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(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 1
BUILDINGONSTUDENTS’PRIORKNOWLEDGE
HowcanIrespondtostudentsinwaysthatimprovetheirlearning?
Introduction
Inquiry-basedteachingassumesthatstudentsdonotarriveatsessionsas‘blankslates’,butasactivelythinkingpeoplewithawidevarietyofskillsandconceptions.Researchshowsthatteachingismoreeffectivewhenitassessesandusespriorlearningsothattheteachingmaybeadaptedtotheneedsofstudents(Black&Wiliam,1998).Priorlearningmaybeuncoveredthroughanyactivitythatoffersstudentsopportunitiestoexpresstheirunderstandingandreasoning.Itdoesnotrequiremoretesting.Forexample,itcantaketheformofasinglewrittenquestiongivenatthebeginningofasessiontoelicitarangeofexplanationsthatmaythenbediscussed.Thisprocess,oftenreferredtoasformativeassessment,maybedefinedas:
"…allthoseactivitiesundertakenbyteachers,andbytheirstudentsinassessingthemselves,whichprovideinformationtobeusedasfeedbacktomodifytheteachingandlearningactivitiesinwhichtheyareengaged.Suchassessmentbecomes‘formativeassessment’whentheevidenceisactuallyusedtoadapttheteachingworktomeettheneeds."(Black&Wiliam,1998para,91)
Thismoduleconsidersthedifferentwaysthiscanbedoneandfocusesonthefollowingquestions:
• Howcanproblemsbeusedtoassessperformance?• Howcanthisassessmentbeusedtopromotelearning?• Whatkindsoffeedbackaremosthelpfulforstudentsandwhichareunhelpful?• Howcanstudentsbecomeengagedintheassessmentprocess?
ActivitiesActivityA: Introducingformativeassessment....................................................................................3ActivityB: Teachers'ownexperiencesofformativeassessment.......................................................4ActivityC: Principlesforformativeassessment.................................................................................6ActivityD: Analysestudents’responsestoproblem-solvingtasks....................................................8ActivityE: Analysestudents’responsestoconcept-focusedtasks..................................................10ActivityF: Observeformativeassessmentinaction........................................................................12ActivityG: Planandreportbackonanassessmentlesson..............................................................13ActivityH: Considertheeffectsoffeedbackonstudentlearning....................................................14Acknowledgement:Inpreparingthismaterial,weacknowledgethepermissionsgivenbytheBowlandCharitableTrusttoadapttheprofessionaldevelopmentresources,BowlandMaths,thattheyhadpreviouslyfundedustoproducefortheUK.Thisincludesmanyofthehandoutsandmostofthevideoextracts.AdditionalresourceswerealsoadaptedfromImprovingLearninginMathematics,agovernmentfundedprogramintheUK.Theoriginalsourcesare:Swan,M;Pead,D(2008).Professionaldevelopmentresources.BowlandMathsKeyStage3,BowlandTrust/DepartmentforChildren,SchoolsandFamilies.ObtainableintheUKfrom:http://www.bowlandmaths.org.uk.Swan,M;(2005).ImprovingLearninginMathematics,challengesandstrategies,DepartmentforEducationandSkillsStandardsUnit.ObtainableintheUKfromhttp://tlp.excellencegateway.org.uk/pdf/Improving_learning_in_maths.pdf
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 2
Relatedresearchtohelpyourplanningforthismodule.Black,P.,&Wiliam,D.(1998).Insidetheblackbox:raisingstandards
throughclassroomassessment.King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:
http://shop.gl-assessment.co.uk
Thisshortbookletoffersasummaryoftheextensiveresearchliteratureintoformativeassessment.Itshowsthatthereisclearevidencethatimprovingformativeassessmentraisesstandards,andoffersevidenceshowinghowformativeassessmentmaybeimproved.Thisbookletisessentialreadingforallteachers.
Black,P.,&Harrison,C.(2002).Workinginsidetheblackbox:Assessmentforlearningintheclassroom.King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:
http://shop.gl-assessment.co.uk
Inthisbooklet,theauthorsdescribeaprojectwithteachersinwhichtheystudiedpracticalwaysofimplementingformativeassessmentstrategiesandtheeffectthishadonlearning.Thesectiononfeedbackandmarking(pages8-9)areparticularlyrelevanttothismodule.
