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(c) Centre for Research in Mathematics Education, University of Nottingham 2010 1 TACKLING UNSTRUCTURED PROBLEMS ‘Do I stand back and watch, or intervene and tell them what to do?’ Introduction In most mathematics and science classrooms, students are provided with structured tasks and are told precisely which techniques to deploy. Students learn by following instructions. Problems and situations that arise in the world are not usually like this. Rather than being exercises in the use of a particular skill or concept, real-world problems require students to make simplifications, model situations, choose appropriate knowledge and processes from their 'toolkit', and test whether their solution is "good enough" for the purpose in hand. It seems logical that if students are to learn to use their skills autonomously in their future lives, they will need some opportunities to work on less structured problems in their classrooms. This unit compares structured and unstructured versions of problems and considers the demands and challenges unstructured problems present to students and teachers. Activities Activity A: Revising structured problems ....................................................................................... 2 Activity B: Compare structured and unstructured problems ......................................................... 4 Activity C: Consider strategies for offering help ............................................................................. 6 Activity D: Observe and analyse a lesson ....................................................................................... 8 Activity E: Plan a lesson, teach it and reflect on the outcomes ...................................................... 9 Further reading ................................................................................................................................ 10 References ....................................................................................................................................... 10 Acknowledgement: This material is adapted for PRIMAS from: Swan, M; Pead, D (2008). Professional development resources. Bowland Maths Key Stage 3, Bowland Trust/ Department for Children, Schools and Families. Available online in the UK at: http://www.bowlandmaths.org.uk. It is used here by permission of the Bowland Trust.

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Page 1: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(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.

Page 2: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 2

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.

Page 3: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 3

Handout1:Structuredproblems

Page 4: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 4

ACTIVITYB:COMPARESTRUCTUREDANDUNSTRUCTUREDPROBLEMS

Timeneeded:10minutes

Handout2containsunstructuredversionsofthesametasksthatwereusedinactivityA.

• Comparethelessstructuredversionsoftheproblemswiththestructuredversions.• Whatdecisionshavebeenlefttothestudents?• Whatpedagogicalissueswillarisewhenyoustarttouseunstructuredproblemslikethis?

Someimmediateissuesthatteachersraiseare:

• Unstructuredproblemsaremoredifficult.• Itismoredifficulttoplanalessonwiththeseproblems.• Studentsmaynotevenknowhowtogetstartedonthem.Willwethereforeneedto

structurethemanyway?• Studentswillnotnecessarilyusewhatwehavetaughtthem.• Ifweofferhelptooquickly,studentswillsimplydowhatwesayandnotthinkfor

themselves.• Studentswillgenerateagreatervarietyofapproachesandsolutions.• StudentsmayneedreassurancethatitisOKtotryadifferentapproachorreachadifferent

conclusion.Handout3containssomenotesonthesolutionstothethreeproblems.

Page 5: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 5

Handout2:Unstructuredversionsoftheproblems

Page 6: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 6

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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(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 7

Handout4: Practicaladviceforteachingproblemsolving

Page 8: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 8

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.

Page 9: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 9

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.

Page 10: TACKLING UNSTRUCTURED PROBLEMS · 2017-10-29 · Mason, J., Burton, L. and Stacey, K. (1982) Thinking Mathematically, London: Addison-Wesley The book that inspired so much of the

(c)CentreforResearchinMathematicsEducation,UniversityofNottingham2010 10

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.


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