBarufaldiforallIlearnedintheirclassesat theuniversityoftexas.throughtheirmasterfulteaching,igainedmeaningful insightsintolearning,teaching,professionaldevelopment,researchdesigns,andthe challengesinherentineducation.thefeedbackireceivedfromthemwas invaluable. Iwanttoexpressmyappreciationtothesixteacherswhoallowedmeto observetheirclassesandwhosharedwithmetheirreflectionsofteaching.ihave comeawaywithadmirationandrespectfortheirdepthofknowledge,creativityin teaching,andthekindnessandrespecttheydemonstratetotheirstudents. MythankstoMichaelKellermanwhoreadandcopyeditedthefinaldraftand enhanceditsreadabilityandtonamakshinamakshiforherhelpincreating graphicsandreadingdrafts. v
MychairandcolleaguesatTexasStateUniversity SanMarcoswerea constantsourceofsupportandencouragement.tocolleaguesterrymccabefor sharinghisloveofteachingandracquetball,toalejandrasortofortranslatingforms intospanishonamoment snotice,toalexwhite,samuelobara,stanwayment, BryanNankervis,andallthoseIranintointhehalls,myappreciationforaskingand gentlyremindingmetokeepfocusedonmakingprogress. EachsummerIfoundrenewalontheNorthCarolinabeachsurroundedby Elaine,mymother in law,andmyhusband ssiblings,david,tom,leo,susanand theirfamilies.theirlove,encouragement,andbrightoptimisminthepowerthat individualscanaccomplishgreatthingshavealwaysbeenasourceofinspiration. TocousinSarahWarshauerFreedman,thankyouforthewalkonthe CarolinabeachjustasIwasponderingaboutadissertationtopic.Ourconversation thenandyoursuggestionssinceaddedtowhathascometobe.tomybrothers, YoshihiroandJiroKawaguchi,thankyouforyourunconditionallove.Mydear friends,mimirosenbush,lisalefkowitz,lilliandegand,diannmccabe,deanna Badgett,RobertGonzalez,andStephenRedfield,thankyouforyourfriendship whichhasbeenconstantandenduring. Tomychildren,Amy,Nathan,Lisa,andJeremy,thankyouforbringingsuch joy,laughter,andrichnessintomylife.iamgratifiedasiseeyoupursuingyour dreamswiththesamedetermination,enthusiasm,caring,andsenseofhumorthat youhavepossessedsinceyouwereveryyoung. vi
Myhusband,Max,hasbeenmybiggestsupporter;providingchallenges, inspiration,andcomfort.thequestionsheasked,theeditshemade,the encouragementhegaveallkeptmethinkinganewandmoredeeply.itisthanksto workingwithteachersandstudentsthroughmathworksthatihadanidealsetting toconductmyresearch. Finally,tomyparents,MotohiroandSuzukoKawaguchi,whonamedme, scholarlychild, Ithankthemforalwaysencouragingmetodomyverybest, whetherinmathematics,music,orsportsandforinstillinginmeadeeploveof learning. vii
TheRoleofProductiveStruggleinTeachingandLearning MiddleSchoolMathematics HirokoKawaguchiWarshauer,Ph.D. TheUniversityofTexasatAustin,2011 Supervisor:SusanEmpson Students strugglewithlearningmathematicsisoftencastinanegativelight. Mathematicseducatorsandresearchers,however,suggestthatstrugglingtomake senseofmathematicsisanecessarycomponentoflearningmathematicswith understanding.inordertoinvestigatethepossibleconnectionbetweenstruggle andlearning,thisstudyexaminedstudents productivestruggleasstudentsworked ontasksofhighercognitivedemandinmiddleschoolmathematicsclassrooms. Students productivestrugglereferstostudents efforttomakesenseof mathematics,tofiguresomethingoutthatisnotimmediatelyapparent (Hiebert& Grouws,2007,p.287)asopposedtostudents effortmadeindespairorfrustration. Asanexploratorycasestudyusingembeddedmultiplecases,thestudy examined186episodesofstudent teacherinteractionsinordertoidentifythekinds andnatureofstudentstrugglesthatoccurredinanaturalisticclassroomsettingas studentsengagedinmathematicaltasksfocusedonproportionalreasoning.the viii
studyidentifiedthekindsofteacherresponsesusedintheinteractionwiththe studentsandthetypesofresolutionsthatoccurred. Theparticipantswere3276 th and7 th gradestudentsandtheirsix mathematicsteachersfromthreemiddleschoolslocatedinmid sizetexascities. Findingsfromthestudyidentifiedfourbasictypesofstudentstruggles:getstarted, carryoutaprocess,giveamathematicalexplanation,andexpressmisconception anderrors.fourkindsofteacherresponsestothesestruggleswereidentifiedas situatedalongacontinuum:telling,directedguidance,probingguidance,and affordance.theoutcomesofthestudent teacherinteractionsthatresolvedthe students struggleswerecategorizedas:productive,productiveatalowerlevel,or unproductive.thesecategorieswerebasedonhowtheinteractionsmaintainedthe cognitiveleveloftheimplementedtask,addressedtheexternalizedstudent struggle,andbuiltonstudentthinking. Findingsprovideevidencethatthereareaspectsofstudent teacher interactionsthatappeartobeproductiveforstudentlearningofmathematics.the struggle responseframeworkdevelopedinthestudycanbeusedtofurther examinethephenomenonofstudentstrugglefrominitiation,interaction,toits resolution,andmeasurelearningoutcomesofstudentswhoexperiencestruggleto makesenseofmathematics. ix
TableofContents ListofTables... xiii ListofFigures...xiv Chapter1:Rationale...1 Introduction...1 StruggleandLearning...2 StruggleandTask...3 StruggleandTeaching...4 ResearchQuestions...5 StudyDesign...6 Chapter2:ConceptualFramework...8 Introduction...8 OverviewofConceptualFramework...9 NatureofMathematics...12 RoleofStruggleinLearningMathematics...14 LearningMathematicsByDoing...14 ModelofStruggle...19 ProductiveStruggleinLearning...20 ResearchConnectsStruggleandConceptualLearning...21 NatureandTypesofTasksthatSupportProductiveStruggle...25 ImportanceofMathematicalTasks...25 TaskFramework...27 LevelsofCognitiveDemand...28 ModelingStruggleandTasks...30 KindsofTasksthatSupportProductiveStruggle...32 Teacher sresponsetostruggle...35 ResponsesthatSupplyInformationtoStudents...39 ResponsesthatConnecttoStudents PriorKnowledge...40 x
ResponsesthatClarifytheStudentStruggle...42 ResponsesthatQuestionStudents Thinking...43 ResponsesthatBuildStudentAgency...46 Summary...50 Chapter3:Methodology...53 Participants...54 Procedure...56 DataCollection...56 DataAnalysis...60 CodingStruggle...62 CodingTasks:TaskDescriptions...63 CodingTasks:ByLevelsofCognitiveDemand...66 CodingTeacherResponse...70 CodingResolutionoftheStudents Struggle...72 Trustworthiness...73 Chapter4:Results...77 Overview...77 Tasksimplementedintheclassrooms...79 Students Struggle...81 Descriptionandexamples...81 DiscussionofStudentStruggle...89 TeacherResponse...94 Overviewofteacherresponsecategories...94 DefiningTeacherResponseTypes...95 DescriptionsandImpactonThreeDimensions...100 1.Telling...100 2.DirectedGuidance...105 3.ProbingGuidance...115 4.Affordance...123 xi
