©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 1

SELFANDPEERASSESSMENT

HowcanIuseassessmenttoimprovelearning?

Introduction“...self-assessmentbystudents,farfrombeingaluxury,isinfactanessentialcomponentofformativeassessment.Whereanyoneistryingtolearn,feedbackabouttheireffortshasthreeelements—thedesiredgoal,theevidenceabouttheirpresentposition,andsomeunderstandingofawaytoclosethegapbetweenthetwo.Allthreemusttoadegreebeunderstoodbyanyonebeforetheycantakeactiontoimprovetheirlearning.”(Black&Wiliam,1998)ThisisparticularlytruewhenthefocusoftheassessmentisontheprocessesinvolvedinIBL.Manystudentsdonotunderstandtheirnatureandimportanceinmathematics.Ifastudent’sgoalisonlytoget‘therightanswer’,thenshewillnotattendtothedeeperpurposesofthelesson.Thismoduleencouragesdiscussionofthefollowingissues:

• HowcanwehelpstudentstobecomemoreawareofIBLprocesses,andtheirimportanceinproblemsolving?

• HowwecanencouragestudentstotakemoreresponsibilityfortheirownlearningofIBLprocesses?

• Howcanstudentsbeencouragedtoassessandimproveeachother’swork?

ActivitiesActivityA: ExplorehowstudentsmaybemadeawareofIBLProcesses............................................2ActivityB: Considerhowstudentscanlearnfromsamplework........................................................4ActivityC: Discusshowstudentscanlearnfromassessingtheirownwork.......................................6ActivityD: Plantousepeerandself-assessmentstrategies...............................................................8ActivityE: Discussstrategiesfordifferentiation...............................................................................10Suggestedfurtherreading....................................................................................................................13

Acknowledgement:Inpreparingthismaterial,weacknowledgethepermissionsgivenbytheBowlandCharitableTrusttoadapttheprofessionaldevelopmentresources,BowlandMaths,thattheyhadpreviouslyfundedustoproducefortheUK.Thisincludesmanyofthehandoutsandvideoextracts.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

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 2

ACTIVITYA: EXPLOREHOWSTUDENTSMAYBEMADEAWAREOFIBLPROCESSES

Timeneeded:20minutes.

Inquiry-basedlearning(IBL),ashasbeensaidinearliermodules,isaboutengagingstudents'curiosityintheworldandtheideasthatsurroundthem.DuringMathematicsandSciencelessons,wewantstudentstomovefrombeingpassivereceiversoffactualknowledgetoactiveengagementinusinginquiry-basedlearningprocesses.Thismeansusingtheirconceptualknowledgetotacklenew,unfamiliarproblems,insimilarwaystothoseusedbyscientistsandmathematicians.Studentswillbegintoobserveandposequestionsabouttheworldaroundthem.Whenthesequestionsaretoocomplex,theywillengageinamodelingprocess:simplifyandrepresentthesituation;analyzedata;interpretandevaluatefindings,communicatefindingswithothers(Seefigure)andreflectontheoutcomes.Formanystudents,thesearenewandunfamiliarlearninggoalsandwillrequireashiftinthewaystudentsapproachtheirlearning.Thisactivitybeginstoconsidersomewaysinwhichstudentsmaybehelpedtoappreciatethesenewlearninggoalsandthusbegintoconsidertheirvalueandimportance.

Handout1presentsanumberofsuggestionsmadebyteachersthatcouldhelpstudentsbecomemoreawareofIBLprocesses.

• Discusstheadvantagesanddisadvantagesofeachsuggestion.• Canyouthinkofanyotherwaysofmakingstudentsmoreawareoftheseprocesses?

Thismoduleconsiderssomeofthesesuggestionsinmoredepth.

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B5! #72!8/C2<>0?!@D@<2!

K1'!(/**,A!B,J'.!*'F*'.'()!.)/)'.!,+!)1'!4,7'D5(=!F*,0'..I!!!K1'!A57'!B,J'.!7'.0*5B'!)1'!/0)5,(.!)1/)!4,:'!+*,4!,('!.)/)'!),!)1'!('J)I!!

Simplify and represent

• identify a problem • simplify and represent the problem • select information, methods and tools !

Situation

Mathematical model

Analyse and solve

• make connections with what you already know • visualise; draw diagrams • systematically change variables • look for patterns and relationships • make calculations and keep records • make conjectures and generalisations • use logical, deductive reasoning !

Solution

Interpret and evaluate

• form conclusions, arguments and generalisations • consider appropriateness and accuracy • relate back to the original situation Is the solution good enough? !

Communicate and reflect

• communicate and discuss findings effectively • consider alternative solutions • consider elegance, efficiency and equivalence • look for connections to other problems !

Report

NO

YES

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 3

Handout1: HelpingstudentstobecomeawareofIBLlearninggoals

1.UsingaposterorhandoutMakeapostershowingthegenericlistofprocessesanddisplaythisontheclassroomwall.Refertothishabitually,whilestudentsworkonunstructuredproblems,sothattheybecomemoreawarethatyourgoalsforthelessonareforthemtobecomemoreabletosimplifyandrepresent,analyseandsolve,interpretandevaluate,communicateandreflect.

2.Creatingtask-specifichintsBeforethelesson,preparesometask-specifichintsthatapplyIBLprocessestotheparticularprobleminhand.Whenstudentsarestuck,givethemtheappropriatehinteitheronpaperororally.Forexample,youcouldask:Canyouuseatableorgraphtoorganisethisdata?”;“Whatisfixedandwhatcanyouchangeinthisproblem?”;“Whatpatternscanyouseeinthisdata?”.

