Improving the Quality of Elementary Mathematics Student ... · PDF fileof Elementary Mathematics ... pre-service teachers to be reflectivemathematics teachers who plan and imple-

Hillary S. Hertzog & Nancy O’Rode

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Teacher Education Quarterly, Summer 2011

Improving the Qualityof Elementary Mathematics

Student Teaching:Using Field Support Materials to Develop

Reflective Practice in Student Teachers

By Hillary S. Hertzog & Nancy O’Rode

Introduction Thisarticlepresentsan“evidence-based”programimprovementeffortthatsoughttostrengthenstudentteachers’implementationofsubject-specificpedagogyforteachingmathematicsinaK-8multiplesubjectteachereducationprogram.Wereporttheprocessofhowweusedaresearch-basedapproachtogatherevidenceabout“statusquo”ofthemathematicsstudentteachingcomponentthatpreparedelementarylevelteachers,changesthatweremadeintheprogramtobetterpreparepre-service teachers tobereflectivemathematics teacherswhoplanandimple-

Hillary S. Hertzog and Nancy O’Rode are professors in the Michael D. Eisner College of Education at California State University, Northridge.

ment effective subject-specific pedagogy, and howwemeasuredlevelsofeffectiveness.Specifically,weinvestigatedwhethermentoringstrategiesandmateri-alsdesignedtoengagestudentteachersinapplyingaspects of mathematical knowledge for teaching(MKT)duringthelessonplanning/teaching/feedbackcycleofstudent teachingwould impactpre-serviceteacherreflectivepracticeandteachingperformance.In addition,we studied howwe should change the

Improving the Quality of Elementary Mathematics Student Teaching

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supervisionprocessusedtodevelopthereflectivethinkingofstudentteachersastheyengagedinteachingmathematics.

Theoretical Framework

What Student Teachers Need to Know about Teaching Mathematics Recentstudieshaveidentifiedtheneedforimprovedclassroompracticesinteachingmathematics as a condition for improvingK-12pupil achievement inmathematics(Ball,Hill,&Bass,2005;Ma,1999;NationalMathematicsAdvisoryPanel,2008;Stigler&Hiebert,1999).Toimproveclassroompractices,pre-servicemethodsclassesshouldfocusnotjustonteachinggeneralmethodsofinstruction,butshouldengagepre-serviceteachersinlearninghowtosuccessfullyteachsubjectmattercontentusinghighlyspecificstrategiesthatarespecializedtothatdiscipline(Shulman,1987).Wefocusedonthreeimportantcomponentsinthecurrentliteraturebaseonteachingmathematicstoguideusindevelopingapre-servicepreparationprogramthatdevelopsmathematicalproficiencyinteachers:(1)DeborahBallandhercolleagues’workonMathematicalKnowledgeforTeaching,(2)processstandardsformulatedbytheNationalCouncilofTeachersofMathematics(NCTM),and(3)theNationalResearchCouncil’s(NRC)workonmathematicalproficiency. Therearefourcommonthemesintheseworks:(1)ProblemSolving—beingabletoposegoodmathematicalquestionsandproblemsthatareproductiveforstudents’learning;(2)Explanations—communicatingmathematicalideas,justi-fyingreasoning,interpretingstrategiesofothers,andrespondingproductivelytoquestions;(3)Representations—carefullychoosingthebestdiagrams,examples,symbols,formaximumunderstanding;and(4)MathematicalConnections—mak-ingexplicithowmathematicalideasarerelatedtoeachotherandappliedtotherealworld(Ball,Hill,&Bass,2005;Hill,Rowan,&Ball,2005;NCTM,2000;NRC, 2001).Our elementary teacher preparationprogramchose to emphasizeproblemsolving,explanations,representations,andmathematicalconnectionsasfourimportantsubject-specificstrategiesthatstudentteachersneedtobeabletoimplementtoeffectivelyteachmathematics.

Applying What Student Teachers Need to Knowabout Teaching Mathematics to the Student Teaching Experience

Foryearsnow,teacherpreparationprogramshavebeenchallengedwithre-forminghownewteachersarepreparedforteaching(CarnegieTaskForce,1986;Darling-Hammond,1999).Thestudentteachingexperiencehasbeenidentifiedasoneofthemostinfluentialfactorsinpreparingbeginningteachers(Koehler,1988;Lemma,1993).ZeichnerandConklin(2008),intheirdescriptionofcharacteristicsofexemplaryteachereducationprograms,citedimensionsoffieldexperiencesthatcancontributetoaprogram’ssuccess,includingtheneedtocloselyconnectsuper-visionduringstudentteachingtocontentofcoursessothatfacultyandcurriculum


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experiencesreflectonevisionofteachingandlearning.Inaddition,earlyresearchfoundthatfewstructuresexistedtosupportcooperationofsupervisors,teachersandteachercandidates(Guyton&McIntyre,1990).Laterreviewsofresearchpointedtothelackofquantifiableorqualitativedatathatcoulddemonstratewhetherthestudentteachingcomponentofaprogramwasproducingmorethoughtful,reflec-tiveteachers(McIntyre,Byrd,&Foxx,1996).Goodprogramsshouldintegratetheinstructionofcandidateswithcontextualpracticewithpupilsandconnectlearningaboutteachingtotheactualproblemsofteachingpractice(Darling-Hammond&Bransford,2005). Recenteffortstoexaminethedevelopmentofreflectivepracticeinpre-serviceteachershavebeeninfluencedbythosewhohaveappliedtheoriesoflearningtotheprocessoflearningtoteach.BorkoandPutman(1996)frametheprocessthroughacognitivelens,assertingthat“learningisanactiveconstructiveprocessthatisheavilyinfluencedbyanindividual’sexistingknowledgeandbeliefsandissituatedinparticularcontexts”(pp.674-675).Asteachereducatorswecanimpactelementsofthatcontextandthecomponentsthatpre-serviceteachersusetoconstructtheirknowledge.Ithasbeenarguedthatcarefullydesignedstudentteachingexperiencescan“helpnovicesgobeyondhavingexperiencestohelpingthemlearnfromtheirexperiences”(Rosaen&Florio-Ruane,2008,p.709)andthatthiscandevelopapre-serviceteacher’sabilitytoassessasituation,makejudgments,creategoals,chooseacourseofactionandreflectonitssuccess.Takingintoaccountthecogni-tivefactorsthatsignificantlyinfluencethedevelopmentofapre-serviceteacher’sthinkingandtheformationofhabitsofmindthatthenovicewilltakeintotheirprofessionalpractice,weshouldcarefullydesignstudentteachingexperiencesthatactivatethosethinkingprocesses.Wemustconsiderthepotentialinfluencethatuni-versitysupervisors,supervisingteachers,learners,andfocusedprogrammaterialscanhaveonthedevelopmentofpre-serviceteachers’reflectivethinking.

