learning concepts through inquiry

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LEARNINGCONCEPTSTHROUGHINQUIRY

IntroductionThisunitconsidershowtheprocessesofinquirybasedlearningmaybeintegratedintotheteachingofMathematicsandSciencecontent.Often,thesetwoaspectsoflearningarekeptseparate:weteachcontentasacollectionoffactsandskillstobeimitatedandmastered,and/orweteachprocessskillsthroughinvestigationsthatdonotdevelopincorporateimportantcontentknowledge.Theintegrationofcontentandprocessraisesmanypedagogicalchallenges.Theprocessesunderconsiderationhereare:observingandvisualising,classifyingandcreatingdefinitions,makingrepresentationsandtranslatingbetweenthem,findingconnectionsandrelationships,estimating,measuringandquantifying,evaluating,experimentingandcontrollingvariables.Assomehavepointedout,thesearedevelopmentsofnaturalhumanpowersthatweemployfrombirth(Millar,1994).Tosomeextent,weusethemunconsciouslyallthetime.Whenthesepowersareharnessedanddevelopedbyteacherstohelpstudentsunderstandtheconceptsofmathematicsandscience,studentsbecomemuchmoreengagedandinvolvedintheirlearning.Thisunithasmanyactivitieswithinit-toomanyforonesession.Itissuggestedthatthisunitisusedasamenu,fromwhichprofessionaldevelopmentproviderscanchoose.Itishowever,importantthatparticipantsaregivenanopportunitytotryoutsomeoftheseactivitiesintheirlessonsandtoreportbackontheoutcomes.

ActivitiesActivityA: Observingandvisualising..............................................................................................2ActivityB: Classifyinganddefining.................................................................................................4ActivityC: RepresentingandTranslating........................................................................................6ActivityD: Makingconnections......................................................................................................8ActivityE: Estimating....................................................................................................................10ActivityF: Measuringandquantifying.........................................................................................12ActivityG: Evaluatingstatements,resultsandreasoning............................................................14ActivityH: ExperimentingandControllingvariables....................................................................17ActivityI: Planalesson,teachitandreflectontheoutcomes.....................................................19Furtherreading................................................................................................................................20References.......................................................................................................................................20

Acknowledgement:ThemoduleshavebeencompiledforPRIMASfromprofessionaldevelopmentmaterialsdevelopedbytheShellCentreteamattheCentreforResearchinMathematicsEducation,UniversityofNottingham.ThisincludesmaterialadaptedfromImprovingLearninginMathematics©CrownCopyright(UK)2005bykindpermissionoftheLearningandSkillImprovementServicewww.LSIS.org.uk.

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ACTIVITYA:OBSERVINGANDVISUALISING

Timeneeded-30minutesTheprocessesofobservingandvisualisingarenaturalhumanpowersthatwehavefrombirth.Observationisprimarilyaboutwhatwecanseeandnoticedirectly,whereasvisualisationconcernswhatwecanimagineandtransformmentally,inour'mind'seye'.Thecontentionhereisthatthesepowersareoftenunder-usedinclassrooms,atleastpartlybecausewedon'tusetasksthatrequiretheuseofthesepowersfortheirsuccessfulcompletion.Theactivitiespresentedhereareintendedtobejustexamplesofthreewaysofharnessingstudents'powersofobservationandvisualisation.Theseareonlyexamples;alternativesareeasilyfoundatanylevelofdifficulty.Inthelefthandcolumnoftheworksheetweoffergenericdescriptionsoftheactivities,whileintherighthandcolumnweofferaspecificexample.Thesearediscussedbriefly,below.

• WorkonsomeoftheactivitiesonHandout1.• Shareyouobservationsandmentalimages:

-howdidyou'see'theobjectdifferently?-whatdidyounoticeorsingleoutforattention?-whatdidyoutrytomanipulatementally?

• Trytodevelopanactivityusingoneofthesetypesforuseinyourownclassroom.Trytodeviseexamplesthatforcestudentstoobservepropertiescarefully,andthatwillcreatediscussionaboutdefinitions.