Black,P.,Harrison,C.,Lee,C.,Marshall,B.,&Wiliam,D.(2003).Assessmentforlearning:Puttingitintopractice.Buckingham:OpenUniversityPress.
ThisbookgivesafulleraccountoftheearlierbookletsInsidetheblackboxandWorkinginsidetheblackbox.Itdiscussesfourtypesofaction:questioning,feedbackbymarking,peer-andself-assessmentandtheformativeuseofsummativetests.Thesectiononfeedbackandmarking(pages42-49)isparticularlyrelevanttothismodule,whilethesectiononpeerandself-assessment(pp49-53)isrelevantforthenextCPDmodule.
Hodgen,J.,&Wiliam,D.(2006).Mathematicsinsidetheblackbox.
King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:
http://shop.gl-assessment.co.uk ThisbookletappliestheabovefindingsspecificallytoMathematics.
ItconsiderssomeprinciplesforMathematicslearning,choiceofactivitiesthatpromotechallengeanddialogue,questioningandlistening,peerdiscussion,feedbackandmarking,andselfandpeerassessment.Thisbookletisessentialreadingforallmathematicsteachers.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 3
ACTIVITYA:INTRODUCINGFORMATIVEASSESSMENT
Minimumtimeneeded:10minutes.
Thedifferenttypesandpurposesofassessment.Inviteparticipantstodiscussthefollowingissues:
• Whydoyouassessstudents?• Whatdifferentpurposesdoyourassessmentsserve?Makealist.
Theirlistofreasonsmightinclude:diagnosingdifficulties;celebratingachievement;motivatingstudents;selectingstudentsforclasses;maintainingrecordstokeepteachersandparentsinformedofprogress;toassessteachingmethods.Tosummarize,therearetwomainpurposesofassessment:
• Summativeassessment-tosummariseandrecordoverallachievementattheendofacourse,forpromotionandcertification.Most‘highstakes’testsandexternalexaminationsaredesignedforthispurpose.Summativeassessmentisalsousedtoevaluatetherelativeeffectivenessofaparticularcourse,teachingmethod,orevenaninstitution.
• Formativeassessment–torecogniseachievementsanddifficultiesatthebeginningorduringacourse,sothatteachersandstudentscantakeappropriateaction.Thistypeofassessmentformsanintegralpartofalllearning.
Thepotentialofformativeassessmenttoimprovelearning.Brieflymentiontheresearchevidencethatsetsoutthecaseforformativeassessment.ThisissummarizedbyBlackandWiliaminseveralaccessiblepublicationsforteachers(seeopposite),mostofwhicharefreelydownloadableontheinternet.Theseresearcherssetouttofindoutwhetherornotimprovingformativeassessmentimproveslearning.“Wecheckedmanybooksandnineyears'worthofmorethan160journals,andearlierreviewsofresearch.Thisprocessyielded580articlesorchapterstostudy.Wepreparedareviewusingmaterialfrom250ofthesesources.All…studiesshowthat…strengthening…formativeassessmentproducessignificant,andoftensubstantial,learninggains.Thesestudiesrangeoverages,acrossseveralschoolsubjects,andoverseveralcountries…”(BlackandWiliam,1998).1Thismodulewillexaminetheimplementationofformativeassessment,basedonthisandotherresearch.Asecondmodulewillexploretheroleofselfandpeerassessment.
1PaulBlackandDylanWiliam,"AssessmentandClassroomLearning,"AssessmentinEducation,March1998,pp.7-74.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 4
ACTIVITYB:TEACHERS'OWNEXPERIENCESOFFORMATIVEASSESSMENT
Minimumtimeneeded:10minutes.
Whatdoteachersknowabouttheirstudentsandwhatconsequentialactiondotheytake?Askparticipantstoworkinpairs,consideringthefollowingquestions.
• Thinkoftwostudentsinyourclass,onewhoisparticularlystrongandonewhoisfindingtheworkverydifficult.Takeitinturnstodescribethestudents'strengthsanddifficultiestoyourpartner,inasmuchdetailaspossible.
