acer research conference proceedings (2010)

Conference Proceedings

Contents

Foreword� v

Keynote�papers� ProfessorDavidClarke 3

Speaking in and about mathematics classrooms internationally: The technical vocabulary of students and teachers.

MrPhilDaro 8Standards, what’s the difference?: A view from inside the development of the Common Core State Standards in the occasionally United States.

ProfessorKayeStacey 17Mathematics teaching and learning to reach beyond the basics.

ProfessorPaulErnest 21The social outcomes of school mathematics: Standard, unintended or visionary?

Concurrent�papers� ProfessorRobynJorgenson 27

Issues of social equity in access and success in mathematics learning for Indigenous students.

ProfessorTomLowrie 31Primary students’ decoding mathematics tasks: The role of spatial reasoning.

ProfessorJohnPegg 35Promoting the acquisition of higher order skills and understandings in primary and secondary mathematics.

AssociateProfessorRosemaryCallingham 39Mathematics assessment in primary classrooms: Making it count.

DrDavidLeigh-Lancaster 43The case of technology in senior secondary mathematics: Curriculum and assessment congruence?

AssociateProfessorJoanneMulligan 47Reconceptualising early mathematics learning.

ProfessorPeterSullivan 53Learning about selecting classroom tasks and structuring mathematics lessons from students.

MrRossTurner 56Identifying cognitive processes important to mathematics learning but often overlooked.

AssociateProfessorRobertReeve 62Using mental representations of space when words are unavailable: Studies of enumeration and arithmetic in Indigenous Australia.

ProfessorMerrilynGoos 67Using technology to support effective mathematics teaching and learning: What counts?

DrShelleyDole 71Making connections to the big ideas in mathematics: Promoting proportional reasoning.

DrSueThomson 75Mathematics learning: What TIMSS and PISA can tell us about what counts for all Australian students.

Poster�presentations� 81

Conference�program� 85

Crown�Conference�Centre�map�and�floorplan� 89

Conference�delegates� 93

Research Conference 2010

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Research Conference 2010 Planning CommitteeProfessorGeoffMastersCEO,ConferenceConvenor,ACER

DrJohnAinleyDeputyCEOandResearchDirectorNationalandInternationalSurveys,ACER

ProfessorKayeStaceyProfessorMathematicsEducation,UniversityofMelbourne

DrDavidLeigh-LancasterMathematicsManager,VictorianCurriculumandAssessmentAuthority

MrRossTurnerPrincipalResearchFellow,ACER

MsKerry-AnneHoadDirectorACERInstitute,ACER

MsLyndaRosmanManagerProgramsandProjects,ACERInstitute,ACER

Copyright©2010AustralianCouncilforEducationalResearch

19ProspectHillRoadCamberwellVIC3124AUSTRALIA

www.acer.edu.au

ISBN978-0-86431-958-6

DesignandlayoutbyStaceyZassofPage12andACERProjectPublishing

EditingbyCarolynGlascodineandKerry-AnneHoad

PrintedbyPrintImpressions

Foreword

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Geoff�MastersAustralian Council for Educational Research

GeoffMastersisChiefExecutiveOfficerandamemberoftheBoardoftheAustralianCouncilforEducationalResearch(ACER)–roleshehasheldsince1998.

HehasaPhDineducationalmeasurementfromtheUniversityofChicagoandhaspublishedwidelyinthefieldsofeducationalassessmentandresearch.

ProfessorMastershasservedonarangeofbodies,includingtermsasfoundingPresidentoftheAsia-PacificEducationalResearchAssociation;PresidentoftheAustralianCollegeofEducators;ChairoftheTechnicalAdvisoryCommitteefortheInternationalAssociationfortheEvaluationofEducationalAchievement(IEA);ChairoftheTechnicalAdvisoryGroupfortheOECD’sProgrammeforInternationalStudentAssessment(PISA);memberoftheBusinessCouncilofAustralia’sEducation,SkillsandInnovationTaskforce;memberoftheAustralianNationalCommissionforUNESCO(andChairoftheCommission’sEducationNetwork);andmemberoftheInternationalBaccalaureateResearchCommittee.

Hehasundertakenanumberofreviewsforgovernments,includingareviewofexaminationproceduresintheNewSouthWalesHigherSchoolCertificate(2002);aninvestigationofoptionsfortheintroductionofanAustralianCertificateofEducation(2005);anationalreviewofoptionsforreportingandcomparingschoolperformances(2008);andareviewofstrategiesforimprovingliteracy,numeracyandsciencelearninginQueenslandprimaryschools(2009).

ProfessorMasterswastherecipientoftheAustralianCollegeofEducators’2009CollegeMedalinrecognitionofhiscontributionstoeducation.

Research Conference 2010isthefifteenthnationalResearchConference.Throughourresearchconferences,ACERprovidessignificantopportunitiesatthenationallevelforreviewingcurrentresearch-basedknowledgeinkeyareasofeducationalpolicyandpractice.Aprimarygoaloftheseconferencesistoinformeducationalpolicyandpractice.

Research Conference 2010bringstogetherkeyresearchers,policymakersandteachersfromabroadrangeofeducationalcontextsfromaroundAustraliaandoverseas.Theconferencewillexploretheimportantthemeofteachingandlearningmathematics.Theconferencewilldrawtogetherresearch-basedknowledgeabouteffectiveteachingandlearningofmathematicsandexploreapproachestoteachingthatdevelopthemathematicalproficiencyofstudentsandcatchtheirinterestinmathematicsfromtheearlyyearsthroughtopost-compulsoryeducation.

Wearesurethatthepapersanddiscussionsfromthisresearchconferencewillmakeamajorcontributiontothenationalandinternationalliteratureanddebateonkeyissuesrelatedtotheeffectiveteachingandlearningofmathematics.

WewelcomeyoutoResearchConference2010,andencourageyoutoengageinconversationwithotherparticipants,andtoreflectontheresearchanditsconnectionstopolicyandpractice.

ProfessorGeoffNMastersChiefExecutiveOfficer,ACER


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Keynote papers

Teaching�Mathematics?�Make�it�count:�What�research�tells�us�about�effective�teaching�and�learning�of�mathematics

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Speakinginandaboutmathematicsclassroomsinternationally:Thetechnicalvocabularyofstudentsandteachers

David�ClarkeUniversity of Melbourne

DavidClarkeisaProfessorofEducationandtheDirectoroftheInternationalCentreforClassroomResearch(ICCR)attheUniversityofMelbourne.Overthelast15years,ProfessorClarke’sresearchactivityhascentredoncapturingthecomplexityofclassroompracticethroughaprogramofinternationalvideo-basedclassroomresearch.TheICCRisuniqueinthefacilitiesitoffersforthemanipulationandanalysisofclassroomdataandprovidesthefocusforcollaborativeactivitiesamongresearchersfromChina,theCzechRepublic,Germany,HongKong,Israel,Japan,Korea,NewZealand,Norway,thePhilippines,Portugal,Singapore,SouthAfrica,Sweden,theUnitedKingdomandtheUnitedStatesofAmerica.UnderProfessorClarke’sdirectiontheICCRhasdevelopedasystemforweb-mediated,secure,high-speeddataentry,retrievalandanalysisonaninternationalscale(videoPortal).Othersignificantresearchhasaddressedteacherprofessionallearning,metacognition,problem-basedlearning,andassessment(particularlytheuseofopen-endedtasksforassessmentandinstructioninmathematics).Currentresearchactivitiesinvolvemulti-theoreticresearchdesigns,cross-culturalanalysesandthechallengeofresearchsynthesisineducation.ProfessorClarkehasover120researchpublications,including8books,35bookchapters,41refereedjournalarticles,and39refereedpapersinconferenceproceedings.

AbstractThispresentationtakespatternsoflanguageuseastheentrypointfortheconsiderationofdiscoursesinandaboutthemathematicsclassroom.Thesepatternsoflanguagetaketheformofdiscoursesperformedwithinmathematicsclassroomsaroundtheworldandamongtheinternationalmathematicseducationcommunityaboutthemathematicsclassroom.Cross-culturalcomparisonsrevealhowdiscoursesinandaboutthemathematicsclassroomhavedevelopedindifferentcultures.ResearchisusedtoexploretheroleofspokenlanguageinmathematicsclassroomssituatedinAsianandWesterncountries.Inconceptualisingeffectivelearning,researchers,teachersandcurriculumdevelopersneedtolocateproficiencywithmathematicallanguagewithintheirframeworkofvaluedlearningoutcomes.Further,differentcultures,employingdifferentlanguages,havechosentonameandthereforeprivilegedifferentclassroomactivities.Researchisreportedintohowlanguageisandmightbeusedtodescribetheeventsofmathematicsclassroomsindifferentcultures.Researchandtheorisingundertakeninandaboutthosemathematicsclassroomsmustbesensitivetotheparticipants’conceptionsofclassroompractice,asperformedinclassroomdiscourseandasexpressedintheprofessionaldiscourseofmathematicseducatorsinthosecommunities.

Presentation summaryClassroomdiscourse(andprofessionaldiscourseaboutclassrooms)isaformofsocialperformanceundertakenwithinaffordancesandconstraintsthatcanbebothculturalandlinguistic.Thenatureofthesediscourses,asperformedinmathematicsclassrooms,providesakeyindicatorofpedagogicalprinciplesunderlyingclassroompracticeandthetheoriesoflearningonwhichtheseprinciplesareimplicitlyfounded.Thediscoursesaboutmathematicsclassroomsgiveexpressiontothesepedagogicalprinciplessometimesexplicitlyandsometimesthroughembeddingprivilegedformsofpracticeinthenamingconventionsbywhichthemathematicsclassroomisdescribed.Fromresearchundertakeninclassroomssituatedindifferentcultures,itappearsthatbothmathematicaldiscourseandprofessionaldiscoursetakedifferentformsandaredifferentlyvaluedindifferentcommunities.Thispresentationdrawsonandconnectsresearchintothesetwodiscourses.

The spoken mathematics studyResearchwasconductedintothesituateduseofmathematicallanguageinselectedmathematicsclassroomsinternationally.Themajorconcernofthisstudywastodocumenttheopportunityprovidedtostudentsineachclassroomfortheoralarticulationoftherelativelysophisticatedmathematicaltermsthatformedtheconceptualcontentofthelessonandtodistinguishoneclassroomfromanotheraccordingtohowsuchstudentmathematicaloralitywasaffordedorconstrainedinbothpublicandprivateclassroomcontexts.

Thisresearchwasundertakenasasub-projectwithintheLearner’sPerspectiveStudy,inwhichdatagenerationused


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threevideocameras,supplementedbythereconstructiveaccountsofclassroomparticipantsobtainedinpost-lessonvideo-stimulatedinterviews.Thecompleteresearchdesignhasbeendetailedelsewhere(Clarke,2006).Fortheanalysisreportedhere,theessentialdetailsrelatetothestandardisationoftranscriptionandtranslationprocedures.Sincethreevideorecordsweregeneratedforeachlesson(teachercamera,studentcameraandwholeclasscamera),itwaspossibletotranscribethreedifferenttypesoforalinteractions:(i)wholeclassinteractions,involvingutterancesforwhichtheaudiencewasallormostoftheclass,includingtheteacher;(ii)teacher–studentinteractions,involvingutterancesexchangedbetweentheteacherandanystudentorstudentgroup,notintendedtobeaudibletothewholeclass;and(iii)student–studentinteractions,involvingutterancesbetweenstudents,notintendedtobeaudibletothewhole

class.Allthreetypesoforalinteractionsweretranscribed,althoughtype(iii)interactionscouldonlybedocumentedfortheselectedfocusstudentsineachlesson.Wherenecessary,alltranscriptswerethentranslatedintoEnglish.

Theanalysisdeterminedthenumberofutterancesoccurringinwholeclassandteacher–studentinteractionsinasequenceoffivelessonsfromeachoftheclassroomsstudied(atotalof105lessonsfrom21classroomsinBerlin,HongKong,Melbourne,SanDiego,Seoul,Shanghai,SingaporeandTokyo),togetherwiththefrequencyofpublicstatementofmathematicaltermsand,inaseparateanalysis,thenumberofutterancesandspokenmathematicaltermsinthecontextofstudent–student(ratherthanpublic)interactions.Anutterancewastakentobeasingle,continuousoralcommunicationofanylengthbyanindividualorgroup(choral).Privatestudent–studentinteractionsweredistinguishedfrom

wholeclassorteacher–studentinteractions,bothofwhichwereconsideredtobepublicfromthepointofviewofthestudent.

Theaveragenumberofpublicutterancesperlessonprovidesanindicationofthepublicoralinteractivityofaparticularclassroom.Figure1distinguishesutterancesbytheteacher(lightgrey),individualstudents(black)andchoralresponsesbytheclass(e.g.inSeoul)oragroupofstudents(e.g.inSanDiego)(darkgrey).Anyteacher-elicited,publicutterancespokensimultaneouslybyagroupofstudents(mostcommonlybyamajorityoftheclass)wasdesignateda‘choralresponse’.Lessonlengthvariedbetween40and45minutesandthenumberofutteranceshasbeenstandardisedto45minutes.EachbarinFigure1representstheaverageoverfivelessonsforthatclassroom.Figure2showsthenumberofpubliclyspokenmathematicalterms(asdefinedearlier)

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Figure�1:Averagenumberofpublicutterancesperlessoninwholeclassandteacher–studentinteractions(publicoralinteractivity)


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perlesson,averagedoverfivelessonsforeachclassroom.

Theclassroomsstudiedcanbealsodistinguishedbytheusemadeofthechoralrecitationofmathematicaltermsorphrasesbytheclass.Thisrecitationincludedbothchoralresponsetoateacherquestionandthereadingaloudoftextpresentedontheboardorinthetextbook.Themoststrikingdifferencebetweenfirstandsecondstageanalyses(Figures1and2)wasthereversaloftheorderofclassroomsaccordingtowhetheroneconsiderspublicoralinteractivity(StageOne)ormathematicalorality(StageTwo).

Inconsideringstudent-studentutterances,onlyfocusstudents’‘private’utterancescouldberecorded.TheclassroomsinShanghaiandSeoulwerecharacterisedbythealmostcompleteabsenceofthisformofinteraction.Frequencycountswereconstructed

forbothpublicandprivateOralInteractivityandMathematicalOralityandexpressedasperfocusstudentperlesson,effectivelyaveragedoverthespokencontributionsofatleast10studentsperclassroom.Detailedfindingsarereportedelsewhere(e.g.Clarke&Xu,2008).

Itisclearthatsomemathematicsteachersvaluedspokenmathematicsandsomedidnot.Someteachersorchestratedthepublicrehearsalofspokenmathematics,butdiscouragedprivate(student-student)talk(e.g.Shanghai1,2and3),whileotherteachersutilisedstudent–studentmathematicalconversationsasakeyinstructionaltool(e.g.SanDiego2andMelbourne1).Ifthegoalofclassroommathematicalactivitywasfluencyandaccuracyintheuseofwrittenmathematics,thentheteachermayaccordlittleprioritytostudentsdevelopinganyfluencyin

spokenmathematics(e.g.Seoul1,2and3).Ontheotherhand,iftheteachersubscribestotheviewthatstudentunderstandingresidesinthecapacitytobothjustifyandexplaintheuseofmathematicalprocedures,inadditiontotechnicalproficiencyincarryingoutthoseproceduresinsolvingmathematicsproblems,thenthenurturingofstudentproficiencyinthespokenlanguageofmathematicswillbeprioritised,bothforitsownsakeasavaluedskillandalsobecauseofthekeyrolethatlanguageplaysintheprocesswherebyknowledgeisconstructed.DespitethefrequentlyassumedsimilaritiesofpracticeinclassroomscharacterisedasAsian,differencesinthenatureofstudents’publiclyspokenmathematicsinclassroomsinSeoul,HongKong,Shanghai,SingaporeandTokyowerenon-trivialandsuggestdifferentinstructionaltheoriesunderlyingclassroompractice.

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Figure�2:Averagenumberofkeymathematicaltermsperlessoninpublicutterances(wholeclassandteacher–studentinteractions)(mathematicalorality)


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The international classroom Lexicon ProjectTheLexiconProjectisbasedonthepremisethattheinternationaldominanceofEnglishrunstheriskofdenyingresearchers,theoreticiansandpractitionersaccesstomanysophisticated,technicalclassroom-relatedtermsinlanguagesotherthanEnglish,whichmightotherwisecontributesignificantlytoourunderstandingofclassroominstructionandlearning.Theintendedproductofthisresearchisa‘ClassroomLexicon’ofsuchterms,withEnglishdefinitionsanddescriptivedetail,supportedbyvideoexemplars.Suchavideo-illustratedlexiconhasthepotentialtobeamajorresourceinteacherpre-serviceandin-serviceprogramsandtooffernewinsightstoclassroomresearchers.Thelexiconisproducedbyface-to-facenegotiationwithresearchersfrommorethan10countries,throughthecollaborativecodingofaselectionofvideomaterialofmathematicslessonsdrawnfromclassroomsinCeskyBudejovice,HongKong,Melbourne,SanDiego,Shanghai,TokyoandUppsala.Theparticularlessonswerechoseninconsultationwithlocalresearchersineachcountrytoprovideawidevariety

ofdifferentclassroomactivitiesinordertostimulateparticipants’recallofthelargestpossiblenumberofpedagogicalterms.

Itmightbeexpectedthattheinternationalisationofthemathematicseducationcommunitywouldaffordanexpansivere-conceptionofthepracticeofmathematicsteachingreflectiveofthewidediversityofclassroompracticesfoundinmathematicsclassroomsaroundtheworld.Ironically,internationalisationhasstrengthenedtheestablishmentofEnglishasthelinguafrancaoftheinternationalmathematicseducationcommunityandtherebyrestrictedinternationaluseofsomeofthesubtleandsophisticatedconstructsbywhichmathematicsteachersandteachereducatorsinnon-Englishspeakingcountrieswoulddescribeandevaluatethepracticesoccurringintheirmathematicsclassrooms.

Ifanactivityisnamed,itcanberecognisedanditbecomespossibletoask‘howwellisitdone?’and‘howmightitbedonebetter?’Notonlyisanunnamedactivitylessaccessibleforresearchanalysis,butpractisingteachersaredeniedrecognitionofanactivitythatatleastoneculturefeelsissufficientlyimportanttohavebeengivenaspecificname.An

unnamedactivitywillbeabsentfromanycatalogueofdesirableteacheractionsandconsequentlydeniedspecificpromotioninanyprogramofmathematicsteachereducation.Actionsconsideredasessentialcomponentsofthemathematicsteacher’srepertoireinonecountry–forexample,mise en commun(France),pudian(China),ucitelská ozvena(CzechRepublic)ormatome(Japan)–maybeentirelyabsentfromanycatalogueofaccomplishedteachingpracticesinEnglish.Yeteachofthesesamepedagogicalactivitiesmaywellrewardindependentresearch,offeringnovelinstructionalandlearningopportunities(see,forexample,Shimizu,2008).

Mise en commun–awhole-classactivityinwhichtheteacherelicitsstudentsolutionsforthepurposeofdrawingonthecontrastingapproachestosynthesiseandhighlighttargetedkeyconcepts.

Pudian–anintroductoryactivityinwhichtheteacherelicitsstudentpriorknowledgeandexperienceforthepurposeofconstructingconnectionstothecontenttobecoveredinthelesson.

Ucitelská ozvena–the‘teacher’secho’whentheteacher

Figure�3:Videostimuluslayout(keyelementsare:threesynchronizedcameraviews–teachercamera,wholeclasscamera,studentcamera;classroomdialogueinEnglishsubtitles;timecode)


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reformulatesastudent’sanswertoincreaseitsclarityormathematicalcorrectness;ideally,withoutappropriatingthestudent’sintellectualownershipoftheresponse.

Matome–ateacher-orchestrateddiscussion,drawingtogetherthemajorconceptualthreadsofalessonorextendedactivity–mostcommonlyasummativeactivityattheendofthelesson.

We,asresearchers,selectourtheoreticaltoolsbecausetheactionsandoutcomestheyprivilegeresonatewitheducationalvaluesthatwealreadyhold.Theseeducationalvaluesfindtheirembodimentintheformsofclassroomactivitythatourculturehaschosentoname.Thisreproductiveprocesscanonlyamplifyourpre-existingassumptionsregardingwhatistobevaluedandwhatistobediscarded.Research-basedadvocacyofinstructionalpracticerunstheriskofonlyentrenchingthevisionoftheclassroomenshrinedintheresearcher’slanguageandculture.Languagedoesnotjustmediatetheresearcher’scategorisationofwhatoccursintheclassroom.Languagewastherebeforeus,determiningwhichclassroomactivitiesareconceptualisedandenactedbytheparticipants.Further,thetheoriesweconstructareconstrainedtothoseconstructsandrelationshipswearecapableofnaming.Andour‘evidence-based’instructionaladvocacyreproducesthischainofcompoundedconstraints,leadingustoignoreother,potentiallyeffective,instructionalalternatives.

Summative remarksTheprofessionaldiscourseoftheinternationalmathematicseducationcommunityisconstrainedbythedominanceofEnglish.Theclassroomsexperiencedanddescribedbyteachersandresearchersspeakingnon-English

languagesaredifferentclassrooms.Inthesamewaythatthedifferentialpromotionoffluencyinspokenmathematicsindifferentclassroomsaroundtheworldenactsadifferentclassroommathematics,teachers,othereducators,andresearchersindifferentcountrieshaveattheirdisposalverydifferentlinguistictoolsbywhichtoconceptualise,theoriseabout,andresearchthemathematicsclassroom.Ourcapacitytostudy,understandandenactclassroompracticemustbeenhancedratherthanconstrainedbyourgrowinginternationalisation.

ReferencesClarke,D.J.(2006).TheLPSresearch

design.InD.J.Clarke,C.Keitel&Y.Shimizu(Eds.),Mathematics Classrooms in Twelve Countries: The Insider’s Perspective,pp.15–37.Rotterdam:SensePublishers.

Clarke,D.J.,&Xu,L.H.(2008).DistinguishingbetweenmathematicsclassroomsinAustralia,China,Japan,KoreaandtheUSAthroughthelensofthedistributionofresponsibilityforknowledgegeneration:Publicoralinteractivityversusmathematicalorality.ZDM – The International Journal in Mathematics Education, 40(6),963–981.

Shimizu,Y.(2006).Howdoyouconcludetoday’slesson?Theformandfunctionsof‘Matome’inmathematicslessons.InClarke,D.,Emanuelsson,J.,Jablonka,E.,&AhCheeMok,I.(Eds.),(2006).Making Connections: Comparing mathematics classrooms around the world.Rotterdam:SensePublishers.


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Philip�DaroChair, Common Core Standards Mathematics Workgroup USA

Currently,PhilDaroChairstheState-ledCommonCoreStandardsMathematicsWorkgroupintheUSAwhichisdraftingcommonCollegeandCareerReadinessStandardsonbehalfof48USStatesandwasamemberoftheleadwritingteamfortheCommonCoreStateStandards.

PhilisaSeniorFellowforMathematicsforAmerica’sChoicewherehefocusesonprogramsforstudentswhoarebehindandalgebraforall;healsodirectsthepartnershipofUniversityofCalifornia,StanfordandotherswithSanFranciscoUnifiedSchoolDistrictfortheStrategicEducationResearchPartnership(SERP),withafocusonmathematicsandsciencelearningamongstudentslearningEnglishordevelopingacademicEnglish,developsresearchagendaandprojectswhichaddressprioritiesidentifiedintheschooldistrict.

Hehasdirected,advisedandconsultedtoarangeofmathematicseducationprojectsintheUSA.ThemostextensiveandintensiveengagementsincludeNAEPValiditystudies,ACHIEVE,FAM(FoundationsofMathematics)programdevelopmentforAmerica’sChoice,BalancedAssessmentProject(co-Director),MathematicsAssessmentResources(MARS),theElPasoCollaborative(consultant),schooldistrictsandstates,theNewStandardsProject.Fromthemid1980suntilthe1990s,PhilwasthestateDirectoroftheCaliforniaMathematicsProjectfortheUniversityofCalifornia.Hehasalsoworkedwithreadingandliteracyexpertsandpanelsonproblemsrelatedtoacademiclanguagedevelopment,especiallyinmathematicsclassroomdiscourse.

AbstractStandardssequenceaswellasexpresspriority.Onwhatbasis?Learningtrajectoriessequencethroughempiricalinvestigationandtheory.Thesequence,asfarasitgoes,hasempiricalvalidity,butonlysomesequenceshavebeendeveloped.Standards,incontrast,mustchoosewhatstudentsneedtolearnasamatterofpolicy.Thisarticlewilldiscussissuesofsequence,focusandcoherenceinmathematicsstandardsfromtheperspectiveoftheCommonCoreStateStandards(CCSS)forMathematicsintheUnitedStatesofAmerica.

Decisionsaboutsequenceinstandardsmustbalancethepullofthreeimportantdimensionsofprogression:cognitivedevelopment,mathematicalcoherence,andthepragmaticsofinstructionalsystems.Standardsarewrittenasthoughstudentsintheclasshavelearnedapproximately100percentofprecedingstandards.Thisiswildfictioninanyrealclassroom.Thisdifferencebetweenthegenreconventionof‘immaculateprogression’instandardsandthewidedistributionofstudentreadinessinrealclassroomsisadangerousdifferencetoignore.Eachstudentarrivesattheday’slessonwithhisorherownmathematicalbiography,whateverthestudentlearnedontheirpersonaltrajectorythroughmathematics.Aspectaculardiversityofsuchpersonal learning trajectories (PLoTs)facestheteacheratthebeginningofeachlesson.Therearetworelatedmanifoldsinplayduringeachlesson:themanifoldofPLoTs(personallearningtrajectories)intheclassroomandthemanifoldoflearningtrajectories(LTs)thatenablethelearningofthemathematicsbeingtaught.Asrealasthesetrajectories

maybe,neitherisinplainsight.Whatisinplainsightarestandards,tests,textbooksandstudents.

LTsaretoocomplexandtooconditionaltoservedirectlyasstandards.Still,LTspointthewaytooptimallearningsequencesandwarnagainsthazardsthatcouldleadtosequenceerrors.TeachersandstudentsneedtimewithinthelessonandacrosstheunittopullstudentsfromPLoTsalongLTstotheSSTs.Thisrequiresstandardstobewithinreach.

Thetypesoferrorsinthewaystandardsmightbesequencedarereviewed.

Introduction

One sees the difficulty with this standards business. If they are taken too literally, they don’t go far enough, unless you make them incredibly detailed. You might give a discussion of a couple of examples, to suggest how the standards should be interpreted in spirit rather than by the letter. But of course, this is a slippery slope.

Roger�Howe,�Yale,��March�15,�2010��

input�to�common�core�standards

… the “sequence of topics and performances” that is outlined in a body of mathematics standards must also respect what is known about how students learn. As Confrey (2007) points out, developing “sequenced obstacles and challenges for students…absent the insights about meaning that derive from careful study of learning, would be unfortunate and unwise.” In recognition of this, the development of these Standards

Standards,what’sthedifference?:AviewfrominsidethedevelopmentoftheCommonCoreStateStandardsintheoccasionallyUnitedStates


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began with research-based learning progressions detailing what is known today about how students’ mathematical knowledge, skill, and understanding develop over time.

Common�Core��State�Standards,��

2010

Sequence, Coherence and Focus in Standards and Learning Trajectories

Learningtrajectoriessequencelevelsofcognitiveactionsandobjectsthroughempiricalinvestigationandtheory.Asresultthesequencehasempiricalvalidity.However,thequestionofwhatisbeingsequencedisamatterofresearcherchoice,oftendrivenbytheoreticalconsiderationsrelatedtoatrajectoryofinteresttotheresearcher.Someresearchers(ClementsandSarama,2010{thisreport})suggestthesechoicesincludeconsultationwithmathematiciansandeducatorstoobtainvalidfocus.Still,thechoiceofwhatmathematicsgetsresearchattentionisnot,initself,avalidbasisfordecidingwhattoteach.Standards,incontrast,beginwithchoicesaboutwhatstudentsneedtolearnasamatterofpolicy.