Research Questions Asweinvestigatedthequalityofthestudentteachingexperiencesinmath-ematicsinourpreparationprogram,weconductedaresearchstudy.Weposedthefollowingresearchquestionsasweengagedineachphaseoftheresearch:

1.Whatisthe“traditional”focusoftheobservation/feedbackcyclebe-tweentheuniversitysupervisorandstudentteacheranddoesitincludereflectiononsubject-specificpedagogy?(Phase1)

2.Howdoestheuseofsubject-specificfieldguidesinfluencetheobserva-tion/feedbackcyclebetweenuniversitysupervisorsandstudentteachers?(Phase2)

3.Doesreflectiononsubject-specificpedagogyinmathematicsduringstu-dentteachingresultinmoreeffectivemathematicsteaching?(Phase3)


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Methodology Theresearchwasdesignedandledbytheprogramcoordinatorfortheunder-graduatecredentialprogramandthemathmethodsinstructor.InPhase1,aneedsanalysisofthe“traditional”multiplesubjectstudentteachingexperiencewasledbythemathematicsmethodsinstructor/researcher.Arandomsampleofwrittenobserva-tionfeedbacknotescreatedbyuniversitysupervisorsacrossathreeyearperiodwereanalyzedtodeterminethefocusoffeedbackonstudenttaughtlessons.Aconstantcomparativeanalysis(Miles&Huberman,1994)generatedcategoriesandthemesthatcouldbeusedasevidencetodocumentthestatusquooffeedbackgiventostudentteachersduringtheplanning/teaching/feedbackcycleinstudentteaching. InPhase2,aqualitativecasestudyapproach(Patton,1990)wasusedtoprovidein-depthdataconcerninguseofnewlycreatedsupportmaterialsintendedtofocusuniversitysupervisorsandstudentteachersonaspectsofsubject-specificpedagogyduringtheplanning/teaching/feedbackcycle. InPhase3,aquasi-experimentaldesignwasfollowedwhichidentifiedexperi-mentalandcontrolgroupsfromalargergroupofstudentteachers.Theexperimentalgroupusedthenewlydevelopedsubjectspecificfieldmaterialsandworkedwithtraineduniversitysupervisors.Thecontrolgroupcompletedstudentteachingwiththetraditionalmentorshipofuniversitysupervisors.WemeasuredthedevelopmentofmathematicalknowledgeforteachinginbothgroupsbyusingtheLMTassess-mentdevelopedbyHill,Rowan,&Ball(2005).

Data Collection InPhase1,aneedsanalysisofcurrentpracticessoughttogatherevidenceabouttraditionalmentoringpracticesusedduringstudentteachingbyuniversitysupervisors.Writtenobservationnotesrandomlyselectedfromallsubjectareaswereanalyzed.Sixuniversitysupervisors,interestedinimprovingthemathemat-icsteachingofpre-serviceteachers,volunteeredtoworkwiththemathmethodsinstructor/researcher to learn how to code observation notes. Three questionsguidedthePhase1codinganalysis:Whatlessonsubjectsdosupervisorsseemostoften?Whataspectsoflessonsdosupervisorsrecordtohelpteachercandidatesreflectabout theirdevelopingpractice?Towhatextent is subjectmatterand/orsubject-specificpedagogynotedinthewrittenfeedbacktostudentteachers?Thefrequencyofsubjectarealessonsobservedinthissamplewasrecordedaswellasthekindsofwrittenfeedbackgiventothestudentteacheronthelesson.Theuni-versitysupervisorsmetwiththeresearcherstodiscussandtallyallthecategoriesoffeedbackfoundonthewrittenobservationnotes. In Phase 2, based on the findings from the needs analysis indicating thatsubject-specificpedagogywasnotasignificantcomponentoffeedbacktostudentteachers(seeResultssectionbelow),a“mathematicsfieldguide”wasdevelopedforuseduringstudentteaching.Thepurposeoftheguidewastohaveafocusing


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devicethatwouldcausestudentteachersandtheirsupervisorstoconsiderprob-lemsolving,explanations,representations,andmathematicalconnectionsasfourimportant subject-specific strategiesduring the lessonplanning/teaching/reflec-tionprocessenactedduringstudent teaching.Ledby themathematicsmethodsinstructor/researcherwhohadorganizedandledtheneedsanalysisconductedinPhase1,thesixuniversitysupervisorswhoparticipatedintheneedsanalysisdatacollection, togetherwith support fromamathematics subjectmatter professor,conceptualizedthepacketofsupportmaterials.Seventeenfirstsemesterstudentteacherswererandomlyassignedtobesupervisedbythesameuniversitysuper-visorsengagedinthedevelopmentofthefieldguide.Thestudentteacherswererequiredtousetheguideasmathematicslessonswereplanned,implemented,andevaluated.Datacollectedfromthestudentteachersincludedlessonplans,lessonplanreflections,andteachercandidatepost-conferencereflections.Datacollectedfromtheuniversitysupervisorsincludedwrittenobservationnotes,audiotapedfeedbackconferencesandresultsfromaninterviewattheendofthestudentteach-ingperiod.Inaddition,theobservedlessons(approximately3perstudentteacher)wereanalyzedbytheuniversitysupervisorsusingascaledobservationprotocolthattheyweretrainedtouse.Theobservationprotocolaskedobserverstoassigntheobservedlessonascaledcategoryofinstructionlabeledasoneofthefollow-ing: “Ineffective Instruction,” “ElementsofEffective Instruction,” “BeginningStagesofEffectiveInstruction—Low,Solid,High,”and“Accomplished,EffectiveInstruction”(HorizonResearchInc.,2006). InPhase3,102studentteachersweredividedintoexperimental(N=56)andcontrol(N=46)groupsinoneprogramoptioninourcredentialingprogram.Thefirststudentteachingexperiencerequiredthestudentteacherstoteachonlylan-guageartsandmathematicsforanine-weekperiod.Theexperimentalgroupwasclusteredintospecificstudentteachingseminarsectionswheretheyreviewedthesubject-specificpedagogical strategies learned in themathmethodscourseandincludedhowtodesignmathematicslessonswhichfocusedonproblemsolving,explanations, representations andconnections. In the seminar theexperimentalgroupwasintroducedtotheuseofthefieldguideforthelessonplanning/teach-ing/reflectionprocessandexpectedtouseitastheywrotelessonplans.Theywerematchedtothesamesixuniversitysupervisorswhodesignedthefieldguidesandparticipatedinthelessonobservation/feedbackcycleusingthefieldguides. Thecontrolgrouphadbeenintroducedtothesamesubject-specificpedagogi-calstrategiesinthemethodsclass,butengagedonlyinthetraditionalcurriculumofthestudentteachingseminar.Theydidnotusethefieldguideduringstudentteaching,andworkedwithuniversitysupervisorswhohadnotbeentrainedintheuseoftheguideaspartoftheobservation/feedbackcycle.Bothexperimentalandcontrolgroupsdidusethesamelessonplantemplatefordesigningmathematicslessonswhichrequiredwrittenreflectionaftereachlessonregardingproblemsolv-ing,explanations,representationsandconnections.Bothpopulationsmatriculated