DiscussionofTeacherResponses...128 InteractionResolutions...133 TypesofInteractionResolutions...133 InteractionFrameworkandPatterns...135 ExampleTaskWithDifferingResolutions...137 Example4.1:ProductiveStruggle Lowerlevel...137 Example4.2:ProductiveStruggle...144 Example4.3:UnproductiveStruggle...148 DiscussionofInteractionResolutions...151 Chapter5:Conclusion...155 ResearchQuestionsandConclusions...155 Limitation...160 Implication...163 AppendixA:Pre ObservationTeacherInterview(PRTI)...166 AppendixB:Post ObservationTeacherInterview(PSTI)...167 AppendixC:TaskDebrief(TDB)...168 AppendixD:StudentInterview(SI)...169 AppendixE:TaskDifficultySurvey...170 AppendixF:ActivityBooklet...171 AppendixG:Ms.Torres Lessons...189 AppendixH:Samplewarm upproblems...191 References...194 Vita...211 xii
ListofTables Table2.1:Struggleanditsmanifestations...19 Table2.2:ProductiveStruggleintheClassroomInteractionsofTeachingand LearningintheContextofMathematicalActivitiesandTasks...31 Table3.1: CharacteristicsofTeacherParticipants...55 Table3.2: Table3.4: Observedclassfrequencyandhours...56 Activity1:BarrelofFun...67 Table3.5: Activity2:BagsofMarbles...67 Table3.6: Table3.7: Activity3:TipsandSales*...68 Activity4:DetectingChange...69 Table4.1: KindsofStudentStrugglesandtheirPercentFrequencies...82 Table4.2:TeacherResponseSummary...99 xiii
ListofFigures Figure2.1:PreliminaryStruggleandResponseFrameworkinTaskContext49 Figure4.1:Findtheprobabilityoflandingintheunshadedregion....88 Figure4.2: TeacherResponseRange...96 Figure4.3: ProductiveStruggleFrameworkinaninstructionalepisode...135 xiv
Chapter1:Rationale INTRODUCTION Students strugglewithlearningmathematicsisoftencastinanegativelight andviewedasaprobleminmathematicsclassrooms(hiebert&wearne,2003; Borasi,1996;Sherman,Richardson,&Yard,2009).Teachers,parents,educators andpolicymakersroutinelylookforwaystoovercomethe problem,seenasaform oflearningdifficulty,andattempttoremovethecauseofthestrugglethrough diagnosisandremediation(adams&hamm,2008;borasi,1996).fromthisone wouldhardlyexpectthatfocusingonstudents struggleinmathematicscouldbe viewedinapositivelightandasalearningopportunity. MathematicseducatorsandresearchersJamesHiebertandDouglasGrouws suggest,however,thatstrugglingtomakesenseofmathematicsisanecessary componentoflearningmathematicswithunderstanding(hiebert&grouws,2007). Theideathatstruggleisessentialtointellectualgrowthhasalonghistory.Dewey referredtotheprocessofengagingstudentsin someperplexity,confusion,or doubt (1933,p.12)asessentialforbuildingdeepunderstandingwhilePiaget (1960)wroteoflearners struggleasaprocessofrestructuringtheirdisequilibrium towardsnewunderstanding.cognitivetheoristshavereferredtocognitive dissonanceasanimpetusforcognitivegrowth(e.g.festinger,1957)whileothers haveidentifiedexperimentation(polya,1957)andsense making(handa,2003)as importantingredientsforunderstanding.hatano(1988)relatedcognitive incongruitywiththedevelopmentofreasoningskillsthatdisplayconceptual understanding.brownwellandsims(1946)argued,likedewey,thatstudentsmust haveopportunitiesto muddlethrough (p.40)intheprocessofresolving 1
problematicsituationsratherthanconditioningstudentsthroughrepetition.more recently,hiebert&wearne(2003)stated, allstudentsneedtostrugglewith challengingproblemsiftheyaretolearnmathematicsdeeply (p.6). Whilethephenomenonwecallstrugglemaybeinternal,itisalsoobservable inmostclassrooms.inthecontextofclassroominteractions,studentsmayvoice confusionoverdirections,thewordingofaproblem,thequestionbeingaskedor howtodeviseastrategy(polya,1957;lave&wenger,1991).studentsmayvoicea commentsuchas, Idon tgetit.ateachermaydetectstudents misconceptions thatyieldcompetingclaims,uncertainty,andcognitiveconflictinthestudents thinking(zaslavsky,2005).anerrorwhilesolvingaproblemmayleadtoan unreasonableanswerthatpuzzlesastudent(borasi,1996;inagaki,hatano,& Morita,1998).Astudentmaybeveryengagedinworkingonamathematics problembutthenreachanimpasseandget stuck (Burton,1984,p.46).What opportunitiesdotheseinstancesprovideforteaching? STRUGGLEANDLEARNING Struggleanditsconnectiontolearningarecentraltotheissueofhowto strengthenandimprovestudentlearningandunderstandingofmathematics (Hiebert&Grouws,2007).Twokeyfeaturesofclassroommathematicsteaching emergefromresearchthatlinksteachingwithstudents conceptualunderstanding: teachersandstudentsattendexplicitlytoconcepts;and studentsstrugglewithimportantmathematicalideas. (Hiebert&Grouws,2007) Byconceptualunderstanding,HiebertandGrouwsmean themental connectionsamongmathematicalfacts,procedures,andideas (2007,p.380).This 2
isincontrasttoproceduralunderstanding,whichreferstothe accurate,smooth, andrapidexecutionofmathematicalprocedures and intentionallydoesnot includeflexibleuseofskillsortheiradaptationtofitnewsituations (2007,p.380). Teachershaveanopportunitytofacilitatethedirectionthatstudents strugglescouldtake,eitherproductiveorunproductive.bystudents productive struggle,imeanstudents efforttomakesenseofmathematics,tofiguresomething outthatisnotimmediatelyapparent (Hiebert&Grouws,2007,p.287)asopposed tostudents effortmadewithoutdirectionorpurpose. STRUGGLEANDTASK Anexampleofstudents strugglethatcanbeproductiveinlearning mathematicsisgrapplingwithchallengingproblems(hiebert&wearne,2003). Mathematicaltasks,inparticularthosethatplacehighlevelcognitivedemandson studentsincludingmakingsenseoftheproblem,focusingonconceptsandconnections amongconceptsandsharing,explaining,andjustifyingone ssolution(boston& Smith,2009;Hiebert,Carpenter,Fennema,etal,1996;Ball,1993,Doyle,1988), provideaclassroomcontextforstudentstoengageininteractingwithproblems, classmates,andteacherstodeveloptheirconceptualknowledgeandunderstanding (Hatano,1988,Hiebert,1986;Zaslavsky,2005;Goldman,2009;Fawcett&Gourton, 2005).Tasksthatinvolveproblemsolvingcalluponstudents conceptualand proceduralknowledgetoconsideralternativestrategieswhenanapproachdoesnot work,examineone sresourcesandknowledgeuponwhichtobuild,reflectonone s thinking,andexplainandjustifyone ssolutions(nctm,2000;franke,kazemi,& Battey,2007;Kulm,1999;Kulm,Capraro,&Capraro,2007).Engagingstudentsin challengingtasksgivesstudentsopportunitiesto:strugglewithproblems;connect 3