Standards,perforce,sequenceaswellasexpresspriority.Onwhatbasis?Bydesign,atleast,onehopes.Towhatextentcanandhasthedesignofmathematicsstandardsbeeninformedbyresearchandempiricallywellfoundedtheoriesoflearningtrajectories?ThisarticlewillcontemplatethatquestionfortherecentlydevelopedCommonCoreStateStandardsinmathematics,theclosestthisnationhasevercometonationalstandards.Itisaninterestingtalethatleadstofundamental,perhapsveryproductive,questionsaboutstandardsandtrajectories,andtheirconsequencesforinstruction,curriculum,assessmentandthemanagementofinstruction.

Thisarticlewilllookatthegeneralissuesofsequence,focusandcoherenceinmathematicsstandardsfromtheperspectiveoftheCommonCoreStateStandards(CCSS)forMathematics.IwasamemberofthesmallwritingteamfortheCCSS.Assuch,Iwaspartofthedesign,deliberationanddecisionprocesses,includingespeciallyreviewingandmakingsenseofdiverseinputsolicitedandunsolicited.Amongthesolicitedinputweresynthesised‘progressions’fromlearningprogressionsresearchers.

Grade level vs. development

Standardssequenceforgradelevels;thatis,thegranularityofthesequenceisyear-sized.Standardsdonotexplicitlysequencewithingradelevel,althoughtheyarepresentedinsomeorderthatmakesmoreorlesssense.Sometimesthisorderwithingradeiscompelling,thusluringuserstooverinterpretthewithingradepresentationasteachingsequence.

Fromthestart,weencounteraproblematicconvention:standardsarewrittenasthoughstudentshavelearnedeverything(100%)inthestandardsfortheprecedinggradelevels.Noonethinksmoststudentshavelearned100%,butthisgenreconventionforstandardsseemsasensibleapproachtoavoidingredundancyandexcessivelinguisticnuance.Buthowdoesthismeregenreconventiondrivethemanagementofinstruction?Testconstruction?Instructionalmaterialsandtheiradoption?Teaching?Expectationsandsocialjustice?Ah…theletterorthespiritandtheslipperyslope.

Cognitive development, mathematical coherence and pedagogic pragmatics

Decisionsaboutsequenceinstandardsmustbalancethepullofthreeimportantdimensionsofprogression:cognitivedevelopment,mathematical

coherence,andthepragmaticsofinstructionalsystems.Thesituationdiffersforelementary,middleandhighschoolgrades.Inbrief:elementarystandardscanbemoredeterminedbyresearchincognitivedevelopmentandhighschoolmorebythelogicaldevelopmentofmathematics.Middlegradesmustbridgethetwo,bynomeansatrivialspan.

Forexample,theCommonCoreStateStandards(CCSS)incorporateaprogressionforlearningthearithmeticofthebasetennumbersystem.Alogicaldevelopmentmathematicallywouldbeginwithsumsoftermswhichareproductsofasingledigitnumberandapoweroften,includingrationalexponentsfordecimalfractions.Yetnoonethinksthisisthewaytoproceed.Instead,theCCSSforgrade1askstudentsto,

2. Understandthatthetwodigitsofatwo-digitnumberrepresentamountsoftensandones.Understandthefollowingasspecialcases:

a. 10canbethoughtofasabundleoftenones—calleda“ten.”

b. Thenumbersfrom11to19arecomposedofatenandone,two,three,four,five,six,seven,eight,ornineones.…

Therelativeweighttogivecognitivedevelopmentvs.mathematicalcoherencegetsmoretangledwithmultiplication,thenumberlineandespeciallyfractions.Inthirdgrade,theCCSSintroducestwoconceptsoffractions:

1. Understandafraction1/basthequantityformedby1partwhenawholeispartitionedintobequalparts;understandafractiona/basthequantityformedbyapartsofsize1/b.


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2. Understandafractionasanumberonthenumberline;representfractionsonanumberlinediagram.

a. Representafraction1/bonanumberlinediagrambydefiningtheintervalfrom0to1asthewholeandpartitioningitintobequalparts.Recognizethateachparthassize1/bandthattheendpointofthepartbasedat0locatesthenumber1/bonthenumberline.

b. Representafractiona/bonanumberlinediagrambymarkingoffalengths1/bfrom0.Recognizethattheresultingintervalhassizea/bandthatitsendpointlocatesthenumbera/bonthenumberline.

Thefirstconceptreliesonstudentunderstandingofequalpartitioning.JereConfrey(2008)andothershavedetailedthelearningtrajectoryofchildrenthatestablishestheattainabilityofthisconceptoffraction.Yetbyitself,thisconceptisisolatedfrombroaderideasofnumberthat,forthesakeofmathematicalcoherence,areneededearlyinthestudyoffractions.Theseideasareestablishedthroughthesecondstandardthatdefinesafractionasanumberonthenumberline.Thisdefinitionhasalotofmathematicalpowerandconnectsfractionsinasimplewaytowholenumbersand,later,rationalnumbersincludingnegatives(Wu,H.,2007).Simplelookingforward,butmysteriouscomingfrompriorknowledge.

TheWritingTeamofCCSSreceivedwideandpersistentinputfromteachersandmathematicseducatorsthatnumberlineswerehardforyoungstudentstounderstandand,asanabstractmetric,evenhardertouseinsupportoflearningotherconcepts.Thirdgrade,theysaid,isearlyforrelyingonthenumberlinetohelpstudentsunderstandfractions.Wewerewarnedthatasimportant

asnumberlinesareasmathematicalobjectsofstudy,numberlinesconfusedstudentswhenusedtoteachotherideaslikeoperationsandfractions.Inotherwords,includethenumberlineassomethingtolearn,butdon’trelyonittohelpstudentsunderstandthatafractionisanumber.

Thedifferenceinadviceonfractionsonthenumberlinewasnoteasytosortthrough.Intheend,weplacedthecognitivelysensibleunderstandingfirstandthemathematicalcoherencewiththenumberlinesecond.Weincludedbothandusedbothtobuildunderstandingandproficiencywithcomparingandoperationswithfractions.

Doesthenumberlineappearoutoftheblueinthirdgrade?No.Welookedtotheresearchinlearningtrajectoriesformeasurementandlengthtoseehowtobuildafoundationfornumberlinesasmetricobjects(Clements,1999c;Nührenbörger,M.,2001;Nunes,T.,Light,P.,andMason,J.H.1993).TheStandardsfromAsiancountrieslikeSingaporeandJapanwerealsohelpfulinencouragingadeeperandricherdevelopmentofmeasurementasafoundationfornumberandquantity.

ClementsandSarama(2009)emphasizethesignificanceofmeasurementinconnectinggeometryandnumber,andincombiningskillswithfoundationalconceptssuchasconservation,transitivity,equalpartitioning,unit,iterationofstandardunits,accumulationofdistance,andorigin.Byaroundage8,childrencanusearulerproficiently,createtheirownunits,andestimateirregularlengthsbymentallysegmentingobjectsandcountingthesegments.

TheCCSSfoundationfortheuseofthenumberlinewithfractionsin3rdgradecanbefoundinthe2ndgradeMeasurementstandards:

Measureandestimatelengthsinstandardunits.

• Measurethelengthofanobjectbyselectingandusingappropriatetoolssuchasrulers,yardsticks,metersticks,andmeasuringtapes.

• Measurethelengthofanobjecttwice,usinglengthunitsofdifferentlengthsforthetwomeasurements;describehowthetwomeasurementsrelatetothesizeoftheunitchosen.

• Estimatelengthsusingunitsofinches,feet,centimeters,andmeters.

• Measuretodeterminehowmuchlongeroneobjectisthananother,expressingthelengthdifferenceintermsofastandardlengthunit.

Relateadditionandsubtractiontolength.

• Useadditionandsubtractionwithin100tosolvewordproblemsinvolvinglengthsthataregiveninthesameunits,e.g.,byusingdrawings(suchasdrawingsofrulers)andequationswithasymbolfortheunknownnumbertorepresenttheproblem.

• Representwholenumbersaslengthsfrom0onanumberlinediagramwithequallyspacedpointscorrespondingtothenumbers0,1,2,…,andrepresentwhole-numbersumsanddifferenceswithin100onanumberlinediagram.

Thisworkinmeasurementin2ndgradeis,inturn,supportedby1stgradestandards:

• Expressthelengthofanobjectasawholenumberoflengthunits,bylayingmultiplecopiesofashorterobject(thelength


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unit)endtoend;understandthatthelengthmeasurementofanobjectisthenumberofsame-sizelengthunitsthatspanitwithnogapsoroverlaps.Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

ThissequenceintheCCSSwasguidedbythelearningtrajectoryresearch.ThisresearchinformedtheCCSSregardingessentialconstituentconceptsandskills,appropriateageandsequence.Yetthegoalofhavingnumberlineavailableforfractionscamefromtheneedformathematicalcoherencegoingforwardfrom3rdgrade,ratherthanfromlearningtrajectoryresearch.

Instructional Systems and Standards

Perhapsthemostimportantconsequenceofstandardsistheirimpactoninstructionandinstructionalsystems.Thisimpactisoftenmediatedbyhighstakesassessmentswhichwillbedealtwithlater.Twocrucialinstructionissueswillbediscussedthataretoooftenburiedincomfortingcushionsofunexaminedassumptions.Thefirstissueis,howdothestructure,propertiesandbehaviorofmathematicsknowledgeinteractwithinstruction?

Thesecondissuearisesfromthewaystandardsarewritten,asthoughstudentsinthemiddleofgrade5havelearnedapproximately100%ofwhatisinthestandardsforgradek-4andhalfof5.Thisisneverclosetotrueinanyrealclassroom.Thisdifferencebetweenthegenreconventionof“immaculateprogression”instandardsandthewidedistributionofstudentreadinessinrealclassroomshasimportantconsequences.Itmeans,foronething,thatstandardsarenotaliteralportrayalofwherestudentsareorcanbeatagivenpointintime.And,forme,

thenegationof‘can’negates‘should’.Standardsserveadifferentpurpose.Theymapstationsthroughwhichstudentsareleadfromwherevertheystart.

Immaculateprogressionliteralismhascontributedtoconfusionaboutwhat“proficient”meansasatestresult.Moststatetestshave“proficient”cutscoresat60%orless(withguessingallowedonmultiplechoice,[usually4choices],itemsthatmakeupclosetoallofthetest).Thuseventhedistributionof‘proficient’studentslackslargechunksoflearningofthestandards,atleastasassessedbythestandardsbasedtest.

The rough terrain of prior learning where lessons live

Thestandardsbasedcurriculumisasequencethroughthecalendar:yeartoyear,monthtomonth,daytoday.Thinkofthisasahorizontalpathofconceptsandskills.Suchapathcanmatchtextbooksandtests,butneverthedistributionofstudentsinaclassroom.Beneaththesurfaceofthestandardssequencetrajectory(SST)istheunderwaterterrainofpriorknowledge.Eachstudentarrivesattheday’slessonwithhisorherownmathematicalbiography,whateverthestudentlearnedontheirpersonaltrajectorythroughmathematics.Aspectaculardiversityofsuchpersonallearningtrajectories(PLoTs)facestheteacheratthebeginningofeachlesson(Murata,A.,&Fuson,K.C.,2006).

Theteacher,ontheotherhand,bringstothisdiversityanambitionforsomemathematicstobelearned.Themathematicshasalocationinyetanothertrajectory:thelogicalsequenceofideaswhichreflectsthedeductivestructureofmathematics(MTs).Thus,therearethreerelatedmanifoldsinplay:thePLoTs(personallearningtrajectories)intheclassroom,theMTsandthelearningtrajectories(LTs).As

realasthesetrajectoriesmaybe,noneareinplainsight.

…teaching is like riding a unicycle juggling balls you cannot see or count.

Whatisinplainsightarestandards,tests,textbooksandstudents.Ateachercannotactuallyknowthestudents’PLoTs.NorhasresearchmappedtheterritoryofthestandardswithLTs..AndtheMTsarethemselvesamatterofconsiderablechoiceinstartingpoint,andoftenbeyondthemathematicaleducationoftheteacher.Whatisrealishardtosee,whilestandardsflashbrightlyfromeverytest,textandexhortationthatcomestheteacher’sway.

Learningtrajectoryresearchdevelopsevidenceandevidencebasedtrajectories(LTs).EvidenceestablishesthatLTsarerealforsomestudents,apossibilityforanystudentandpossiblymodaltrajectoriesforthedistributionofstudents.LTsaretoocomplexandtooconditionaltoservedirectlyasstandards.Still,LTspointthewaytooptimallearningsequencesandwarnagainsthazardsthatcouldleadtosequenceerrors(seebelow).TheCCSSmadesubstantialuseofLTs,butstandardscannotsimplybeLTs;standardshavetoincludetheessentialmathematics,MTs,whetherweknowanythingaboutitslocationinanLTornot,andstandardshavetoaccommodatethevariationinstudents,ifnotteachers,ateachgradelevel.

Howdoandcouldthesefourtrajectories(LTs,MTsPLoTs,andSSTs)interact?Asystemcouldjustleaveittoindividualteacherstoreckontheoptimizationamongthem.ItcouldimposestrongSSTsaspressureinanaccountabilitysystem,withoutprovidingforPLoTsortakingadvantageofLTs.Itcouldnametheterritorybetweenwhatstudentsbring(PLoTs)andthewhatstandardsdemand(SST)the“achievementgap”,adarkvoidthatonlyexplainsstepsnottakenrather


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thanwhichwaytogo.Itcouldtellteacherstokeepturningthepagesofthetextbookbasedonstandardsaccordingtotheplannedpace,andrelyontheshearforceofexpectationtopullstudentsalong.Atleastthiswouldcreatetheopportunitytolearn,howeverfleetingandpoorlypreparedstudentsmightbetotakeadvantageofit.Whilethisisbetterthandenialofopportunity,itisahollow,ifnotcynical,responsetothepromisestandardsmaketostudents.Shouldn’twedobetter?

Whatwouldbebetter?Somenations,includinghighperformingnations,assumeinthestructureoftheirinstructionalsystemsthatstudentsdifferatthebeginningofeachlesson.Asianclassrooms,K-5,andmostly6-9,followadailytrajectoryofinitiallyprojectingthedivergenceofstudents’development(refractedthroughtheday’smathematicsproblem/s)intotheclassroomdiscourseandpullingthedivergencetowardaconvergentlearningtarget.Thepremiseis:eachlessonbeginswithdivergenceandendswithconvergence.Suchasystemrequiresenoughtimetoachieveconvergenceeachday,enoughtimeonasmallnumberofproblems.Ahurriedinstructionalsystemcannot‘wait’forstudentseachday.Standardsmustrequirelesstolearnratherthanmoreeachyeartomaketimefordailyconvergence.Asystemwhichoptimisesdailyconvergencewillbemorerobustandaccumulatelessdebtintheformofstudentsunpreparedforthenextlesson.Suchdebtcompounds.Unlikethenationaldebt,itdoesnotcompoundquietly,butmakesallthenoisesofchildhoodandadolescencescorned.

Startbyunderstandingthetaskandthenthepeopleinplacewhocandotheirpartstoaccomplishthetask.ThetaskistotakethedomainofPLoTs,thegivenroughterrainofwhatthedistributionofstudentsbring,andtransformthePLoTstoSSTs,giveortake.ThefunctionthatcantakePLoTstoSSTsismappedby

theLTsandMTs.Thatis,LTsandMTscanprovidethemapfromPLoTstoSSTs.Themap,alas,isofaterritorythatisonlypartiallyexplored.Therearestillunknownseasandfearsofseamonstersanddreamsofgoldtofrightenanddistractusfromthevoyage.Still,weknowenoughinelementarygradestodowhatisneededtomakeLTsapartofteacherknowledgeandafeatureintoolsforteachers.

TeachersneedknowledgeofhowLTsworkandthespecificsofLTsthatwillhelpthemunderstandthemostcommonPLoTstheywillfindamongtheirstudents(Murata,A.,&Fuson,K.C.,2006).TheyneedknowledgeoftherelevantMTs.AndtheyneedtoolsthatilluminateratherthanobscurethePLoTs.TheyneedinstructionalprogramsandlessonprotocolsthatposeSSTsasthefinishline,butaccommodatePLoTvariation.TheyneedtimewithinthelessonandacrosstheunittopullstudentsfromPLoTsalongLTstotheSSTs.Thisrequiresstandardstobewithinreach.

ThecrucialissueinthissituationishowwellthestandardsdriventextsandtestsimprovetheperformanceoftheinstructionalsysteminmovingthePLoTsalongtheLTs.ItisquitepossibleforstandardstobeoutofwhackwithLTsandPLoTssothattheydiminishperformance.StandardsareonlyagoodideawhentheyusefullymapunderlyingLTsandMTssotheycanhelpteachersseeandrespondtoPLoTs.IfthesequenceinthestandardsconflictsseriouslywithLTsoraretoofarremovedfromPLoTs,theycansteertheinstructionalsystemsawayfromteachingandlearning,towardstatuesqueposesfacingoutandthesamewasteofchancesinside.

Forexample,theCCSSatgrade7haveastandardforproportionalrelationships.

2. Recognizeandrepresentproportionalrelationshipsbetweencovaryingquantities.

a. Decidewhethertwoquantitiesareinaproportionalrelationship,e.g.,bytestingforequivalentratiosinatableorgraphingonacoordinateplaneandobservingwhetherthegraphisastraightlinethroughtheorigin.

b. Identifytheconstantofproportionality(unitrate)intables,graphs,equations,diagrams,andverbaldescriptionsofproportionalrelationships.

c. Representproportionalrelationshipsbyequations.For example, total cost, t, is proportional to the number, n, purchased at a constant price, p; this relationship can be expressed as t = pn.

d. Explainwhatapoint(x, y)onthegraphofaproportionalrelationshipmeansintermsofthesituation,withspecialattentiontothepoints(0,0)and(1,r)whereristheunitrate.

Thisstandardistheculminationofamanifoldofprogressionsand,itself,thebeginningofmoreadvancedprogressions.PatThompsonhasremarked(2010,advicetostandards)thatproportionalitycannotbeasingleprogressionbecauseitisawholecityofprogressions.Thisstandard,whichstandsalongsideotherstandardsonratiosandrates,explicitlydrawsonpriorknowledgeoffractions,equivalence,quantitativerelationships,coordinategraph,unitrate,tables,ratios,ratesandequations.Implicitly,thispriorknowledgegrowsfromevenbroaderpriorknowledge.ThesequencesupportingthisStandardintheSSTbarelycapturesthepeaksofasimplificationoftheknowledge


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structure.ThecomplexityofthemanifoldofLTsguaranteesthatthedistributionofPLoTsinaclassroomwillhavesplendidvariety.

Whatcouldhelptheteacherconfrontedwiththevarietyofreadiness?Certainlynotpressureto“cover”thestandardsinsequence(SST),keepmovingalongatagoodpacetomakesureallstudentshavean‘opportunity’toseeeverystandardflyingby.PerhapssomeknowledgeoftheLTswouldhelpteachersunderstandthevarietyofPLoTsandwhatdirectiontoleadthestudentsfromwherevertheybeginthelesson.EvenhypotheticalLTscandomoregoodthanharmbecausetheyconceptualizethestudentasacompetentknowerandlearnerintheprocessoflearningandknowingmore(Clements,2004a).Perhapsasystemofproblemsandassignmentswiththediagnosticvalueofrevealinghowdifferentstudentsseethemathematics…howtheythinkaboutit…wheretheyarealongtheLT.Ateacherneedsthethinkingitself,notascorethatevaluatesthethinking.

How do standards express the form and substance of what students learn?

Whatisthenatureofthe‘things’studentslearn?Sometimeswhatiswantedisaperformance,asinlearntorideabike.Standards,instructionandassessmentcanhappilyfocusonthevisibleperformanceinsuchcases.Butoften,inmathematicsanyway,isamentalactiononamentalobject,reasoningmaneuversandrules,representationalsystemsandlanguagesformathematicalobjectsandrelations,cognitiveschemaandstrategies,websofstructuredknowledge,andsocialrepresentations,andsoon.Manyoftheselearnedthingsaresystemsthatinteractwithothersystemsinthinking,knowinganddoing.Standardscannot

expressthiskindofcomplexity;theyrefertosomeobservablesurfaceoflearning.Butthislinguisticconveniencecanleadtologicalfallacieswhenweattributeunwarranted‘thinginess’propertiestowhatweactuallywantstudentstolearn.

Theimportantpointisthatlearnedthingsarenotthingsortopics(names)andnotjuststandards.Asequenceoftopicsorstandardsskimsthesurfaceandmissesthesubstanceandeventheformofasubject.Compare,forexample,theStandard,

• Addandsubtractfractionswithunlikedenominators(includingmixednumbers)byreplacinggivenfractionswithequivalentfractionsinsuchawayastoproduceanequivalentsumordifferenceoffractionswithlikedenominators.For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

towhatthestudentmustactuallyknowanddoto“meet”thestandard(forexample,Steffe,2004,2009;Confreyetal,2008,2009;Wu,2007;Saxeetal,2005).Thestandardgivesagoal,butdoesnotcharacterizetheknowledgeandcompetenciesneededtoachievethegoal.Whilethispointmayseemobvious,itgetslostinthecompressionchamberswheresystemsareorganizedtomanageinstructionforschooldistricts.Devicesareinstalledtomanage“pacing”andmonitorprogresswith“benchmarkassessments”.Thesedevicestreatthegradelevelstandardsastheformandsubstanceofinstruction.Thatis,studentsaretaughtgradelevel“standards”insteadofmathematics.Thisnonsenseisactuallywidespread,especiallywherepressuresto“meetstandards”aregreatest.

Standardsuseconventionalnamesandphrasesfortopicsinasubject.Towhatdotheserefer?

Ifthefieldhadawellunderstoodcorpusofcognitiveactions,situations,knowledgeetc.thenthesenamescouldrefertopartsofthiscorpus.Butthefield,schoolmathematics,hasnosuchwidelyunderstoodcorpus(indeed,itisanimportanthopethatcommonstandardswillleadtocommonunderstandingslikethis).Whatthenamesreferto,ineffect,arethefamiliarconventionsofwhatgoesonintheclassrooms.Thereferencedegeneratestotheoldhabitsofteaching:assignments,grading,assessment,explanation,discussion.Thestandardssay,‘Dotheusualassortmentofclassroomactivitiesforsomecontentthatcanbesortedintothenamesinthestandards.Wewillcallthis“coveringthestandards”withinstructionalactivity.

“Covering”hasaverytenuousrelationshipwithlearning.First,therearemanychoiceswithinatopicaboutfocus,coherencewithinandbetweentopics,whatstudentsshouldlearntodowithknowledge,howskillfultheyneedtobeatwhat,andsoonendlessly.Teachersmakethesechoicesinmanydifferentways.Toooften,thechoicesaremadeinsupportofaclassroombehaviormanagementschemereliedonbytheteacher.Second,differentstudentswillgetverydifferentlearningfromthesameofferedactivity.Third,thequalityofthediscussion,theassignedandproducedwork,thefeedbackgiventostudentswillvarywidelybyteacherworkingundertheblessingofthesamestandard.

Coveringisatbestweak.Whencombinedwithstandardsthataretoofarfromthepriorknowledgeofstudents,andtoomany;thechemistrygetsnastyinahurry.Teachersmoveonwithoutthestudents;studentsaccumulatedebtsofknowledge(knowledgeowedtothem)andopportunitiesforunderstandingthenextchapter,thenextcourseareundermined.


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Theforegoingdiscussionofinstructionalsystemsillustratestheimportance(andpotentialformayhem)insequencingstandards.Whatconstituentsarenecessaryandsufficientaspriorknowledgeforagivenconceptoraction,andhowcantheconstituentsbearrangedtoleaduptothetargetconcept?Thisquestionhasmanylocalanswersthathavetobefittedtogetherintoregionsthatmakesomesense,ifnotharmony.Standardsarefurtherconstrainedbyhowmuchcanbelearnedatanyonegradelevel,andbythecoherencewithinagradelevel.Thesequestionsarenotonlydesignchoices,butpotentialsourcesoferrorwithconsequencesfortheviabilityofinstruction.Thenextsectionsexaminethetypesoferrorsthatcouldmenaceastandardsbasedsystem.

Types of Sequence Errors

Thereareseveraltypesoferrorswithseriousconsequencesforstudentsandteachersinthewaystandardsmightbesequenced.Forexample,acommontypeofsequenceerroroccurswhenaconcept,BdependsonA2versionofconceptA,moreevolvedthantheA1version;StandardshaveonlydevelopedA1.StudenttriestolearnBusingA1insteadofA2.Rate,proportionalrelationshipsandlinearity(B)dependonunderstandingmultiplicationasascalingcomparison(versionA2),butstudentsmayhaveonlydevelopedversionA1conceptofmultiplication,thetotalofthingsinagroupsofbeach.

IntheCCSS,multiplicationisdefinedingrade3asa x b = cmeansagroupsofbthingseachiscthings.Ingrade4,theconceptofmultiplicationisextendedtocomparisonwherec = a x bmeanscisatimeslargerthanb.Ingrade5,theCCSShas:

5. Interpretmultiplicationasscaling(resizing),by:

a. Comparingthesizeofaproducttothesizeofonefactoronthe

basisofthesizeoftheotherfactor,withoutperformingtheindicatedmultiplication.

b. Explainingwhymultiplyingagivennumberbyafractiongreaterthan1resultsinaproductgreaterthanthegivennumber(recognizingmultiplicationbywholenumbersgreaterthan1asafamiliarcase);explainingwhymultiplyingagivennumberbyafractionlessthan1resultsinaproductsmallerthanthegivennumber;andrelatingtheprincipleoffractionequivalencea/b=(n×a)/(n×b)totheeffectofmultiplyinga/bby1.

Ingrade6and7rate,proportionalrelationshipsandlinearitybuilduponthisscalarextensionofmultiplication.Studentswhoengagetheseconceptswiththeunextendedversionofmultiplication(agroupsofbthings)willhavePLoTsthatdonotsupporttherequiredMTs.ThisburdenstheteacherandstudentwithrecoveringthroughLTs.Thiswillbetaxingenoughwithoutillsequencedstandardscausinginstructionalsystemstoneglectextendingmultiplication.

Majortypesofsequenceerrorsfollow:

1. Unrealistic:

a. Toomuchtoofastsogapsinlearningcreatesequenceissuesforstudents,systemcannotdeliverstudentswhoareinsequence.

b. Distributionofpriormathematicsknowledgeandproficiencyinthestudentandteacherpopulationistoofarfromthestandards;nopracticalwaytogetstudentsinagoodenoughsequence.

2. Missingingredient:

a. AisanessentialingredientofB,StandardssequenceBbeforeA.

b. CoherencerequiresprogressionABC,butstandardsonlyhaveAC

c. Termisusedthathasinsufficientdefinitionforthatuse.

3. Cognitiveprematurity:

a. Bdependsoncognitiveactionsandstructuresthathavenotdevelopedyet.

b. Bisatypeofschemaorreasoningsystem,learnerhasnotdevelopedthattypeofschemaorsystem.

c. StudentdevelopsimmatureversionofBandcarriesitforward(see4)

4. Contradiction:

a. CognitivedevelopmententailsABC,mathematicallogicentailsCBA.