• Howdidyoubecomeawareofthesestrengthsanddifficulties?Onwhatevidencedoyoubaseyourjudgements?Testresults?Memoriesoforalresponsesduringlessons?Observationsofthestudentworking?Writtenwork?
• Inwhatwaysdoyourassessmentsofthesestudentsaffectyourlessonplanning?Giveexamples.
Whatdifficultiesdoteacherencounter?IssueparticipantswithcopiesofHandout1:Difficultieswithformativeassessment.
• Howfararethecriticismsonthehandoutvalidinyourcontext?• Ifanyare,thenwhatmaybedoneaboutthem?
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 5
Handout1 Difficultieswithformativeassessment
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1 Black, P., & Wiliam, D. (1998). Inside the black box : raising standards through classroom assessment. London: King's College London School of Education 1998.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 6
ACTIVITYC:PRINCIPLESFORFORMATIVEASSESSMENT
Minimumtimeneeded:20minutes.
IssueparticipantswithacopyofHandout2.Theideaspresentedherearealldrawnfromresearchintoformativeassessment.
• BearinginmindthedifficultiesdiscussedinActivityB,howwouldyousuggestthatyourformativeassessmentpracticesbeimproved?
• DiscusstheprinciplesoutlinedonHandout2.o Whichofthesedoyoucurrentlyuseinyourownteaching?o Whichdoyoufindmostdifficult?Why?
• Whatotherprinciplesdoyouthinkareimportant?
IssuecopiesofHandout3.
"It'sallverywelltellingustoassessourstudents,buthowcanabusyteacherknowwhatisgoingoninside30individualheads?"
• Howwouldyouanswerthisteacher?• Whatstrategiesdoyouhaveforfindingoutwhat
studentsarethinkinginyourlessons?• DiscussthetwosuggestionsshownonHandout3,
andwatchthemoviestoseetheseinaction.• Suggestsomefurtherstrategiesformakingreasoning
moreevident.
ThetwostrategiesdescribedonHandout3andonthemoviesaccompanyingthisactivitymayhelptomakereasoningmore'visible'.Mini-whiteboardsareanindispensableresourceforthefollowingreasons:
• Whenstudentsholdtheirideasuptotheteacher,itispossibletoseeataglancewhateverystudentthinks.
• Duringwholeclassdiscussions,theyallowtheteachertoasknewkindsofquestion(typicallybeginning:‘Showmeanexampleof....’).
• Theyallowstudentsto,simultaneously,presentarangeofwrittenand/ordrawnresponsestotheteacherandtoeachother.
Postersarealsoapowerfulwayofhelpingstudentstoexternalisetheirthinking.Thisusedoesnotrequire'polished','complete','attractive'productsbutrathertheyshouldbeseenasworkingdocuments.Perhapsthesimplestwayofusingaposterisforstudentstosolveaproblemcollaboratively,explainingthethoughtprocessesinvolvedateverystep.Aseconduseofpostersistofindoutwhattheyalreadyknowaboutagiventopic.InthediagramshownonHandout2,theteacheraskedstudentstowritedownalltheyknewabouty=2x-6.Asaclass,thediagramwasdevelopedonthewhiteboard.Studentswerethengivenavarietyofequations(thelevelofchallengewasvariedappropriately)andwereaskedtoproducetheirownposter.Thediscussionenabledtheteachertoassesshowmuchlearnersknewaboutequationsandhowwelltheywereabletolinktheideastogether.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 7
Handout2:Principlesforformativeassessment!! ! !
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(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 8
ACTIVITYD:ANALYSESTUDENTS’RESPONSESTOPROBLEM-SOLVINGTASKS
Minimumtimeneeded:20minutesHandout4presentsthreeproblemstogetherwithfourstudentresponsesoneach.Thetasksare:CountingTrees,CatsandKittens,SecurityCameras.
• Readthroughallthreetasksthenchooseonetaskthatwillbemostsuitableforaclassyouwillsoonteach.Ifyouareworkingonthismoduleinagroup,itwillbehelpfulifeachparticipantchoosesthesameproblem,asthiswillfacilitatethefollow-updiscussion.