facts,procedures,andideas;anddevelopadeeperconceptualunderstandingof mathematics(hiebert&grouws,2007;hiebert&wearne,2003;kahan&wyberg, 2003;Kahan&Schoen,2009). STRUGGLEANDTEACHING Moststudies,however,suggestthatU.S.mathematicsteachingrarelyengages studentsinproductivestrugglewithkeymathematicalideas(e.g.hiebert&wearne, 2003;Rowan,Correnti,&Miller,2002;Stigler,Gonzales,Kawanaka,Knoll,& Serrano,1999).Schoolinstructionisoftenplaguedbyarushforquickanswers (Hiebert,Carpenter,Fennema,etal,1996;Dewey,1933)andfailstogivestudents sufficienttimetoengageinthinkingdeeplyaboutproblems(holt,1982).teachers maydesigntasksthatareintendedtoplacehighlevelsofcognitivedemandon students,butthenallowtaskstodeclineintheirdemandwhenstudentsencounter frustrationordiscomfort(henningsen&stein,1997;romagnano,1994;stigler& Hiebert,2004;Santagata,2005).Forexample,teachersstepinquicklywhenthey observestudentsstrugglingandexplainhowtodotheproblem,leavinglittleofthe challengingmathematicsforthestudentstodo(smith,2000).classroom interactionswhereateachermayresponddismissivelytoastudent squestion, produceananswertoaproblemwithlittlestudentparticipation,orbeunawareofa student sconfusioncanresultinstudentstrugglethatisunproductive.inan analysisofmathematicsclassroominstruction(weiss&pasley,2004;weiss,pasley, Smith,Banilower,&Heck,2003),only15%ofthelessonsobservedwereclassified asprovidingstudentsopportunitiesforthinking,reasoning,andsense making. Empiricalresearchintheareaofstudents struggleandhowitisaddressed productivelyintheclassroomislimited.researchinvolvingthequasarproject 4
(Silver&Stein,1996;Stein&Lane,1996)foundevidenceofincreasesinstudents conceptualunderstandingwhenstudent teacherinteractionsfocusedonfacilitating productivestrugglethroughmathematicaltasksofhighercognitivedemand(stein, Grover,&Henningsen,1996;Stein&Lane,1996).Hiebert&Wearne(1993) demonstratedthatthroughclassroomdiscourseandteacherguidance,students exhibitedstrugglesinmakingsenseofthemathematicsandexpressedtheir emergingunderstandings.researchconductedbyinagaki,hatano,&morita(1998) showedhowstudentsengagedinstrugglingwithconflictingorincorrect mathematicalideasduringclassroominteractionwereabletomakesenseofthe mathematicsandimprovetheirunderstandinginafollow upassessment. Examplessuchastheprecedingstudiessupporttheclaimthatthereisalink betweenteachingthatfacilitatestudents opportunitytoengageinproductive struggleinclassroomcontextsandincreasesinstudents conceptualunderstanding. Inmystudy,Iproposetoexaminethephenomenonofstudents struggletomake senseofmathematicsinthenaturalcourseofmiddleschoolclassroominstruction usinganinquiry basedcurriculum.iwillfocus,inparticular,onstudents struggle withmathematicalconceptsthatismadevisibleinsomewayintheclassroom environment,suchasthroughmistakes,misconceptions,orconfusion;andstruggle thatappearstobeproductiveornon productivetostudentlearning. RESEARCHQUESTIONS Thekindofguidanceandstructureteachersprovidemayeitherfacilitateor underminetheproductiveeffortsofstudents struggle(tarretal,2008;stein, Smith,Henningsen,&Silver,2000;Doyle,1988).Acloseexaminationof interactionsintheclassroombothbetweenteacherandstudentsandamong 5
studentshelpedtorevealthenatureofthestrugglesstudentswerehavingin makingsenseofmathematics.ialsoobservedandanalyzedthefeaturesofteaching andthechoicesteachersmadetoguidethestudentsinwaysthatwereeither productiveornotproductiveindevelopingstudents understandingoftheir problemandthestrategiesandreasoningneededtosolveit. Mystudyfocusedonthefollowingresearchquestions: 1. Whatarethekindsandpatternsofstudents strugglethatoccurwhile studentsareengagedinmathematicalactivitiesthatarevisibletotheteacher and/orapparenttothestudentinmiddle schoolmathematicsclassrooms? 2. Howdoteachersrespondtostudents strugglewhilestudentsareengagedin mathematicalactivitiesintheclassroom?whatkindsofresponsesappearto beproductiveinstudents understandingandengagement? Thepurposeofthisexploratorystudywastoprovidefurtherinsightinto whatstudents productivestrugglelookslikeandhowteachingthatengagesand supportsstudents productivestruggleinmiddleschoolmathematicsclassrooms givesstudentsopportunitiestobuildanddeepen(ortoinhibit)theirconceptual understandingofmathematics. STUDYDESIGN Iobservedtheclassroomsofsixmiddle schoolteacherslocatedinthree differentmid sizedtexascities.theteachersusedthesamemathematicstextbook thatwaswrittentoencourageteacherstoengagestudentsinmathematical exploration,aswellassense makingofmathematicalideasamongstudents (McCabe,Warshauer,&Warshauer,2009). 6
Mystudyidentifiedalltheepisodesduringinstructionwherestudentsmade mistakes,expressedmisconceptions,orclaimedtobelostorconfused,andtowhich teachersresponded.interactionsbetweenstudentsandteachersgenerally advancedtowardsomeresolutionofthestudents difficultiesandattemptsatsensemaking.usinganembeddedcasestudymethodology(yin,2009)withinstructional episodesastheunitofanalysiswithinthelargerunitoftheteachers,iidentifiedand describedthenatureofthestudents struggle.additionally,irecognizedthe instructionalpracticesofteachersthateithersupportedandguidedordidnot supportorguidethestudents sense andmeaning makingofthemathematicsin thelessonepisodes.iusedmyobservationnotes,interviewsofteachersandtarget students,andvideoand/oraudiotapesofclassroomlessonstodescribeandanalyze theinteractivetechniquesandpracticesteachersusedthatfocusedonstudents productivestruggle. 7
Chapter2:ConceptualFramework INTRODUCTION Teachingthatprovidesstudentsopportunitiestostrugglewithimportant mathematicalideashasbeenidentifiedinmathematicseducationresearchasoneof thekeyfeaturesofteachingthatsupportsthedevelopmentofstudents conceptual understandingofmathematics(hiebert&grouws,2007;hiebert&wearne,1993; Stein,Grover,&Henningsen,1996;Borasi,1996).Students learningof mathematicswithunderstandingisviewedascriticalinmeetingthedemandsofthe 21 st century,particularlyinasocietyexperiencingrapidchange,wherepossessing proceduralunderstandingwithoutconceptualunderstandinglimitsflexibilityand creativityinsolvingproblems(nationalmathematicsadvisorypanel,2008;pink, 2006;NCTM,2000;Bransford,Brown,&Cocking,1999;NationalResearchCouncil, 1989).Aportrayalofwhataproductivestudents strugglelookslikesetinthe naturalisticsettingofclassroominstructioncanrevealandprovideinsightintohow aspectsofteachingcansupportratherthanhinderthisinstructionalprocesswhich researchsuggestsisofbenefittostudents understandingofmathematics (Kilpatrick,Swafford,&Findell,2001;Hiebert&Grouws,2007). InmostU.S.middleschoolmathematicsclassrooms,onetypicallyfinds studentsengagedinamathematicslessonswithateacherexplainingaconceptor task,facilitatingaconversation,observingstudents activities,oraddressing studentswhomaybestrugglingwiththeirwork(kawanaka,stigler,&hiebert, 8
