tackling unstructured problems · 2017-10-29 · mason, j., burton, l. and stacey, k. (1982)...

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 1

TACKLINGUNSTRUCTUREDPROBLEMS

‘DoIstandbackandwatch,orinterveneandtellthemwhattodo?’

IntroductionInmostmathematicsandscienceclassrooms,studentsareprovidedwithstructuredtasksandaretoldpreciselywhichtechniquestodeploy.Studentslearnbyfollowinginstructions.Problemsandsituationsthatariseintheworldarenotusuallylikethis.Ratherthanbeingexercisesintheuseofaparticularskillorconcept,real-worldproblemsrequirestudentstomakesimplifications,modelsituations,chooseappropriateknowledgeandprocessesfromtheir'toolkit',andtestwhethertheirsolutionis"goodenough"forthepurposeinhand.Itseemslogicalthatifstudentsaretolearntousetheirskillsautonomouslyintheirfuturelives,theywillneedsomeopportunitiestoworkonlessstructuredproblemsintheirclassrooms.Thisunitcomparesstructuredandunstructuredversionsofproblemsandconsidersthedemandsandchallengesunstructuredproblemspresenttostudentsandteachers.

ActivitiesActivityA: Revisingstructuredproblems.......................................................................................2ActivityB: Comparestructuredandunstructuredproblems.........................................................4ActivityC: Considerstrategiesforofferinghelp.............................................................................6ActivityD: Observeandanalysealesson.......................................................................................8ActivityE: Planalesson,teachitandreflectontheoutcomes......................................................9Furtherreading................................................................................................................................10References.......................................................................................................................................10

Acknowledgement:ThismaterialisadaptedforPRIMASfrom:Swan,M;Pead,D(2008).Professionaldevelopmentresources.BowlandMathsKeyStage3,BowlandTrust/DepartmentforChildren,SchoolsandFamilies.AvailableonlineintheUKat:http://www.bowlandmaths.org.uk.ItisusedherebypermissionoftheBowlandTrust.

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ACTIVITYA:REVISINGSTRUCTUREDPROBLEMS

Timeneeded:20minutes.

Handout1presentsthreestructuredproblems:

• Organisingatabletennistournament• Designingaboxfor18sweets• CalculatingBodyMassIndex

TheseproblemsareofthesametypeasthosetypicallyfoundinmanyMathematicsandScienceClassrooms.Thefirsttwoarepracticalgrouptasksandthethirdisacomputer-basedtask.Theseare,however,structuredsothattheyleadstudentsthroughtheproblems,guidingandmakingdecisionsforthem.

• Workthroughoneofthestructuredproblemscarefully.• Listallthedecisionsthatarebeingmadeforthestudents.• Revisetheproblemssothatsomeofthesedecisionsarehandedbacktostudents.

Thiswillmakethemlessstructured.

Forexample,inOrganisingatabletennistournament,pupilsaretold:

• howtocodetheplayers(A,B,C..etc)• tolistallthematchesthatneedtobeplayed• howtosystematicallyorganisethesematches• howtotabulatetheorderofplay• torememberthatplayerscannotplayontwotablesatonce.

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Handout1:Structuredproblems

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ACTIVITYB:COMPARESTRUCTUREDANDUNSTRUCTUREDPROBLEMS

Timeneeded:10minutes

Handout2containsunstructuredversionsofthesametasksthatwereusedinactivityA.

• Comparethelessstructuredversionsoftheproblemswiththestructuredversions.• Whatdecisionshavebeenlefttothestudents?• Whatpedagogicalissueswillarisewhenyoustarttouseunstructuredproblemslikethis?

Someimmediateissuesthatteachersraiseare:

• Unstructuredproblemsaremoredifficult.• Itismoredifficulttoplanalessonwiththeseproblems.• Studentsmaynotevenknowhowtogetstartedonthem.Willwethereforeneedto

structurethemanyway?• Studentswillnotnecessarilyusewhatwehavetaughtthem.• Ifweofferhelptooquickly,studentswillsimplydowhatwesayandnotthinkfor

themselves.• Studentswillgenerateagreatervarietyofapproachesandsolutions.• StudentsmayneedreassurancethatitisOKtotryadifferentapproachorreachadifferent

conclusion.Handout3containssomenotesonthesolutionstothethreeproblems.

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Handout2:Unstructuredversionsoftheproblems

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ACTIVITYC:CONSIDERSTRATEGIESFOROFFERINGHELP

Timeneeded:30minutes.

Teachersoftenfinditdifficulttoknowwhentogivehelpandwhentoleavestudentsstruggling.Iftheyintervenetooquickly,thenthestudentshavenochancetoexperiencewhatitisliketopursueanunfruitfulidea,ortopuzzleoutasolutionforthemselves.Iftheyintervenetooslowly,thenstudentsbecomefrustrated,boredanddisengaged.Handout4containssomepracticaladvicewhenusingunstructuredproblems.Considerthisadvicecarefully:

• Whichideasdoyounormallyfindmostdifficulttoimplement?Whyisthis?• Isthereanyotheradviceyouwouldaddtothislist?Writeyourownideasatthebottom.