• Tryoutyouractivityandreportbackonit.AlhambraTheAlhambratilingisacomplexrepeatingpatternmadefrommanydifferentshapes.Participantsmaybeaskedtosketchtheindividualtilesthatwentintoconstructingit.Twosmalltileswilldo,asshownbelow.Couldthepatternbemadefromonesmalltile?

CubeofcheeseAskparticipantstodescribealltheshapesthatthey'see'asthecheeseiscut.Initiallyasmalltriangleisformed,butthismaybeofanytype,dependingontheangleoftheknife.Aslargerandlargercutsaremade,participantsmaybesurprisedto'see'allkindsofquadrilaterals,pentagonsandhexagons.Theymaywanttosketchdiagramsandworkonthisfurtherastheydiscuss.Encouragethis,butonlyaftertryingtoworkmentally.

Suspensionbridgecables

Differentwaysofseeingleadtodifferentsequencesandalgebraicexpressions:

Youmayalsobeabletoseethediagramasthedifferenceoftwocubes:

1,7,19,.....

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Handout1:Observingandvisualisingactivities

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ACTIVITYB:CLASSIFYINGANDDEFINING

Timeneeded-30minutesClassificationanddefinitionclearlyplaykeyrolesinScienceandMathematics.Herewearenotonlyconcernedwithlearningclassificationsanddefinitionsdevisedbyothers,butalsowithstudentsengagingintheseprocessestogainanunderstandingofhowScientificandMathematicalconceptscomeabout.Intheseactivities,studentsexamineacollectionof'objects'carefully,andclassifythemaccordingtotheirdifferentattributes.Studentsselecteachobject,discriminatebetweenthatobjectandothersimilarobjects(whatisthesameandwhatisdifferent?)andcreateandusecategoriestobuilddefinitions.Thistypeofactivityispowerfulinhelpingstudentsunderstandwhatismeantbydifferenttermsandsymbols,andtheprocessthroughwhichtheyaredeveloped.

• WorkonsomeoftheactivitiesonHandout2.• Whatkindsof'objects'doyouaskstudentstoclassifyanddefineinyourclassroom?• Trytodevelopanactivityusingoneofthesetypesforuseinyourownclassroom.Tryto

deviseexamplesthatforcestudentstoobservethepropertiesofobjectscarefully,andthatwillcreatediscussionaboutdefinitions.

• Tryoutyouractivityandreportbackonitinalatersession.

Thetypesofactivityshownheremaybeextendedtoalmostanycontext.InMathematics,forexample,theobjectsbeingdecribed,definedandclassifiedcouldbenumerical,geometricoralgebraic.InSciencetheycouldbeorganismsorelements.Theactivityhereisforteacherstotrytoexploretherangeofpossibilities.

SimilaritiesanddifferencesIntheexamplesshown,studentsmay,forexample,decidethatthesquareistheoddoneoutbecauseithasadifferentperimetertotheothershapes(whichbothhavethesameperimeter);therectangleistheoddoneoutbecauseithasadifferentareatotheothersandsoon.Propertiesconsideredmayincludearea,perimeter,symmetry,angle,convexityetc.Inthesilhouettes,studentsmayconsidermanyaspects:wheretheanimalslive,howtheymove,reproduceetc.Participantsshouldtrytodevisetheirownexamples.PropertiesanddefinitionsNoneofthepropertiesbythemselvesdefinesthesquare.Itisinterestingtoconsiderwhatothershapesareincludedifjustonepropertyistaken.Forexample,whenthepropertyis"Twoequaldiagonals"thenallrectanglesandisoscelestrapeziaareincluded-butisthatallthecases?Takentwoatatime,thenresultsarenotsoobvious.Forexample,"Fourequalsides"and"fourrightangles"definesasquare,but"diagonalsmeetatrightangles"and"fourequalsides"doesnot(whatelsecouldthisbe?).CreatingandtestingdefinitionsParticipantsusuallywritearathervaguedefinitionof"polygon"or"bird"tobeginwith,suchas:"Ashapewithstraightedges"oran"animalthatflies".Theythenseethatthisisinadequateforthegivenexamples.Thiscausesthemtoredefinemorerigorously,like"aplanefigurethatisboundedbyaclosedpathorcircuit,composedofafinitesequenceofstraightlinesegments".Definingisadifficultarea,andstudentsshouldrealisethattherearecompetingdefinitionsforthesameidea(suchas"dimension",forexample).Classifyingusingtwo-waytablesTwo-waytablesarenottheonlyrepresentationthatmaybeused,ofcourse,andparticipantsmaysuggestothers.Venndiagramsandtreediagramsarejusttwoexamplesusedbothinscienceandmaths.