• Considerthefourstudentresponses.Whatdoeseachstudent’sresponsetellyouabouthis
orhercapacitytouseeachoftheprocesses:represent,analyse,interpretandevaluate,communicateandreflect?
Handout5offerssomecommentsonstudents’responsestoeachofthetasks.
• Ifyouweretheteacherofthesestudents,whatfeedbackwouldyougivethem,tohelpthemimprovetheirresponses?Trytoframethishelpintheformoforalquestionsyoucouldaskintheclassroom.YoumayfindithelpfultorefertothegenericquestionsgivenonHandout6.
• Watchthevideoofthreeteachersdiscussingthefeedbacktheygaveonthethreeproblems.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 9
Handout4: Assessmenttasksandsampleresponses
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Sample response: Jenny
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Sample response: Woody
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(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 10
ACTIVITYE: ANALYSESTUDENTS’RESPONSESTOCONCEPT-FOCUSEDTASKS
Minimumtimeneeded:20minutesOnHandout7,wepresentfourmathematicaltopicsandsomesamplestudentworkoneachone.Askparticipantstoassesseachresponseandtrytoidentifythereasoningthatliesbehindeachone.
• Whatdoesthestudentappeartounderstand?Whereisyourevidence?• Listtheerrorsanddifficultiesthatarerevealedbyeachresponse.• Trytoidentifythethinkingthatliesbehindeacherror.• Whatfeedbackwouldyougivetoeachstudent?
Therearetwocommonwaysofreactingtopupils'errorsandmisconceptions:(i)Avoidthemwheneverpossible:"IfIwarnpupilsaboutthemisconceptionsasIteach,theyarelesslikelytohappen.Preventionisbetterthancure."(ii)Usethemaslearningopportunities:"Iactivelyencouragepupilstomakemistakesandlearnfromthem.
• Whatisyourview?
Routinepracticeonstandardproblemsdoeslittletohelpstudentsovercomecommonmistakesandmisconceptionsinmathematics.Thisisparticularlytruewhenteacherstrytoavoiddifficultiesarisingbybeginningeachlessonwithexplanationsanddemonstrationsandfollowingthiswithcarefullygradedquestions.Handout8isaone-pageresearchreviewbyAskewandWiliam.ThisdescribessomeresearchthatwasconductedinBritainoverthepast30yearsondealingwithstudenterrorsanddifficulties.Whatdoteachersfeelaboutthedilemmaspresentedinthispaper?-isitpossibletopresentexampleswhererulesdonotwork?-doteacherssharetheviewthatitiscounter-productivetoteachsimplerexamplesbeforemorecomplexexamples?
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 11
Handout7: Assessmenttasksandsampleresponsesforconcepts
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(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 12
ACTIVITYF: OBSERVEFORMATIVEASSESSMENTINACTION
Timeneeded:15minutes.
Inthisactivity,youareprovidedwithvideoextractsofAndrew,DominicandAmyexploringhowformativeassessmentmaybeusedtopromotestudentslearning.TheyareusingthethreetasksfromActivityD.Inanearlierlesson,theseteachershadaskedstudentstositindifferentplacesandattemptoneofthetasksindividually,withnohelp.Theythencollectedintheirstudents'responses,assessedtheworkqualitativelyandpreparedwrittenfeedbackintheformofquestions.Thefilmclipsyouareabouttoseearetakenfromthefollow-uplesson.Studentshavereturnedtotheirnormalplacesandmosthavesolutionsthataredifferenttothoseoftheirpartners.Watchthevideoandconsider:
• Whatdifferentkindsofassessmentcanyousee?• Whatisthepurposeofeachkindofassessment?• Whatdoboththeteachersandstudentslearn?
Inthevideo,youwillsee:
• Andrewexploringhowstudentsrespondtohisfeedbackonthe"countingtrees"problem;• Amylisteningto,thenquestioningindividualsastheytrytosharetheirideasandproduce
jointsolutionstothe"securitycamera"problem;• Dominiclisteningtopresentationsfromstudentsontheirmethodsandreasoningforthe
"catsandkittens"problem;• Amyconcludingherlessonbyaskingstudentstodescribehowtheyhaveusedherfeedback
toimprovetheirwork.
