Copyright by Hiroko Kawaguchi Warshauer 2011

Copyright

by

HirokoKawaguchiWarshauer

2011

TheDissertationCommitteeforHirokoKawaguchiWarshauerCertifiesthat

thisistheapprovedversionofthefollowingdissertation:

TheRoleofProductiveStruggleinTeachingandLearning

MiddleSchoolMathematics

Committee:

SusanEmpson,Supervisor

JamesBarufaldi

EdmundT.Emmer

AnthonyPetrosino

PhilipUriTreisman



by

HirokoKawaguchiWarshauer,B.A.;M.S.

Dissertation

PresentedtotheFacultyoftheGraduateSchoolof

TheUniversityofTexasatAustin

inPartialFulfillment

oftheRequirements

fortheDegreeof

DoctorofPhilosophy

TheUniversityofTexasatAustin

December2011

Dedication

TomyhusbandMaxandchildrenAmy,Nathan,Lisa,andJeremy

v

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.

vi

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.

vii

Myhusband,Max,hasbeenmybiggestsupporter;providingchallenges,

inspiration,andcomfort.Thequestionsheasked,theeditshemade,the

encouragementhegaveallkeptmethinkinganewandmoredeeply.Itisthanksto

workingwithteachersandstudentsthroughMathworksthatIhadanidealsetting

toconductmyresearch.

Finally,tomyparents,MotohiroandSuzukoKawaguchi,whonamedme,

“scholarlychild,”Ithankthemforalwaysencouragingmetodomyverybest,

whetherinmathematics,music,orsportsandforinstillinginmeadeeploveof

learning.

viii



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

ix

studyidentifiedthekindsofteacherresponsesusedintheinteractionwiththe

studentsandthetypesofresolutionsthatoccurred.

Theparticipantswere3276thand7thgradestudentsandtheirsix

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.

x

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

xi

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

xii

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

xiii

ListofTables

Table2.1:Struggleanditsmanifestations ........................................................19

Table2.2:ProductiveStruggleintheClassroomInteractionsofTeachingand

LearningintheContextofMathematicalActivitiesandTasks ...31

Table3.1: CharacteristicsofTeacherParticipants ........................................55

Table3.2: Observedclassfrequencyandhours .............................................56

Table3.4: Activity1:BarrelofFun .................................................................67

Table3.5: Activity2:BagsofMarbles ............................................................67

Table3.6: Activity3:TipsandSales*.............................................................68

Table3.7: Activity4:DetectingChange..........................................................69

Table4.1: KindsofStudentStrugglesandtheirPercentFrequencies .........82

Table4.2:TeacherResponseSummary ...........................................................99

xiv

ListofFigures

Figure2.1:PreliminaryStruggleandResponseFrameworkinTaskContext49

Figure4.1:Findtheprobabilityoflandingintheunshadedregion. .............88

Figure4.2: TeacherResponseRange...............................................................96

Figure4.3: ProductiveStruggleFrameworkinaninstructionalepisode ...135

1

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

2

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

3

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

4

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

5

(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

6

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).

7

Mystudyidentifiedalltheepisodesduringinstructionwherestudentsmade

mistakes,expressedmisconceptions,orclaimedtobelostorconfused,andtowhich

teachersresponded.Interactionsbetweenstudentsandteachersgenerally

advancedtowardsomeresolutionofthestudents’difficultiesandattemptsatsense‐

making.Usinganembeddedcasestudymethodology(Yin,2009)withinstructional

episodesastheunitofanalysiswithinthelargerunitoftheteachers,Iidentifiedand

describedthenatureofthestudents’struggle.Additionally,Irecognizedthe

instructionalpracticesofteachersthateithersupportedandguidedordidnot

supportorguidethestudents’sense‐andmeaning‐makingofthemathematicsin

thelessonepisodes.Iusedmyobservationnotes,interviewsofteachersandtarget

students,andvideoand/oraudiotapesofclassroomlessonstodescribeandanalyze

theinteractivetechniquesandpracticesteachersusedthatfocusedonstudents’

productivestruggle.

8

Chapter2:ConceptualFramework

INTRODUCTIONTeachingthatprovidesstudentsopportunitiestostrugglewithimportant

mathematicalideashasbeenidentifiedinmathematicseducationresearchasoneof

thekeyfeaturesofteachingthatsupportsthedevelopmentofstudents’conceptual

understandingofmathematics(Hiebert&Grouws,2007;Hiebert&Wearne,1993;

Stein,Grover,&Henningsen,1996;Borasi,1996).Students’learningof

mathematicswithunderstandingisviewedascriticalinmeetingthedemandsofthe

21st‐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,

9

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.

OVERVIEWOFCONCEPTUALFRAMEWORKHiebertandGrouws(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

10

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

11

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.

12

NATUREOFMATHEMATICSOverthecourseofhistory,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

13

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

14

myconceptualframework,beginningwiththerolestruggleplaysinlearning

mathematics.

ROLEOFSTRUGGLEINLEARNINGMATHEMATICS

LearningMathematicsByDoingMathematiciansoftenengagein“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‐

15

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

16

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).

17

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

18

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

classroominteractionsaffordstudentswithopportunitiestoparticipateinasense‐

makingactivitythatcanhelpdevelopstudents’thinking(Lave&Wenger,1991;

Fawcett&Gourton,2005).

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ModelofStruggleIintroducethefollowingmodeltoillustratehowIviewstruggle,anduseitas

abaseuponwhichIwillbuildtheothercomponentsofmyconceptualframework.

AsInotedabove,strugglemayormaynotbevisible.Inaddition,students’

strugglemaybepresentorabsentasstudentsengageinmathematicaltasks.Ifthe

struggleispresent,thenitmaybeeitherexternallymanifestedbythestudentand

thusobservableoritmayoccurinternallyandthereforenotbevisibletothe

observer.

Table2.1:Struggleanditsmanifestations

Struggle None Internal External None

Manifestation Tooeasy Independentsense‐making

Visiblesigns Toohard

Inoneextreme,strugglemaybeabsentorminimalbecauseastudent

executesthetaskwithoutdifficulty.Theunderlyingreasonfortheabsenceofthe

strugglemaybeduetothelevelofthetask.Attheotherendofthespectrum,

strugglemaynotbedetectediflittleofthematerialmakessensetothestudentorif

thestudentisdisengagedinthetask.Givingacalculusproblemtomiddleschool

students,forexample,wouldbebeyondthescopeofmostofthesestudents’

understandingandcouldresultinstudentsgivingupratherthanstrugglingthrough

theproblem.

20

Myresearchwillfocusonobservingthevisiblestrugglesastheyare

externalizedinclassroomsandtoexaminethoseactivitiesandinteractionsthat

facilitatestruggleasaproductivepartofmathematicslearningandunderstanding

(e.g.Stein,Grover,&Henningsen,1996;Henningsen&Stein,1997;Schwartz&

Martin,2004).Inthefollowingsection,IelaborateonwhatImeanbyproductive

struggle.

ProductiveStruggleinLearningInthecontextofviewinglearningasagenerativeprocessofmeaning‐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

21

studentswhowerestrugglingindicateabettersenseofwhattodotogetstarted

withaproblem,howtocarryoutprocesses,orwhyaproblemanditssolutionmake

sense.Inothersituations,studentsarebetterabletoreconcileamisconception,

explainorjustifytheirwork,determineanerrorintheirwork,orrecallfactual

informationusefulfortheirtask.Metaphorically,onemayconsideraderailedtrain

putbackontrackoraperson’sdiscoveryofapossiblepassageuponreachingan

impasseorroadblock.

ThisisincontrasttowhatIidentifyasunproductivestruggle,aphenomenon

inwhichstudentswhoshowsignsofstrugglemakenoprogresstowardssense‐

making,explaining,orproceedingwithaproblemortaskathand.Astudentmay

voiceresignationandgiveup,takeupanothertask,orobtainananswerfroma

teacherorstudent,therebyremovingthestrugglebutnotproductivelybuilding

mathematicalunderstanding.

Inthenextsection,Ireviewseveralstudiesofmathematicsclassroomsthat

supporttheclaimthatproductivestrugglesleadtostudents’developmentofgreater

conceptualunderstanding.

ResearchConnectsStruggleandConceptualLearningResearchershavelookedatavarietyofstudents’attemptstomakesenseof

mathematicsthatinvolvedsomedifficulty:whenstudentswrestlewithproblems

usingmultiplestrategies(Carpenter,Fennema,Peterson,Chiang,andLoef,1989),

undertaketasksofhighcognitivedemand(Stein,Grover,&Henningsen,1996),or

22

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

23

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#).

24

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

25

nextsection,Iexaminethenatureandtypesofmathematicaltasksthathelp

facilitatestudents’productivestrugglesthroughinteractionandactiveengagement

amongstudentsandteachers.

NATUREANDTYPESOFTASKSTHATSUPPORTPRODUCTIVESTRUGGLE

ImportanceofMathematicalTasksTasksareacentralpartofateacher’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.

26

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.

27

TaskFrameworkTasksofvaryingcognitivedemandsproducedifferentresultsinstudent

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.

28

LevelsofCognitiveDemandSteinetal.,(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;

29

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.

30

(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.

ModelingStruggleandTasksInordertosituateproductivestruggleasapossibleoccurrencein

interactionsamongteacher,students,andmathematicalcontent,Iexpandthemodel

ofstruggleintroducedearliertoincludethelevelsofimplementedtasksascontext

andsettingfortheclassroominteractionsandstudents’struggle.

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Table2.2:ProductiveStruggleintheClassroomInteractionsofTeachingandLearningintheContextofMathematicalActivitiesandTasks

StruggleCognitiveLevelofImplemented

Tasks

NoneTooeasy

InternalIndependentsensemaking

ExternalVisiblesigns

NoneToohard

Memorization

Procedureswithout

connections

Procedureswithconnections

“Doingmathematics”

Thetask‐strugglemodelwillrelatethenatureofstudents’struggleandthe

taskcontextinwhichitoccurs.Ataskofhighercognitivedemandmayprovoke

minimalstruggleforsomestudentswhoareabletoformulateappropriate

strategiesandcarryoutthetaskorsolvetheproblemwithoutsignsofstruggle.In

general,however,tasksofhighercognitivedemandwouldmostlikelyprovide

greaterincidencesofstruggle(Stein,Grover,&Henningsen,1996).Astudentmay

alsostrugglewithataskoflowcognitivedemand,suchasfindingaleastcommon

denominatorifthestudenthasforgottentheprocedureforfindingleastcommon

multiples.Therefore,instudyingvariousformsofstruggle,itisinstructivetoalso

32

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.

KindsofTasksthatSupportProductiveStruggleTasksthatevokeuncertaintyforthelearnersuchascompetingclaims,

unknownpathwaysorquestionableconclusions,andnon‐readilyverifiable

outcomesplacesignificantlymorecognitivedemandonthelearnerandasaresult

canfostermathematicalunderstandingandmeaningfullearning(Zaslavsky,2005).

Zaslavsky’s(2005)studyhighlightedtheimportanceofanappropriateclassroom

33

settingandtherolethatsocialinteractionsandtheexchangeofpersonalviewpoints

andpreferenceshaveinutilizinguncertaintyasaforceinlearningmathematics.

Othertasksthatexplorecontradictions,investigateerrors,orexamine

misconceptionsserveasusefullearningactivitieswhileaddressingstrugglesthat

studentshaveinresolvingtheirmistakesormisunderstandings(Hiebertetal,1997;

Lampert,2001;Kazemi&Stipek,2001).Theinteractionscreateapressfor

conceptuallearningbygivingstudentsopportunitiestoreconceptualizeaproblem,

explorecontradictionsinsolutions,andpursuealternativestrategies(Kazemi&

Stipek,2001;Borasi,1996;Townsend,Lannin,&Barker,2009).

Classroominstructionthatrequiresstudentstoexplaintheirsolutionsto

problems,whethercorrectorincorrect,cangivestudentsopportunitiestoargue

theirpointsofviewandfortheirpeerstostruggleinunderstandingorquestioning

others’thinking(Inagaki,Hatano,&Morita,1998).Similarly,havingstudents

describeandexplainalternativestrategies,askingstudentsmorequestionsand

providingstudentswithtimetoexplaintheirresponsesrevealstoteacherswhere

studentsarestrugglingwithemergingideas(Hiebert&Wearne,1993).Tasksbased

oninventionorprojectsoftenrequirestudentstobothconnecttotheirprior

knowledgeasastartingpointandtoseeknovelsolutionpathsthatarenot

straightforward.Studentswhoengagedinsense‐makingandstruggleintheprocess

ofexperimentation,consideration,andrejectionofideas,werebetterpreparedfor

34

futurelearningandshowedpositiveeffectsonstudentlearning(Schwartz&Martin,

2004;Barron,etal,1998).

Thosestudentsengagedintasksthatwereset‐upandimplementedwith

high‐levelofcognitivedemandshowedthehigheststudentlearninggainsinstudies

suchastheQUASARProject(Silver&Stein,1996;Stein&Lane,1996).Whilethe

QUASARresearchersobservedthathigh‐leveltasksdonotguaranteehigh‐level

studentengagement,theyalsonotethatlow‐leveltasksalmostneverresultinhigh‐

levelengagement(Smith&Stein,1998).Inastudyontechnologicaltasks,Borchelt

(2007)foundthatamongtheeightcodingcategoriesoftasksthatemergedinhis

data,thecategoryoffrustration,wherestudentsexperienceanxietyandinsecurity

inapproachestoproblemsolving,isoneofthecharacteristicsofatasksupporting

thehighestlevelofcognitivedemand.

ThefindingsfromtheQUASARProject(HenningsenandStein,1997)

stronglysuggestthatinorderforstudentsto“domathematics”,theclassroommust

provideanenvironmentwherestudentscanengageinworthwhileandhigh‐level

activitiesinwhichstrugglingwithproblemsandtasksisanexpectedpartofthe

dailyroutine.Theirresearchidentifiedthosesupportfactorsincluding:tasksbuilt

onstudents’priorknowledge;appropriateamountoftime;high‐levelperformance

modeled;sustainedpressureforexplanationandmeaning;scaffolding;studentself‐

monitoring;andtheteacherdrawingconceptualconnections,allofwhichhelped

maintainengagementofstudentsatahigh‐levelofthinkingandreasoning.The

35

supportfactors“relatedtotheappropriatenessofthetaskforthestudentsandto

supportiveactionsbyteachers,suchasscaffoldingandconsistentlypressing

studentstoprovidemeaningfulexplanationsormakemeaningfulconnections”(p.

546).

Thus,aclosestudyoftheinteractionsbetweenteacherandstudentsinwhich

aformofstudents’struggleoccursduringengagementinamathematicalactivityis

ofvitalimportanceinexaminingwhetherandhowthestrugglecanbeproductively

resolvedornot.Iwillnowaddressthethirdcomponentofmyconceptual

frameworkandreportonstudiesthatidentifysupportiveactionstakenbyteachers

duringtaskenactments.Iwilldiscuss,inparticular,thoseteacherresponseslinked

tothekindsofstrugglethatoccurinthemidstofmathematicalactivities.

TEACHER’SRESPONSETOSTRUGGLE

Thethirdcomponentofmyconceptualframeworkfocusesontheroleof

teacher’sresponseinfacilitatingstudents’struggleinaproductivemanner.As

studentsandteachersparticipateintheenactmentoftasksintheclassroom,

studentsengageintheprocessofmakingsenseofthesetasks.Aswellplannedas

thetasksmaybe,studentscanencounterdifficultyduringvariousstagesofthe

lessonenactmentprocessfromitsintroductionanddevelopmenttoitsclosure.The

externalizationofstudents’strugglecanengagetheclassroomcommunity,oratthe

veryleasttheteacher,insomeresponseaction.Myconceptualframeworkwas

informedbystudiesthatfocusedoninteractionsamongtheclassroomparticipants

36

andexaminedthekindsofsupportandguidancetheinteractionsaffordedin

resolvingthestruggles.Ontheonehand,explicitactionsandmovesbyteachersor

peerscanworktobuildcommunityunderstandingandresolvestudents’struggle

withoutdeprivingstudentsoftheopportunitytothinkforthemselves.Ontheother

hand,theurgebyteacherstohelpstrugglingstudentscanresultinloweringor

removingthecognitivedemand(Henningsen&Stein,1997)bysuchactionsas

tellingstudentstheanswer(Chazan&Ball,1999),directingthetaskintosimpleror

mechanicalprocesses(Stein,Grover,&Henningsen,1996),orgivingguidancethat

funneledstudents’thinkingtowardsananswerwithoutbuildingnecessary

connectionsormeaning(Woodetal,1976;Herbel‐Eisenmann&Breyfogle,2005).

Teachers’supportisdictatedbythecontext,situation,studentneedsand

theirownbeliefsandknowledge.Responsesthathelpstudentsmakeconnections

orsupplysomeneededinformationmaybeanecessarypartofsupportingstudents’

productivestruggle.Mystudyattemptstodelineatebetweenteachers’actionsthat

directstudents’struggleproductivelytowardsense‐making,asinunderstandingthe

what,how,orwhyofthetaskasopposedtoactionsthatcoulddirectstudents’

struggleunproductivelytowardsanansweroraprocedurethattranspireswithout

studentsalsoseeingthemathematicalconnectionsormeaningtotheiractions.

Researchershaveinvestigatedthekindsofscaffolding:thediscourse,

questioning,andmotivationalstrategiesusedbyteachersintheirinteractionwith

studentsthatsupportstudentlearning(e.g.Lampert,1990;O’Connor&Michaels,

37

1996;Dweck,1986;Anghileri,2006;Williams&Baxter,1996).Mysynthesisof

existingresearchonteachingandlearninginteractionsfocusesonexaminingthe

natureofteachers’responsesandthewayssuchactionscanhelpmanageaspectsof

theinteractionsthateitherdirectlyorindirectlysupportproductivestruggleasa

valuablepartoflearning.Iusethefollowingtypesofteacherresponsesthatwill

guidemydataanalysis.

• Supplyinformation:

o Giveanswer

o Remindrelevantaspect

o Suggestmethodortechnique

o Giveexample

o Evaluatecorrectness

o Modelmethodortechnique

• Connecttostudents’priorknowledge:

o Refertoexampleofpreviousworkrelatedtocurrenttask

o Suggestanalogyandcomparisonofconcepts

o Providevisualrepresentation

o Modifyorabandontheproblem

• Attendtoandclarifythestruggle:

o Statetheproblemorsharestrugglewithothersintheclass

38

o Revoicestudent’sstrugglebyreframing,refocusing,orrephrasing

withgreaterclarity

• Askguidingquestions:

o Supportandbuildonstudentideas

o Considersimplercase

o Refocusstudentsonpartsofthetask

• Askprobingquestions:

o Elicitfromstudentswhatisknown,whattheyseek

o Pressforclarityandarticulationofquestionandreasoning

o Examinepossiblemisconceptions

o Providereflectivetoss

• Provideencouragementandagency:

o Acknowledgestudents’thinkingandselectportionsusefultostruggle

o Examineerrorsasvaluabletolearning

o Affordmoretimeforthinking

Thenatureofinteractioninaclassroomistheunpredictabilityofwhatcan

happenbetweenteachersandstudentsinthemoment‐to‐momentworkingsof

mathematicalactivity,particularlyasteachersattempttobalancethecomplexities

andconstraintsofclassroomsettings(Kennedy,2005).Astudentstrugglingwithan

aspectofamathematicaltaskmayseeksomeformofsupportinordertoresolvethe

tensionthatconfrontshimorher.Theimpactofsupportingstudentsinlearning

39

canvaryfromonelearnertoanotherandwhattheycometoknowasmathematics

(Gresalfi,2004;Ball&Bass,2003).Bestintentionedlessonscangivewaytothe

immediateneedsofthestudentatthemomentofenactment(Steinetal,2000;

Wells,1996)andthechoicesteachersmakeinresponsetothesituationcancreate

differentlearningopportunities“accordingtothepedagogicpurposestheyhavein

mindatparticularmoments”(Haneda,2004,p.181).Forexample,intheirfindings

fromtheQUASARProject,Stein,Grover,andHenningsen(1996)reportedthat

teachers’useofscaffoldingservedasoneofthesupportfactorsthathelped

maintainthestudentsengagementwithataskatahigh‐level.However,inmany

instances,theteachers’responsesprovidedsomuchinformationthatthecognitive

demandofthetaskwasreducedalongwithstudents’opportunitytostruggle

productivelywiththemathematics.Theinteractionsbetweenteachersand

students,therefore,requireconstantbalancingofchallengesandsupportasthe

tasksunfold(Mariani,1997;Michell&Sharpe,2005).Inwhatfollows,Ielaborate

onthecategoriesproposedaboveforteachers’responsesthataddressstudent

strugglesanddescribetheirpossibleintendedpurpose.

ResponsesthatSupplyInformationtoStudentsOnceateacherobservesastudentstruggle,theteachermakesadecision

regardinganactiontotake.Onecategoryofteacherresponseistoprovide

informationtothestrugglingstudent.Throughthistypeofaction,thestudentsmay

thenbemoretask‐enabledthanpriortotheinteraction(Maybin,Mercer&Stierer,

40

1992).Thetypeofinformationmayrangefromgivingtheanswertotheproblem,

showingstudentshowtosolvetheproblem(Smith,2000),givingsufficienthintsor

addingkeymissingpiecestohelpstudentscontinuetheirwork.Theresponsemay

beacorrectiontostudents’workorstatementoraselectionofthecorrectanswer

fromseveralchoicesthestudentsarestrugglingover(Santagata,2005).Itcould

consistofprovidingstudentsameaningfulcontexttoanabstractconcept(Anghileri,

2006,p.42).Teachersmayrespondbysupplyingthestudentsausefultechniqueor

methodorareminderofaresourcesuchasthetextorworkontheboardthatmay

promptsomeusefulconnections.Ateachermayfurthersupplyanexamplethat

illustratesormodelsausefulstrategythatprovidessupportofmathematicalideas

oranalyticscaffoldingforstudents(Williams&Baxter,1996).Inshort,theteacher

tellsstudentssomeinformationthatappearstobeusefulforimplementingthetask,

thoughintheprocessmayaffectitslevelofcognitivedemand.

ResponsesthatConnecttoStudents’PriorKnowledgeStudieshavehighlightedtheimportantrolethatpriorknowledgeplaysin

students’learning(e.g.Piaget,1952,1962;Rittle‐Johnson,2009,Rittle‐Johnson&

Koedinger,2005;Bransford,Brown,&Cocking,1999).Teachersmaymodifythe

problem,intermsofcognitivedemand,orevenabandontheactivityiftheyfindthe

taskinappropriate(Stein,Grover,&Henningsen,1996).Teacher’sresponsewith

examplesthatconnectstudents’thinkingwiththeirpriorknowledgecangive

studentsusefulstrategiesandskillsinapproachingtheirpresenttask.These

41

examplesmayrelateconceptscoveredinthepastwiththecurrentworkstudents

aredoing.Teachers’useofanalogiesisanotherstrategythattargetsconnecting

students’currentstrugglewithelementsoftheirpriorknowledge(Richland,

Holyoak,&Stigler,2004).Analogyresponsecanhelpstudentsrelatetheircurrent

situationwitharelatedconceptthatthestudentknowsbutdidnotthinktoconnect

duetosurfacedifferences.Theanalogybuildsonstudents’priorknowledgeand

movesthemforwardintheirsense‐makingofthenewconcept.Forexample,

supposestudentsareaddingtwoalgebraicfractions, 1x2 yand 1

xy2,andareunclear

whattodo.Ateachermayremindthestudentsaboutworkingwithnumerical

fractionswithunlikedenominatorssuchas 112 and 118 .Theintentoftheanalogy

thenistoencouragestudentstousestrategiestheyalreadyunderstandandapply

thesethentothenewideaswithwhichtheyarestruggling.Theteacher’suseofan

analogyhelpsstudentsconnectandextendtheirpriorknowledgeaswellasconnect

proceduralusagetomathematicalconceptssuchastheadditionoffractions

(Richland,Holyoak,&Stigler,2004).

Additionalsupportandinformationforconceptsorproceduresmaybe

neededwhenstudentsstrugglewithmakingaconnectionsbetweensuchexamples

asfindingtheleastcommondenominatorforalgebraicproblemsandnumerical

problems.Teacher’sresponsewithanappropriatevisualrepresentationcanserve

toextendamethodusedforthenumericaltasktothealgebraictask.Forexample,

considerthecaseoffindingaleastcommonmultiplefor12and18intheexample

42

above.AVenndiagramcanbeusedtorepresenttwosetscontainingfactorsforthe

numbers12and18withthecommonfactorsof2and3intheintersectionofthe

twosets.Thisrepresentationgivesavisualizationofhowthefactorsof12and18

aredistributedandshared.Ingeneral,Kaput(2001)notedthatusingavisual

representationgivesstudentsandteachersadditionalwaystopresent,share,and

storethemathematicalobjectsandrelationshipsbeyondverbalandsymbolic

representations.Byencouragingstudentstothinkofvisualmodels,teachershavea

moreinformedwayofrespondingandguidingstudentsastheysometimestakea

“zig‐zagroute”towardunderstanding(Lakatos,1976).

ResponsesthatClarifytheStudentStruggleAnothertypeofresponsereportedintheliteratureistoaskthestudentto

restatetheproblemfortheteacherortosharetheproblemorquestionthestudent

ishavingtotheclass.Thisdiscoursestrategyisintendedtobringstudentsinto

intellectualsocializationandmaintaintheirthinkingandwaysofactingsothat

otherparticipantsinthelearningcommunitycanjointlyowntheirstruggle.Askew

&Wiliam(1995)arguedthataprocessofcooperativelyfiguringthingsout

determineswhatcanbesaidandunderstoodbybothteacherandstudents.A

productiveresolutiontounderstandingcanresultfromastruggleforshared

meaning.

O’ConnorandMichaels(1993)reportedteachers’useofrevoicingasatool

thatcanhelpstudentsclarifysolutionsorproblems,remindthemofaconnectionto

43

priorknowledge,animatetheparticipantsintheinteraction,alignthetask

expectation,andshareinreformulatingorreframingtheproblem(p.328).

Revoicingandingeneralrebroadcastingthestrugglehasthepotentialtobean

effectiveteacherresponsetoproductivestruggle,particularlywhenstudentsareat

animpasseandareunabletomakeprogresswithaproblem.

ResponsesthatQuestionStudents’ThinkingRespondingtostudentsstrugglebyaskingquestionsservesvarious

purposes.Aspartofadiscourseinteractionbetweenteacherandstudents,

questionscangivedirectiontostudents’thinkingandopportunitiesforstudentsto

organizeideasastheyengagewithatask(Sorto,McCabe,Warshauer,&Warshauer,

2009).Whenastudentexhibitsstruggle,questionsthatelicitareasonedguess

ratherthana“savage”guess(Polya,1945)couldinformtheteacherofthestudent’s

difficulty.Questions,particularlywhencarefullysequencedtodevelopandbuildon

students’ideas,canthusservetoassessthestudents’thinkingandsupportand

directthem(Cazden,2001).However,questionscanalsoservetoreducecognitive

demandiftheemphasisisplacedmerelyonrightorwronganswersorfactualrecall.

Questionscandirectstudentstorestatetheirproblem,clarifyandarticulate

theirmeaningorrestructuretheiremergingideas.Respondingwith“whydoyou

thinkthat?;“whatdidyoudothere?”;or“howdidyougeta5here?”mayserveto

connectstudents’workwiththeirthinkingwhilerefocusingstudentsonimportant

mathematicalpointsthattheymayhavemissed(Anghileri,2006).Havingstudents

44

verbalizetheirthinkingcanhelpthemdevelopwaysofmathematically

communicatingandexplainingtoothersaswellasrevealpossiblemisconceptions

thatwouldhaveotherwisegoneunnoticed(deBock,Verschaffel,&Janssens,2002;

Forman&Ansell,2002;O’Connor&Michael,1993,1996).Inaddition,bycarefully

questioningandlisteningtoaspectsofstudents’responses,theteachercanmake

carefulselectionsandbuilduponstudents’ideasandthinking.