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Handout2:Classifyinganddefining

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ACTIVITYC:REPRESENTINGANDTRANSLATING

Timeneeded:20minutes

MathematicalandScientificconceptshavemanyrepresentations;words,diagrams,algebraicsymbols,tables,graphsandsoon.Itisimportantforstudentstolearnto'speak'theserepresentationsfluentlyandtolearntotranslatebetweenthem.Itishelpfultothinkofeachcellinthegridbelowasatranslationprocess.Sometranslationsaremorecommonthanothersinclassroomactivities.Forexample,weoftenaskstudentstomovebetweentablesandgraphs.Thisislabelledas'plotting'.

from\to words pictures tables graphs formulaewords pictures tables plotting graphs formulae

• Whichrepresentationsdoyouusemostofteninyourclassroom?• Whichtranslationprocessesdoyouemphasisemost?Whichreceivelessattention?• DiscusstheexamplesshowninHandout3.

Asparticipantsworkontheactivitiestheymaybegintorealisethatsomeofthesearelesscommonintheirclassroom.Somenotesoneachactivityaregivenbelow:JobtimesThewordsdescribeaninverseproportion,suchasthefollowing.Numberofpeople 1 2 3 4 5 6Timetakenin

hours24 12 8 6 4.8 4

RollercoasterAsuitablegraphisshownbelow.Itisinterestingtonotehowdifficultsomestudentsfindthis,particulalywhentheymisinterpretthegraphasapictureofthesituation.

Wordsandformulae

Studentsenjoytryingtoconstructtheseandmakingthemasdifficultaspossible!

TablesandgraphsThisparticularexamplefocusesongraphsketchingratherthangraphplotting.

Tournament

Thediagramshowsthestructureofthesituation.Therearen2-ncells.

PenguinsTheweightisproportionaltovolume,thendimensionalanalysisshouldsuggestthatweightisproportionaltothecubeoftheheightifthepenguinsaregeometricallysimilar.Thisturnsouttobeareasonableassumptionandanapproximatemodelis: w=20h3wherehistheheightinmetres,andwistheweightinkg.

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Handout3:Translatingbetweenrepresentations

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ACTIVITYD:MAKINGCONNECTIONS

Timeneeded:20minutes

TheactivityinHandout4isintendedtoencouragestudentstodiscussconnectionsbetweenverbal,numeric,spatialandalgebraicrepresentations.Forthefollowingactivity,participantsshouldworkinpairsorthrees.Theyshouldbeginbycuttingoutthecards.

• CutoutthesetofcardsonHandout4.• TakeitinturnstomatchCardSetA:algebraexpressionswiththeCardSetB:verbal

descriptions.Placepairsofcardsside-by-side,faceuponthetable.Ifyoufindcardsaremissing,createtheseforyourself.

• Next,matchCardsetC:tablestothecardsthatyouhavealreadymatched.Youmayfindthatatablematchesmorethanonealgebraexpression.Howcanyouconvinceyourselforyourstudentsthatthiswillalwaysbetrue,whateverthevalueforn?

• Next,matchCardsetD:areastothosecardsthathavealreadybeengroupedtogether.Howdothesecardshelpyoutoexplainwhydifferentalgebraexpressionsareequivalent?

• Discussthedifficultiesthatyourstudentswouldhavewiththistask.

Thefinalmatchingmaybemadeintoaposter,ashasbeendonehere.Thenextactivityencouragesparticipantstocomparetheirownthinkingwithanepisodeoflearningfromtheclassroom.Thestudentsonthe5minutevideocliparealllowattaining16-17yearsoldwhohavehadverylittleunderstandingofalgebrapreviously.

• Watchthevideoclip.• Whatdifficultiesdothestudentshavewhileworkingonthistask?• Howistheteacherhelpingstudents?