5. Missingconnection:BisaboutordependsonconnectionbetweenX-Y,butX-Yconnectionnotestablished.

6. Interference:

a. BdependsonA2versionofA,moreevolvedthanA1version;StandardshaveonlydevelopedA1.StudenttriestolearnBusingA1insteadofA2.

b. BbelongsnestledbetweenAandC,butDisalreadynestledthere.WhenlearningBisattempted,Dinterferes.

7. Cameo:

a. Bislearnedbutnotusedforalongtime.ThereisnoCsuchthatCdependsonBforalongtime.Bmakesacameoappearanceandthengetslostinthelandoffreefragments.

8. HardWay:

a. CneedssomeideasfromB,butnotallthedifficultideasandtechnicaldetailsthatmakeB


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takemoretimethanitisworthandmakeithardforstudentstofindtheneededideasfromB,soCfails.

b. TherearemultiplepossibleroutestogetfromAtoE,standardstakeanunnecessarilydifficultroute

9. Aimless:

a. Standardspresentedasliststhatlackcomprehensibleprogression.

Types of Focus and Coherence Errors

Theissuesoffocusandcoherenceinstandardsdeservesmoreattentionthanwewillgiveithere.Nonetheless,learningtrajectoriesinteractwithcoherenceandfocusinstandards.Thefollowingarecriticaltypesoferroroffocusandcoherence:

1. Sprawl:

a. Milewide,inchdeep.Collectionofstandardsdilutestheimportanceofeachone.

b. Standardsdemandmorethanispossibleintheavailabletimeformanystudentsandteachers,soteachersandstudentsforcedtoeditonthefly.Thisistheoppositeoffocus.

c. Standardsarejustlistswithoutenoughorganisationalcuesinrelationtohierarchyofconceptsandskills

2. Wronggrainsize

a. Thegranularityistoospecificortoogeneral.Theimportantunderstandingisatacertainlevelofspecificitywherethestructureandthecognitivehandlesare,morespecificormoregeneral;grainsizewillnotmatchuptopriorknowledge,mentalobjectsandactionsonthem(seeAristotleEthics:thechoiceofspecificityisaclaim

thatshouldbeexplicitanddefended.)

b. Toofine:complexideasarechoppedupsothemainideaislost;thecoherencemaybeevoked,butnotilluminated.Alignmenttransactionsintestconstruction,materialsdevelopmentmissthemainpointbut‘cover’theincidentals.Studentscanperformtheverticallinetestbutdonotknowwhatafunctionisorhowfunctionsmodelphenomena.

c. Toobroad:includeswhateverandfocusesonnothinginparticular.

3. Wrongfocus

a. Focusonanswergettingmethods,oftenmnemonicdevices,ratherthanmathematics.

4. Narrowfocus

a. Justskills,orjustconceptsorjustprocess;orjusttwooutofthree.

5. Prioritiesdonotcohere:

a. Fragmentsthathavelargegapsbetweenthem;

b. grainsizetoofine

6. Congestion:

a. Somegradelevelsarecongestedwithtoomuchtobelearned;densityprecludesfocus

b. B,C,Dareallbeinglearnedatonce,butcognitiveactionsneededforlearningcanonlyhandleoneortwoatatime.OnlyBCandCDarelearned,buttheessentialpointislearningBCDandthesystemBC-BD-CD.

7. Inelegance:

a. AXBYCZisequivalenttoABCandwastedtimeandcognitionon–X-Y-Z.

8. Waste:

a. InvesttimeandcognitiononBandBisnotimportant.

9. Resolutionofhierarchy:

a. Thehierarchalrelationshipbetweenstandardsisnotexplicated.Detailsareconfusedwithmainideas.

b. Thehierarchyofstandardsdoesnotexplainrelationshipsamongideas,itjustcollectsstandardsintocategories.

10.Excessivelyliteralreading:

a. Thiserrorisinthereadingasmuchasthewriting;itleadstofragmentedinterpretationofthesubject,losingthecoherencebetweenthestandards.

b. Readingindividualstandardsasindividualingredientsofatest.whentheexplicitgoalistohavetheingredientscookintoacake,tastingtheuncookedingredientsisapoormeasureofhowthecaketastes(althoughitisrelated).Thegoal,asstatedinthegradelevelintroductionsandthepracticesstandardsisforthestudentstocook.

What are Standards?

Standardsarepromises.Standardspromisethestudent,“Studyandlearnwhatishere,doyourassignmentsandwepromiseyouwilldowellonthetest.”Weneedtestsandexaminationsdesignedtokeepthatpromise.Weneedschoolsystemsdesignedtokeepthepromises.

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Nührenbörger,M.(2001).Insightsintochildren’srulerconcepts—Grade-2students’conceptionsandknowledgeoflengthmeasurementandpathsofdevelopment.InM.V.D.Heuvel-Panhuizen(Ed.),Proceedings of the 25th Conference of the International Group for the Psychology in Mathematics Education, 3,447-454.Utrecht,TheNetherlands:FreudenthalInstitute.

Nunes,T.,Light,P.,andMason,J.H.(1993).Toolsforthought:Themeasurementoflengthandarea.Learning and Instruction, 3,39-54.

Park,J.,&Nunes,T.(2001).Thedevelopmentoftheconceptofmultiplication.Cognitive Development, 16,763-773.

Peterson,P.L.,Carpenter,T.P.,&Fennema,E.H.(1989).Teachers’knowledgeofstudents’knowledgeinmathematicsproblemsolving:Correlationalandcaseanalyses.Journal of Educational Psychology, 81,558-569.

Sarama,J.,&Clements,D.H.(2002).Building Blocksforyoungchildren’s

mathematicaldevelopment. Journal of Educational Computing Research, 27(1&2),93-110.

Sarama,J.,&Clements,D.H.(2009a).Early childhood mathematics education research: Learning trajectories for young children.NewYork:Routledge.

Saxe,G.,Taylor,E.,McIntosh,C.,&Gerhart,M.(2005).RepresentingFractionswithstandardnotions:Adevelopmentalanalysis.Journal for research in Mathematics Education. 36 (2),137-157.

Sherin,B.,&Fuson,K.(2005).MultiplicationStrategiesandtheAppropriationofComputationalResources.Journal for Research in Mathematics Education., 36(4),347-395.

Simon,M.A.(1995).Reconstructingmathematicspedagogyfromaconstructivistperspective.Journal for Research in Mathematics Education, 26(2),114-145.

Steffe,L.P.(2004).”Ontheconstructionoflearningtrajectoriesofchildren:Thecaseofcommensuratefractions”.Mathematical Thinking and Learning. 6(2),129-162

Steffe,L.P.&Olive,J(2009).Children’s fractional knowledge.Boston,Springer.

Wilson,P.H.(2009).Teachers’ Uses of a Learning Trajectory for Equipartitioning.NorthCarolinaStateUniversity,Raleigh,NC.

Wu,H.(2007),Fractions, decimals and rational numbers”, http://math.berkeley.edu/-wu/


17

Mathematicsteachingandlearningtoreachbeyondthebasics

Kaye�StaceyUniversity of Melbourne

KayeStaceyisFoundationProfessorofMathematicsEducationattheUniversityofMelbourneandtheleaderoftheScienceandMathematicsEducationcluster.Sheworksasaresearcher,primaryandsecondaryteachereducator,supervisorofgraduateresearchandasanadvisertogovernments.Shehaswrittenmanypracticallyorientedbooksandarticlesformathematicsteachers,aswellasproducingalargesetofresearcharticles.ProfessorStacey’sresearchinterestscentreonmathematicalproblemsolvingandthemathematicscurriculum,particularlythechallengeswhicharefacedinadaptingtothenewtechnologicalenvironment.SheiscurrentlyamemberoftheAustralianResearchCouncilCollegeofExperts.Herresearchworkisrenownedforitshighengagementwithschools.HerdoctoralthesisfromtheUniversityofOxford,UK,isinnumbertheory.ShehasbeenthemathematicsexpertontheAustralianAdvisoryCommitteefortheOECDPISAprojectsinceitsinceptionandisnowChairofitsinternationalMathematicsExpertGroup.KayeStaceywasawardedaCentenaryMedalfromtheAustraliangovernmentforoutstandingservicestomathematicaleducation.

AbstractThepurposeofthispresentationistopaintabroadbrushpictureofthechallengeofprovidingmathematicsteachingthatencourageslearningthatgoesbeyond‘thebasics’.ThepresentationfocusesonmathematicalreasoningandsuggestswaysinwhichitcanbegivenamoresecureplaceinAustralianmathematicsclassrooms.Twostudiesarereported,bothofwhicharosefromconcernaboutthe‘shallowteachingsyndrome’evidentinmanyAustralianclassroomswherethereisverylittlemathematicalreasoninginevidence.OnestudyexaminedYear8textbooks,findingthatveryfewpresented‘ruleswithoutreasons’andtakenoverallgenerallypresentedagoodarrayofexplanationsinvolvingreasoningofseveraldistincttypestohelpstudentsunderstandwhyresultsweretrue.Itwasevident,however,thattheseexplanationsweregenerallyonlyusedtojustifytherule,andwerenotcalleduponinanywayonceitwasestablished.Asecondstudyinterviewedabout20leadersinmathematicseducationtoexploretheiropinionsontheshallowteachingsyndrome(most–butnotall–feltitwasarealeffectofdisturbingprevalence),andtheteachingofmathematicalreasoningandproblemsolving.ThepresentationincludessomesuggestionsforstrengtheningtheplaceofmathematicalreasoninginAustralianclassroomsandthenewAustraliancurriculum.

Introduction

Thepurposeofthispaperistopaintabroadbrushpictureofthechallengeofprovidingmathematicsteachingthatencourageslearningthatgoesbeyond‘thebasics’.ThepaperfocusesonmathematicalreasoningandsuggestswaysinwhichitcanhaveamoresecureplaceinAustralianmathematicsclassrooms.

Becauseoftheirabstractness,learningabouttheobjectswithwhichmathematicsisconcernedisdifficult.Becausemathematicsisadoingsubject,transformingandcombiningtheseobjectsiscentral,sodevelopingtherelevantskillstoahighdegreeoffluencyiscentral.Thedifficultyofthelearningisheightenedbythehierarchicalnatureofmathematics,whereskillisbuiltonskillandconceptisbuiltonconcept.Nowonderthatlearning‘thebasics’(theconcepts,theskillsandhowtousetheminstandardwaystosolveproblemsthatrelatedirectlytoreal-worldsituations)caneasilyfillallthetimeinschooldevotedtomathematics.Listingtheconcepts,theskillsandtheirdirectapplicationscouldalsoeasilyfillawholenationalcurriculum.

Importantasthecontentaboveis,anddespitethetendencyforittoappeartodefinewhatmathematicsis,mathematicsisonlypartiallydescribedbysuchconcepts,skillsandstandardapplications.Thelessvisibleaspectofmathematicsisitsprocessside(howmathematicsisdone)whichforthepastnearly20yearshasbeenlabelled‘WorkingMathematically’inAustralia.Inthepresentation,IwillgiveabriefoverviewofthevariouswaysinwhichthisstrandhasbeentreatedinAustralianmathematicsinthepast,leadinguptothecurrentfirstcycleoftheAustraliancurriculum.HeretheelementsofWorkingMathematicallymostclearlyappearastwoofthefourproficiencystrands:problemsolvingandreasoning.Neitherofthesestrandsseemstobeyetoperationalisedasclearlyaswillberequiredifteachersaretobeencouragedtopayseriousattentiontothem.Thispresentationwillpresentideasonthedevelopmentofthereasoningstrand.

Reasoninginmathematicsisacognitiveprocessoflookingforreasonsandlookingforconclusions.Tolearnmathematics,studentsneedtolearn


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aboutthereasonswhichothershavefoundtosupportconclusions(forexample,whytheanglesumofanytriangleis180degrees)andtheyalsoneedtoengageintheirownreasoningbothwhenworkingonwhatPolyacalls‘problemstoprove’and‘problemstofind’.Thesetwosidesareconnected.Learningaboutthereasoningofexpertsshouldassistinfosteringyourownreasoningabilities;itshouldestablishafeelingthatmathematicsmakessenseandisnotjustasetofarbitraryrules;andmoregenerally,itshoulddemonstratetheuniquelydeductivecharacterofmathematics.

IwillreportontworelatedstudiesthatarerelevanttothequestionofhowstudentsinYear8learnaboutreasoning.Thestartingpointforboththesestudiesisaninternationalstudy,theTIMSS1999videostudy,whichanalysedarandomsampleofYear8Australianlessonsandcomparedthemwithlessonsfromsixothercountries.Thevideostudy(http://www.acer.edu.au/research;http://www.lessonlab.com/timss1999)revealedmanypositivefeaturesofAustralianclassrooms.However,theAustralianmathematicslessonsdisplayedaclusteroffeatureswhichIcallthe‘shallowteachingsyndrome’(Stacey,2003):apredominanceoflowcomplexityproblems,whichareundertakenwithexcessiverepetition,andanabsenceofmathematicalreasoningandconnectionsinclassroomdiscourse.Togivejustoneexample,only2percentoftheproblemsolutionspresentedbyteachersorstudentsintheAustralianlessonsdemonstrated‘makingconnections’,i.e.showedsomelinkingbetweenmathematicalconcepts,factsorprocedures.

Thefirststudy(Stacey&Vincent,2009)examinedthewayinwhichtextbookspresentexplanationsofmathematicalresults.Itisoftenreportedthatsecondaryteachingisdominatedbytextbooks,andsoitwasofinterestto

ustoseethenatureofthereasoningthattheydisplayandpromote.Thestudy’sfocuswasonexplanationsofwhyimportantmathematicalresultsaretrue,notexplanationsofwhatorhow(e.g.WhatdoesNNWmean?,Howdoyoumakeastem-and-leafplot?).Thesewhyexplanationsinvolvemathematicalreasoningatitsbest.

Inthesecondstudy,alsocarriedoutwithDrJillVincent,weinterviewedabout20mathematicseducationleadersaroundAustraliatoexploretheirresponsestothenotionoftheshallowteachingsyndromeandtheplaceofelementsofworkingmathematics(includingreasoning)inclassroomteaching.Theywereeducationdepartmentofficers,mathematicsassociationleadersandtextbookwriters.Althoughthesamplewastoosmalltodrawfirmconclusions,therewerefewobviousdifferencesinresponsesbyemploymenttype,althoughtheeducationdepartmentofficersweremoreawareofsystemlevelinitiativesandthedauntingscaleofthetaskofreachingallschoolswithin-depthassistance.

Forthetextbookstudy,weselectedninepopulartextbooksfromfourAustralianstates,andwithinthatchoseseventopicswheretherewasaresultofmathematicalimportancethatneededsomejustificationorproof.Examplesincludetheanglesumoftriangles,multiplicationoftwonegatives,theareaofacircleandtherulefordivisionoffractions.Foreachtopicandeachtextbook,weexaminedalltheexplanationsoftheresultpresentedexplicitlyintheexplanatorytextortheassociatedelectronicmaterialdevotedtothattopic.Theexplanatorytexttypicallyoccupiedhalfapage,butsometimesonlyoneortwolines.Weaskedthe20mathematicseducationleaderswhethertheythoughttheamountofclassroomreasoninghadchangedsincethe1999study.Theintroductionofbetterelectronic

resourceswastheonlyreasongivenmorethanonceforsuggestingthattheremighthavebeenpositivechange.

Thefirstobservationfromthetextbookstudyisthatmathematicalresultsareestablishedusingavarietyofdifferentmodesofreasoning.Mostofthetextbooksmadesomeattempttoexplaineveryruleratherthansimplypresenting‘ruleswithoutreason’.Textbooks,andgoodlessons,buildanunderstandingofmathematicalresultsbyofferingarangeof‘didacticexplanations’,includingbutnotrestrictedtoage-appropriateversionsof‘proper’mathematicalproofs.Thephrasedidactic explanationdoesnotimplyaverbaldemonstrationprovidedbytheteacherortextbookinacolloquially‘didactic’manner,butisintendedtorecognisethattherearemanyusefulexplanationsforstudentsinadditiontoformalproofs.Adidacticexplanationmaybeevidentthroughguideddiscovery,useofamanipulativemodel,adatagatheringactivity,orateacherpresentation.

Manytextbooksprovidemorethanoneexplanationforaresult.Whilemultiplemathematicalproofsofaresultareinasenseredundant(onegoodproofsufficestoprove),inteachingitisbeneficialtooffermultiplewaysofestablishingthesameresult.Sevendifferentmodesofexplanationswereidentified.Inafewcases,resultsareprovedbydeductionusingageneralcase,inawaythatcloselyapproximatesstandardmathematicalproofs,althoughatalowlevelofformality.Deductivereasoningisalsoevidentinotherways.SincestudentsatYear8donotspeakalgebrafluently,deductionisoftennotfromageneralcase,butfromaspecialcasethatisintendedtobegeneral.So,forexample,studentslearnedthatmultiplyingtwonegativesresultsinapositivebycleverlyextendingthe5timestabletonegativeintegers.Suchexpectationthatstudentswillseethegeneralintheparticularisvery


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commoninallmathematicsteaching(e.g.demonstratinghowtocarryoutanalgorithm),butthetextbooksdidnotdrawanyattentiontotheneedtothinkofthespecificcaseinageneralway.Thisisonesimplewayinwhichstudents’appreciationoftheuniquefeaturesofmathematicalreasoningcouldbeimproved,evenbeforetheyhavetheformalmathematicallanguagetodealwithitwell.

Didacticexplanationsusinginductivereasoningthatismoreappropriatetosciencethanmathematics,arecommon.Sometimesaruleisconfirmedbyshowingthatinspecificinstancestherulewouldgivethesameresultascouldbepredictedfromamodel(forexample,theresultofsharingaquarterofapizzabetweenthreepeoplecouldbeshowntobethesameastheanswerobtainedbyfollowingtheto-be-learnedrule).Atothertimes,studentsmeasureorcounttoempiricallydiscoverarulefromdata,suchastheanglesumofatriangleis180degrees.Inafewinstances,thetextbooksmadeitclearthattestingafewcaseswasnotanadequatemathematicalproof,butthiscouldcertainlybedonemoreoftentoimprovestudentawarenessofreasoning.Manyoftheempiricalactivitiesseemtoustohavesubstantialpedagogicalvalue(asnotedabove,havingmultiplemethodsaddstolearning),buttextbookscouldcommentthattheirroleisinmathematicaldiscoveryratherthaninproof.

Insomecases,the‘explanations’madenocontributiontodevelopingmathematicalthinkingatall.Sometimes,therewassimplyastatementorappealtoauthority(e.g.Euclidoracomputer),andothersdiscussedloosequalitativeanalogieswhichmayhavehadsomemnemonicvaluebutwerenotmodellingthemathematicalessence.

Lookingovertheresults,itwasclearthatthesetextbooksgenerallypaidreasonableattentiontomathematical

reasoninginexplanations,anditisdoesnotseemthatprevalenceof‘textbook’teachingisanadequateexplanationforthelackofreasoningevidentinAustralianclassroomsinthevideostudy(althoughrelatedfactorssuchasaprevalenceoflowcomplexityproblemsinthetextbookscertainlycontribute).However,apartfromofferingexamplesofreasoning,therewerefewinstancesofinstructioninmathematicalreasoning.Amongstthe69instancesexamined,oneexceptionwasthattwotextbooksexplicitlyrejectedmeasuringforfindingtheanglesumofatriangleinfavourofadeductiveproof.Intheotherexception,atextbookmentionedthatanexplanationpresentedforaspecificcasecouldalsobeappliedinallothercases,explicitlypointingtothegeneralitythatwasrequired.Attentiontoinstructioninreasoning,andtopointingoutkeyelementsofreasoning,wouldenrichthedidacticexplanationsgiven.

Wefoundthatthenatureofthereasoningdependsontheresultbeingexplained.Alltextbookshadatleastonedeductiveexplanationoftheformulafortheareaofatrapezium,butonlyhalfcontaineddeductiveexplanationsfortheanglesumofatriangle.Thenatureofthereasoningalsovariesfromtextbooktotextbooksincedifferentbooksarewrittenwithdifferentstudentaudiencesinmind.Intheinterviewstudy,oneofthemostcommonexplanationsforallfeaturesoftheshallowteachingsyndromewasthedifficultyofprovidingsuitablematerialofthisnaturetoamixedabilityclass.Overcomingthisdifficultyisnotassimpleassomepeopleclaim.

Inthetextbooks,explanationsweregenerallyverycurtailedandusuallyomittedbasicreasoning(forexample,statingthatafindingaboutaspecificcasealsoappliesingeneral).Hencetheexplanationsareunlikelytostandalone,andstudentsmustrelyonteacherstoelaborate.Itisunlikelythatall

teacherscanpresenttheseelaborationsfromthematerialprovided,sothisfindingfurtherhighlightstheoftencitedneedforteacherstopossesssufficientlystrongmathematicalknowledgeanddeepmathematicalpedagogicalcontentknowledge.Thishighlightsanotherstrongthemeoftheinterviewstudy,wheremanyoftherespondentsexpressedstrongconcernthatteachersteachingout-of-fieldneededconsiderablymoresupporttodoagoodjobontheworkingmathematicallythemes.

ForestablishingafirmerplaceformathematicalreasoninginAustralianclassroomsthanithasatpresent,Isuggestthefollowing.

1 Althoughallaspectsofworkingmathematicallyaretaughtduringengagementwiththecontentofmathematics,thisdoesnotmeanthattheyshouldnoteverreceiveexplicitattention.Thisappliesatthelevelofclassroomtasks,classroomdiscourse,unitplanningandcurriculumdescription.Inclassroomteaching,asinthetextbooks,therearemanyopportunitieswhereinstructioninreasoningissimpletoadd.

2 Adescriptionisneededofadevelopmentalpathinmathematicalreasoningacrossthegrades,thatwouldgiveteachers,textbookauthorsandcurriculumwritersasenseofwhattypeofreasoningtheycanexpectandencourageateachlevelandinwhatdirectionsstudents’reasoningshouldbedeveloped.Thiscouldnotbeasspecificasinthecontentstrands,butitcouldstillbehelpfulindevelopingasharedvocabulary,cleargoalsandexpectations.

3 Guidanceforteachersbeprovidedontheusefulnessofdidacticexplanations,thedistinction(insomecases)withage-appropriateproof,andwaysofevaluatingthem.


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4 Themajorpurposeofexplanationsinthetextbooksseemedtobetoderivearuleinpreparationforusingitintheexercises,ratherthantogiveexplanationsthatmightbeusedasathinkingtoolinsubsequentproblems.Changingthispracticecouldgivereasoningmoreprominence.

Acknowledgement

IthankthesurveyparticipantsforgenerouslygivingtheirtimeandsharingtheirexpertiseandacknowledgethefinancialsupportoftheAustralianResearchCouncilDiscoveryGrantDP0772787‘TheShallowTeachingSyndromeinSchoolMathematics’forpartofthiswork.

ReferencesPolya,G.(1945)How to solve it.

Princeton,NJ:PrincetonUniversityPress

Stacey,K.(2003).Theneedtoincreaseattentiontomathematicalreasoning.InH.Hollingsworth,J.Lokan&B.McCraeTeaching Mathematics in Australia: Results from the TIMSS 1999 Video Study.(pp119–122).Melbourne:ACER.

Stacey,K.,&Vincent,J.(2009).ModesofreasoninginexplanationsinAustralianeighth-grademathematicstextbooks.Educational Studies in Mathematics, 3,271–288.


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Thesocialoutcomesoflearningmathematics:Standard,unintendedorvisionary?

Paul�ErnestUniversity of Exeter, UK

PaulErnestisemeritusprofessorofphilosophyofmathematicseducationatExeterUniversity,UnitedKingdom,visitingprofessorinOsloandTrondheim,Norway,andadjunctprofessoratHopeUniversity,Liverpool,UnitedKingdom.Hismainresearchinterestsconcernfundamentalquestionsaboutthenatureofmathematicsandhowitrelatestoteaching,learningandsociety.HehaslecturedandpublishedwidelyonthesesubjectsandhismostcitedbooksareThe Philosophy of Mathematics Education,Routledge,1991,andSocial Constructivism as a Philosophy of Mathematics,SUNYPress,1998.In2009hewaskeynotespeakerattheworldclassPME 33researchconferenceinGreece.ProfessorErnestfoundedandeditsthePhilosophy of Mathematics Education Journal,accessedviahttp://www.people.ex.ac.uk/PErnest/.Recentspecialissueshavefocusedonmathematicsandsocialjustice,andmathematicsandart.

AbstractWhyteachmathematics?Whyshouldstudentsinschoollearnmathematics?Whatareourintendedaimsandtheoutcomesofteachingandlearningmathematicsinschool?TooffermyanswerstothesequestionsIfinditusefultodistinguishthreegroupsofaims/outcomes:

1 Standardaimsofschoolmathematics–whataregenerallyagreedtobethebasicorstandardreasonsforteachingthesubject?

2 Unintendedoutcomesofschoolmathematics–arethereunexpectedandunintendedoutcomesoftheprocessforsomeorallstudents?

3 Visionaryaimsforschoolmathematics–whatdoweasmathematicseducatorswishtoseeasbothaimsandoutcomesofschoolmathsteaching/learning?Whatnewemphaseswouldenhanceourstudentsandindeedsocietybeyondwhatwedonow?

The standard aims of school mathematics

Thesearebasicandfunctionalgoalsthataimtodevelopthefollowingcapabilities:

1.� Functional�numeracy

Thisinvolvesbeingabletodeploymathematicalandnumeracyskillsadequateforsuccessfulgeneralemploymentandfunctioninginsociety.Thisisabasicandminimalrequirementforallattheendofschooling,excludingonlythosefewwithsomepreventativedisability.

2.� Practical,�work-related�knowledge

Thisisthecapabilitytosolvepracticalproblemswithmathematics,especially

industryandwork-centredproblems.Thisisnotnecessaryforall,forthedepthandtypeofproblemsvaryacrossemploymenttypes,andmostoccupationsrequiringspecialistmathematicsalsoprovidespecialisttraining.However,astrongcasecanbemadeforschoolprovidingthebasicunderstandingandcapabilitiesuponwhichfurtherspecialistknowledgeandskillscanbebuilt.

3.� Advanced�specialist�knowledge

Thisknowledge,learnedinhighschooloruniversity,isnotanecessarygoalforalladults,butsuchadvancedstudyleadstoahighlynumerateprofessionalclass,asexistsinFrance,Hungary,etc.,whereallstudentsstudymathematicstoaround18yearsofageminimum.Advancedspecialistknowledgeisneededbyaminorityofstudentsasafoundationforabroadrangeoffurtherstudiesatuniversity,includingSTEMsubjects,aswellasmedicalandsocialsciencestudies.Clearlythisoptionmustbeavailableinanadvancedtechnologicalsociety,andindeedmorestudentsshouldbeencouragedtopursueit,butitshouldnotdominateordistorttheschoolmathematicscurriculumforall.

Thesethreecategoriesconstituteusefulornecessarymathematicsforallorsome,primarilyforthebenefitofemploymentandsocietyfromaneconomicperspective,aswellassustainingmathematicsandmathematicalintereststhemselves.Theyalsobenefittherecipientstudentsintermsoffunctioninginsociety,workandfurtherstudy.

Unintended outcomes of school mathematics

Whatcouldtheunintendedoutcomesofschoolmathematicsbe?WhatIhaveinmindarethevalues,attitudesandbeliefsthatstudentsdevelopduring


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theiryearsofschoolingthatarenotplannedorintended,outcomesofwhatisknownasthe‘hidden’curriculumofschooling.Theseconcernbeliefsaboutthenatureofmathematics,aboutwhatisvaluableinmathematics,andaboutwhocanbesuccessfulinmathematics.Thesebeliefsinclude:

• Mathematicsisintrinsicallydifficultandinaccessibletoallbutafew.

• Successinmathematicsisduetofixedinheritedtalentratherthantoeffort.

• Mathematicsisamaledomain,andisincompatiblewithfemininity.