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 13
ACTIVITYG:PLANANDREPORTBACKONANASSESSMENTLESSON
Minimumtimeneeded: 30minutesbeforethelesson 20minutesforthepre-lessonassessment 30minutestopreparefeedback 60minutesforthelesson 15minutesforreportingback
PlanningthelessonPlanyourownlessonusingoneoftheproblems.Planatimeforstudentstotackletheproblemontheirownwithouthelp.Planhowyouwillassessthiswork,givefeedbackandconductafollowuplesson.Collectsamplesofstudents'worktoshowhowtheirthinkinghaschanged.Thesewillbediscussedatthefollow-upsession.Tohelpyourplanning,youmaynowliketowatchthe10-minutevideothatshowsAndrewteachingtheCountingTreesproblemfromActivityD.HeisfollowingthelessonplanonHandout9.
Andrew'slesson
Thepatternofactivitiesonhandout9isasfollows:
• Givetheproblembeforethelessonandaskstudentstoattemptit.(20minutes)
• Collectintheworkandpreparesomeconstructive,qualitativefeedback.
• Inthefollow-uplesson,reintroducetheproblemtotheclass.(5minutes)
• Studentsworkalone,respondingtothefeedbackusingminiwhiteboards.(5minutes)
• Studentsworkinpairstoimprovetheirsolutions.(10minutes)• Studentssharetheirapproacheswiththeclass.(15minutes)• Studentscontinuewiththeproblemorextendtheproblem.(20
minutes)
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! ReportingbackonthelessonAfteryouhavetaughtthelesson,reflectonwhathappenedwithagroupofcolleagues.Takeitinturnstosharestoriesoftheassessmentstrategiesyouused.
• Howdidyoucollectandassessevidenceofyourstudents'work?• Whatdidyoulearnfromthisevidence?• Whatdidstudentslearnfromthefollow-uplesson?• Whataretheimplicationsforyoumathematicsteachingmoregenerally?
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 14
ACTIVITYH:CONSIDERTHEEFFECTSOFFEEDBACKONSTUDENTLEARNING
Timeneeded:20minutes.
Sofarwehavefocusedontheteachers’roleinprovidingassessmentfeedbacktostudents.Inthisactivitywewillconsidertheusestudentsmakeofdifferenttypesoffeedbackandtheimpactthishasontheirlearning.WatchthevideoofAndrew'sstudentsastheydiscusstheimpactofassessmentfeedbackontheirlearning.
• Whichoftheircommentsstrikeyouasparticularlyperceptiveandimportant?
• Whataretheimplicationsoftheircomments?
Handout10presentssomeresultsofresearchfromBlackandWiliam(1998)intotherelativemeritsoffeedingbackassessmentinformationtostudentsindifferentforms.Inparticularitcomparestheeffectsoffeedingbackquantitativeinformationintheformofmarks,levelsandrankingswiththeeffectsofofferingqualitativeinformationintheformofspecific,content-focusedfeedback.Comparethestudents'commentswiththeresearchquotesgivenonHandout9.Thedangersofgivingmarks,levels,rewardsandrankings
• Whataretheimplicationsofthisforyourpractice?• Whatwouldhappenifyoustoppedgivingmarksorlevelsonpupils’work?• Whyaresomanyteachersresistanttomakingthischange?
Theadvantagesofgivingclear,specific,content-focusedfeedback
• What are the implications of this for your practice? • Does this kind of feedback necessarily take much longer to give?
Researchshowsthatstudentsbenefitmostfromfeedbackthat:• Focusesonthetask,notongradesorscores.• Isdetailedratherthangeneral.• Explainswhysomethingisrightorwrong.• Isrelatedtoobjectives• Makesclearwhathasbeenachievedandwhathasnot• Suggestswhatthestudentmaydonext• OffersspecificstrategiesforimprovementConcludethismodulebydiscussingsomewaysofapplyingwhatyouhavelearnedinthisPDmoduletotheothermathematicslessonsthatyouteach.
• Howcouldyouinvolvepupilsinimprovingyourassessmentpractices?
(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 15
Handout10:Theeffectsoffeedbackonstudentlearning
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