introducing formative assessment handouts

FormativeAssessmentinScienceandMathematicsEducation

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INTRODUCINGFORMATIVEASSESSMENTHANDOUTSFORTEACHERS

Contents

Handout1:Theimportanceofformativeassessment....................................................................2Handout2:Difficultieswithformativeassessment........................................................................3Handout3:Principlesandstrategiesforformativeassessment.....................................................5Handout4:Problem-solvingintheclassroom................................................................................7Handout5:Countingtrees..............................................................................................................8Handout6:Catsandkittens..........................................................................................................12Handout7:Suggestionsforquestions..........................................................................................16Handout8:Formativeassessmentinaction.................................................................................17Handout9:Theeffectsoffeedbackonstudentlearning.............................................................18Handout10:Aformativeassessmentlessonplan........................................................................20Handout11:Exchangingexperiences...........................................................................................23

Acknowledgement:

Inpreparingthismaterial,weacknowledgethepermissionsgivenbytheBowlandCharitableTrusttoadapttheprofessionaldevelopmentresources,BowlandMaths(www.bowlandmaths.org),thattheyhadpreviouslyfundedtheUniversityofNottinghamtoproducefortheUK.Thisincludesmanyofthehandoutsandmostofthevideoextracts.AdditionalresourceswerealsoadaptedfromImprovingLearninginMathematics,agovernmentfundedprogramintheUKandfromtheMathematicsAssessmentProjectfundedbytheBillandMelindaGatesFoundation.

©2010CentreforResearchInMathematicsEducationUniversityofNottingham,adaptforFaSMEdbyAfricanInstituteforMathematicalSciencesSchoolEnrichmentCentreandUniversityofDuisburg-Essen.

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Handout1:Theimportanceofformativeassessment

Therearetwomainpurposesofassessment:

• Summativeassessment-tosummariseandrecordoverallachievementattheendofacourse,forpromotionandcertification.Most‘highstakes’testsandexternalexaminationsaredesignedforthispurpose.Summativeassessmentisalsousedtoevaluatetherelativeeffectivenessofaparticularcourse,teachingmethod,orevenaninstitution.SummativeassessmentissometimescalledAssessmentofLearning.

• Formativeassessment-torecogniseachievementsanddifficultiesatthebeginningorduringacourse,sothatteachersandstudentscantakeappropriateaction.Thistypeofassessmentformsanintegralpartofalllearning.FormativeassessmentissometimescalledAssessmentforLearning.

Thepotentialofformativeassessmenttoimprovelearning

Thereisawealthofresearchevidencethatsetsoutthecaseforformativeassessment.ThisissummarisedbyBlackandWiliaminseveralaccessiblepublications(seebelow),mostofwhicharefreelydownloadablefromtheInternet.Theseresearcherssetouttofindoutwhetherornotimprovingformativeassessmentimproveslearning.

“Wecheckedmanybooksandnineyears'worthofmorethan160journals,andearlierreviewsofresearch.Thisprocessyielded580articlesorchapterstostudy.Wepreparedareviewusingmaterialfrom250ofthesesources.All…studiesshowthat…strengthening…formativeassessmentproducessignificant,andoftensubstantial,learninggains.Thesestudiesrangeoverages,acrossseveralschoolsubjects,andoverseveralcountries…”(BlackandWiliam,1998)1.

Recommendedreading

Black,P.,&Wiliam,D.(1998).Insidetheblackbox:raisingstandardsthroughclassroomassessment.King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:

http://shop.gl-assessment.co.uk

Thisshortbookletoffersasummaryoftheextensiveresearchliteratureintoformativeassessment.Itshowsthatthereisclearevidencethatusingformativeassessmentraisesstandards,andsuggestshowformativeassessmentcanbeimproved.Thisbookletisessentialreadingforallteachers.

1PaulBlackandDylanWiliam,"AssessmentandClassroomLearning,"AssessmentinEducation,March1998,pp.7-74.

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Handout2:Difficultieswithformativeassessment

Theresearchliteraturesuggeststhatformativeassessmentpracticesarebesetwithproblemsanddifficulties,asoutlinedbrieflybelow.