Finally,participantsmaybegintoconsiderhowthistypeofactivitymaybeappliedtorepresentationsthattheyteach.

• Deviseyourownsetofcardsthatwillhelpyourstudentstranslatebetweendifferent

representationsthatyouareteaching.

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Handout4: RepresentingandmakingconnectionsEachgroupofstudentsisgivenasetofcards.Theyareinvitedtosortthecardsintosets,sothateachsetofcardshaveequivalentmeaning.Astheydothis,theyhavetoexplainhowtheyknowthatcardsareequivalent.Theyalsoconstructforthemselvesanycardsthataremissing.Thecardsaredesignedtoforcestudentstodiscriminatebetweencommonlyconfusedrepresentations.

Swan,M.(2008),ADesignerSpeaks:DesigningaMultipleRepresentationLearningExperienceinSecondaryAlgebra.EducationalDesigner:JournaloftheInternationalSocietyforDesignandDevelopmentinEducation,1(1),article3.

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ACTIVITYE: ESTIMATING

Timeneeded:20minutes.

Estimationproblemsinvolvestudentsmakingassumptions,thenworkingwiththeseassumptionstobuildchainsofreasoning.Itisoftenthecasethat,individually,studentsfeelunabletocopewithsuchproblemsbutwhentheyworkcollaboratively,theyaresurprisedathowmuchknowledgetheycanbuildon.

• Inpairsorsmallgroups,worktogetheronthetreesproblemonHandout5.• Wheneachgrouphasproducedareasonedanswer,takeitinturnstoexplainyour

solutions,describingalltheassumptionsyouhavemade.• Inwhichsolutiondoyouhavemostconfidence?Whyisthis?

Thefollowingshowsjustoneapproachthatteachershaveadopted:

1. Estimatethenumberofteachersinthecountry.2. Estimatethesizeoftheaveragefamily.3. Estimatethevolumeofatypicalnewspaper.4. Assumingthateachfamilybuysonenewspaperperday,estimatethetotalvolumeofnewspaper

consumedperday.5. Estimatetheradiusandheightoftheuseablepartofasuitabletree.6. Calculatethevolumeofthetrunk.7. Assumingthatthetotalvolumeofthetrunkisconvertedintonewsprint,useyouranswersfrom(4)

and(6)toestimatetherequirednumberoftrees.

Thefollowingdatawassuppliedbytheforestrycommission,andmayprovideausefulindependentcheck:"Theexampleassumesthatthewholetreegoesforpaper.Inrealityonlythesmallerendwouldbeused.Approximately2.8kgofwoodwillmake1kgofnewsprint.1cubicmetreofwood,freshlyfelled,assuppliedtoapulpmill,weightsabout920kg.ThisisbasedontheSitkaspruceandistheaveragethroughouttheyear.Atthetimeoffellingattheageof55years,eachtreewillhaveavolumeof0.6cubicmetres,includingthebark.Thediameterat1.4metresfromthegroundwouldbe27cm."

• Makealistofestimationproblemsthatwouldbeaccessibletooneofyourclasses.• Discusshowyoumightorganisealessonbasedonanestimationproblem.

Apossiblelistofquestionsmightbe:• Howmuchdoyoudrinkinoneyear?• Howmanyteachersarethereinyourcountry?• Howlongwouldittakeyoutoreadoutallthenumbersfromonetoonemillion?Wouldthis

bedifferentindifferentlanguages?• Howmanypeoplecouldstandcomfortablyinyourclassroom?• Howmanytimesdoesaperson'shearbeatinoneyear?• Howmanyexercisebooksdoyoufillinyourschoolcareer?• Howmanypetdogsarethereinyourtown?

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Handout5 Estimating

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ACTIVITYF: MEASURINGANDQUANTIFYING

Timeneeded:20minutes.

Oursocietycreatesandusesmeasuresallthetime.Wecreatemeasuresforfundamentalconcepts(e.g.length,time,mass,gradient,speed,density)andmorecomplexsocialconstructs(e.g.academicability,wealth,inflation,jobperformance,qualityofeducation,sportingprowess,physicalbeauty).ScientistsandMathematiciansdevisemeasuresinordertoseekpatterns,relationshipsandlaws.Politiciansusemeasurestomonitorandcontrol.Alleducatedcitizensshouldrealisethatmanyofthesemeasuresareopentocriticismandimprovement.