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toasecondstudentteachingexperienceinwhichtheytaughtallsubjectareasfornineweeks,butdidnotusethefieldguidesaspartofthesupervisionprocess. At theculminationof the twostudent teachingexperiences,student teach-ersinbothgroupswereassessedusingtheLearningMathematicsforTeaching(LMT)SurveydevelopedbyHill,Rowan,andBall(2005).TheLMTSurveyforNumbersandOperationsconsistsofthreesubsectionswhichcanbeadministeredseparately.TheyincludeKnowledgeofContent,KnowledgeofStudentsandCon-tent,andKnowledgeofPatterns,Functions,andAlgebra.Forthepurposesofthisresearch,thesub-sectionwhichassessesKnowledgeofStudentsandContentwasusedtoexaminegrowthofbothcontentandhowstudentslearncontentduringstudent teaching experiences. In addition, lesson planswithwritten reflectionsanduniversitysupervisorwrittenfeedbacknoteswererandomlyselectedfrom10studentteachersineachgroupandanalyzedtodeterminetheamountandqualityofreflectivecommentsmadeaboutsubject-specificpedagogy.

Results

Phase 1—Needs Analysis Themajorityoflessonsintherandomlyselectedsampleoflessonobservationsnoteswerefromlanguagearts(30%)andmathematics(24%),withscience(16%),socialstudies(9%),andart,health,ormusiclessonscomprisingapproximately7%ofthelessons.(Theuniversitysupervisorsweresurprisedtofind14%ofthelessonobservationnotesheldnoclueastothesubjectareaobservedandimmediatelysuggestedthatthesubjectareaobservedshouldbewrittenonallwrittenfeedbacknotes.)Sixmainthemeswereidentifiedbysupervisorsinthecomparativeanalysisofthe200setsofwrittenobservationnotesasfocalpointsforreflectionduringobservedlessonsandareshowninTable1.

Table 1:Emerging Themes from Classroom Observation Notesas Categorized by University Supervisors

Categories Subcategories

1.ClassroomManagement studentbehavior,positivereinforcement, classroomorganization

2.LessonPlanning instructionalstrategies,contentstandards

3.LessonImplementation modeling,explanations,pacing,sequencing

4.StudentEngagement studentactivity,involvement

5.Assessment formal,informalquestioning

6.Professionalism punctuality,dress


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Missingfromthewrittensupervisors’observationnoteswerereferencestospecificpedagogicaltechniquesforasubjectarea,subject-specificquestioning,anddepthorqualityofsubjectmatterinlessons.StrategiesforworkingwithspecialneedsstudentsandaccommodationsforEnglishLearnerswerealsomissingfromthesupervisors’notes. Theresultsoftheneedsanalysisconfirmedthataspectsofthestudentteachingobservation/feedbackcycleneededtochangetobetterfocusstudentteachersandsupervisorsonsubjectspecificpedagogy.Withleadershipprovidedbymathematicsmethodsfacultyandmathematicsdepartmentfaculty,theuniversitysupervisorscreatedsupportmaterialsthatcouldbeusedduringlessonobservationsthattheynamedtheField Guide for Mathematics Lessons.Theguidewasintendedtofocusstudentteachersanduniversitysupervisorsonaspectsofmathematicsthatwerebeingemphasizedinmathematicscontentandmethodscourses:problemsolving,explanationsofmathematicalideas,useofvariedrepresentations,andmathematicalconnections. Componentsoftheguideincludeda“reminder”sheetforstudentteacherstousewhenplanningandrehearsingtheteachingofthelesson(calledStudentTeacher’sFieldGuideforPlanningandTeachingMathematicsLessons),a“reminder”sheetfor university supervisors to usewhen observing a lesson (called Supervisor’sFieldGuide toObservingMathematicsLessons),andaLessonReflectiongridthatfocusedthestudentteachersanduniversitysupervisorsonindentifyingvaluedcomponentsofthelessonsduringlessondebriefing(SeeAppendixA.)

Phase 2—Use of Field Materials during Student Teaching Analysisof theuseof thefieldguidesfocusedontheperformanceofsev-enteen student teachers. Overall, changes in lesson plans, written observationnotes,feedbackconferences,andwrittenreflectionswerenotedbytheuniversityresearchers.Emphasisontheuseofthesematerialsimpactedthestudentteachingcycleasevidencedbyartifactsgatheredfromthesupervisorsandstudentteachers.Changesaredescribedbelow. Therewasachangeinthewaythattheuniversitysupervisorsinteractedwiththestudentteachersregardingtheplanningandteachingofmathematicslessons.Interviewswith theuniversity supervisors indicated that all sixwere impactedbyhavingthefieldguidematerialstofocustheirobservationoflessons.Thesixsupervisors indicated that useof theguide changed their assessmentof lessonplans,impactedwhattheywerewatchingforduringmathematicslessonobserva-tions,changedthewaytheyrecordedwrittenobservationnotes,andsignificantlychanged the focus and tone of the post-observation feedback conference. Onesupervisordescribedtheprocessthatallsupervisorsappearedtofollow,reportinginaninterviewthat“havingaonepageguidethatIcouldhaveonthedeskwithmeremindedmewhattolookforduringthelessonandtointegratethatfeedbackinmynotestothestudentteacherandconversationsafterthelesson.”


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Analysisofwrittenobservationnotesdemonstratedthelevelofchangereportedininterviews.Writtenobservationnotesformathematicslessonsarchivedfromprevioussemestersforeachuniversitysupervisorparticipatinginthestudywerecomparedtothenotesthateachsupervisormadeforthecurrentsemesterwhileusingthefieldguidematerials.Table2showsthecategoriesusedforanalysisofobservationnotesandtherelativefrequencyofcommentsspecifictomathematicsbeforeandaftertheuseofthefieldguidesforthegroupofsixsupervisors.