Thinkingaboutandairingemergingideasinarelativelyrisk‐free

environmentgivestudentsopportunitiesnotonlytoclarifytheirthinkingbutalso

addcoherencetotheirthinkingthroughtheactofsaying(Wells,1999).Whilea

teachermayrespondwithevaluativecommentsasinatypicaldiscoursepattern

suchasInitiation‐Response‐Evaluation,withholdinganevaluativecommentand

respondingwithafollow‐upsimilartothereflectivetossreferredtobyvanZeeand

Minstrell(1997)encouragedstudentstoreflectonanddevisewaystoovercome

theirstruggle.Inherdissertationresearch,Pierson(2008)focusedonexploring

discoursepatternsofmiddleschoolmathematicsstudentsandteachersthat

incorporatedasfollow‐upanexpectationor“prospectiveness”ofstudentsto

respond,muchlikeareflectivetoss,ratherthananevaluativecomment.Theuseof

probingquestionsandquestionsthatdemandedintellectualworkresultedina

moreproductiveexchangeandincreasedstudentlearningthanthosequestionsthat

didnot(Pierson,2008).Follow‐upquestionsthatrequireastudent‐generated

explanationcanstimulatethestudents’priorknowledgewhileconnectingtonew

45

conceptsthatmustbeassimilatedintothestudents’existingschema(Piaget,1952,

1962).AsWebb(1991)noted,“thiscognitiverestructuringmayhelptheexplainer

tounderstandthematerialbetter,aswellashelphimorherrecognizegapsin

understanding”(p.368).

Questionsthatprobestudents’thinkingormisunderstandingcanprovide

studentsadditionalinformationthathelpsdirecttheirthinking.Inacasestudyby

WilliamsandBaxter(1996),theteachers’andstudents’useofprobingquestionsin

adiscourse‐orientedclassroomproductivelysupportedstudentsthroughepisodes

ofconfusionandsense‐making.Onefindingfromthisstudy,however,suggeststhat

thequestionsanddiscoursemustbemeaningfulinorderforthestudentstolearn

fromtheinteraction.

Questionscanalsostimulateconsiderationofotherpointsofviewby

redirectingthequestiontootherstudents.Theresponseoftheteachershouldbe

relativetothedifficultiesthestudentsarehavingandthekindsandsequencesof

questionsforwhichtheyseekclarification,description,explanation,justification,

interpretation,andreason(vanZee&Mistrell,1997).Whatisimportantthroughout

thisprocessofrespondingtostudents’struggleisthenecessaryencouragementand

supportthathelpstudentscontinuetoengageinthetask,seekaresolution,andnot

giveup.Oneshouldnotethatstudentbeliefsaboutself‐efficacy(Bandura,1997;

Pajares,1996),thatistheperceptionthatonehasthecapacitytoachieveasetgoal,

46

playsaroleinthepersistencewithwhichastudentiswillingtostruggle,asnotedin

thefindingsregardingproblemsolvingbyPajares&Miller(1994).

ResponsesthatBuildStudentAgencyTheestablishedteacher‐studentrelationshipplaysanimportantroleinwhat

studentscometovalueintheirinteractionswiththeteacher.Statementssuchas

“you’reontherighttrack”canpromptastudent’ssenseofagencyandconfidenceto

continueorrevisetheirideas(Doerr,2006):however,aresponsesuchas“you’re

wayoff”servestorejectthestudent’seffortasawholewithoutsalvagingany

portionofit(vanZee&Minstrell,1997;O’Connor&Michaels,1996).Theformer

responsecangivestudentssomesenseofcompetenceandprovidesmotivationfor

persistingintheireffortwhilethelatterprovideslittleencouragementfortheir

effort.Similarly,aresponseof“nicejob”positionsastudent’scontributionas

competent.Teacherresponsescancapitalizeonincorrectanswersasimportant

contributionsthatacknowledgethestudentsascompetentandfurnishesinsightto

understanding(Gresalfi,Martin,Hand,&Greeno,2009).

Studiessuggestthatmotivationisanaffectivefactorthatplaysacriticalpart

inhowstudentsandteachersengageandinteractmeaningfullyintasks(Ames,

1992;Dweck,1986).Gresalfietal.(2009)refertotheconstructionofstudent

competenceandagencyasavaluablepartofhowstudentstakeupopportunitiesto

participateandlearnintheclassroom.Strugglecandissipateunproductivelyif

studentsdisengagefromtheirtaskoractivity.Therefore,thestudent‐teacher

47

interactionsinaclassroomenvironmentcanservetocontributeorhinderstudents’

willingnesstopersistandstruggle(Ecclesetal.,1993)aswellastoconstructtheir

senseofagencyinaproductivestruggle(Gresalfietal.,2009).

Forexample,thewaymistakesarehandledinstudent‐teacherinteractions

caninfluencestudents’motivationandlearningperformance(Ames&Archer,

1988).Whilebehavioristtheoriesoflearningviewmistakesasobstaclesfor

learningandabehaviortobeavoided(Skinner,1958),aconstructivisttheoryviews

mistakesasatoolforlearningandasopportunitiestofacilitatestudents’meta‐

cognitiveawareness(Palincsar&Brown,1984).

Researchhasshownthatexposinganddiscussingerrorsandmisconceptions

improveslearning(Borasi,1994).Eggleton&Moldavan(2001)notethatbyhelping

studentsconfronttheirerrorsandresolvetheincongruity,themistakescanbeseen

asasourceoflearningandsense‐making.Participationintheprocessof“doing

mathematics”andthewillingnesstouseerrorsaslearningopportunitiesrather

thanobstaclestomakingsenseofmathematicalideasleadstoconceptuallearning

intheclassroom.Whenteacherresponsescanchangestudent’sstatementof“Idon’t

getit”toastatementof“Idon’tgetityet”thestruggleshowssignsofproductively

movingthestudentforwardinhisorherengagementwiththemathematics

(Eggleton&Moldavan,2001).

Ateacherresponsethat:(1)allowsmoretimeontask,(2)acknowledgesthe

students’effortandcompetenceparticularlyinthefaceofdifficulty,and(3)

48

increasesthequalityofengagementwithoutloweringthecognitivedemand,are

moreapttoencouragethestudenttopersistdespitethestudentstruggles(Ames,

1992;Anderman&Maehr,1994).Atthesametime,thestudents’self‐theorycan

interactwiththeteachers’supportstructuresandaffectthemotivationthestudents

bringtotheirtaskengagement(Weinert&Kluwe,1987;Sullivan,Tobias,&

McDonough,2006).Aresponsethatemphasizescompetition,socialcomparison,

andabilityself‐assessment(Ecclesetal,1993)reflectsanorientationthatstudies

suggestdirectsstudentstooptforeasiertasksorgiveuptoavoidfailure(Ames,

1992;Dweck&Leggett,1988;Harter,1981).Therefore,teachersmustcomplement

theabovecomponentswithappropriateattributionstoenablestudentstoconfront

possiblefailuresinthefuture.Whenteachersexpressthestudents’struggleas

naturalinsolvingproblemsandtheireffortconstructive,studentsgainintrinsic

support.Studentsmaintainengagementandpersistencewhenteachers’responses

acknowledgestudents’competence,effortandinvolvementintheintellectualwork

demandedofthetasksanddonotfocusonjust“therightanswer”totheproblemor

task(Holt,1982;Dweck,2006,1986).

Suchateachers’stancecanencouragestudentsto“wanttosucceedonthis

task”(Ecclesetal.,1993,p.564)fortheirownlearningandnotforhowtheyare

perceivedbyothersintermsofsuccessorfailure(Dweck,2006;Holt,1982).When

studentsarestrugglingtomakesenseofaproblem,timeaffordedcanmakethe

differencebetweenstrugglethatisproductiveandstrugglethatisnot.Important

49

tooisthelengthanddepthoftheresponseusedbytheteacher(Pierson,2008;

Maloch,2002).Therushforquickanswersattimesliketheseareadetrimentto

allowingstudentstheopportunitytoclarifyandarticulatetheiremergingideas,

addressgapsintheirunderstanding,andlistentootherviewpoints(Kawanaka,

Stigler&Hiebert,1999).

FrameworkforProductiveStruggle

InowextendthemodelofTasksandStruggleproposedearlierand

incorporatethenatureandpatternsofTeacherResponses.Iwilltaketheteacher

responsedataandanalyzethemusingthefollowingthreedimensionsofhowthe

teacher’sresponses(1)maintainthecognitivedemandofthetask,(2)respond

directlytothestudents’struggles,and(3)buildonstudents’thinking.Research

abovesuggeststheseelementsareimportantinhowproductivelytheinteraction

canberesolved.

Figure2.1:PreliminaryStruggleandResponseFrameworkinTaskContext

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50

Asanexploratorystudy,myproposedframeworkwasbasedonthe

literatureandmypreliminaryobservationsfromclassroomvisits.Myresearchgoal

andthepurposeofmystudywastogatherdatafrommyfieldobservationsthat

wouldrevealwithgreaterclaritythenatureofteacherresponsesthatsupportthe

kindsandpatternsofstudents’strugglethatoccurwhilestudentsareengagedin

mathematicalactivitiesandtofurtherrefinemyframework.

Inanyclassroom,itisverypossiblethatstudentswillshowsignsof

strugglingwithagiventaskatanystageoftaskenactment.Iproposedtoclassify

students’struggleinthegeneralcategoriesof“whattodo?”“howtodoit?”and

“whydoesitmakesense?”thatIrefinedfrommyresultsandnotedthelevelof

cognitivedemandthetaskimposedonthestudentsastheyenactedthem.Ithen

lookedforthoseteachers’responsesthatappearedtobeproductiveinstudents’

understandingandengagementaswellasdocumentedthosethatappeared

unproductive.

SummaryMystudywillfocusonthefollowingresearchquestions:

1. Whatarethekindsandpatternsofstudents’strugglethatoccurwhilestudents

areengagedinmathematicalactivitiesthatarevisibletotheteacherand/or

apparenttothestudentinmiddleschoolmathematicsclassrooms?

51


mathematicalactivitiesintheclassroom?Whatkindsofresponsesappeartobe

productiveinstudents’understandingandengagement?

Mystudydocumentedthephenomenonofstudents’productivestruggleasit

occurredinmiddleschoolmathematicsclassroomsasaprocessthatwasfirst

observedwhenstudentsexpressedformsofperplexity,doubt,uncertainty,or

conflictwhileengagedinworkingonatask,activity,orproblem.Whatthenensued

wastheinteractionofstudentsandteachertoaddressthestruggleproductively(or

not)throughactionsofthetypesindicatedinFigure2.1.Inthosecaseswherethe

strugglesweredirectedproductivelyasaresultoftheseinteractionsandresponses,

thestudentswouldproceedto

• makesenseoftheproblemstatementandunderstanditsmeaningand

goal:or

• organizeconceptsordevelopandrefinestrategiestowardssolving

theproblemorexecutingthetask

inordertomovetheirunderstanding,reasoning,andsense‐makingofthe

mathematicsforwardinaccomplishingthegoalofthetask.

Inaddition,mystudyidentifiedthelevelofcognitivedemandmadebythe

mathematicaltasks,activities,orinstructionwithinwhichstudents’strugglewas

externalizedandmadepublicintheclassroom.Inthatcontext,Iobservedtheways

teacherresponsesprovidedguidance,motivation,andadditionalinstructionto

52

supportthestudents’struggle.Theinteractionofteachingandlearningthat

occurredatthesesitesofstrugglewentindirectionsthatwereproductiveandat

othertimesnot.Ifstudents’engagementinproductivestruggleofmathematical

conceptscanindeeddeepenstudents’conceptualunderstanding,thenexamining

thoseinteractionsoflearningandteachingthatmanageandfacilitatestudent

strugglesproductivelycaninformeducatorsofitsnatureandvalueforteachingand

learning.

53

Chapter3:Methodology

Myresearchisanexploratorycasestudyusingembeddedmultiplecasesin

ordertostudytheroleofproductivestruggleinlearningandteachingmathematics.

Specifically,theresearchquestionsIaddressinmystudyare:







Mygoalwastostudyclassroominteractionsinnaturalisticsettingsand

documenthowteacherssupportstudentswhoshowsignsofstruggleinlearning

mathematicswhileengagedinsomemathematicalactivity.Usinganembeddedcase

studymethodology(Yin,2009)withinstructionalepisodesasunitofanalysiswithin

thelargerunitofteachers,Iidentifiedanddescribedthenatureofthestudent

strugglesandtheinstructionalpracticesofteachersthatsupported,guidedordidn’t

guidethestudents’senseandmeaning‐makingofthemathematicaltasksinthe

lessonepisodes.Iusedmyfieldnotes,teacherandstudentinterviews,andvideo

and/oraudio‐recordedclassroomlessonstodescribeandanalyzetheinteractions

andpracticesteachersusethatfocusonstudents’productivestruggle.Theintentof

54

thisstudyistocontributetothisunderstandingbyexaminingandidentifyingthose

aspectsoftheteachingandlearninginteractionskeytoproductivestruggle.

PARTICIPANTSTheparticipantswere6thand7thgrademiddleschoolstudentsandtheir

teachersfromthreemiddleschoolslocatedinmid‐sizeTexascities.Theteachers

taughtthestudentsusingthesamemathematicstextbook,MathematicsExploration

part2(ME2)(McCabe,Warshauer,&Warshauer,2009)astheirprimarytextduring

the2009‐2010schoolyear.Amongothergoals,thetextbookwaswrittento

encourageteacherstoactivelyengagestudentsinmathematicalinquiry.The

teachershadalsoreceivedongoingprofessionaldevelopmentthroughouttheschool

yearbytheauthorsincludingtheresearcheronthetextbookimplementation.A

largerpilotprojecthad16teachersutilizingtheME2textbook.Fromthatpoolof

teachers,Iinvitedtwoteacherstoparticipateinmyresearchfromeachofthree

middleschoolsites,allsixofwhichagreed.Thisselectionwasnotrandombut

basedonpriorclassroomobservationsIhadmadeoffourofthesixpilotteachers.I

hadnotedtheteacher‐studentinteractionsthatweretakingplaceinthesefour

classroomsandhowstudentswereencouragedtoengageinclassroomdiscourse

anddevelopandexpresstheirideasduringmathematicalactivity.Ihad

correspondedwiththeothertwounobservedteachers.Theyhadsharedreflections

oftheirlessonsduringtheyearthatsuggestedhowtheyvaluedencouraging

55

students’engagementabouttasks,particularlychallengingonesinwhich“figuring

thingsout”wasimportant.

Theteachersfromtwoofthesitestaught7thgradestudentsandtheother

twoteachersfromthethirdsite,amagnetmiddleschool,taught6thgradestudents.

InTable3.1,thecharacteristicsandeducationalinformationregardingtheteachers

aregiven

Table3.1: CharacteristicsofTeacherParticipants

Teachers Gender Ethnicity Certification

Yearsteachingatthisgradelevel

Totalyearsteaching

Ms.Norris(site1)

Female White Grades:6‐12 14yearsin7thgrade

18years

Ms.Torres(site1)

Female Hispanic Grades:6‐12 19yearsin7thand8thgrades

19years

Ms.George(site2)

Female White Grades:EC‐8 2yearsin7thgrade

12years

Mr.Baker(site2)

Male White Grades:5‐8 2yearsin7thand8thgrades

2years

Ms.Harris(site3)

Female White Grades:6‐12 6yearsin6thgrade

8years

Ms.Fine(site3)

Female White Grades:4‐8 2yearsin6thand7thgrades

2years

56

PROCEDURE

DataCollectionIobservedeachteacherteachingsixtoeightclassesinaone‐weekperiod

witheachclassrangingfrom60minutesto90minutes.Theobservationswere

carriedoutinMay,2010andthefrequency,duration,andscheduleinformationare

showninTable3.2below.TheschoolforthelasttwoteachersusedanA/B

schedule.Ithereforeobservedtwooftheclassesontwoseparatedaysbutobserved

twoofMs.Harris’classesandoneofMs.Fine’sclassesonlyonce.

Table3.2: Observedclassfrequencyandhours

Teachers(site#)

#classesobservedperteacher

#timeseachclassobserved

Durationofeachclass

Totalhoursofobservation

Totalnumberofstudents

Ms.Norris(#1)

2 3 1hour 6hours 56

Ms.Torres(#1)

2 3 1hour 6hours 54

Ms.George(#2)

2 4 1.5hours 12hours 47

Mr.Baker(#2)

2 4 1.5hours 12hours 29

Ms.Harris(#3)

4 1or2 1.5hours 9hours 82

Ms.Fine(#3)

3 1or2 1.5hours 7.5hours 59

Iobserved39classsessionsamongthesixteachersand327studentsfora

totalof52.5observationhours.Theclasssizeofthe15differentclassesranged

57

from13to29studentswithsomevariationsinclasssizewhenabsencesoccurred

duringtheobservationperiod.Theaverageclasssizewasapproximately18

studentswithamedianandmodeof22students.

Table3.3providesstudentdemographicinformationforthethreeschools

andreflectsthestudentpopulationfortheparticularclassesthatIobserved.

Table3.3: StudentDemographics*

Site#

School White Hispanic Other(AfricanAmerican,Asian,NativeAmerican)

EconomicallyDisadvantaged

LimitedEnglishProficiency

#1 Mid‐sizecitysouthernTexas777studentsGrades6th‐8th

6.7% 89.7% 3.6% 52.9% 6.7%

#2 Mid‐sizecitywestTexas687studentsGrades7th&8th

33.5% 55.9% 10.6% 53.6% 6%

#3 Mid‐sizecitycentralTexas800StudentsGrades6th‐8th

56% 23%

21%

20% 0%

(*Fromhttp://ritter.tea.state.tx.us/perfreport/src/2010/campus.srch.htmlandschooldirectoratsite#3)

58

Observationsconsistedofvideofootageofeachteacher’smathematicsclass

usingonestationarycamerafocusedontheclassroomandanothermobilecamera

thatIusedtocaptureinteractionswhenteachersrespondedtostudents’struggle.

Thepurposeofusingthestationarycamerawastocapturetheoverallnatureofthe

classroomintermsofactivities,classroomengagement,andactionstakingplacein

theclassroomsimultaneouslywithaparticularstudent’sstruggle,whichmightnot

becapturedonthemobilecamerathatfocusedonthisparticularteacher‐student

interaction.Ialsokeptfieldnotesoftheclassroomactivitywhennotusingthe

videocamera,andwrotereflectionnotesoftheclassroomobservationsaftereach

class.

Attheendofmosttasksoractivities,thestudentsfilledoutaresponsesheet

indicatingtheirperceptionofthedifficultylevelofthetask(SeeAppendixE).Some

classesranoutoftimeandthesurveyscouldnotbeadministered.Iobservedthe

actionsofthestudentsduringclass,particularlythosethatshowedsignsofstruggle

andthennotedtheirtasksurveyresponsestodeterminetheirperceptionofthe

task.Forexample,astudentmayhaveindicatedataskasveryhardonthesurvey

butshowednosignsofstruggleinclass.Anotherstudentmayhaveconsistently

viewedataskaseasybutshowedsignsofstruggle.Thedatagatheredwasusedto

comparethecognitivelevelofthetasksasdesignedtothecognitivelevelofthetask

asperceivedbythestudent.Inaddition,thesurveyresponsescouldindicatesome

patternthatrelatesstudents’perceptionoftheirtasktotheirstrugglebehaviorin

59

class.Ihypothesizedthatthosestudentswhoviewedataskasdifficultwouldbe

thosewhowouldexhibitsomeformofstruggleinenactingthetask.

Pre‐andpost‐projectinterviewsofeachparticipatingteacherwere

audiotapedandtranscribed.Thepurposeofthesemi‐structuredpre‐interviews

wastolearnwhataretheteachers’viewsofmathematicslearningandhowstudents

cometolearnmathematics,howtheyviewstrugglingstudents,ideasofwhat

students’strugglelookslike,howtheygenerallymanagethestrugglewhenitoccurs

intheclassroom,andwhytheychoosethosekindsofactions.Bysemi‐structured,I

meanthatkeyquestionswereneitherspecificallyformattednorsequencedand

followedtheparticipantsresponsesastheyprogressedinanopen‐endedmanner

(Fontana&Frey,2005).Iaskedtheteacherstothinkofsomeexamplesofhow

studentsmightstruggle,whattheywoulddoinresponse,andhowtheactionhelps

students(Fennema,Carpenter,Franke,Levi,Jacobs,&Empson,1996)(See

AppendixA).Thisgavemeateacher’sperspectiveofwhattobelookingfor,as

studentsappearedtostruggleinclass.

Thepurposeofthesemi‐structuredpost‐interviewswastoaskteachersto

elaborate,explain,discuss,andreflectonwhathappenedduringspecificepisodesof

student‐teacherinteractionswherestrugglewasvisible,andwhytheychosetheir

actions.(SeeAppendixB).Asaformoftriangulation,thiswastoclarifyand

reconciletheintentoftheteachers’actionswithmyinterpretationofwhatI

observed.Theteachers’explanationsoftheseclassroomsnapshotsinformwhat

60

theyvalueandwhatunderliestheresponsestheymake.Ialsometbrieflywiththe

teachersaftereachclasstodebriefandfollow‐uponanyquestionsIhadregarding

theepisodesjustobservedwhiletheinteractionswerestillfreshintheteachers’

andresearcher’sminds(SeeAppendixC).

Finally,Isingledoutoneortwotargetstudentswhoexhibitedstruggle

duringtheclassroomactivitiesandconductedbriefinterviewsimmediatelyafter

class.Thecriteriaforchoosingthetargetstudentsincludedvisiblestruggle,

durationofengagementinthestruggleoveratask,interestinginteractionwiththe

teacheroranotherstudent,andsomeindicationthattheirstrugglewasproductive.

Theinterviewswereintendedtofindouthowthestudentsweredealingwiththeir

struggle,howtheirclassroominteractionsfacilitatedtheirthinkingabouttheir

struggle,andwhattheyfelttheywerelearningandunderstanding(SeeAppendix

D).Thiswasagainintendedtocheckmyobservationsandinterpretationswith

whatthestudentsreportedabouttheirexperienceswiththeirstruggle.I

documentedanystudentworkontheboardoronpaperthatillustratedstudents’

struggleasadditionaldataforpurposesoftriangulation(Cohen,Manion,&

Morrison,2000).

DataAnalysisAsanexploratorycasestudy,thegoalofmydataanalysiswastoidentify,

examine,anddescribethenatureandkindsofstudents’strugglethatoccurred

61

whenstudentswereengagedinmathematicalactivityandthenatureandkindsof

teacherresponsesthatdirectedstudents’struggleproductively(orunproductively).

Iviewedallthevideofootageandcreatedanexcerptfileofvideoclipsof

instructionalepisodesguidedbyErickson’s(1992)methodsforanalyzingvideo

data.Aninstructionalepisodeforthepurposesofmystudyconsistedofa

classroominteractionaboutamathematicaltaskthatwasinitiatedbyastudent

strugglethatwasinsomewayvisibletoateacheroranotherstudentwhether

voiced,gestured,orwritten.Ifollowedthesequenceofmovesinresponsetothe

studentstruggle,whichinmostcaseswereteacherresponses.Insomecases,a

discussionamongorbetweenstudentsensuedintheinteraction.Anepisode

conclusionwasmarkedinseveraldifferentways:(1)thestudentacknowledgesby

wordoractionunderstandingorisabletocompletehis/hertask;(2)thestudent

overcomesahurdleorimpasseandcontinuestomoveoninattemptinghis/her

task;(3)thestudentcontinuestostrugglebuttheteacherhasmovedon;or(4)

thereisashiftbytheteachertoadifferenttaskwithnoresolutiongivenbythe

studentnordemandedbytheteacher.

Teacherandstudentinterviewsaftereachclassweretranscribedandusedto

provideadditionalcorroborationandexplanationoftheobservedphenomenonof

productivestruggleintheclassroomsandtotriangulatetheobservationdata(Yin,

2009).

62

Thetranscriptsoftheclassobservationsandinterviewswerecodedusing

theopen‐codingprocess(Strauss&Corbin,1990)toidentifyandanalyze(1)the

kindsofstrugglethatoccurred,(2)thelevelofcognitivedemandwithinwhicheach

struggleoccurredand(3)thenatureandkindofresponseseachteachermadetothe

students’struggle.Idescribeeachcodingingreaterdetailbelow.

CodingStruggle Onestudentinitiatedanepisodewithanexternalizationofstrugglemade

visibletotheteacherorpossiblytoanotherstudent.Iinitiallytriedtocapture

elementsofeachofthe186episodesastowhatwasthenatureofthestrugglethat

wasbeingvoiced.Ifoundthemesemergingwhereinstudentstriedtodetermine

whattodowiththetaskortheproblem.Otherswonderedhowtoproceedwith

somesteptheycouldnotcarryoutandothersstruggledwithwhatappearedtobe

ananswerbutcouldnotexplainwhytheiranswerwasorwasn’tcorrect.Afterthis

firstiteration,itappearedtherewereoverlapsinthecodesandlackofclarityabout

theclassificationsthatfailedtoalignwithhowstudentsvoicedtheirstruggle.For

example,studentswouldsay,“Idon’tknowwhattodo.”Otherswouldsay,“Idon’t

knowhowtodotheproblem.”ThoughtheyusethekeyclassifyingwordsthatIhad

considereddistinct,namelythe“what”andthe“how”,itseemedinlookingattheir

workandthestageatwhichtheyvoicedtheirstrugglethatthenatureofthe

struggleappearedessentiallythesame.Ithusconsideredexaminingthestruggles

asproceduralversusconceptualinnature,buttheseclassificationsweretoobroad

63

foranalysis.Ididanotheriterationoftheepisodesusingcodesinformedby

literatureonproblemsolving(Schoenfeld,1987;Kulm&Bussmann,1980;Polya,

1957;de‐Hoyos,Gray,&Simpson,2004).Iexaminedstrugglesoccurringatvarious

stagesoftheprocessofsolvingproblems:formulation,implementation,andsense‐

makingandverification.Thisclassificationhadpromise,butthesense‐making

strugglecouldhaveoccurredduringformulationoratimplementation,andIfelt

compelledtoexaminemydataonceagain.

Whilealltheabovecategorizationscouldbejustifiablyused,Iwentbackto

myfindingsyetagainandconsideredwhatteachersmightseeasthestruggle.I

arrivedatthefourcategoriesreportedasfindingsinchapter4.

Teachersgenerallyhaveaveryshortspanoftimetorespondtostudent

actions(Kennedy,2005).Thiscouldincluderespondingtoastudentwhoindicates

struggleoveratask.Intheirownanalysisofastudent’sstruggle,teachersmost

likelyassessedandidentifiedwhatappearedmostprominentlyasatypeofstruggle.

CodingTasks:TaskDescriptions Thesixteacherswereaskedtoimplementlessonsfromasetofactivitiesthat

Isubmittedfortheirconsiderationpriortomyobservationdates.Idesignedthe

activitieswiththelevelofcognitivedemandinmindusingseveralsourcesthatIfelt

werealignedwiththecurriculumgoalsandvisionoftheMathExplorationPart2

(ME2)textbooktheteachersusedduringthegivenacademicyear(August2009

untilMay2010).Inordertoobservestudents’struggleacrossthespectrumof

64

learners,Ifeltthetasksneededtobechallengingbutaccessibleusingwhatwould

buildonstudents’priorknowledge,andappropriatelyalignwiththecontentand

pedagogythestudentswereaccustomedtoduringtheschoolyear.Thetasksalso

neededtobedesignedsothatstudents’workwouldbevisibletotheteachersin

orderforteacherstoformativelyassesstheirstudents’thinkingandseeevidenceof

possiblestudents’struggleevenwhenstudentswerenotinclinedtoindicatetheir

strugglesopenly.

KnowingthatIwouldbeobservingthegivensetofstudentsanywherefrom

oncetofourtimes,theactivitybookletcontainedfourdifferentactivitiesthatwere

brokendownintotaskssuitableforclassesthatrangedfrom60minutesto90

minutesinlength.Whilethetaskswithinanactivitybuiltoneachother,teachers

hadflexibilityinendingthelessonbeforeallthetaskswerecompleted(Iusethe

termactivityhereasasetoftasksfocusedaroundaparticularcontextandtaskasa

problemthatinvestigatedamathematicalaspectwithinthatcontext.)Threeofthe

fouractivitiesfocusedondevelopingadeeperconceptualunderstandingof

proportionalrelationships.Proportionalreasoningisaprimaryfocalpointacross

themiddleschoolband(TEA,2005,NCTM,2000,2005;Schielack,Charles,

Clements,etal,2006).Itisconsidered"thecapstoneofchildren'selementaryschool

arithmetic;...itisthecornerstoneforthemathematicsthatistofollow."(Lesh,Post,

&Behr,1988,p.94).Proportionalrelationshipsbuildonstudents’understandingof

fractionsasstudentsdeveloptheconceptsofratiosandratesinthecontextsof

65

numbers,algebraicreasoning,measurement,geometry,andprobability.Piaget

consideredproportionalreasoningtobeasignificantconceptualshiftfroma

concreteoperationallevelofthoughttoaformaloperationallevel.(Piaget&Beth,

1966).