Definition

WithinFaSMEdformativeassessmenthasbeen‘defined’asadeliberateprocessofgatheringinformationaboutstudents’understandingsandusingthisinformationtoadaptteaching.However,thereisstillagenerallackofclarityaboutwhatformativeassessmentactuallyis.Forexample,therearesome(particularlypublishers)whousethetermtorefertoaninstrumentsuchasadiagnostictest.Inreality,eachoftheseperspectivesisperhapsanoversimplification;theprocessperspectiveunderplaystheimportanceoftheinstrumentandtheinstrumentdefinitionappearstoignoreprocess.Further,definitionsofformativeassessmentgenerallydonottakeintoaccounttheimportanceofthesubjectarea(e.g.mathematicsorscience)wheninfactformativeassessmentwithindifferentsubjectareascanlookverydifferent.

Difficultandtime-consuming

Doingformativeassessmentwellisdifficult.Whilstitisrelativelyeasytogatherdataintermsof,forexample,students’responsestoquestionsinclassroomorstudents’selfreportedconfidencelevels,itismuchmoredifficulttointerpretthisdataintermsofthestudents’actualunderstanding.Thisrequirestheteachertoconsidercarefullyresponsesofthestudentsintermsoftheirownknowledgeaboutmathematics,thecurriculum,priorexperienceoftheclassandthelearningtrajectoryoftheindividualstudentandtomakeaninferenceaboutthestudent’sunderstanding.Theteacherthenneedstoprovideusefulfeedbacktoeachstudent,whichagainrequiresconsiderableexpertiseandthought.Itisalmostimpossibleforteacherstoprovidedetaileddescriptivefeedbackforeachstudentmorethanoccasionally.Providingmeaningfulfeedbackistime-consumingandrequiressignificantandsustaineddedicationfromtheteachers.

Planningandcoverage

Formativeassessment,bydefinition,requirestheteachertoactontheinformationheorshereceivesfromthestudents,usuallybyadaptingteaching.Thismightmeanabandoningpreviousplansforthelessonandmakein-the-minutedecisionsaboutwhattodonext.

Teacherssometimescomplainthat,iftheyareseriousaboutreactingtothedatatheycollectabouttheirstudents’understandings,thentheymayhavetodwelllongeroncertaintopicareasthanenvisaged,whichmeansthattheymaynothavetimetocoverthewholesyllabus.

Accountability

Inmanycountries,schoolsworkwithinacultureofaccountabilityandseniormanagersrequireevidenceofsomekindofobjectivemeasurementofstudents’performanceateachlevel.Formativeassessment,byitsfleetingnature,doesnotprovidethatkindofaccountability.Thismeansthatschoolmanagershavelimitedwaysinwhichtheycanmonitorformativeassessmentandhencetheyfeeluncomfortableaboutteachersusingit.

Expectations

Manyparentsdonotknowwhatformativeassessmentis.Inmanycultures,parentstendtoexpectformal(summative)testsandgradesthroughouttheschoolyear.

Toalargeextent,aformativeassessmentclassroomrequirescommitmenttotheprocessbystudentsaswellasteachers.Studentsneedtotakeresponsibilityfortheirownlearningbyrespondingtofeedbackfromtheteacherortheirpeers.Inmanyclassrooms,studentsarereluctanttodoso,wantingrathertobetoldwhattodoandexpectinggradesandmarks.

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Howfararethedifficultiessummarisedherevalidinyourcontext?

Ifanyare,thenwhatcouldbedoneaboutthem?

“Iknowitmakessensetoassessstudentsaswegoalong,buthowcanI,inthemidstofalesson,knowwhateachofmy30studentsisthinking?”

Howwouldyouanswerthisteacher?

