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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
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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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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