ThefullsetofactivitiesisincludedinAppendixF.Abriefdescriptionand

intendedobjectiveoftheactivitiesfollow:

1. TheBarrelsofFunactivityfocusedondistinguishingbetweenadditive

andpercentdifferencesofliquidquantitiescontainedintwodifferent

sizedcontainers.Theobjectivewastoencouragestudentstoreadthe

taskproblemscarefullytodeterminewhatquantitieswerebeingasked,

useproportionalreasoningtosolvetheproblemsandtohaveagraphical

representationthatconnectedthenumericalvaluetoavisualmodel.All

6taskscontainedinthisactivitywereimplementedbyfiveofthesix

teachers.

2. TheBagsofMarblesactivityagainfocusedondevelopingstudents’

understandingofproportionalreasoning,thistimeinadiscrete

probabilitycontext.Allofthefivetasksinthisactivitywereimplemented

bythreeofthesixteachers.

3. TheTipsandSalesactivityincludedsixtasksthatusedpercents,algebraic

expressionsandequationstoexploreproportionalrelationshipsinretail

contextsoftaxes,tips,anddiscounts.Twoofthetasksrequired

66

determiningpopulationsizegivenapercent.Theobjectiveoftheactivity

wastohavestudentsexplorealgebraicwaysofexpressingquantities

withvariablesscaledbypercents.Fourofthesixteachersimplemented

fromtwotosixofthetasksinthisactivity.

4. Thefourthandfinalactivity,DetectingChange,wasimplementedbyonly

oneofthesixteachers.Thetwotasksinthisactivitywereintendedfor

studentstoobservechangeinageometricpatternthatcouldthenbe

articulated,generalized,andwrittenalgebraicallyandrepresented

graphically.Thereweremanypossiblepatternsthatcouldbeobserved

includingsomethatwerelinearandothersthatwerenon‐linear.

Inaddition,Ms.TorresusedexercisesfromtheME2probabilitysectionforher

classes.ThosetasksareincludedinAppendixG.

CodingTasks:ByLevelsofCognitiveDemand IusedtheMathematicsTasksFramework(Stein,Grover,&Henningsen,

1996)tocodetheenactedtasksthatservedasthecontextfortheinstructional

episodes.Theintendedandenactedtaskswereidentifiedasoneofthefourlevelsof

cognitivedemand:Level1‐Memorization;Level2‐Procedureswithoutconnections

toconceptsormeaning;Level3‐Procedureswithconnectionstoconceptsand

meaning;andLevel4‐Doingmathematics.Iusedtheteacher’sintendeddailylesson

tasksandidentifiedthelevelofcognitivedemandinthosetasksbelow.

67

Table3.4: Activity1:BarrelofFun

Supposewehavea48gallonrainbarrelcontaining24gallonsofwateranda5gallonwaterjugcontaining3gallonsofwater.Task IntendedLevelofCognitive

Demand1.1Whichcontainerhasmorewater?

3

1.2Whichcontainerissaidtobefuller?Explainyouranswer.

3

1.3Usethecoordinategridbelowtodrawapictureofthetwocontainersandtheirwaterlevel.Youmayleteachsquarerepresent1gallonandshadeinthepartrepresentingthewater.Doesitmatterwhatshapeyoumakethesecontainers?

3

1.4Howmanygallonsofwaterwouldneedtobeinthe5‐gallonjugsothatithasthesamefullnessasthe24gallonsinthe48‐gallonbarrel?

4

1.5Ifwedrainagallonofwaterfromeachcontainer,doesthischangeyouransweraboutwhichcontainerisfuller?Explain.

4

1.6Howmanymoregallonsofwaterdoweneedtocatchinthebarrelinordertohavethesamefullnessinthebarrelaswehaveinthejug?Explain.

4

Table3.5: Activity2:BagsofMarbles

Therearethreebagscontainingredandbluemarblesasindicatedbelow: Bag1hasatotalof100marblesofwhich75areredand25areblue. Bag2hasatotalof60marblesofwhich40areredand20areblue. Bag3hasatotalof125marblesofwhich100areredand25areblue.Task IntendedLevelofCognitive

Demand2.1Eachbagisshaken.Ifyouweretocloseyoureyes,reachintoabag,and

3

68

removeonemarble,whichbagwouldgiveyouthebestchanceofpickingabluemarble?Explainyouranswer.2.2Whichbaggivesyouthebestchanceofpickingaredmarble?Explainyouranswer

3

2.3HowcanyouchangeBag2tohavethesamechanceofgettingabluemarbleasBag1?Explainhowyoureachedthisconclusion.

4

2.4HowcanyouchangeBag2tohavethesamechanceofgettingabluemarbleasBag1ifBag2mustcontain60totalmarbles?

4

2.5ConsideronlyBags1and2.MakeanewbagofmarblessothatthisbaghasagreaterchanceofgettingabluemarblethanBag1butlessofachanceofgettingabluemarblethanBag2.Explainhowyouarrivedatthenumberofblueandredmarblesforyournewbag.

4

Table3.6: Activity3:TipsandSales*

Task IntendedLevelofCognitiveDemand

3.1Supposearestaurantbillis$X.Writeanexpressionforthetipon$Xusinga15%tiprate.Whatisthetotalamountyouwouldpaytherestaurant?

3

3.2Supposeagenerouscustomerusesa20%tiprateonabillof$X.Writeanexpressionforthetipon$Xusinga20%tiprate.Whatisthetotalamountthiscustomerpaystherestaurant?

3

3.3If40%ofagroupof35studentsparticipateinathletics,howmanyofthese35participateinathletics?

3

3.4AnothergrouphasNstudentsand40%ofthemparticipateinathletics.

3

69

WriteanexpressionusingNforthenumberofstudentswhoparticipateinathleticsfromthisgroup.3.5Writeanexpressionforthenumberofstudentswhodonotparticipateinathletics.

3

3.6Apairofpantsregularlycosts$40butisonsaleat25%offtheregularprice.Howmuchwillyoupayforthesesalespants,withoutcomputingtax?Explainhowyougotyouranswer.

3

3.7Ashirtregularlycosts$Sandisonsaleat25%offtheregularprice.Writeanexpression,usingS,fortheamountofdollarsdiscounted.Writeanexpressionthatrepresentshowmuchyouwillpay,disregardingtax.

3

3.8AnMP3playerisonsalefor$60aftera20%discount.Whatwastheoriginalprice?Whatwastheamountofthediscount?

4

*Iincludeonlythosetasksinthisactivitythatteachersimplemented.

Table3.7: Activity4:DetectingChange

Inthefigure,asthestepschange,whatalsochanges?Task IntendedLevelofCognitive

Demand4.1Describewhatyouobservechangesasthestepsincrease.Recordtheseobservations.

4

4.2Selectonechangethatyouobservedanddescribethechange.WhathappensinStep4?WhathappensinStep5?WhathappensinStep10?WhathappensinStepn,fornapositiveinteger?Useatable,graph,andanequationtodescribethechangesthatyounotice.

4

70

Ingeneral,thecognitivedemandforthefouractivitiesandallthetasks

exceptforfourofthe27implementedtaskswereatthehigher‐order,levels3and4.

Othertasksusedbyteachersconsistedofwarm‐upproblems,someofwhich

werepreparatorytypeproblemsfortheTexasAssessmentforKnowledgeandSkills

(TAKS)andnotnecessarilyconnectedtothelesson(e.g.findthesurfaceareaofa

circularcylinder,convertagivenamountofpesostodollarswithagivenconversion

rates,ormultiplytwomixedfractionalnumbers).OthersproblemswerenotTAKS

formatted,suchasaproblemtofindthesurfaceareaofarectangularprismwitha

circularcylindercutout(SeeAppendixH).Icodedtheseadditionaltasksas

implementedbytheteacherataleveltwocognitivedemandexceptfortheproblem

onthesurfacearea,whichwasatalevelthree.

CodingTeacherResponseInmythirditerativeexaminationoftheinstructionalepisodesfollowinga

roundofexcerptingepisodesandroundsofidentifyingstudents’struggles,I

characterizedtheinitialturnintheteacherresponsebythepreliminarycategoriesI

previouslyproposedinmyStruggle‐Responseframework:supplyinformation;

connecttostudents’priorknowledge;addressthestruggle;askguidingquestions;

askprobingquestions;provideencouragementandagency,andothersthatIhad

notaccountedfor,suchasconnecttostudentthinkingorevaluatework.The

teacherresponsesgenerallyelicitedsomeactiononthepartofthestudent,anda

sequenceofmovesofvaryinglengthsgenerallyfollowedbetweenstudentand

71

teacheroramongstudents.Theepisodethereforeconsistedofasequenceofthe

teacherresponsesfittingintothepreliminarycategoriesIhadcoded.

Inthenextiterationofexaminingtheepisodeinteractions,Ilookedforthe

overalldirectionandthrustoftheteacherresponses.Thiswasanattemptto

characterizetheintentoftheresponsesasawholeintheepisodeandnotjustbythe

teacher’sinitialresponsetothestudentstruggle.Ihadoriginallycodedeach

teacherresponsemovethatcomprisedthesequencesinanepisode.Forexample,

theinitialresponsebyateachermayhavebeentoconnecttothestudent’sthinking.

Suchamovecouldbefollowedbyrebroadcastingtotheclasstheproblemthatwas

causingthestudent’sstruggleandthenaskingguidingquestionstothestudentor

theclasswhilealsoprovidinginformationusefulforthetask.

Althoughaninitialteacherresponsetothestudentstrugglehascertain

characteristics,suchaselicitingstudentthinking,Iconcludedthatitisthesetof

responsesequencesintheinteractionthatprovidesdirectionandsupportforthe

students’struggle,whetherproductivelyornot.Therefore,itisintheseresponse

sequencesduringtheepisodesthatIbegantoobservethemesinthepatternsof

teacherintentandpurposeinhis/herresponses.Usingelementsofgrounded

theory(Strauss&Corbin,1990),Iidentifiedfeaturesofteacher’sresponsesthat

occurredinthe186episodes.Ifoundamethodofclassifyingthesepatternsalonga

continuum.Thisallowedmetoseearangeofcharacteristicsthatthenleadmeto

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identifyfourgeneralcategories(Glesne&Peshkin,1992;Strauss&Corbin,1990),

whichIdescribeindetailinthenextchapter.

Thefinalcodinginvolvedexaminingwherethestudentappearedtobe

headingintermsofhis/herresolutionofstruggleattheconclusionoftheepisode.I

calledthistheresolutiontothestudents’struggle.

CodingResolutionoftheStudents’StruggleTheepisodeisconsideredendedwhen:(1)thestudentacknowledges

understandingbywordoractionorisabletocompleteshis/hertask;(2)the

studentovercomesahurdleorimpasseandcontinuesattemptinghis/hertask;(3)

thestudentcontinuestostrugglebuttheteacherhasmovedon;or(4)thereisa

shiftbytheteachertoadifferenttaskwithnoresolutiongivenbythestudentnor

demandedbytheteacher.

Icoderesolutionsinthreecategories:productive,productiveatalowerlevel,

orunproductive.Inaproductiveresolution,characteristicsincludedsomeifnotall

ofthefollowing:thestudentsolvedtheproblemattheintendedlevelofcognitive

demand,explainedasolutiontoothers,connectedtoanalogousproblems,gave

justificationorreasontoasolution,expressedconfidenceinhis/herwork,

continuedtoworkatthesamelevelofcognitivedemand,orcorrectederrorsor

misconceptionusinghis/herthinkingashe/sheengagedincontinuingtoworkon

theproblemortask.

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Itis,however,possiblethatthestudentcontinuedwiththetaskbutthatthe

teacherorotherstudentsdidasignificantamountofthinkingforthestruggling

studentorthatthetaskwasalteredwithdecreasedcognitivedemand.Thistypeof

resolutiontostudent’sstruggleIclassifyasproductivebutatalowerlevelbecause

theintellectualeffortexpectedofthestudentindoingtheproblemortaskwas

removedorsimplifiedbytheteacherorbyanotherstudent.Thevalueofthe

students’struggleisdiminishedwiththestudentsnotbeinggiventheopportunity

torelyuponhis/herresourcestomakeconnectionsamongmathematicalconcepts.

However,thestudentmaynothavemadeanystridesinworkingthetaskwithout

thistypeofintervention.

Finally,anepisodecanendwiththestudentgivingup,continuingtobe

confused,and/orunabletofigurehowtodothetask,orunderstandwhysomething

works.Iclassifythisresolutionasanunproductivestruggle.Intime,thestudent

maycometoseeorovercomewhatmaynothavemadesenseattheendofthe

episode.However,forthepurposesofmystudy,thestruggleappeared

unproductive.

Inthenextchapter,Ireportthefindingsfrommyinvestigationusingthis

ProductiveStruggleFrameworktostructuremyanalysis.

TRUSTWORTHINESS Asanexploratorystudy,myinvestigationofstudentstruggleslackeda

predeterminedsetofcategoriesbywhichIcouldanalyzemydata.Literatureon

74

qualitativeresearch,casestudy,andgroundedtheoryprovidedguidanceasIbegan

mycodingandanalyzingofcollecteddata(Miles&Huberman,1994;Yin,2009;

Strauss&Corbin,1990)throughnumerousiterations,searchingforthemes,

patternsandexplanations.Whilethedataanalysisisaninterpretiveprocess,Itried

toguardagainsttheintroductionofsubjectivebiasandtomakeinterpretationsand

judgmentsbasedonsubstantiveevidenceandtotriangulatethedataforagreement

andconsensus(Creswell,2003;Lincoln&Guba,1985;Patton,1990).More

specifically,ItranscribedallthevideoobservationsIhadmadeandhadachanceto

gainasenseoftheoverallcorpusofcollecteddataagainfromwhichIwouldbegin

myanalysis.Inaddition,Iexaminedthepreandpostinterviews,student

interviews,anddebriefinginterviewsthatIhadtranscribedforme.Ialsostudied

myfieldnotesandexaminedthestudentsurveysoftaskdifficulty.Thesewereall

usedtoclassifystruggles,responses,andresolutionsthatIproposedfrommy

findings,aprocessthatwasbothgradualandrepetitive,numberingatleastthree

timeswiththestruggles,twicewiththeteacherresponses,andanotherthreetimes

fortheresolutions.

Theteachershadallusedatextbookco‐authoredbymycolleaguesandme

andhadworkedwithmeinprofessionaldevelopmentduringtheyear.Itherefore

knewtheteachersandrespectedthekindofteachingthatappearedtonaturally

includeopportunitiesforstudentstomakesenseofthemathematics.Itriednotto

influencetheirclassroomsetting,practices,orstudentsasmuchasastrangerwith

75

twocamerascouldpossiblyavoid.Alltheteachersintroducedmetotheirstudents

atthestartofmyobservationperiodandthestudentsforthemostpartseemedto

carryonwiththeirclassactivitiesdespitetheintrusivenessofhavingavideo

cameranearbytocapturetheirconversations.

TheactivitiesimplementedwereallfrommaterialIprovidedteachers,and

theirappropriatenessfor6thand7thgradeclasseswereconfirmedbyinputfrom

colleaguesinthemathematics/mathematicseducationdepartmentatmyuniversity

aswellasfromtheparticipatingteachers.

Irequestedtwoindependentreaders,oneamathematicianandonea

mathematicseducationgraduatestudenttotakeasampleof20episodesto

determinehowconsistentmycodeswerewiththeircodingofstruggles,responses,

andresolutions.Thiswastoremovebias,toconfirmthevalidityofmypropositions,

andtoreachalevelofconfirmabilityandconsistency(Mathison,1988;Lincoln&

Guba,1985).Wereached90%consistencyafterseveraldiscussionsand

refinementsofcodingandclassifications

Inordertodevelopgreatercredibilitytomyinterpretationoftheobserved

studentteacherinteractions,Irequestedateacherparticipanttoexamineepisodes

andtoprovidefeedbackonmyfindingsofstruggle,responses,andresolutions.

Afterreading12episodes,includingsomewithherstudents,herobservations

appeared83%consistentwithmycodingandanalysis.Theconversationwehad

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providedinsightintoherinterpretationsandgavemeadditionalconfirmationtomy

preliminaryanalysis.

Iaskedanothermathematicseducatorandmathematiciantoreadthefinal

documentforcommentsandfeedback.Theperspectiveofmystudyremainsan

interpretationofmyobservationsbutIattemptedtopresentthemwith

methodologicalrigorbasedontheliterature,asIunderstoodthem.

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Chapter4:Results

OVERVIEW

Inthischapter,Iwilldescribethenatureofthestrugglesthatstudentshadin

theclassroom,thewaysthatteachersresponded,andthenclassifythewaysthat

thesestruggleswereresolved.Whenstudentsworkedonmathematicaltasksinthe

contextofclassroomsettings,manyoftheminvariablyhadquestions,expressed

uncertainty,anddisplayedstrugglethattheyrevealedtotheirteacher,totheir

fellowclassmates,orsilentlythroughgesturesthatsuggesteddiscomfortordoubt.

Inmycollecteddata,consistingoftranscriptionsbothofvideofootageand

interviewswithteachersandstudents,Iidentified186casesofstruggle.

Thesestudentstrugglescametotheteachers’attentionthroughteachers’

actionsastheywalkedaroundtheclassroom,lookedoverstudentpapersandboard

work,listenedtostudentdiscussions,andrespondedtorequestsforindividualhelp.

Teacherresponsestothesestudentstruggleswerealsovariedastotheirextent,

apparentintent,andpurpose.Justasamini‐dramahasacontext,namelyaplotin

whichconflictortensionarises,adevelopment,andthenaresolutionandsome

conclusion,mystudyexaminedepisodesinwhichstudents’strugglesarosein

responsetoaselectedsetofmathematicaltasksandwasresolvedinsomeway.The

searchforaresolutionbecameajointeffortofstudentsandteachersasobservedin

theirclassroominteractions.Asuccessfulconclusiontoamini‐dramaisgenerallya

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resolutionofthetension.Asuccessfulresolutiontostudents’struggle,Icontend,is

notjusttorelievethetensionbyremovingitssource,buttorespondtothestudents’

struggleinawaythathelpsstudentsbetterunderstandthemathematicsinvolvedin

theirtask.Theproductiveaspectoftheinteractioniscontainedinthestruggle

itself,ifsupportedskillfullybytheteacher,todeepenstudents’mathematical

understandingandtoachievethelearninggoalsofthetask.

Iexaminedthedifferentwaysthatthesemini‐dramasintheclassroomswere

enactedandresolvedinordertobetterunderstandwhatelementsofteachingwere

atplaythatindicatedsupportofstudents’learning.

Myresultsareorganizedbythethreemaingoalsofmyresearch:

1. Todeterminethekindsofstudents’strugglesthatoccurredwhilestudentswere

engagedinmathematicaltasks,andtheirnature;

2. Todeterminethekindsandnatureofteachers’responsesmadeinthecontextof

theinteractionbetweenstudentsandteachersduringstudentengagementin

mathematicaltasks;and

3. Toclassifythewaysthattheseinteractionsresolved.

Inmyanalysis,fourmaintypesofstruggleemergedasstudentsengagedin

mathematicaltasks.Thestrugglescenteredaboutstudents’attemptsto:

1. Getstarted

2. Carryoutaprocess

3. Giveamathematicalexplanation

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4. Expressamisconceptionorerrors

Myfindingsshowedfourmainwaysthatteachersrespondedtostudent

strugglessituatedalongacontinuumthatincludestelling,directedguidance,probing

guidance,andaffordance.Finally,studentstruggleswereresolvedinthreetypesof

outcomes:productive,productiveatalowerlevel,andunproductive.

Thischapterwillfollowtheoutlinedevelopedabove.ThestructurethatI

developedwastheresultofmyclassroomobservations.AsIgothroughthis

framework,Iwillexplainthedifferentcategoriestogetherwithspecificexamples

thatillustrateeachcategoryfollowedbyadiscussionontheabovegoalsofmy

research.Ibeginwithabriefreportonhowthetaskswereimplementedduringmy

datacollectiontogivecontexttothestrugglesthatoccurred.

TASKSIMPLEMENTEDINTHECLASSROOMS Inmyobservations,teacherssetupthetaskswithreasonablefidelitytothe

suggestedteacherguideprovidedwiththestudentactivityset.Theteacherguide

includedasuggestedlessonsequencethatallowedstudentstimeforindividual

work,groupwork,andthenwholeclassdiscussion.Studentsforthemostpart

enactedthetasksinasimilarmanner,thoughthestructureoftheobservedclasses

wasnotidenticalintermsofseatingarrangements,lessonintroductionbyteacher,

andclasstimeallottedforthetasks.Thestudentsgenerallybeganbyworking

individuallyforaboutfiveminutespertask.Studentsthendiscussedtheirwork

withapartnerorwithasmallgroupofthreetofourstudentsforanotherfive

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minutesorsowhiletheteacherwalkedaroundtheseatedstudentsandlistened.

Somediscussionsweremoreanimatedthanothers,andatsometablesstudentshad

littletosharewitheachother.Teacherslistenedorengagedinaskingstudents

questions.Forexample,oneteacherwentaroundthetablesduringstudent

discussionsandstampedasheetofpaperoneachsmallgroup’stabletoindicate

thatthestudentsatthetablewereengagedindiscussingtheirproblemsandtheir

solutions.Somestudentsraisedtheirhandstogaintheteacher’sattentionandto

indicateaquestionorproblem.

Teachersalsoimplementedlowercognitivedemandtasksinwarm‐up

activitiesatthebeginningofsomeoftheirlessons.Thesetaskswereincludedinmy

analysis.Teachers’taskintroductionsrangedfromframingthetaskswithascenario

linkingtheideaofproportionalthinkingtonon‐mathematicalcontextssuchasthe

“BiggestLoser”show,tothosethatsimplyhadastudentreadeachtasktodetermine

ifthewordingwasclearforallthestudents.Thestudentsthenwentabouttheir

taskforthemostpartontheirown,thoughthevideoclipsfromthestationary

cameracaughtsomestudentstalkingamongstthemselvespresumablyaboutthe

task,thoughthisisnotconfirmed.

Afterattemptingtheproblemsontheirown,thestudentsdiscussedtheir

workasasmallgrouporasawholeclassatwhichtimedifferentquestionsand

strugglessurfacedthathadnotoccurredduringtheindividualworktime.

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STUDENTS’STRUGGLE

Descriptionandexamples Iusedthe186episodetranscriptionstoidentifyandanalyzepatternsof

studentstruggles.Ikeptinmindtheperspectiveoftheteachersastheyobserved

andinteractedwiththeirstudentsengagedinmathematicaltasks,namelywhatthey

wouldseeandhearfromtheirstudents.Iexaminedtheepisodesthroughatleast

threeiterations,consideringpossiblecodes,refiningthembyusinganopen‐coding

process(Strauss&Corbin,1990),andconferringwithtwoindependentreadersfor

inter‐raterreliabilityofmycodes,untilIreachedover90%agreement.Myfinal

codinggroupedthekindsofstudentstrugglesintothefollowing4types.Struggle

typesincludedstudentattemptsto:

1. Getstarted

2. Carryoutaprocess

3. Giveamathematicalexplanation

4. Expressmisconceptionsanderrors

InTable4.1below,Isummarizethecharacterizationsofthe4typesof

studentstrugglesobservedandthefrequencyoftheiroccurrences.

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Table4.1: KindsofStudentStrugglesandtheirPercentFrequencies

Kindofstruggles Descriptions Frequency%

(186total)1.Getstarted • Confusionaboutwhatthetaskis

asking• Claimforgettingtypeofproblem• Gestureuncertaintyandresignation• Noworkonpaper

24%

2.Carryoutaprocess

• Encounteranimpasse• Unabletoimplementaprocessfrom

aformulatedrepresentation• Unabletoimplementaprocessdue

toitsalgebraicnature• Unabletocarryoutanalgorithm• Forgetfactsorformula

33%

3.Giveamathematicalexplanation

• Justifytheirwork• Explainprocessbywhichananswer

isobtained• Givereasonsfortheirchoiceof

strategy• Expressuncertaintyandinabilityto

findwordstoexplain• Makesenseoftheirwork

30%

4.Expressmisconceptionanderrors

• Misconceptionrelatedtoprobability• Misconceptionrelatedtofractions• Misconceptionsrelatedto

proportions.

13%

Inthesectionthatfollows,Iwilldescribeeachstruggleingreaterdetail.

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1.GetStarted

44ofthe186or24%oftheobservedepisodesinvolvedstudentstrugglesat

thestartoftheirtasks.Thesestrugglesalloccurredasstudentswereattempting

tasksofhigher‐levelcognitivedemand.Intheiruncertaintyabouthowtoget

startedwithatask,studentsvoicedconfusionaboutwhatthetaskwasaskingthem

todo(“Ikindofunderstandit…butI’malittleconfused”);claimedtheydidn’t

rememberdoingproblemsofthistypethoughitappearedvaguelyfamiliartothem

(“Ihaveabsolutelynoidea….Idon’trememberthatfar”);calledforhelp(“Mr.Baker,

Ineedhelp.”);gestureduncertaintyandresignation(looks,thinks,sitsbackand

thensays,“Idon’tknow.”);orhadnoworkontheirpaper.

Students’strugglestogetstartedwiththeirtasksalignedtoissuesof

recognizing,analyzing,andunderstandingthegoalofthetaskandcomingupwitha

plantoachievethegoal.Forexample,astudentinMs.George’sclasswasuncertain

howtowriteanexpressionforthediscountedpriceforanitemcosting$Snowon

saleat25%off.Thestudent’scomments(S)expressuncertaintyinsortingthrough

andanalyzingtheexpressionsthatarerelevantfortheproblem.Sheisalsounable

toconnectthealgebraicproblemtoanumericalexampleshejustcompletedtosee

howthegoalsarerelated.Theteacherresponse(T)attemptstogetthestudent

startedwithwhatthestudentalreadyknowsandattemptstoconnectthatprocess

tothecurrenttask.

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1. S: Iknowtheamount.Ican’tfigureoutwhattodo.Iknowthatequals

that.Andthe$10(anearliernumericalexample)butIdon’tknow

like….(trailsoffinthought).

2. T: Okay,Tellmewhatyoudidwiththe$10(makingareferencetothe

earliertask).

3. S: Isubtracteditfromtheregularprice.

2.CarryOutaProcess

56ofthe186documentedstrugglesor30%representedstrugglesby

studentsattemptingtocarryoutaprocessinordertoachievethegoalofthetask.

Studentswhoencountereddifficultyincarryingoutaproceduredemonstratedor

voicedsomeplanforachievingthegoalofthetaskbutencounteredanimpasse.

Theseimpassestendedtorevolvearoundaninabilitytoimplementaprocess,some

moreroutinethanotherssuchassolvingforanunknowninaproportionor

convertingafractiontoapercent.Otherissuesincludedmistakesmadesuchthat

theprocessnolongermadesense;failuretocarryoutaprocedurethatseemed

moreconfoundingtostudentswhenthetaskwasalgebraicinnature;ordifficulty

recallingaformulaoritsuserelevanttothetasksuchasthesurfaceareaofa

cylinder.

Inoneexample,astudentappearedtohaveaplanbutreachedanimpasse

whenthenumberstotheproblembecamemoredifficult.Thetaskcomesfromthe

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BarrelofFunactivity,whereintask1.5thestudentswereaskedtocomparethe

fullnessoftwocontainers,namelythewaterjugandarainbarrel,whenagallonof

waterwasremovedfromeach.AstudentinMs.Fine’sclasshaddeterminedthatthe

fullwaterjugwasnow or40%full.Shewas,however,unabletodetermine

whattodowiththerainbarrelthathadbeen fullbutwasnow full.Shesaidto

theteacher,“Ineedhelp…Idon’tknow,firstIthoughtIwouldtrytogetit(the48in

thedenominator)ascloseto100aspossiblesoImultiplieditbytwo.”butthis

studentultimatelyreachedanimpasseincarryingoutherplan.