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Handout3:Principlesandstrategiesforformativeassessment

Formativeassessmentmaybedefinedas:"…allthoseactivitiesundertakenbyteachers,andbytheirstudentsinassessingthemselves,whichprovideinformationtobeusedasfeedbacktomodifytheteachingandlearningactivitiesinwhichtheyareengaged.Suchassessmentbecomes‘formativeassessment’whentheevidenceisactuallyusedtoadapttheteachingworktomeettheneeds."(Black&Wiliam,1998)

Maketheobjectivesofthelessonexplicit2

Sharetheobjectiveswithstudentsandfromtimetotimeaskstudentstoproduceevidencethattheycanachievetheseobjectives.

“MakeupanexampletoshowmethatyouknowandunderstandPythagoras’theorem.”“Thislessonwasaboutyoudecidingwhatmethodstouse.Showmewhereyoudidthis.”

Studentsmayfinditdifficulttoappreciatethatsomelessonsareconcernedwithunderstandingconcepts,whileothersaremoreconcernedwithdevelopingmathematicalpractices.

Makingobjectivesexplicitdoesn’tmeanwritingthemontheboardatthebeginningofthelesson,butratherreferringtothemexplicitlywhilestudentsareworking.Iftheobjectivesaretodevelopmathematicalpractices,theninplenarysessionsaskstudentstoshareandcompareapproaches,ratherthananswers.

Assessgroupsaswellasindividualstudents

Groupactivitiesallowmanyopportunitiestoobserve,listen,andquestionstudents.Theyhelptoexternalisereasoningandallowtheteachertoseequicklywheredifficultieshavearisen.Moreover,workingingroupshelpsstudentstopeer-assessandfunctionasinstructionalresourcesforoneanother.

Watchandlistenbeforeintervening

Beforeinterveninginagroupdiscussion,waitandlisten.Trytofollowthelineofreasoningthatstudentsaretaking.Whenyoudointervene,beginbyaskingthemtoexplainsomething.Iftheyareunsuccessfulthenaskanotherstudenttohelp.

Usedivergentassessmentmethods(“Showmewhatyouknowabout...”)

Convergentassessmentstrategiesarecharacterisedbyticklistsandcan-dostatements.Theteacherasksclosedquestionsinordertoascertainwhetherornotthestudentknows,understandsorcandoapredeterminedthing.Thisisthetypeofassessmentmostusedinwrittentests.

Divergentassessment,incontrast,involvesaskingopenquestionsthatprovidestudentswithopportunitiestodescribeandexplaintheirthinkingandreasoning.Thesequestionsallowstudentstosurpriseus-theoutcomeisnotpredetermined.

Changeteachingtotakeaccountofassessment

Aswellasprovidingfeedbacktostudents,goodassessmentfeedsforwardintoteaching.Beflexibleandpreparedtochangeyourteachingplansinmidcourseasaresultofwhatyoudiscover.

2Theseprinciplesareadaptedfrom:ImprovingLearninginMathematics,DepartmentforEducationalandSkills,2005

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Giveconstructive,usefulfeedback

Researchshowsthatrespondingtostudents’workwithmarksorlevelsisusuallyineffectiveandmayevenobstructlearning:itresultsinstudentscomparingmarksorlevelsanddetractsfromthemathematicsitself.Instead,usequalitativeoralandwrittencommentsthathelpstudentsrecognisewhattheycando,whattheyneedtobeabletodoandhowtheymightnarrowthegap.

Fivekeystrategies

Research3suggeststhatformativeassessmentusesfivekeystrategies:

1) Clarifying/Understanding/Sharinglearningintentionsandcriteriaforsuccess,2) Engineeringeffectiveclassroomdiscussionsandotherlearningtasksthatelicitevidenceof

studentunderstanding,3) Providingfeedbackthatmoveslearnersforward,4) Activatingstudentsasinstructionalresourcesforoneanother,5) Activatingstudentsasownersoftheirownlearning.

Whichoftheseprinciplesandstrategiesdoyouuseinyourownteaching?

Whicharemostdifficulttoimplement?

Whatotherprinciplesorstrategieswouldyouaddtothelist?

3 Wiliam,D.,&Thompson,M.(2007). Integratingassessmentwithlearning:whatwill ittaketomakeitwork?InC.A.Dwyer(Ed.),TheFutureofAssessment:ShapingTeachingandLearning(pp.53-82).Mahwah,NJ:Erlbaum.

