learning concepts through inquiry
TRANSCRIPT
©2010UniversityofNottingham 1
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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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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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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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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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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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.