3.AskingstudentstoassessprovidedsamplesofworkAfterstudentshaveworkedonatask,presentthemwithsomeprepared,sampleresponsesfromotherstudents.Thesesolutionsprovidealternativestrategiesstudentsmaynothaveconsideredandmayalsocontainerrors.Askstudentstopretendtheyareexaminers.Thestudentsrankorderthesesolutions,alongwiththeirownresponse,givingexplanationsastowhytheythinkoneresponseisbetterthananother.

4.Usingprepared‘progressionsteps’Studentsevaluatesampleresponsesasin(3)above,butthistimeyoualsoprovidethemwithpreparedprogressionstepsthathighlighttheIBLprocesses.Studentsusethesetoevaluatethework.Endthelessonbysharingwhathasbeenlearnedfromthisprocess.

5.Askingstudentstoassesseachother’sworkAftertacklingataskinpairs,studentsexchangetheirwork.Eachpairofstudentsisgiventheworkofanotherpair.Studentsmakesuggestionsforwaysofimprovingeachsolutionandsticktheseontheworkusing“sticky”notes.Thesecommentsarepassedbacktotheoriginators,whomustthenproduceafinal,improvedversionbasedonthecommentsreceived.Thisisamorechallengingstrategyfortheteacherthan(3),astheissuesthatarisewillbelesspredictable.

6.StudentsintervieweachotherabouttheprocessestheyhaveusedWhenstudentshavefinishedworkingonatask,askthemtogetintopairs.Eachmemberofapairinterviewshisorherpartnerabouttheirapproachandtheprocessestheyhaveusedwhileworkingonthetask.Theteachermayprovidesomepre-preparedquestionstoassistinthis.Afternotingdownthereplies,studentschangeroles.Suitablequestionsmightbe:• Whatapproachdidyoutake?• Whichprocessesdidyouuse(fromaprovidedlist)?• Howcouldthisworkbeimproved?• Whatcouldyouhavedonedifferently?• Istherestillsomethingyouareconfusedby?

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 4

ACTIVITYB:CONSIDERHOWSTUDENTSCANLEARNFROMSAMPLEWORK

Timeneeded:30minutesOnepowerfulstrategyforenablingstudentstoappreciatedifferentlearninggoalsistoaskthemtoassesstheworkofothers.Thisroleshifthasseverallearningadvantages:

• Itencouragesstudentstoconsideralternativemethods.InmanyMathematicsandSciencelessons,studentsareonlypresentedwithonemethodfordoingeachtask.Theydonotthereforecometoappreciatethestrengthsandweaknessesofalternativeapproaches.

• Itencouragesstudentstoconsidermethodsthattheywouldhavenothavenormallychosen.Whensolvingmathematicalproblems,forexample,researchshowsusthatmanystudentsdonotchoosetousealgebraorgraphicalmethods.

• ItenablesstudentstoseethepurposeofIBLtasksmoreclearly.Manystudentsjustconsiderthepurposeofthelessonasto‘gettherightanswer’.Inassessingwork,particularlyagainstprovidedcriteria,studentsareencouragedtoappreciatetherelativequalitiesofdifferentmethods.

Thisactivityinvolveswatchingavideoclipofalessoninwhichsecondarystudentsassessstudentworkthathasbeenprovidedbytheteacher.ThisworkwaschosentorepresentfivedifferentapproachestotheproblemTextMessaging,onHandout2.Beforethelesson,thestudentshadbeenaskedtoattempttheproblemindividually,withouthelp.Inthisfollow-uplesson,studentsfirsttrytocomprehendthesampleworkthentheyevaluateit.

Beforewatchingthevideoclip,dothetaskTextMessagingyourself,andconsiderthesampleworkwithagroupofcolleagues,ifpossible.• WhatIBLprocessesareevidentinthesamplework?• Anticipatetheissuesthatwillarisewhenthissampleworkisassessedbystudents.

Nowwatchstudentsastheyassessthesamplework,andthengoontoimprovetheirownwork.• Whatdoaspectsoftheprovidedworkdostudentsattendto?• Whatcriteriadostudentsuseastheyassessthesamplework?• Whatarestudentslearningfromthesamplework?

Teacherssometimescommentthatsomestudentsattendmoretotheneatnessofthesampleworkthantothequalityandcommunicationofthereasoningemployed.Otherteachersareconcernedthatstudentswilluncritically'copy'samplework.• Howdoyourespondtotheseconcerns?• Whatcriteriawouldyouuseforchoosingsampleworktousewithstudents?

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 5

Handout2: Anassessmenttaskandstudentresponses

TextMessaging

1. Howmanytextmessagesaresentiffourpeopleallsendmessagestoeachother?2. Howmanytextmessagesaresentwithdifferentnumbersofpeople?3. Approximatelyhowmanytextmessageswouldtravelincyberspaceifeveryoneinyour

schooltookpart?4. Canyouthinkofothersituationsthatwouldgiverisetothesamemathematical

relationship?

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!!"#$%&'(%)*+&!

&!

Sam’s answer

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Chris’s answer

Lily’s answer

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Marvin’s answer

!

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 6

ACTIVITYC:DISCUSSHOWSTUDENTSCANLEARNFROMASSESSINGTHEIROWNWORK.