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Handout4:Problem-solvingintheclassroom

Situationsandproblemsarisingintherealworldareusuallymessyandcomplicated.Toaddressthem,andcometosomesortofsolution,weneedtosiftthroughtheinformationgiventochoosewhatisneeded,simplifyandmodelsituations,selectappropriateknowledgeandprocessesandtestwhethersolutionsappeartobereasonable.

Classroomproblemsolvingtasksaimtohelpstudentslearntheskillstheywillneedintherealworld.Theyoftenprovemoredifficultthantheindividualelementsofmathematicalcontentwouldsuggest,asstudentsarerequiredtochooseandcombinetechniquesinnon-routineways.Thereareusuallyarangeof‘goodenough’answers,makingitdifficulttoassessproblem-solving.However,formativeassessmentapproachescangivetheteacheragoodideaofthestudents’understanding.

Thefollowingphasesofproblemsolvingprovideausefulwayofanalysingstudents’workonthesetasks:

• Formulatequestions,chooseappropriaterepresentationsandtechniques.

• Reasonlogically,constructhypothesesandarguments,computeaccurately.

• Interpretandevaluateresultsobtained.

• Communicateandreflect.

Handout5:CountingtreesandHandout6:Catsandkittenspresentexamplesoftwotasks,eachwithfoursampleresponses.

Readthroughbothtasksthenchooseonetaskthatwillbemostsuitableforaclassyouwillsoonteach.Ifyouareworkingonthismoduleinagroup,itwillbehelpfulifeachparticipantchoosesthesametask,asthiswillmakethefollow-updiscussioneasier.

Considerthefourstudentresponses.Thinkaboutwhateachstudent’sresponsetellsyouabouthisorhercapacitytouseeachoftheprocessesrequiredinproblemsolving:represent,analyse,interpretandevaluate,communicateandreflect?Considerwhatyoucaninferfromtheirresponsesaboutwhattheydoanddonotunderstand.

Thehandoutsoffersomecommentsonstudents’responsestoeachofthetasks.Theyprovideaspaceundereachcommentforyoutosuggestquestionsyoucouldaskeachstudenttohelpthemimprovetheirresponses.Trytoframethishelpintheformoforalquestionsyoucouldaskintheclassroom.

YoumayfindithelpfultorefertothegenericquestionsgivenonHandout7:Suggestionsforquestions.

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Handout5:Countingtrees

CountingTrees

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Thisdiagramshowssometreesinaplantation.

Thecircles ● showoldtreesandthetriangles ▲ showyoungtrees.

Tomwantstoknowhowmanytreesthereareofeachtype,butsaysitwouldtaketoolongcountingthemall,one-by-one.

1. Whatmethodcouldheusetoestimatethenumberoftreesofeachtype?Explainyourmethodfully.

2. Onyourworksheet,useyourmethodtoestimatethenumberof:

• Oldtrees

• Youngtrees

Note:thistaskisalsodiscussedintheprofessionaldevelopmentmoduleStudentsbecomingassessors.

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Sampleresponse:Laura

Lauraattemptstoestimatethenumberofoldandnewtreesbymultiplyingthenumberalongeachsideofthewholediagramandthenhalving.Shedoesnotaccountforgapsnordoessheappeartorealisethatthereareanunequalnumberoftreesofeachkind.

WhatquestionscouldyouaskLaurathatwouldhelpherimproveherresponse?

Sampleresponse:Woody

Woodyusesasampleoftwocolumnsandcountsthenumberofoldandyoungtrees.Hethenmultipliesby25(halfof50columns)tofindanestimateofthetotalnumber.

WhatquestionscouldyouaskWoodythatwouldhelphimimprovehisresponse?

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Sampleresponse:Jenny

Jennyrealisesthatsamplingisneeded,butshemultipliesthenumberofyoungtreesandoldtreesinthelefthandcolumnbythenumberoftreesinthebottomrow.Sheignoresthecolumnswithnotreesinthebottomrow,sohermethodunderestimatesthetotalnumberoftrees.Shedoes,however,takeaccountofthedifferentnumbersofoldandnewtrees.