Thissecondexampleillustratesthedifficultystudentshaveincarryingouta

processwhenalgebraicexpressionswereinvolved.Intask3.7,studentsinMs.

Fine’sclasswereaskedtowriteanexpressionfortheamountdiscountedandthe

amountonewouldpayifashirtcosting$Swasonsaleat25%off.Moststudents

wereabletowritethediscountamountusingtheexpression0.25S,butstruggles

occurredwhenstudentsattemptedtowriteanexpressionforthenewsalespriceof

theshirt.Astudentwaspuzzled,“Umm(pause)Sminus(pause)no(pause)25

timesS(pause)25timesSminusS?”Afterafewminutes,thestudenttriedagain,

“Aaahh(pause)IthinkIknowhowtodoit.Isit0.25dividedbyS?”Otherstruggles

relatedtothistaskaredescribedlaterinthesectionthatcoversstruggleover

misconceptionanderrors.

Strugglestocarryoutaprocesswereindicativeofthedifficultystudents

haveconnectingproceduretoconcepts.Mistakeswerejustoneofthecausesofthis

3

5

2

5

24

48

23

48

86

typeofstruggle,particularlyifstudentscouldnotlocatethemistakesordidnot

evenrecognizethatamistakehadoccurred.Manyoftheseprocessmistakeswere

broughttotheforefrontfordiscussioninlargepartduetotheopportunities

teachersprovidedforstudentstosharetheirworkordiscusswhattheyweredoing

(Fawcett&Gourton,2005).

3.GiveaMathematicalExplanation

Students’strugglestoexplaintheirworkaccountedfor30%or57ofthe186

observedstruggles.Theytendedtooccurinthelatterpartoftheenactedtaskwhen

studentshadachancetoengageingettingstartedwiththetaskandcarryingouta

planofexecutionpartiallyorevencompletely.Inorderforstudentstocomplete

eachtask,theywereexpectedtoexplaintheirworkandtheirsolutionsinwriting

andinmanyinstancestoeachotherortotheclass.Studentsstruggledtoverbalize

theirthinkingandtogivereasonsfortheirstrategiesforthesekindsoftasks.

Forexample,someofthestudentsvoicedtheirstrugglestoexplaintheir

workintask1.4.Theproblemaskedstudentstodeterminetheamountofwaterthe

five‐gallonwaterjugneededtobeasfullasthe48‐gallonrainbarrelwith24gallons

ofwater.Havingfoundananswer,onestudentresponded,“Idon’tknowhowto

explainit,it’sjustkindalike(pause)Idon’tknowhowtojustifyit.”Anotherstated,

“IknowwhatI’mthinking,Ijustcan’tshowtheexactway.”Manyoftheseinstances

ofstrugglewouldnothavesurfacediftheteacherhadnotgonearoundtoquestion

87

thestudentsabouttheiranswersorifthestudentsdidnothavetheopportunityto

sharetheirworkwiththeirsmallgroups.Listeningtoothersexplaintheirwork

promptedstudentstoquestioneachother’sworkandalsojustifytheirthinking.

4.ExpressMisconceptionandErrors

24ofthe186or13%ofthestudentstrugglesoccurredamongstudents

dealingwithamisconceptionorerrors.Amisconceptionwasnotaninstancewhere

acarelessmisconnectioninthinkingledtostudent’sconfusionandpossibleerror,

butratheramoredeep‐seatedsituationwheremisconceptionswereusedasabasis

forsolvingproblems.Strugglesarosewhenthestudent’sthinkingwaschallenged

ordidnotmakesensetoothers.Amisconceptioninoneprobabilitysettingcaused

astudenttoapplyajustificationthatworkedinasimplercase,sayofonecoin,but

didn’tinamorecomplexcase,sayfortwoormorecoins.Forexample,Ms.Torres

implementedaCoinTossactivitywhereoneofthetaskswastoexplainwhythere

wasa50%chanceofgettingaheadandatail(HT)oratailandahead(TH).The

studentusedtheone‐cointossjustificationtoexplainhisanswer:“becausethecoin

hastwosidesandoneofthemisheadsandtheotherpartistails.”

Inanotherclass,astudentspokeupaboutaprobabilitytaskthathad

seeminglybeenresolvedbythewholeclass.Thetaskwastodeterminethe

probabilityoflandingintheunshadedregioninfigure4.1below.Theisosceles

triangleiscontainedinarectanglewithlength4unitsandwidth6units.

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3units3units

4units

Figure4.1:Findtheprobabilityoflandingintheunshadedregion.

Thestudentasked,“whenit’scombinedtogether,woulditequal orwoulditstill

be ?”Themisconceptionofinterpretingtwopartsoutofthree,eventhoughthe

partswerenotallequal,causedthestudenttostrugglewiththeanswerthathad

beentakenassharedinclass.

Othermisconceptionsoccurredregardingdistinctionsofgallonamountand

percentamount.Onesuchexampleoccurredintask1.6whereastudent

interpreted1%oftheliquidcontentsofacontainerasequaltoonegallonina48‐

galloncontainer.Task1.6hadaskedstudentstodeterminehowmuchwatermust

beaddedtothe48‐gallonrainbarrelwith24gallonstobeofthesamefullnessas

thefive‐gallonwaterjugwiththreegallonsofwater.Theparticularstudent

correctlyconvertedtherainbarrelwith fullofwateras50%fullandthewater

jugwith fullofwateras60%.She,however,incorrectlyconcludedthatthe10%

differenceinthepercentageswasequivalenttoa10‐gallondifference.Shesaid,

2

3

1

2

24

48

3

5

89

“Shadein10moresquares(eachsquarewastorepresent1gallon)…it’slike80%.”

Thestudentthengesturedconfusionandfellsilent.

DiscussionofStudentStruggle Priorempiricalresearchonstudentstruggleshasbeenlimitedandhasfocused

onexaminingtheiroccurrencesinthesettingofawholeclassdiscussionwithout

examiningindetailthenatureofeachindividualstudents’struggles(e.g.,Inagaki,

Hatano,&Morita,1998;Santagata,2005).WhileBorasi(1996)andZaslavsky

(2005)havelookedatstrugglesstudentshavewitherrors,misconceptions,and

uncertainties,moststudieshavebeengeneralinreferencetostruggle(e.g.Carter,

2008,Hiebert&Wearne,2003).

Mystudyusedproportionalrelationshipsascontextintheimplementedtasks

becausetheseconceptsareanimportantpartofmiddleschoolmathematicsand

becausestudentsmustbegivenopportunitiestomakesenseofimportantideasin

mathematicsandtoseeconnectionsamongtheseideas(Boaler&Humphreys,

2005).Proportionsareoftentreatedasproceduralcomputationalproblemsthat

involvefindingmissingvaluesusingatechniquesuchas"cross‐multiplication"

(Heinz&Sterba‐Boatwright,2008).However,asamilestoneinstudents'cognitive

development(Cramer&Post,1993),theconceptofproportionalreasoning

demandsamuchdeeperconceptualunderstandingofdynamictransformations,

structuralsimilarities,andequivalencesinmathematics(Lesh,Post,&Behr,1988).

Learningthisdifficultwatershedconceptwillnecessarilyinvolvestruggle.

90

Thesetasksonproportionalreasoningindeedgaverisetostudentstruggles

duringvariousstagesofthetaskimplementation.Asexpectedfrompriorstudies,

strugglesasexpressedbythestudentswereoftenverygeneralwithstatements

suchas,“Idon’tgetit”,“Itdoesn’tmakesense”,and“Idon’tknowwhattodo”

(Carter,2008).

TheProductiveStruggleFrameworkIusedidentifiedthefourkindsofstruggle

(getstarted,carryoutaprocess,giveamathematicalexplanation,andexpress

misconceptionanderrors)inordertobetterinformteachersabouttheirstudents’

thinkingandastheyconsiderappropriateinstructionalsupportsthatcanhelp

studentsdirecttheirstrugglesproductively.Thesetypesofstudents’struggles

initiatethestudentsinthecultureofdoingmathematicsanddramatizetheparallels

towhatmathematiciansencounterin“doingmathematics”.Forexample,amongthe

11activitiesmathematicianandmathematicseducatorBass(2011)identifiesas

integraltodoingmathematicsareexploringandexperimentingwiththecontextand

processes,modelingandrepresentingthecontext,connectingproblemsandideas

withanalogiesandreflections,opportunisminfollowingyournosewithideasand

pursuingwherethemathematicsseemstobeleading,andconsultingwithexperts

andfriendsaboutthemathematics.

Myfindingsshowstudentsstrugglingintheactofdoingmathematicsasthey

seektofindstrategiesandrepresentationswithproportionalrelationshipsand

attempttofollowthroughwithaplan,knowingthatwhenoneplandidnotwork

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theyhadtoreconsiderotherapproaches(Schoenfeld,1992;Hiebertetal,1996).

Studentsstruggledtoexamineandexplainthesolutiontheyhadproducedand

connecttohowitpertainedtotheoriginalproblem.Intheirengagementofvarious

tasks,studentsvoicedconfusionoverwhattodoastheytriedtounderstandthe

problem,howtodoacomputation,oruseanalgorithmsuchasaproportional

computationsorrationalnumberrepresentationconversionssuchasfromfractions

topercents.Othersstruggledtomanipulatealgebraicexpressionswithuncertainty

whileotherstudentsstruggledtoexplainandmakesenseofanswers,processes,

andotherpeople’sexplanations.Thesefindingsalsoaligncloselytothefour

componentsofproblemsolvingPolyahadproposedinhisbook,HowtoSolve

(1957),namelytounderstandtheproblem,deviseaplan,carryouttheplan,and

lookback.Myfindingsconfirmthatmanyofthestudents’strugglesoccurredat

thesesitesasstudentsattemptedtostarttheproblem,formulateaplanandcarryit

out,andthentrytoexplaintheirsolutions.

Beingabletoexecuteaproceduredidnotguaranteethatstudentscould

solveataskthatinvolvedprocedureswithconnectionordoingmathematics

(Boaler,1998).Fortasksthatincludedalgebraicrelationships,itwascommonthat

studentswouldindicatetheirstrugglemostoftenatexplainingwhatthe

expressionsmeant(Carraher,Carraher,&Schliemann,1987).Thestruggleandthe

subsequentdeclineinthecognitivelevelofthetask,insomeoftheepisodes,may

havebeenduetoaninappropriatenessoftaskforaparticulargroupofstudents

92

(Henningsen&Stein,1997).Inapost‐classinterview,Mr.Bakermentionedthatthe

algebraicnatureoftasks3.1through3.5created,“…themoststrugglesandthemost

frustrationasfarastheydidn’tknowwheretobegin.”

Therelativelyhighincidenceofstruggleswithuncertaintiesorconfusionto

getstartedwithataskinsomeclassesascomparedtootherclassessuggeststhat

theclassasawholehadvaryinglevelsoftoleranceforgrapplingwithaproblem.

Someofthestudentswentofftaskbysocializingwitheachotherorbecame

disruptive.Othersceasedworkingontheirtask.Otherclasseshadsignificantly

moreproceduralstrugglesandfarfewerexplanationstrugglesorvisaversa.In

theselatterclasses,aclassroomnormandcultureseemedtobeinplacewhere

studentshadopportunitiestodiscusstheirsolutions,andinthecourseof

explanationtriedtoreasonandcommunicatetheirthinkingmathematically(Cobb,

Wood,&Yackel,1993).Thenatureofstudentstrugglessometimesseemedtobe

relatedtothesociomathematicalnormsthatwereinplaceineachclass.Inother

words,struggleswerenotonlycognitiveinnature.

Ialsofoundthatstruggleswouldnothavesurfacedifnotforthe

opportunitiesteachersgavestudentstosharetheirworkandtoreachconsensus

abouttheirsolutionsandtheirworkwithinsmallgroups.Intheirinterviews,

teachersmentionedthebenefitsofhavingstudentsshareandexplaintheirworkso

thatnotonlycanstruggleariseinclass,itcanhelpotherstudentswhowere

strugglingseeapproachesandstrategiesthatcouldsupporttheirownthinking.

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Furthermore,teacherspointedoutthatstudentstryingtoexplaintheirmathematics

couldalsorevealstudents’struggletomakesenseoftheirworkeveniftheyappear

tobeabletodotheproblemontheirpaper.

Myfindingsconfirmthatoccurrencesofstruggledependedonstudents’

engagementintheprescribedtasksthatchallengedthemandhadsomeelementof

difficulty.Whethertheytriedtoformulateaplan,explorepossiblestrategies,

reconsiderinitialattempts,orexplainnotonlytheirownthinking,butmakesense

ofthethinkingofothersastheydiscussedtheirworkinclass,theinitiationofthe

students’actofexternalizingtheirstrugglepromptedteachersorotherstudentsto

respond.

Assessingtheexplicitkindsofstrugglethatconfrontstudentscaninform

teachingthatresponds,supports,andguidesthestudentswithgreaterspecificityto

theparticularstudentstruggles.Inaddition,thestudentscanself‐regulatetheir

ownlearningbynotingtheaspectsoftheproblemtheyareunabletoaddressorthe

progresstheyaremakinginaccomplishingtheirtask(Pape,Bell,&Yetkin,2003;

Butler,2002).Myframeworkisintendedtoinformteachingthatcanbettersupport

studentlearningandtoalsoraiseawarenessinthestudentsthattheirstrugglemay

notnecessarilybeovertheentiretaskbutperhapsoveraparticularaspect.Inthis

way,thegeneralstruggleismademorespecificandappearsmoremanageableto

thestudentswhenfocusisplacedonanalyzingthemathematicalproblemandnot

onlyonthestudents’inabilitytogettotheanswer.

94

TEACHERRESPONSE Myfindingsshowthatduringthe39videotapedandobservedclasses,

teachersrespondedtoanaverageoffivetosixstudentstrugglesineachclassperiod

thatresultedininteractionsbetweenstudentsandteachers.Theseinteractions

rangedfromafewminuteslongtoover15minutes.Duringthe60or90‐minute

classperiods,theteachersbeganeitherwithawarm‐uptaskormoveddirectlyinto

thelessontaskswhilealsotakingattendanceandmakinggeneralannouncements

aboutupcomingactivitiesfortheendofthesemester,suchasfieldtripsandfinal

projectduedates.Themajorityoftheclasstime,however,wasspentonthe

mathematicaltasks.

Overviewofteacherresponsecategories Usingprinciplesofgroundedtheory(Strauss&Corbin,1990),Ifoundthat

teachersrespondedtostudentsstrugglein4mainways.Iclassifytheteacher

responsesas:

1. Telling

2. DirectedGuidance

3. ProbingGuidance

4. Affordance

Inthissection,IwillexplainhowIdevelopedthesecategoriesthenprovide

morein‐depthdescriptionsofeachtypewithexampleepisodesthatcapturethe

natureandperceivedteacher’sintentintheseresponses.Again,thiswillbethrough

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thelensoftheclassroomobservationsofstudent‐teacherinteractions.Iwillclose

thesectiononteacherresponseswithadiscussion.

Inmycodingofteacherresponses,Iusethenotionofteacherresponsenotas

asingleutterancebutasasequenceofmovesmadebyteachersduringtheir

interactionswithstudentsthataddressedstudentstrugglesinsomeway.The

sequenceofteachermovesconsistedofquestions,follow‐upstatements,and

suggestionsdirectedtowardmanagingandresolvingstudents’struggle.

DefiningTeacherResponseTypesTelling

Thepatternsofteacherresponsessuggestedacontinuumalongwhichthe

responsescouldbeclassified.Atoneendofthecontinuumwerethoseteacher

responsesthatsuppliedstudentsneededinformationtohelpaddresstheir

struggles.Icalledthistypeofresponsetelling.Functionally,thisclosed‐ended

approachmovedthestudentsforwardincompletingtheirtasksbyprovidingwhat

theteachersperceivedtobeneededinformation(Kennedy,2005).Theseteacher

responsesdiminishedtheintensityofthestudent’sstrugglewithinterventionsthat

reducedthecognitivedemandforthestudentsandtherebyloweredthecognitive

demandoftheintendedtask.

Affordance

Ontheotherendofthecontinuumweremoreopen‐endedtypesofteacher

responsesthataddressedthestudents’thinkingandsuggestedkeyideasfor

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studentstobuilduponwhilealsoprovidingadditionaltimeforstudentstoworkand

discusstheirideaswithoutrushingthemtowardsaresolution.Ilabeledthis

teacherresponsetypeaffordancesasteachersprovidedstudentsopportunityand

timeforfurtheractionandinteraction(Gaver,1996)withoutloweringtheintended

cognitivedemand.

Figure4.2: TeacherResponseRange

DirectedGuidance

Withinthesetwoextremes,Iidentifiedteacherresponsesthatappeared

eitherteacher‐drivenorstudent‐driven.Thedirectedguidanceresponsesfellcloser

totheclosed‐endedsideofthecontinuum,andservedtoguidestudentsina

directionthattheteacherperceivedhelpful.Thistechniqueattimesredirectedthe

studentawayfromthestudent’soriginalideas.Themovesservedtoexpeditethe

student’sprogresstowardcompletingthetaskbysuggestingmethodsorconcepts

theteachersthoughtwereappropriate.Ininstanceswherethestudentswereata

lossastohowtocarryouttheirtask,teacherssoughtwaystoprovidesomeideaor

meanstoconnecttostudents’priorknowledgewithoutlosingmomentumin

keepingstudentsengagedwiththetask.

!"#$%&'%(&%&)*+,--".*/(0#12345#( 6-%('%(&%&)*700#1&*6--#14,(54.

97

ProbingGuidance

ThefourthtypeofteacherresponseIlabelprobingguidance.Thisiscloserto

theopen‐endedapproachinbeingresponsivetothestudents’thinking,probingfor

theirideas,suggestingmathematicalconceptsorproceduresthatrelatedtoand

builtonthestudents’thinking.Theintellectualeffortneededtotackletheproblems

restedwiththestudents,buttheresponsesservedtoclarify,connect,orconfirm

ideasthestudentspresented,andwerethereforemadevisiblethroughtheteacher

responses.

InFigure4.3below,Ireportmyfindingsusingtheteacherresponse

classificationsalongthiscontinuum.

Figure4.3: TeacherResponseContinuum

Ratherthandiscretejumpsfromonecategorytotheother,Iobserveda

continuousrangeofresponseswithdegreesofinformation,directing,probing,or

affordanceprovidedbytheteacher.Responseshadvaryingdegreesofprobingor

directingresponseswithcharacteristicsthatattimeswerehybridsofsomeprobing

andsomedirecting,asIwillreportinthissection.Insomeinstancesthecognitive

demandofthetaskasoriginallyconceivedchangedintheimplementationsothat

!"##$%& '(()*+,%-".$*"-/"+012$+,%-" 3*)4$%&012$+,%-"

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whilesomeresponsespreservedthecognitivedemandlevelasinanaffordance

response,othersloweredthetask’scognitivedemand.

Inmyanalysis,Ifocusedonhowthefollowingthreedimensionsofthe

student‐teacherinteractionswereaffectedbytheteacherresponsetypesthatI

observed:

• Thelevelofcognitivedemandofthemathematicaltask;

• Theattentiontothestudent’sstruggle;and

• Thebuildingonstudent’sthinking.

Thesethreedimensionswerechosenbasedonmyconceptualframework.

Teacherresponsesandinteractionshavetheabilitytoaffectthelevelofcognitive

demandinresponsetostudentstrugglesovertheimplementedtasks(e.g.Stein,

Smith,&Henningsen,1996).Second,theinteractionandteacherresponsesmay

takevaryingstancestowardattendingtothestudent’sstruggleaspartoflearning

withunderstanding(e.g.Borasi,1994).Thirdly,thefocusandbuildingonstudent’s

thinkingduringtheinteractioncanaffectstudent’sunderstandingofmathematics

(e.g.Doerr,2006).

Thefollowingtablesummarizesmyfindingsofthefourteacherresponses.

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Table4.2:TeacherResponseSummary

TeacherResponse

Characterizations Frequency Dimensions

1.Telling • Supplyinformation

• Suggeststrategy• Correcterror• Evaluatestudent

work• Relatetosimpler

problem• Decreaseprocess

time

27% 1a)CognitiveDemand:

• Lowered1b)AttendtoStudentStruggle:

• Removestruggleefficiently.

1c)BuildonStudentThinking:

• Suggestanexplicitideaforstudentconsideration

2.DirectedGuidance

• Redirectstudentthinking

• Narrowdownpossibilitiesforaction

• Directanaction• Breakdown

problemintosmallerparts

• Alterproblemtoananalogy


• Loweredormaintainedfromintended

2b)AttendtoStudentStruggle:

• Assesscauseanddirectstudent


• Usetobuildonwithteacherideas

3.ProbingGuidance

• Askforreasonsandjustification

• Offerideasbasedonstudents’thinking

• Seekexplanationthatcouldgetatanerrorormisconception

• Askforwritten


• Maintained3b)AttendtoStudentStruggle:

• Question,encouragestudent’sself‐reflection

3c)BuildonStudent

100

workofstudents’thinking

Thinking:• Useasbasisfor

guidingstudent4.Affordance • Askfordetailed

explanation• Buildonstudent

thinking• Pressfor

justificationandsense‐makingwithgrouporindividually

• Affordtimeforstudentstowork.


• Maintainedorraised

4b)AttendtoStudentStruggle:

• Acknowledge,question,andallowstudenttime


• Clarifyandhighlightstudentideas

InthefollowingsectionIdescribeinmoredetailtheteacherresponsesand

theirimpactonthethreedimensions,namelythelevelofcognitivedemand,the

attentiontostudentstruggle,andtheuptakeonstudentthinking.

DescriptionsandImpactonThreeDimensions

1.Telling Inatellingresponse,theteachersevaluatedthestudentstatusinrelationto

thetaskandthenprescribedsufficientinformationneededforthestudentsto

overcomethestruggle.Thedirectionoftheinteractionwasdominatedbyteachers’

thinkingandthestudents’rolewastotakeuptheteachers’suggestions.Thegoalof

theinteractionappearedfocusedonstudentsarrivingatthecorrectanswerforthe

taskwithanefficientmethod.Theapparentefficiencyofteachingdecreasedthe

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timethestudentsmayhaveneededinordertoconnecttheirthinkingtothe

suggestedideasinordertogettotheunderlyingissuesthatcausedthestudents’

struggle.Forexample,inanalgebraicsetting,astudentstrugglingwithavariable

asked,“It’sanynumberyoumakeup?”andMr.BakerandMs.Georgevoiced

similarresponsesascapturedinMr.Baker’sresponse,“We’renotgoingtomakeup

anothernumber,we’regoingtousex.”Theresponsewasanexplicitdirective

regardingthetaskbutdidnotaddressapossibleissuesrelatedtothesourceofthe

student’sstruggle,whichseemedtobeovertheuseofavariable.

1a)CognitiveDemand

Akeyfeatureofthetellingresponsesincludedchangingtheproblemfeatures

oftenintheformsofsimplificationandsupportinconnectingthesimplifiedversion

ofthetasktotheoriginaltask,essentiallydoingsomeoftheintellectualworkforthe

student.Forexample,Ms.Fine’sresponsetoastudentstrugglingwithwritingan

expressioninvolvingpercentsandvariableswas,“Let’sjustnotthinkaboutthe

0.25;let’ssayIhaveanumberandIwant50%ofthat.Itwouldbe0.5right?Isn’t

50%0.5?Wouldyouturnaroundandputa1infrontofit[referringtowriting

0.25xandnotthe1.25xasthestudenthadwritten]?”Thecognitivedemandofthe

problemwasdecreasedduetotheinformationsuppliedandstrategiessuggested

withstudent’seffortsminimized.

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1b)AttendtotheStruggle

Theinteractionsinthetellingresponsefocusedonefficientlygetting

studentsbackontrackafteraderailment.Forexample,whenstudentswere

strugglingwithaproblemthatcouldhavebeenresolvedwithanexampleandan

equationthatwaswrittenontheboard,Mr.Bakermadereferencetoitbystating,

“Wecanuseaformulawehadabovethatmaybehelpfulforustosolvethisone.So

whatdidwepayandhowdidtheygetit?Theyprobablydidthesamething

here…okay?Solet’scombinethis.”Theteacherrespondedinawaythat

acknowledgedthedifficultythestudentseemedtobehavingbutpointedouta

strategythroughanexamplethatdirectlyhelpedthestudentinresolvingthe

difficulty.

1c)BuildonStudent’sThinking

Informationwasoftensuppliedorstrategiessuggestedwithoutbuildingon

students’thinking.Forexample,Ms.Georgesuggestedthefollowingquestions:“Do

yourememberhowtosetupaproportion?Whichratioareyougoingtouse?Can

yousetitupasafraction.”Thesequestionsprovidesignificantlymoreguidance

withteachersdoingmuchoftheworkforthestudentsindevisingastrategyforhow

tocarryoutthetask.Thetellingresponseseemedtobedrivenbyaneedtomove

studentsfurtheralongincompletingataskandinsodoingdisregardorfailtobuild

onstudents’thinking.Theteachertookovertheroleofperformingthetaskor

suggestedastrategythatputthestudentinasecondaryrole.AninterviewwithMr.

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Bakercorroboratesthisperspectivewhenherelatedthat,“Withteachingand

helpingsomanykids,it’skindoflikeamatteroftime.Whatwillgetmetogetthem

tounderstandtomoveonthefastest.”Studentswerenotgivenasmuchtimetoair

theirthinkingbecausetheteachers’directionandpacingincarryingoutthetask

dominatedtheinteraction.

Thefollowingepisodeisanexampleofthekindofinteractionthat

incorporatedatellingresponsethatmadeexplicitthedirectionintendedbythe

teacherbutinsodoinglessenedthecognitivedemandofthetask,theconnection

withthesense‐makingeffortsexpendedbythestudentinhisstruggleandthe

effortstobuildonstudent’sthinking.

EpisodeT1:

Thesettingforthisepisodewasaboutdeterminingthetipforabillusing

algebraicexpressions.Thestatementfortask3.1isasfollows:Supposethebillis$x.

Writeanexpressionforthetipon$xusinga15%tiprate.Whatisthetotalamount

youwouldpaytherestaurant?

StudentsinMr.Baker’sclasswerehavingdifficultygettingstarted

withtheuseofalgebraicrelationshipstoformulateanexpressioninvolvinga

variable.

4 S: It’sanynumberyoumakeup.

5 T: We’renotgoingtomakeupanothernumber.

We’regoingtousex.

104

Mr.Bakermakesitclearwhatistobeused,thoughthestudentseems

uncertainabouttheroleofthevariable.

6 S: Oh,yeah.Butxisthenumberwewant,right?

7 S1: No.xislikeingeneral…

8 S: Iknow.Youcanputinwhateverforx.Right?

9 S1: No,no

10 T: You’rejustgetting…butitcouldbewhatever.

ThestatementbySsuggestsanemergingunderstandingoftheuseofa

variableinthissetting,butsuggestsastruggleformeaningaboutwhatheis

supposedtodowiththevariable.

11 S2: Soit’sxplusxtimes0.15equalsx?

12 T: Well,insteadof$5[asusedinanearlierexample],

we’regoingtousex.

Again,Mr.Bakergaveexplicitinstructionbytellingthestudenthowthex

wastobeusedinplaceofthenumericalvaluethatwasstillontheboard.

13 S3: How?

14 T: We’renotsolving,we’rejustsayingwhat’stheprocess,

what’stheprocess.

15 Class:Oh.

Thestudentsweretoldwhatneededtobeusedinplaceofthe$5from

anearlierproblem.Thestudentsdidnotseemgenuinelycertainhowto

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makethatconnectionorworkwithanexpression,thoughthistypeof

strugglewouldnotbeunusualfor7thgradestudentswithlimitedexposureto

algebraictermsandconcepts.

Insummary,thetellingteacherresponsesattemptedtosupply

information,suggeststrategies,correcterrors,evaluatestudentwork,relate

aproblemtoasimplerone,ordecreaseprocesstimeforstudentsinorderto

completethegiventask.

2.DirectedGuidance Teacherresponsesthatprovideddirectedguidancetostudentsoccurred

mostfrequentlyandwithgreatsimilarityacrossalltheteachers.Onenotable

characteristicofdirectedguidanceresponseswashowtheygenerallybeganwith

teacherassessmentofwhatstudentsknewandwhattheystillneededtodo.