Timeneeded:30minutesOftenwhenstudentshavefinishedapieceofwork,theywanttomoveon.Theydon’twanttore-examineit,polishit,orpresentitsothatotherpeoplecanunderstandandfollowtheirreasoning.Onthevideo,twoteachers,EmmaandShane,asktheirstudentstoassessandimproveeachother'swork.Tohelpthemdothis,theyprovidesomestructuredframeworks.EmmausestheGoldenRectanglestaskandhascollatedaselectionofherownstudents'workonthistaskintoaposter.ShehasalsosimplifiedtheassessmentframeworkonHandout3forusewithherstudents.Inthelesson,sheasksgroupsofstudentstoassesstheworkontheposterusingthesimplifiedframework.Theheadingsonherframeworkare:“Represent”,“Analyze”,“Interpret”,“Communicate”.Thesecorrespondtothephasesinthemodelingdiagram(showninActivityA).Onthevideo,studentsmaybeheardreferringtoa"trafficlights"schemethatEmmausesinherMathematicslessons.Here,'Green'meansthatstudentsunderstand,while'Red'meansthattheydonot.‘Amber’liesinbetween.ShaneusedtheCountingTreestaskandhaspreparedalessstructuredsheettohelphisstudentsassesseachother'swork.Thissheetcontainsthequestions:Didtheychooseagoodmethod?;Wastheirreasoningcorrect?;Aretheirworkingsaccurate?;Aretheirconclusionssensible?;Wastheirreasoningeasytofollow?;Whatdidyoulikeabouttheirwork?;Whatwouldyouliketoseenexttime?

Familiarizeyourselfwiththetasks.WatchthevideoextractsofShane'sandEmma'slessons.

• Whatobservationsdostudentsmakeabouteachother'swork?• Howmightthishelpthemtoimprovetheirownwork?

CompareEmma'ssimplifiedprogressionstepswithShane'slessstructuredsheet.

• Whataretheadvantagesanddisadvantagesofeachmethodforhelpingstudentstoreflectonandimprovetheirwork?

Comparetheuseofworkfromwithinthestudents'ownclasstotheuseofthesampleresponsesusedinActivityB.

• Whataretheadvantagesanddisadvantagesofeachmethod?

TheassessmentframeworksmayhelpstudentstodevelopanawarenessofhowIBLprocessesrelatetoparticulartasks,andrecognizehowtheycanimprovetheirresponses.Forthestepstobeusedinthiswaythelanguagewillneedtobeadaptedtotheclassandspecific'answers'willneedtoberemoved.Teachershavecommentedthatstudentsaremoreabletobecriticalofsampleresponsesthataretakenfromsourcesoutsidetheclassroom,whentheycannotbeidentified.Whengivingfeedbacktomembersoftheirownclass,personalrelationshipscomeintoplay.Studentsdonotfeelsoabletocriticizetheworkoffriends.Classroomculturesmayneedtobedevelopedwhereideasandworkmaybecriticizedwithoutindividualsfeelingthreatenedandexposed.

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 7

Handout3: Twoassessmenttaskswithassessmentframeworks

Goldenrectangles

CountingTrees

AssessmentframeworkforGoldenrectangles

AssessmentFrameworkforCountingTrees

Involving pupils in self and peer assessment Handout 2

r4 © 2010 Bowland Charitable Trust Page 15 of 20

2 A Bowland assessment task and five sample responses

Counting Trees

! ! ! ! ! ! ! ! ! " ! ! " ! " ! ! " ! " ! " ! ! ! ! ! ! ! " " ! " ! ! ! " ! "" " ! ! " ! ! " " " ! ! ! ! ! ! " ! ! ! " ! " " ! ! ! ! ! " ! " ! " " "! ! ! " " ! ! ! ! ! ! " ! ! ! ! ! " ! " ! " " " ! ! " ! ! ! ! ! " " " ! ! " ! ! ! ! " !" ! " ! " " ! ! ! ! ! ! " ! ! ! ! " ! " ! ! ! " ! " ! " ! ! ! ! ! ! !! " " " " ! ! ! ! ! " " " ! " " ! " ! ! " ! ! " ! ! ! " ! ! " ! ! " ! !! ! " ! ! " " " ! ! " ! ! ! ! ! ! ! ! " ! ! ! ! ! ! ! ! " ! ! ! ! " " ! ! " " " !! ! ! ! ! ! " ! ! ! " ! ! ! ! " " ! ! " ! ! " ! ! " ! ! " " ! " ! ! ! ! !! " ! ! ! ! ! ! ! ! " " ! ! " " " " " ! ! ! " ! ! ! ! ! " ! ! ! ! ! !! ! ! " ! ! ! " " ! ! ! ! ! " " " ! ! ! " ! " ! ! ! " " ! " " ! " ! " "" " ! " ! ! " " " " " ! ! " ! " " ! ! ! ! ! ! ! ! " ! ! ! ! ! " " ! ! " " ! ! !" ! ! ! ! ! ! ! ! ! ! ! ! ! " " " ! " " ! " " ! " ! ! ! ! " " " ! ! ! ! ! " ! " !! ! ! " ! " " ! ! ! " " " ! ! ! ! " ! " ! ! ! " ! ! ! ! ! " ! ! " ! ! ! " ! !! " " ! ! ! ! ! ! ! " ! " ! ! " " ! " ! ! " ! ! ! ! " ! " ! ! " ! ! " "

! ! ! ! ! ! " " ! ! ! ! " ! ! " " ! " ! " " " " " " ! ! " ! ! ! ! ! ! " " " !" " ! " " ! ! " ! " " ! ! ! " " " ! ! ! ! ! ! ! " ! ! " ! ! ! " ! " ! !