WhatquestionscouldyouaskJennythatwouldhelpherimproveherresponse?

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Sampleresponse:Amber

Amberchoosesarepresentativesampleandcarriesthroughherworktogetareasonableanswer.Shecorrectlyusesproportionalreasoning.Shechecksherworkasshegoesalongbycountingthegapsinthetrees.Herworkisclearandeasytofollow.

WhatquestionscouldyouaskAmberthatwouldhelpherimproveherresponse?

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Handout6:Catsandkittens

Hereisaposterpublishedbyanorganisationthatlooksafterstraycats.

Workoutwhetherthisnumberofdescendantsisrealistic.

Herearesomefactsyouwillneed:

Make sure your cat cannot have kittens.

Cats can’t add but

they do multiply!

In just 18 months, this female cat

can have 2000 descendants.

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Sampleresponse:Alice

Alicechosetorepresentthetaskusingatimeline.Shehasonlyconsideredthenumberofkittensfromtheoriginalcat.Thecomputationrequiredisaccurate.

WhatquestionscouldyouaskAlicethatwouldhelpherimproveherresponse?

Sampleresponse:Wayne

Wayneappearstofavouraminimalistapproach!Hestartswithwhatwouldbeatimeconsumingpictorialrepresentationwhichhethenabandonsinfavourofanumericalrepresentation.

WhatquestionscouldyouaskWaynethatwouldhelphimimprovehisresponse?

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Sampleresponse:Ben

Benhasdecidedtodrawa‘cattree’,andtriestocontrolfortime(withsomeerrors).Thecommunicationisreasonablyclear,allowingthereadertofollowtheargument,butthefinalanswerof9846isnotexplainedanddoesnotfollowfromthereasoning,since,again,onlythekittensfromtheoriginalcatareconsidered.Thenumberofkittensperlitterismadeexplicit.

WhatquestionscouldyouaskBenthatwouldhelphimimprovehisresponse?

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Sampleresponse:SallyandJanet

Twostudentsworkedonthistask,discussingandsharingtheirmethods.Theyusedaspreadsheet.

Wethink2000isabitmuchin18monthsbecauseevenifeachlitterwas6andnothingdiestherewouldbe1860thoughthatroundsto2000somaybeitsOK.Thecatpeoplewantownerstohavetheircatsnewturedsothattheyusethebiggernumbersothatpeoplesaythatisalotofcatsandrushtothevets.

SallyandJanetusedaspreadsheettocontrolforbothtimeandmultiplicationandtheirmethodisclearandeffective.

WhatquestionscouldyouaskSallyandJanetthatwouldhelpthemimprovetheirresponse?

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Handout7:Suggestionsforquestions

Formulatequestions,chooseappropriaterepresentationsandtools.

Whatquestionsmightyouaskaboutthissituation?Howcanyougetstartedonthisproblem?Whattechniquesmightbeusefulhere?Whatsortofdiagrammightbehelpful?Howcanyousimplifythisproblem?Whatisknownandwhatisunknown?Whatassumptionsmightyoumake?

Reasonlogically,constructhypothesesandarguments,computeaccurately

Wherehaveyouseensomethinglikethisbefore?Whatisfixedhere,andwhatcanyouchange?Whatisthesameandwhatisdifferenthere?WhatwouldhappenifIchangedthis..tothis...?Whatwillyoudowhenyougetthatanswer?Thisisjustaspecialcaseof...what?Canyouformanyhypotheses?Canyouthinkofanycounterexamples?Whatmistakeshaveyoumade?Canyousuggestadifferentwayofdoingthis?Whatconclusionscanyoumakefromthisdata?Howcanyoucheckthiscalculationwithoutdoingitallagain?Whatisasensiblewaytorecordthis?