Table 2:Analysis of University Supervisor Written Observation NotesBy Category before and after Use of Field Guides

Mathematics Before After Classroom Practices Before AfterComments Comments

ProblemSolving 3.6% 8.9% ClassroomManagement 26.3% 10.7%

Explanations 5.1% 31% LessonPlanning 5.1% 3.6%

Representations 9.5% 13.1% Implementation 25.5% 13.1%

Mathematical 5.8% 8.3% StudentEngagement 11.7% 6.0%Connections

Other 2.9% 2.1% Assessment 3.6% 2.4%

Table 3:Sample University Supervisor Observation NotesBefore and after Use of Field Guides

Supervisor –Written Feedback Before Same Supervisor-Written Feedback After

GoodexplanationfromGroup1. “Let’smakeittaller.”Missedanopportunity toexplaintostudentswhythefacewouldnotbe asquareifwemadethecubetaller.

Yourquestionsreallymadethemthink.Ilikedthewaythatyouletthemsolvetheproblem themselvespriortogivingthemtheanswers.

Considerusingtheoverheadprojector.Youmighthavemodeledthemakingofthe rectangleontheoverhead.

Howwillyouassesstheirwork? Assessedpriorknowledge-youhandledthiswell. Coveredmanyunderstandings(bargraph, coordinatepairs,axes,circlegraph,etc).

Thoroughclosuretolesson. Theclosureisthemostdifficulttobring togetherbecauseyoudon’tknowwhatchildren aregoingtosay.Bringingoutthemathematical thinkingofstudentsratherthan‘telling’is somethingtoworkon.


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Afterusingthe Field Guide for Math Lessons,thedatashowedanincreaseintheuseofcommentsspecifictotheteachingofmathematicsbytheuniversitysupervisorsintheirwrittenobservationnotesofstudentteacherlessons.Furtheranalysesrevealedthatthemeannumberofcommentsaboutmathematicsincreasedfrom4.1to16.4whentheguideswereused,whereasthemeannumberofcommentsongeneralclassroompracticedeclinedfrom11.1to9.4withtheuseofthefieldguides.The“value-add”ofusingtheguideisshownbyanincreaseinthenumberofcommentsforspecificmathematicspedagogywithrelativelylittlechangeinthenumberofcommentsaboutgeneralclassroompractice.Approximately10morecommentsonaveragewerewerenotedwhensupervisorsusedthe guide. Relativefrequenciesofthesupervisorcommentsdepictonewaytodocumentthechangesinthefeedbackcycle.Table3illustrateshowsupervisorsinthestudyusedspecificmathematicallanguagetocommunicatetheirconcernsaboutthelessonsandfocustheirobservationsmorepreciselyonthemathematicsinthelesson. Supervisorsusedmathematicaltermsandconceptstofocusteachercandidate’sthinkingaboutwhathappenedinthelessonorwhatteachercandidatesmighthavedonetopushpupil’smathematicalthinking.Overall,thesupervisionnoteswrittenwhileusing thefieldguidematerials show that supervisorsgavemoredetailedsuggestionsandcomplimentsspecificto teachingmathematics,whichincludedmathematicalterms,processes,conceptsandpupilexplanations.Boththequantityandqualityofuniversitysupervisors’commentstostudentteachersaboutmath-ematicsteachingincreasedafteruniversitysupervisorsusedtheguide. Immediatelyfollowinglessonobservations,theuniversitysupervisorswereaskedtocompleteanobservationinstrumentthataskedthemtoratethequalityofvariouscomponentsof themathematics lessons,basedon training theyhadreceivedonuseoftheprotocolpriortothestudentteachingsemester.DataforalloftheobservedlessonsaredepictedinTable4. Inaddition,feedbackconferencesbetweenthestudentteachersanduniversity

Table 4:Distribution of Observation Protocol Scoresfor Mathematics Lessons

Category Number of Observed Lessons Scored at this Level

Level1:IneffectiveInstruction 4Level2:ElementsofEffectiveInstruction 8Level3:BeginningStagesofEffectiveInstruction-Low 7Level3:BeginningStagesofEffectiveInstruction-Solid 10Level3:BeginningStagesofEffectiveInstruction-High 5Level4:Accomplished,EffectiveInstruction 8

TotalLessonsObservedusingProtocol 42


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supervisorswereaudiotapedandanalyzed.Acasestudyapproachwasusedbytheuniversityresearcherstolinktogetheralldatathatwascollectedfromstudentteachersandtheiruniversitysupervisors.Eachcasewasassigneda“levelofimple-mentation”basedonthreesetsoflessonplans,supervisorobservationsofteaching,observationnotesandstudentteacherreflectionsaboutthelessons.Acomparisonoftwocasesgavedescriptiontothelevelofimplementationofsubject-specificpedagogyandthequalityofreflectionthattookplacebetweenuniversitysupervi-sorsandstudentteachers.

Karen’s “Accomplished Instruction” ThefirstgradersinKaren’sstudentteachingclassroomhadbeenworkingonadditionandsubtractionstoryproblemsandcontinuedthatworkthedayherlessonwasobservedbytheuniversitysupervisor. Thechildrenhadbeenexploringnum-bersbyformingcombinationsofnumbersto12(i.e.,8+4=12,7+5=12,6+6=12,etc.)forthefirst10weeksoftheschoolyearandthislessoncameatthecloseoftheunitonnumbersense.

From Karen’s Lesson Plan. Aftermodelingastoryproblem,Karenplannedtogivetheclassaproblemtoworkonaboutadding5and6.Anexcerptfromthatplanexposedherthinking:

Presenttheproblem:

Teacher:(storyonchartpaper)Iwascleaningtheclassroomtheotherday.Ifound5pencilsonthefloorunderthistable.Ifound6morenexttothesink.

Teacher:Whatcanyoutellmeaboutwhathappenedinthestory?

Teacher:NowIwantyoutosolvethisproblemforme.Writedownthenumbersentencethatyoucameupwithandtheanswer.Iwantyoutowritehowyougotyouranswer.Youcanusewords,pictures,andnumbers.

AtClosure:

Teacher:Cansomeonewithasilenthandtellmehowtheyknowthatit is theanswer?Howdidyoufigureitout?(Ifchildrensay,“Ijustknow,”askthemhowtheycouldexplainittosomeonewhodoesn’tknowitbyheart.)

Karenwaspreparedtoelicitthechildren’sunderstanding;sheplannedtoprobefurtherifstudentscouldnotexplainhowtheyknewtheanswer.