" " ! " ! ! " ! " " ! " " ! " ! ! ! " " ! " ! ! " ! " " " ! ! ! ! " ! "! ! ! " ! ! ! ! ! ! ! " ! ! " " ! ! " ! ! ! ! " ! " " " ! " " ! " ! " ! ! ! ! ! " !

! ! ! ! ! ! " ! ! ! ! " ! ! " ! " ! " " " " " " ! ! ! ! ! ! ! ! ! ! ! " ! ! ! ! " ! ! !" ! ! ! ! ! ! " ! ! ! ! ! " ! " " ! ! ! ! ! " ! ! " ! ! ! " " ! ! ! " ! ! ! ! ! !" ! ! ! " ! " ! ! ! " " " " ! ! " ! " " " ! ! " " ! ! ! ! " ! ! ! ! ! " ! ! ! ! !

" ! " ! " ! ! " " " " ! ! " " ! " ! ! ! ! ! " ! ! " ! " ! ! ! ! " ! " " " " ! ! !! ! " " " " ! ! ! " " " ! " ! ! ! " " " ! ! " ! " ! " ! ! ! " ! " ! " " ! " " ! ! ! ! !! ! " " " ! ! ! ! ! ! " " ! ! " ! " " ! ! ! ! " ! " ! ! ! " ! " ! ! ! " !

! ! " ! " " ! " ! ! " ! ! ! " " ! ! ! ! ! " ! ! " ! ! ! " ! ! ! ! " !! ! ! " " " " " " ! ! " ! " " ! ! " ! ! ! ! " " " ! " ! ! " ! "! " ! " ! ! ! ! ! ! " ! ! " " " ! ! " ! " ! " " ! " " " ! ! ! ! ! ! ! ! " !

" ! ! ! " ! " ! ! ! " ! ! ! " " " ! ! " ! ! ! ! " " ! " ! " ! ! ! ! ! ! ! "" ! ! ! ! ! ! " ! ! ! " ! ! ! ! " " " ! " " " ! ! ! ! " ! ! ! " ! !" ! " " " " ! " ! ! " ! ! " " ! " ! ! " ! " " ! ! ! " ! ! ! " " ! ! ! ! ! " " ! !! ! " " " ! ! ! ! ! " ! ! ! ! ! " " ! " ! " ! " ! ! " " " ! " " " ! ! ! ! ! " ! ! !! ! ! " " ! ! ! ! ! " ! ! ! " ! ! ! ! ! " ! ! ! ! ! " ! " ! " " ! " " " ! " ! !! ! " ! ! ! " ! " " " ! ! ! ! ! ! ! ! ! ! ! " " " ! ! " ! " " ! ! " ! " ! " " ! !

" ! ! ! ! " ! ! ! ! ! " ! ! ! " " ! " ! " ! ! ! " " ! ! ! ! " " ! ! ! " " " " " ! "! " ! ! ! ! " ! ! ! " " ! ! ! ! ! ! ! ! ! ! " ! ! ! ! ! ! ! ! ! ! " " " " ! !! ! " ! " ! " ! ! ! " ! ! ! ! ! " ! ! ! " ! ! ! ! ! ! ! ! ! ! " ! ! " ! ! " ! ! ! "! ! " ! ! ! ! " ! " ! " ! " ! ! " ! ! " ! ! ! ! ! ! ! ! ! ! ! " ! ! !

! " " ! ! ! ! ! " ! ! " " ! " ! ! " ! ! ! ! ! ! ! ! " ! ! ! ! ! ! ! " !! ! ! ! ! ! ! " ! ! ! " ! ! " ! " " " " ! ! ! " ! " ! ! ! ! ! " ! ! ! ! " ! !! ! ! ! " " ! ! ! ! ! " " ! ! " ! " ! ! " ! ! ! ! ! " ! " " " ! ! ! !! ! ! " ! ! " ! ! " ! ! " ! " ! ! " ! " ! ! " ! " ! ! " ! " ! " " ! ! ! ! "" ! ! ! ! ! ! " ! " ! ! ! ! ! ! ! ! ! ! " ! " ! ! ! ! ! ! ! ! ! " " ! " ! " " ! " ! !! ! ! ! ! ! ! " ! ! ! " ! ! ! ! ! " " ! " ! " ! ! ! ! ! ! " " ! ! ! ! ! ! " ! " " !! ! ! ! ! ! " " ! ! ! ! " ! ! ! ! ! " ! " ! ! " " ! ! ! ! ! ! ! ! ! ! " ! " ! " "! " ! " " ! ! " " ! ! " ! ! ! " ! ! ! ! ! ! ! " " ! ! ! " " ! " ! ! " ! ! ! ! ! ! !

! ! " " ! ! ! ! " ! ! " ! ! ! " ! ! ! ! " ! " ! " ! ! " ! " ! ! " ! " ! ! ! ! ! !! " ! " ! ! " " ! " ! ! ! ! ! ! ! ! ! ! ! ! ! ! " ! ! ! ! ! ! " ! ! ! ! ! ! !

! ! " " ! ! ! ! ! " ! ! ! " ! " ! " " ! ! " " ! " ! " ! ! ! " ! " ! " ! ! ! ! !" ! ! ! " ! " ! ! ! ! ! ! ! ! ! " ! " " " ! " " ! " ! " ! ! ! ! " " " " ! !! " ! ! " ! ! " " " ! ! ! ! " " ! ! " " ! ! ! " " " " ! " " " " " ! ! ! " "" " ! ! ! " ! ! ! ! ! " " ! " " ! ! ! ! " ! " " ! " ! ! ! " ! ! " This diagram shows some trees in a plantation. The circles ! show old trees and the triangles ! show young trees. Tom wants to know how many trees there are of each type, but says it would take too long counting them all, one-by-one.