1999).Theseactivitiesandinteractionsarenotnecessarilymutuallyexclusive eventsandoftenoccurconcurrentlyalongwithnon mathematicalactivitiesthatadd timeandcomplexitytoclassroomroutinessuchastakingattendance,pickingup homework,orestablishingrulesandsocialnorms(kennedy,2005).whilestudents mayappeartoprogresstowardsorachievethelesson sintendedlearningobjectives withoutdifficulty,moreoftenthannot,studentsvoicetheirconfusion, misunderstanding,oracontradictionintheirthinkingandsense makingthat requirestheteachertorespond.whatisobservableinmanyclassrooms,andthus servesastheprimaryfocusofmystudy,isthisphenomenonwecallstudent struggles.mystudywillinvestigatethoseaspectsofstudentstrugglesthatbecome productiveinstudents understanding. OVERVIEWOFCONCEPTUALFRAMEWORK HiebertandGrouws(2007),intheSecondHandbookonResearchon MathematicsTeachingandLearning,usedthetermstudents struggle tomeanthat studentsexpendefforttomakesenseofmathematics,tofiguresomethingoutthatis notimmediatelyapparent (p.387).they donotusestruggletomeanneedless frustrationorextremelevelsofchallengecreatedbynonsensicaloroverlydifficult problems orthefeelingsofdespairthatsomestudentscanexperiencewhenlittle ofthematerialmakessense (p.387).thisstruggleoccursinthecontextof students solvingproblemsthatarewithinreachandgrapplingwithkey mathematicalideasthatarecomprehensiblebutnotyetwellformed (p.387).in 9
otherwords,struggleisaparticularkindofphenomenonthatmayoccurasstudents engageinamathematicalactivityorproblemthatischallengingbutreasonably withinthestudents capabilities,possiblywithsomeassistance.thesekindsof difficulties,namelythestrugglesthatpushthestudentsintheirthinking,canplayan importantroleindeepeningstudents understandingifdirectedcarefullytowarda resolution(hiebert&grouws,2007). Asacognitiveprocess,astudent sstruggletomakesenseofmathematicscan beviewedasinternaltothelearner.ontheotherhand,students strugglemaybe visibletoanobserverwhenstudentsexternalizethedifficultytheyareexperiencing. Theoriesoflearninghaveincorporatedbothkindsofstruggle. Otherresearchersandlearningtheoristshavearguedthataconnection existsbetweenstudentengagementinastruggletomakesenseofmathematical ideasanddeeperunderstandingoftheunderlyingconcepts(piaget,1960;dewey, 1926;Inagaki,Hatano,&Morita,1998;Stein,Grover,&Henningsen,1996).From this,iusethenotionofstruggleasacomponentofstudents engagementin mathematicalactivity.thestrugglemaytakeondifferentformsdependingonthe levelofstudentthinkingdemandedbytheactivity. Strugglemaytaketheformof:studentsarguingovercompetingclaims;or expressingtheiruncertaintyoverquestionableprocessesorconclusions(inagaki, Hatano,Morita,1998;Zaslavsky,2005;Hoffman,Breyfogle,&Dressler,2009);or simplyshuttingdowninthefaceoffrustration(dweck,1986).theseinstances 10
provideopportunitiesforteacherstorespondtoandsupportstudents struggles productively.researchsuggests,therefore,thatstudentsmaystrugglewith decidingwhatconceptsorprocedurestouseinsolvingaproblem,determininghow toproceedinacalculationorexplaininghowsomethingworks,orunderstanding whyaconclusionfollows.strugglemaytaketheformofstudentsvoicingconfusion inawhole classdiscussionorseekingclarificationfromtheteacherinaone on one setting(inagaki,hatano,&morita,1998;borasi,1996;santagata,2005). Myconceptualframework,therefore,isbuiltonthreemaincomponents: 1. Theroleofstruggleinlearningmathematicswithunderstanding 2. Thenatureandtypesofmathematicaltasksandtheirrelationshipto students struggle 3. Thewaysteachers respondtostudents struggleinclassroom interactions. Becausemystudyaboutstruggleisinthecontextoflearningmathematics withunderstandingandtheinfluenceofteachingonthedevelopmentofthat understanding,itisimportanttoconsiderwhatconstitutesthenatureof mathematicsandwhatitmeanstoengageinandbecompetentinthediscipline (Schoenfeld,1988).Ifirstpresentmyviewofthenatureofmathematicsandthen elaborateonandreviewtheliteratureconcerningthethreecomponentsofmy conceptualframework. 11
NATUREOFMATHEMATICS Overthecourseofhistory,differingperspectiveshaveresultedfromthe question:whatismathematics?theplatonists viewsuggestsmathematicsisabout discoveringtruthsandideasthatexisteternally,whiletheformalists view mathematicsasasetofrulesoraxiomsfromwhichtheoremsarelogically developed(hersh,1997).hershandothermathematiciansandmathematics educatorstakeamorehumanisticposition,viewingmathematicsasasocialactivity (Freudenthal,1991;Hersh,1997;Bass,2005).Mystudyusesthisperspectiveof mathematicsasasocialphenomenon,wherepeoplecreateobjectsandstudythe patternsandrelationshipsoftheseobjectswithinasocialculture(hersh,1997; White,1993;NCTM,2000;AAAS,1993). Ialsotaketheviewthatmathematicsisadynamicdisciplinethatinvolves exploringproblems,seekingsolutions,formulatingideas,makingconjectures,and reasoningcarefullyandnotastaticdisciplineconsistingonlyofastructuredsystem offacts,procedures,andconceptstobememorizedorlearnedthroughrepetition (Schoenfeld,1992;Hiebertetal,1996;Romberg,1994). Observationsaboutquantitativeandspatialpatternsandrelationshipslead mathematicianstoaskquestions,andmakeinquiries,generalizations,claims,and predictions.theinferencesandpossibleexplanationsinmathematicsthenarethe conjecturesandtheoremsthataremadethroughobservedpatternsand connections.whatisuniquelymathematicalisthenotionofaproofthatservesto communicate,explainandprovideaconvincingargumentforanidea,aproperty,a 12