From Karen’s Observed Lesson—University Supervisor Feedback Notes.AfterstudentsweregiventimetofindandrecordtheiranswerstheymetontheruginfrontofKaren.Theuniversitysupervisorcapturedthegroupconversationthattookplace:

Karen (Student Teacher):Howdidyouknowthat5and6is11?

Student M:BecauseIknowthat6and6is12andonelessis11.


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Karen:Thankyou.Isawalotofpeoplethathaddifferentwaysoffiguringthatout.

Student E:Youhave6andaddfivemore.Thenyoucountitall.1,2,3,4,5,6,and7,8,9,10,11.

Student K:If5and5equals10,then6and5hastoequal11,becauseyouareaddingonemore.

Student S:Ifyouknow4and6is10,thenyoujustaddonemorebecauseitis5andnot4.

Thesupervisorwrote“Karen,youkeptanemphasisonstudents’ thinking.Youkeptaskingforvariousstrategiesinsolvingtheproblemandputseveralwaystorepresentstudentthinkingontheboard.…Wonderfullessoninbringingoutstudentsensemakingandacceptingmanywaystoworkonaproblem.”

From the Audiotape of Karen’s Post-Observation Conference.Thesupervi-sorusedtheField Guide for Math Lessons“LessonReflectionGrid”tofocusthediscussionofthelesson(SeeAppendixA).AtranscriptofthediscussionshowedthatKarenbegantheconferencediscussingwiderangingtopicsabouttroublesometraitsofindividualchildren,waysthatchildrencount,gainsinstudentknowledge,andclassroommanagement.Thediscussionbecamemorefocusedwhenthesuper-visorsuggestedusingtheLessonReflectionGrid.ThefollowingtranscriptfromtheaudiotapeshowshowKaren’stalkchangedfromvaguenotionsofchildren’sunderstandingtoafocusedandexplicitdescriptionofstudentwork:

Karen:OverallIreallylikedhowtheywereabletoaccomplishthelesson.Andreallyunderstandandthattheycouldgivemesomereallygoodanswers.Andhowyoucouldfigureouttheanswer.

Supervisor:Sonowlet’sreviewourLessonReflectionGrid.AndIhaveonethatwecanactuallyfillout.

Supervisor: So…for problem solving?What evidenceof success is there forproblemsolving?

Karen:Thelessonintheverybeginningwasactuallygivingthemcombiningandseparatingproblems.Theyhadtocomeupwiththeirownnumbersentencesanddrawingstoreflecttheiranswer.Andfinishthat.Andthentheyhadtocomeupwiththeirownproblemandsolvethose.

Karen:Andthegame[playedwithdicebeforethetask]wasevenproblemsolving.Howdoyouseewhatthesetwodierollandseeifthesumisonthegameboard.Ormore,“HowdoIstrategicallyplacethemtogetfiveinarow?”

Supervisor:Right,right.Ok,solet’sjotdownafewthings.

Karen:Also,thestudents,whentheywerewritingtheirownproblems,didn’thavetoaskmehowtostarttheproblem.Theyjustwrotesomeliketheonestheysaw.


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Thiswasnew.Thestudentsmightbethinking,“IfIhavetowritemyown,I’llputmyownthings,myownnames,myownobjectsintheproblem.”

TheaudiotapecapturedKarenandhersupervisor’scontinuedconversationastheycompletedtheLessonReflectionGridandthatendedthefeedbacksession. Apatternoffocusedandexplicitdiscussionaboutmathematicsemergedinthedata;specifictalkaboutmathematicscontentandpedagogypermeatedthefeed-backcyclebetweenthesupervisorandthestudentteacher.Inthepost-observationconference,Karenbegantotalkinvaguetermsabout“somereallygoodanswers.”AftertheLessonReflectionGridwasintroduced,sheusedmathematicaltermsforproblemstructures,identifiedthreewaysinwhichchildrenwereproblemsolvingandnewways inwhich the childrenwere engaged in doingmathematics.TheanalysisofKaren’slessonplans,writtenfeedbackfromtheuniversitysupervisor,and the transcriptsof thepost-observationconferencesdemonstrate thatuseofthefieldguideinfluencedtheplanning/teaching/reflectionofthestudentteacher.Additionally,basedontheobservationprotocol,thesupervisorratedthelessonasa“Level4:Accomplished,EffectiveInstruction”lesson.

Wanda—“Beginning Elements of Effective Instruction” Wanda’sfirststudentteachingexperiencewasinafifthgradeclassroom.Uni-versitysupervisorfeedbacknotesfromalessonobservedduringanintegerunitdemonstratedthatclassroommanagementwasaproblemforWanda.Forthesecondobservation,sheplannedtointroduceanewunitonstatisticsandgraphing.

From Wanda’s Lesson Plan.ThelessonplanshowedhowWandawantedthestudentsto“beabletoread,interpret,andmakelinegraphs.”Theplanproposedthefollowingquestions:

Teacher:Cansomeonereminduswhatalinegraphshows?Thinkaboutthelinegraphweconstructedyesterday.

Teacher:Linegraphscanshowmanymorekindsofdata. (Showgraph).ThisgraphshowsthechangeinpopulationdensityintheUSbetween1940and1990.Howwouldyoureadthisgraph?

Teacher:Trendisanotherwordforpattern.Wecanusethetrendorpatterntomakepredictionsaboutthedata.Whatpredictioncanbemadeaboutthisdata?

Teacher:Whatinformationisgivenonthex-axis?They-axis?Whatistheintervalforthisgraph?Lookattheyaxis,whatisthedifferencebetweeneachnumber?Whataboutthexaxis?

Teacher:Lookatthegraphtotellinformation.Whatisthepopulationdensityin1960?Howwouldwefindthatinformation?Lookatthex-axisandthey-axis.Thepointhascoordinates(1960,51).

Theremainderofthelessonplanincludedadditionalconvergentquestioningthat


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askedforinformation.Itdidnotincludeproblemsolvingquestionsandnofurtherdatatrendswereexamined.Explanationsbystudentswerenotproposedorantici-patedintheplan,andconnectionswereonlyhintedat.

From Wanda’s Observed Lesson—University Supervisor Feedback Notes. Thesupervisor’sscriptednotescapturedwhathappenedinthelesson.

Wanda:We’regoingtolearnmoreaboutlinegraphs.We’velearnedaboutchangeindistance,andtemperatureovertime.Wecanlookatchangeinpopulationandchangeinheightovertime.Thisgraphshowspeopleandyears.Whatinfoisonthexaxis?

Student A:Years.

Wanda:Andonthey-axis?

