Copyright
by
HirokoKawaguchiWarshauer
2011
TheDissertationCommitteeforHirokoKawaguchiWarshauerCertifiesthat
thisistheapprovedversionofthefollowingdissertation:
TheRoleofProductiveStruggleinTeachingandLearning
MiddleSchoolMathematics
Committee:
SusanEmpson,Supervisor
JamesBarufaldi
EdmundT.Emmer
AnthonyPetrosino
PhilipUriTreisman
TheRoleofProductiveStruggleinTeachingandLearning
MiddleSchoolMathematics
by
HirokoKawaguchiWarshauer,B.A.;M.S.
Dissertation
PresentedtotheFacultyoftheGraduateSchoolof
TheUniversityofTexasatAustin
inPartialFulfillment
oftheRequirements
fortheDegreeof
DoctorofPhilosophy
TheUniversityofTexasatAustin
December2011
Dedication
TomyhusbandMaxandchildrenAmy,Nathan,Lisa,andJeremy
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
TheRoleofProductiveStruggleinTeachingandLearning
MiddleSchoolMathematics
HirokoKawaguchiWarshauer,Ph.D.
TheUniversityofTexasatAustin,2011
Supervisor:SusanEmpson
Students’strugglewithlearningmathematicsisoftencastinanegativelight.
Mathematicseducatorsandresearchers,however,suggestthatstrugglingtomake
senseofmathematicsisanecessarycomponentoflearningmathematicswith
understanding.Inordertoinvestigatethepossibleconnectionbetweenstruggle
andlearning,thisstudyexaminedstudents’productivestruggleasstudentsworked
ontasksofhighercognitivedemandinmiddleschoolmathematicsclassrooms.
Students’productivestrugglereferstostudents’“efforttomakesenseof
mathematics,tofiguresomethingoutthatisnotimmediatelyapparent”(Hiebert&
Grouws,2007,p.287)asopposedtostudents’effortmadeindespairorfrustration.
Asanexploratorycasestudyusingembeddedmultiplecases,thestudy
examined186episodesofstudent‐teacherinteractionsinordertoidentifythekinds
andnatureofstudentstrugglesthatoccurredinanaturalisticclassroomsettingas
studentsengagedinmathematicaltasksfocusedonproportionalreasoning.The
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).
19
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.
31
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
2. Howdoteachersrespondtostudents’strugglewhilestudentsareengagedin
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:
1. Whatarethekindsandpatternsofstudents’strugglethatoccurwhilestudents
areengagedinmathematicalactivitiesthatarevisibletotheteacherand/or
apparenttothestudentinmiddleschoolmathematicsclassrooms?
2. Howdoteachersrespondtostudents’strugglewhilestudentsareengagedin
mathematicalactivitiesintheclassroom?Whatkindsofresponsesappeartobe
productiveinstudents’understandingandengagement?
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
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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
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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
76
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
80
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.
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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.
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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.
!"#$%&'%(&%&)*+,--".*/(0#12345#( 6-%('%(&%&)*700#1&*6--#14,(54.
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ProbingGuidance
ThefourthtypeofteacherresponseIlabelprobingguidance.Thisiscloserto
theopen‐endedapproachinbeingresponsivetothestudents’thinking,probingfor
theirideas,suggestingmathematicalconceptsorproceduresthatrelatedtoand
builtonthestudents’thinking.Theintellectualeffortneededtotackletheproblems
restedwiththestudents,buttheresponsesservedtoclarify,connect,orconfirm
ideasthestudentspresented,andwerethereforemadevisiblethroughtheteacher
responses.
InFigure4.3below,Ireportmyfindingsusingtheteacherresponse
classificationsalongthiscontinuum.
Figure4.3: TeacherResponseContinuum
Ratherthandiscretejumpsfromonecategorytotheother,Iobserveda
continuousrangeofresponseswithdegreesofinformation,directing,probing,or
affordanceprovidedbytheteacher.Responseshadvaryingdegreesofprobingor
directingresponseswithcharacteristicsthatattimeswerehybridsofsomeprobing
andsomedirecting,asIwillreportinthissection.Insomeinstancesthecognitive
demandofthetaskasoriginallyconceivedchangedintheimplementationsothat
!"##$%& '(()*+,%-".$*"-/"+012$+,%-" 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
35% 2a)CognitiveDemand:
• Loweredormaintainedfromintended
2b)AttendtoStudentStruggle:
• Assesscauseanddirectstudent
2c)BuildonStudentThinking:
• Usetobuildonwithteacherideas
3.ProbingGuidance
• Askforreasonsandjustification
• Offerideasbasedonstudents’thinking
• Seekexplanationthatcouldgetatanerrorormisconception
• Askforwritten
28% 3a)CognitiveDemand:
• Maintained3b)AttendtoStudentStruggle:
• Question,encouragestudent’sself‐reflection
3c)BuildonStudent
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workofstudents’thinking
Thinking:• Useasbasisfor
guidingstudent4.Affordance • Askfordetailed
explanation• Buildonstudent
thinking• Pressfor
justificationandsense‐makingwithgrouporindividually
• Affordtimeforstudentstowork.
11% 4a)CognitiveDemand:
• Maintainedorraised
4b)AttendtoStudentStruggle:
• Acknowledge,question,andallowstudenttime
4c)BuildonStudentThinking:
• 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.
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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.
2b)AttendtotheStruggle
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.”
2c)BuildonStudent’sThinking
Teachersaskedquestionsinadirectedguidanceresponsethattriedto
establishwhatthestudentswerethinkingandthenaskedthestudentstoclarifyand
confirmwhattheyknewandwhatwasnotclear.Thisseemedtoinformthe
teachersofthekindsofsupportthatcouldguidetheirstudentsandhelpresolve
theirstruggles.
Anothercharacteristicofdirectedguidancewasatendencyfortheteachers
toredirectthestudents’thinkingtowardsanactionthatwouldfunctioninamore
helpful,efficient,orcorrectwayasperceivedbytheteacher.Theguidancefocused
onteachers’strategies,procedures,orunderstandingofthetasksthatstudents
couldthentakeupasopposedtobuildingonstudents’ideas.Forexample,teachers
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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
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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].
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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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3b)AttendtotheStruggle
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.
3c)BuildonStudent’sThinking
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.
4b)AttendtotheStruggle
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.
4c)BuildonStudent’sThinking
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
134
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
3
5
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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?
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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
2
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.”
144
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.
3
5
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.
2
5
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
147
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.
148
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
153
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:
1. Whatarethekindsandpatternsofstudents’strugglethatoccurwhilestudents
areengagedinmathematicalactivitiesthatarevisibletotheteacherand/or
apparenttothestudentinmiddleschoolmathematicsclassrooms?
2. Howdoteachersrespondtostudents’strugglewhilestudentsareengagedin
mathematicalactivitiesintheclassroom?Whatkindsofresponsesappeartobe
productiveinstudents’understandingandengagement?
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?
169
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
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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$50.15atarestaurantthatincludedthemealand
an18%tip.Whatwasthepriceofthemealalone?(Havestudentstrythisontheirownandthenshareasacall.)(LetX=mealprice..18X=tiponthemealX+.18X=expressionforamountmompays50.15=amountmompaysX+.18X=50.15
X(1+.18)=50.151.18X=50.15X=50.15/1.18X=42.50isthepriceofthemealalone
183
StudentActivitySheets:TipsandSalesTipping
1. (Task3.1)Supposearestaurantbillis$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
192
193
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.