• Mathematicsisanabstracttheoreticalsubjectdisconnectedfromsocietyandday-to-daylife.

• Mathematicsisabstractandtimeless,completelyobjectiveandabsolutelycertain.

• Mathematicsisuniversal,value-freeandculture-free.

Everyoneofthesebeliefsiswrong,andmanyofmywritingsoverthepast30yearshavebeendevotedtoshowingthis(Ernest1991).Thegoodnewsisthatagrowingnumberofresearchersandteachershavecometorejectthesebeliefs.Furthermore,theiracceptancehasalwaysvariedgreatlybycountryandculture,soforexampleAsiancountriestypicallysubscribetothebeliefthatmathematicalsuccessisduetoeffortratherthanintrinsicability.

Thebadnewsisthatsuchbeliefsarestillheldbymanystudentsandparents.Suchbeliefsarestillcommunicatedthroughpopularimagesofmathematicswidespreadinsocietyandthemedia,andintheimageofmathematicspresentedinsomeclassrooms.

Onewidespreadoutcome,althoughfarfromuniversal,isthatmanystudentsdevelopnegativeattitudesaboutmathematicsandabouttheirownmathematicalcapabilities.Aswehave

learntfromsport,attitudesarevitaltosuccess,andforstudentsalackofconfidenceintheirmathematicalabilitiesbecomesaself-fulfillingprophecy–afailurecycle(Figure1).

Poorconfidenceandmathsself-concept;possible

mathsanxiety

➚ ➘

Failureatmathematical

tasks

Reducedpersistence&learning

opportunitiesmathsavoidance

Figure�1:Thefailurecycle

Takeanotherexample.Despiteprogress,mathematicsisstillwidelyseenasamaledomain,andalthoughgirlsnowequalboysinmathematicalachievementat16yearsofageorso,toomanywomenstilldoubttheirownabilitiesandchoosenottopursuemathematicsrelatedstudiesorcareersafterthisage,

Inmyview,values,images,beliefsandattitudesaboutmathsunderliemanyofthedifferencesinlearningoutcomesobservedacrossdifferentgroupsofstudentsdefinedintermsofsex,socio-economicstatusandethnicity.Forexample,inAustralia,mathematicsperformanceofIndigenousAustralianscanlagovertwoyearsbehindthatofnon-Indigenousstudents(QueenslandStudiesAuthority,2004).ButafullaccountofsuchinequalitiesrequiresmorecomplexexplanationsinvolvingsuchnotionsasBourdieu’sculturalcapitalandstructuralinequalitiespresentinsociety,aswellasthemathsrelatedmisconceptionsdiscussedhere.

Visionary goals for school mathematics

Thetraditionalmathematicscurriculumisdefinedintermsofmathematical

contentanditsuse.InsteadIwanttomoveawayfromcontentandproposeaimsformathematicsthatareempoweringandbroadeningforstudents.Studentsshoulddevelop:

4 Mathematicalconfidence

5 Mathematicalcreativitythroughproblemposingandsolving

6 Socialempowermentthroughmaths(criticalcitizenship)

7 Broaderappreciationofmathematics.

Thesefouraimsarelessdirectlyutilitariansincetheyaremoretodowithpersonal,culturalandsocialrelevance,althoughultimatelyIbelievetheyhavepowerfulincidentalbenefitsforsociety,aswellasforindividualstudents.

4.� Mathematical�confidence

ElevatingthistoanaimshouldcomeasnosurprisegiventheimportanceIattachtoattitudesaspartoftheincidentaloutcomesofschoolmathematics.Mathematicalconfidenceincludesbeingconfidentinone’spersonalknowledgeofmathematics,feelingabletouseandapplyit,andbeingconfidentintheacquisitionofnewknowledgeandskillswhenneeded.Thisisthemostdirectlypersonaloutcomeoflearningmathematics,ituniquelyinvolvesthedevelopmentofthewholepersoninaroundedway,encompassingbothintellectandfeelings.Effectiveknowledgeandcapabilitiesrestonfreedomfromnegativeattitudestomathematics,andthefeelingsofenablementandempowerment,aswellasenjoymentinlearningandusingmathematics.Theselatterleadtopersistenceinsolvingdifficultmathematicalproblems,aswellaswillingnesstoacceptdifficultandchallengingtasks.MatchingbutinvertingthefailurecycleIdiscussedabove(seeFigure1)isthevirtuous,upwardlyspirallingsuccesscycle(seeFigure2).


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Pleasure,confidence,senseofself-efficacy,motivation

inmaths

➚ ➘

Successatmathstasksandmaths

overall

Effort,persistence,

choiceofmoredemandingtasks

Figure�2:Thesuccesscycle

Thiscycleisoneoftheintrinsicmechanismswhichdrawsustothepleasuresofsuccessandself-enhancementlikealightdrawsamoth.Indeedwecanpotentiallyturnafailurecycleintoasuccesscyclebysubtractingriskandmakingsuccessachievable.Inschoolthismeansreducingtheimportanceofexaminationsandpayingmoreattentiontothequalityofstudentlearningexperiences.

Inmyviewthisdomainofattitudes,beliefsandvaluesisoneofthemostimportantpsychologicaldimensionsoflearningmathematicsandweneedtopaymuchmoreattentiontoitinschool.Seeminglyinsignificantincidentscanswitchalearneronoroffmathematics,andweneedtobemoresensitivetothisinourteaching.

5.� Mathematical�problem�posing�and�solving

Mathematicsistoooftenseenasanon-creativeandmechanicalsubject,butdeployingmathematicalknowledgeandpowersinbothposingandsolvingproblemsistheareaofgreatestpotentialforcreativityinschoolmaths.Studentschoosewhichmodelsandapproachestouseintheirsolutions.Problemsolvingiswidelyendorsed,buttoooftenfocusedonroutineproblems.Trueproblemsolving,thecreativeuseofmathematics,requiresnon-routineproblems,inwhichnewmethodsandapproachesmustbecreated.Problemposing,thearticulationandformulation

ofquestionsandproblemstobesolved,hasbeenmoreneglectedinmaths.Butitenablestheseeingofmathematicalconnectionsbetweensuperficiallydiversequestionsandtopics,andtheframingofquestionsbyanalogy.Itinvolvesseekingmodelsfordifferentaspectsoflifeormathematicalpatternsasdiscoveredorchosenbystudentsthemselves.Thisiswherefullcreativityflowersthroughstudentchoicesateverystage:problemormodelformulation,thechoiceofmethodstoapply,andtheconstructionofsolutions.

6.� Social�empowerment�through�mathematics

Contrarytopopularbelief,mathematicsisapoliticalsubject.Mathematicsshouldbetaughtinordertosociallyandpoliticallyempowerstudentsascitizensinsociety.Itshouldenablelearnerstofunctionasnumeratecriticalcitizens,abletousetheirknowledgeinsocialandpoliticalrealmsofactivity,forthebettermentofboththemselvesandfordemocraticsocietyasawhole.Thisinvolvescriticallyunderstandingtheusesofmathematicsinsociety:toidentify,interpret,evaluateandcritiquethemathematicsembeddedinsocial,commercialandpoliticalsystemsandclaims,fromadvertisements,suchasinthefinancialsector,togovernmentandinterest-grouppronouncements.Economicsisappliedmathematicsandthisisthemainlanguageofpolitics,powerandpersonalfunctioninginsociety.Everycitizenneedstounderstandthelimitsofvalidityofsuchusesofmathematics,whatdecisionsitmayconceal,andwherenecessaryrejectspuriousormisleadingclaims.Ultimately,suchacapabilityisavitalbulwarkinprotectingdemocracyandthevaluesofahumanisticandcivilisedsociety.

CriticalcitizenshipthroughmathematicsisamajortopiconitsownandtheCriticalMathematicsEducation

movementhasspringuptodealwiththeoryandpracticeinthisarea.TherearemanyrelevantpublicationssuchasSkovsmose(1994),Ernest(2001)andthespecialissueofThe Philosophy of Mathematics Education Journal forthcomingsummer2010.

7.� Appreciation�of�mathematics�

Thelastofmyproposedsevenaimsorcapabilitiesisthedevelopmentofmathematicalappreciation.Thereisananalogybetweencapabilityversusappreciationinmathematics,ontheonehand,andthestudyoflanguageversusthatofliterature,ontheother.Mathematicalcapabilityislikebeingabletouselanguageeffectivelyfororalandwrittencommunication,whereasmathematicalappreciationparallelsthestudyofliterature,concernedwiththesignificanceofmathematicsasanelementofcultureandhistory,withitsownstoriesandculturalpinnacles,sothattheobjectsofmathematicsareunderstoodinthisway,justasgreatbooksareinliterature.

Theappreciationofmathematicsitself,anditsroleinhistory,cultureandsocietyingeneral,involvesanumberofdimensionsandroles,includingthefollowing.

• Havingasenseofmathematicsasacentralelementofculture,artandlife,presentandpast,whichpermeatesandunderpinsscience,technologyandallaspectsofhumanculture.Thisextendsfromsymmetryinappreciatingelementsofartandreligioussymbolism,tounderstandinghowmodernphysicsandcosmologydependonalgebraicequationssuchasEinstein’sE=mc2.Itmustincludeunderstandinghowmathematicsisincreasinglycentraltoallaspectsofdailylifeandexperience,throughitsimportincommerce,economics(e.g.,thestockmarket),telecommunications,ICT,and


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theroleitplaysinrepresenting,codinganddisplayinginformation.However,itmustberecognisedthatmathematicsisbecominginvisibleasitisbuiltintothesocialsystemsthatbothcontrolandempowerusinourincreasinglycomplexsocietiesandlives.

• Beingawareofthehistoricaldevelopmentofmathematics,thesocialcontextsoftheoriginsofmathematicalconcepts,itssymbolism,theoriesandproblems.Theevolutionofmathematicsisinseparablefromthemostimportantdevelopmentsinhistory,fromancientsocietiesinMesopotamia,Egypt,IndiaandGreece(numberandtaxandaccounting,geometryandsurveying)viamedievalEuropeandtheMiddleEast(algorithmsandcommerce,trigonometryandnavigation,mechanicsandballistics)tothemodernera(statisticsandagriculture-biology-medicine-insurance,logicanddigitalcomputing-media-telecommunications).Thisincludesbeingawareofethnomathematics,whichstudiesinformalculturallyembeddedmathematicalconceptsandskillsfromculturesaroundtheglobe,bothruralandurban,pastandpresent.

• Havingasenseofmathematicsasauniquediscipline,withitscentralbranchesandconceptsaswellastheirinterconnections,interdependencies,andtheoverallunityofmathematics.Thisincludesitscentralrolesinmanyotherdisciplinesasappliedmathematics.Aftermanyyearsspentstudyingmathematicslearnersshouldhavesomeconceptionofmathematicsasadiscipline,includingunderstandingthatthereismuchmoretomathematicsthannumberandwhatistaughtinschool.

• Understandingthewaysthatmathematicalknowledgeisestablishedandvalidatedthroughproofisalsoimportant,aswellthelimitationsofproof.Ibelievethisshouldincludeintroductiontothephilosophyofmathematics:understandingthattherearebigquestionsandcontroversiesaboutwhethermathematicsisdiscoveredorinvented,aboutthecertaintyofmathematicalknowledgeandaboutwhattypeofthingsmathematicalobjectsare.Beingawareofsuchcontroversiessupportsamorecriticalattitudetothesocialusesofmathematics,aswellaswithstandingattributionsofcertaintytoanythingmathematical.

• Learnersshouldgainaqualitativeandintuitiveunderstandingsomeofthebigideasofmathematicssuchaspattern,symmetry,structure,proof,paradox,recursion,randomness,chaos,infinity.Mathematicscontainsmanyofthedeepest,mostpowerfulandexcitingideascreatedbyhumankind.Theseextendourthinkingandimagination,aswellasprovidingthescientificequivalentofpoetry,offeringnoble,aesthetic,andevenspiritualexperiences.

Aretheseaimsconcerningappreciationfeasibleforschool?Evenbigideaslikeinfinitycanbeappreciatedbyschoolchildren.Manyaninterested8-year-oldwillhappilydiscusstheinfinitesizeofspace,orthenever-endingnatureofthenaturalnumbers.

Inmathematicsweareprivilegedtohavearound2000hoursofcompulsoryschooltimeovertheyears–surelywecanaffordtospendsometimeonthesevisionaryaims–theyhavethepotentialtohelpbuildmoreconfidentandknowledgeablestudentsandcitizens,anddareIsayit,abettersociety?

ReferencesBourdieu,P.andPasseron,J.C.(1977)

Reproduction in Education, Society and Culture,London:Sage.

Ernest,P.(1991).The Philosophy of Mathematics Education.London:FalmerPress.

Ernest,P.(2001).‘CriticalMathematicsEducation’.InGates,P.(Ed.),Issues in mathematics teaching,pp.277-293.London:Routledge/Falmer..

QueenslandStudiesAuthority.(2004).Overview of statewide student performance in aspects of literacy and numeracy: Report to the Minister for Education and Minister for the Arts.Brisbane,QLD:QueenslandStudiesAuthority.

Skovsmose,O.(1994).Towards a philosophy of critical mathematics education.Dordrecht:Kluwer.

Philosophy of Mathematics Education Journal (2010).SpecialIssueonCriticalMathematicsEducation,no.25,Summer2010.Accessedfromhttp://people.exeter.ac.uk/PErnest/May2010

Concurrent papers


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Robyn�JorgensenGriffith University

RobynJorgensenisProfessorofEducationatGriffithUniversity.ProfessorJorgensenhasworkedintheareaofequityinmathematicseducationformorethantwodecades.Herworkexploreshowthesocial,politicalandculturalcontextscontributetotheexclusionofsomestudentsastheycometolearnschoolmathematics.Theparticularfociofherworkhavebeenintheareasofsocialclass,geographicallocation(ruralandremote)andIndigenouscontextsandlearners.SherecentlytookleavefromtheuniversitysectortoworkwithAnangucommunitiesinCentralAustralia.TheimmersioninthelivedworldsofremoteAboriginaleducationhasprovidedkeyinsightsintothedeliveryofWesterneducationinremoteAustralia.

AbstractOnWesternmeasuresofeducationperformance,suchasNAPLAN,studentslivinginremoteareasofAustraliaareover-representedinthetailofperformance.ThegapbetweenIndigenousandnon-Indigenouslearnersinnumeracywidensasstudentsprogressthroughschool(ACARA,2009).Thispresentationexploresthecontextwithinwhichthisgapiscreatedandofferssomesuggestionstoteachers,educationalresearchersandpolicymakersonreasonsforthisgap,butalsoonhowthegapmaybeaddressed.

Introduction

ProvisionofqualitylearningforIndigenouslearners,particularlyforstudentswhosehomecultureisstillverystrongandnotcontiguouswithWesternculture,remainsanelusivechallenge.DevelopingqualitylearningenvironmentsforIndigenousstudentsrequiresaholisticapproachtopracticeandpolicy.Keepingmathematicseducationisolatedfromthecomplexmilieuinwhichlearningoccursfailstoincorporateandaddressthecompetingdemandsfacedbyteachersandeducationproviders.InthissessionIconsiderthreekeyelementsthatimpactonmathematicsteachingandlearning:attendance,language/cultureandmathematics.Allofthesevariablesimpactonhowteachersandeducationsystemsplanforqualitylearning.

Inthemodelproposedinthispresentation,Iwishtoextendthethinkingofmathematicseducatorstoencourageagreaterawareness,recognitionandembodimentofthewiderissuesthatshape,constrainandenablemathematicslearning.Withoutconsiderationoftheseothervariables,thefieldofmathematicseducationisimpoverishedandunable

toaddressthesystemicmarginalisationofIndigenousAustralians.Ifthefieldcontinuestoresearchandtheoriseaboutmathematicseducationdivorcedfromtherealityoftheteachingcontext,thefieldwillremainimpoverishedandunabletoaddressthesystemicfailureofgenerationsofIndigenouslearners.

PlanningforLearning

Attendance

MathematicsLanguage/culture

Figure�1:Planningforlearningmathematics

TodevelopamoreholisticsenseoftheissuesofteachingmathematicsinsomeofthemostdisadvantagedcontextsintheAustralianeducationallandscape,Iproposeamodelthatincorporates,butisnotlimitedto,anumberofkeyissuesimpactingonthedevelopmentofqualitylearningforIndigenousstudents.InthispaperIcontendthatwithoutregularattendanceandsubsequentengagementinmathematicslearning,theissuesofcultureandlanguagemustalsobeconsideredaspartofthenexusofmathematicseducation.Failuretodoso,willresultinthecontinuedpracticesthathaveforgenerationsdealtfailuretotoomanystudents.

Attendance

Attendanceisthemostchallengingaspectofeducationdeliveryinremotecommunities.Theneedtoattend(andengage)isperhapsthebiggestchallengeforteachers–ofmathematicsandothersubjects–increatingqualitylearning.Thepressureonschoolstohavegoodattendancefiguresmeansthatthereisarangeoftechniquesusedtorecordstudentattendance.Typicallystudentsmayappeartobemarkedas

IssuesofsocialequityinaccessandsuccessinmathematicslearningforIndigenousstudents


28

attending,buttherealityisthattheymayhaveappearedforonlyashorttimeintheday.Assuch,attendancefiguresareoftensignificantlyinflatedintermsoftherealnumberofstudentsattending.Thisrollingattendancepresentsuniqueproblemsfortheteachingofmathematics.Notonlyisattendanceirregularoveraperiodoftime,butalsoovertheday.Assuch,bothshort-termandlong-termplanningarecompromised.

AscanbeseeninTable2,forsecondaryIndigenousstudents,attendanceratesatschooldecreaseswiththelevelofremoteness.Similartrendsoccurforprimaryschoolstudents.Forexample,for17-years-oldslivinginmajorcities,44percentofIndigenousstudentsattendschool.Incontrast,only16percentof17-year-oldIndigenousstudentslivinginremoteareasattendschool.

Teachermoraleisseriouslycompromisedbypoorattendance.Neversureiftherewillbe1or2studentsor20students,teachersarerequiredtobeprofessionalandprepareasiftherewillbeafullcontingentofstudentsattending.However,thepoorattendanceisreflectedinlearning

outcomessothatforanycohortofstudents,thevarianceinperformancelevelsisconsiderable.Thismakesplanningforlearningcomplexandunpredictable.Thefrustrationcausedtoteachersbynon-orirregularattendancehasadevastatingeffectformanyteachersontheirsenseofidentity.Asoneteachercommented,‘Ididnotspendfouryearstrainingtohaveaclasswithnostudentsturningup.’

Withoverallpoorattendance,teachersinremoteareasarefacedwithsubstantiveissuesinhowtoaddressthesignificantgapsinlearning.WhilethereisaconsiderablepushfromIndigenouseducatorssuchasChrisSarra(1995)tohavehighexpectationsoflearners,thisgoalcanbesomewhatmisplaced.Theissuesaroundattendancemeansthatwhiletheteachersmayholdhighexpectationsoflearninginmathematics,thelevelsofachievementandunderstandingsarequitelimitedforstudents.Thismakesthehighexpectationsmantradifficultduetotheverylimitedachievementandneedforbackfillingofmathematicalideas.ThegapsformanyIndigenouslearnersareprofound.Manybasicconceptsand

Table�1:SecondaryschoolattendancebyIndigenousstatusandage,2006

Age�in�Years Indigenous�% Non-Indigenous�%

15 73 89

16 55 81

17 36 66(Source:ABS,2010)

Table�2:Secondaryschoolattendancebyremotenessareabyage,Indigenouspersons,2006

Age�in�years

Major�cities

Inner�regional

Outer�regional

RemoteVery�

remote

15 % 77 77 76 67 53

16 % 60 58 60 49 34

17 % 44 38 37 29 16(Source:ABS,2010)

understandingsarenotevident,soholdinghighexpectationsmaybeaworthyideal,thepracticalramificationsforsecondary-agedstudentsrequiresaprimarylevelofwork.Thisrendersthe‘highexpectations’asmisplacedintermsofbenchmarkingactivities.

Language and culture

Inmanyremoteareas,homecultureisstillastrongpartofthelifeworldsofIndigenousstudents.Theseculturalactivitiesimpactonlearninginmanyways.First,culturaleventscandemandtimeoutofschool.InCentralAustralia,Men’sBusinessmayrequiremanyyoungfellastobeoutofschoolforamonthormore,aswellastheimpactonthecommunitymembersthroughwhichMen’sBusinessisundertaken.Otherculturalevents,suchasSorryBusiness,similarlyimpactonattendance.InNorthernArnhemlandtherehavebeenmovestoshiftschooltermstoallowfortheextendedculturalactivitiesoverthewetseasonwhichmaygoforseveralmonths.Collectively,theseeventstakepriorityoverschooling,thusresultinginsubstantiveperiodsofmissedschool.

Atamorelocallevel,cultureimpactsontheinteractionsinclassrooms.Thismaybeinthewaythatthestudentsinteractwiththeteacherand/orcommunity.Thestylesofinteractionandquestioningareoftendifferentfromthoseofmainstreameducation.Forstudentscomingintoschool,thereisaneedtoconstitutetheirIndigenoushabitustoenablethemtoaccessthedialogicpatternsinorderto‘crackthecode’ofclassroompractice.Forexample,posingquestionsinclassrooms–suchas‘Whatisthesumof15and23?’–ismetwithabarrageofanswers.Studentsplayadifferentgametotheteacher.Whiletheteacher’sgameisoneinwhichhe/sheisseekingthestudentstoaddtwonumbersandcometoatotalof38,thestudents’gameisoneofrespondingwithany


29

answer.Thesetwodialogicpatternsarequitedifferentingoalsothatthereisconsiderablescopeformisrecognitionoftheoutcome.

Languageandcultureareintrinsicallyintertwinedsothatthecultureisrepresentedthroughlanguage.Asthelanguagegameaboveindicates,thegoalsoftheteachersmaybedifferentfromthoseofthestudentsbutthesegoalsareintrinsicallyinterwovenwiththecultures.InPitjantjatjara,languageuseisveryfrugalsothatthereisoftenlittlesaidandwhatissaidisverycontracted.Thelanguagestructureisonewithbrevityinspeech.Thisisevidentinthelanguagedevelopedwithinthecontextofdesertpeople.

Prepostions

InPitjantjatjara,therearelessthan10prepositions,whereasEnglishhasmorethan60.Ifthelanguageofmathematicsisconsideredinconcertwiththepedagogicrelaywhereconceptsaretaught/learnedthroughlanguage,theuseofprepositionsincomingtolearnmathematicsisprofound.Ashasbeenarguedelsewhere(Zevenbergen,2000,2001),comingtolearnmathematicsisheavilyassociatedwiththeuseofprepositions.Howonelearnsnumbersenseisthroughcomparisonsandplace.Considerthefollowingstatements–Whichnumberisbigger than4?;Whichnumberis2more than6?;Whichnumbercomes before3?;Whichnumbercomes after11?Theselittlewordsaresignificantinhowstudentslearnthevalueandorderofnumbers.

ImaginethedifficultiesofIndigenouslearners,whooftenhavehearingproblems,differentiatingbetweenoffandof.InPitjantjatjaraforexample,thereisno‘f’sound,sotermssuchas‘football’ispronouncedas‘pootball’.Intryingtohearthedifferencebetweenoffandofwhenthereisnosoundinthehomelanguagewouldbeverydifficult.Yet,inmathematics,these

differencesinmeaningaresignificant.Ashasbeenidentifiedinotherlearnersofmathematics(Zevenbergen,Hyde,&Power,2001),theskillslearntinreadingtextsmeanthatskimmingisawelldevelopedstrategy,yetinmathematicsthehighlycontractedlanguagemeansthatsuchastrategyisverymisplaced.

Temporality

ManyIndigenousculturesliveinthehereandnowsothatlong-termplanningisaforeign/elusiveconcept.YetplanningunderpinsmuchofWesternthought.Thereareconsiderableexamplesofhowthenon-planningofIndigenouspracticesandeventsareatloggerheadswithWesternwaysofthinking.Theneedtoplanalongtripinthedesertisundertakenwithastrongsenseofgravityasitcanmeanlifeanddeath.Yet,formanyIndigenouspeople,thetripisoneofopportunityasthesenseoflifeanddeathisnotasparamountduetotheirintimateknowledgeofthedesertandsurvival.Thesetwoverydifferentworldviewsimpactontheprimarygoalofmuchofwhatistaughtinschoolsandthehomecultures.

Mathematics

Indrawingtogetherabsenteeismandculture,theimpactonmathematicsbecomesobvious.Inremotecommunities,thereisalackofnumberandtextsothatimmersioninnumberisdifficultinremotecommunities.SomeofthefundamentalassumptionsmadeinWesternworldviewsareverydifferentfromthoseofthebush.Travellingalongadirtroadmaybemeasuredinkilometres,withparticularmarkersatparticulardistances.However,travelinoutbackroadsismarkedbyothersignificantbearings–suchalandmarksorman-mademarkersratherthanaparticulardistance.Similarly,thequalityofroadsatapointintimeismoreprofoundthanthedistancetobetravelled.These

differencesmakeforverydifferentassumptionsthatunderpinlearningactivities.

Inmanyremotecommunities,theabsenceofnumberintheirworldviewsisobvious.Theneedfornumberisrelativetotheregion.AsWittgenstein(1953)arguedstrongly,ourknowledgesystemsderivefromandareshapedbythelanguagegamesthatareplayedoutinaparticularsystem.Theneedfornumberinremoteareasislimited.Forcoastalmobs,wheretradingwasmorelikelyakeenersenseofnumberismorerelevant,butthisisnotthecaseinremoteareas.Manystudentsdonotknowtheirageorbirthday;fewhavephonesinthehome;streetsarenotnamedornumbered;thereisnoneedforlargenumbers.Theirlifeworldsshapetheneedfornumber(orothermathematicalideas/concepts).

WhilenumbermaynotbeastrongaspectofmanyIndigenouscultures,thesenseofspaceisacute.InacomprehensivestudyofYolngulifeworlds,WatsonandChambers(1989)documentedthecomplexwaysinwhichlandwassigned.ForYolngu,thelandwasmarkedbyculturalandhistoricalevents.Theselandmarkswere‘sung’toyoungergenerationswhointernalisedthesestoriesandsodevelopedasenseoftheirland.ThesestoriesaremarkedlydifferentfromthoseofWesternconventions,yetservetomakestrongconnectionstotheland.

Planning for quality learningInordertocreateenvironmentsthatsupportaccessandsuccessinschoolmathematicsforIndigenouslearners,thethreekeyfactorsthathavebeenidentifiedinthispapermustbeconsideredinconcertwithanemphasisonplanningforlearning.Thelearningisforbothteachersandstudents.Therealityforteachinginremoteareasis


30

thattheteachingforceispredominantlyearlycareerteacherswhohavehadlittleornoexposuretoremoteeducation,toworkingwithIndigenousstudentsandcommunitiesandtoteachingasaprofession.Collectivelytheseexperiencescontributetotheidentifieddifficultieswithretainingteachersinremoteareas.Thehighturnoverratescanbeseentobeindicativeofthechallengesofremoteeducation.ThisclaimisnotnewandtheissueshavebeenrecognisedforsometimeascanbeseenintheHumanRightsandEqualOpportunitiesCommissionreport:

…schoolsmaysufferfromhighteacherturnover,alackofspecialistservices,arestrictedrangeofcurriculumoptionsandahighproportionofyounginexperiencedteachers.