1. What method could he use to estimate the number of trees of each type? Explain your method fully.

2. On your worksheet, use your method to estimate the number of:

(a) Old trees (b) Young trees

Involving pupils in self and peer assessment Handout 2

r4 © 2010 Bowland Charitable Trust Page 14 of 20

Progression in key processes

Representing Analysing Interpreting and

evaluating Communicating

The student draws one or two rectangles with a perimeter of 100m.

The student works out the areas of their rectangles correctly.

The student draws several rectangles but not a square and the justification is incorrect or omitted.

The work is communicated adequately, but there are gaps and/or omissions.

Draws several rectangles.

Calculates the areas of their rectangles and attempts to come to some generalisation.

Realises that different shapes have different areas but comes to incorrect or incomplete conclusion.

The work is communicated clearly and the reasoning may be followed.

Draws several, correct rectangles for an adventurer working alone and for 2 working together. May draw far too many rectangles.

Calculates the areas correctly and finds that a square is best for 1 adventurer and that 2 working together do better than alone.

Attempts to give some explanation for their findings.

The work is communicated clearly and the reasoning may be easily followed.

! !

! P

rogress ! !

!

Draws an appropriate number of rectangles and collects the data in an organised way.

Calculates the correct areas, finds that a square is best for 1 adventurer and that 2 working together do better than alone. Finds a rule or pattern in their results.

Gives reasoned explanations for their findings.

Explains work clearly and may consider other shapes.

Involving pupils in self and peer assessment Handout 2

r4 © 2010 Bowland Charitable Trust Page 20 of 20

Progression in key processes

Representing Analysing Interpreting and

evaluating Communicating and reflecting

Chooses a method, but this may not involve sampling. E.g. Counts all trees or multiplies the number of trees in a row by the number in a column.

Follows chosen method, possibly making errors. E.g. Does not account for different numbers of old and young trees or that there are gaps.

Estimates number of new and old trees, but answer given is unreasonable due to method and errors.

Communicates work adequately but with omissions.

Chooses a sampling method but this is unrepresentative or too small. E.g. tries to count the trees in first row and multiplies by the number of rows.

Follows chosen method, mostly accurately. E.g. May not account for different numbers of old and young trees or that there are gaps.

Estimates number of new and old trees, but answer given is unreasonable due mainly to the method.

Communicates reasoning and results adequately, but with omissions.

Chooses a reasonable sampling method.

Follows chosen method, mostly accurately.

Estimates a reasonable number of old and new trees in the plantation. The reasonableness of the estimate is not checked. E.g. by repeating with a different sample.

Explains what they are doing but explanation may lack detail.

! !

! P

rogress ! !

!

Chooses an appropriate sampling technique.

Follows chosen method accurately. Uses a proportional argument correctly.

Deduces a reasonable number of old and new trees in the plantation. There is some evidence of checking the estimate. E.g. Considers a different sampling method.

Communicates reasoning clearly and fully.

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 8

ACTIVITYD:PLANTOUSEPEERANDSELF-ASSESSMENTSTRATEGIES

Minimumtimeneeded: 30minutesbeforethelesson 20minutesforthepre-lessonassessment 30minutestopreparefeedback 60minutesforthelesson 15minutesforreportingbackInthisactivity,participantsplanandcarryoutalessoninwhichstudentsbecomeassessorsoftheirownwork(TrackA),orassessorsofworkthatweprovideforthem(TrackB).Itishelpfulifsomeparticipantschoosetodoeachtrackandthentheycanbecompared.AsamplelessonplanforeachtrackisprovidedonHandout4.• PlanwhenyouwillallowstudentstimetotackleanIBLtask,individuallyorinpairs,without

yourguidance.• Planalessoninwhichstudentsrevisitthetaskandassessotherstudents'work-eitherwork

fromtheirclassmatesorfromsomesampleresponsesthatyouwillprovide.MakesurestudentshaveanopportunitytodiscusstheimportanceoftheIBLProcesses,andsufficienttimetorevisetheirownworkinthelightofthecomments.

Ifyouareworkingonthismodulewithagroup,itishelpfulifeachparticipantchoosesthesameassessmenttask,asthiswillfacilitatethefollow-updiscussion.Afterthelessonmeettogethertofeedbackonwhathappened.Takeitinturnstodescribeyourexperiencesofusingselfandpeerassessment.

• Howdidyourstudentsperformonthetask,unaided?• Howdidstudentsassesstheprovidedresponsesandtheworkoftheirpeers?

Whataspectsdidtheyattendto?• Howdidstudentsmakeuseoftheassessmentframeworks?

DidthesehelpstudentstounderstandIBLprocesses?• Howwelldidstudentsreacttoandusetheevidencetoimprovetheirownwork?• Whataretheimplicationsofthislessonforyourfuturelessons?

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 9

Handout4: AlessonplaninwhichstudentsareassessorsThefollowingsuggestionsdescribeonepossibleapproachtoself-andpeer-assessment.Studentsaregivenachancetotackleaproblemunaided,tobeginwith.Thisgivesyouachancetoassesstheirthinkingandtoidentifystudentsthatneedhelp.Thisisfollowedbyformativelessoninwhichtheycollaborate,reflectontheirworkandtrytoimproveit.