Brunerusesthemetaphorofscaffoldingtodescribethestructuringthatateacherprovides(D.Wood,Bruner,&Ross,1976).Theteacherencouragesstudentstoasmuchastheyarecapableofunaidedandonlyprovidestheminimumofsupporttohelpthemsucceed.Thissupportmayinvolvereducingtheirchoices,drawingattentiontoimportantfeaturesthroughquestioning,orevenattimesdemonstratingwhattodo.Inhisworkwithyoungchildren,Wood(1988)categoriseddifferentlevelsofscaffolding,fromlessdirectivetomoredirective:givinggeneralverbaladvice,givingspecificverbalinstructions,breakingtheproblemdown,demonstratingasolution.Woodalsointroducedtworulesofcontingency:

"Anyfailurebyachildtosucceedinanactionafteragivenlevelofhelpshouldbemetbyanimmediateincreaseinhelporcontrol.Successbyachildthenindicatesthatanysubsequentinstructionshouldofferlesshelpthanthatwhichprecededthesuccess,toallowthechildtodevelopindependence."Wood(1988)

Theimportantideahereisthatscaffoldingshouldberemovedasthestudentbeginstocope,otherwiseitreinforcesdependency.

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Handout4: Practicaladviceforteachingproblemsolving

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ACTIVITYD:OBSERVEANDANALYSEALESSON

Timeneeded:30minutes.

Thetwovideoclipsshowstudentsworkingwiththeunstructuredversionsofthesameproblemsyouhaveworkedon.WatchMichelleusingtheOrganisingatabletennistournamentproblem.Asyouwatchthevideo,consider:

• Howdidtheteacherorganisethelesson?Whatphasesdiditgothrough?• Whatresourcesdidtheteacherhaveavailable,andwhenweretheseused?• Whywerestudentsexpectedtoworkinpairs/smallgroups?• Howdidtheteacherintroducetheproblemtostudents?• Whatdifferentapproacheswerebeingusedbystudents?• Howdidtheteachersupportthestudentsthatwerestruggling?• Howdidtheteacherencouragethesharingofapproachesandstrategies?• Whatdoyouthinkthesestudentswerelearning?

AfterwardsyoumayenjoywatchingthesecondvideoclipofJudith'slessonwheresheaskstheclasstodesignaboxtohold18sweets.

Michellebeginsthelessonbyintroducingtheproblemandalsobyexplaininghowthestudentsshouldworktogether.Thestudentsarethengiven3-4minutestowritetheirindividual,initialideasofhowtheywilltackletheproblem.Thishelpsthemtoformulatetheirideasandhavesomeideastocontributetothegroupdiscussions.Studentswerethengivenafewminutestodiscusstheirideas.Thevideoshowsstudentsgettingtogripswiththeproblemandtheconstraintsinvolved.Someforexamplerealisethat4tableswillnotbeneeded.Atthispoint,Michelleintroducestheresourcesthatareavailableforthemtouse.Sheemphasisesthattheyshouldnotrushtheproblemsolvingandthatsomemaynotgetfinished.Shesaysthattheimportantthingistothinkabouttheapproaches.Whilegroupswork,Michellefirstlistens,thenintervenestopromotedeeperthinking:

"You'vefoundoneapproach.You'vefoundthatitisnotgoingtoquitework,isit?Sowhatdoweneedtodofromthere?"

"Re-readtheproblem.Lookatthelasttwosentences."

Studentsuseawiderangeofrepresentationsandresourcestosolvetheproblem.Someusedtables,whileothersusedcounters.Thesemethodsaresharedinthefinalwholediscussion.

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ACTIVITYE: PLANALESSON,TEACHITANDREFLECTONTHEOUTCOMES

Timeneeded:

• 15minutesdiscussionbeforethelesson• 1hourforthelesson• 15minutesafterthelesson

Chooseoneofthethreeproblemsthatyoufeelwouldbeappropriateforyourclass.Discusshowyouwill:

• Organisetheclassroomandtheresourcesneeded.• Introducetheproblemtopupils.• Explaintopupilshowyouwantthemtoworktogether.• Challenge/assistpupilsthatfindtheproblemstraightforward/difficult.• Helpthemshareandlearnfromalternativeproblem-solvingstrategies.• Concludethelesson.

Ifyouareworkingonthismodulewithagroup,itwillbehelpfulifeachparticipantchoosesthesameproblem,asthiswillfacilitatethefollow-updiscussion.Nowyouhavetaughtthelesson,itistimetoreflectonwhathappened.

• Whatrangeofresponsesdidpupilshavetothiswayofworking?Didsomeappearconfident?Didsomeneedhelp?Whatsortofhelp?Whydidtheyneedit?

• Whatsupportandguidancedidyoufeelobligedtogive?Whywasthis?Didyougivetoomuchortoolittlehelp?

• Whatdifferentstrategiesdidpupilsuse?Sharetwoorthreedifferentexamplesofpupils'work.

• Whatdoyouthinkpupilslearnedfromthislesson?

Ifthereistime,youmayalsoliketowatchthevideosoftheteachersastheyreflectontheirownlessonswiththeTabletennisandSweetboxproblems.

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FURTHERREADING

TheseminaltextforaskingpupilstothinkmathematicallyMason,J.,Burton,L.andStacey,K.(1982)ThinkingMathematically,London:Addison-WesleyThebookthatinspiredsomuchoftheresearchintoproblemsolvingheuristics(orwhattodowhenyouare'stuck')Polya,G.(1957)HowtoSolveIt:ANewAspectofMathematicalMethod,(2ndEd)PenguinScience.REFERENCES

Wood,D.(1988).HowChildrenThinkandLearn.OxfordandCambridge,MA:Blackwell.Wood,D.,Bruner,J.,&Ross,G.(1976).Theroleoftutoringinproblemsolving.Journalofchild

psychologyandpsychiatry,17,89-100.