(CommonwealthSchoolsCommission,1975:75–79)

ComingintoremotecontextstoteachIndigenousstudentswhoseattendanceisoftenlow,whohavegapsintheirmathematicalunderstandings,whosecultureandlanguagesaresignificantlydifferentfrommainstreamschools,createsasetofchallengesthatneedtobeaddressed.Teachersneedtodevelopskillsthatwillenablethemtolearntoplanandadapttothesecircumstances.Appropriateaccesstosuchskilldevelopmentiscriticalifsuccessfulchangeistobeimplemented.However,thismustalsobeconsideredwithintheconstraintsimposedbyeconomics,geographyandavailableresourcesforsuchskilldevelopment.Furthercompoundingtheissueofprofessionaldevelopmentistheriskofinvestmentinstaffwherethereisahighturnover.

PlanningforqualitylearningmusttakeintoconsiderationthesemultiplefactorsinordertoenableaccessandsuccessforIndigenouslearners.Neophyteandestablishedteachersneedtobeableto

developinnovativemodelsofplanningfordiversityinlearningneedsanddemandsofremoteeducation.WorkingwithintheexistingdominantparadigmswillnotyieldtheoutcomesrequiredforsuccessfulIndigenouseducationparticipationand/oroutcomes.

ReferencesAustralianBureauofStatistics.(2010).

Indigenous statistics for schools. http://www.abs.gov.au/websitedbs/cashome.nsf/4a256353001af3ed4b2562bb00121564/be2634628102566bca25758b00116c3d!OpenDocumentAccessedMay15,2010.

AustralianCurriculum,AssessmentandReportingAuthority.(2009).http://www.naplan.edu.au/reports/national_report.html.AccessedMay15,2010.

Sarra,C.(2005).Strongandsmart:ReinforcingaboriginalperceptionsofbeingaboriginalatCherbourgstateschool.UnpublishedPhD:MurdochUniversityhttp://wwwlib.murdoch.edu.au/adt/browse/view/adt-MU20100208.145610

Stokes,H.Stafford,J.&Holdsworth,R.(unknown).Rural and Remote school education: A survey for the Human Rights and Equal Opportunity Commission.Melbourne:YouthResearchCentre,UniversityofMelbourne.http://www.hreoc.gov.au/pdf/human_rights/rural_remote/scoping_survey.pdf.AccessedMay12,2010.

Watson,H.&Chambers,W.(1989)Singing the land, Signing the land.Geelong:DeakinUniversityPress.

Wittgenstein,L.(1953).Philosophical investigations.Oxford:Blackwell

Zevenbergen,R.(2000).‘Crackingthecode’ofmathematicsclassrooms:Schoolsuccessasafunctionoflinguistic,socialandculturalbackground.InJ.Boaler(Ed.),Multiple perspectives on mathematics teaching

and learning(pp201–223).Westport,CT:Ablex.

Zevenbergen,R.(2002).Mathematics,socialclassandlinguisticcapital:Ananalysisofamathematicsclassroom.InB.Atweh&H.Forgasz(Eds.),Social-cultural aspects of mathematics education: An international perspective(pp.201–215).Mahwah,NJ:Erlbaum.

Zevenbergen,R.,Hyde,M.,&Power,D.(2001).Language,arithmeticwordproblemsanddeafstudents:Linguisticstrategiesusedbydeafstudentstosolvetasks.Mathematics Education Research Journal,13(3),204–218.


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Tom�LowrieCharles Sturt University

TomLowrieisDirectoroftheResearchInstituteforProfessionalPractice,LearningandEducation(RIPPLE)atCharlesSturtUniversity.ProfessorLowrie’spreviouspositionsincludedworkingasaprimaryschoolclassroomteacher,teachingmathematicseducationandresearchmethodcoursestoundergraduateandpostgraduatestudentsatCSUandworkingwithclassroomteachersoncurriculumframeworks.PreviousadministrativepositionsincludebeingtheHead,SchoolofEducationandactingDeanoftheFacultyofEducationatCSU.

AsubstantialbodyofProfessorLowrie’sresearchisassociatedwithspatialsense,particularlystudents’useofspatialskillsandvisualimagerytosolvemathematicsproblems.Hehasco-authoredMathematics for children: Challenging children to think mathematically(nowinitsthirdedition)andhasbeentheEditoroftheAustralian Primary Mathematics Classroom Journal.ProfessorLowrie’scurrentresearchprojectsincludeAustralianResearchCouncilgrantswhichexamineyoungstudents’abilitytodecodeinformationgraphicsinmathematicsandMathematicsinthedigitalage:ReframinglearningopportunitiesfordisadvantagedIndigenousandruralstudents.

AbstractRepresentationisanimportantaspectofmathematics.Inrecentyearsgraphicsrepresentationshavebecomeincreasinglywidespreadassocietycomestotermswiththeinformationage.Althoughthemathematicscurriculahavenotvariedtoanyrecognisabledegreeinthepastdecadeorso,theassessmentproceduresassociatedwithmathematicseducationcertainlyhave.Thispresentationhighlightsthechangingnatureofstudents’spatialreasoningastheyengagewithdifferenttypesofmathematicsrepresentations.Acaseispresentedwhichdescribestheshiftfromstudents’useofencodingtechniquestorepresentmathematicalideastoanincreasingrelianceonstudentsdecodinggraphicalrepresentationsconstructedbyothers.ThepresentationanalysesanumberofstudentworksamplesastheywerevideotapedcompletingassessmentitemsfromtheNationalAssessmentPlanforLiteracyandNumeracy(NAPLAN).Implicationsfromthestudyincludetherecognitionthatstudentsneedtoacquiredifferentspatial-reasoningskillswhichallowthemtoconsider(andnavigate)alltheelementsofamathematicstask,includingspecificfeaturesofagraphicandthesurroundingtext.

Introduction

Althoughmathematicscurriculahaschangedlittleinthepasttenyearsthewayinwhichmathematicalideasarerepresentedandcommunicatedhasshifteddramatically.Untilrecently,mostmathematicstasksthatprimary-agedstudentswererequiredtosolvewereheavilywordbased,whereasthecurrentpractice,frombothcurriculumandassessmentperspectives,istohavemoregraphicsembeddedintotaskrepresentation(Lowrie&Diezmann,2009).Thisisunsurprisinggiventheincreaseduseofgraphicsin

societyandtheincreasingchallengeofrepresentingburgeoningamountsofinformationinvisualandgraphicforms.Theamountofinformationatanindividual’sdisposalandtheextenttowhichthisinformationcanbemanipulatedanddirectedtowardspecificpurposeshasalsoincreased(e.g.,thedetailedinformationavailableforweatherforecasts).Fromayoungage,childrenareexposedtovisualformsofcommunicationwithmoreintensityandengagement,whetherplayingcomputergames,navigatingwebpages,orinterpretingtherichdesignfeaturesofmoretraditionalpictorialrepresentations,andasaconsequencedifferentformsofsensemakingarerequired.

Withineducationcontextsincreasedattentionhasbeengiventotheroleofrepresentationinschoolmathematics(e.g.,NationalCouncilofTeachersofMathematics[NCTM]Yearbook,2001).Mathematicalrepresentationshavealwaysbeenviewedasanintegralcomponentoftheideasandconceptsusedtounderstandandengagewithmathematics(NCTM,2000);however,thestructureoftheserepresentationscontinuetoevolve.InthispresentationIarguethatthenatureanddegreeofinfluencemathematicalrepresentationshaveonteachingandlearningcontextshavechangedandthesechangeshaveemergedalmostunnoticed.

Representationstendtofallundertwosystems,namelyinternalandexternalrepresentations.Internalrepresentationsarecommonlyclassifiedaspictures‘inthemind’seye’(Kosslyn,1983)andincludevariousformsofconcreteanddynamicimagery(Presmeg,1986)associatedwithpersonalised,andoftenidiosyncratic,ideas,constructsandimages.Externalrepresentationsincludeconventionalsymbolicsystemsofmathematics(suchasalgebraicnotationornumberlines)orgraphicalrepresentations(suchasgraphsandmaps).

Primarystudentsdecodingmathematicstasks:Theroleofspatialreasoning


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Althoughthesetwosystemsdonotexistasseparateidentities(Goldin&Shteingold,2001),thereissomescope(andbenefit)forthinkingofthesetwoformsofrepresentationindifferentways.Internalrepresentationsofteninvolvetheprocessofencodinginformation.Encodinggenerallyoccurswhenstudentsconstructtheirownrepresentationsinordertosolveatask.Encodingtechniquesincludedrawingdiagrams,visualisingandspatialreasoning.Thesetechniquesprovidestudentswiththeopportunitytounderstandalltheelementsofanygivenprobleminawaythatismeaningfultothem,forexample,drawingacircleanddividingitintosegmentsinordertobetterunderstandafractionproblem.Bycontrast,decodingtechniquesareusedtomakesenseofinformationwithinagiventask,whentheinformationhasbeenrepresentedvisuallyforotherstosolve,forexample,interpretingamaptodeterminethecoordinatepositionofaspecificstreetcrossing.Tenyearsago,ahighproportionofmathematicstaskswereword-problembasedandteachersexplicitlytaughtheuristicswhichincluded‘drawadiagram’,or‘imaginetheproblemscene’.Theseapproachesrequiredencodingofinformation.Currently,ahighproportionoftaskshaveadiagramembeddedintherepresentation.Asaconsequence,itishardforstudentstothinkbeyondthediagramtoconstructrepresentationalmeaningandthusapproachestoproblemsolvingnowaremorelikelytorequiredecodingskills.

Thispresentationconsidersthechangingnatureofmathematicsrepresentationinclassroompractices,andanevolutioninstudentengagement–wherestudentsareincreasinglyrequiredtodecodeinformationbutatthesametimearelesslikelytoexperiencesituationsinwhichtheyarechallengedtoencodemathematicsideasandrepresentations.

Mandatoryassessmentpractices,suchastheNationalAssessmentPlanforLiteracyandNumeracy(NAPLAN)(MCEETYA,2009),fosterthischangeinstudentinformationprocessing.ThestructureandnatureofNAPLAN-liketaskspromotedecoding,especiallyinsituationswherestudentsarerequiredtogenerateamultiple-choicesolution.Ourstudies(e.g.,Lowrie&Diezmann,2009)haveshownthatstudentsarereluctanttoactuallydrawontheirtestbookletswhentheycompletequestionsintheNAPLAN.Otherformsofencoding,includinginternalrepresentations,areseldomevokedsincetheanswertothequestionsgenerallyappearonthepageandthisthusreducesthelikelihoodofstudentsutilisingotherformsofimagery.Moreover,thetypesofquestionsposedtypicallyrequirestudentstodecodeinformationfromthegraphicsembeddedinthetask.Byprovidingagraphicalrepresentationtoscaffoldthinking,awholenewsetofskillsandpracticesisbroughttothefore.Thecapacitytointerpretvariousformsofinformationisnowrequiredforstudentstosolvetasksandtheseskillsetsarequitedifferenttothoseneededwhenencodinginformation.

Encoding and decoding information in mathematics

WithcolleaguesIhavebeeninvestigatingstudents’encoding(Lowrie&Logan)anddecoding(Diezmann&Lowrie,2008;Lowrie&Diezmann,2007;Logan&Greenlees,2008)skillsastheysolvemathematicstaskscommonlyusedasassessmentitems.Theworkonencodinghasfocusedontheextenttowhichstudentsutilisepicturesordiagramstomakesenseoftasksandtheextenttowhichtheyevokeimagerytocontextualisetheproblem.Thestudiesthatinvestigatestudents’decodingskillshaveconsideredtheextenttowhichchildrenmakesenseofinformationgraphicsthat

havedifferentpurpose,structureandorientation.

Oneofourcurrentinvestigations(Lowrie&Logan)hassetouttoconsidertheinfluenceencodinganddecodingprocesseshaveonprimary-agedstudents’mathematicalthinkingastheycompletetasksintheNAPLAN.Grade3and5students(N=45)whosatthe2010NAPLANwereinterviewedonthe2009NAPLANbeforeattemptingthisyear’spaper.StudentswerevideotapedastheysolvedthetasksandexplainedtheirsolutionstotenitemsfromtherespectivegradeNAPLANtests.Theinterviewprotocolencouragedthestudentstoverbalisetheirthinkingandtorepresenttheirthinkinginwaystheyfeltappropriate(i.e.,writingdownnumbersordrawingapicture).Thesemi-structuredinterviewallowedstudentstheopportunitytoreflectuponanexperiencethatisotherwiseonlyaquantitativemeasureofperformance.

Representation and sense making with graphic-based tasks

Ofthe75itemsacrosstheGrade3andGrade5tests,fewitemswouldbeclassifiedastraditionalword-basedproblems.Infact,only13ofthe35Grade3items(37%)and15ofthe40Grade5items(38%)didnotcontainagraphicwithinthetask.Moreover,only15items(20%)acrossthetwotestswouldbeconsideredtraditionalwordproblems.Thestudentsseldomutilisedencodingskillstosolvethetasks,especiallyinternalrepresentationslikedrawingadiagramandconstructingpersonalimagesorrepresentations.Whenstudentsdidconstructsuchrepresentations,theywerealmostentirelyontasksforwhichagraphicwasnotembeddedwithinthetask(seeFigure1).Thus,whenataskcontainedanexternalgraphicrepresentation,


33

studentswereunlikelytocreateapersonalisedinternalrepresentationaspartoftheirsensemaking.

WithregardtoFigure1,thestudentdrewcirclestorepresentthecakesandenclosedeachgroupoffivecircleswithasquaretorepresentabox.Hethenproceededtokeepatally(inhishead)ofthenumberof‘cakes’hehadrepresenteduntilhereached34.Hethenarguedthat7boxeswererequired.Thistypeofprocedurerepresentsacommonencodingtechniqueutilisedbystudentstosolvewordproblems.

Giventhehighproportionofthetasksineachtestcontaininggraphics,itwasnotsurprisingthatstudentsfrequentlyutiliseddecodingtechniquestosolvethetasks.Inthesesituations,thestudentsdidnothaveanymarkingsandthusdidnotdrawdiagramsorpicturestoscaffoldtheirunderstandings.Inrelationtothestudentsdecoding(seeFigure2),thegraphicsgenerallyhadanimportantparttoplayinthetasksolution.Insomesituations,thegraphicmerelyprovidedacontextforthetask;however,inmostsituations,theinformationcontainedwithinthegraphicwasindeedinfluential.

Figure�1:Exampleofastudentusinganencodingtechnique

Figure�2:Anexampleofataskthatrequiresdecodingusingspatialreasoningandmentalimagery

Figure�3:Thesametaskrepresentedintheorientationthestudentusedtosolvetheitem

WithregardtoFigure2,thestudentlocatedthepositionofthelibraryasthestartingpoint.Inordertocompletethetask,thestudentrotatedthemaptotheright(seeFigure3)asawayofensuringshecouldfollowthesubsequentdirections.Thismeantshewasfacingthelibraryasopposedtostandinginfrontofthelibrary.ShethenturnsrightalongHighStreet,whichisinfactleftofthelibrary.Consequently,sheansweredthistaskincorrectly.Shehadherhandsonthepagefollowingtheroutewithherfingersassheproceededtoworkoutthetask.Thisexamplehighlightsthenecessityofcorrectlydecodingthegraphic(inthisinstanceamaptask)inordertogenerateanappropriatesolution.

Thepresentationwillprovideanumberofexampleswhichhighlightthewayschildrenencodeandinparticular,decodegraphicalrepresentationsinmathematicstasks.

ImplicationsSeveralpracticalimplicationsemergefromthestudy.


34

• Themovementawayfromtraditionalword-basedproblemsolvinglimitsstudents’opportunitiestoutiliseencodingtechniquestomakesenseofmathematicsideas.Iftheseencodingskillsarenotencouragedandpromotedelsewhere,students’generalreasoningskillswillberestrictedsincesuchtechniquesarenecessarywhenstudentsencounternovelorcomplexproblems.

• Conversely,theintroductionofmathematicstasksrichingraphicsrequiresadifferentskillbase.Explicitattentionneedstobegiventospecifictypesofgraphicssincetheyhavedifferentstructureandconventions.Teachingmap-basedgraphics,forexample,requiresdifferentapproachesandtechniquesthangraph-basedgraphics.Indeedbargraphsandlinegraphsrequirespecificandindependentattention.

• Giventheincreasingrelianceofgraphicsinsociety,itisnotsurprisingthatgraphicrepresentationsholdaprominentplaceincurrentformsofassessment.Andsinceassessmenttendstoinfluenceandevendrivepractice,thewayinwhichmathematicsideasandconventionsarerepresentedimpactgreatlyonteachingpracticesandstudentlearning.

• Studentsarerequiredtodecodeexternalrepresentationwithmoreregularitythantheprocessofevokinginternalrepresentationsthroughencoding.Althoughbothrequirehighlevelsofspatialreasoning,mostrepresentationsarenow‘teacher’generatedratherthanstudentconstructed.

• Studentsneedtoacquiredifferentspatial-reasoningskillswhichallowthemtoconsideralltheelementsofatask,includingspecificfeaturesof

agraphicandthesurroundingtext,whensolvingmathematicstasks.

ReferencesDiezmann,C.M.,&Lowrie,T.(2008).

Assessingprimarystudents’knowledgeofmaps.InO.Figueras,J.L.Cortina,S.Alatorre,T.Rojano,&A.Sepúlveda,(Eds.),Proceedings of the Joint Meeting of the International Group for the Psychology of Mathematics Education 32, and the North American chapter XXX(Vol.2,pp.415–421).Morealia,Michoacán,México:PME.

Goldin,G.,&Shteingold,N.(2001).Systemsofrepresentationsandthedevelopmentofmathematicalconcepts.InA.A.Cuoco(Ed.),The roles of representation in school mathematics(pp.1–23).Reston,VA:NationalCouncilofTeachersofMathematics.

Kosslyn,S.M.(1983).Ghosts in the mind’s machine.NewYork:Norton.

Logan,T.,&Greenlees,J.(2008).Standardisedassessmentinmathematics:Thetaleoftwoitems.InM.Goos,R.Brown&K.Makar(Eds.),Navigating currents and charting directions.Proceedingsofthe31stannualconferenceoftheMathematicsEducationResearchGroupofAustralasia,Vol.2,pp.655–658.Brisbane,QLD:MERGA.

Lowrie,T.,&Logan,T.(2007).Usingspatialskillstointerpretmaps:Problemsolvinginrealisticcontexts.Australian Primary Mathematics Classroom, 12(4),14-19.

Lowrie,T.,&Diezmann,C.M.(2007).Solvinggraphicsproblems:Studentperformanceinthejuniorgrades.The Journal of Educational Research, 100(6),369–377.

Lowrie,T.,&Diezmann,C.M.(2009).Nationalnumeracytests:Agraphictellsathousandwords.Australian Journal of Education, 53(2),141–158.

MinisterialCouncilonEducation,Employment,TrainingandYouthAffairs[MCEETYA](2009).National assessment program: Literacy and numeracy. Grade 3 and 5 Numeracy.Retrieved6February6,2010from:http://www.naplan.edu.au/tests/naplan_2009_tests_page.html

NationalCouncilofTeachersofMathematics.(2000).Principles and standards for school mathematics.Reston,VA:Author.

Presmeg,N.C.(1986).Visualisationinhighschoolmathematics.For the Learning of Mathematics, 6(3),42–46.


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John�PeggUniversity of New England

JohnPeggbeganhiscareerasasecondarymathematicsteacher.CurrentlyheisProfessorandDirectoroftheNationalCentreofScienceICTandMathematicsEducationforRuralandRegional(SiMERR)AustraliaattheUniversityofNewEngland,Armidale.SiMERRprogramsidentifyandaddressimportanteducationalissuesof(i)specificconcerntoeducationinruralandregionalAustralia,and(ii)nationalconcerntoeducatorsacrossAustraliabutensuringruralandregionalvoicesarestronglyrepresented.

Hisworkisfarranging,andisparticularlyknowninternationallyandnationallyforitscontributiontotheory-basedcognitionresearchinmathematicseducationandassessment.Recentlyhehasbeeninvolvedinmanylarge-scalenationallysignificantprojectslinkedto:underachievingstudentsinliteracyandbasicMathematics,statewidediagnostictestingprogramsinscience,developmental-basedassessmentandinstruction,thevalidationoftheNSWprofessionalteachingstandards,andtheÆSOPstudyinvestigatingfacultiesachievingoutstandingstudentlearningoutcomes.

AbstractWhatdowemeanbyhigher-orderskills?Howdostudentsdevelophigher-orderskills,andutiliseabstractideasorconcepts?Howcanwepromotetheacquisitionofhigher-orderunderstandingsinaclassroomsituation?Thissessionconsidersthesequestionsandthereasonsforthedifficultiesandchallengesteachersfaceinaddressingtheneedtopromotehigher-orderunderstandingsintheirstudents.Theresearchreporteddrawsondatafromthreelarge-scalelongitudinalstudiescarriedoutwithprimaryandsecondaryteachers.Theapproachesareconsistentwithrecentresearchfindingsoncognitionandbrainfunctioning,andprovideinsightintohowsuchskillsaredevelopedinstudents.Participantswillconsiderpracticalwaystocreateconditionsthatincreasethelikelihoodofhigher-orderskillsandunderstandingsintheirstudents.

Introduction

Thereislittleevidenceofsystematicuseofcognitive-basedresearchtoinfluencewide-scalecurriculumdevelopments,ortheirassociatedassessmentandinstructionpractices(Pegg&Panizzon,2001).Significantly,andcentraltothispaper,ifassessmentandteachingpracticesaretoimprove,thensuchpracticesmustrestontheoreticalbasesforlearningwhichprovideuseableinformationtoteacherstoguidetheirthinkingandsubsequentteachingactions(Pellegrino,Chudowsky,&Glaser,2001).

Further,anytheoreticalpositionadoptedmustbeempiricallybasedandnotsimplyrelyon‘logic’foritsrationale.Thetheorymustofferteacherstheopportunitytoachievethesynchronisationofthethreearmsofcurriculum–assessment,pedagogy,andsyllabuscontent–thusachieving

‘constructivealignment’(Biggs,1996).ItisthepositionoftheauthorthattheSOLO(StructureoftheObservedLearnedOutcome)model(Biggs&Collis,1982;1991;Pegg,2003)meetstheserequirementsandprovidesatheoreticalunderpinningforassessmentandinstructiondecisionstakenbyteachers.

Theideasreportedheredrawondatafromthreelarge-scalelongitudinalstudies,involvingtheSOLOframework,withprimaryandsecondaryteachersinNSW.Thispaperdrawsfromthesestudiesideasassociatedwiththedevelopmentofhigher-orderskillsandunderstandings.TheuseofSOLOemphasisestheintegralroleassessmentpracticesplayaspartofnormalclassroomactivitywiththeinformationobtainedbeingusedtoinform,monitorandpromotestudentlearning(Black&Wiliam,1998).

Thefindingsofthesestudiesillustrateddramaticallythevaluesuchaframeworkplayswhengroupsofteachersinterpretedstudentresponsestoassessmenttasksandplanhowresponsiveinstructionmightproceed.WithoutaframeworksuchasSOLO,teacherscouldofferlittleguidanceonhowtheymightdecideconsistentlyandacrossarangeofactivitieswhetherassessmentitemswereappropriate,whetherstudentresponsestoassessmentitemswereadequate,whatskillsandunderstandingsstudentspossessed,andwhereinstructionmightbedirectedmostprofitablyinthefuture.

Inthispaperweconsider:Whatismeantbyhigher-orderskills?Howwillstudentsacquirehigher-orderskillsandutiliseabstractideasorconcepts?Inwhatwayscanwepromotetheacquisitionofhigher-orderskillsandunderstandingsinaclassroom?

Promotingtheacquisitionofhigher-orderskillsandunderstandingsinprimaryandsecondarymathematics


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Higher-order skills and understandings

Whatdowemeanbyhigher-orderskillsandunderstandings?Probablythebest-knowndescriptionisofferedbyBloom’sTaxonomy,namedaftertheleaderofthegroupofacademicsin1956thatreleasedtheTaxonomy of Educational Objectives.TherearesixcategoriestoBloom’sTaxonomy.Theseare:knowledge,comprehension,application,synthesis,analysisandevaluation.Knowledgeandcomprehensionareseenasimportantlower-levelskillsandareconcernedwithrememberinginformationandbasicunderstanding.Higher-orderskillsinvolveapplication(usingknowledge),analysis,synthesisandevaluation.

WhileBloom’sTaxonomyhascomeunderincreasingcriticismleadingtoreview(Andersonetal.,2001),thebasicideasstillofferhelptoteachers,inadvanceoftesting,toidentifyassessmentitemsthattargetdifferentcategoriesofquality.Theissuehereisthatthecategoryofaparticularquestiondoesnotusuallyprovideinsightintothelevelofastudent’sresponse.

SOLOadoptsadifferentposition,namely,that‘thereare“natural”stagesinthegrowthoflearninganycomplexmaterialorskill’(Biggs&Collis,1982,p.15).Themodelseekstodescribethisgrowthsequencethroughaseriesofmodesofunderstandingandlevelsofperformancewithinthesemodes.SOLOlevelsprovideteacherswithaconvenientwaytolabelportionsofthecontinuumforpracticalpurposes.

SOLO model

TherelevanceofSOLOtohigher-orderfunctioningisthatitisanempiricallyverifiableassessmentframeworkdesignedforuseinclassrooms.Overthepast30years,SOLOhasbuiltasubstantialempiricalbaseinvolvingnumerousresearchstudiesresultingin

manyhundredsofpublishedarticles.SOLOisamodelforcategorisingtheresponsesofstudentsintermsofstructuralcharacteristics.

ThefocusoftheSOLOcategorisationisoncognitiveprocessesratherthantheendproductsalone.Thetaskoftheteacheristoanalysethepatternofideaspresentedbythestudent.SOLOfacilitatesthesuccessfulcompletionofthistaskbyprovidingabalancebetweenstructuralcomplexityandcontent/context.InSOLO,developmentisdependentuponthenatureorabstractnessofthetask(referredtoasthemode)andaperson’sabilitytohandle,withincreasedsophistication,relevantcues(referredtoasthelevelofresponse).

SOLOcomprisesfivemodes of functioningreferredtoassensori-motor,iconic,concretesymbolic,formalandpostformal.Learningcanoccurinoneofthesemodesorbemulti-modal.Withineachmodeareseriesofthreelevelsofresponse.Aunistructuralresponseisonethatincludesonlyonerelevantpieceofinformationfromthestimulus;amultistructuralresponseisonethatincludesseveralrelevantindependentpiecesofinformationfromthestimulus;andarelationalresponseisonethatintegratesallrelevantpiecesofinformationfromthestimulus.ThesethreelevelscompriseaU-M-Rcycleofdevelopment.

Havingachievedarelationallevelresponseinonecycle,studentsmovetothenextlevelthatrepresentsanewunistructurallevelinanewcycle.Thisenhancedunistructuralresponserepresents(i)aconsolidationofthepreviousrelationalresponseintoasinglemoresuccinctformwithinthesamemode,or(ii)anewunistructuralresponsethatnotonlyincludesallrelevantpiecesofinformation,butalsoextendstheresponsetointegraterelevantpiecesofinformationnotin

thestimulusthataretypicalofthenextmodeofunderstanding.

ThestrengthoftheSOLOmodelisthelinkingofthehierarchicalnatureofcognitivedevelopmentthroughthemodesandthecyclicalnatureoflearningthroughthelevels.Eachlevelprovidesbuildingblocksforthenexthigherlevel.SOLOalsoprovidesteacherswithacommonandsharedlanguagethatenablesthemtodescribeinameaningfulwaytheirobservationsofstudentperformance.Thisisparticularlyimportantwhenteacherstrytoarticulatedifferencesbetweenlower-orderandhigher-orderskillsandunderstandings.

SOLO and higher-order functioning

Themostcommonmodesforinstructionforprimaryandsecondarymathematicsaretheconcretesymbolicmode(becomingavailableonaverageabout5–6yearsofage)andtheformalmode(becomingavailablearound15–16yearsofage).InSOLOthelevelsareorderedwithinamode,withstudentsenteringthefieldpickingupsingleaspects,thenmultiplebutindependentaspects,andfinallyintegratingtheseseparateaspectsintoacohesivewhole.