Beforethelesson 20minutes

Beforethelesson,perhapsattheendofapreviouslesson,askstudentstoattemptoneoftheassessmenttasks,Textmessages,GoldenrectanglesorCountingTrees,ontheirown.Studentswillneedcalculators,pencils,rulers,andsquaredpaper.

Theaimistoseehowableyouaretotackleaproblemwithoutmyhelp.Therearemanywaystotackletheproblem-youchoose.Theremaybemorethanone'rightanswer'.Don’tworryifyoucannotunderstandordoeverythingbecauseIamplanningtoteachalessononthisnextinthenextfewdays.

Collectinstudents’workandreviewasampleofit.Lookcarefullyattherangeofmethodsstudentsuseandthequalityofthereasoning.Trytoidentifyparticularstudentswhohavestruggledandwhomayneedsupport.Alsolookoutforstudentsthathavebeensuccessful.Thesemayneedanextensionactivitytochallengethemfurther.

Re-introducetheproblemtotheclass 5minutes

Beginthelessonbybrieflyreintroducingtheproblem:

DoyouremembertheproblemIaskedyoutohaveagoatlasttime?Todaywearegoingtoworktogetherandtrytoimproveyourfirstattempts.Evenifyougotmostofitrightfirsttime,youwilllearnsomethingbecausetherearedifferentwaystotackletheproblem.

Atthispoint,choosebetweentheTrackAorTrackB.Eitherdecidetoletstudentsassessandimprovetheirownwork,orofferthemtheprovidedsamplesofworktoassess.Therewon’tbetimeforboth!

TrackA:Studentsassesstheirownwork

TrackA:Studentsassessandimprovetheirownwork-15minutes

Askstudentstoworkinpairsorthreesandgiveeachgroupalargesheetofcardandafelt-tippedpen.Giveeachgroupbacktheirinitialattemptsattheproblem.

I want you to look again at you answers but this time, work as a group. Take it in turns to describe your attempt to the rest of the group. After each suggestion, the others in the group should say what they like about your method and also where they think it can be improved. After you have all done this, I want you to work together to produce a better answer than you did separately. Make a poster showing your best ideas. It doesn’t have to be beautiful, but it should show you thinking.

Goroundtheroom,listening,assessingtheirthinkingandmakingappropriateinterventions.Listenspecificallytostudentsthatstruggledwiththetaskwhentheyworkedalone,andofferthemsupport.Ifstudentshavesucceededandtheirworkiscorrect,provideoneoftheplannedextensions.

TrackA:Studentsexchangeandcommentoneachothers’work-15minutes

Askstudentstoexchangetheirposterswithanotherpairandissueeachgroupwithacopyofthe“progressionsteps”frameworkforthetask–onethatiswritteninstudent-friendlylanguage.

On a separate sheet of paper, write comments on: •Representing: Did they choose a good method? •Analysing: Is the reasoning correct – are the calculations accurate? •Interpreting: Are the conclusions sensible? •Communication: Was the reasoning easy to understand and follow?

Astheydothis,goroundencouragingstudentstoreadtheworkcarefullyandcommentonthepointsmentioned.Youmayneedtohelpthemunderstandwhatthe‘progressionsteps’mean.Whenstudentshavecommentedonthework,onepersonfromthegroupshouldtakethepostertothegroupthatproducedit,andexplainwhatneedstobedonefortheworktobeimproved.

TrackA:Studentsimprovetheirownwork-5minutes

Givegroupsalittletimetoabsorbthecommentsandtimetofurtherimprovetheirideas.

TrackA:Plenarydiscussiononapproachesandchanges-15minutes

Towardstheendofthelessonholdadiscussionontheapproachesusedandthechangesthathavebeenmade:

What changes have you made to your initial work? Why is it now better than it was before?

Collect in the work and assess how the thinking has improved.

TrackB:Studentsassessprovidedsamplework

TrackB:Studentsassessprovidedsamplework-15minutes

Giveoutthesamplestudentwork.

These samples of work were taken from another class. I want you to imagine that you are their teacher. This work may give you ideas you haven’t thought of. It is also full of mistakes! I want you to comment on each of the following themes: •Representing: Did they choose a good method? •Analysing: Is the reasoning correct – are the calculations accurate? •Interpreting: Are the conclusions sensible? •Communication: Was the reasoning easy to understand and follow?

Inthisway,studentswillbecomemoreawareofwhatisvaluedintheirwork–theKeyProcessesofrepresenting,analysing,interpretingandcommunicating.

Listentotheirdiscussionsandencouragethemtothinkmoredeeply.Encouragestudentstosaywhattheylikeanddislikeabouteachresponseandaskthemtoexplaintheirreasons.

TrackB:Studentsassesssampleworkusing“progressionsteps”-10minutes

Afterstudentshavehadtimetorespondfreely,issueeachgroupwithacopyofthe“progressionsteps”frameworkforthetask–onethatiswritteninstudent-friendlylanguage.

This framework may give you further ideas. Where would you put the work on the framework?

TrackB:Plenarydiscussionofthesamplework-15minutes

Projecteachpieceofsamplestudentworkontheboardandaskstudentstocommentonit:What can we say about this piece of work? Share some of the comments you wrote. What did you think of the methods they chose? Which method did you like best? Why was this? Did you find any mistakes in their work? Do you agree with their conclusions?

TrackB:Workinginpairs:Studentsimprovetheirownwork-10minutes

Nowusingwhattheyhavelearned,askstudentstoworktogethertoimprovetheirownsolutions.Astheydothis,asstudentstoexplaintheirthinking.