Student C:Peoplepersquaremile.

Wanda:What’stheinterval?(pause)Whoremembersinterval?(studentsshakeheadsnegatively).

Wanda:Whatwasthepopulationdensityin1960?

Student G:50.

Wanda:Inwhatyearwerethereabout44peoplepersquaremile?

Student G:1957.

Wanda:Arethereanyquestionssofar?

Thesupervisorwrotethefollowingnotes:

Howmuchinformationaboutthegraphcouldbeexplainedbystudents?Howcouldyouguidestudentstoreadthegraph,explainthegraphandaskquestionsaboutwhatthedatais“saying”inthegraph?Doyouthinkthestudentscouldasktheirownquestionsabouttrendsthatareinthegraph?Whenyouaskaquestionthatasksforfactualdata,andyougetheadsnodding,whatdoesthattellyouaboutthelevelofthinkingthatishappeningatthispointinthelesson?Let’sthinkabouthowtorestructurethequestion/answerinteractiontopromotemoreproblem-solvingandstudentexplanation.

From the Audiotape of Wanda’s Post-Observation Conference. Thesupervisoraudiotapedthepost-observationfeedbackdiscussionandusedtheLessonReflectionGridtostructuretheconversation.Aportionofthetranscriptfromtheconferencefollows:

Supervisor:Whenwethinkaboutthelevelofproblemsolvingandstudentexpla-nationsinthislesson,whatareyourthoughts?

Wanda:Tobehonest,I’mworkingsohardtotrytokeeptheclasswithmethatIdon’thavetimetothinkaboutproblemsolvinginthelesson.IwastryingtoaskquestionstodifferentstudentsaroundtheroomsoIwouldn’tlosethem.Theyjust


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don’tseeminterestedinpayingattentionmorethanfiveminutes.IjustwanttoknowthatI’mgettingthemtoreadagraph.

Supervisor:Whataboutgraphing—doyouthinkitmightbeinterestingenoughtothemtokeeptheirattention?

Wanda:Theyprobablywouldbebetteroffcreatinggraphs,buthowdoImakeithappeninsuchawaythatIdon’tloseseveralstudents?Ifweweretocreategraphs,thenIwouldhavetoletpairsorsmallgroupsmakethegraphsonchartpapersowecouldallseethemtoanalyzethegraph,thedata,andthetrends.Idon’tthinkIwanttoputthemallovertheroomwithchartpaper.I’dnevergetthembackagain.

Supervisor:Soitsoundslikeyou’remostconcernedaboutthematerialsyou’dbeputtingintheirhands,butifwecouldsolvethatproblem,doyouthinktheywoulddomoreproblem-solvingwithgraphsiftheyhadtocreateone?

Wanda:Ofcourse,andtheirexplanationswouldprobablybemoreextensive,butItriedhavingthemworkinpairslastweekanditterrible.Ihadtore-teachthelessonthenextdayattheboard.

Basedontheobservationprotocol,thesupervisorratedthelessonasa“Level2:ElementsofEffectiveInstruction”lesson,perhapsbecausetherewerestudentworkproductsthatwereplannedandimplementedthatmatchedalessonobjective. Incompletingareviewofallofthedatafromthislesson,itwasevidentthatWandawas aware of themathematics strategies that she should be using, andshecouldseetherelationshipbetweencomponentsofamathematicslesson(e.g.representations and explanations), but shedecided that shewasnot competentenoughatclassroommanagementandconductingdiscussionstoallowthosekindsofinteractionstohappen.Instead,Wandachosetoaskconvergentquestionsthatrequiredcontrolledanswers, andmaintainedcontrolbynot allowing toomuchstudentinput.Useofthefieldguidewasnotsuccessfulatimpactinginstructionandreflectionbecauseofissueswithclassroommanagement. Overall,theresultsfromthisphaseoftheresearchindicatedthatuseofsubject-specificsupportmaterialsduringstudentteachingfocusedtheuniversitysupervisorsandstudentteachersoncriticalcomponentsofeffectivemathematicsinstruction.StudentteachersdemonstratedvariedabilitytoimplementidentifiedstrategiesandtheuseoftheLessonReflectionGridoftenhelpedtodiscernwhichkindsofstrate-giesweredifficultforthestudentteachers.Perceivedqualityofimplementationwasimpactedbygeneralclassroommanagementskillsaswellasspecificmanagementskillsneededtoimplementsomeofthesubject-specificstrategiessuchasstudentexplanations.Managementabilityalsoimpactedthefocusofreflectivediscussionbetweenuniversitysupervisorsandstudentteachers.


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Phase 3—Use of Field Guides to Foster Teacher Reflectionand Pedagogical Growth

Inthefinalphaseofthestudy,all102studentteachersinbothexperimental(N=56)andcontrol(N=46)groupswereaskedtoreflectuponeachmathematicslessontheytaughtduringthefirstsemesterofstudentteachingbycompletingthe“Reflection”sectionofthelessonplanformatwhichdirectedtheirreflectionintothefourcategoriesofmathematicssubjectspecificpedagogy(problemsolving,explanations, representations, connections). Both groups turned in those planswhichwerethenkeptfordatacollection.Lessonplansfromarandomsampleoftenstudentteachersineachgroupwereanalyzedforreflectivecommentsmadeafter each lessonwas taught.Thenumberof comments for each categorywasanalyzedtoseewhetherthestudentteachersinbothgroupsdifferedinthequantityofreflectivecommentstheymade.Table5showsalargedifferencebetweentheexperimentalandcontrolgroupsinthenumberofreflectivecomments. Thequalityofreflectionthatthestudentteachersproducedforthelessonplansalsodiffered.Table6showsa representativesampleof the typesof reflectionsproducedbythestudentteachersfromeachgroup.Manyofthereflectionswrit-tenbytheexperimentalgroupwerefocusedonstudentunderstandingandstudentlearning;writtenreflectionsdiscussedtheamountandfocusonpupil’smathemati-calreasoningandjustificationinthelesson.Pupilexperienceswerecentraltothethoughtsoftheseteachercandidatesastheythoughtaboutthemathematicslessonthattheyplannedandimplemented.Manyofthecommentswerespecificratherthangeneralandgaveadetailedaccountofwhataspectsofthelessonwereeffec-tiveandineffective. Analysisofthenumberofreflectivecommentsandthecontentofthelessonreflectionswrittenbytheexperimentalgroupdemonstratedtheimpactoftheuseofthefieldmaterials.Anemphasisonproblemsolvingandexplanationswasclearlyevident.Thecontrolgroup teachercandidates’ lessonplans indicated that theirreflectionaboutthelessonsfocusedonteachingratherthanonpupilunderstanding