Teachersthenincorporatedthisinformationintheirresponsestostudentsby

suggestingstrategiesthatappearedtobeknowntothestudents,directingor

narrowingdowntheiractions,orredirectingthestudentstowardsareasoning

basedontheteachers’formulation.Forexample,whenastudentstruggledwhile

tryingtocarryoutaprocess,Ms.Georgeinquiredofthestudent,“Let’stalkthisout.

Whatdoesthatgiveus?”,toassesswherethestudentwasintheproblem.Another

characteristicincludedguidingstudentsinbreakingtheproblemintosmallerparts.

Ms.Finesuggested,“Justseparateitinyourheadokay.Whatistheamountthatis

thediscount?Justtellmethediscountamount.”Thistypeofguidancehelped

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studentsunpacktheirthinkingandfocusonelementsoftheproblemthatthe

studentsmaynothaverecognizedaskeytothesolution.Otherteachersresponded

byprovidingabridgeacrossanapparentgapthestudentencounteredinsolvinga

problembyprovidingamorefamiliartypeofproblemsuchasanumericalonethat

wasmoreaccessibleforthestudenttobetterunderstandtheunderlyingprocesses

involvedinanumericalproblem.

2a)CognitiveDemand

Teachersusedtheirthinkingintheirdirectedguidanceresponseasthe

primaryguidingfocustosupporttheexploratoryandself‐monitoringaspectsof

problemsolving.Bydirectingthestudent’sthinking,thecognitivedemandofthe

taskintheenactmentasexperiencedbythestudentdecreasedbyvaryingdegrees

fromtheintendedlevel.Teachersoftencutshorttheopportunitiesforstudentsto

grapplewiththeformulationandimplementationofastudents’planbymakinga

suggestion.Forexample,astudentinMr.Baker’sclasswasuncertainhowtobegin

task2.4,fromtheBagofMarblesactivity,todeterminehowmanyredandblue

marblesshouldbeplacedinabaginordertohavethechanceofpickingoneblue

marblethatisbetweenbag1with chanceandbag2with chance.Mr.Baker

firstreiteratedtheproblemandengagedthestudentinastrategy:

16 T:Trytofindacombinationofmarblesredandblue,liketheydid,ifyouhad

awholebunch.Youhavetomakeacombinationthatwillfitwithachance

25

100

20

60

107

rightinhere[pointingbetweenthegraphicsofBag1andBag2onthe

worksheet].Okay,youknowBag1haswhatpercentchance?

17 S: Wait,

18 T: Whichis25%,right?Bag2hasawhatpercentchance?

19 S: percentchance

20 T: or…

21 S: 33.3%

22 T: Okay,well,youknowthat[writingallofthisonthestudent’spaper].

Givemeabag,givemeanumberofredandblueyouseehowtheyhave

differentnumbers[pointingtotherepresentationontheworksheet]?Give

mearedandbluethatwouldfitinbetweenhere,thatwouldbe

somewherebetween0.25and0.33.

23 S: 29

24 T: Sohow,giveme…[studentnearbysays30]30.Howwouldyou

make…

25 S: Isaid29.

26 T: Okay,howwouldImakeabag?WhatwouldIneed?

27 S: Whatdoyoumean?

28 T: HowwouldImakeit29%chanceofablue?

29 S: Inwhichbag?

1

4

1

3

1

3

108

30 T: Thisonerighthere.Yousaidyouwant0.29right?That’sthenumber

you’regoingfor?

31 S: [nodsyes]

32 T: Whatcombinationofmarblesdoweneedtodotogetthat?You

wanted30likeyousaid[speakingtotheotherstudent].Whatcombinationof

marblesdoweneed?

33 S: Actually,30soundseasier.I’mgoingwith30.

Intheaboveinteraction,theboldedteacherresponsesdirectedthestudent

withastrategyforhowtoapproachtheproblemsothattheoriginallevelfour

cognitivedemanddeclinedtoamoreprocedurallevelofdeterminingacombination

ofmarbleswith30%chanceofblue.Thereisstillworktobedone,butasthe

interactioncontinued,theteachersuggested:“Iwouldsayit’seasiertoworkwith

10’sor100’s…”andguidedthestudenttowardsthinkingof30%as30outofa100

implying30bluemarblesoutof100total,whichfurtherreducedthecognitive

demand.


Interactionsthatinvolvedadirectedguidanceresponsetriedtoestablishthe

natureofthestudents’struggle.StatementssuchasMs.Torres’statement:“Soyou

thinkit’s oryouknowit’s ?”triedtodeterminewhethertoaskforevidence

thattheanswerwas ortoaskwhythestudentthoughtitwasaparticularvalue.

Ms.Norrisaskedastudent,“Whydoyousaythe5gallon?”heretryingtodetermine

3

4

3

4

3

4

109

thereasonbehindastudent’sanswertobringthestudents’thinkingoutintheopen.

Inanothersetting,sheaskedastudent,“Well,whatisthat?”toinquireabouta

representationthatastudenthaddrawnandtohaveherexplaininwordsthe

natureofthegraphicaboutwhichthestudentwasstruggling.Todeterminewhat

studentswerethinkingastheystruggledwiththeanswerstotheirproblem,Ms.

Harrisaskedonestudent,“What’snotcorrect?”ashecouldnotreconcilehisanswer

withothermembersofhisgroup.Inanotherinstancewhenastudentstruggledto

implementaprocedure,Ms.Harrisasked,“Andsohowdidyougettheotherpart?”.

Thestudentresponded,“FirstIhadthisasmyanswerbutthenIrememberedthis

soIaddedinthat.”


Teachersaskedquestionsinadirectedguidanceresponsethattriedto

establishwhatthestudentswerethinkingandthenaskedthestudentstoclarifyand

confirmwhattheyknewandwhatwasnotclear.Thisseemedtoinformthe

teachersofthekindsofsupportthatcouldguidetheirstudentsandhelpresolve

theirstruggles.

Anothercharacteristicofdirectedguidancewasatendencyfortheteachers

toredirectthestudents’thinkingtowardsanactionthatwouldfunctioninamore

helpful,efficient,orcorrectwayasperceivedbytheteacher.Theguidancefocused

onteachers’strategies,procedures,orunderstandingofthetasksthatstudents

couldthentakeupasopposedtobuildingonstudents’ideas.Forexample,teachers

110

wouldbedirectinsuggestingaproblemsetupsuchasMs.Torres’questiontoa

student,“Canyoutellmewhatthatproportionwouldlooklike?“,orstatementsthat

directedanactionsuchas,“Theybothcancel.Okay,socancelthem.“madebyMs.

Harrisindirectingastudentwhoappeareduncertainofaprocesstoexecute.Other

teacherswouldattempttoclarifythemeaningofaproblemandasksomequestions

thatcouldhelpstudentsformulateaplan.Thequestionsdidnotalwaysoccurin

suchquicksuccessionasMs.Harrisexpressedhere,“Howcouldyouwritethis?

What’sreallythere?Doweknow?(pause)That’swhatwewanttofigureout.”The

teacher’sintentappearedtobeanattempttoenablethestudentstogetstartedby

focusingonunpackingthewordingandinformationintheproblemandclarifying

theimportantquestionintheproblem.Duringotherimplementationstruggles,

teacherssuchasMr.Bakerinthefollowingepisodepressedforstepsinaprocedure,

questioningstudents,“Whatdidwedo?Youmultipliedwhat?Whatdoyoureally

needtodohere?”Thereappearstobeanimpliedrightstepthatthestudentshould

considerdoinghereandMr.Bakerattemptedtoguidethestudentsinthatdirection.

Ms.Norris’sresponseinline36belowillustrateshowateachercandirecta

studentwhoisstrugglingwithaformulationtogetstarted.Shewantedthestudent

tothinkofcomparativefullnessasopposedtomerelyatthequantity.Whenthe

studentlookeduncertainandstated,“Idon’tknow,”Ms.Norrisresponded,

34 T: Okay.Well,howfullisthiscontainer[pointingtotherainbarrel].

35 S: 24

111

36 T: Outof

37 S: 48[lookingatTwithaquestioninglookforapproval].

Directingstudentstowardaparticulartechniquesuchasproportions,useof

fractions,orpercentsoccurredfrequently.Forexample,Ms.Georgewouldask

students,“Areyoustillsettingupaproportion?”orinanotherinstance,“Canyoutell

mewhatthatproportionwouldlooklike?”Thesequestionssuggestaprocedurethe

teachersthoughtwouldhelpstudentscreateaconceptualrepresentationinorderto

formulateandimplementaproblemwithwhichtheywerestruggling,suchasin

tasks1.5and1.6.

Directedguidancewasoftenusedasaformofresponsetostudents’

strugglingwithalgebraictasksinthetipsandsalesactivity.Thestudentsstruggled

withtheirunderstandingofvariablesandexpressionsthatpromptedteachersto

providemoresupport,knowingthestudentshadlimitedexposureand

understandingofalgebraicrepresentation.Ms.Georgeaskedastudent,“Whatis

0.4Nsaying?Whatisthatsaying?“asshetriedtoredirectastudentaddressingtask

3.5fromusingaproportiontousingascalingofthetotalbythegiven40%.With

anotherstudent,shetriedtorespondtoastudenttryingtoformulateanalgebraic

expressionfortask3.7bydeterminingwhatthestudentunderstoodofan

expression,“Let’stalkthisout.If25%ofS,0.25S,whatdoesthatrepresent?”This

wasacommontheme,oftryingtohavestudentsarticulatethemeaningofan

112

expression.Ataconceptuallevel,itwasnotapparentthatthestudenthadaclear

ideawhattheirvariablesrepresented.

Thefollowingepisodeexemplifiesaspectsofteacherdirectedstrategiesand

methodsthatguidedastudentinresolvingherstruggletocarryoutataskasthe

student(S)reachedanimpasseinsolvingtask1.5.Ms.Fine(T)beganby

questioningthestudent(S)inline39inordertodeterminewhatthestudent

understoodoftheproblem.Thiswasthegeneralpatternfornotonlythedirected

guidance,butfortheprobingteacherresponsesaswell.Thedistinctive

characteristicsofadirectedguidanceresponseasillustratedinthefollowing

responsesequenceiswhatappearstobetheteacher’sintenttoredirectthe

student’sthinkingtowardstheteachers’thinkingaboutthetaskduringthecourse

oftheinteraction.Infact,theteacher’sformulationandimplementationiswhat

guidedthestudent’sactionsasseeninline49.IincludethetranscriptofepisodeD1

asanexampleofhowtheteacher’shints,centeredaroundtheteacher’s

implementationplan,directedthestudenttocarryoutthework.

EpisodeD1:

38 S: Igotthisfar.

39 T: Sowhat’stheactualquestion?Howdidthatgoforso….

40 S: Youdrainagallonofwater.Itwas 35soitbecame 2

5becauseyou

drainedagallonofwater,right?

41 T: Sowhatpercentis 25?

113

42 S: It’s40%

43 T: Yeah,butwhatabouttheotherone?Howdiditspercentagechange?

44 S: What?Thisone?That’stheoneI’mstumpedon.Ineedhelp.

45 T: Okay,howdowegofromthistoapercentage?

46 S: Idon’tknow.FirstIthoughtIwouldtrytogetitascloseto100as

possiblesoImultiplieditby2.

47 T: Okay

48 S: Whichis4offof…

49 T: Okay,what’stheotherwaywedidthis…

50 S: [Shakesherheadno.]

51 T: Youdon’tremember?It’sbeenawhile.

52 S: It’sbeenawhile.

53 T: Butifyouhadyourcalculatorwouldyoubeabletosolvethat?

54 S: [Showshercalculator.]

55 T: Okay.Youcangoaheadandusethat.What’sthenormalwaywedo

that?[PointingtoS’spaper.]

56 S: Proportions?

57 T: Fromafractiontoapercent.

58 S: Ohyeah,toadecimal

59 T: Yeah.

60 S: Yougofromadecimalto[inaudible].

114

61 T: No

62 S: Ohno,wait,yougolongdivideit.Oh…

63 T: But,I’mnotgoingtomakeyoudothat.Youcanuseyourcalculator.

64 S: HowdoI?It’s23by48right?

65 T: Exactly

Theboldedresponsessuggesttheteacher’sthinkingasthedrivingforcein

theimplementationofthetaskwhilethestudentprovidesnominalconfirmation

abouttheproblemsolvingprocess.

Insummary,directedguidanceresponsesredirectedstudentthinking

towardstheteacher’sthinking,narroweddownpossibilitiesforaction,directedan

action,brokedownproblemsintosmallerparts,oralteredproblemstoan

analogousonesuchasfromanalgebraictoanumericalone.Whilethe

characteristicsoftheseresponseshavesomesimilaritieswiththetellingresponses,

theinteractionsdemandedstudentstocommunicatetheirthinkingandtoremain

engagedindoingtheproblem,evenifthedevisedplanwasmoreattributabletothe

teacher’sthinkingthanthestudents’.

Inowdescribetheprobingguidancetypeofresponsethatfocusedthe

studentsbacktotheirthinkingandideasevenmorethanwithdirectedguidancein

orderforstudentstounderstandandbuildonthem.

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3.ProbingGuidance Teachers’useofprobingguidancemadestudents’thinkingvisibleandwas

usedasthebasisforaddressingthetaskasopposedtodirectedguidance,wherethe

teachersfocusedontheirthinkinginordertoaddressthetask.Theteacherhadto

expendefforttohavestudentsarticulate,insomeway,theirthinkingwhetherina

verballyorwrittenform.Forexample,teacherswouldaskforreasonsand

justification,couchingthequestionwithoutintimidatingthestudent,suchasMs.

Torresquestion,“Ican’tquiteunderstandhowyougotfromthe3coinsandthe

chanceofgettinganycombination….canyouexplainthattome?Sometimesgood

feelingsareverydifficulttoexplain.”

Probingguidanceresponsesdidnotoccurasfrequentlyasthedirected

responseswith28%oftheteacherresponsesascomparedwith35%directed

guidanceresponsesand27%tellingresponses.Someindicationofstudentthinking

andworkhadtobecommunicatedormadevisibleinorderfortheteachertobuild

uponit.Iftheteacherdidnotperceivetherewassomethingtobuildon,ratherthan

trytofindoutwherethestudentswereintheirthinking,theteacherwould

generallyprovidesomeformofscaffolding,eitherasatellingordirectedguidance

responsethatwouldprovidemoreexplicitdirectionforthestudent.

3a)CognitiveDemand

Theintendedcognitivedemandofthetasksweremaintainedbytheuseof

probingguidanceresponsesbecausetheinteractionsbuiltonstudent’sthinking,

supportingitwhileattimespressingformoreexplanation,elaboration,or

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justification,whichhelpedreinforceratherthandiminishthosemathematical

processesimportantinproblemsolving.Forexample,Ms.Harrisaddresseda

studentuncertainaboutthemeaningofaprobleminordertogetstarted.Whenthe

studentasked,“Isn’t(problem)BjustlikeA?”Ms.Harrisresponded,“What’s

differentaboutit.Readthequestion.What’sdifferentaboutit?”andplacedthe

intellectualeffortbackonthestudenttoexamineandconsider.

Anotherexamplecomesfromtask3.7,thetipsandsalesactivitywhichreads

asfollows:“Ashirtregularlycosts$Sandisonsaleat25%offtheregularprice.

UsingS,writeanexpressionfortheamountofdollarsdiscounted.Alsowritean

expressionthatrepresentshowmuchyouwillpay,disregardingtax.”Ms.George

noticedincorrectsolutionsonvariousstudentspapersandoneofthestudents,

“Okay,tellmewhatthatmeans[whensheseesastudentwith0.25S–Sratherthan

thecorrectformS–0.25S].Shefurtherprobedthestudent,“Subtractit[repeating

thestudent’sresponse].Whatdoyoumeansubtractit?”.Thenshesuggestedtothe

studentanumericalversionforthestudenttoconsider,“Whatkindofnumber

wouldyougetifItoldyouthatSwas$12?Plugin$12forSandtellmewhatyou

get.”Thecognitivedemanddidnotdiminish,thoughtheteacherprovidedguidance

forthestudentstoconsideranexampleinordertohavethemreevaluatetheir

understandingoftheirproblemandpossiblyidentifyforthemselvesthesourceof

theirstruggle.

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Attimes,students’ideaswerenotyetwellformulatedandteachers

requestedthatstudentswritedownwhattheywerethinking,asMs.Fineinsisted,

“Showme.Writedownwhatyouhave”inwhatappearedtobeanattemptto

identifythestudents’struggle.Andevenwhenstudentshadwrittenwork,teachers

wouldaskforthemeaningoftheirworkasdidMs.George,“Youtellme,whatdoes

thatmean?”orMr.Baker,“Soyoufound$10.Allright.Whatisthat$10telling

you?”,againtogettotheunderlyingreasonfortheirexternalizedstruggle.The

teacherssoughtanexplanationforwhatwassaid,whatwaswritten,orsometimes

whatwasnotsaidwhenastudentwasreluctanttovoicehisorherthinking.

Inamovethatsuggestedtheteacherswerepromotingstudentstoself‐

monitortheirthinking,action,orwork,teachersaskedstudentstorepeatwhatthey

saidorrepeatedwhatthestudentssaidwithaquestioningtone.Thistechniquewas

oftenusedtoclarifyastudents’misconceptionthroughreasoningaboutwhatthe

studentssaidandwhethertheirideasoundedreasonable.Forexample,when

studentswereconfusedabouttheaxesfortheindependentanddependentvariable

inagraphingproblem,Ms.Harrisconnectedthediscussiontowhatstudentsmay

havebeenexposedtoinotherdisciplinesandasked,“Inscience,whatvariabledo

youusuallyputonthex‐axis?”Inanotherinstance,Ms.Georgeaskedastudentwho

thoughthemadeamistake,“Youmessedup?Why?Whatdidyoudothattellsyou

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[that]youmessedup?”andattendedtothestudent’sstruggleaswellasraisedthe

student’smetacognitivelevelofawarenessinsolvingproblems.


Inordertoelicitstudent’sthinkingwhenastudentwasuncertainwhetheran

answerwasrightorwrong,Ms.Fine,forexample,probedthestudentbyasking,

“Whydon’tyouthinkit’sright?Whatwereyouthinkingherewhenyoudidthat?”

Theresponsedidnotattempttoevaluatethestudent’sanswerbutpressedthe

studentforfindingreasonsforhisorherconclusion.

Thefollowingepisode,P1,illustratesateacherresponsethatconsistedof

movesthatsupportedstudents’thinkingbydisplayingitintheforefrontand

directingthestudentsthroughtheirstrugglebybuildingontheirwork.Thefocusin

probingguidanceistotakeupstudents’thinkingandguidethemtowardbetter

understanding.Thedifferencebetweenthisformofguidanceanddirectedguidance

istokeepthestudents’reasoningondisplay(Pierson,2008)andtheteacher’s

thinkinginthebackdrop,thoughtheteachertriestoprovidesupportappropriate

fortheparticularstudents.

Aprominentcharacteristicintheprobingguidanceresponsewastohold

students’accountablefortheirreasoningandsense‐making.Thefollowingexample

ofateacherresponsecontinuestheexampleoftheresponseMs.Georgeusedfor

task3.7thatbuiltonstudent’sthinkingandprobedfordemonstrationsof

understanding.

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EpisodeP1:

Ms.Georgenoticedonmanyofherstudents’papersthatthesalespricewas

writtenas0.25S–S.Inthisinteraction,Ms.GeorgeprobedAmy(A)toexplainthe

workonherpaperbutrefrainedfromsayingtherewasamistake.Theprobing

questionstoAmyaskedforanexplanationandjustificationfortheworkas

constitutedonherpaper,madearestatementoftheexplanationforconfirmation,

andthenaskedthatherprocessbetestedforverification.

66 T: Okay.Tellmewhatthatmeans.[Shesees0.25S–Swrittenon

Amy’spaperwhileNathan’sreadsS‐0.25S].

67 A: Itmeansthatyoutimesitbythepercent,whichis0.25,andthenyou

havetosubtractitandthat’swhatyouhavetopay.

68 T: Subtractit.Whatdoyoumeansubtractit?

69 A: Subtractthetotalfromit.

Ms.GeorgeconfirmedthatindeedtherewasanerrorinbothAmy’s

statementandherwrittenworkandfocusedontheexpressionthatAmyhad

written.

70 T: Soyou’regoingtofindthediscount,0.25timesSisthediscount.

Onceyougetyourdiscount,you’regoingtosubtractthediscountandthe

total.

71 A: [Nodsherheadinagreement.]

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Ms.GeorgeofferedanumericalexampleandaskedAmytotestherlineof

reasoning.ShethenworkedwithAmyandhergroupconsistingofNathan(N)and

Lisa(L).

72 T: WhatkindofnumberwouldyougetifItoldyouthatSwas$12.

Plugin$12forSandtellmewhatyouget.[WhileAmydoeshercomputation,

anotherstudent,Nathan,inthegrouphandedhispapertoTandTlookedit

over.]Now,whatI’dlikeyoutodo,isIwouldlikeyoutoshowmeWinterms

ofonecondensedS,withSwithsomething.Whatwouldthat[inaudible]…

LisaalsopartofAmy’sgrouprespondedtoMs.George’squestionabove.

73 L: …$9

74 T: Whatwouldbe$9?[AskingL].

75 L: [inaudible.]

76 T: IfSwere$12.

77 A: Howdidyouget9?

Amywasattentivetotheconversationtakingplace,andaskedaboutwhere

$9camefrom.Ms.GeorgeinvitedLisa,inthegroup,toexplaintoAmy.

78 T: Iwouldlikeyoutoexplaintoherhowyougot9.[AskingLisato

explaintoAmy.]

79 A: [Lookingpuzzled]Howdidyouget9?

80 T: Holdon.Whatdidyoudo?[AskingAmy.]

81 A: [Checkshermultiplication.]

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82 T: Okay.Youjustdid12times0.25.Youdidthisstep[pointingout

Amy’s0.12Sportionofher0.12S–Sexpression].Okay.Youhaven’tfinished.

Nowminusthisstep[pointingtothe–Sportionoftheexpression].SoSis12

[andasAmydoeshersubtraction,TtakesAmy’spaper]letmeshowyou

something.That’showshegot9,butletmeshowyousomething.Yougot3

forthis.9–12.[Writingthembelowthe0.12S–S]Whatdidyoudoinyour

expression?Youtookwhat?

Ms.Georgetriedtousethenumericalexampletoillustratetheerrorinthe

problemsetup.Amywaslookingintentlyatthewrittenwork.

83 A: [LookingatT’swork] Amyrealizedhererror.Shethenrewroteherexpressiononherpaper.84 T: Thediscountand….whatdoyoumean,youflippedit.

85 A: [Inaudible.]

86 T: Yes.That’swhyIgaveyouanumbertotrytoseeit.Youcan’ttake

adiscountandthensubtractthetotal.Yousubtractthediscountfromthe

total.

87 A: Oh.

88 T: Alwaysmakesureyouputyourtotalfirst,ifyou’regoinginthat

direction.Good.

Ms.Georgefocusedthestudent’sthinkingonthemechanicsofwhatwas

happeningthroughanumericalexample.Withtheexample,thestudentwasableto

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makesenseoftherelationshipofthetermsandtoreconsiderhowtocorrecther

expressionMs.Georgebuiltonthestudent’sthinkingandmaintainedthecognitive

levelofthetaskbychangingthefeaturesoftheproblemandseeingifaneasierand

non‐algebraicformulationoftheproblemwouldhelpthestudenttoseethe

connection.

Insummary,probingguidanceresponsesconsistentlyreverttostudents’

thinkingbybuildingontheirthinkingandaskingforexplanations,reasons,and

justifications.Questionsaskedbyteacherswereopen‐endedforstudentsto

consider,discuss,andrespondsometimesamongsmallgroupsorinwholeclass

discussions.Inmostinstancestimewasgivenstudentstoconsiderthequestion

aloneorasagroup.Thistimeintervalsrangedfromasshortas20secondstoclose

to15minutes.

Finally,incontrasttothepreviousthreetypesofteacherresponses,the

affordanceresponsesgavestudentsopportunitiestofurtherexploretheirthinking

andtodiscusstheirideaswithotherstudents.Thestruggleresolutionwas

thereforenotasapparentorevenachievedinoneclasssetting.Thepersistenceand

intellectualefforttoaddressthetask,however,stillremainedsquarelywiththe

student.Whilesimilartoprobingguidance,theaffordanceresponsesencouraged

studentstocontinueinvestigatingwithevenlessguidancefromtheteacher.

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4.Affordance Affordancetypeofteacherresponsesprovidedopportunitiesforstudentsto

continuetoengageinthinkingabouttheproblemandbuildingontheirideasbut

withlimitedinterventionbytheteacher.Theseresponsesoccurredin11%ofthe

episodes,farlessfrequentlythantheotherthreetypesofteacherresponses.Putting

themathematicalworkbackonthestudentswithoutdisengagingthemfromtheir

taskbecauseoftheirstrugglesproducedvaryingresults.Someinteractionswere

richandsometimesresultedinheateddiscussionsamongstudentsinsmallgroups

overdifferencesinanswers,strategies,procedures,ormisconceptions.These

discussionsweremoreproductivewhenstudentsverbalizedtheirideas,listenedto

eachotherandhadsomemeansofillustratingordemonstratingtheirworkin

relationtotheirclaimsasmembersintheirgroupsoftendemandedevidence.At

othertimes,however,whenstudentswereaffordedmoretimeandguidedlessby

theirteachers,theyfailedtomakeprogressanddisengagedfromworkingontheir

taskandsimplygaveup.

4a)CognitiveDemand

Anaffordanceresponseoftenhintedatpossibleformulationsor

implementationsforstudentstousebutleftthetasksintactwiththeexpectation

thatthestudentswouldcontinueworkingthroughtheirstruggle.Forexample,Ms.

Norrisasked,“Doesthissoundfamiliartothewaterquestionwedidyesterday?

Workitoutwithyourgroup.”

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Teacherswereexplicitinencouragingstudentstocontinuetheireffortsand

engagementintheirtasks.Forinstance,anaffordanceresponsebyMs.Torres

suggestedtoastudentstrugglingwithuncertaintyaboutanexplanation,“Isthatthe

samereason?...I’llletyouponderonthatokay?”Inanotherepisode,Ms.Harris

approachedasmallgroupwithdifferentanswerstotask1.5,“Justifyyouranswer.

Afteryou’vedoneyourwork,thenwriteasentenceyesornoandwhy.Andmake

sureyoujustifyitsomewayeitherwithsomemathorapictureorsomeway….NowI

wantyoutogoaheadandcompareyouranswersatyourtable.”Ms.Harristhen

listenedtothediscussionamongthefourstudentsatatablebutdidnotintervene

withwhatbecameaheatedargumentasstudentsstruggledtoconvinceeachother

oftheiranswers.Thecognitivedemandremainedatalevelfourasintendedinthe

task.


Acharacteristicaffordanceresponsewasforteacherstoconfrontwhat

appearedtobestudentstrugglesandthroughtheinteractionseektoclarifythe

studentideasanddemandrigorintheirexplanation.Forexample,Ms.Torresasked

herstudenttoconsiderhisthinkingthatsheperceivedwasonamisconceptionby

stating,“Buthowdoyouknow?HowdoyouknowthatplayerChada50%chance

ofwinning?I’llletyouponderonthat,okay?”andaffordedthestudentmoretimeto

considerthebasisofhisthinking.Thisapproachproducedbothstrugglesthat

reachedaresolutionduringtheclassperiodandothersthatwereleftunresolvedat

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leastinoneclassperiod.Whatseemedimportantwastoacknowledgethestruggle

thestudentswerehavingandprovidesomeinsightintoinvestigatingthekindsof

questionsthatcouldleadthemtobetterunderstandtheproblem.


Theteachersfocusedonthestudents’effortstoseeiftheirideasworked

ratherthanevaluateiftheywererightorwrong.Studentsoftenappearedto

restrainthemselvesfromofferingtoomuchandwithheldinformationthatmight

haveresolvedthestudentstruggleswithgreatereasebutwouldhavedeprivedthe

studentstheopportunitytousetheirowneffortstoovercometheirstruggles.