Itistheanswerscodedattheunistructuralandmultistructurallevelsthatareseenaslower-orderresponses.Herethestudentsrecallsingleormultipleideas,knowbasicfacts,andareabletoundertakeroutinetasksbyapplyingstandardalgorithms.

Higher-orderskillscommenceattherelationallevel.Thisarisesthroughtheabilitytointegrateinformationandmakepersonalconnectionsresultinginusingthisknowledgeinrelatedbutnewareas.Herestudentsareableto:demonstratesomeflexibilityintheirwork;undertakeproblemswithoutrelyingonstep-by-steplearntalgorithms;seenovelconnectionsnot


37

previouslytaught;haveanoverviewoftheconceptunderconsiderationandhowdifferentaspectsoftheconceptarelinked;showinsight–abletoundertake‘new’questions;andprovidereasonableevidenceofunderstanding.

Therelationallevelresponseisaprecursortomoreabstractthinkingthatoccursinthesubsequentmode(theformalmode)wherestudentsareabletoworkwithrelationshipsbetweenconceptsastheirthoughtprocessesbecomemoreabstractandtheymoveawayfromtheneedforconcretereferents.Theyareabletoformulatetheirownhypotheses,developtheirownmodels,workintermsofgeneralprinciples,andconstructtheirownmathematicalarguments.

Ideas about cognitive architecture

WhatdeterminestheSOLOlevelsforparticularstudents?Theanswerseemstoencompasssixmainideas.Theseare:generalcognitiveabilitiesofthestudent;familiarityofthecontent;presentationofthetask;degreeofinterestormotivationofthestudent;amountofrelevantinformationthatcanberetainedsimultaneouslyforthistask;andtheamountofinformationprocessingrequiredforasolution.

Theselasttwopointsareparticularlyimportanttothisdiscussionastheyleadtothenotionofworking memory.Workingmemoryisatheoreticalconstructandisusuallydefinedastheabilitytoholdinformationinthemindwhiletransformingormanipulatingit.Workingmemoryisusedtoorganise,contrast,compare,orworkoninformation.Workingmemoryislimitedincapacityandduration.Aswebecomemoreexpertinatask,ourworkingmemorycapacitydoesnotincreasebutitdoesbecomemoreefficient.

Thereissomeconjectureabouttherelationshipbetweenworkingmemoryandbothshort-termandlong-term

memory.Thecurrentconsensusisthatworkingmemoryandshort-termmemoryaredistinct.Short-termmemoryisassociatedwithinformationthatisheldforshortperiodsoftimeandreproducedinanunalteredfashion.Long-termmemoryiswherepermanentknowledgeisstoredforlongperiodsoftime.Individualsaccessandworkonthisstoredknowledgethroughtheirworkingmemory.

Implications�for�learning�I

• Humanintelligencecomesfromstoredknowledgeinlong-termmemory,notlongchainsofreasoninginworkingmemory.

• Skilledperformanceconsistsofbuildingchainsofincreasinglycomplexschemasinlong-termmemorybycombiningelementsconsistingoflow-levelschemasintohigh-levelschemas.

• Aschemacanholdahugeamountofinformationasasimpleunitinworkingmemory.

• Higher-orderprocessingoccurswhenthereis‘sufficientspace’inworkingmemorysothatappropriateschemascanbeaccessedfromlong-termmemoryandworkedupon.

Implication�for�learning�II

• Improvedautomaticityinfundamental/basicskills,suchascalculating,atlowerlevelsfreesupworkingmemoryresourcesforprocessinghigher-orderskillsandunderstandings.

• Deliberatepracticeattheunistructurallevelreducesthedemandsofworkingmemoryontheseconcepts.

• Ifattheunistructurallevel,workingmemorydemandsarereduced,thegrowthofmultistructuralresponsesisfacilitated.

• Freeingupofresourcesatlowerlevelsallowsstudentstofocusoninherentlyattention-demandinghigher-ordercognitiveactivities.

Implications�for�learning�III

• Attheunistructuralandmultistructurallevelsrelevantinformationcanbe‘taught’inthetraditionalsense.

• Attherelationallevel,‘teaching’inatraditionalsenseisproblematicasstudentsneedtodeveloptheirownconnections–theirownway.

• Languagedevelopmentisimportantindevelopingstudents’understandingandreducingworkingmemorydemandsatthemultistructurallevel–establishingastrongbasisforrelationalresponses.

• Studentscanrespondbyroteatrelationallevelswithoutunderstandingandhencegivetheimpressionofhavingattainedhigher-orderskills.

Implications for teaching

Oncestudentscanrespondconsistentlyatthemultistructurallevel,withappropriatelanguageskills,teachersshouldfocusoncreatinganenvironmenttopromoteSOLOrelationalresponses.Suchanapproachencouragesstudentstointegratetheirunderstandingofindividualideasandseeconnectionsandelaborationsnotpreviouslymet.Attemptingnon-routineproblemsisoneimportantwayinachievinghigh-orderskillsandunderstandingsas,ingeneral,thesequestionsrequireatleastrelationalresponses.Generally,withnon-routinequestions,therearenoprescribedalgorithmicapproaches.

Examplesofhowtogeneratesuchenvironmentsincludeprovidingstudentswith:


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• theanswertoaproblemandhavingthemgeneratequestions,i.e.,reversibility

• moreinformationthanthequestion/problemrequires

• lessinformationthanthequestion/problemrequires.

Conclusion

Higher-orderskillsandunderstandingsaremoredifficulttolearnandtoteach,astheyrequiremorecognitiveprocessinganddifferentformsofinstruction.Suchskillsandunderstandingsareprizedastheyallowknowledgetobeownedbytheindividualand,hence,appliedinnovelwaystodifferentsituations.Teachersshouldorchestrate,attheappropriatetimes,environmentsforhigher-ordermathematicalthinkingactivitiestotakeplaceonthesyllabuscontentbeingcoveredinclass.

Forthesuccessfuldevelopmentofhigher-orderskillsandunderstandings,activitiesofinstructionandassessmentneedtobecloselyintertwined.Inparticular,formaltestingandinformalformativeassessmentsneedtoinformteaching.Consideringassessmentsthiswaywillhelpteachersunderstandwherestudentsareintheirlearningjourney,andbetterfacilitatethefocusofinstructiontomeettheactualneedsofstudents.

Importantinthismovementfromlower-ordertohigher-orderskillsandunderstandingsistheuseofanevidence-basedcognitiveframework.ThispaperadvocatestheSOLOmodelasonesuitableframework.Withsuchamodel,teachershaveattheirdisposalsignpostsalongacontinuumofcognitivedevelopment.Oneobviousconsequenceisthatsuchaframeworkhelpsexplainwhenitismostappropriatetoaddresshigher-orderskillsandunderstandings,andwhentoconsiderdifferentinstructional

strategiesasstudentsmovethroughlevelsacquiringnewknowledge.

AnimplicationoftheSOLOhierarchyisthathigher-orderskillsandunderstandingsinthemathematicsclassroomarebuiltupontheacquisitionoflower-orderskillsandunderstandings.Theyhaveasymbioticassociationinwhich:(i)therelationallevelrepresentsthestartofhigher-orderfunctioning;and(ii)theunistructurallevelrepresentshigher-orderfunctioningforanearliergrowthcycleandatthesametimethebeginningoflower-orderfunctioninginthecurrentcycle.

Finally,workingfromadevelopmentalcognitiveperspective,suchastheSOLOmodel,exposes as fanciful and counter productive‘commonsense’expectationsofteachers:‘thatalmostallthetimetheirstudentsshouldbeengagedinhigher-orderthinking’.

ReferencesAnderson,L.W.,Krathwohl,D.R.,

AirasianP.W.,Cruikshank,K.A.,Mayer,R.E.,Pintrich,P.R.,Raths,J.,&Wittrock,M.C.(Eds.)(2001).A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives. AddisonWesleyLongman.

Biggs,J.(1996).Enhancingteachingthroughconstructivealignment.Higher Education, 32,347–364.

Biggs,J.,&Collis,K.(1982).Evaluating the quality of learning: The SOLO taxonomy.NY:AcademicPress.

Biggs,J.,&Collis,K.(1991).Multimodallearningandthequalityofintelligentbehaviour.InH.Rowe(Ed.),Intelligence: Reconceptualization and measurement(pp.56–76).Melbourne:ACER.

Black,P.,&Wiliam,D.(1998).Assessmentandclassroomlearning.Assessment in Education, 5(1),7–74.

Bloom,B.S.,(Ed.)(1956).Taxonomy of educational objectives: The classification of educational goals: Handbook I, cognitive domain.NewYork:Longman.

Pegg,J.(2003).AssessmentinMathematics:adevelopmentalapproach.InJ.M.Royer(Ed.),Cognition and mathematics teaching and learning.NewYork:InformationAgePublishing.

Pegg,J.,&Panizzon,D.(2001).Determininglevelsofdevelopmentinoutcomes-basededucation:Niceidea,butwhereistheresearch-baseforthedecisionstaken?PaperpresentedattheAmericanEducationalResearchAssociationConferenceinSeattle,on10–14thApril,1–5.

Pellegrino,J.W.,Chudowsky,N.,&GlaserR.(Eds.)(2001).Knowing what students know: The science and design of educational assessment.Washington:NationalAcademiesPress.


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Rosemary�CallinghamUniversity of Tasmania

RosemaryCallinghamisAssociateProfessorinMathematicsEducationattheUniversityofTasmania.ShehasanextensivebackgroundinmathematicseducationinAustralia,atschool,systemandtertiarylevels.Herexperienceincludesclassroomteaching,mathematicscurriculumdevelopmentandimplementation,large-scaletestingandpre-serviceteachereducationwithintwouniversities.ShehasworkedonprojectsinHongKongandNorthKorea,aswellasstudiesinmanypartsofAustralia.Herresearchinterestsincludestatisticalliteracy,mentalcomputationandassessmentofmathematicsandnumeracy,andteachers’pedagogicalcontentknowledge.

AbstractMuchhasbeenwrittenaboutassessmentoflearning,assessmentforlearningandassessmentaslearning.Thesethreeconceptionsofassessmentareexaminedinrelationtoprimarymathematics.DrawingonresearchfromAustraliaandoverseas,effectivepracticesinmathematicsassessmentintheprimaryclassroomareidentifiedandtheimplicationsforteachingandlearningconsidered.

Introduction

Assessmentpracticehasbeenanongoingfocusofeducationalresearchforoveraquarterofacentury.Inthattimenewtoolshavebeendevelopedandthecurriculumfocushasshiftedtotheoutcomesofthelearningprocess(Black&Wiliam,2003).Thepromiseofraisingstudents’learningoutcomesthroughtargetedassessmentstimulatedAustralianandothereducationsystemstointroducelarge-scaleandcostlyassessmentprogramssuchasNAPLAN,aspartofa‘pressureandsupport’approachtoeducationalreform(Fullan,2000).Despitethisactivity,thepromiseofimprovedoutcomesfromchangedassessmentpracticeshasnotbeenachievedonalargescale(Stiggins,2007).

Inthispaper,aspectsofqualityassessmentpracticeinprimarymathematicsareexplored,basedonlocalandinternationalresearch.Assessmentisregardedasmorethanthetaskormethodusedtocollectdataaboutstudents.Itincludestheprocessofdrawinginferencesfromthedatacollectedandactinguponthosejudgementsineffectiveways.Suchactionsmayoccuratmanylevels,butthekeyfocusconsideredhereistheschooland,particularly,theclassroom.Theassessmentfocusmaybesummativeinnatureprovidingasnapshotintimeofmathematicalcompetenceorachievement.

Alternatively,itmaybeformativeandusedtochangeteachingandlearningapproaches.

ConsiderthisscenarioobservedinaTasmanianprimaryschool:

Theteachersaremeetingingradeteams.Theyaresharingthe‘bigbooks’aboutmathematicsthatthechildrenintheirclasshaveproduced.Thediscussioncentresonwhatthebooksdemonstrateaboutthechildren’sunderstanding,andwhattheteachersneedtodotomovethatforward.Inthediscussion,teacherscomparetheworksamplesandmakejudgementsabouttheirownandotherteachers’students.Theyreferfrequentlytothestatecurriculumdocuments,NAPLANresults,theschoolpoliciesand‘throughlines’thathavebeendevelopedcollaborativelytoensureacommonlanguageandfocusacrosstheschool.Thesethroughlines,alongwithspecificstrategiesforcomputation,areprominentineveryclassroom.Bytheendofthemeeting,allteachershaveacommitmenttosomeactionfortheirclass,andtoincreasetheschoolfocusonspecificaspectsofmathematicsatwhichthestudentsappearedtodolesswellontheNAPLAN.Thisschoolisinamiddle-lowersocio-economicrangeandisoneofthemostsuccessfulinthestateonNAPLANnumeracy,particularlywhenvalue-addedmeasuresareconsidered.

Thepicturepaintedaboveisofarealschoolinwhichmathematicsassessmentisusedproductively.Theteacherswereusingacomplexmixofassessmentinformationtodevelopteachingplans.NAPLANdatawasdiscussedtoidentifywhere,asaschool,therewereidentifiedstrengthsandweaknesses.ThisuseofNAPLAN

Mathematicsassessmentinprimaryclassrooms:Makingitcount


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assessmentdataprovidedaformativefunctionataschoollevel.Theworkthatstudentshadproducedintheirclassroomswasbeingusedbothformativelyandsummatively.Teachersreferredtothecurriculumstandardstomakejudgementsabouttheirstudents’progressionandunderstanding,moderatingtheirdecisionsagainstworksamplesfromotherteachers’classroomsthroughdeepprofessionaldiscussion.Theseconversationssupportedteachersinmakingchoicesfortheirownclassrooms.

Theclassroomisthepowerhouseoflearning.Teachersmakeadifference(Hattie,2009)andeffortstoimprovestudents’learningoutcomesmustfocusonteacherpractice.Itisimpossible,however,totalkaboutassessmentdivorcedfrompedagogy.Theapproachthattheteacherusesunderpinsthequalityandnatureoflearningintheclassroom(Wiliam&Thompson,2007).Suchapproachesincludetheuseofassessmentforlearning–identifyingastudent’s‘readinesstolearn’(Griffin,2000)sothatplannedlearningexperiencesaremaximallyeffective.Thenotionofassessmentforlearningimpliesthatteacherswillnotonlybeabletoidentifywhatstudentscando,butalsowhatactivitiesandlearningexperiencesneedtobeplannedtodevelopstudents’thinking.

Assessment for learning

Whatdoesthislooklikeinpractice?Firstataskisneededthataddressesthedesiredmathematicalconceptandalsoprovidesforawiderangeofdifferentlevelsofunderstanding.Teachersthenpredictlikelyresponses,andmaybegrouptheseintocategoriesofsimilarunderstanding.Thefinalaction,andthisisthekey,istodevelopstrategiesforextensionforeachlevelofunderstanding.Thefirstoftheseactions,providingatask,isrelativelyeasy.Thereisanabundanceofqualitymaterialavailabletoteachers–the

difficultyischoosingwhattouse.Thesecond,predictinglikelyresponses,isalsoonethatteacherscandorelativelywell,andisnowsupportedbyaplethoraofworksamplesandexamplesfrompublishers,educationsystemsandprofessionalbodies.Identifyingwhattodonext,however,isdifficult(Wiliam,2000a).

Recentworkonidentifyingandmeasuringteachers’mathematicalpedagogicalcontentknowledge,however,indicatesthatalthoughprimaryteacherscanrecogniseandpredictstudents’responsestoquestions,bothcorrectandincorrectones,theyhaveconsiderabledifficultyinidentifyingthenextstepstotaketodevelopstudents’understanding(Watson,Callingham,&Donne,2008a,2008b).

Forexample,oneprimaryteacherparticipatinginastudyrelatingtodevelopingstudents’statisticalunderstandinginresponsetoaquestionshowinginformationaboutmarketshareamonglargesupermarketsusingapiegraphthataddeduptomorethan100percent,suggestedthatstudentsmightrespondinthefollowingways:

*WhatpercentageoftheretailmarketColeshas.*Somemightnotice(a)thatitdoesn’taddupto100%,*(b)61%shouldbemorethanhalfthegraph,*(c)thewholegraphisinaccurate(notmeasuredusingaprotractoretc.)

Inherresponsetothefollow-upquestion,‘Howwould/couldyouusethisitemintheclassroom?Forexample,howwouldyouintervenetoaddresstheinappropriateresponses?’,thesameteacheranswered‘Asacriticalliteracy/mathsactivity’.Althoughthisteacherdemonstratedadepthofunderstandingofthemathematicsinvolved,andaboutwhatherYear6studentsmightdo,shewasunableorunwillingtosuggestanyrealfollow-upactivity.

Assessment as learning

Ifteachersfinditdifficulttoarticulatemeaningfulactivitiesthatwouldmovetheirstudentsforward,whatdoesthissuggestaboutassessmentaslearning,thatisassessmentcompletelyindistinguishablefromthelearningactivity?Suchassessmentisinformal,undertakenaspartoftheteacher’s‘normal’activity.Itofteninvolvesateacherrecognisinga‘teachablemoment’andactingonthis.Forexample,inaKoreankindergartenclasschildrenwereusingblockstoexplorethenumberninebyputtingthemintogroupsoffiveandfour.Onegirlhadtakentenblocksandhadorganisedtheseintotwogroupsoffive.Theteachernoticedthisandsetupthenexttasktorearrangetheblocksintogroupsofsixandthree.Thisnextstepprovidedthechildwiththechancetoself-correct,andsheputtheextrablockbackintothecontainer.Clearlytheteachermadeanassessmentofthechildandgaveanimmediateresponsethatprovidedfeedbacktoherinawaythatchangedheractions.ItseemsthatthiskindofteachingactivitymeetstherequirementsindicatedbyBlackandWiliam(1998)foreffectivefeedback.

Classroomassessment,bothassessmentforandaslearning,reliesondialoguebetweenthechildandtheteacher(Callingham,2008).Primaryteachersknowthisandwhenaskedaboutwhattheywoulddowiththeirstudentsoftenreplyintermsofthequestionstheywouldposeorthediscussionstheywouldhave.Teachersinthestatisticsstudywereasked,forexample,howtheywouldrespondtoachildwhohadreadapictographabouthowchildrencametoschoolandhadgiventheincorrectresponse‘Bike,becausethemajorityofboysridetoschool’.AtypicalresponsewasthisonefromaSouthAustralianprimaryteacher:

That’sinterestingisn’tit?Iwouldbeaskingwhathisreasoning


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behindthatwouldbeandobviouslyhewouldsay,wellthey’reallboysandTom’saboy,thereforehewillcometoschoolbecausethat’swheremostoftheboyscomealong.AndIwoulddiscusswiththatchild,andtalkabouthisreasoningwhyhediscountedthebus,car,walkingandtrain.Whatwasthereasoningbehindyoudiscountingthefactthathecouldn’tcomebybus,car,walkortrain?AndthatwouldbehowIwouldmovehimforward.

Teachersperceivethiskindofactivityastheprocessofteaching,ratherthanfeedbackfromassessment,andthisperceptionhasimplicationsforprofessionallearning(Callingham,Pegg&Wright,2009).

Assessment of learning

Sofartherehasbeenlittleinthisdiscussionabouttheplaceofsummativeassessment:assessmentoflearning.Inrecentyearsitseemsthatteachershaverejectedthenotionofsummativeassessment.Biggs(1998),however,arguedthatithasanimportantplaceinclassroomassessment,andshouldbeseenaspartofacomprehensiveassessmentplan.Headvocated,forexample,usinggradedportfoliosasan‘information-rich’formofsummativeassessmentandsuggestedthatwhetheranassessmentwassummativeorformativewaslargelyamatteroftiming.Assessmentoflearningdoesnothavetobetest-based,andworksamplesthatdemonstrateastudent’smathematicalunderstandingareaffirmingandpowerfuldemonstrationstothechild,andothers,ofwhatheorshehaslearned.ThetwoworksamplesshowninFigure1,forexample,demonstratetwokindergartenstudents’attemptstocopyapattern.Thechildwhoproducedthetopexampleappearstounderstandthatthedesignhastorunacrossthepage,butdoesn’tpay

Figure�1:Kindergartenchildren’sattemptsatcopyingapattern

attentiontotheorderofthesymbols.Thebottomexample,however,ordersthesymbolsbutappearstobereadingthepatternfromrighttoleft,makingamistakeasthepatternrunsontoasecondline.Ifthesesampleswerecollectedattheendofateachingsequence,theyperformasummativefunction,providingarecordatonepointintimeofwhatachildcando.Incontrast,collectedduringateachingsequence,thesametaskcouldprovideformativeinformationhelpingtoinformtheteacher’splanning.

Assessment in the primary mathematics classroom: Making it count

Assessmentisarguablythemostpowerfulelementinteachingandlearning.Qualityassessmentcanprovideinformationtostudents,teachers,parentsandsystemsineffectiveandusefulways.Tobehelpful,however,itmustbebroadranging,collectingavarietyofinformationusingarangeoftasksbefore,duringandafterateachingsequence.

Tomakeassessmentcount,thefocusofprofessionallearningforprimarymathematicsteachersmightneedtoshift.Ratherthandevelopingteachers’mathematicalcontentknowledge,changingpedagogicalapproachesthroughrichmathematicaltasks,or

applyingmodelssuchastheNSWQualityTeachingmodel,moreproductiveprofessionallearningmightbefocusedonaddressingstudents’specific,identifiedlearningneeds,usingthemanyworksamplesnowavailableandaskingthequestion‘wheretonow’?

Mathematicslearningisidiosyncratic–notwochildrenlearnmathematicsinthesameway.Itisalsonon-linear–proceedinginjumpsasagroupofideascoalesceintoanewcognitiveframework.Assessmentneedstoaccommodatethesevariationssothatfeedbacktostudentscandirectlychangewhattheydo,suchasthesubtlefeedbackgivenbytheKoreanteacherdescribedearlier.Educatingteachersabouteffectivefeedback,however,maybemoreefficaciouswithinapedagogicalperspectivethanonethatisdirectedatassessment.

Perhapsthetimehascometostopworryingaboutthenatureoftheassessmentactivity,itssummativeorformativepurposeandthepoliticalendsforwhichtheinformationmay,ormaynot,beused.Instead,alleducatorsneedtoget‘backtobasics’andrememberthatitisqualityteachers,makingrapidprofessionaljudgementsontheruninbusyclassroomsthatcreatethe‘meaningsandconsequences’(Wiliam,2000b)thataffectchildren’s


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interestandinvolvementinmattersmathematical.

ReferencesBiggs,J.(1998).Assessmentand

classroomlearning:Aroleforsummativeassessment?Assessment in Education: Principles, Policy & Practice, 5(1),103–110.

Black,P.,&Wiliam,D.(1998).Assessmentandclassroomlearning.Assessment in Education, 5(1),7–74.

Black,P.,&Wiliam,D.(2003).‘Inpraiseofeducationalresearch’:Formativeassessment.British Educational Research Journal, 29(5),623–637.

Callingham,R.(2008).Dialogueandfeedback:Assessmentintheprimarymathematicsclassroom.Australian Primary Mathematics Classroom, 13(3),18–21.

Callingham,R.,Pegg,J.,&Wright,T.(2009).Changingteachers’classroompracticethroughdevelopmentalassessment:Constraints,concernsandunintendedimpacts.InR.Hunter,B.Bicknell,&T.Burgess(Eds.),Crossing divides(Proceedingsofthe32ndannualconferenceoftheMathematicsEducationResearchGroupofAustralasia),8pp.[CDROM].PalmerstonNorth,NZ:MERGA.

Fullan,M.(2000).Thereturnoflarge-scalereform.Journal of Educational Change,1,5–28.

Griffin,P.(2000).Students! Take your marks, get set, learn. Identifying ‘Readiness to Learn’ as a benefit of outcomes based education.KeynoteaddressdeliveredattheEducationQueenslandMountGravattSymposium,AssessmentandReportinginanOutcomesFramework,July17,2000.

Hattie,J.A.C.(2009).Visible learning: A synthesis of meta-analyses relating to achievement.Abingdon:Routledge.

Stiggins,R.(2007).Assessmentthroughthestudent’seyes.Educational Leadership, 64(8),22–26.

Watson,J.M.,Callingham,R.,&Donne,J.(2008a).Establishingpedagogicalcontentknowledgeforteachingstatistics.InC.Batanero,G.Burrill,C.Reading&A.Rossman(2008).Joint ICMI/IASE Study: Teaching Statistics in School Mathematics. Challenges for Teaching and Teacher Education.ProceedingsoftheICMIStudy18and2008IASERoundTableConference.Monterrey:ICMIandIASE.Online:www.stat.auckland.ac.nz/~iase/publicatons

Watson,J.M.,Callingham,R.,&Donne,J.(2008b).Proportionalreasoning:Studentknowledgeandteachers’pedagogicalcontentknowledge.InM.Goos,R.Brown,&K.Makar(Eds.),Navigating currents and charting directions. Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia(Vol.2,pp.555–562).Adelaide:MERGA.

Wiliam,D.&Thompson,M.(2007).Integratingassessmentwithinstruction:Whatwillittaketomakeitwork?InC.A.Dwyer(Ed.)The future of assessment: Shaping teaching and learning.Mahwah,NJ:LawrenceErlbaumAssociates.

Wiliam,D.(2000a).Integrating formative and summative functions of assessment.PaperpresentedtotheWGA10fortheInternationalCongressonMathematicsEducation9,Makuhari,Tokyo.Availablefromhttp://www.dylanwiliam.net/

Wiliam,D.(2000b).Themeaningsandconsequencesofeducationalassessments.Critical Quarterly, 42(1),105–127.


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David�Leigh-LancasterVictorian Curriculum and Assessment Authority (VCAA)

DavidLeigh-LancasteristheMathematicsManagerattheVictorianCurriculumandAssessmentAuthority(VCAA),formerHeadofMathematicsP–12atKingswoodCollege,Victoria,andhastaughtsecondarymathematicsforabout20years.DuringthistimeDrLeigh-Lancasterhasbeenextensivelyinvolvedincurriculumdevelopment,teacherprofessionallearning,resourcedevelopment,examinationsettingandmarkingandthedevelopmentandverificationofschool-basedassessmentinmathematics.Hehaslongstandinginterestsinmathematicallogic,computabilitytheory,foundationsofmathematics,historyandphilosophyofmathematicsandmathematicseducation,thenatureofmathematicalinquiry,curriculumdesignandteaching,learningandassessmentinmathematics.DrLeigh-Lancaster’sresearchinterestsfocusonmeta-mathematicseducation,theinterfacebetweenmathematicsandschoolmathematics,andthenotionofcongruencebetweencurriculum,assessmentandpedagogy–inparticularwithrespecttotheroleofenablingtechnology.

AbstractThispaperoutlineshowcurriculumandassessmentcongruenceconsiderationshavebeenaddressedinthecontextoftheincorporationofcomputeralgebrasystem(CAS)technologyintoVictorianseniorsecondarymathematicscurriculumandassessment,inparticularexaminations,overtheperiod2000–2010.Theroleofsomerelatedresearchisdiscussed.