patternorrelationshiptoothers(hersh,1993).whilenotionsofproofsuggesta formallystructuredargument,theimportantpartofprovingistomakethe mathematicalideashumanlyunderstandableandverifiable(thurston,1994).thus, theroleofproofwilldependontheaudience,sothatinmiddleschoolclassrooms, forexample,aspectsofexplaining,verifying,communicating,andeven systematizingmathematicsinitiatethestudentsintheprocessofmathematical justification(knuth,2002). Intheprocessofproving,newmathematicscanbecreatedordiscovered (devilliers,1999;knuth,2002);thisdemonstratesthatmathematicsisahuman activityinvolvingbothcreativityandimagination.theseactivitiesalsoinclude makingconjectures,seekingwarrants,findingrelationships,andpursuingideasthat maybedestinedforfailurebutrevealnewstrategyoptionsandalternatives. Mathematiciansconfrontnewideas,untriedstrategies,andunknownsolutionsby acknowledgingthatalongwithfailure,grapplingwithandevenstrugglingwith waystosolveproblemsispartoftheprocessof doingmathematics (Holt,1982; Polya,1957;Hiebert&Grouws,2007). Thenatureofmathematicsisthereforedefinednotjustbyfactual, procedural,andconceptualknowledge,butalsobyarangeofprocessesthat constitutedoingmathematics(kilpatrick,swafford,&findell,2001;hiebert& Grouws,2007;NCTM,2000).Fortheremainderofthischapter,Iusethiscontextof whatlearninganddoingmathematicsmeanstodescribethethreecomponentsof 13
myconceptualframework,beginningwiththerolestruggleplaysinlearning mathematics. ROLEOFSTRUGGLEINLEARNINGMATHEMATICS LearningMathematicsByDoing Mathematiciansoftenengagein tryingtofigurethingsout and grappling withproblems astheyinvestigateproblemswithsolutionsnotyetknowntothe investigatorortothegeneralmathematicscommunity.similarly,students learning ofmathematicscanbeconceivedasparallelingthisprocess,wherestudentsengage inexploringproblemsthattheyneitherunderstandnorknowhowtodo.learning mathematicswithunderstandingthenincludesengagingin doingmathematics throughaprocessofinquiryandsense making(schoenfeld,1992;lakatos,1976) thatbynecessityinvolvesstudents expendingefforttofigureoutsomethingthatis notimmediatelyapparent (Hiebert&Grouws,2007)i.e.toexperiencestruggle (Brown,1993).Cobb(2000)suggeststhatbyengagingin doingmathematics, withstruggleasacomponent, studentsactivelyconstructmeaningasthey participateinincreasinglysubstantialwaysinthere enactmentofestablished mathematicalpractices (p.21).asanexample,arnoldross,scholar, mathematician,teacher,andfounderoftherossmathematicsprogramatohiostate University,encouragedhisstudentsto thinkdeeplyofsimplethings, amottostill usedinhisprogramtopromotemathematicalexploration,inquiry,andsense making(retrievednovember4,2009,fromhttp://www.math.ohio 14
state.edu/ross/rossbrochure09.pdf.)encouragingstudentstoparticipateintheir meaning makingsignifiesstudentsareaffordedopportunitiestothinkdeeplyabout problemsandtoacceptstruggleaspartoftheprocessoflearningmathematics. Sometheoriesoflearningincorporatetheconceptofstruggleasacognitive processinternaltothelearnerandothersexaminestruggleasacomponentof learninginasocialsettingasanobservablepartofparticipationinclassroom activity.whilethefocusofmystudyistoexaminetheexternalizedformsof strugglethatoccurintheclassroomsettingthroughasocialcognitivelens,bywhich Imeanboththepersonalconstructionsandsocialinteractionswhichplayimportant rolesinstudentlearning(cobb,yackel,&mcclain,2000),iaminformedbystudies inboththecognitiveandsocialculturaltheoriesoflearning.inthefollowingsection, Idescribethepertinenttheoriesandstudiesofmathematicslearningthatinclude formsofstruggle. CognitiveStruggleinTheoriesofLearning Overthelastcentury,learningtheorieshavereferredtoconceptsakinto struggleanditsconnectiontolearningwithunderstanding.forinstance,dewey (1910,1926,1929,and1933)madereferencestoaprocessofengagingstudentsin someperplexity,confusion,ordoubt (1910,p.12).Inthissetting,Deweyreferred toaparticularthoughtprocesshecalledreflectivethinkingthatinvolved anactof searching,hunting,inquiring,tofindmaterialthatwillresolvethedoubt,settle,and disposeoftheperplexity (p.12).accordingtodewey(1929),schoolinstruction 15
plaguedbyapushforthe quickanswer shortcircuitsthenecessaryfeelingof uncertaintyandinhibitsthesearchforalternativemethodsofsolution.brownwell andsims(1946)argued,likedewey,thatstudentsshouldbegivenopportunitiesto muddlethrough (p.40)theprocessofresolvingproblematicsituationsratherthan conditioningstudentsthroughrepetition. Festinger s(1957)workinthetheoryofcognitivedissonancereferredtothe notionofcognitiveperplexityasanimpetusforcognitivegrowth.morerecently, Hatano s(1988)extensiveresearchinbothmathematicsandscienceeducation relatedcognitiveincongruitywiththedevelopmentofreasoningskillsthatdisplay conceptualunderstanding.themathematicianpolya(1957)wroteextensively aboutproblem solvingandtheprocessbywhichonesolvesproblems.inhowto SolveIt,Polyawrote,...andifyousolveitbyyourownmeans,youmayexperience thetensionandenjoythetriumphofdiscovery (1957,p.v).Thetension,as describedbypolya,inlearninghowtosolveproblemscanbeviewedasafeeling thataccompaniesthestruggletomakeconnectionsamongmathematicalfacts, procedures,andideas.thisdescriptionisconsistentwithpiaget snotionof workingtowardsequilibriumornewunderstandingwhendisequilibriumis introducedthroughanewproblem.learnersrestructuretheirconceptual frameworkorschematoreachcognitiveequilibriumbyincorporatingtheirnew understanding(piaget,1960;carter,2008). 16
Ibasetheconceptofstudents struggleonthetheorythatstudentsdevelop conceptualunderstandingbymaking thementalconnectionsamongmathematical facts,procedures,andideas (Hiebert&Grouws,2007).JustasPiaget(1960)used thetermdisequilibriumtorefertocognitiveconflictbetweenconceptionsalready heldbythelearnerandnewideasandexperiences,incorporatingnewknowledge wouldtheninvolvechallenginglearners currentthinkingandcreatingnew connections(glaser,1984). ObservableStruggleinLearning Inusingasocialconstructivistperspectiveoflearning,Iacknowledgethat bothpersonalconstructionsandsocialinteractionsplayimportantrolesinstudents comingtounderstandmathematics(cobb,yackel,&mcclain,2000).ideally, studentslearningmathematicswithunderstandingoccursintheclassroomas studentsengageintheprocessofexploringproblems,lookingforpatterns,making conjectures,sharingstrategies,connectingmultiplewaysofrepresentingconcepts, explainingthroughreasonedandlogicalarguments,andquestioningoutcomesand conclusionsatbothpersonalandsociallevels(yackel&cobb,1996;schoenfeld, 1988).However,studiessuggestclassroomenvironmentsoftenfallshortofthe idealsettingto domathematics (Schoenfeld,1988).Amoretypicalclassroom environmentisamixtureof doingmathematics withmoretraditionalclassroom settingsthatinvolvestudentsobservingasteachersdemonstrateandexplainways todocertaintypesofproblemsandthenhavingstudentspracticeproblemsusing 17