Thefollowingepisodecapturedcharacteristicsoftheaffordanceresponse

wherestudentsweregivenopportunitytoconsidertheproblemfurtherwithtime

fordiscussionandindependentthinking,knowingtheircontinuedeffortandwork

wouldbeseenasworthwhile.

EpisodeA1:

AgroupoffourstudentsinMs.George’sclasswereworkingontask3.3:

Giventhreebagscontainingredandbluemarbles,Bag1with75redand25blue;Bag

2with40redand20blue;Bag3with100redand25blue.HowcanyouchangeBag2

tohavethesamechanceofgettingabluemarbleasBag1?Explainhowyoureached

thisconclusion.

Astudent,Jeremy(J),istentativeinhowtodothislevelthreetask.89 J: Add5tobag2?

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90 T: Well,Idon’tknow.Canyou?Youwanttochangebag2tohavethe

samechanceofgettingabluemarbleasinbag1….okay,wellwhydon’t

youtestit.

Here,Ms.Georgetakesupthestudent’sthinkingandencouragesJeremyto

testhishypothesis,whichistohavethestudent“do”mathematics.

91 J: Thatwouldgiveme…thatwouldn’twork.Ihavetotake10awayfrom

that.Sothat’dbe 3060oronehalf.

92 T: Whatwouldthatdoforyou?I’mjustalittleconfusedastowhat

you…Sothisiswhichbag?...markitforme…andyoumayhavetolookatthe

marblesinthereandseewhatyouhavetochange.Okay?Justthinkabout

thatforamoment.Okay.(LeavesJtowork,asheseemsengagedwiththe

problem.).

Ms.Georgetookupwhatthestudentwasthinkingasaninitialresponsebut

didnotevaluatewhethertheanswerwasrightorwrong.Instead,theteacherasked

forwhetherthestudent’ssolutionwaspossibleandreiteratedthegoalofthetask.

Thestudentthenhadtoevaluatehisownanswer.Theteachersupportedthe

studentbyaskingclarifyingquestionsandhighlightingwhatappearedtobethe

resultofhisorherlineofthinking.Thissuggestedthattheprocessesof

conjecturing,testing,andfollowinguponone’sideasareimportant.

Theseactionsbyteachers“pressed”(Kazemi&Stipek,1997)studentsat

variousleveltoreason,justify,andconnecttotheirthinking.Thestudentand

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teacherinteractionsthatensuedlastedfromafewminutesofdiscourseinsome

casestoanentireclassperiod.Intheselattercases,thediscussionsevolvedinto

teachingopportunitiesdirectedtothewholeclass.Forexample,Ms.Torres

addressedthewholeclasswhenasignificantnumberofstudentswerestruggling

withaconceptinprobabilitytodeterminethechanceofgettingatailandahead

whentwocoinsweretossed.Sheconnectedthestudents’struggletotheirprior

knowledgebycreatingasamplespaceforthisexperiment.Herexpectationswere

notonlyaboutthestudents’computationoftheprobabilitybutthestudents’ability

toexplainwhereandhowtheygotthenumbers.Otherteachersusedthistechnique

ofaddressingthewholeclassoveraspectsofstrugglethatsurfacedwhile

interactingwithindividualsorasmallgroupofstudents.Throughobserving

studentsduringthesephasesofproblemsolving,theteacherswereabletolisten

andrespondtostudentsworkingontheirtasks.Whenstrugglesoccurredat

multiplesites,someoftheteachersresortedtoawholeclasspresentationor

discussion.Forexample,inobservingseveralofherstudentsstruggling,Ms.Fine

wenttothefrontoftheclassandstated,“I’mgoingtogoovernumberonebecause

we’rehavingalotoftrouble;confusionhere.Okay?“

Insummary,affordanceresponsesaskedstudentsforexplanationswith

detailsofstrategies,procedures,ortheirthinkingandpressedforstudentsto

considerfurthertheirjustificationandsense‐makingoftheseproblemswhetheron

theirownorwithgroups.Acriticalcomponentofthistypeofinteractionresponse

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wastoprovidestudentstimetoattendtotheirthinkingandsomemotivatingreason

tocontinuetoworkontheirtask.Theteachershadtomonitortheprogressofthe

students,however,asmomentumindoingthemathematicscouldgetlostifthe

studentscouldnotnavigatebeyondtheirstruggle.

DiscussionofTeacherResponses Studieshaveshownthatteachersimplementavarietyofmovesintheir

interactionwithstudentsthatisdictatedbythesituation,needsofthestudents,and

theirownbeliefsandcontentknowledge(Anghileri,2006;Haneda,2004;Stein,

Grover,andHenningsen,1996;Dweck,1986;Kennedy,2005).Theseactions

includeconnectingtostudents’priorknowledgeandbuildingonstudentthinking,

questioningstudentsinordertoprobeandclarify,andpressingstudentswith

intellectualworkinordertomaintainthecognitivedemandofthetask.Theprior

researchaddressedmanyissuesaboutteachingthatinformedinstructional

practices.Theseactions,however,havenotallbeensynthesizedtoexaminehow

theycouldbeusedspecificallytosupportstudentstruggles,particularlyina

productivemanner.TheframeworkthatIdevelopedwasbasedonthenotionthat

teachersusearangeofactionsastheyinteractwithandrespondtowhatstudents

aresaying,writing,anddoinginanattempttoaddresstheirstruggles.Acontinuum

modelcapturesthebroadsetofpracticestheteachersimplementthatisresponsive

tothestudentactionswithanunderlyingintent,purpose,andfunction.

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Whilesomeresponsescastawidenetalongthecontinuum,teachersoften

demonstratecertaingoalsanddirectionsintheirinteractionwiththestudent.I

noticedresponsesthatprovidedclarityforstudentsthroughnarrowingafieldof

examinationwhileothersrestrictedthefieldtotheextentthatitputthestudents’

ideasoutoftherangeofconsideration.Thesecancreateverydifferentlearning

opportunitiesforthestudents.Forexample,inaninteractionwithastudentand

Ms.Norrisovertask1.5,astudentrespondedthatthefullnesswasnow“23outof

48”intherainbarrel.Ms.Norriswentontoaffirmthestudent’sresponse,“sothat’s

right,buthowdoyougetthatit’smorethan40%?Howdoyougettothepercent?”,

wherebyMs.Norrisnarrowedthefieldofexaminationwithoutputtingthestudent’s

ideaoutofconsideration.Incontrast,Ms.Fine’sresponseoverthesametask

includedthestatement,“Okay,what’stheotherwaywedidthis…”whichrefersnot

tothestudent’slineofthinkingbuttowhattheteacherhadinmind.

ThethreedimensionsthatIusedtoanalyzetheteacherresponses,namely

maintainingthecognitivelevelofthetask,addressingthestudents’struggle,and

buildingonstudents’thinking,providethelensesthroughwhichonecanexamine

theproductivedirectionthestudent‐teacherinteractionsaretakingaboutthe

studentstruggles.Theyalsoprovideameanstogaugethepossibleoutcomesthat

result.

Myfindingsshowthatteachersmustconstantlystrikeabalancebetween

tryingtosustainstudentengagementandmaintainingthecognitivedemandofthe

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task(Kennedy,2005).Weseethatinvaryingdegrees,theteacherresponses

provideddirection,hints,corrections,andsuggestionswhenstudentswereataloss.

Theinteractionsrevealtheteacher’sroleintryingnottooverwhelmthestudents

whopossessvaryinglevelsoftoleranceforpersistenceandfrustration.Attimes,

intenseinteractionsamongteachersandstudentssignaledaneedtomoveonwith

thetaskwithoutgivingthestudents’possibleneededtime.Similartothefindingsin

theQUASARstudy(Stein,Grover,andHenningsen,1986),Ifoundthatwhen

teachersfocusedonthestrugglingstudents,theyalsoriskedlosingthefocusand

engagementoftherestoftheclass.Theteachers,therefore,appearedtofocuson

twolevels:firstontheimmediacyofhowtobestaddressthestrugglingstudentwith

thetask’sgoalsandcognitivedemand;andatthesecondlevel,themanagementof

therestoftheclasswhowerenotengagedorfinishedwiththeirtask.Myanalysis

findsthatdespitethesechallenges,someteacherschosetoaffordstudentstime;to

question,probe,clarify,interpret,orconfirmstudents’thinking;andtoprovide

opportunitiesfordiscussionamongclassmates.Thesefactorscontributedto

keepingtheintellectualworkofthetaskssquarelywiththestudents.

Inspiteofeffortsbyteacherstokeepstudents’thinkingvisibleandthefocal

pointoftheintendedtask,ifthestudentsbecamestymiedorshowedsignsof

frustrationorlackofresources,theteacherresponsesthenattemptedtobalancethe

probingquestionswithencouragement,andthekindsofguidancethatwouldkeep

thosestrugglingstudentsengagedwhilefocusedonattendingtotheirstruggle.

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Knowledgeoftheirstudentsinfluencedhowteachersrespondedtotheirstudentsas

notedinaninterviewwithMs.Georgewhereshementionedinherpostinterview

thecasesoftwoofherstudents;thefirsttypewasanexampleofasilentstruggler

andthesecondsheviewedasfairlystronginmathematicalability.

Interview1:

I’malwaysafraidI’mgoingtomissthatonewhodoesn’ttalkawholelot…Imisswhenhestrugglessometimesbecausehe’ssoquiet….andsometimesIforgetthosequietones…Ihavetowalkuptoandsayhowareyoudoingorjustreallychecktheirworkbecausetheywillneveraskmeforhelp.

Interview2:

[Noticestudent]overhere,howfrustratedhegotwhenhedidn’tunderstandwhatthedifferencewasbetweenthequestions?Well…hewantedmetotellhim.Andhewantedmerealquicktosaywellit’dbethesameright,right?AndIwenttolookatthequestionsandthatjustmakeshimsomad.

Manyoftheteacherresponsescorrespondedtopriorresearchthathas

showntheimportantrolequestioninghasingivingdirectiontostudents’thinking

andorganizingtheirideas(Sorto,McCabe,Warshauer&Warshauer,2009;

Anghileri,2006;WilliamsandBaxter,1996).Teacherresponsesgenerallyconsisted

oftryingtodeterminethenatureofthestrugglesothatsuchquestioninghelped

teachersassesstheirstudents’thinking(Cazden,2001).Myfindingsshowthatmost

teachersresponsestostudentstrugglesbeganwithanassessmentofwherethe

studentswereintheirtask,whetherbyaskingtohearaboutorseethestudents’

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writtenwork.AcommonresponsewassimilartoMs.Torres,whoaskedastudent

strugglingtoexplainhissolution,“Ican’tquiteunderstandhowyougot[there]…can

youexplainthattome?”Thisformativeassessmentwouldinformboththestudent

whoappearedunsureandtheteacherwhowasuncertainhowtoproceedwiththe

interventionandsupportthestudent.

Studieshaveshowntherolepriorknowledgeplaysinconnectingnew

knowledgetostudents’workingknowledgeasstudentsengageinmathematical

tasks(Bransford,Brown,&Cocking,1999;Rittle‐Johnson,2005;Richland,Holyoak,

&Stigler,2004).Inmyfindings,teacherresponsesalsomadereferencestomethods

andconceptsstudentshadbeenexposedtoandmadeanalogiesthatrelatedtheir

currentproblemstoproblemsthathad“easiernumbers”orwerenumericalrather

thanalgebraicaswiththetasksinactivity3thoughttoberoutine.Responsesto

studenterrorsandmisconceptionsservedtohighlighttheimportantvaluethe

teachersplacedonreasoningandsense‐makingasteachersconfrontedthestudents

withthinkingthatmayhaveleadtomistakesoruncertainty(Eggleton&Moldavan,

2001;Borasi,1994).Theseinteractionsabouterrorsgavestudentsopportunitiesto

revisetheirthinkingandnotdismisstheefforttheyexpendedbyacknowledging

aspectsthatcontributedtotheprocessofproblemsolving(Gresalfi,Martin,Hand,&

Greeno,2009).

Whatisimportanttonoteisthatwhenteacherresponsesmaintainedthe

coherenceofthetaskgoalandthestudentstrugglesrequiredtoachieveit,the

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responsesshowevidenceofsupportforthestudentsintheirintellectualeffortsand

apursuitofthetask’sobjectiveswithoutsimplificationorremovingthechallenge

fromthestudents.Threeprimaryfactorsappeartoinfluencetheteacherresponses:

theaccountabilityoftheexpendedefforttheteachersexpectsstudentsto

demonstrateasshowninhowteachersprobestudents’thinkingandaffordtime;the

valuethatteachersplaceonaccomplishingthetaskinrelationtothevalueof

studentsformulating,implementing,andmakingsenseofthetaskasshowninhow

theyusedirectedguidance;andtheefficiencybywhichtheteachersperceivethe

taskneedstobeenactedbythestudentsasseeninhowtheyutilizetelling

responses.

INTERACTIONRESOLUTIONS Inthissection,IwillfirstdescribethreetypesofinteractionresolutionsthatI

documented:productive,productiveatalowerlevel,andunproductive.Second,I

reportonsomepatternsofstudent‐teacherinteractionsandthestruggleresolution

framework.Third,Iuseasanexampleonetaskduringwhichsimilarstruggles

occurredandpresentsampleepisodesthatcapturethedifferentinteraction

resolutionsthatoccurredasaresultofthevaryingelementsinthestudent‐teacher

interactions.Finally,Iclosewithadiscussiononinteractionresolutions.

TypesofInteractionResolutions Iidentifiedthoseresolutionsasproductiveifthey(1)maintainedthe

intendedgoalsandcognitivedemandofthetask;(2)supportedstudents’thinking

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byacknowledgingeffortandmathematicalunderstanding;and(3)enabledstudents

forwardinthetaskexecution.Myfindingsshowthat42%ofthestrugglesfulfilled

allthreeofthesecriteria.

Iclassifiedasproductiveatalowerlevelthoseresolutionsthatwere

productiveinpoints(2)and(3)abovebutthatloweredthecognitivedemandofthe

intendedtask.40%ofthestudentstrugglesresolvedatalowerlevel.Anoticeable

wayinwhichthecognitivedemanddecreasedinvolvedtheredirectionofthe

studentstowardparticularmethodsorstrategiessuggestedbytheteacherandnot

bypursuingstudents’thinking.Anotherwaywasbysimplifyingtheproblemsor

supplyinginformationthatthestudentscouldhaveworkedandobtainedontheir

own.Iclassifythisasstillbeingproductivebecausethestudentsremainedengaged

inthemathematicalactivity,thoughatalowerlevel.

Icategorizedstrugglesasunproductiveifstudentscontinuedtostruggle

withoutshowingsignsofmakingprogresstowardsthegoalsofthetask,reacheda

solutionbuttoataskthathadbeentransformedduringtheinteractiontoa

proceduralonethatsignificantlyreducedthetask’sintendedcognitivedemand,orif

thestudentssimplystoppedtrying.18%ofthestudentstrugglesresolved

unproductively.Becausemyobservationssuggestthatstudentstrugglesdonot

necessarilyresolvethemselvesinhour‐longlessonsorwithintheactivity,whatmay

appearasanunproductiveresolutionmayhavebecomeproductivehadmoretime

beenavailabletosupportthestudent’sstruggles.Mydataalsosupportthefindings

135

ofotherstudies(e.g.Sullivan,Tobias,McDonough,2006)thatfoundsomestudents

seemedtodeliberatelynotengageinthetaskorgetfrustratedandshutdown.

Teachers,therefore,havemuchtoaddressintheclassroomtobalancetheissuesof

engagementofallstudentswithcognitivelydemandingtaskswhileatthesametime

respondtostrugglesamongstudentswithdifferentlevelsoftolerance,motivation,

andpersistencetowardthetasks.

InteractionFrameworkandPatterns Inordertoanalyzeanepisodewithit’sbeginning,middle,andend,Iusedmy

earliercategoriesoftheexternalizedstrugglestoaccountforthebeginningofthe

episode,theinteractionthatensuedwiththeteacherresponseandthestudent

uptakesinthemiddle,andfinallythepatternsofwhatsignaledanendingtothe

episode.Ianalyzedtheresolutionsusingarevisedframeworkfromchapter2.

Figure4.3: ProductiveStruggleFrameworkinaninstructionalepisode

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136

Onenoticeablepatternofanepisodeendingwaswhenastudentstatedthe

correctanswertoataskproblem.Thestudent’sanswerappearedtoresolvethe

student’sstruggleandtoconcludetheepisode.Asteacherandstudentsinteracted

withquestionsandresponses,thestudentswouldgivetheanswerinasuccinct

formsuchas,“The24”intask1.1.Theteacherresponded,“Doyouseewhy?”to

whichthestudentresponded,“Iseewhynow.”Thistypeofstatementendedthe

interactionaboutatask,thoughitwasnotalwaysaccompaniedbyevidenceof

understandingofthemathematicsbythestudent.

Asecondpatternthatbecameevidentamongepisodesacrossthethree

teachingsiteswasthecommonpointsatwhichstudentstrugglesoccurredduring

thesametasks.Studentshadsimilarissuesofstrugglingtogetstartedwith

problemssuchasintask2.3,anopen‐endedquestiontocomeupwithabagof

marbleswithacertainratioofbluetoredmarbles.Strugglestoimplementtasks3.1

and3.7werecommonasstudentsworkedwithalgebraicexpressions.Intasks1.1,

1.2,and1.5,studentsstruggledoversimilarpointsincarryingoutprocessesand

explainingtheiranswers.

Athirdpatternsuggeststhatinlookingatthesimilarstruggles,thekindsof

teacherresponsescreateddifferentoutcomesintheproductivequalitiesofthe

struggleresolutions.Keepinginmindtheexploratorynatureofmystudy,I

examinedmydatawiththefollowingquestion:Givenataskwithinwhichsimilar

137

strugglesoccurred,domyfindingsshowthatcertainkindsofteacherresponseslead

toparticulartypesofresolutionsofthestudents’struggle?

ExampleTaskWithDifferingResolutions Iusetask1.5oftheBarrelofFunactivitytoillustratehowataskthatelicited

similarkindsofstrugglesresultedindifferentresolutions.Thefirstepisodebegan

withastudentstrugglingoveraninabilitytoexplainherapproachtotask1.5.The

student‐teacherinteractioninvolveddirectedguidanceandresolvedinaproductive

struggleatalowerlevel.Theothertwoepisodeswillincludeoneproductive

struggleresolutionandoneunproductivestruggleresolutions.

Example4.1:ProductiveStruggle–Lowerlevel Theepisodebeganwhenastudent,Nora(N),sittinginagroupoffourwas

approachedbyMs.Norris(T)andwasunabletoexplainheranswer.

93 N: Welllike,soliketomakeabettercomparison[pointingtoboth

graphicalrepresentations]IgavelikeIgavethemthesamenumberof

thingsandlike[gesturingwithhandsinaflutter]likeIdon’tknowhow

toexplainit,it’sjustkindalike…[pause]

Norahaddrawnarepresentationofthetwocontainerswithwateronagrid

sheetprovidedbasedonapercentagefilledratherthanallowingeachsquareto

represent1gallonassuggestedintheinstructionsfortask1.4.Asaresult,both

containersappearedtobethesamesize,namely100%depictedas10vertical

squareswith5shadedforthe =50%filledrainbarreland6shadedforthe =24

48

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60%waterjug.ThefollowinginteractionwaspromptedbyMs.Norris’response

afterobservingNora’sworkandthestrugglethatshehadexternalized.In

supportingNora’sstruggle,Ms.NorrisprobedNora’sthinking,validatedher

attempt,andnotedpossibleshortcomingsoftherepresentationNorahadonher

paper,namely,herinabilitytoaccuratelyandquantifiablyseeandstatethe

differenceinchangeusingtherepresentationsonherpaper.SheaskedNoraabout

drawingthecontainersasoriginallysuggested.

94 T: Doyouthinkyou…andthat’swhatwewant,wewanttoseea

comparisonofthetwosothey’reequal.Myquestionis,I’mwondering,do

youthinkyoucouldhavedrawna48‐gallonoverhere?(Pointingtoanopen

partofthegraphpaper.)

95 N: Yeah,Ihaditoverhere[pointingtoaportionofthegraphpaper]

96 T: Where?

97N: AndIerasedit.

98 T: Why?

99 N: SoIcoulddoitthisway.

100 T: Soyoudidhalflikethis[pointingtohercurrentgraphicof5shaded

outof10].Nowwhathappensifyoutakeonegallonoutofhere,how

wouldyoushowmethat?

101N: [Usesherthumbtocoveruponesquareofthegraphicforthe5outof

10shaded‐representingthe24outof48gallons].

139

102 T: It’skindofawhat?

103 N: Idon’tknow.

104 T: Isitaccurateorareyoukindofguessingandestimating?

105 N: Kindofguessing[withagiggle].

106 T: Sodoyouthinkmaybeifyouhaddrawnitasa48andshaded24you

wouldhavebeenabletoadjustthatalittlebiteasier?

107 N: Iguess.

Ms.NorrisdoesnotinvalidateNora’sworkbutsuggeststhatverifyingand

justifyinghersolutionisproblematicwiththecurrentrepresentation.Ms.Norris

continuedtopushNora’sthinkingwithquestionsthatcouldleadhertowardsaline

ofreasoningandausefulrepresentationthatwouldthenrevealtheimportant

featuresoftheproblem.

108 T: Igetwhereyou’redoingthis[pointingtothehalfgraphic]butjust

becausethey’rebothshowingahalfandthisisalittlebitmorethanahalf,

youdon’tknowforsureexactlywhatahalfissoyouwanttodrawitto

scale.

109 N: Sodoesitordoesitnot?

110 T: Doesitmatter?Doesitchangewhatcontainerisfuller?

111 N: No.

112 T: Okay,sohowareyougoingtojustifyitforme?

140

Nora’sfrustrationbecamemoreapparentandMs.Norrisrefocusedonthe

question.Shethenofferedasuggestionaboutapossiblenumericalwaytoviewthe

task,asNora’scurrentrepresentationhadnotbeenhelpful.

113 N: Idon’tknowhowtojustifyit.

114 T: Youdon’tknow…hmmm.Dowehaveanynumbersthatweknow

aboutthatwecanwrite?

115 N: Yeswedo.

116 T: Whatnumbersdowehave?

117 N: 48.

118 T: Gotthat.Whataboutthat48?

119 N: 24outof48andthen3outof5.

120 T: Okaybutwhathavewegothere?

121 N: 2outof5and23outof48.

122 T: Let’swritethosedownsothatwecanbethinking.

Ms.NorrisdirectedNoratoconsidertherelevanceofthegiveninformation

byrecordingthemforfurtherexamination.ThisgaveNoraapossiblepathfor

carryingoutaprocedure.Ms.NorrisleftNora’ssidebutreturnedafteraminuteof

talkingwiththethreeothergirlsinNora’sgroup.ThisprovidedtimeforNorato

continuethinkingabouttheproblem.WhenMs.NorrisreturnedtocheckNora’s

progress,shefoundthatNorahadnotmademuchprogressandwasshowing

furthersignsoffrustration.Shethenaskedaboutawayofconsideringthefullness

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withanexpectationofjustification.Here,Noraindicateddecimalsasanalternative

representationthatmightprovehelpful.Ms.Norrismadeanacknowledgementof

thatideaandagainleftNoratoherworkagain,affordingthestudentstimeand

opportunityfordiscussionandsense‐making.

123 T: Youstilldon’tknow.Wellwhatdidyouwritedown?Talkwithin

yourgroupaboutthisproblem.[Announcestothewholeclassasshe

walkovertoNagain.]

124 N: [Motionswithfingerstowardsherwritingonthepaper.]

125 T: [LookingatM’swork]Justoneless.Youthinkthesame…youstill

thinkthis5‐gallonjugisfuller?

126 N: [Looksatherpapersbutdoesnotrespond.]

127 T: You’rejustbasingthisonaguess.

128 N: Maybe.

129 T: Wellhowcanyouproveittome?Doyouhavesomethoughtson

that?

130 N: Idon’t.[Exasperatedandpullsherhair.]

131 T: Youdon’t?What’sanother,what’sanotherwayIcouldexpress

exceptasafraction?What’sanotherwaytowrite?

132 N: Asadecimal.

133 T: [TapsN’sforearminacknowledgement,thenpointstothepaper]…do

youthinkwecouldwrite inoneofthoseotherformatstolookat

2

5

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5

142

it?Becausethenthey’dbeonthesamebasis,right?Okay,solet’stry

thatandwe’llcomeback[leavesNandgoestoanothertable].

Afterafewmoreminuteshadelapsed,Ms.NorrisreturnedtoseewhatNora

haddone.DuringMs.Norris’absence,anothermemberofNora’ssmallgroup,S2

seemedtohaveawayofcomparingthefullnessofthetwocontainersinawaythat

madesensetoNora.Shethenpreparedtousetheseideastowriteherjustification

onherpaper.WhenMs.NorrisobservedthesatisfiedNora,shesuggestedthat

Nora’sworkwasattributabletoS2.Nora,soundingindignant,statedthatshedid

indeedunderstandtheproblemnowandimpliedthatshewaspartofthegroup

effort.Ms.NorrisinsistedthatNoramustjustifyhersolution,assheunderstoodit.

Thismoveconcludedtheepisodeasthetaskshifted.

134 T: [ReturnstolookatN’spaper].Shemusthavedoneagoodjobof

convincingyou[pointingtoS2].Iwanttoseethatworkstill.

135 N: Ihavemyowntrainofthought.

136 T: Okay.S3,didyouwritethat?Thatitchanged.Okay.Iwanttosee

justification,notbecauseS2saidso.

137 N: Nooooo,wealreadywentthroughitasagroup.[Ratherindignantlyto

T.]

138 T: Okay[acknowledgingthenmovestoanothergrouptoinquirewhat

theygot.]

143

TheresolutiontoNora’sstruggleoccurredoveranextendedtime.Ms.

Norrisleftthegroupseveraltimessothatthegroupmemberscoulddiscussand

sharetheirideasandwork.Therewereothergroupsthatwerealsostrugglingover

thistask.Ms.Norris’intentseemedtobetoallowthestudentstoworktogetherand

tolimitherguidance.OnestudentinNora’sgroup,S2,wasabletoexplainher

solutioninawaythatmadesensetohergroupmembers.Inexplainingtoher

group,S2ineffectplayedtheroleofteacher.Whilethecognitivedemandwas

decreasedforNora,sheseemedtotakeownershipofherunderstandingofthe

problembyherpersistenceintryingtomakesenseofbothherworkandthework

ofothersinhergroupandtherebyproductivelyresolvedherstruggle.

Thefollowingexamplecapturesaproductiveresolutionatahighlevel

becauseitmaintainedthehighcognitivedemandofthetaskandthetaskresolution

wasachievedthroughthestudent’sengagementandintellectualeffort.This

examplealsoservestoillustrateafairlycommonoccurrencewherestudentsdidnot

showstruggleuntilaquestionposedbyateacherorotherstudentscreatedan

uncertaintyorconfusionvoicedbythestudents.ThisepisodeinMs.George’sclass

beganwithDrew,whogavehisanswerfortask1.5withnoindicationthatitwas

incorrect,“Iputnobecausetherewouldbethesameasbeforebecauseyouhave

takenagallonfromboth.”

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Example4.2:ProductiveStruggle Ms.Georgewasfacedwithastudent(Drew)whohadanincorrectanswer

stemmingfromamisconceptioninproportionalreasoning.Ms.Georgeresponded

toDrew’sanswerwiththefollowingprobingquestionstosolicitstudent’sthinking:

“Okay.Soshowmewhatthatwouldlooklike.Showmewhatyourgallonswould

looklike.Ifyoutakeagallonfromeach,whatareyoulookingat?”WhenDrew(D)

showedhisworkonhispaper,Ms.George(T)usedittoquestionandconfirmhis

work.

139 T: Okay.Sowhatyou’retellingme(pointingtoDrew’sworkonhis

paper),youhave23gallonsoutof48and2gallonsoutof5,thatyou’restill

goingtohave2gallonsoutof5willbe…

140 D: Lower,lessfull.

141 T: Butdidn’tyoutellme wasmorefull.

142 D: Wait,wait.

143 T: Sowoulditchange?

144 D: Oh.Okay.Yes.Yesbecause,see…

145 T: Becausewhy?

Ms.GeorgesoughtconfirmationfromDrewabouthisstatementandprobed

himtogiveanexplanationforhisansweraswellasprovideamathematicalwayto

verifyhisclaim.