Interpretandevaluateresultsobtained

Howcanyoubestdisplayyourdata?Isitbettertousethistypeofchartorthatone?Why?Whatpatternscanyouseeinthisdata?Whatreasonsmighttherebeforthesepatterns?Isthisapproachgoinganywhere?Canyougivemeaconvincingargumentforthatstatement?Thisisjustaspecialcaseof...what?Doyouthinkthatanswerisreasonable?Why?Howcanyoube100%surethatistrue?Convinceme!WhatdoyouthinkofAnne'sargument?Why?Whichmethodmightbebesttousehere?Why?

Communicateandreflect

Whatmethoddidyouuse?Whatothermethodshaveyouconsidered?Whichofyourmethodswasthebest?Why?Whichmethodwasthequickest?Wherehaveyouseenaproblemlikethisbefore?Whatmethodsdidyouuselasttime?Wouldtheyhaveworkedhere?Whathelpfulstrategieshaveyoulearnedfornexttime?

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Handout8:Formativeassessmentinaction

Writingthesortsofcommentsandquestionsonstudents’workyoudidinActivityDistime-consuminganddifficult.

Inthevideotheteachersdiscusshowtheymadecomments.

Towhatextentdotheircommentsresonatewithyourexperience?

Inthesecondvideo,thethreeteachersaskstudentstorespondtothefeedbacktheyhavewrittenandthentoworkinsmallgroupstoconstructajointsolutiontotheproblem.

Inthevideo,youwillsee:

• Andrewexploringhowstudentsrespondtohisfeedbackonthe"countingtrees"problem;

• Amylisteningto,thenquestioning,individualsastheytrytosharetheirideasandproducejointsolutionstothe"securitycamera"problem;

• Dominiclisteningtopresentationsfromstudentsontheirmethodsandreasoningforthe"catsandkittens"problem;

• Amyconcludingherlessonbyaskingstudentstodescribehowtheyhaveusedherfeedbacktoimprovetheirwork.

Whatdifferentkindsofassessmentcanyousee?

Whatisthepurposeofeachkindofassessment?

Whatdoboththeteachersandstudentslearn?

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Handout9:Theeffectsoffeedbackonstudentlearning

Inthevideo,Andrew’sstudentsdiscusstheirresponsestothefeedbackcommentstheirteachergave.

Whichoftheircommentsstrikeyouasparticularlyimportant?

Whataretheimplicationsoftheircomments?

ReadthefollowingtwoextractsfromBlackandWiliam(1998)4andrespondtothequestionsthatfollow.

Thedangersofgivingmarks,levels,rewardsandrankings

“Wheretheclassroomculturefocusesonrewards,‘goldstars’,gradesorplace-in-the-classranking,thenpupilslookforthewaystoobtainthebestmarksratherthanattheneedsoftheirlearningwhichthesemarksoughttoreflect.Onereportedconsequenceisthatwheretheyhaveanychoice,pupilsavoiddifficulttasks.Theyalsospendtimeandenergylookingforcluestothe‘rightanswer’.Manyarereluctanttoaskquestionsoutoffearoffailure.Pupilswhoencounterdifficultiesandpoorresultsareledtobelievethattheylackability,andthisbeliefleadsthemtoattributetheirdifficultiestoadefectinthemselvesaboutwhichtheycannotdoagreatdeal.Sothey‘retirehurt’,avoidinvestingeffortinlearningwhichcouldonlyleadtodisappointment,andtrytobuilduptheirself-esteeminotherways.Whilstthehigh-achieverscandowellinsuchaculture,theoverallresultistoenhancethefrequencyandtheextentofunder-achievement.”

Whataretheimplicationsofthisforyourpractice?

4Black,P.,&Wiliam,D.(1998).Insidetheblackbox:raisingstandardsthroughclassroomassessment.London:King'sCollegeLondonSchoolofEducation1998.

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Whatwouldhappenifyoustoppedgivingmarksorlevelsonpupils’work?

Whyaresomanyteachersresistanttomakingthischange?

Theadvantagesofgivingclear,specific,content-focusedfeedback

“Whatisneededisacultureofsuccess,backedbyabeliefthatallcanachieve.Formativeassessmentcanbeapowerfulweaponhereifitiscommunicatedintherightway.Whilstitcanhelpallpupils,itgivesparticularlygoodresultswithlowachieverswhereitconcentratesonspecificproblemswiththeirwork,andgivesthembothaclearunderstandingofwhatiswrongandachievabletargetsforputtingitright.Pupilscanacceptandworkwithsuchmessages,providedthattheyarenotcloudedbyovertonesaboutability,competitionandcomparisonwithothers.