thedemonstratedmethods(stigler&hiebert,1999).whilestudents strugglemay ariseinawidespectrumofclassroomenvironments,studiessuggestthatsettings thatarerisk freewherestudentscanexternalizetheirstruggleandwhere consequencesof wrong answersarenotseenasfailuresbutratheropportunities toexplore,grow,andlearnservetobettersupportandmotivatestudentstopersist andstruggle(holt,1982;borasi,1996;carter,2008).theinteractionofthe studentswiththeteacherscanplayacriticalroleinhowstudentsperceivethevalue oftheirstruggle. AVygotskianperspectiveunderscorestheimportanceoftheclassroomasa sitewheretheinterrelationshipoftheinternalmentalfunctioningofthelearnerand thesocialinteractionsthatoccuramongstudentsandteachershelpdirectlearners struggletowardsunderstanding(vygotsky,1978,1986).theroleofproofand justificationisanexampleofakeymathematicalpracticethatmustbeunderscored inthepromotionofmathematicalunderstanding(hanna,2000;knuth,2002;maher &Marino,1996;Thurston,1994).Forexample,studentsmakemistakesanda teacherusestheseinstancesassitesforlearningandasopportunitiesforstudents toquestion,explain,justify,andevenextendtheirideaswiththeirpeers(sherin, Mendez,&Louis,2000;Hoffman,Breyfogle,&Dressler,2009;Borasi,1996).Such classroominteractionsaffordstudentswithopportunitiestoparticipateinasensemakingactivitythatcanhelpdevelopstudents thinking(lave&wenger,1991; Fawcett&Gourton,2005). 18
ModelofStruggle IintroducethefollowingmodeltoillustratehowIviewstruggle,anduseitas abaseuponwhichiwillbuildtheothercomponentsofmyconceptualframework. AsInotedabove,strugglemayormaynotbevisible.Inaddition,students strugglemaybepresentorabsentasstudentsengageinmathematicaltasks.ifthe struggleispresent,thenitmaybeeitherexternallymanifestedbythestudentand thusobservableoritmayoccurinternallyandthereforenotbevisibletothe observer. Table2.1:Struggleanditsmanifestations Struggle None Internal External None Manifestation Tooeasy Independent sense making Visiblesigns Toohard Inoneextreme,strugglemaybeabsentorminimalbecauseastudent executesthetaskwithoutdifficulty.theunderlyingreasonfortheabsenceofthe strugglemaybeduetothelevelofthetask.attheotherendofthespectrum, strugglemaynotbedetectediflittleofthematerialmakessensetothestudentorif thestudentisdisengagedinthetask.givingacalculusproblemtomiddleschool students,forexample,wouldbebeyondthescopeofmostofthesestudents understandingandcouldresultinstudentsgivingupratherthanstrugglingthrough theproblem. 19
Myresearchwillfocusonobservingthevisiblestrugglesastheyare externalizedinclassroomsandtoexaminethoseactivitiesandinteractionsthat facilitatestruggleasaproductivepartofmathematicslearningandunderstanding (e.g.stein,grover,&henningsen,1996;henningsen&stein,1997;schwartz& Martin,2004).Inthefollowingsection,IelaborateonwhatImeanbyproductive struggle. ProductiveStruggleinLearning Inthecontextofviewinglearningasagenerativeprocessofmeaning making andmathematicsasadynamicdiscipline,studentandteacherengagementsin mathematicalactivitiesarepossiblesitesforstudentstruggles.theroleofstudent struggleinsupportinganddirectingstudentlearningcanbeexaminedfromthis perspective.productivestruggleisthenaphenomenonthatoccursinaclassroom interactionbetweenteachersandstudentsasstudentsattempttomakesenseof mathematicsand tofiguresomethingout,thatisnotimmediatelyapparent (Hiebert&Grouws,2007,p.287).Itmaybefirstobservedwhenstudentsexpress formsofperplexity,doubt,uncertainty,orconflictwhileengagedinworkingona task,activity,orproblem.whaticallproductivestruggleisaphenomenonthat directstheprocessofstudents struggletowardsunderstanding,reasoning,or sense makingofthemathematicswithpossiblesupportfromtheteacherorpeers andgivesstudentsasenseofagencyindoingmathematics(kilpatrick,swafford,& Findell,2001).Inotherwords,therearesignsofproductivestrugglewhen 20
studentswhowerestrugglingindicateabettersenseofwhattodotogetstarted withaproblem,howtocarryoutprocesses,orwhyaproblemanditssolutionmake sense.inothersituations,studentsarebetterabletoreconcileamisconception, explainorjustifytheirwork,determineanerrorintheirwork,orrecallfactual informationusefulfortheirtask.metaphorically,onemayconsideraderailedtrain putbackontrackoraperson sdiscoveryofapossiblepassageuponreachingan impasseorroadblock. ThisisincontrasttowhatIidentifyasunproductivestruggle,aphenomenon inwhichstudentswhoshowsignsofstrugglemakenoprogresstowardssensemaking,explaining,orproceedingwithaproblemortaskathand.astudentmay voiceresignationandgiveup,takeupanothertask,orobtainananswerfroma teacherorstudent,therebyremovingthestrugglebutnotproductivelybuilding mathematicalunderstanding. Inthenextsection,Ireviewseveralstudiesofmathematicsclassroomsthat supporttheclaimthatproductivestrugglesleadtostudents developmentofgreater conceptualunderstanding. ResearchConnectsStruggleandConceptualLearning Researchershavelookedatavarietyofstudents attemptstomakesenseof mathematicsthatinvolvedsomedifficulty:whenstudentswrestlewithproblems usingmultiplestrategies(carpenter,fennema,peterson,chiang,andloef,1989), undertaketasksofhighcognitivedemand(stein,grover,&henningsen,1996),or 21
mustexplaintheirthinking(hiebertandwearne,1993).studentsfromthese studiesshowedhigherlevelsofperformanceandgainsintheirmathematics assessments.however,notmanyresearchershavedirectlystudiedthe phenomenonofproductivestruggleasihaveframedit;thekindsofstrugglethat mayoccuratvariousstagesofataskwhenstudentsencounterdifficultyfiguringout howtogetstartedorcarryouttheirtask,areunabletopiecetogetherandexplain theiremergingideas,orexpressanerrorinsolvingaproblem. MoredirectlyrelatedtomyinvestigationisastudybyJapaneseresearchers Inagaki,Hatano,andMorita(1998)thatexaminedstudentssharingtheircorrectas wellasincorrectanswersanddemonstratingtheirconfusionalongwiththeir emergingunderstanding.theresearchersexaminedwhole classstudent to student interactionsoffourth andfifth gradestudents.theclassroomdiscussionfocused onstudents sharingtheirsolutions,bothcorrectandincorrect.theteacherdidnot intervenetoidentifythecorrectnessoftheanswers.rather,chosenstudent presenterswereresponsibleforjustifyingtheirsolutionsontheboardtotheclass andtheirclassmatescouldquestionsolutionsthatconfusedthemordidnotmake sense.recallingstruggle, tomeanthatstudentsexpendefforttomakesenseof mathematics,tofiguresomethingoutthatisnotimmediatelyapparent, (Hiebert& Grouws,2007,p387),thediscussionthatfollowedshowedstudentsstrugglingto explaintheirsolutionortomakesenseoftheanswergivenbytheirclassmate.the studentsthenhadtodecideforthemselveswhatmadesensefromthegiven 22