146 D: Theywouldbethesameasbeforebecauseyou’retakingagallonfrom

both.

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145

147 T: Butthey’renotgoingtobethesame.Yousaid,yesbecausethey

wouldnotbethesame.They’reaskingyou,woulditchange?Youjusttold

meitchanged.

148 D: Itwouldnot.[Eraseshisanswer.]Okay.Yes,theywouldnotbethe

same.Yes,theywouldnotbethesameastheywerebefore.[Lookingintently

athispaper.]

149 T: Okay.Whichoneareyoutellingmeisfuller?

150 D: .Butisn’tthatfullernow?[LookingupatTquestioningly.]

151 T: Whywouldthatonebefullernow,doyouthink?

152 D: Becausetheotheroneisn’tahalf.

153 T: Okay,it’snothalf.Thatmeansit’snotfuller?

154 D: It’smorethan0.5now.

155 T: Tellmethatonemoretime.

Thereisalotofconfusiononthepartofthestudent,andMs.Georgeasked

Drewtorepeatwhathehadsaidtoclarifyhisstanceandreasoning.Atthesame

time,Ms.Georgetriedtoslowdownthepaceofthedialogueandnotshow

impatienceforananswer.

156 D: Isaidit’smorethanahalfnow.Becausethisoneisnolongerahalf

because[squintsandthinks]youhavetosubtract,youhavetakenawaya

gallon.

157 T: Okay.

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146

158 D: Andthisoneisnolongersix‐tenthsbecauseyouhavetakenawaya

gallon.Andthisonewouldnolongerbeahalf[inaudible]itwouldn’tbethe

same …[takesabreath].IknowwhatI’msaying[inaudible]…

159 T: Bepatient.[GivesDtimeandlistensintently.]

Ms.GeorgewatchedasDrewsetupalongdivisionprocess.ShesaidtoDrew,

“Getyourpercentsandcallmebackover.Keepworking,”andthenlefthissideas

Drewcarriedouthiscomputation.Ms.Georgeofferedencouragementinwhat

appearedtobeconstructiveengagementonthepartofDrew.Drewhadtoaskhis

groupmatetolethimkeepworking,thencalledMs.Georgeback.

160 D: Backoff[whenhisneighborlooksathispaper].…Okay.Ihaveit.

[CallsouttoT].Ring,ring,ring,ring[makesabelllikesound].

161 T: [ComesovertoDrew’sside]Whathaveyougot?I’mhere.Gladyou

called.

162 D: NowIhavegotthepercentage,drumrollplease.

163 T: [Tapsonhisdesk]go.

164 D: Thisoneisnow41%.Thisis40%.

165 T: Whichone’sfuller?

166 D: The48‐gallonbarrel.

EvenasDrewfoundawaytoexplainhisconclusion,Ms.Georgeprobedhim

withafollowupquestiontoseeifDrewmightbeabletoprovideindicationsof

deeperunderstandbeyondthecomparisonofpercentages.Shealsoseemedto

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addressthemisconceptionofequalquantityremovalnotnecessarilyleadingto

equalpercentageremovalandwhythecontainersizemattered.Thoughthe

studentsdonotrigorouslystatetheanswerforher,Ms.Georgegavethestudentsan

opportunitytothinkaboutandexplaintheconceptualnatureoftheproblem.

167 T: Beforeyousaid…whydoyouthinkitchangedonyou?Whydoesit

change?Justbyonegallon?

168 D: Becausethisonewasnothalf.

169 S1: Becausehereismoregallons,thepercentwoulddroplikeless.

170 D: Likesoyeah,whathesaid,moregallonsthenthepercentwoulddrop.

171 T: Thepercentwouldnotbeasbig.Right.Good.It’sgoodteameffort.

Theaboveresolutionisanexampleofaproductivestruggleinwhichthe

cognitivelevelwasmaintainedasMs.George’spressedthestudentstofurther

reflectandattempttomakesense.Sheprobedthestudentthenletthestudenthave

timetoconsiderthequestionsthatwereposed.Throughthatprocessofreflection

andopportunitiestoexplaintheirthinking,Iconjecture,thestudent’slevelof

understandingmayhavebeendeepened.

Thefollowingepisodeillustratesanunproductiveresolution.Incomparison

totheintellectualworkdemandedofMs.George’sstudentsinaproductivestruggle,

weseetheuseofatellingresponseinwhichthetask’slevelofcognitivedemand

wasloweredsignificantlywithelementsofanalysisandexaminationofsolution

strategiesundertakenbytheteacherratherthanthestudents.Astudent,S1,inMr.

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Baker’sclasswasunabletoexplainherwork.Theteacherresponseinvitedanother

studenttohelpmakeanexplanation.

Example4.3:UnproductiveStruggle172 S1: Thebarrel.Ididn’tgetit. Mr.Baker’s(T)responsewastoincludeLily(L)whohadworkedonthetask

withS1whenS1wasunabletojustifyheranswer.Theresponsefailedtoaddress

thenatureofS1’sstruggleandputemphasisonarrivingattheanswer.

173 T: Youdidn’tgetit.Nowwho’syourpartner?Lily?Lily,canyouhelpher

out?Whatdidyousay?

174 L: Isaidthe48‐gallon[inaudible].

175 T: Because…sayitonemoretime…because…

176 L: Becauseifyoutakeoutagallonfromeach…

177 T: Soyou’resayingthebarrelwouldbefullerorthejugwouldbefuller?

178 L: Thebarrel.

Lilywasabletogivethecorrectanswerbutdidnotexplainorelaborateon

whyshedecidedthatthisanswerwascorrect.Instead,Mr.Bakerprovidedthe

explanationtotheclasswhiletheclassremainedquietandunresponsive.The

studentsmayhaveunderstoodhisexplanationbutfromthelackofresponsesand

attentivenessfromtheotherstudents,onecannotconcludethattheywouldbeable

tojustifytheiranswer.Insomecases,teachersmaygivesolutionsthatthey

themselvesunderstandwithoutmakingsurethatthestudentsdo.

149

179 T: Thebarrel.Okay.Sowhenwe’retalkingaboutit,ifyoutake,ifyou

haveajuganditcanonlyhold5gallons,isthatthatmuchcomparedtothe

barrelthatcanhold48?

180 Class: [Noresponse.]

181 T: Notthatmuch,isit?Butyoutakeagallonoutofthatjug,isthatgoing

tomakequiteabitofadifference?

182 S3: Yeah.

Thisstudent’sresponse,however,wasnotindicativeoftheclass’

understanding,buttheteachercontinuedwiththerecitation,usingtheresponseto

assumesense‐makingbythestudentsatlarge.

183 T: Butifyoutakeagallonoutofthebarrel,doesitmakequiteas

muchofadifference?

184 Class: [Noresponse].

185 T: Notquiteasmuch,doesit?Sowhatyoucandoisyoucantakeand

makefractionsso,letmeseeifIcangetthishere.Imadeafractionthatsaid,

thebarrel,excusemethejugisnow2outof5gallonsfull,right.Andthe

barrelis23outof48gallonsfull.AndthenIcancomparefractionsand

what’seasierformetodowhenIcomparefractionsistochangethem

intodecimals.Sothisoneisgoingtobe0.4,thisoneisgoingtobealittlebit

biggerthanthat,justalittlebitbiggerthanthat.Sowhichoneismorefullor

fuller?[Noreaction.]Thebarrelorthejug?

150

186 S3: Thebarrel.

187 T: Justalittlebit.Becauseitmadesuchabigdifferencetakingagallon

outofthelittlejug.Okay.MovingontoF(task1.6).

Mr.Bakershiftedtoanothertaskaftergivinghisexplanation,therebyending

theepisode.Theepisodefailedtoengagestudentswiththetaskortoprovidethe

justificationneededtoexplaintheanswer.Thestudent’sstrugglewasnot

supportedbutratherwasdirectedbytheteacherswhousedarecitationformat.

Theresponsedidlittletobuildonthestudent’sthinkingortoinvolveherinthe

problemsolvingprocess.Furthermore,theimpreciseuseofthemathematical

languagebytheteacherfailedtomodeltherigorthatwasexpectedinthetask

design.Thisisanexampleofanunproductiveresolutiontostudents’strugglewith

atellingtypeofteacherresponse.

Insummary,theoutcomesforthestudents’struggleandtheresolutionsto

theinteractionwereamixofthosethatwereproductivebymaintainingthehigh

levelofcognitivedemandofthetask,buildingonthestudent’sthinking,and

attendingtothestruggleasaprocessthatthestudentscouldworkthroughwhile

otherswerelessproductivewhenthedirectedguidanceloweredthelevelof

cognitivedemand.Thelatterresolutionsincludedthoseteacherresponseswith

moreinformationgiventothestudent,strategiesandideassuppliedbytheteacher,

andlessintellectualeffortdemandedofthestudents.Finally,unproductive

interactionresolutionsresultedinthestudentsnotindicatingaclearunderstanding

151

ofthetasknoranabilitytomakeprogressintacklingtheproblem.Inaddition,

thoseinteractionswhereteacherresponsesfailedtotapintoorsupportstudents’

thinking,tookoverthechallengingaspectsofthetaskfromthestudents,or

simplifiedtaskstoprocedureswithoutconnectionsresultedinanunproductive

resolution.

DiscussionofInteractionResolutions Myanalysisofstudentstruggleresolutionsshowedthatoutcomescould

differdespitethecommontasksthatservedascontextfortheinteractionsprovoked

bythestudentstruggles.Theprocessleadingtowardsaresolutionappearsfar

morecomplexthanjustrelatingstruggletoresponsewhenwetakeintoaccountthe

uniquenessofthestudentsandtheirpriorknowledge,fluencywithskills,

dispositiontowardsdoingmathematics,andtheirlevelofmotivation.Whatworks

foronestudentmaynotalwaysworkforanother(Gresalfi,2004).Whiletheroleof

thetaskistogivecontextforstudentengagementinmathematics,theroleof

studentengagementandtheinstructionalpracticesteachersbringtotheinteraction

isvitaltosupportingstudentlearning.AstheNationalResearchCouncil(Kilpatrick,

Swafford,&Findell,2001,p.315)asserts:

Ourreviewofresearchmakesplainthattheeffectivenessofmathematicsteachingandlearningdoesnotrestinsimplelabels.Rather,thequalityofinstructionisafunctionofteachers’knowledgeanduseofmathematicalcontent,teachers’attentiontoandhandlingofstudents,andstudents’engagementinanduseofmathematicaltasks.

152

Consistentwithpriorresearch,myfindingsshowthatwithinagiventask,the

natureofteacherresponsesthataddressthecognitivedemandofthetaskandthe

timeaffordedthestudentstoworkareimportantfactorsinhowproductively

interactionscanresolve(Haneda,2004;Steinetal,2000).Secondly,theinteraction

resolutionsdependonwhatthestudentsbringtothetaskintermsoftheirprior

knowledgeandtheirwillingnesstoengageintheproblem(Bransford,Brown,&

Cocking,1999;Dweck,1986).Athirdfactorthataffectedtheresolutionwasthe

structuralconstraintofclasstimeandclassroomdynamic.Timeconstraintsposeda

challengeforteachersastheyattemptedtobringclosuretoataskintheirallotted

classtimebutconflictedwithattemptstoaddressstudentswhocontinuedto

struggleoveraspectsofthetask.Teachersalsohadtobalanceaddressingthe

strugglesofsomeoftheirstudentswiththerestlessnessoftheotherstudentsthat

hadbecomedisengagedorhadalreadycompletedtheirtask.Thisandtheothertwo

factorsareconsistentwithpreviousstudies(Stein,Grover,&Henningsen,1996;

Henningsen&Stein,1997;Lampert,1990;Kennedy,2005)thatpertainto

challengestoinstructionalpractices.

Thosepracticesthatareattentivetothestudents’thinkingandutterances

andthatincludethestudents’fullparticipationwithoutimposingtheteacher’s

thinkingmaintainedthechallengeforthestudentstodomathematics.Thechoices

teachersmakecansupportorminimizestudenteffort.Sometimesthesupplyof

informationmaybeappropriateinordertoachieveamoreimportantgoalofthe

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task.Thesearethechoicesteachersmakethatprioritizethegoalofthetaskwith

theefficiencyorproductivenessoftheprocessusedtoreachthegoal.Inknowing

theirstudentsbytheendoftheschoolyear,interviewsfromtheteacherssuggest

thattheyalsotookintoaccountthestudents’capacityforandinclinationtoward

persistencewiththetask.

Animportantaspectoftheinteractionresolutionthatsurfacedinthe

transcriptepisodeswasthesociomathematicalnormsthatappearedtobeinplace

inthevariousclasses.Theclassroomculture,environment,andnormswerewell

establishedbytheendoftheschoolyear,withclearexpectationsofhowstudents

discuss,question,makeassertions,justify,andmakemeaningofmathematics.The

rangeofstudentengagementandbehaviorevenwithinoneteacher’sdifferent

classesbecameapparentasthestudentsvoicedvariouslevelsofdeferenceand

acceptanceofteacherstatements,explanationandjustificationofmathematical

processes,andeffortontheirtasksbeforerequestinghelpfromtheirteacher.

Inorderforstruggletoberecognizedandresolvedinclassroomdiscussions

insmallgroupsorasawholeclass,studentsmusthaveopportunitiestoactively

engageinarticulatingtheirthinkingandgiveitshape.Thesediscussions,attimes

richandrobustandatothertimesconfrontationalandill‐formed,canalso

supplementteacherresponsesthatmaynotbeabletosupportallthestrugglesthat

occurduringtheclasstime.Acollaborativegroupdynamiccangivestudents

opportunitiestoconnecttoother’sthinking,clarifytheirownthinkingandsupport

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others’inresolvingtheirstruggle.Ifstudentsandteachersaretoengagein

interactingaboutstrugglesthatoccurwhenstudentsworkonmathematics,these

expectationsofdoingmathematicsmustbepartofthesociomathematicalnormsof

theclass(Ellis,2011).

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Chapter5:Conclusion

RESEARCHQUESTIONSANDCONCLUSIONS Thephenomenonofstudentstruggleisoftenviewednegativelyasa

symptomofalearningproblemthatteachingshouldtrytopreventratherthan

utilizeforthepurposeofstudentlearning(Hiebert&Wearne,2003;Borasi,1996).

Somemathematicseducators,researchers,andtheoreticians,however,havewritten

aboutaspectsofstudentstruggleaspotentiallybeneficialandpromisingtoward

learningmathematicswithunderstanding(Hiebert&Grouws,2007;Hiebert&

Wearne,2003).Idesignedmystudytofocusonexaminingstudentstruggleswith

thegoalofgainingabetterunderstandingofthenatureofthestudentstrugglesand

thekindsofteacheractionsthatguidethestrugglestowardaresolution.Thestudy

examinedstudentstrugglesastheyoccurrednaturallyinmiddleschool

mathematicsclassroomsinthecontextofstudentsworkingontasksofhigher

cognitivedemand.Findingsfrommyexploratorycasestudyprovidedescriptionsof

whatstudentstruggleslooklike,evaluatehowteachersrespondtothesestruggles,

andpresentevidencethatthereareaspectsofstudent‐teacherinteractionsthat

appeartobeproductiveforstudentlearningofmathematics.TheProductive

StruggleFrameworkIdevelopedisusedtoexaminethephenomenonofstudent

strugglefrominitiationtointeractionandtoresolution,andcanbeusedinfuture

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studiestomeasureanddeterminetheoutcomeoflearningthatoccurredasaresult

ofthestruggleprocessstudentsexperienced.

Theresearchquestionsthatguidedmyinvestigationandtheempiricaldata

gatheredprovideinsightintothoseaspectsofstudentactions,teacheractions,and

thecontextsoftheinteractionsthatresolvestudentstrugglesmoreproductively

thanothers.

Bywayofreview,myresearchquestionswere:







Myconclusionisbasedonmyfindingsfromtheanalysesreportedinchapter

four.Ifirstsummarizemyfindingsinrelationtothetworesearchquestionsand

thenelaborateonmyconclusionregardingaspectsofproductivestruggle.

1. Iidentifiedfourkindsofstudentstrugglesthatoccurredwhilestudentswere

engagedinmathematicaltasks.Thestrugglescenteredonactionsthat

studentsattemptedbutappearedunabletocompletesuccessfullywithout

someformofintervention.Thesefourstrugglesasdescribedinchapterfour

are:

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• Getstarted

• Carryoutaprocess

• Giveamathematicalexplanation

• Expressmisconceptionanderrors

2. Teacherresponsestothestudentstruggleswereoffourtypesofvarying

gradationsalongacontinuum:

• Telling

• DirectedGuidance

• ProbingGuidance

• Affordance

Myanalysisoftheteacherresponsecategoriesfocusedontheireffectonthe

cognitivedemandoftheintendedtask,howtheyaddressedthestudentstruggleas

voicedduringtaskimplementation,andhowtheybuiltonstudentthinking.

Findingsshowedthatthecognitivedemandofthetasksgenerallydecreased

inthetellingtypeofresponses,weretoalesserdegreeinthedirectedguidance,and

weremaintainedintheprobingguidanceandaffordancetypes.Thestudent

strugglesweredirectlyaddressedindirectedandprobingguidancebutlesssoin

tellingwiththeteacheraddressingmoresothetopicoverwhichthestruggle

occurredandnotasmuchthespecificwayinwhichthestudentwasstruggling.The

affordanceacknowledgedthestruggle,anddidsowitharesponsethatallowedthe

studentmoretimeforconsideration.Thetellingtypeofresponse,andagaintoa

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lesserdegreethedirectedguidanceresponse,focusedontheteacher’sapproachto

solvingtheproblemoverwhichthestudentwasstruggling.Theprobingguidance,

however,soughttoaddressthestudent’sapproachandthenattemptedtobuildand

guideusingthatapproach.Theaffordancetypeprovidedthestudentsmoretimeto

consideranddiscussthestudent’sapproachwithgreaterindependence.

Whilethekindsofstudentstrugglesasindicatedin(1)aboveinitiatedthe

episodesofinteractionwithteachersorotherstudents,thestudent’sactionsthat

contributedtoaproductiveresolutionincludedthestudent’swillingnesstoattempt

todotheproblem.Studentscouldnotachievethemathematicalobjectivesofthe

giventaskswithoutsomeformofengagementinthetasks.Ifstudentsencountered

difficultyaftertheyattemptedtheproblem,theythenhadtoinitiatearequestfor

helporaskaquestionthatwouldmakeitvisibletoothersthattheywere

attemptingtocompletethetask.Communicatingtheirunderstandingorshowing

theirworkinsomewaywaskeytobetteraddressingthestrugglethestudents

experienced.Anothercriticalelementwasastudent’swillingnesstopersistand

remainengageddespitethedifficultieswiththeirtaskastheyattemptedtoaddress

theirownstruggle.

Theproductiveinteractionswereinterplaysofthestudentactionsof

persistence,questioning,andcommunicatinginattemptingtheirtasks.These

interactionsweresupportedbytheteacherorpossiblyotherstudent’sactions

whichbuiltonthestudent’sworkbyaskingclarifyingquestionsregardingtheir

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struggle,supportingthestudent’seffortsandthinking,pressingforrigor,and

providingsufficienttime.

Thecontextfortheinteractionsthatsupportedthestudentstruggleswerein

theengagementoftasksofhighercognitivedemandthatrequiredstudentsto

deepentheirthinkingaboutthemathematicswithemphasisonboththeconcepts

andprocessesinvolved;inprovidingindividualworktimeforeachstudentto

examinetheproblemandvaluetheeffortexpendedbyeachstudent;insharing

workwithotherstudentssothattheycouldbroadentheirperspectiveonthinking

aboutthetaskproblems;andindiscussingthetaskquestionsandsolutionswiththe

teacherandwiththeclasstofostercommunication,explanations,andjustifications

ofthemathematics.

Theaspectsoftheinteractionsthathelpeddirectandsupportstudent

strugglesproductivelyandtowardstudentunderstandingofmathematicscouldbe

viewedthroughthejointstudentactions,teacheractions,andthephysicaland

culturalcontextsestablishedbythenormsintheclass.Theencouragementto

communicatewithteacherresponsessuchas,“Tellmewhatyoumean”and“Talk

aboutitsomemore”orinsistenceonsense‐makingwith“Whyisthat?”provided

opportunitiesforstudentstoelaborateonwhattheyunderstoodandperhaps

clarifiedthesourceoftheirstruggles.Responsesthatencouragedcontinuedeffort

suchas,“Trythat”and“Well,whatifyoudo…”gavepositivereinforcementfor

engagementwithoutstudentworryingaboutwhethertheresultwasrightorwrong.

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Teacherscoulddemonstratetheirowninquiryabouttheproblemsbythinking

aloudandmodelingaprocessofengagingintheproblems.Anappropriatetempo

fortheinteractionthatdidnotrushtheprocessorresorttoshortcutspromotedthe

sensethatunderstandingboththeproblemandtheprocesswasmoreimportant

thanjustfindingaquickwaytofindingtheanswer.

Posingproblemsofhighcognitivedemandgavethestudentsopportunitiesto

think,reason,andproblem‐solveinwaysthatmeantthestudentshadtothink

deeplyabouttheproblemsandnotjustfindroutinemethodstoapply.The

appropriatelevelofdifficultycontributedtoasettingforstudentstograpplewith

ideasandchallengedstudentstomakeanattemptwiththebeliefthatwitheffort

andpossiblysomeassistance,theycouldsolvetheproblem.Thetasksthatthe

studentsengagedinduringmystudyfocusedonproportionalreasoning.These

taskswereintendedtoencouragestudentstotakethebasicknowledgetheyhad

aboutproportionalconceptsandextendthatunderstandingincontextsofhigher

cognitivedemand.Insodoing,manystudentsstruggledtoperformthetasks.They

builtontheirknowledgeanddevelopednewwaystoextendandusethatknowledge

todeepentheirconceptualunderstandingofthemathematicsthattheywere

learningwithadifferentperspective.

LIMITATION Inordertogaininsightintothekindsofteachingpracticesthatsupport

conceptualunderstanding,mystudywaslimitedtoasampleofsixteachersatthree

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sites.Thesample,however,reflectsthestatepopulationanddemonstrates

importantaspectsoftheteachingpracticesthatencouragedstudentstoengagein

tasksdespitethestrugglesthattheyexperiencedandtosupportadispositionthat

fostereddoingmathematics.Itwasbeyondthisstudy’sintenttodirectlymeasure

thestudentlearningoutcomesasaresultofthestudentstruggles,howeverIbelieve

thekeyprinciplesofthestudentteacherinteractionsthatsupportstudentstruggles

productivelymaybeastartingpointtoexaminethekindsoflearningthattakes

place,andifadispositionfordoingmathematicsispositivelyaffectedintheprocess

ofworkingthroughstruggle.

Thisisnottosaythatstruggleshouldbeincorporatedatalllevelsoflearning.

Infact,instructionalpracticesshouldhaveenoughvariationandflexibilityto

incorporatethoseopportunitiesforstudentstostrugglebutalsotoacknowledgethe

satisfactionandfuninherentintheirhardworktolearnmathematics.Teacherscan

findbalancebetweenthosemomentswhenstrugglingisproductiveandothertimes

whenitmaybeunnecessaryandcounterproductive,particularlyifthelearninggoal

is,forexample,skillfluency.Thesejudgmentsarebestmadebyteacherswhocan

sensewhatismosteffectiveandappropriatefortheclassroomenvironmentandthe

intendedlearninggoalsatthetimeofinstruction.

Theroleofproductivestruggleexaminedinmystudywaslimitedtoa6thand

7thgradeband.Furtherresearchthatexaminestheroleofstruggleinothergrade

levels,orwithcertainethnicgroupsmaygiveinsightintootherpossiblekindsof

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strugglethatoccurandthemoreeffectivemeansofsupportingthesestruggles

productively.Differenttypesofcurriculaorvariationsininstructionalpractices,

withtheiremphasisandexpectationsinstudentengagement,mayalsoaffecthow

struggleplaysoutininteractionswiththeimplementedtasksandthestudent

learningthatisachieved.

Itmaybearguedthatallowingstudentstostruggleinlearningmathematics

wouldhavenegativeeffects,asitwouldnotcontributetodeepeningstudents

understandingofmathematics.Studentsmayfeelthatstrugglingtodomathematics

wouldshowothersthattheycannotdothemathematicswitheaseandthattheyare

notsmart(Ames&Archer,1988;Dweck,2000).Bynotattemptingtodothe

problem,studentswouldavoidtheissueofstrugglebecausenoeffortisexpended.

Butavoidingstrugglemaybyextensionmeanavoidingdoingmathematicsthatis

difficultandchallenging.Seekingstrategiesforsolvingdifficultproblems,

communicatingthoseissuesofdifficultyinordertoovercomethem,andlookingat

alternateperspectivesofotherstudentsaremissedopportunitiesindoing

mathematics.Disengagementinsolvingproblemsindeedisnotanintended

consequenceofprovidingstudentswithtasksofhighercognitivedemand.Teaching

practicesmustthereforeincludecarefulconsiderationoftheappropriatenessof

tasks(Henningsen&Stein,1997).

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IMPLICATION Ioutlinethreepotentialimplicationsofmystudyfortherolestudent

struggleplaysinlearningmathematics.Thefirstimplicationisthecritical

importanceofinstructionalpracticesthatsupportstruggleintheclassroom.Under

thisinstructionalpracticeheading,Iincludethewaystasksareimplementedbythe

teacherintheclassroomandthekindsofsupportthatareprovidedstudents

throughdiscourse,resources,time,andopportunitiesforstudentstowork

independently,withclassmates,andwiththeteacher.Professionaldevelopmentfor

in‐serviceteachersandpre‐serviceeducationthatmakesstudentstrugglesexplicit

asanimportantcomponentoflearningwithunderstandingandaddresseseffective

instructionalpracticesthatfacilitatethestudentstruggles’productiveresolution

couldbeofbenefittobothteachingandstudentlearning.Struggleisoftenviewed

asundesirableforlearningwithanaccompanyingperspectivethatlearning

occurredif“studentsgotitrightaway”butlesssoif“studentsreallyhadahard

time”.Teachertrainingmustmakeclearhowstrugglemayinitiateanopportunity

forlearningtotakeplace,particularlywhenstudentsengageintasksofhigher

cognitivedemand.Effectiveinstructionalstrategiesshouldbedevelopedtoguide

andsupportstudentsinresolvingtheirstruggleswithoutdeprivingthemofthe

intellectualeffortrequiredbythetask.Teachingmustdemonstratebalance,

flexibility,restraint,andsensitivitytothestudents’strugglesandtheircapacityfor

persistence.Forthis,teachersmustbringtobeartheirknowledgeoftheirstudents

andalsoassessthestudents’inthemoment‐by‐momentinteractions.

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Asecondimplicationistoraiseawarenessamongschooladministrators,

schoolboards,parents,andstakeholdersofeducationoftheroleproductive

struggleplaysindeepeningunderstandingandthatbysupportingtheadoptionof

curriculathatincorporatetasksofhighercognitivedemandandpromoteproblem

solving,itensuresawayofdevelopingstudentswhoarewillingtograpplewith

difficulttasksandtothinkcreativelyaboutsolvingproblems.Curriculadesignedto

havestudentsfollowdemonstratedexamplesandthatminimizeintellectualeffortin

problemsolvingfosterinstructionalpracticesthatworktoavoidstudentstruggles

(AAAS,1989).Guidanceisthereforeneededforselectionofcurriculathatdevelop

notonlyskillproficiencybutalsoworktodevelopconceptualunderstanding

throughtasksofhighercognitivedemand(Kilpatrick,Swafford,&Findell,2001).

Thethirdimplicationisthatinordertogainamorecompleteunderstanding

ofproductivestruggle,studiesmayuseothermathematicalconcepts,suchasrates

andratiosoralgebraicreasoning,asthetaskconcepts.Myexaminationofstruggle

focusedaboutstudentengagementwithproportionalreasoningtasks.Wouldthe

roleofproductivestruggleappeartobedifferentinteachingandlearningwhenthe

contextisdifferent,orarethereidentifiablecommoncharacteristicsofproductive

struggleforlearningingeneral?Inaddition,Ihavenotaddressedthekindsof

strugglesstudentsmayhavethatarenotmadevisibletotheteachers.Towhat

extentdostudentsstruggleontheirownandwhendotheydecidetoseekhelp?In

whatwaysdostudentsmanagetoresolvetheirstrugglesontheirown?Tothatend,

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Ihopethisstudyofproductivestrugglecontributestoidentifyingandhighlightinga

componentofteachingandlearningthatprovidesstudentswithopportunitiesto

buildanddeepentheirconceptualunderstandingofmathematics.