Insummary,themessagecanbestatedasfollows:Feedbacktoanypupilshouldbeabouttheparticularqualitiesofhisorherwork,withadviceonwhatheorshecandotoimprove,andshouldavoidcomparisonswithotherpupils.”

Whataretheimplicationsofthisforyourpractice?

Doesthiskindoffeedbacknecessarilytakemuchlongertogive?

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Handout10:Aformativeassessmentlessonplan

Thefollowingsuggestionsdescribeonepossibleapproachtoaformativeassessmentlessononproblemsolving.First,studentsaregivenachancetotackleaproblemunaided.Thisgivesyouachancetoassesstheirthinkingandtoidentifystudentsthatneedhelp.Thisisfollowedbyformativelessoninwhichtheycollaborate,reflectontheirworkandtrytoimproveit.

Beforethelesson 20minutes

Beforethelesson,perhapsattheendofapreviouslesson,askstudentstoattemptoneoftheassessmenttasks,CountingTrees,CatsandKittensorSecurityCamerasontheirown.Studentsmayneedcalculators,pencils,rulers,andsquaredpaper.

Theaimistoseehowableyouaretotackleaproblemwithoutmyhelp.

Youwillnotbetoldwhichbitsofmathstouse.

Therearemanywaystotackletheproblem-youchoose.

Theremaybemorethanone'rightanswer'.

Don’tworryifyoucannotunderstandordoeverythingbecauseIamplanningtoteachalessononthisnextinthenextfewdays.

Makesurethatstudentsarefamiliarwiththecontextoftheproblem.

CountingTrees

• Doesanyoneknowwhatatreeplantationis?

• Howisaplantationdifferentfromanaturalforest?

• Theplantationconsistsofoldandnewtrees

• Howmightthearrangementoftreesinaplantationdifferfromthatofanaturalforest?

CatsandKittens

• Thisisapostermadebyacats’charity,encouragingpeopletohavetheircatsspayedsotheycan’thavekittens.Theactivityisaboutwhathappensifyoudon’thaveyourcatspayedandwhetherthestatementontheposteriscorrect.

• Isitrealisticthatonefemalecatwouldproduce2000descendantsin18months?Youaregivensomefactsaboutcatsandkittensthatwillhelpyoudecide.

RemembertoshowyourworkingsoIcanunderstandwhatyouaredoingandwhy.

Collectintheirworkandprovideconstructive,qualitativefeedbackonit.Thisshouldfocusongettingstudentstothinkandreason-aKeyProcessagenda.Don'tgivegrades,scoresorlevels!Writeonlyquestionsbelowtheirwork.Focusfeedbackonsuchissuesas:

Representing: Canyouthinkofadifferentwayoftacklinghisproblem?Whatsortofdiagrammightbehelpful?Whatassumptionshaveyoumade?

Reasoning:Howhaveyougotthisresult?Haveyoucheckedyourcalculations?Whatwouldhappenif...?

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Interpreting:Howcanyoutesttheaccuracyofyourestimate?Whatothersamplecouldyouhavechosen?

Communicating:Ifinditdifficulttofollowyourthinkinghere.Canyoupresentyourreasoningsothatsomeoneelsecanfolloweverystep?

Trytoidentifyparticularstudentswhohavestruggledandwhomayneedsupport.Alsolookoutforstudentsthathavebeensuccessful.Thesemayneedanextensionactivitytofurtherchallengethem.

ResourcesneededforthelessonYouwillneedthefollowingresources:

• Onecopyoftheproblemsheetperstudent

• Miniwhiteboards

• Largesheetsofpaperformakingpostersandfelt-tippedpens

• Calculatorsandrulers

CountingTrees

• Spare,largecopiesofthetreespictureforgroupstoworkontogether.