explanationsandjustifications.findingsfromthisstudyshowedthatengagingin sense makingofsharedsolutions,bothcorrectandincorrect,resultedinimproved understandingofmathematicscontent. Thereareadditionalcasestudiesofclassroomsthataddsupporttotheclaim thatteachersengagingstudentsinproductivestrugglewithimportantmathematics buildsstudents conceptualunderstanding(ball,1993;fawcett,1938;heaton, 2000;Lampert,2001;Schoenfeld,1985).Forexample,Carter(2008)foundgreater persistenceinproblemsolvingamonghersecond grademathematicsclasswhen shecreatedalearningenvironmentthatacknowledgedstruggleasanexpectedpart oflearning.amottousedincarter sclassresemblesaquotemadeatavery differenttimeandcontextbyabolitionistandorator,frederickdouglass(1857),"if thereisnostruggle,thereisnoprogress.theclassmottousedincarter sclass, If youarenotstruggling,youarenotlearning (p.136),emphasizestheimportanceof studenteffortandpersistenceinlearning.furthermore,confusionwasacceptedas astateonegoesthrough,ratherthanapermanentstate. Inaseven yearstudyofminorityandlow incomestudentsinnewark,new Jersey,RobertaSchorr,aRutgers educationresearcher,foundevidencethat studentsbecomeengagedandsuccessfulinmathematicswhenallowedtostruggle withchallengingmathproblems, thereisahealthyamountoffrustrationthat s productive (Yeung,B.(2009,September10).RetrievedonDecember29,2009, fromwww.edutopia.org/math underachieving mathnext rutgers newark#). 23
Severalstudiesoutsideofmathematicseducationprovideevidenceof conceptuallearningasanoutcomeofstruggle.aresearchstudybyrobertbjork (1994)reviewedcognitivetrainingstudiesandfoundthatthosetraineeswho experienceddifficultiesmasteringtargetedskillsdevelopeddeeperormoreuseful competenciesintheend.theprocessofovercomingdifficultiesandobstacles seemedtoprovokethinkingthatledtoamoregeneralizableandtransferable learning.bjorkreasonedthatthisistheresultoflearnershavingtoconstructtheir understandingbyconnectingtowhattheyalreadyknew,therebylearningcontent andskillsmoredeeply. Inanotherstudy,CaponandKuhn(2004)foundthatinlearningnew businessconcepts,thembastudentswhoattemptedtosolveproblemsratherthan justlisteningtoalectureanddiscussioncouldmoreeffectivelyexplainarelated concept.theresultssuggestthatteachingthatincludedtasksofactiveengagement suchasworkingonsolvingproblemspromotedadeeperconceptualunderstanding thanthosethatmadeonlypassivedemandsonstudents. Descriptionsoftasksprovidenotonlyacontextbutalsoalinkbetween learningandteaching.inparticular,thestrugglesstudentsexperienceare generatedwithinthecontextofclassroomactivityaroundtasksthatplacedifferent demandsonstudents cognitiveprocesses.thestudents experienceinthe classroomoftasksofvaryingcognitivedemandcanproducedifferentresultsin theirlearning(hiebert&wearne,1993;stein,grover,&henningsen,1996).inthe 24
nextsection,iexaminethenatureandtypesofmathematicaltasksthathelp facilitatestudents productivestrugglesthroughinteractionandactiveengagement amongstudentsandteachers. NATUREANDTYPESOFTASKSTHATSUPPORTPRODUCTIVESTRUGGLE ImportanceofMathematicalTasks Tasksareacentralpartofateacher sinstructionaltoolkit,andwhat students learnisoftendefinedbythetaskstheyaregiven(christiansen&walther, 1986).Inordertomovestudentstowarddevelopingadeepconceptual understandingofmathematics,classroomteachingmustincorporateopportunities forstudentstograpplewithmeaningfultasks(lampert,2001;nctm1991; Schoenfeld,1994).Inaddition,studentsmustbegivenopportunitiestomakesense ofimportantideasinmathematicsandtoseeconnectionsamongtheseideas (Boaler&Humphreys,2005). Tasksdefinetheactivitiesstudentsengageinandprovidestudentssocial experiencestoparticipateinactivenegotiation,sense making,andreasoningthat areinternalizedashighermentalprocessesthroughenactment(vygotsky,1962, 1978;Rogoff&Wertsch,1984;Wertsch,1998;Bakhtin,1982).Whatisimportant inthetaskandclassroomactivityistheworkthestudentsarerequiredtodo(doyle, 1988).Theteachersdefinenotonlytheproductsstudentsaretoproducebutalso theprocessesandresourcesstudentsmayuse,andthenormsbywhichthe students workareevaluated. 25
Mathematicseducatorsandresearchersvoicesimilarpointsofview regardingtasks.henningsenandstein(1997)stated,inregardtofindingsintheir workwiththequasarproject,afive yearstudyofmathematicsreforminurban middleschools, thenatureoftaskscanpotentiallyinfluenceandstructuretheway studentsthinkandcanservetolimitortobroadenstudents viewsofthesubject matterwithwhichtheyareengaged (p.546).krainer(1993)asserted, powerful tasksareimportantpointsofcontactbetweentheactionsoftheteacherandthoseof thestudent (p.68).studiesshowthatmathematicaltasksatstagesofconception, selection,set up,implementation,andexecutionbytheteacherandthenthe enactmentandinteractionbystudentsandteacherplayedcriticalrolesinthefocus, demand,andvalueofwhatstudentslearnedasmathematics(smith&stein,1998; Schoenfeld,1992;Doyle,1983;Hiebert&Wearne,1993).InAddingItUp(Kilpatrick, Swafford,&Findell,2001),theauthorsstatethat, tasksarecentraltostudents learning,shapingnotonlytheiropportunitytolearnbutalsotheirviewofthe subjectmatter (p.335).nctm(2000)andsimon&tzur(2004)bothpointto mathematicaltasksasthekeypartoftheinstructionalprocessthatprovidestools forpromotingthelearningofparticularandimportantmathematicalconcepts. Itisinstructivewhenstudyingvariousformsofstruggletoalsoexaminethe taskcontextandsituationthatengagesandsupportsthestudents learning preciselybecausetaskshelpshapestudents cognitivegrowthandtheprocessesby whichstudentsconstructtheirunderstanding. 26