Table 5: Number of Comments in Each Category Foundin Mathematics Lesson Reflectionsby Two Sample Groups of Teacher Candidates

Category Experimental Group Sample Control Group Sample (N=10) (N=10)

ProblemSolving 32 1Explanations 39 8Representations 40 19Connections 33 13

Total 144 41


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asevidencedbythelargenumberof“I”statementsintheirreflectivestatements.Generalitiesratherthanspecificitycharacterizedthecommentsmadeinthecontrolgroup’slessonreflections.Overall,studentunderstandingwasnotalargefocusofreflectionfortheteachercandidatesinthecontrolgroup. Didchangesinthesupervisionprocessincreasethestudentteachers’abilitytousemathematicalknowledgeforteaching(MKT)intheirlessons?UsingtheLearningMathematicsforTeaching(LMT)Survey(Hill,Rowan,&Ball,2005;Hill,Ball,&Schillings,2008),allstudentteacherswereassessedat theendofthemathematicsmethodsclassandagainattheendofthesecondoftwostudentteachingexperiences.TheLMTsurveyreportsscoresasz-scoresandwasnormedforimplementationwithapopulationofpracticinginserviceteachers(Hill,Rowan,&Ball,2005).Areportedz-score indicates thedeviationfromthemeanofallinserviceteacherswhotooktheexamaspartoftheprocessusedforestablishingreliabilityandvalidityfortheLMTinstrument.Ascoreof1.0indicatesascore1standarddeviationabovethemeanwhencomparedtoinserviceteachers.Ascoreof-1.0indicatesascore1standarddeviation belowthemeanwhencomparedtoinserviceteachers.UsingthescoresfromNumberandOperationsKnowledgeofStudentsandContent componentof theassessment,describedasan importantaspectofMKT,allowedustocompareexperimentalandcontrolgroupgrowthinMKTskills.

Table 6:Examples of Written Reflections from Two Sample Groupsof Teacher Candidates after Teaching a Mathematics Lesson

Experimental Group Sample Control Group Sample(Student Understanding Focus) (Teacher Action Focus)

“Whenstudentsbegantosetasidethe “WhenIbegantomakearelationshipamountthateachitemcost,theysaw betweenthegameandlesson,therelationshipbetweenmoneyand Iheardalotof‘aaahhs’.”price.” “IthinkthevarietyofmanipulativesIgave“Studentshadtofigureoutwhat themhelpedtokeepthemengagedintheactivity.”wouldhappeniftheyusedtheirpersonalfootmeasuretobuilda “TomorrowIwillgivethemanotherhouse.” opportunityfordiscoveringequivalent fractionswiththefractionbarhandout”“Somestudentswerebored,somakingthelessonmoreinteractive “Iwantedtobringreallifetostudents”wouldhelpnexttime.” “Myexplanationsduringguided“Itgotalittlecomplicatedwhenthe practicefacilitatedtheassignment.”4thgradershadtomeasuretheshapesandthenmaketheirownshapes.”


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Scoresfromthetenrandomly-selectedexperimentalgroupstudentteachersandthetencontrolgroupstudentteachers,whosereflectionsweanalyzedindetail,werecompared.At-testofthescoresshowedthattherewasnosignificantdiffer-enceforeachgroupaftercompletingthemathematicsmethodsclass.However,attheendofstudentteaching,whentheteachercandidatesinbothgroupswereassessedagain,theexperimentalgroupscoredsignificantlyhigherthanthecontrolgroupontheNumberandOperationsKnowledgeofStudentsandContentconstruct(p=0.0597).Theyscoredapproximatelyone-thirdofastandarddeviationhigherthanthecontrolgroup(z=0.373)ontheLMT,whichisanoteworthyeffectsize. When the MKT scores for all of the student teachers in the experimental(N=56)andcontrol(N=46)groupswerecompared,similarresultswerefoundtothosereportedabovefortherandomlyselectedsample.Theaveragegainfortheexperimentalgroup(n=56),inz-scoresreportedfrommathematicsmethodsclasstotheendofstudentteachingwasanincreaseof0.314,similartotherandomlyselectedgroupoftenstudentsfromtheexperimentalgroup.Forthecontrolgroup(n=46)therewas,infact,asmalllossof-0.015.Amatchedpairt-testfortheex-perimentalgroup(p=0.0007)andcontrolgroup(p=0.412)showsthattherewasastatisticallysignificantgaininknowledgefortheexperimentalgroup,buttherewasnostatisticallysignificantgain(orloss)forthecontrolgroup. Returningtoourresearchquestionwhichasked“Doesreflectiononsubject-specificpedagogyinmathematicsduringstudentteachingresultinmoreeffectivemathematicsteaching?,”welearnedthattheresultsofthisresearchindicatethatpurposefullydirectingstudent teachersanduniversity supervisors to focusandreflectonsubject-specificpedagogyduringthestudentteachingcomponentofapre-serviceprogramcanpositivelyimpactplanning,teachingandreflectionaboutthosepedagogicalelements.Theteaching/observation/feedbackcycleexpandstoincludemorespecificreflectionaboutthequalityofsubjectmatterlearninginles-sons.Usingfieldmaterialsthatfocussupervisorsandstudentteachersiseffectiveincreatingmoreknowledgeableteachers.BasedontheresultsoftheLMTsurvey,engagingindeeperreflectionaboutsubject-specificpedagogyimprovesstudentteachers’knowledgeofsubjectmatterandtheirunderstandingofwhichstrategieswillbemoreeffectiveintheirteachingofmathematics.