CatsandKittens

• Asupplyofgraphpaperorsquaredpaper(ifrequested)

Re-introducetheproblemtotheclass 5minutesBeginthelessonbybrieflyreintroducingtheproblem:

DoyouremembertheproblemIaskedyoutohaveagoatlasttime?

IhavehadalookatyourworkandIhavewrittensomecommentsatthebottomofit.

Todaywearegoingtoworktogethertryingtoimproveontheseinitialattempts.

First,onyourown,carefullyreadthroughthequestionsIhavewrittenonyourwork.Useyourmini-whiteboardstonotedownanswerstothesequestions.

Itishelpfultoaskstudentstowritetheirideasonalargesheetofpaperorminiwhiteboardusingfelt-tippedpen.Thishelpsyoumonitortheirworkandalsohelpsstudentstosharetheirideaslaterinthelesson.

Studentsworkalonerespondingtoyourfeedback 5minutesAllowthestudentssometimetoreflectonyourcommentsandwritetheirresponses.

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Studentsworkinpairstoimprovetheirsolutions 10minutesAskstudentstonowworkinpairsorthrees.GiveoutalargesheetofA3(atleast)paperandafelt-tippedpentoeachgroup.

NowIwantyoutoshareyourworkwithapartner.Takeitinturnstoexplainhowyoudidthetaskandhowyounowthinkitcouldbeimproved.NowIwanteachpairtoworktogether,comparingtheirideasandthefeedbackIhavegiven.Together,Iwantyoutotrytoproduceananswertotheproblemthatisbetterthaneachofyoudidseparately.

Goroundtheroom,listening,assessingtheirthinkingandmakinginterventionsaskingstrategicquestions.Consultacopyoftheprogressionstepsfortherelevantproblemanddecidewhichquestionswouldbemostappropriateformovingtheirthinkingtowardshigherlevelsofperformance.Usestrategicquestionslike:

Whatisknownandwhatisunknown?Whatareyouaskedtofindout?Howcanwesimplifythisproblem?Whatassumptionshaveyoumade?

Studentssharetheirapproacheswiththeclass 15minutesAskstudentstopresenttheirideasandapproachestotheclass.Focusontheirmethodsratherthantheiranswers.Astheyrespond,usetheprogressionstepstoassesstheirresponses.Inparticular,focusonthequalityofthereasoningandcommunication.

"Wedecidedtocountthedifferenttypesoftreesalongeachside,thenmultiplythesenumberstogether.""Wedrewatimelinealongthetopofthepaperandthendrewcatsunderneathtoshowwhentheygavebirth."

Asstudentspresenttheirideas,askotherstudentstocommenton:• Representing:Didtheychooseagoodmethod?• Analysing:Isthereasoningcorrect–arethecalculationsaccurate?• Interpreting:Aretheconclusionssensible?• Communication:Wasthereasoningeasytounderstandandfollow?

Studentscontinuewiththeproblem/oranextensionoftheproblem 20minutesEncouragestudentstoreturntotheproblemandcontinueworkingonitusingsomeoftheideasthathavebeenshared.Iftheyhavealreadyproducedagoodsolution,eitheraskthemtofindanalternativemethod,amoreconvincingreason,ortoexploreanextension.

CountingTreesIfInowshowedyouaverylargejarofSmarties,howcouldyouestimatethefractionthatarered?Writedownyoumethod.Canyouusewhatyoulearnedfrom“CountingTrees”?CatsandKittensCanyoufindasimpler,moreelegantwayofpresentingyourcalculationsto“CatsandKittens”?Canyouuseadiagramofsomekind?

Collectexamplesofstudents’workforthefollow-updiscussion.Trytoassesshowmuchstudentshavelearnedfromthesharingsession.

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Handout11:Exchangingexperiences

Whatworkedwellandlesswellintermsofformativeassessment?

Whatdidyoulearnaboutthestudents’understandingfromtheinitialactivity?

Howdidthestudentsrespondtoyourfeedback?

Whatworkedwellandlesswellintermsofproblemsolving?

Whataretheimplicationsofthislessonforyourfutureteaching?