TaskFramework Tasksofvaryingcognitivedemandsproducedifferentresultsinstudent learning(hiebert&wearne,1993),dueinparttothedifferentexperiencesstudents haveintheclassroom.researchersalsosuggestthattasksdesignedtoprompt higher orderthinkingaremorelikelytoproducedeeperconceptualunderstanding thantasksdesignedtoofferskillspractice(doyle,1988;hiebert&wearne,1997). Bycognitivedemand,Imeanthesortofstudentthinkingthatthetaskdemands (AmericanEducationalResearchAssociationResearchPoints,2006.Retrieved January5,2010from http://www.aera.net/uploadedfiles/journals_and_publications/research_points/r P_Fall06.pdf).Raisingthelevelofdemandonstudents cognitiveprocessesmay thereforeresultingeneratingmorestrugglewithinthecontextofclassroom activity. Iuseataskframeworkbasedoncognitivedemand(Stein,Smith,Henningsen, andsilver,2000)inordertogainaclearerpictureofthekindsoftaskswherethese productivestrugglesoccur.thequasarresearchers(silver&stein,1996)created amathematicaltasksframeworkthatfirstsituatesmathematicaltasksinthree stagesasitunfoldsintheclassroomsetting:(1)asdesignedbythecurricular material,(2)asset upbyateacher,and(3)asimplementedbystudents.the frameworkthenanalyzestasksatfourlevelsofcognitivedemand(smith&stein, 1998).Inthefollowingsection,IdescribethelevelsofcognitivedemandIwilluse inmystudy,basedonthemathematicaltasksframework. 27
LevelsofCognitiveDemand Steinetal.,(1996)identifiedfourlevelsofcognitivedemand.Fromlowestto highesttheyare:memorization,procedureswithoutconnectionstoconceptsor meaning,procedureswithconnectionstoconceptsandmeaning,and doing mathematics. Isummarizethecharacteristicsofeachlevelbelow: Memorization o involveseitherreproducingpreviouslylearnedfacts,rules,formulas, ordefinitionsorcommittingfacts,rules,formulas,ordefinitionsto memory;and o involvesverysimilarreproductionofpreviouslyseenmaterial. Procedureswithoutconnectionstoconceptsormeaning o arealgorithmic; o havenoconnectiontotheconceptsormeaningthatunderliethe proceduresbeingused;and o arefocusedonproducingcorrectanswersratherthandeveloping mathematicalunderstanding. Procedureswithconnectionstoconceptsormeaning o focususeofproceduresforpurposesofdevelopingdeeperlevelsof understandingofmathematicalconceptsandmeaning; o usuallyrepresentedinmultiplewayswithconnectionsamong multiplerepresentations; 28
o suggestexplicitlyorimplicitlypathwaystofollowthatarebroad generalproceduresthathavecloseconnectionstounderlying conceptualideasasopposedtonarrowalgorithmswithconceptsthat arenottransparent;and o engagewithconceptualideasthatunderlietheproceduretocomplete thetasksuccessfully. Doingmathematics o requirescomplexandnon algorithmicthinking; o requiresexplorationandunderstandingthenatureofmathematical concepts,processes,orrelationships; o demandsself monitoringorself regulationofone sowncognitive processes; o requiresaccesstorelevantknowledgeandexperiencesandmake appropriateuseofthem; o requiresanalysisoftaskandexaminetaskconstraintsthatmaylimit possiblesolutionstrategiesandsolutions;and o requiresconsiderablecognitiveeffortandmayinvolvesomelevelof anxietyforthestudentbecauseoftheunpredictablenatureofthe solutionprocessesthatarerequired. 29
(SmithandStein,1998withacknowledgementbytheauthorstoworksbyStein, Grover,andHenningsen,1996;Stein,Lane,andSilver,1996;NCTM,1991;Resnick, 1987;Doyle,1988). Alearningenvironmentthatprovidesstudentsopportunitiestostruggle withmathematics,ihypothesize,engagesstudentsathighlevelsofcognitive demand.inparticular,thosetasksinvolving doingmathematics becausethey requirenon algorithmicandcomplexthinking,haveagreaterlikelihoodofcausing struggleamongthestudent.thecognitiveeffortrequiredatthelevelof procedureswithconnectiontoconceptsandmeaning couldalsogeneratestruggle asstudentsmakesenseofthetask,makeconnectionstotheirpriorknowledge,and formulatestrategiesinordertocompletetheirtask.tasksatthelowerlevelcan generateothertypesofstrugglesuchasforgettingausefulalgorithmorinabilityto executeacalculation. ModelingStruggleandTasks Inordertosituateproductivestruggleasapossibleoccurrencein interactionsamongteacher,students,andmathematicalcontent,iexpandthemodel ofstruggleintroducedearliertoincludethelevelsofimplementedtasksascontext andsettingfortheclassroominteractionsandstudents struggle. 30
Table2.2:ProductiveStruggleintheClassroomInteractionsofTeachingand LearningintheContextofMathematicalActivitiesandTasks Struggle CognitiveLevel ofimplemented Tasks None Internal Tooeasy Independent sense making External Visiblesigns None Toohard Memorization Procedures without connections Procedureswith connections Doing mathematics Thetask strugglemodelwillrelatethenatureofstudents struggleandthe taskcontextinwhichitoccurs.ataskofhighercognitivedemandmayprovoke minimalstruggleforsomestudentswhoareabletoformulateappropriate strategiesandcarryoutthetaskorsolvetheproblemwithoutsignsofstruggle.in general,however,tasksofhighercognitivedemandwouldmostlikelyprovide greaterincidencesofstruggle(stein,grover,&henningsen,1996).astudentmay alsostrugglewithataskoflowcognitivedemand,suchasfindingaleastcommon denominatorifthestudenthasforgottentheprocedureforfindingleastcommon multiples.therefore,instudyingvariousformsofstruggle,itisinstructivetoalso 31
examinethetaskcontextandsituationthatengagesandsupportsthestudents learning. Thesourceofthestrugglemayhaveabasisinmathematicalconceptsand procedures,suchastheaboveexampleofforgettinghowtofindtheleastcommon multiple.othersourcesmayincludestrugglesrecallingmathematicalterminology, theculturalcontextoftheproblem,ortheenglishlanguageitself(secada,1992, Khisty&Morales,2004).Suggestedinthismatrixoftasksbystruggleisazoneof proximaldevelopment(zpd)orthegreyzoneofvariousshadingsindicatedintable 2.2,whereteacheractionandresponsecanprovidetheneededsupporttomovethe studentsforwardintheirunderstanding(vygotsky,1962;1978;wertsch,1985).by linkingthekindsofteacherresponsestotheformsofvisiblestudentstruggles occurringinclassroominteractions,wecanrelatetheroleofteachingthatsupports thestrugglestowardproductiveresolutions. Inowdescribestudiesthathaveincludedaspectsofstudents struggleinthe contextoftasksandrelatedinteractions. KindsofTasksthatSupportProductiveStruggle Tasksthatevokeuncertaintyforthelearnersuchascompetingclaims, unknownpathwaysorquestionableconclusions,andnon readilyverifiable outcomesplacesignificantlymorecognitivedemandonthelearnerandasaresult canfostermathematicalunderstandingandmeaningfullearning(zaslavsky,2005). Zaslavsky s(2005)studyhighlightedtheimportanceofanappropriateclassroom 32