Max, tell me what you have done to improve your own solution. Collectexamplesofstudents’workforthefollow-upsession.Trytoassesshowmuchstudentshavelearnedfromthesharingsession.

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ACTIVITYE: DISCUSSSTRATEGIESFORDIFFERENTIATION

Timeneeded:20minutesReflectonyournormalteachingpractices.Whenyouassessclasses,youbegintorealizetheconsiderableindividualdifferencesinstudentsandtheyhaveverydifferentlearningneeds.Somestudentsneedmoresupport,whileothersneedagreaterchallenge.

• Howdoyounormallydealwithrangeofdifferentlearningneedsofyourstudents?• DiscusstheadvantagesanddisadvantagesofthefourstrategiesshownonHandout5.• CompareyourviewswiththecommentsgivenonHandout6.

Handouts5and6: MeetingtheneedsofallstudentsAssessmentrevealsthatallstudentshavedifferentlearningneeds.Howdoyourespondtothisinyournormallessons?Discussandnotedowntheadvantagesanddisadvantagesofeachapproach.Addyourownideasunderneath.

Differentiatebyquantity?Whenstudentsappearsuccessful,youprovidethemwithanewproblemtodo.............................................................................................................................................

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Differentiatebytask?Youtrytogiveeachstudentaproblemthatismatchedtotheircapability.............................................................................................................................................

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Differentiatebyoutcome?Youuseopenproblemsthatencourageavarietyofpossibleoutcomes.............................................................................................................................................

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Differentiatebylevelofsupport?Yougiveallstudentsthesameproblem,butthenofferdifferentlevelsofsupport,dependingontheneedsthatbecomeapparent.............................................................................................................................................

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Differentiatebyquantity?Whenstudentsappearsuccessful,youprovidethemwithanewproblemtodo.Thisapproachiscommon,butitleadstostudentsviewingthecurriulumasalistofproblemstodo,ratherthanprocessestoacquire.Thisapproachwillnotpromotereflectiononalternativemethodsfordoingaproblem-differentwaysofrepresenting,analysing,interpretingandcommunicating.Differentiatebytask?Youtrytogiveeachstudentaproblemthatismatchedtotheircapabilities.Buthowdoesoneknowifaproblemissuitable?Wecanonlymatchaproblemtoastudentifwehaveaprofoundunderstandingofboth.Ourviewoftheproblemisusuallybasedonourownwayofdoingit–andtheremaybemanyotherapproaches.Wealsohaveanimperfectandoftenprejudicedviewofstudents’capabilities.Wesoeasilyjudgestudents’‘mathematicalability’bytheirabilitytocarryoutroutineprocedurestheyhaverecentlybeentaught.Problemsolvingrequiresadifferentsetofskillsandmayresultindifferentstudentsperformingwell.Thisapproachalsocreatesmanagementdifficultiesasdifferentproblemsareusedwithdifferentstudents.Thisreducespossibilitiesforwholeclassdiscussionsandsharingknowledge.Differentiatebyoutcome?Youusemoreopenproblemsthatencourageavarietyofpossibleapproachesandoutcomes.Thisapproachrequiresproblemsandsituationsthatallowforsuchavarietytoemerge.TheBowlandproblemsarelikethis,buttheydomakeconsiderabledemandsonstudentswhoareunfamiliartoproblemsolving.Manyteacherscommentthatassoonasstudentsbegintostruggle,theywantto‘leapin’,‘takeover’andstructuretheproblem,sothatstudentshaveclearstepstofollow.Thistendencyunderminestheverypurposeofthelesson–todevelopstudents’abilitytouseKeyProcessesinanautonomousway.Ontheotherhand,toolittleguidancemayresultinprolongedfailureandfrustration.Someteachersthereforemakeitarulethatstudentsshouldalwayshelpandshareideaswitheachother,beforeaskingforhelpfromtheteacher.Differentiatebylevelofsupport?Yougiveallstudentsthesameproblem,butthenofferdifferentlevelsofsupport,dependingontheneedsthatbecomeapparent.Thisapproachavoidsmanyofthedifficultiesdescribedabove.Thesupportmaybebyotherstudents,orbytheteacher-orally,orinwrittenform.Inthelessonswehavesuggested,theteacherasksthestudentstoattemptwhattheycanunaided,thentheyareofferedthesupportoftheirpeersasideasandapproachesaresharedanddiscussed.Iffurthersupportisneeded,thentheteachermaysupplythisthroughquestionsthatcausestudentstoattendtoparticularfeaturesoftheproblem,orthroughmorespecifichints.Timingsuchhelpiscritical.Oneoftheimportantgoalsofproblemsolvingistoallowstudentstheexperienceofstrugglingwithaproblemforsometimeandexperiencingthesenseofachievementthatarriveswhentheproblemhasbeenovercome.Ifwehelpstudentstooquickly,werobthemofthisexperience.

ThestrategiessuggestedonHandout5are:

• Differentiatebyquantity?Whenstudentsappearsuccessful,youprovidethemwithanewproblemtodo.