Conclusions Thegrowingresearchbaseontherelationshipofsubject-specificpedagogicalskillsandpupilachievementcompelsustoconsiderhowtoinfusethisknowledgebaseintopre-servicemultiplesubjectpreparationprograms.Wearegivenlittletimetopreparenoviceswhomightconsiderhowsubject-specificstrategiescanhaveaprofoundimpactonstudentlearning.Weareaskedtodevelopgeneralpedagogicalskills,suchasmanagementstrategiesforguidingdiscussions,butwealsoneedtohelpourpre-serviceteachersunderstandhowthosestrategiesarechallengedby


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morecomplexsubject-specificstrategiessuchaselicitingmathematicalthinkingfromlearners.Inaddition,wemustacknowledgingthateachelementarypre-serviceteacherwillhavepersonalaffinitiesforsomesubjectareasandmaylackdepthofunderstandinginotherswhichcanimpacttheirmotivationtoimprovetheirsubject-specificpedagogicalskillsandtheyneedastructuredexperiencetodevelopskillsin“weak”subjectareas. Inaddition toconsidering theneedsof thepre-service teacher,weneedtothinkabouttheroleoftheuniversitysupervisorinthedevelopmentofsubject-specificreflectivethinking.Usingfieldmaterialstofocusuniversitysupervisorsisaworthwhilestrategy,butconsiderationmustbegiventothefactthatsupervi-sors,likepre-serviceteachers,willhaveparticularaffinitiesfordifferentsubjectareasandmaybemoreorlesssuccessfulatpromotingreflectivethinkingaboutsubject-specificpedagogy.Developmentofsubject-specificpedagogicalskillsdur-ingstudentteachinginmultiplesubjectclassroomssuggeststhatothermodelsofsupervisionmayneedtobeconsideredwhichallowsubjectmatterspecialiststomentorstudentteachers,ratherthandependingonthetraditionalrelationshipofoneuniversitysupervisortoonestudentteacher.Additionally,ourresearchaddstothegrowingresearchaboutthecriticalrolethatthecooperatingteacherplaysinthedevelopmentofreflectivepracticeofsubject-specificpedagogy(Borko&Mayfield,1995;Griffin,1989;Shantz&Ward,2000).Wehavebeguntoworkwithsmallgroupsofteacherswhohostourstudentteacherstodeterminehowtostructureafocusonsubject-specificpedagogythatincludestheclassroomteacher.Finally,weneedtoapplywhatwehavelearnedinourfocusedworkonimprovingmathematicsteachingtotheotherimportantsubjectareasthatourteachercandidateswillberesponsiblefor.

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Darling-Hammond,L.&Bransford,J.(2005).Preparing teachers for a changing world. Report of the Committee on Teacher Education of the National Academy of Education.SanFrancisco:JosseyBass.

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Griffin,P.(1989).Teachingtakesplaceintime,learningtakesplaceovertime.Mathematics Teaching, 126(1)12-13.

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Hill,H.,Ball,D.L.,&Schilling,S.G.(2008).Unpackingpedagogicalcontentknowledge:Conceptualizingandmeasuringteachers’topic-specificknowledgeofstudents.Journal for Research in Mathematics Education,39(4),372-400.

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Appendix

Student Teacher’s Field Guidefor Planning and Teaching Mathematics Lessons

Planning Teaching

Problem 1.Doestheteacherpose 1.DidstudentsunderstandSolving tasksthatarebasedon thetaskorproblem? significantandworthwhile 2.Didtheclassroomculture mathematics(bigideas)? encourageadiversityofideas 2.Willactiveparticipation andmultiplepathways ofallstudentsbe tosolveaproblem? encouragedandvalued? 3.Arestudentsactivelyengaged Howwillyouprovide insolvingtheproblem? forstudentswithspecial needs? 3.Doestheplanallow adequatetimeforchildren toinvestigatetheproblem? 4.Howwilltheteacher encouragestudentsto generateideas,questions, andconjectures?

Explanations 5.Istheclassroomsetup 4.Werestudentsallowedto sothatchildrencandiscuss reflectontheirthinkingand easily? verbalizetheirexplanations 6.Doestheplanallow beforewritingthemdown? adequatetimeforchildren 5.Canstudentsclearlystate toreporttheirsolutions/ whattheyknow? strategiesandexplainhow theyknowtheiransweris correct? 7.Howwilltheteacher askforconsensusinresolving differencesbetweenstudent explanationsandlookfor clarificationofstudentideas?


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Student Teacher’s Field Guidefor Planning and Teaching Mathematics Lessons

(continued)

Planning Teaching

Representations 8.Arematerialsavailable 6.Dostudentslinkwords, forstudentstousein visuals,number,and/or explainingtheirideasand manipulativestoexplaintheirideas? solvingtheproblemsuch asvisuals,manipulatives, diagrams,charts,tables, calculators,etc? 9.Whatvariousways willtheimportant mathematicalideasinthe lessonberepresented?

Connections 10.Howwillconnections 7.Didtheteacherconnectstudent’s bemadetostudent’sprior priorknowledge? knowledge? 8.Wasclosuretothelessonprovided 11.Isacontextprovided whichvalidatedstudents’ideas forthetask? andreviewedwhatstudentslearned? 12.Whattypeofclosure 9.Didstudentsunderstand isplannedsothatstudents thebigideasofthelesson? canmakeconnectionsto othermathematicalideas? Tootherreal-world applications?

AppendixContinuedonNextPage


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Supervisor’s Field Guide for Observing Mathematics Lessons

Did you observe the student teacher planning and implementing a lesson with:

Problem Studentsactivelyengaged NotesonProblemSolving:Solving insolvingaproblem? •Problemsolvingmeansengagingin ataskforwhichthesolutionisnot knowninadvance(i.e.,notpractice) •Studentsshouldbebuildingnew knowledgethroughproblemsolving •Studentsratherthanteachersare doingthemathematics

Explanations Studentsmakingsenseof NotesonExplanations: themathematicsby •Teacheracceptsvariousmethods presentingtheirsolutions tosolveaproblem andexplainingtheir •Studentsandteacherlistento, reasoning? respondto,andquestiononeanother •Studentsareallowedtoreflecton theirthinkingandverbalizetheir explanationsbeforewriting themdown •Teacherasksforconsensusin resolvingdifferencesandlooks forclarificationofstudent’sideas

Representations Studentsandteachers NotesonRepresentations: usingmultiplewaysto •Pictures,words,numbers,models, representasolution? symbols,concreteobjects,tables, charts,diagrams,calculators, ortechnologyareused •Studentslinkwords,pictures, numbers,forexample,toexplain theirideas

Connections Teacherandstudents NotesonConnections: makingconnections? •Teachersetsthestagetouse student’spriorknowledgeand providesacontextforthelesson •Teacherandstudentsconnect mathematicalideasandapplyideas toothersubjectareas •Teachersallowadequatetime toprovideclosuretothelesson

AppendixContinuedonNextPage


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Mathematics Lesson Reflection Grid

Name_______________________ Date_______

Providespecificexamplesfromthemathematicslessontaughtforeachoftheboxesinthegrid.Feelfreetochangethespacinginthegridorcontinueyourreflectiononanothersheet.

Evidence of Success Missed Opportunities/Next Steps

ProblemSolving:Studentsactivelyengagedinproblemsolving

Explanations:Studentsmakingsenseofmathbyexplainingreasoning,questioningideas

Representations:Multiplewaysusedtorepresent/communicateideas

Connections:Mathematicalconnectionsmadebyteacher&students

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