• Whatkindsofmeasuredoyoumeetinyoureverydaylife?MakealistonHandout6.• Whatkindsofmeasuredoyourstudentsexperience?

OnHandout6,twotypesofactivityaresuggestedforstudents.

• Workonthemeasuringslopetasktogether.• Trytoarriveataconvincingexplanationastowhyheightofstep÷lengthofstepisthe

bettermeasureforslope.• Canyouthinkofotherexamplesofalternativemeasuresforthesameconcept?

Theratioheightofstep÷lengthofstepisbetterthanthedifferenceheightofstep-lengthofstepbecausetheratioisdimensionless.Thismeansthatifyougeometricallyenlargeastaircase,theratiowillnotchange,whereasthedifferencewill.Thefinalactivitysuggestsdevisingameasureforaneverydayphenomenon.Participantsmayliketostartthisbythinkingabout"compactness":

Overrecentyears,geographershavetriedtofindwaysofdefiningtheshapeofanareaofland.Inparticular,theyhavetriedtodeviseameasureof'compactness'.Youprobablyhavesomeintuitiveideaofwhat'compact'meansalready.Ontherightaretwoislands.IslandBismorecompactthanislandA.'Compactness'hasnothingtodowiththesizeoftheisland.Youcanhavesmall,compactislandsandlargecompactislands.

• Drawsomeshapesandputtheminorderofcompactness.• Trytoagreewhatismeantbytheterm.• Isarea÷perimeteragoodmeasureofcompactness?Whyorwhynot?• Trytodeviseseveralwaysofmeasuringcompactness.Trytomakeyourmeasuresrange

from0to1,where1isgiventoashapethatisperfectlycompact.• Afterwards,compareyourideaswiththoseusedbygeographersonHandout7.

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• Finally,considerothereverydayphenomenaandconsiderhowyouwouldmeasurethem(returntoHandout6).

Handouts6and7 Measuringandquantifying

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Reference:Selkirk,K(1982)PatternandPlace-AnIntroductiontotheMathematicsofGeography,CambridgeUniversityPress.

ACTIVITYG:EVALUATINGSTATEMENTS,RESULTSANDREASONING

Studentsthatareactivelylearningareconstantlychallenginghypothesesandconjecturesmadebyothers.Theactivitiesconsideredherearealldesignedtoencouragethiskindofbehaviour.AskparticipantstoworktogetheringroupsoftwoorthreeusingtheactivityofHandout7.Inthisactivity,youaregivenacollectionofstatements.

• Decideonthevalidityofeachstatementandgiveexplanationsforyourdecisions.Yourexplanationswillinvolvegeneratingexamplesandcounterexamplestosupportorrefutethestatements.

• Inaddition,youmaybeabletoaddconditionsorotherwiserevisethestatementssothat

theybecome‘alwaystrue’.

• Createsomestatementsthatwillcreateastimulatingdiscussioninyourclassroom.

Thiskindofactivityisverypowerful.Thestatementsmaybepreparedtoencouragestudentstoconfrontanddiscusscommonmisconceptionsorerrors.Theroleoftheteacheristopromptstudentstoofferjustifications,examples,counterexamples.Forexample:Payrise:"OKyouthinkitissometimestrue,dependingonwhatMaxandJimearn.CanyougivemeacasewhereJimgetsthebiggerpayrise?Canyougivemeanexamplewheretheybothgetthesamepayrise?"Areaandperimeter:

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"Canyougivemeanexampleofacutthatwouldmaketheperimeterbiggerandtheareasmaller?""SupposeItakeabiteoutofthistriangularsandwich.Whathappenstoitsareaandperimeter?"Rightangles.Canyouprovethisisalwaystrue?BiggerfractionsYouthinkthisisalwaystrue?Canyoudrawmeadiagramtoconvincemethatthisisso?Whathappenswhenyoustartwithafractiongreaterthanone?

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Handout7:Always,sometimesornevertrue?

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ACTIVITYH:EXPERIMENTINGANDCONTROLLINGVARIABLES

Timeneeded:40minutes.