IntroductionTherelationshipbetweencurriculumandassessmentiscentraltodiscourseinmathematicseducation.Itisafocusofcloseattentionintheseniorsecondaryyearswherethereisastrongconnectiontomattersofcertificationandpathwaysintopost-secondaryeducation,trainingandwork.Akeyaspectofmathematicsistheroleoftechnologyinworkingmathematically.Howthisisreflectedinseniorsecondarymathematicscurriculumandassessmentisoneofthebigissuesofourtime,especiallyasvarioussoftwareandhand-helddevicesthatsupportandintegratepowerfulnumerical,statistical,graphical,geometricandsymbolicfunctionalityhavebecomereadilyavailableforwidespreaduseinschoolmathematics.Thenotionofcongruenceisusedhereasametaphorforeffective alignmentbetweentheuseoftechnologyasanenablingtoolinthecurriculumanditsuseinrelatedassessment.Thetermtechnologywillbeunderstoodtoindicateasynergybetweenanartefactandtheknowledge and understandingofhowitcanbeusedasatoolforapurpose.Relevantresearchincludesphilosophicalstudiesormeta-analysesofbeliefsandvalues(see,forexample,Bishop,2007;Ernest,1991),rationales,policies,trialsandpilotstudies(see,forexample,Stacey,McCrae,Chick,Asp&Leigh-Lancaster,

2000)andstrategiesandprocessesthatleadtocertaindirectionsandapproachesbeingtakenwithinandacrossjurisdictions.There-energisingofdiscussionsontheroleofdigitaltechnologiesintheschoolmathematicscurriculumarisingfromtheemergingAustraliannationalcurriculuminitiativeisagoodexampleofacontemporarycontextfortheseconsiderations(ACARA,2009).

Ithasbeencommontoassociatemathematicalfunctionalitywithcertaindevices;forexample,numericalwithscientificcalculators;statisticalwithspreadsheetbasedapplications;geometrywithdynamicgeometrysoftware;graphingwithgraphicscalculators;andsymbolicmanipulationwithcomputeralgebrasystems(CAS).Theseassociationshavebeenusedasthebasisofjurisdictionspecificationsforproscribed,permittedorprescribedtechnologyaccessinformalassessment,especiallyexaminations.Overthepasthalf-decadetheyhavebecomelessdistinctivewithmultiplefunctionalitiesavailableonasingleplatform,forexampleCASIOClasspadorTexasInstrumentsNspirehand-helddevicesandgeneralpurposeCASsoftwaresuchasMapleandMathematica.Thesetechnologiescanalsobeusedfordevelopingdocumentsthatintegratetextwith‘live’mathematicalcomputations(calculations,tables,graphs,diagrams,symbolicexpressions)andaspresentationtools.

Intheircomplementaryrelationship,curriculumandassessmentarekeyindicatorsofeducationalbeliefs,valuesandpreferences;forexample,whatis,orisnottobedone,andhowitmaybedone,byandforwhom,andinwhatcontexts.Ifcurriculumistosaywhatstudentsshould,asaconsequenceoftheirlearning,knowandbeabletodo(concepts,skills,processesandthelike)andassessmentisthemeansbywhich

Thecaseoftechnologyinseniorsecondarymathematics:Curriculumandassessmentcongruence?


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judgmentsaremadeaboutprogressandachievement,thenacurriculumthatsetsexpectationsfortheactiveuseoftechnologyasanenablingtoolforworkingmathematicallyrequirescongruentexpectationsandpracticesforassessment.Thisistypicallyinformedbyinter-jurisdictionbenchmarkingresearchofcurriculumand/orassessmentroutinelycarriedoutbyeducationauthoritiesaspartofthedevelopment–evaluation–reviewcycle(see,forexample,Coupland,2007).

A brief historical background

Overthepastfewdecades,varioustechnologieshavebeenusedinseniorsecondarymathematicscurriculaandrelatedYear12finalexaminationsinVictoria.Whiledifferentmodelshavebeenusedtodesignanddevelopthesecurricula,therehavebeenessentiallythreemaintypesoffinalyearmathematicscourses:

• apracticallyorientedstatisticsanddiscretemathematicscourse(e.g.networks),oftenwithabusiness/financialmathematicscomponent/option

• amainstreamfunction,algebra,calculusandprobabilitycourse

• anadvancedmathematicsfunctionsandrelations,algebra,calculus,vectors,complexnumbers,differentialequationsandmechanicscourse(thiscourseassumesconcurrentorpreviousstudyofthemainstreamcalculusbasedcourse).

InVictoria,from1993thesehavebeencalledFurtherMathematics,MathematicalMethods/MathematicalMethodsCASandSpecialistMathematicsrespectively,andtheircorrespondingassumedtechnologiesforexaminationsareshowninTable1.

Table�1:Assumedtechnologyforendofyear12finalexaminationsinVictoriafrom1970

Stage Assumed�technology�for�end�of�Year�12�examinations�in�Victoria

Pre-1978 Four-figurelogarithmtablesand/oranapprovedsliderule.

1978–1996

Scientificcalculator.Until1990therewasasingle3-hourexamination.From1991thereweretwo1½-hourexaminations.

1997 Scientificcalculator–approvedgraphicscalculatorpermittedbutnotassumed.

1998–1999

ApprovedgraphicscalculatorassumedforMathematicalMethodsandSpecialistMathematics(bothexaminations).ScientificcalculatorwithbivariatestatisticalfunctionalityorapprovedgraphicscalculatorassumedforFurtherMathematics(bothexaminations).

2000–2005

ApprovedgraphicscalculatorforFurtherMathematics,MathematicalMethodsandSpecialistMathematics(bothexaminations).

ApprovedCAS(calculatororsoftware)forMathematicalMethodsCASpilotstudy,2002–2005(bothexaminations).

2006–2009

ApprovedgraphicscalculatororCASforFurtherMathematics(bothexaminations).

MathematicalMethodsandMathematicalMethods(CAS)werealternativebutlikestudieswithacommontechnologyfreeExamination1(worth40marks)andaseparatetechnologyassumedExamination2(worth80marks),witharound70%–80%commonmaterial,approvedgraphicscalculatorassumedforMathematicalMethodsExamination2,approvedCASassumedforMathematicalMethods(CAS)Examination2.

SpecialistMathematics–technologyfreeExamination1.ApprovedgraphicscalculatororCASassumedforExamination2(technologyactivebutgraphicscalculator/CASneutral).

2010–2013

ApprovedCASorgraphicscalculatorassumedforFurtherMathematics(bothexaminations).

MathematicalMethods(CAS)andSpecialistMathematicseachhavea1-hourtechnologyfreeexamination.

MathematicalMethods(CAS)andSpecialistMathematicseachhavea2-hourtechnologyactiveexamination.AnapprovedCAS(calculatororsoftware)istheassumedtechnology.

2014�and�beyond

(Draft)Australiancurriculumhasfourseniorsecondarymathematicsstudies:Essentialmathematics(CourseA);Generalmathematics(CourseB);Mathematicalmethods(CourseC)andSpecialistmathematics(CourseD),currentlyunderconsultation.Ifthingsproceedwell,2014couldbethefirstyearofimplementationinVictoria.Assessmentremainstheprovinceofstatesandterritoryjurisdictionsfortheinterim.


45

TheextenttowhichatechnologysuchasCASisactivelyusedincurriculum,pedagogyandassessmenthasmuchvariationacrossjurisdictions(see,forexample,Leigh-Lancaster,2000).AcurriculummayspecifyexpectedstudentuseofCASinworkingmathematically,whileprecluding,permittingorassumingitsuseincomponentsofschool-basedorexaminationassessment.Decisionsaboutpossibleorrequireduse(ornot)mayrestwiththeclassteacher,orbepartlyorwhollyprescribedbytherelevantauthority.WithrespecttotheuseofCASinexaminationassessment,itmaybethecasethattheuseoftechnologyisprecludedforsomecomponents(CollegeBoardAPCalculus,Denmark,Sweden,andVictoria,WesternAustralia,NewZealand)andpermitted(CollegeBoardAPCalculus,Sweden)orassumed(Denmark,Victoria,WesternAustralia,NewZealand)forothercomponents.OtherjurisdictionspermitbutdonotrequireCASforallexaminationassessment(France,Tasmania).Somejurisdictionsdonothaveexternallysetexaminations,withonlyschool-basedassessment(OntarioCanada,Queensland),buthaveacurriculumthatexplicitlyincorporatestheuseofCASwhileteachersdecidelocallywhattechnologyistobeusedinassessment(typicallywithatleastgraphicscalculatorfunctionalityassumed).AsummaryofjurisdictionswhichpermitorrequirestudentaccesstoCASforsomecomponentsoftheirseniorsecondarycurriculumandassessmentcanbefoundatComputerAlgebrainMathematicsEducation(seeCAME,2010).Thustherewillbemultipleassessmentmodels,andtheirefficacywithrespecttotheaimsofthecorrespondingcurriculumisarichareaforresearch.

Mathematical Methods – Mathematical Methods (CAS) 2006–2009

TheVictorianmodelfortrialling,developmentandimplementationofMathematicalMethods(CAS),hasbeensubstantiallyinformedbyexperienceandexpertisefromotherjurisdictions–theCollegeBoard,Denmark,France,AustriaandSwitzerland.Itis,however,quiteunique.Victoriaistheonlyjurisdictiontohavemovedfromanestablishedstudy,MathematicalMethods(1992–2009)toconcurrentpilotingofarelatedequivalentandalternativestudy,MathematicalMethodsCAS(2001–2005);thenconcurrentimplementationofbothfullyaccreditedstudiesasequivalentbutalternative(2006–2009)withatransitiontotheCASversionreplacingthe‘parent’versionofthestudyfrom2009(Units1and2–Year11level)and2010(Units3and4–Year12level).Duringtheconcurrentimplementationphase,bothstudieshadacommontechnologyfreeexamination;andeachhaditsowntechnologyassumedexaminationwith70%–80%questionscommontothetwopapers.ThefirstphaseoftheVCAAMathematicalMethods(CAS)pilotstudywasfoundedintheworkoftheComputerAlgebraSystem–CurriculumAssessmentandTeaching(CAS-CAT)project(2000–2002)anAustralianResearchCouncilgrantfundedresearchprojectpartnershipbetweentheVCAA,theUniversityofMelbourne,andcalculatorcompanies.Theexpandedpilot(2001–2005)alsoincorporatedtheuseofCASsoftware.

Questionsofinterestincludeconsiderationofmatterssuchaspotentialandactualcurriculumgains,theperceivedandactualimpactofregularstudentaccesstoCASonstudentfacilitywithtraditional‘by-hand’skills,changesinteacherpedagogyandstudentapproachestoworkingmathematically,useoftechnologywithrespectto

gender,andperformanceofthetwocohortswithrespecttoassessmentinconcurrentadvancedmathematicsstudy–SpecialistMathematics.TheperformanceofthetwocohortsoncommonassessmentitemsinexaminationshasbeenmonitoredcloselybytheVCAAandreportedinAssessmentReports(see,forexample,VCAA,2010a,2010b)andpapers(see,forexample,Evans,Jones,Leigh-Lancaster,Les,Norton&Wu,2008).

Facilitywithtraditional‘by-hand’skillsisanareaofsomeinterest–meanscoredataonthetechnologyfreeExamination1for2006–2009consistentlyindicatethat,ingeneral,theMathematicalMethods(CAS)cohortperformatleastaswellastheMathematicalMethodscohortonrelatedquestions.Inparticularfor2009(wherethesizeofthecohortswasaround7000–8000),thedistributionofstudentscoresforeachcohortacrossthemarkrangefrom0to40showsthatatthetopend,theperformanceofthetwocohortsisessentiallythesame;attheverybottomend,theperformanceoftheMathematicalMethods(CAS)cohorttendstobebetter,whilefromthelowtohighmarkrangetheMathematicalMethods(CAS)cohortconsistentlyachievesaslightlyhigherscorethantheMathematicalMethodscohort.ThispatternpersistswhenthedataiscontrolledforgeneralmathematicalabilityusingtheMathematics,ScienceandTechnologycomponentoftheGeneralAbilityTest(whichhasmoderatecorrelationwithrespecttostudyspecificability)conductedinthemiddleofthesameyear.WhenExamination1resultsareusedtocontrolforabilityoncommonExamination2extendedresponsequestions(thatis,technologyindependentorgraphicscalculator/CASfunctionalityneutral)comprising21itemsforascoreof35marksoutofatotalof80marks,asimilarpatternisobserved,asshowninFigure1.


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assessment and qualifications.London:CAME.RetrievedMay25,2010fromhttp://www.lkl.ac.uk/research/came/curriculum.html

Coupland,M.(2007).A critical analysis of selected Australian and international mathematics syllabuses for the post-compulsory years of secondary schooling.Sydney:BoardofStudies.

Ernest,P.(1991).Philosophy of Mathematics Education.London:Falmer.

Evans,M.,Jones,P.,Leigh-Lancaster,D.,Les,M.,Norton,P.,&Wu,M.(2008).The2007CommonTechnologyFreeExaminationforVictorianCertificateofEducation(VCE)MathematicalMethodsandMathematicalMethodsComputerAlgebraSystem(CAS).InM.Goos,R.Brown&K.Makar(Eds.),Navigating currents and charting directions.Proceedingsofthe31stannualconferenceoftheMathematicsEducationResearchGroupofAustralasia,UniversityofQueensland,Brisbane(pp.331–336).Brisbane:MERGA.

Leigh-Lancaster,D.(2000).Curriculum and assessment congruence – Computer Algebra Systems (CAS) in Victoria.Ohio:OhioStateUniversity.RetrievedMay25,2010fromhttp://www.math.ohio-state.edu/~waitsb/papers/t3_posticme2000/leigh-lancaster.pdf

Stacey,K.,McCrae,B.,Chick,H.,Asp,G.,&Leigh-Lancaster,D.(2000).Research-ledpolicychangefortechnologically-activeseniormathematicsassessment.InJ.Bana&A.Chapman(Eds.),Mathematics Education Beyond 2000.Proceedingsofthe23rdannualconferenceoftheMathematicsEducationResearchGroupofAustralasia(pp.572–579).Freemantle:MERGA.

VictorianCurriculumandAssessmentAuthority.(2010a).Mathematical Methods Examination 1 Assessment Report 2009.Melbourne:VCCA.RetrievedMay25,2010fromhttp://www.vcaa.vic.edu.au/vcaa/vce/studies/mathematics/methods/assessreports/2009/mm1_assessrep_09.pdf

VictorianCurriculumandAssessmentAuthority.(2010b).Mathematical Methods Examination 2 Assessment Report 2009.Melbourne:VCCA.Retrieved25May2010fromhttp://www.vcaa.vic.edu.au/vcaa/vce/studies/mathematics/cas/assessreports/mmcas2_assessrep_09.pdf

non-CAS CAS

Raw Score on Maths Methods exam 1 (short answers) (CAS and non-CAS groups)

Ave

rage

Raw

Sco

re o

n ex

am 2

com

mon

item

s (e

xten

ded

answ

ers)

Scores on Maths Methods exam 1 and exam 2 by CAS and non-CAS groups 2009

0

5

10

15

20

25

30

35

0 5 10 15 20 25 35 4530 40

Figure�1:AveragescorewithrespecttoExamination1(technologyfree)score

Thisisperhapsnotsurprising–thereisanaprioriargumentthatuseofCASasanenablingtechnologywhichprovidesnumerical,graphicalandalgebraicrepresentationoffunctionsandrelations(andcanmovesmoothlybetweentheserepresentations)affordsadditionalsupportforlearningcomparedtotechnologythatprovidesforonlynumericalandgraphicalrepresentationsuchasagraphicscalculator.Ifonewishestodevelopstudentfacilitywiththeproductrulefordifferentiation(fg)′=fg ′+gf ′thenthisisassistedbybeingabletoreadilygenerateandanalysecorrectpatterns,forexample,movingfromthegeneralformoftheproductruletoaformwherefisleftundetermined,andavarietyofspecificfunctionrulesforgused,totheformwheretheruleoffisspecified,forexampleexandthesamevarietyofspecificfunctionrulesused.

Inthiscontext,evaluationofthederivativecanberelateddirectlytothegradientofthetangenttothegraphoftheproductfunctionataparticularpointandrepresentedgraphically.Wheredynamicfunctionalityisalsoutilised,thegraphofthecorrespondingderivative

function,andthetableofvaluesforthederivative,canbegeneratedtogether.Studentscouldthenemploythistocomparetheirperceptionofthegradientofthefunctionacrossitsdomain(andsubsetsofthedomain)withwhattheyareseeingasthepointatwhichthederivativeisbeingevaluatedismovedalongthecurvethatformsthegraphofthefunction.Naturally,thegeneralresultisestablishedbyaproofofsuitablelevelofformalityforthestudentcohort.

ReferencesAustralianCurriculumAssessmentandReportingAuthority.(2009).The Shape of the Australian Curriculum: Mathematics.Melbourne:Author.RetrievedMay25,2010fromhttp://www.acara.edu.au/verve/_resources/Australian_Curriculum_-_Maths.pdf

Bishop,A.J.(2007).Valuesinmathematicsandscienceeducation.InU.Gellert&E.Jablonka(Eds.)Mathematisation demathematisation: Social, philosophical and educational ramifications(pp.123–139).Rotterdam:SensePublishers.

ComputerAlgebrainMathematicsEducation.(2010).Some senior secondary mathematics CAS active/permitted curriculum,


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Joanne�MulliganMacquarie University

JoanneMulliganisanAssociateProfessorofEducationandAssociateDirectoroftheCentreforResearchinMathematicsandScienceEducation(CRiMSE)atMacquarieUniversity,Sydney.Herbackgroundineducationalpsychology,primaryteachereducationandmathematicseducationpsychologyiscombinedwithearlyteachingandadministrativeexperienceinNSWprimaryschools.Overthepast25yearsherresearchhasfocusedprimarilyonthedevelopmentandassessmentofnumberconceptsandprocesses,wordproblems,multiplicativereasoning,andpatternandstructurewith4-to9-year-olds.Shehasmadeasignificantcontributiontolarge-scaleAustraliangovernmentandstate-fundednumeracyprojectssincethe1990s(e.g.,CountMeInToo;CountingOn;theNumeracyResearchinNSWPrimarySchools’Project;theEarlyYearsNumeracyResearchProject(Victoria)andtheMathematicalThinkingofPreschoolersinRuralandRegionalAustralia(DEST).ShehasalsocontributedtothedevelopmentandanalysisofnumeracyitemsintheNSWBasicSkillsTestingProgramandqualityassessmenttasksfortheNSWQualityTeacherProgram.

AschiefinvestigatorofacurrentARCDiscoveryproject,herresearchaimstoreconceptualisetraditionalviewsandpracticesofearlymathematicaldevelopmentandlearning.AssociateProfessorMulliganhasdevelopedarangeofinterview-basedassessmentinstrumentsbasedonframeworksoflearningthatenablein-depthanalysisofmathematicalgrowth.Hertechniqueshavepotentiallysignificantimplicationsforaddressingstudents’learningdifficulties.Currentresearchencompassesarangeofprojectsfocusedonearlymathematicaldevelopmentandprofessionallearningsuchastheroleoftechnologicaltools,theuseofchildren’sliterature,preschoolers’mathematicalpatterningandmathematicseducationinIndigenousearlychildhoodcontexts.SheisalsocurrentlyleadingaNSWDETproject,EnhancingSuccessinMathematics(ESiM),focusedonmiddleschooling.

AbstractOverthepastdecadeasuiteofstudiesfocusedontheearlybasesofmathematicalabstractionandgeneralisationhasindicatedthatanawarenessofmathematicalpatternandstructureisbothcriticalandsalienttomathematicaldevelopmentamongyoungchildren.Mulliganandcolleagueshaveproposedanewconstruct,AwarenessofMathematicalPatternandStructure(AMPS),whichgeneralisesacrossmathematicalconcepts,canbereliablymeasured,andiscorrelatedwithstructuraldevelopmentofmathematics.

Acurrentlargeevaluationstudywasdesignedandimplementedtomeasureanddescribeyoungchildren’sstructuraldevelopmentofmathematicsinthefirstyearofschooling,Reconceptualising Early Mathematics Learning: The Fundamental Role of Pattern and Structure.AninterventionwasimplementedtoevaluatetheeffectivenessofthePatternandStructureMathematicalAwarenessProgram(PASMAP)onkindergartenstudents’mathematicaldevelopment.Fourlargeschools(twofromSydneyandtwofromBrisbane),16teachersandtheir316studentsparticipatedinthefirstphaseofatwo-yearlongitudinalstudy.ThispaperprovidesanoverviewofthebackgroundstudiesthatinformedthedevelopmentofPASMAP,describesaspectsoftheassessmentandintervention,andprovidessomepreliminaryanalysisoftheimpactofPASMAPonstudents’representationsofstructuraldevelopment.

Introduction

Oneofthemostfundamentalchallengesformathematicseducationtodayistoinspireyoungchildrentodevelop‘mathematicalminds’andpursuemathematicslearninginearnest.Currentresearchshowsthatyoungchildrenaredeveloping

complexmathematicalknowledgeandabstractreasoningmuchearlierthanpreviouslyconsidered.Arangeofstudiespriortoschoolandinearlyschoolsettingsindicatethatyoungchildrendopossesscognitivecapacitieswhich,withappropriatelydesignedandimplementedlearningexperiences,canenableformsofreasoningnottypicallyseenintheearlygrades(e.g.,Clarke,Clarke,&Cheeseman,2006;Papic,Mulligan,&Mitchelmore,2009;Perry&Dockett,2008).

Ontheotherhand,findingmoreeffectivewaysofestablishingtherootcausesoflearningdifficultiesinmathematicsisakeyconcern.Thegapbetweenachieversandnon-achieversinmathematicsbeginsinearlychildhoodandbecomeswiderasstudentsgrowolder,andthereisstillinsufficientresearchevidenceandlittleconsensusabouttheunderlyingcausesofunderachievement.Despiteinitiativesandreformsinmathematicseducationmanychildrendonotseemtoaccessthedeepideasandkeyprocessesthatleadtosuccessbeyondschool.

ThePatternandStructureProject,initiatedin2001,aimstomeetthischallengethroughadifferentapproachtomathematicslearning,beginningwithveryyoungchildren,thatreachesbeyondbasicnumeracytoonethatcultivatesmathematicalpatternsandrelationships.Overthepastdecade,asuiteofstudiesfocusedontheearlybasesofmathematicalabstractionandgeneralisation,hasfoundthatanawarenessofmathematicalpatternandstructureisbothcriticalandsalienttomathematicaldevelopmentamongyoungchildren.Mulliganandcolleagueshaveproposedanewconstruct,AwarenessofMathematicalPatternandStructure(AMPS),whichgeneralisesacrossmathematicalconcepts,canbereliablymeasured,andiscorrelatedwithincreasinglydevelopedstructuralfeaturesofmathematics(Mulligan&Mitchelmore,2009).Findingreliable

Reconceptualisingearlymathematicslearning


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andconsistentmethodsfordescribingthegrowthofchildren’smathematicalstructuresandrelationships,andutilisingchildren’sideastodevelopquantitativereasoningatanoptimumage,whentheyareeagertolearn,iscentraltothisproject.

What is pattern and structure?

Amathematicalpatternmaybedescribedasanypredictableregularity,usuallyinvolvingnumerical,spatialorlogicalrelationships.Inearlychildhood,thepatternschildrenexperienceincluderepeatingpatterns(e.g.,ABABAB…),spatialstructuralpatterns(e.g.,geometricalshapes),growingpatterns(e.g.,2,4,6,8,…),unitsofmeasureortransformations.Structurereferstothewayinwhichthevariouselementsareorganisedandrelatedincludingspatialstructuring(seeMulliganetal.,2003).Structuraldevelopmentcanemergefrom,orunderliemathematicalconcepts,proceduresandrelationshipsandisbasedontheintegrationofcomplexelementsofpatternandstructurethatleadtotheformationofsimplegeneralisations.Forexample,recognisingstructuralfeaturesofequivalence,4+3=3+4mayreflectthechild’sperceivedsymmetricalstructure(seeMulligan&Mitchelmore,2009).

Background

Thereisincreasingevidencethatstructuraldevelopmentiscrucialtomathematicalreasoningandproblem-solvingamongyoungchildren.Failuretoperceivepatternandstructuremayalsoprovideanexplanationforpoormathematicalachievement.Earlyassessmentof,andinterventioninmathematicslearning,isconsideredpreventativeoflaterlearningdifficulties(Clements&Sarama,2009;Wright,2003).Thequality,scopeanddepthofboththeteachingandassessmentof

earlymathematicsarenowregardedascriticaltofuturesuccessinthesubject(Thomson,Rowe,Underwood,&Peck,2005).

Research on pattern and structure

Researchonearlymathematicslearninghasoftenbeenrestrictedtoananalysisofchildren’sdevelopmentallevelsofsingleconceptssuchascounting,buthasnotprovidedinsightintocommonunderlyingprocessesthatdevelopmathematicalgeneralization(Mulligan&Vergnaud,2006).However,recentinitiativesinearlychildhoodmathematicseducation,forexample,theBuildingBlocksProject(Clements&Sarama,2009),theBigMathsforLittleKidsProject(Ginsburg,Lee&Boyd,2008)andtheMathematicsEducationandNeurosciences(MENS)Projectprovideframeworkstopromote‘bigideas’inearlymathematicsandscienceeducation(vanNes&deLange,2007).

Thistrendisreflectedintheincreasingbodyofresearchintoyoungchildren’sstructuraldevelopmentofmathematicsandearlyalgebraicreasoning.Algebraicthinkingisthoughttodevelopfromtheabilitytoseeandrepresentpatternsandrelationshipssuchasequivalenceandfunctionalthinkingfromtheearlychildhoodyears(Papic,Mulligan,&Mitchelmore,2009;Warren&Cooper,2008).Researchinnumber(Hunting,2003;Mulligan&Vergnaud,2006;Thomas,Mulligan&Goldin,2002;vanNes&deLange,2007;Young-Loveridge,2002),patterningandreasoning(Clements&Sarama,2009;English,2004),spatialmeasurement(Outhred&Mitchelmore,2000;Slovin&Dougherty,2004),andearlyalgebra(Blanton&Kaput,2005;Carraher,Schliemann,Brizuela,&Earnest,2006;Warren&Cooper,2008),haveallshownhowprogressinstudents’mathematicalunderstandingdependsonagraspofunderlyingstructure.Significantconcentrationsofnew

researchwithyoungchildrenfocusedondatamodelingandstatisticalreasoningalsoprovideanintegratedapproachtostudyingstructuraldevelopment(e.g.,English,2010;Lehrer,2007).

The Pattern and Structure Project

Earlystudiesonthestructureofmultiplicationanddivision(Mulligan&Mitchelmore,1997),thenumbersystem(Thomas,Mulligan,&Goldin,2002),andareameasurement(Outhred&Mitchelmore,2000)focusedonanalysinganddescribingstructuraldevelopmentinstudiesof5-to12-year-olds.Furtherresearchonchildren’srepresentationsofmathematicsfoundthatalackofstructuralawarenessimpedesmathematicaldevelopmentandrelatestopoorrepresentationalcapacity.Lowachieversconsistentlyproducedpoorlyorganisedrepresentationslackinginstructure,whereashighachieversusedabstractnotationswithwell-developedstructures.Essentially,low-achievingstudentsdidnotfocusonstructuralfeatureswhenlearningmathematics(seeMulligan,2010).

Asuiteofstudiesthatfollowed,thePatternandStructureProject,indicatedthatyoungchildrenwhounderstandtheunderlyingstructureofonemathematicalconceptarealsolikelytoperceivethestructureunderlyingotherquantitativeconcepts,andcanlearntoabstractandgeneraliseconceptsatanearlyage.Theassessmentoffirstgradersfoundtheirresponsestoarangeofmathematicaltaskscouldbecategorisedintofourstagesofstructuraldevelopment–pre-structural,emergent,partialandstructural,withafifthstage,advancedstructural,addedwiththeprogressionofhigh-achievingstudents(Mulligan&Mitchelmore,2009).Thestudent’sstageofstructuraldevelopmentwashighlyconsistent


49

overallandreflectedtheirlevelofmathematicalunderstanding.