Todomathematicsistodotheworkofthinkingaboutandengagingwith

conceptsthatareattimesconcreteandatothersabstract.Tomakesenseofthese

conceptsrequireseffortandsometimesstruggle.Inthewordsofpsychologist,

Csikszentmihalyi(1990),“Thebestmomentsusuallyoccurwhenaperson’sbodyor

mindisstretchedtoitslimitsinavoluntaryefforttoaccomplishsomethingdifficult

andworthwhile.Optimalexperienceisthussomethingthatwemake

happen….opportunities,challengestoexpandourselves….suchexperiencesarenot

necessarilypleasantatthetimetheyoccur.”Therewardfortheeffortofstruggling

withmathematicsmaybeasteptowardsthegoalofgainingadeeperunderstanding

ofthemathematicsaccompaniedbyasenseofself‐satisfaction.

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AppendixA:PreObservationTeacherInterview(PRTI)

1. Thinkofthreeexamplesinyourclassroomwhereyouobservedastudentstrugglingwhilehe/sheworkedonamathematicalactivityortask.Iwillaskyoutorespondtothefollowingquestionspertainingtoeachofyourexamples.a. Ineachcase,describethetaskeachstudentwasengagedindoing.e.g.

whatwasthestudenttryingtodo?Beasspecificaspossibleaboutthemathematicalobjectiveofthetask,thelevel,andtheintendedactivity.

b. Whatdidthestudentdothatcausedyoutonoticethathe/shewasstruggling?Whatwasthestruggleabout?

c. Howdidyourespondtoeachofthesestudents’struggle?d. Whatwereyourreasonsforrespondinginthatway?e. Howdoyouthinkyouractionsaffectedthestudents?(e.g.helpfulinwhat

way;noimpact;worsened;talkaboutthelearningandtheeffectonthestruggle).Whydoyouthinkso?

f. Didthestudents’responsetoyouractionsurpriseyouorwasitwhatyouexpected?Whydoyouthinkso?

g. Wouldyoudoanythingdifferentlyinteachingthelesson?

2. Howdoyouthinkstudentslearnproportionalreasoning?

3. Whatdoyouthinkyourroleisinsupportingtheminlearningaboutproportionalreasoning?

4. Ingeneral,whataresomefactorsthataffecthowyourespondto

studentsinclass?

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AppendixB:PostObservationTeacherInterview(PSTI)

1. Considertwoepisodesthatyourecallteachingthisweekwherestudents’struggleoccurred.Inastimulatedrecallsessionwewillexaminetwovideoclipsoftheteachers’choosingandonevideoclipthattheresearcherchooses.h. Describethetasksandthekindsofstrugglesyounoticede.g.whatwere

thestudentstryingtodo?Beasspecificaspossibleaboutthemathematicalobjectiveofthetask,thelevel,andtheintendedactivity.

i. Whatdidthestudentsdothatcausedyoutonoticethatthestudentswerestruggling?

j. Howdidyourespondtoeachofthesestudents’struggle?k. Whatwereyourreasonsforrespondinginthatway?l. Howdoyouthinkyouractionsaffectedthestudents?(e.g.helpfulinwhat

way;noimpact;worsened;talkaboutthelearningandtheeffectonthestruggle)

m. Didthestudents’responsetoyouractionsurpriseyouorwasitwhatyouexpected?

n. Wouldyoudoanythingdifferentlyinresponsetothestudents’struggle?o. Howsatisfiedwereyouwiththislesson?p. Whatdoyouthinkthestudentsgotoutofthislesson?

2. Isthereanotherinstanceofastudentstrugglethatyouwanttotalk

about?

3. Otherthanthekindsofstruggleyoumentionedabove,whatotherformsofstruggleinmathematicsdidyouobservethisweek?Orinyourpastteaching?

a. Talkaboutthestudents’struggleinwhichyoufeelyourresponses

andactionswerenotparticularlyhelpfulforstudentslearningandunderstanding.

b. Talkaboutthoseinstanceswhereyouthinkyouractionswereparticularlyhelpfulandproductivetoyourstudents.Whydoyouthinkyouractionswerehelpful?

4. Whatvaluedoyouthinkthereisinallowingstudents’tostrugglein

learningmathematics?

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AppendixC:TaskDebrief(TDB)

1. Whatareacoupleofinstancesofstudents’strugglethatoccurredintoday’sclass?

2. Whatareyourthoughtsandobservationsaboutwhathappened?

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AppendixD:StudentInterview(SI)

1. Didanypartofthetaskthatyoudidinclasstodayseemhard?Pleaseexplain

2. Describewhatwashardorconfusing.

3. Howdidyoudealwiththehardpart?

4. Isthereanythingoranyoneinclassthathelped?Whoorwhatwashelpfulandinwhatway?

5. Whataresomewaysthatlearningmathematicscanbehard?Howdoyouusuallydealwiththatkindofstruggle?

6. Whataresomewaysthatyoufindteacherstobehelpfulwhenyouareconfused,arestuck,orfindsomethingdifficult?

170

AppendixE:TaskDifficultySurvey

Theleveloftoday’slesson(ortask)was:______VeryHard_____Hard______Justright______Easy______VeryEasyforme.Name:____________________________________________

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AppendixF:ActivityBooklet

Activity1:BarrelsofFunSupplementalActivitiesforMathExplorationsPart2BasedontheIngenuityandInvestigationfromsection8.5page286(Materials:Graphpaperwith1cmgrids;pencil;calculator(optional))

I. ObjectiveStudentswillbeabletosolveproblemsinvolvingproportionalrelationships.

II. MotivatingProblem

Supposewehavea48gallonrainbarrelcontaining24gallonsofwateranda5gallonwaterjugcontaining3gallonsofwater.

A. Whichcontainerhasmorewater?B. Whichcontainerissaidtobefuller?

(Havestudentsworkindividuallyfirstsothateachstudentshasthoughtaboutthequestionsandperhapsgrappledwithmakingsenseofitontheirownorpossiblybyaskingtheteacherquestionsbutnotyethaveawholeclassdiscussion,unlessthereisaneedforclarification.)Useasheetofgraphpapertodrawapictureofthetwocontainersandtheirwaterlevel.Youmayleteachsquarerepresent1gallonandshadeinthepartrepresentingthewater.Doesitmatterwhatshapeyoumakethesecontainers?Discussyouranswerandexplanationwithyourgroup.(Havestudentsshareideaswithagroupof3or4students.)Haveeachgroupdecidewhattopresentandhow.)(Students’discussionsmayhaveincludeddiscussionthatthe5gallonjugwasfullerbecausetheratioofthevolumeofwaterinthejug,whichis3/5,isgreaterthantheratioofvolumeofwaterinthebarreltothevolumeoftherainbarrel,whichis24/48=½.Whilethefractionsareclose,theyarenotequivalent.Whatdoes3/5representforthe5gallonjug?Whatdoes½=24/48representfortherainbarrel?)

III. ReflectionA. Eachgroupreportstheiranswerandexplanation.

(Lookformultiplewaysthatstudentsuserepresentationstosolvetheproblem.)

B. Whyisthecontainerwithlesswatersaidtobefuller?

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C. Whatmathtoolsdidthestudentsusetohelpunderstandtheproblem?(e.g.Table,picture,fractions?)

D. Didthestudentsuseatable?Didthestudentsusefractionstoexplainyouranswer?How?

(Havestudentscomparedifferentwaysthattheproblemwasexplained,howsomewayshaveadvantages,wholikesaparticularpresentationorexplanation,andhowconvincedaretheywiththeexplanation.)E. Howmanygallonsofwaterwouldneedtobeinthe5gallonjug

sothatithasthesamefullnessasthe24gallonsinthe48gallonbarrel?(Havestudentsworkindividuallyfirstonthispartthendiscussasaintheirgroupbriefly,thenasawholeclasstogetinput.)

IV. FurtherExploration

A. Ifwedrainagallonofwaterfromeachcontainer,doesthischangeyouransweraboutwhichcontainerisfuller?Explain.

B. Howmanymoregallonsofwaterdoweneedtocatchinthebarrelinordertohavethesamefullnessinthebarrelaswehaveinthejug?Explain.

(Havestudentsworkindividuallythendiscussasagroupbriefly,thenasawholeclass.)(Wesaythecontainershaveproportionallythesameamountofwateriftheratiosoftheamountsofwatertothecapacityofthecontainerareequivalentfractions.Letx=theamountofrainneededtomakethebarrelhavethesamefullness3/5=partofthejugthatiswater(24+x)/48=fractionofthebarrelthatwouldbewaterifweaddedxgallonsofrain.Thesemustbethesameifthetwocontainerswillhaveproportionallythesameamountofwater.Thus,3/5=(24+x)/48.Insolvingforx,wehave(3)(48)=(24+x)(5)or144=120+5x.24=5xorx=24/5=4.8gallons.)

V. ExtensionA48gallonrainbarrelcontains18gallonsofwater.A5gallonwaterjugcontains2gallonsofwater.A. WhichcontainerhasmorewaterB. Whichcontainerissaidtobefuller?

(Inthiscase,wehave18/48=3/8fullofwaterinthebarreland2/5fullofwaterinthebarrel.Thisisveryclose,andstudentsmayusecommondenominatorstocomparethetwofractions.3/8=15/40and2/5=16/40.Because2/5greaterthan3/8,thejugismorefullofwaterthanthebarrel.)

173

C. Ifwedrainagallonofwaterfromeach,doesthischangetheanswertowhichisfuller?Explainyouranswer.(Therainbarrelwillhave17/48waterandthejugwillhave1/5water.Thestudentsmaywishtousedecimalsorcommondenominatorstocompare.17/48=.35416666666….while1/5=.2.Drainingchangesthefullnesstothebarrelbeingmorefull.)

StudentActivitySheets:BarrelsofFunSupposewehavea48‐gallonrainbarrelcontaining24gallonsofwateranda5‐gallonwaterjugcontaining3gallonsofwater.

A. (Task1.1)Whichcontainerhasmorewater?

B. (Task1.2)Whichcontainerissaidtobefuller?Explainyouranswer.

C. (Task1.3)Usethecoordinategridbelowtodrawapictureofthetwocontainersandtheirwaterlevel.Youmayleteachsquarerepresent1gallonandshadeinthepartrepresentingthewater.Doesitmatterwhatshapeyoumakethesecontainers?

174

D. (Task1.4)Whyisthecontainerwithlesswatersaidtobefuller.Explain.

E. (Task1.5)Howmanygallonsofwaterwouldneedtobeinthe5‐gallonjugsothatithasthesamefullnessasthe24gallonsinthe48‐gallonbarrel?

F. (Task1.6)Ifwedrainagallonofwaterfromeachcontainer,doesthischangeyouransweraboutwhichcontainerisfuller?Whyorwhynot?

G. (Task1.7)Howmanymoregallonsofwaterdoweneedtocatchinthebarrelinordertohavethesamefullnessinthebarrelaswehaveinthejug?Explain.

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Activity2: BagsofMarblesSupplementalActivitiesforMathExplorationsPart2Section12.3ProbabilityOrchestratingDiscussions:FivepracticesconstituteamodelforeffectivelyusingstudentresponsesinwholeclassdiscussionthatcanpotentiallymaketeachingwithhighleveltasksmoremanageableforteachersbySmith,Hughes,Engle,andSteininMay2009MathematicsTeachingintheMiddleSchool(Materials:Graphpaperwith1cmgrids;pencil;calculator(optional))

VI. ObjectiveStudentswillbeabletosolveproblemsinvolvingproportionalrelationshipsinprobability

VII. PriorKnowledge

Supposeabaseballteamismadeupof6boysand3girls.Eachpersonwriteshisorhernameonapieceofpaperandputsitinahat.Thecoachdrawsonepieceofpaperfromthehat.Whichnameismorelikelytobedrawnfromthehat,aboy’snameoragirl’sname?Whydoyouthinkthat?Whatisthechancethatthenamewillbeaboy?Whatisthechancethatthenamewillbeagirl?Explain.(Dotheaboveofasimilarbackgroundcheckasalaunchtomakesuretheideasofprobabilityfromsection12.3aresecure.)

VIII. MotivatingProblemTherearethreebagscontainingredandbluemarbles.Thethreebagsarelabeledasshownbelow.Bag1: 75redand25blueforatotalof100marblesBag2: 40redand20blueforatotalof60marblesBag3: 100redand25blueforatotalof125marbles

176

Eachbagisshaken.Ifyouweretocloseyoureyes,reachintoabag,andremoveonemarble,whichbagwouldgiveyouthebestchanceofpickingabluemarble?Justifyyouranswer.(Havestudentsworkindividuallyontheproblemandobservedifferentapproaches.)(Oncethestudentshaveallhadanopportunitytomakesenseofandcomeupwithtentativeideasforthesolution,havethemworkinsmallgroupstosharetheirideas.)Discussyouranswerandexplanationwithyourgroup.(Students’discussionsmayincludediscussionthatbag1is1/4blue,bag2is1/3blueandbag3is1/5blue.Othersmayusepercents:bag1is25%blue,bag2is331/3%blue,bag3is20%blue.Othersmayargueincorrectlythatbag1andbag3havemorebluemarblesthanbag2soNOTbag2.SomeareasofconfusionmaybeinlookingatratiosofBluetoRedratherthanBluetothetotalBlue+Red,thoughtheydoprovidesomeinformation.)

IX. ReflectionF. Eachgroupreportstheiranswerandexplanation.G. Explainwhythisbaggivesyouthebestchanceofpickingablue

marble?Youmayusethediagramaboveinyourexplanation.H. Whatmathtoolsdidyouusetohelpunderstandtheproblem?

(e.g.Table,picture,fractions?)I. Didyouuseatable?Didyouusefractionstoexplainyour

answer?How?

X. FurtherExplorationC. Whichbaggivesyouthebestchanceofpickingaredmarble?

Explainwhy.D. HowcanyouchangeBag2tohavethesamechanceofgetting

abluemarbleasBag1?Explainhowyougotyouranswer.(Thestudentsmaywishtoaddmarblesofeithercolor,forexampleif20redballsareaddedtoBag2thenthechanceofgettingablueis20/80=¼.

E. HowcanyouchangeBag2tohavethechanceofgettingablueasBag1ifBag2mustcontain60totalmarbles?(15blueand45redmakesachanceofgettingabluetobe15/60=¼)

XI. Extension

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ConsideronlyBags1and2.MakeanewbagofmarblessothatthisbaghasagreaterchanceofgettingabluethanBag1butlessofachanceofgettingabluethanBag2.Explainhowyouarrivedatthenumberofblueandredmarblesforyournewbag.

(Studentsmaytryaddingorsubtractingquantitiesineitherorbothbags.Youmaymonitortheireffortsandaskwhateffecttheirchangeshave.Havethemexplainorshowwhatischanging.Somewaysthatthisnewbagcanbeobtainedinclude:Taking25/100=¼inBag1and20/60=1/3inBag2anddetermininganumberbetweenthetwo.Studentsmayfindacommondenominatorsuchas12,24,etc.andfindtheequivalentfractionsfor¼=3/12and1/3=4/12.Whilethesetwofractionsmakeitdifficulttoseewhatliesbetweenthem,theequivalentfractions¼=6/24and1/3=8/24wouldleadto7/24asacandidateforthenewbag.Namely,abagwith24marblesofwhich7areblueand17arered.Anotherwayistolookatthedecimalrepresentationfor¼=.25and1/3=.3333….Adecimalsuchas.3=3/10isbetween.25and.3333…..soabagwith10marbles,ofwhich3areblueand7redwouldalsowork.Thestudentsmaycomeupwithotherinterestingcompositions.Infact,with¼and1/3,thefraction(1+1)/(3+4)=2/7isafractionbetween1/3and¼!)

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StudentActivitySheets:BagsofMarblesTherearethreebagscontainingredandbluemarblesasshownbelow

Bag1 Bag2 Bag3 75red 40red 100red 25blue 20blue 25blue Total100marbles Total60marblesTotal125marbles

1. (Task2.1)Eachbagisshaken.Ifyouweretocloseyoureyes,reachintoabag,andremoveonemarble,whichbagwouldgiveyouthebestchanceofpickingabluemarble?Explainyouranswer.

2. (Task2.2)Whichbaggivesyouthebestchanceofpickingaredmarble?Explainyouranswer.

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3. (Task2.3)HowcanyouchangeBag2tohavethesamechanceofgettinga

bluemarbleasBag1?Explainhowyoureachedthisconclusion.

4. (Task2.4)HowcanyouchangeBag2tohavethechanceofgettingabluemarbleasBag1ifBag2mustcontaining60totalmarbles?

5. (Task2.5)ConsideronlyBags1and2.MakeanewbagofmarblessothatthisbaghasagreaterchanceofgettingabluemarblethanBag1butlessofachanceofgettingabluemarblethanBag2.Explainhowyouarrivedatthenumberofblueandredmarblesforyournewbag.

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Activity3:TheAlgebraofTipsandSalesSupplementalActivitiesforMathExplorationsPart2section8.5

XII. ObjectiveStudentswillbeabletosolveproblemsinvolvingproportionalrelationshipsusingalgebraicexpressionsandequations.

XIII. PriorKnowledge

Inrestaurants,weoftenincludeatipof15%to20%oftheamountofthebill.Forexample,incomputingtips,whataresomewayswecandeterminehowmuchtotiptoincludeusinga15%rateifyourrestaurantbillis$40?Explainhowyougotyourtipamountandthestrategythatyouused.(Somestudentsmayconvert15%toadecimal,.15andmultiplyto40.Othersmaytake40andmultiplyby15/100toget$6.Somemaytake10%of40toget$4andthentakehalfofthattoget$2,whichwouldbe5%of40,addthe4tothe2toget$6.)

XIV. Problemusingalgebraicexpressions.

A. TIPPING(Havestudentsworkindividuallyfirstthenshareasawholeclass.)

a)Supposethebillis$X.Writeanexpressionforthetipon$Xusinga15%tiprate.Whatisthetotalamountyouwouldpaytherestaurant?(.15Xistheamountofthetip.Totalamount=X+.15X=X(1.15))

b)Supposeagenerouscustomerusesa20%tiprateonabillof$X.Writeanexpressionforthetipon$Xusinga20%.Whatisthetotalamountthiscustomerpaystherestaurant?(.2Xistheamountofthetip.Totalamount=X+.2X=X(1.2))

B. ExtensiontoExample1in8.5

a) If40%ofagroupof35studentsparticipateinathletics,howmanyofthese35participateinathletics?(Havestudentsdotheseindividuallyfirstbeforesharingasaclass.)

b) AnothergrouphasNstudentsand40%ofthemparticipateinathletics.WriteanexpressionusingNforthenumberofstudentswhoparticipateinathleticsfromthisgroup.

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c) Writeanexpressionforthenumberofstudentswhodonotparticipateinathletics.

(Encouragestudentstosayandwriteinwordswhattheexpressionsshouldsay.Thenhavestudentswritetheexpressionsalgebraically.)

C. SalesProblem

a)Apairofpantsregularlycosts$40butisonsaleat25%offthe

regularprice.Howmuchwillyoupayforthesesalespants,withoutcomputingtax?Explainhowyougotyouranswer.(400.25(40)=40(10.25)=400(.75)=30Studentsmaywishtodothemultiplicationof0.25by40firstandthensubtract10from40.Butaswewillseeinthenextpart,itisusefultosubtractthedecimalsfirst.)

b)Ashirtregularlycosts$Sandisonsaleat25%offtheregular

price.Writeanexpression,usingS,fortheamountofdollarsdiscounted.Writeanexpressionthatrepresentshowmuchyouwillpay,disregardingtax?

(.25S=discountamount.S.25S=S(1.25)=S(.75)=salesprice)

XV. ReflectionJ. Eachgroupreportstheiranswerandexplanation.

K. Didanyoneuseavisualrepresentationtoexplainthealgebraic

expression?

XVI. FurtherExplorationA. Anmp3playerisonsalefor$60aftera20%discount.Whatwas

theoriginalprice?Whatwastheamountofthediscount?(Dothisproblemasawholeclassandmodelthealgebraicsetupwritingclearlywhateachexpressionrepresents.Theorganizationandformatcanhelpstudentsmakesenseoftheequationthatevolves.LetX=originalprice.0.2X=discounttaken.Notestudentsmaysay.2(60)X0.2X=saleprice60=saleprice

182

Thelasttwolinesbothrepresentthesalepricesotheymustbeequaltoeachother.X0.2X=60X(10.2)=X(0.8)=0.8X=60X=60/0.8=75=theoriginalprice.

B. Acomputerisonsalefor$420aftera30%discount.Whatwastheoriginalpriceofthecomputer?Whatwastheamountofthediscount?(Havethestudentsworkonthiseitherindividuallyorinsmallgroups.LetX=originalprice.0.3X=discountAgain,studentsmaysaythediscountis.3(420).X.3X=expressionforthesaleprice420=salepriceSoX.3=420X(1.3)=420.7X=420X=420/.7X=600istheoriginalprice

C. Yourmompays$50.15atarestaurantthatincludedthemealand

an18%tip.Whatwasthepriceofthemealalone?(Havestudentstrythisontheirownandthenshareasacall.)(LetX=mealprice..18X=tiponthemealX+.18X=expressionforamountmompays50.15=amountmompaysX+.18X=50.15

X(1+.18)=50.151.18X=50.15X=50.15/1.18X=42.50isthepriceofthemealalone

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StudentActivitySheets:TipsandSalesTipping

1. (Task3.1)Supposearestaurantbillis$X.Writeanexpressionforthetipon$Xusinga15%tiprate.Whatisthetotalamountyouwouldpaytherestaurant?

2. (Task3.2)Supposeagenerouscustomerusesa20%tiprateonabillof$X.Writeanexpressionforthetipon$Xusinga20%tiprate.Whatisthetotalamountthiscustomerpaystherestaurant?

PartoftheCrowd

1. (Task3.3)If40%ofagroupof35studentsparticipateinathletics,howmanyofthese35participateinathletics?

2. (Task3.4)AnothergrouphasNstudentsand40%ofthemparticipateinathletics.WriteanexpressionusingNforthenumberofstudentswhoparticipateinathleticsfromthisgroup.

3. (Task3.5)Writeanexpressionforthenumberofstudentswhodonotparticipateinathletics.

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OnSale

1. (Task3.6)Apairofpantsregularlycosts$40butisonsaleat25%offtheregularprice.Howmuchwillyoupayforthesesalespants,withoutcomputingtax?Explainhowyougotyouranswer.

2. (Task3.7)Ashirtregularlycosts$Sandisonsaleat25%offtheregularprice.Writeanexpression,usingS,fortheamountofdollarsdiscounted.Writeanexpressionthatrepresentshowmuchyouwillpay,disregardingtax.

Discounts

1. (Task3.8)Anmp3playerisonsalefor$60aftera20%discount.Whatwastheoriginalprice?Whatwastheamountofthediscount?

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Activity4:DetectingChangeBasedon“LinearandQuadraticChange:AProblemfromJapan”byBlakeE.PetersoninMathematicsTeacherofOctober2006.

XVII. ObjectiveStudentswillbeabletousetables,graphs,andalgebraicexpressionstodescribegeometricpatterns.

XVIII. MotivatingProblemInthefigure,asthestepschanges,_____________alsochanges.

Whatattributeschangeasthestepincreases?Havestudentsworkindividuallytowritedownwhattheyobservechanging.Thenbringthestudentstogetheranddiscussasawholeclass.Therearemanyobservationsthatcanbemade,includingperimeter,height,width,sizeofenclosingrectangle,numberof“toothpicks”,numberofverticaltoothpicks,numberofhorizontaltoothpicks,numberofsquares,numberofsegments,lengthoflongestline,numberofrectangles,etc.Recordtheseobservations.Asstudentsgeneratealistofanswers,questionsofclarificationneedtobeposed,suchas“Whatdoyoumeanby“toothpicks”,“Whatdoyoumeanbynumberofsquares?”Oncealistofchangingattributesisidentified,askstudentstoworkingroupstoworkondescribingthechangesinoneattribute.Whathappens

186

inthenthstep.Youmaywishtoassigngroupsofstudentstotheirfavoritechange.Haveeachgrouphaveatleasttwowaystorepresenttheirobservedchange.Ifpossible,encouragetable,graph,andequationtodescribethechangethattheynotice(i.e.numerical,visual,andsymbolicrepresentations)Havegroupsofstudentssharetheirworkbypresentingthepatterntheyobservedandthevariousrepresentationstheyusedtodescribethepattern,tothenthstep,ifpossible.Somepossiblepatternsthestudentswillobserve:Linearchange:1. Lengthofthebase

a. Step1:1b. Step2:3c. Step3:5d. Step4:7e. Stepn:2n–1

2. Heighta. Step1:1b. Step2:2c. Step3:3d. Step4:4e. Stepn:n

3. Perimetera. Step1:4b. Step2:1+2+3+2+2=10c. Step3:1+2+2+5+2+2+2=16d. Step4:1+2+2+2+7+2+2+2+2=22e. Stepn:1+2(n‐1)+(2n‐1)+2n=6n‐2

Quadraticchange1. Thetotal1x1blocksorareainsquareunits:

a. Step1:1b. Step2:1+3=4c. Step3:1+3+5=9d. Step4:1+3+5+7=16

187

e. Stepn:1+3+5+7+…..+(2n–1)=n22. Thenumberofhorizontaltoothpicks

a. Step1:1+1=2b. Step2:1+3+3=7c. Step3:1+3+5+5=14d. Step4:1+3+5+7+7=23e. Stepn:1+3+5+7+9+…..+(2n–1)+(2n–1)=n2+2n–1

3. Thenumberofverticaltoothpicksa. Step1:2or1+1b. Step2:2+4or1+2+2+1c. Step3:2+4+6or1+2+3+3+2+1=12d. Step4:2+4+6+8or1+2+3+4+4+3+2+1=20e. Stepn:2+4+6+….2n=2(1+2+3+4+….n)=n(n+1)

4. Thetotalnumberoftoothpicksa. Step1:numberofhorizontalplusnumberofverticals1+1+2b. Step2:1+3+3+2+4=1+2+3+4+3=13c. Step3:1+3+5+5+2+4+6=1+2+3+4+5+6+5=26d. Step4:1+2+3+4+5+6+7+8+7=43e. Stepn:1+2+3+….+2n+(2n‐1)=2n(2n+1)/2+2n‐1=

2n2+n+2n–1=2n2+3n–1

188

StudentActivitySheet:DetectingChange

(Task4.1)Usingthefigurebelow,describewhatyouobservechangesasthestepsincrease.Recordtheseobservations.

(Task4.2)Selectonechangethatyouobservedanddescribethechange.WhathappensinStep4?WhathappensinStep5?WhathappensinStep10?WhathappensinStepn,fornapositiveinteger?

Useatable,graph,andanequationtodescribethechangethatyounotice.

189

AppendixG:Ms.Torres’Lessons

190

(2)ProbabilityandGeometryTask

1. Findtheprobabilityoflandinginthenon‐shadedregion.Explainyouranswer.3units 3units

4units

Findtheprobabilityoflandingintheshaded

2. Findtheprobabilityoflandingintheshadedregion.Explainyouranswer.

2 in.

4 in.

191

AppendixH:Samplewarmupproblems

194

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Vita

HirokoKawaguchiWarshauerwasborninKyoto,Japanandcompletedherfirst

gradeinJapanbeforeimmigratingtoChicagoin1960withherfamily.After

graduatingfromLakeViewHighSchoolin1970,sheattendedtheUniversityof

ChicagoandreceivedherBachelorofArtsdegreeinMathematicsin1974.She

continuedhergraduatestudiesatLouisianaStateUniversitywhereshereceivedher

MasterofSciencedegreeinMathematicsin1976.From1976to1979,Hiroko

taughtmathematicsatLSUasaninstructor.Hirokojoinedthemathematics

departmentatTexasStateUniversity‐SanMarcosin1979whereshecontinuesto

teach.Sheandherhusbandhavefourchildren.

Emailaddress:[email protected]

Thisdissertationwastypedbytheauthor.

Copyright by Hiroko Kawaguchi Warshauer 2011

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