Twoactivitiesarepresentedhere.Oneinvolvestheplanningofanexperiment,theotherinvolvesacomputerappletthatispresentedwiththisunit.StartbydiscussingthefirsttwosituationsonHandout8.

• ChooseoneofthescientificquestionsshowninDevisingafairtest.• Workontheexperimentaldesigninasmallgroup.• Ofteninscienceclassrooms,theteacherdesignstheexperimentsandstudentscarrythem

out.Handingovertheexperimentaldesigndecisionspresentsmanychallengesforbothteachersandstudents.Forexample,studentsmayaskforequipmentthatyoudonothavereadilyavailable.Whatotherchallengesarethere?Makealist.

NowaskparticipantstoconsiderthefinalproblemBodyMassindex.

• WorkontheBodyMassIndexprobleminpairs,usingthecomputerapplet.• Notedownthemethodyouadopt.• Nowwatchthevideoclipshowingalessonwithstudents.

o Howdidtheteacherorganisethelesson?Whatphasesdiditgothrough?o Whydoyouthinksheorganiseditthisway?o Howdidtheteacherintroducetheproblemtostudents?o Whatdifferentapproacheswerebeingusedbystudents?o Howdidtheteachersupportthestudentsthatwerestruggling?o Howdidtheteacherencouragethesharingofapproachesandstrategies?o Whatdoyouthinkthesestudentswerelearning?

Itiseasytofindtheboundariesatwhichsomeonebecomesunderweight/overweight/obeseifonevariableisheldconstantwhiletheotherisvariedsystematically.Theboundariesoccurat: BMIUnderweight Below18.5Idealweight 18.5-24.9Overweight 25.0-29.9Obesity 30.0andAbove

Inordertofindouthowthecalculatorworks,itisbettertoforgetrealisticvaluesforheightandweightandsimplyholdonevariableconstantwhilechangingtheothersystematically.Forexample,ifstudentsholdtheheightconstantat2metres(notworryingifthisisrealistic!),thentheywillobtainthefollowingtableand/orgraph:Weight(kg) 60 70 80 90 100 110 120 130BMI 15 17.5 20 22.5 25 27.5 30 32.5 Underweight Idealweight Overweight Obese FromthisitcanbeseenthatthereisaproportionalrelationshipbetweenweightandBMI.(Ifyoudoubleweight,youdoubleBMI;HereBMI=Weight/4)

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Handout8: Experimentingandcontrollingvariables

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

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ACTIVITYI: PLANALESSON,TEACHITANDREFLECTONTHEOUTCOMES

Timeneeded:

• 15minutesdiscussionbeforethelesson• 1hourforthelesson• 15minutesafterthelesson

Chooseoneoftheproblemsinthisunitthatyoufeelwouldbeappropriateforyourclass.Discusshowyouwill:

• Organisetheclassroomandtheresourcesneeded.• Introducetheproblemtostudents.• Explaintostudentshowyouwantthemtoworktogether.• Challenge/assiststudentsthatfindtheproblemstraightforward/difficult.• Helpthemshareandlearnfromalternativeproblem-solvingstrategies.• Concludethelesson.

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

• Whatrangeofresponsesdidstudentshavetothetask?Didsomeappearconfident?Didsomeneedhelp?Whatsortofhelp?Whydidtheyneedit?

• Whatdifferentscientificprocessesdidstudentsuse?Sharetwoorthreedifferentexamplesofstudents'work.

• Whatsupportandguidancedidyoufeelobligedtogive?Whywasthis?Didyougivetoomuchortoolittlehelp?

• Whatdoyouthinkstudentslearnedfromthislesson?

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FURTHERREADING

Swan,M(2005)ImprovingLearninginMathematics:ChallengesandStrategies,DepartmentforEducationandskillsanddowloadablefrom:

http://www.nationalstemcentre.org.uk/elibrary/resource/1015/improving-learning-in-mathematics-challenges-and-strategies

REFERENCES

Millar,R.(1994).Whatis'scientificmethod'andcanitbetaught?InR.Levinson(Ed.),TeachingScience(pp.164-177).London:Routledge.

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

psychologyandpsychiatry,17,89-100.