• Differentiatebytask?Youtrytogiveeachstudentaproblemmatchedtohercapability.• Differentiatebyoutcome?Youuseopenproblemsthathaveavarietyofpossibleoutcomes.• Differentiatebylevelofsupport?Yougiveallstudentsthesameproblem,butofferdifferent

levelsofsupport,dependingontheneedsthatbecomeapparent.Thefirsttwooftheseapproachesareunhelpful,particularlywhendevelopingIBLprocesses,forthereasonsidentifiedinHandout6.IBLtasksare'open'inthesensethattheyencourageavarietyofapproaches.Theirdifficultyisnotmerelyrelatedtotheirapparent'content',butisalsorelatedto

©CentreforResearchinMathematicsEducation,UniversityofNottingham2011 11

thefamiliarityofthecontext,thecomplexityofinformationwithintheproblem,theconnectionsthatneedtobemade,thelengthofthechainsofreasoningrequired,andsoon.HelpingstudentsthatstruggleAswellasfindingthetaskschallenging,studentsmayfindthewholeideaofselfandpeerassessmentdifficult.Theyarebeingaskedtoreflectonthemethodsandprocessesthattheyandothershaveused.ThinkagainaboutyourlessonsusingtheIBLtasks.

• Howmightyouhelpthosewhostrugglewiththetask?• Howcanyouhelpthosewhostrugglewiththewholeideaofpeerassessment?

Teachershavefoundthatwhenstudentsgetstuckwithatask,thentheymaybeconsiderablyhelpedby:

• discussingtheirdifficultywithapartner(notnecessarilytheirneighbour);• lookingatexamplesofotherstudents'work(howeverrough)-thesewillsuggestsnewways

toaccessandapproachthetask.Assoonastheteachergivesdetailedguidanceonwhattodo,thestudentsareunabletomakestrategicdecisionsforthemselves.Suchguidanceshouldthereforeonlybegivenasalastresort,afterstudentshavebeenallowedtostruggleandhelpeachother.Wehavefoundthatmoststudentsenjoyandvalueselfandpeerassessment.Somehowever,maybeunusedtorevisitingtasksandreflectingonearlierworkandmaynotthereforeappreciatethevalueofdiscussingdifferentsolutionmethodsindepth."WhenIknowtheanswer,whatpointisthereindiscussingtheproblemfurtherandlookingatotherpeople'swork?"Suchstudentspreferto'geton'andtacklenewtasks.Wehavefoundthatitisimportanttocarefullyexplainthepurposeofpeerassessmenttostudentsmeetingitforthefirsttime.StretchingstudentsthatsucceedSomestudentsmayhavedoneverywellattheproblems,evenattheverybeginning.Othersmayhaveworkedwellandfinishedquickly.Itisagoodideatoplanforsucheventualities.

Thinkbacktoyourownlesson.

• Whenstudentssucceeded,howdidyouextendtheirthinking?• Whatalternativeapproachestothetaskdidyou,orcouldyousuggest?• Whatextensionstothetaskdidyou,orcanyousuggest?

Evenifstudentssucceedintheproblems,theycanstilllearnagreatdealbyrevisitingthem.Studentsmaybeencouragedto:

• findalternativeormoreelegantwaysofrepresentingandtacklingthetask;• makeuptheirownvariantsorextensionstotasks• devisetheirown"progressionsteps",todeveloptheirunderstandingofKeyProcesses.

Youmaybeliketosuggestyourownpossibleextensionstothetasks.Forexample:

• TextMessaging:Howlongwouldittaketospreadapieceofnewsaroundtheschoolifeachpersonsendsatextmessagetofourotherpeople?

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• CountingTrees:Whatmethodwouldyouuseifyouwereaskedtoestimatethenumberofbeansinajar?

• Goldenrectangles:Supposetheadventurerswereonlygiventhreestakeseach?(Thetaskwouldneedtoberenamed:Goldentriangles).

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SUGGESTEDFURTHERREADING

Black,P.,&Wiliam,D.(1998).Insidetheblackbox:raisingstandardsthroughclassroomassessment.King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:http://shop.gl-assessment.co.uk

Thisshortbookletoffersasummaryoftheextensiveresearchliteratureintoformativeassessment.Itshowsthatthereisclearevidencethatimprovingformativeassessmentraisesstandards,andoffersevidenceshowinghowformativeassessmentmaybeimproved.Thisbookletisessentialreadingforallteachers.

Black,P.,&Harrison,C.(2002).Workinginsidetheblackbox:Assessmentforlearninginthe

classroom.King'sCollegeLondonSchoolofEducation.NowpublishedbyGLAssessment:http://shop.gl-assessment.co.uk

Inthisbooklet,theauthorsdescribeaprojectwithteachersinwhichtheystudiedpracticalwaysofimplementingformativeassessmentstrategiesandtheeffectthishadonlearning.Thesectiononpeer-assessmentandself-assessment(pages10-12)areparticularlyrelevanttothismodule.

Black,P.,Harrison,C.,Lee,C.,Marshall,B.,&Wiliam,D.(2003).Assessmentforlearning:Puttingit

intopractice.Buckingham:OpenUniversityPress.

ThisbookgivesafulleraccountoftheearlierbookletsInsidetheblackboxandWorkinginsidetheblackbox.Itdiscussesfourtypesofaction:questioning,feedbackbymarking,peer-andself-assessmentandtheformativeuseofsummativetests.Thesectiononpeerandself-assessment(pp49-53)isparticularlyrelevanttothismodule.

Hodgen,J.,&Wiliam,D.(2006).Mathematicsinsidetheblackbox.King'sCollegeLondonSchoolof

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

ThisbookletappliestheabovefindingsspecificallytoMathematics.ItconsiderssomeprinciplesforMathematicslearning,choiceofactivitiesthatpromotechallengeanddialogue,questioningandlistening,peerdiscussion,feedbackandmarking,andselfandpeerassessment.Thisbookletisessentialreadingforallmathematicsteachers.Pages9-10areparticularlyrelevanttothismodule.