ThePatternandStructureMathematicsAwarenessProgram(PASMAP)wasthendevelopedtoraisestudents’awarenessofpatternandstructurethroughavarietyofwell-connectedpattern-elicitingexperiences.Studieshaveincludedanextensive,whole-schoolprojectacrossKindergartentoYear6;twoyear-long,designstudiesinYears1and2;andanintensive,a15-weekempiricalevaluationofanindividualisedprogramwithasmallgroupofkindergartenchildren(seeMulligan,2010).

Inrelatedstudies,Papicfoundthatpreschoolerswhoareprovidedwithopportunitiestoengageinmathematicalexperiencesthatpromoteemergentgeneralisation(aninterventionprogram)arecapableofabstractingcomplexpatternsbeforetheystartformalschooling(Papic,Mulligan,&Mitchelmore,2009).

Thesestudiesindicatethatyoungchildrencanlearncomplexmathematicalconceptsveryquicklyandeffectivelybyfocusingoncrucialfeaturesofmathematicalpatternandstructure;visualmemory,constructingandrepresentingstructuresindependentlyofmodels,andthearticulationof‘samenessanddifference’wascentraltothisprocess.However,thesefindingsalsosupportedthoseofearlierstudiesinthatlowachieversfailedtoperceivestructureeveninsimplemathematicalformssuchasthepropertiesofasquare.

Reconceptualising Early Mathematics Learning

ThisnewstudywasdesignedtoevaluatetheeffectivenessofPASMAPonstudents’mathematicaldevelopmentinthefirstyearofformalschooling.Apurposivesampleoffourlargeprimaryschools,twoinSydneyandtwoinBrisbane,representing

316studentsfromadiverserangeofsocio-economicandculturalcontexts,participatedintheevaluationthroughoutthe2009schoolyear.Twodifferentmathematicsprogramswereimplemented:ineachschool,twokindergartenteachersimplementedthePASMAPandtwoimplementedtheirstandardprogram.ThePASMAPframeworkwasembeddedintothestandardkindergartenmathematicscurriculum.Aresearcher/teachervisitedeachteacheronaweeklybasisandequivalentprofessionaldevelopmentforbothpairsofteacherswasprovided.Incrementalfeaturesoftheprogramwereintroducedbytheresearchteamgradually,atapproximatelythesamepaceandwithequivalentmentoringforeachteacher,overthreeschoolterms.

Allstudentswerepre-andpost-testedwith I Can Do Maths(ICDM)(Doig&deLemos,2000);frompre-testdatatwo‘focus’groupsoffivechildrenineachclasswereselectedfromtheupperandlowerquartiles,respectively.These160studentswerepre-andpost-interviewedusinganewversionofa20-itemPattern and Structure Assessment (PASA).Intervention-baseddataincludedobservationnotes,digitalrecordingsoftheirlearningexperiencesandarangeofworksamples.Studentprofilesoflearningaimto(i)describethe‘tracked’developmentalpathway(s)oftheirmathematicalconceptsandprocesses,(ii)analysethequalityoftheunderlyingstructuralcharacteristics,(iii)describesalientfeaturesorrelationshipsbuiltbythestudentbetweencomponentsorconcepts,and(iv)provideevidenceofemergentgeneralisationsandreasoningtosupportthese.

The Pattern and Structure Mathematics Awareness Program Intervention

Theprogramisinnovativeinitsconceptualframeworkandtheway

learningexperiencesarescaffolded,wherechildrenareencouragedtoseekoutandrepresentpatternandstructureacrossdifferentconceptsandtransferthisawarenesstootherconcepts.Itfocusesonfundamentalprocessessuchassimpleandcomplexrepetitions,growingpatternsandfunctions,unitisingandmultiplicativestructurealsocommontounitsofmeasure;spatialstructuring,thespatialpropertiesofcongruenceandsimilarity,andtransformation(seeMulligan,Mitchelmore,English,&Robertson,2010).Emphasisisalsolaidoncountingthroughpatternsandmeasures,thestructureofoperations,equivalenceandcommutativity.

Discussion

PreliminaryanalysisindicatesthatbothgroupsofstudentsmadesignificantprogressinmathematicslearningoutcomesasdescribedbythestatesyllabusandmeasuredbytheICDMtest.ItwasnotexpectedthatsignificantdifferenceswouldbefoundbetweenPASMAPandregularstudentsonpre-andpost-testsscoresonthisstandardisedmeasure.However,initialanalysisofqualitativedata,trackingofthe‘focus’students,indicatedmarkeddifferencesbetweengroupsinstudents’levelofstructuraldevelopment(AMPS).StudentsparticipatinginthePASMAPprogramshowedhigherlevelsofAMPSthantheregulargroup,madeconnectionsbetweenmathematicalideasandprocesses,andformedemergentgeneralisations.Someofthemoreablestudentsusedoneaspectofpatternandstructuretobuildnewandmorecomplexconcepts.Graduallytheseconnectionsbecamemorelikesystemsoflearningthathadcommonstructuralfeatures.GoldininhisworkwithThomasandcolleaguesreferstotheseasautonomouspowerfulsystemsthatbecomeindependentovertime(Thomas,Mulligan,&Goldin,2002).


50

Someexemplarsofstudents’developingstructuralfeaturesarenowdescribed.Studentsusedtenframecardstopromotethestructureoften,spatialandcountingpatterns,groupingandadditioncombinations.Asanassessmenttask,theywererequiredtodrawtheframefrommemory,describehowtheydidthisandwhytheframewasused.Figures1to6showtypicalexamplesoftenframesthathavebeendrawnbysixindividualsatthesamepointinthelearningsequence.Eachfigurereflectsdevelopmentalfeaturesofstudents’awarenessanduseofthestructureoftheten-frame:theuseof2-wiseor5-wisepatterns(quinary-basedstructure),theuseofco-linearity(rowandcolumnstructure)andtheconstructionofadditionpairs.Figures1to3shownorecognitionofthestructureoftheten-frameanditsfacility,althoughthesestudentswereusingtenframesregularly;thesestudentshadpoorAMPSacrossarangeoftasks.Figure4showsawarenessofthepatternoffivesandFigures5and6strongstructuralfeatures.

Inanothertaskthechildrenhadtorecalltheiruseofpatterncardsdepictingthepatternofsquaresi.e.,1,2×2,3×3,4×4,5×5squaregridcards.Thispatternwaslinkedtoprior

useofsimplegridpatternsintroducedearlyintheprogramandthecountingpatternsofmultiples.Figures7,8and9showattemptstodrawthepatternfrommemory,butthestructureofincreasinglylargersquaresisnotgeneralisedandthenumberofunitsiscountedoraddedonindividually.Figure9showsunitsalignedbutextendeduni-dimensionally;thisisaddingacolumnratherthanrecognisingthemultiplicativestructure.Figure10showsthestudent’sstructuraldevelopmentofthepatternofincreasinglylargerarraysassquaresusingthealignmentofthe‘growingsquares’.Healsoexplainsthenumericalsequenceasmultiplicative.

Implications

Oneoutcomeoftheprojectistovalidatealternativedevelopmentalpathsforyoungchildren’smathematicslearning.Ultimatelythisresearchmayprovidebetterpathwaysforthosechildrenwhomaybepronetodifficultiesinlearningmathematics;thatis,thosewholackAMPS.Tracking,describingandclassifyingchildren’smodels,representationsandexplanationsoftheirmathematicalideas,andanalysingthestructuralfeaturesofthisdevelopmentarefundamentallyimportant.Ourstudies

indicatethatconsistentmethodsforanalysingstudents’AMPSareindeedpossibleandthisprocessprovidesarichbasisforassessingandscaffoldingstudents’mathematicaldevelopment.Ourgoalisareliable,coherentmodelforcategorisinganddescribingstructuraldevelopmentwithalignedpedagogicalframeworks.

IntheforthcomingAustralianNationalCurriculum(ACARA,2010),NumberandAlgebrastrandsarealignedwithProblemSolvingandReasoningProficiencies.‘Analgebraicperspectivecanenrichtheteachingofnumber…andtheintegrationofnumberandalgebra,especiallyrepresentationsofrelationshipscangivemoremeaningtothestudyofalgebrainthesecondaryyears.Thiscombinationincorporatespatternand/orstructureandincludesfunctions,setsandlogic’.Further,theintegrationofmeasurementandgeometry,andstatisticsandprobabilitybringsnewopportunitiestodevelopastructuralapproach.TheproposedPASMAPwillenableprofessionalstodevelopandevaluateanewapproachwithflexibility–onethatintegratespatternsandstructuralrelationshipsinmathematicsacrossconceptssothatamoreholisticoutcomeisachieved.

Figure�1:�Pre-structuralimageof‘tallbuildingswithbridges’.

Figure�2:�Emergentstructuralimagesofsingleunits.

Figure�3:�Emergentstructuralimagesof‘singleanddouble’frames.

Figure�4:�Partialstructureshownby2x5unequalunits.

Figure�5:�Partialstructure:alignedsingleunitstenframestructure.

Figure�6:�Structuralfeaturesshowing5-wisepattern.


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Mathematicslearningforthefuturewillrequireyoungchildrentoreasonmathematicallyincreativeandflexiblewaysinordertosolvemulti-disciplinaryproblems.Focusingonpatternandstructuremaynotonlyleadtoimprovedgeneralisedthinking,butcanalsocreateopportunitiesfordevelopingcognitivecapacitiescommensuratewiththeabilitiesofyounglearnersandthedemandsofmathematicslearningforthefuture.

ReferencesAustralianCurriculum,Assessmentand

ReportingAuthority.(2010).Shape of the Australian curriculum: Mathematics.http://www.acara.edu.au/verve/_resources/Australian_Curriculum_-_Maths.pdf

Blanton,M.,&Kaput,J.(2005).Characterizingaclassroompracticethatpromotesalgebraicreasoning.Journal for Research in Mathematics Education, 36,412–446.

Carraher,D.W.,Schliemann,A.D.,Brizuela,B.M.,&Earnest,D.(2006).Arithmeticandalgebrainearlymathematicseducation.Journal for Research in Mathematics Education, 37,87–115.

Clarke,B.,Clarke,D.,&Cheeseman,J.(2006).Themathematicalknowledgeandunderstandingyoungchildrenbringtoschool.Mathematics Education Research Journal, 18(1),78–103.

Clements,D.,&Sarama,J.(2009).Learning and teaching early maths: The learning trajectories approach. NY:Routledge.

Doig,B.,&deLemos,M.(2000).I can do maths. Melbourne:ACER

Ellemor-Collins,D.&Wright,R.,(2009).Structuringnumbers1–20:Developingfacileadditionandsubtraction,Mathematics Education Research Journal, 21(2),50–75.

English,L.D.(2010).Modelingwithcomplexdataintheprimaryschool.InR.Lesh,P.Galbraith,C.R.Haines,&A.Hurford(Eds.),Modeling students’ mathematical modeling competencies: ICTMA 13.Springer.

English,L.D.(2004).Promotingthedevelopmentofyoungchildren’smathematicalandanalogicalreasoning.InL.D.English(Ed.),Mathematical and analogical reasoning of young learners.Mahwah,NJ:LawrenceErlbaum.

Ginsburg,H.P.,Lee,J.S.,&Boyd,J.S.(2008).Mathematicseducationforyoungchildren:Whatitisandhowtopromoteit.Social Policy Report, 22(1),3–11and14–23.Availableonlinefrom:http://www.srcd.org/spr.html

Hunting,R.(2003).Part–wholenumberknowledgeinpreschoolchildren.Journal of Mathematical Behavior, 22(3),217–235.

Lehrer,R.(2007).Introducingstudentstodatarepresentationandstatistics.InJ.Watson&K.Beswick(Eds.),

Mathematics: Essential for learning, essential for life(Proceedingsofthe30thannualconferenceoftheMathematicsEducationResearchGroupofAustralasia,Hobart,Vol.1,pp.22–41).Adelaide:AAMT.

Mulligan,J.T.,Prescott,A.,&Mitchelmore,M.C.(2003).Takingacloserlookatyoungstudents’visualimagery.Australian Primary Mathematics Classroom, 8(4),23-27.

Mulligan,J.T.,Mitchelmore,M.C.,English,L.,&Robertson,G.(inpress).ImplementingaPatternandStructureMathematicsAwarenessPrograminkindergarten.Shaping the future of mathematics education,(Proceedingsofthe33rdannualconferenceoftheMathematicsEducationResearchGroupofAustralasia),Fremantle,WA:MERGA.

Mulligan,J.T.,&Mitchelmore,M.C.(2009).Awarenessofpatternandstructureinearlymathematicaldevelopment.Mathematics Education Research Journal, 21(2),33–49.

Mulligan,J.T.,&Vergnaud,G.(2006).Researchonchildren’searlymathematicaldevelopment:Towardsintegratedperspectives.InA.Gutiérrez&P.Boero(Eds.),Handbook of research on the psychology of mathematics education: Past, present and future(pp.261–276).London:SensePublishers.

Figure�7:�Emergentstructure:patternofsquaresusingsingleunits

Figure�8:�Partialstructure:patternofsquaresusingequal-sizedunits;lackofstructureof‘square’

Figure�9:�Partialstructure:patternofsquareslimitedto5x5

Figure�10:�Structuralresponseshowingpatternandarraystructure


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Mulligan,J.T.(2010).Theroleofrepresentationsinyoungchildren’sstructuraldevelopmentofmathematics.Mediterranean Journal of Mathematics Education, 9(1),163–188.

Outhred,L.N.,&Mitchelmore,M.C.(2000).Youngchildren’sintuitiveunderstandingofrectangularareameasurement.Journal for Research in Mathematics Education, 31,144–167.

Papic,M.,Mulligan,J.T.,&Mitchelmore,M.C.(2009).Thegrowthofmathematicalpatterningstrategiesinpreschoolchildren.InM.Tzekaki,M.Kaldrimidou,&H.Sakonidis(Eds.),Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education(Vol.4,pp.329–336).Thessaloniki,Greece:PME.

Perry,B.,&Dockett,S.(2008).Youngchildren’saccesstopowerfulmathematicalideas.InL.D.English(Ed.),Handbook of international research in mathematics education(2nded).NY:Routledge.

Slovin,H.,&Dougherty,B.(2004).Children’sconceptualunderstandingofcounting.InM.J.Høines&A.B.Fuglestad(Eds.),Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education(Vol.4,pp.209–216).Bergen,Norway:PME.

Thomas,N.,Mulligan,J.T.,&Goldin,G.A.(2002).Children’srepresentationsandcognitivestructuraldevelopmentofthecountingsequence1–100.Journal of Mathematical Behavior, 21,117–133.

Thomson,S.,Rowe,K.,Underwood,C.,&Peck,R.(2005).Numeracy in the early years.Melbourne:AustralianCouncilforEducationalResearch.

Young-Loveridge,J.(2002).Earlychildhoodnumeracy:Buildinganunderstandingofpart–whole

relationships.Australian Journal of Early Childhood, 27(4),36–42.

vanNes,F.,&deLange,J.(2007).Mathematicseducationandneurosciences:Relatingspatialstructurestothedevelopmentofspatialsenseandnumbersense.The Montana Mathematics Enthusiast, 2(4),210–229.

Warren,E.,&Cooper,T.J.(2008).Generalisingthepatternruleforvisualgrowthpatterns:Actionsthatsupport8yearoldsthinking.Education Studies in Mathematics. 67(2),171–185.

Wright,R.J.(2003).MathematicsRecovery:Aprogramofinterventioninearlynumberlearning.Australian Journal of Learning Disabilities, 8(4),6–11.

AcknowledgementsTheresearchreportedinthispaperwassupportedbyAustralianResearchCouncilDiscoveryProjectsgrantNo.DP0880394,Reconceptualising early mathematics learning: The fundamental role of pattern and structure.TheauthorsexpresstheirthankstoDrCoralKemp;researchassistants–NathanCrevensten,SusanDaley,DeborahAdamsandSaraWelsby;participatingteachers,teachersaides,studentsandschoolcommunitiesfortheirgeneroussupportofthisproject.


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Learningaboutselectingclassroomtasksandstructuringmathematicslessonsfromstudents

Peter�SullivanMonash University

PeterSullivanisProfessorofScience,MathematicsandTechnologyEducationatMonashUniversity.Heistheauthoroftheshapepaperforthenewnationalmathematicscurriculum,editoroftheJournal of Mathematics Teacher Education,forfouryearswasamemberoftheAustralianResearchCouncilCollegeofExperts,andispresidentoftheAustralianAssociationofMathematicsTeachers.

AbstractAspartofalargerproject1,students’viewsontheirpreferencesforparticulartypesofmathematicaltasksweresought,aswellashowtheydescribetheiridealmathematicslesson,andtheirresponsestospecificallypreparedtasksfromsequencesoflessons.Thestudentshadparticularviewsaboutbothtasksandlessonsandwereabletoarticulatetheirviews.Teacherswoulddowelltoseektofindoutthetypesoftasksandlessonsthatparticularstudentsprefer,andtobemoreexplicitaboutwhattheyareintendingtodoineveryoneoftheirlessons.

Introduction

Therearemanysetsofrecommendationsaboutcharacteristicsofeffectiveteaching,whicharegenerallycompiledtheoretically,orfromsurveys,orfromdescriptionsofexemplaryteachers(seeClarke&Clarke,2004;Hattie&Timperley,2007;EducationQueensland,2010).Theresearchsummarisedhereattemptedtoexaminetheviewsofstudentsonthetypesoftaskstheyvalue,andthestructureoflessonsthattheyprefer.

Whiletherehavebeenmanystudiesseekingstudents’attitudes,values,beliefsandmotivation,theapproachreportedherealignswithZananddiMartino(2010)whoarguedthatemphasisshouldmovefrommeasuringattitudestodescribingthem.Theyarguedformorenarrativeapproachestodescribingstudentattitudes,including

1 TTMLisanAustralianResearchCouncilfundedresearchpartnershipbetweentheVictorianDepartmentofEducationandEarlyChildhoodDevelopment,theCatholicEducationOffice(Melbourne),MonashUniversityandAustralianCatholicUniversity.BarbaraClarkeandDougClarkewerealsoresearchersontheproject.

withlargesamples,withthegoalofunderstandingbehaviour.

ThisresearchperspectivealsoadoptedasimilarperspectivetothatofDaniels,KalkmanandMcCombs(2001),whoarguedthateventhoughstudentsareabletoarticulatecoherentviewsonissuesofpedagogytheyareseldomaskedtodoso,andthatstudentsareparticularlyabletocommentonclassroomandschoolenvironments.Allen(2003)similarlyarguedthattherehasbeentoolittleattentiontostudents’perspectivesofaspectsofteachingandclassorganisation.Itisrecognisedthatteachinginvolvesmuchmorethanfindingwaystopresentthecontent,andisconnectedtorelationships,studentself-regulation(Dweck,2000)andmotivation(Middleton(1995),soitisrelevanttoseekstudents’perspectivesontheseissues.

Intermsofseekingstudents’viewsabouttaskstheprojectchosetofocusdatacollectionontheextenttowhichtheyfelttheylearned,andwhethertheylikedparticulartypesoftaskssincetheseseemedtobemaindeterminantsoftheirdecisionsonengagement.Inthepilotingofourinstrumentswefoundthatthestudentswereabletorespondtobothtypesofpromptswithoutrequiringfurtherclarification.Ourapproachwastoseeksomeresponsestopredeterminedscalesaswellassomefreeformatnarrativesbythestudentstoallowtheirrealconcernstoemerge.Wecollectedthreecomplementarysetsofdata,givingabreadthoftypesofdataandthereforegreaterinsightsintotheviewsofstudents.Thethreeseparatedatasetsarenotpresentedhereduetospacelimitationsbutwillbepresentedintheworkshop.Asummaryofthefindingsaredescribedinthefollowingsections.


54

Responses of students to predetermined prompts about tasks and pedagogies

Asurveywasdesignedtogatherresponsesonaspectsoflessonsandtasksfromacross-sectionofstudents.Aswellasseekinginformationonvariousaspectsoflessons,wealsoincludedspecificitemsaskingstudentstocomparedifferenttypesoftasksandtoindicatetheirpreferences.

TheitemsongeneralaspectsofpedagogywereadaptedfromClarkeetal.(2002)andSullivanetal.(2009),andtheitemsontaskswerewrittenspecifically.Therewere930studentsin96classesacross17schoolswhocompletedthesurvey.

Tosummarisetheresultsfromthesurvey,itseemsthatateachofthesemiddleyears’levelsthereisarangeofstudentsatisfactionandconfidence,andteachersshouldbeawareoftheviewsofeachoftheirstudents.Italsoseemsthatteachersmakeadifferencetostudents’responsesandteachersneedsupportnotonlytofindoutstudents’levelsofsatisfactionandconfidence,butalsoonstrategiestoaddressnegativeresponses.Eachofthetasktypespresentedwerelikedmostbysomestudents,andlikewiseeachofthetypeswasratedastheonefromwhichtheycanmostlearn;thissuggeststhatteachersneedtousealltypesoftaskintheirteaching.Arelatedissueisthatstudentsmayneedsupporttogainbenefitsfromtasksthattheydonotlikeordonotfeelthattheycanlearnfrom.Itseemsimportantthatteachersmakestudentsawareofthepurposeoftasksandwhatitistheteachersarehopingthestudentswilllearnfromthem.Thestudentsseemtoliketasksthatareeasyyetfeeltheylearnbestfromtasksthatarechallenging.Ofcourse,wewouldhopethatstudentscanalsolearnfromtaskstheyfindeasy,andliketasksthatarechallenging.Again,itmaybeimportantforteacherstoillustrateor

emphasisetheroleofthetasksandthenatureofthechallengetheyoffer.

Narrative descriptions of students’ perceptions of characteristics of desired mathematics lessons

Usingadifferentapproach,wealsosoughtinsightsintostudents’perceptionsofthedesiredcharacteristicsofmathematicslessonsthroughtheirnarrativeresponses.Itwashopedinthiswaytogaininsightsintothewaysstudentsdescribedtheirdesiredcharacteristics,ratherthanbyratinglessoncharacteristicspreparedbyus.Wedidthisthroughopen-endedresponsestoparticularpromptsontheoverallsurvey.

Insummary,themainimpressionfromtheirresponsesistheirdiversity,andthereareclearlymanywaysinwhichstudentsrespondtolessons.Thereweretwotrendsintheirlessondescriptionsof,ononehand,studentsrecallingeffectiveteachingofacontenttopic,whereastherewereotherswhorememberedinterestingaspectsofthepedagogy.Inexplainingtheirchoiceoflesson,themaincategoryofresponsesrelatedtofun,butlearningsomethingnewwasalsofrequentlycited.Wenotethatthedescriptionsofhatedlessonsalsoreferredtoparticulartopics.Sowhilerecognisingthatsomestudentsdislikesometopics,teachersareadvisedtofocusonthestudents’learningofcontent,andtochooseinterestingandfunwaystoengagestudentsinthatlearning.

Students’ essays on their ‘ideal maths class’

Wealsosoughtstudents’viewsonlessonsandteachingthroughaparticularpromptseekingnarrativeresponses.Weaskedthestudentsintwooftheschoolsthatcompletedalessonsequencetowriteanessay,theparticularpromptofwhichwas:

Writeastoryaboutyouridealmathsclass.Writeaboutthesortsofquestionsorproblemsyouliketoanswer,whatyouliketobedoingandwhatyouliketheteachertobedoinginyouridealmathsclass.

Theintentionwastogaininsightintowhatthestudentsrecalledabouttheirmathematicsclasses,anditcanbeassumedthattheseresponsescanbetakenasindicativeofthelessonfeaturesthatthestudentslike.Thefollowingisanexampleofatypicalstudent’sessay,presentedasitwaswritten:

Myfavoritemathswouldstartwitha10minintroductionweretheteacherexplainsthegametoallofusandstillallowingtimeforquestions.Thegameswouldbe2+peopleforacompetitionandpeoplewillsplitintogroupsandwillorganizewhoplayswho5mineveryonewillbeplayingatalltimesunlessthereisanoddamountofpeoplewewillplayfor25min.attheendoftheLessonthegroupswillfigureoutwhowasthewinnerandpeoplecansharewhattheyLearntLikedandstrategiestheyused.Sharingisfor10min.formysecondoptionIwoulddoreallifeproblemsLike250gramsofsugarfor$10.50or750gramsfor$33.15.Ilikereallifeproblemsbecausetheycouldhelpmeonedayanditssetoutdifferentlythanmath.forthistheexplanationisfor5minthisisbecauseyoudon’tneedtoexplaintherules.

Inthisresponsethereweretwokeyelements:theuseofagame,andtheuseofreal-lifeproblems,buttherealimplicationisthatthisisindicativeofthedetailthatstudentsusedtodescribetheidealclass.

Insummary,itseemsthattheresponsestothispromptaboutanideallessonseemeddependentontheteacher.In


55

synthesisingtheresponses,studentslikedlessonsthatusedmaterials(althoughthesewerenotstructuredmaterials),wereconnectedtotheirlives,involvedgames,werepracticalwithsomeemphasisonmeasurement,inwhichtheyworkedoutside,withthemethodofgroupingbeingimportant,andoverhalfofthestudentsclaimtoliketobechallenged.Aninterestingresultwasthat,contrarytoexpectations,manystudentsclaimedtolikehelpfromtheteacheronlyafteraperiodofeffort.

ConclusionItisclearthatthereismuchthatcanbelearnedfromtheresponsesofstudents.Thestudentswhorespondedtotheseinstrumentsareclearlyawareofaspectsofteaching,includingthoseaspectsthataresubtle.Whilemostoftheircommentsarenotsurprising,theydoendorsestronglymanyofthepedagogiesthatsometeachersseemreluctanttoadopt.Oneclearimplicationistheneedforteacherstouseavarietyoftasksandlessonstructures,arecommendationthatonesuspectshasparticularsignificanceforsecondaryteachers.Anotherimplicationisthat,sincenotalltasksorlessonscanbethosepreferredbystudents,teachersneedtomakeeffortstoexplainthechoiceoftaskanditspurpose,andtoexplainthegoalofparticularpedagogiesthattheymightuse.

ReferencesAllen,B.(2003).Pupils’perspectiveson

learningmathematics.InB.Allen&S.Johnston-Wilder(Eds.),Mathematics education: Exploring the culture of learning (pp.233–241).London:RoutledgeFalmer.

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project: Final report.AustralianCatholicUniversityandMonashUniversity.

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Ross�TurnerAustralian Council for Educational Research

RossTurnermanagesACER’sInternationalPISAproject,coordinatingtheACERteamandinternationalconsortiumpartnerstomeettherequirementsofACER’scontractwiththeOECD.HehasfilledthisroleasaPrincipalResearchFellowsince2007,andbeforethatprovidedgeneralleadershipandmanagementtothePISAprojectandtootherACERprojectsasaSeniorResearchFellowsince2000.Rossalsoprovidesleadershipinthemathematicsarea,havingledPISAmathematicsframeworkandtestdevelopmentandbeingresponsibleforPISAmathematicsimplementationthroughouthistimeatACER.

For13yearspriortothatRosswasemployedinvariousrolesattheVictorianBoardofStudies.Hewassecondedin1987tocontributetoredevelopmentofthemathematicscurriculumandassessmentarrangementsintheVictorianCertificateofEducation.HewasappointedtothepositionofManager,Mathematicsin1989andledtheimplementationoftheVCEmathematicsstudy.HewasappointedasManager,ResearchandEvaluationin1993.InthatrolehemonitoredannualVCEoutcomes,andoversawdevelopmentandimplementationofstatisticalproceduresemployedintheprocessingandreportingofVCEdata.

Identifyingcognitiveprocessesimportanttomathematicslearningbutoftenoverlooked

AbstractThispresentationintroducesasetofmathematicalcompetenciesthatdeservetobegivenmoreattentioninourmathematicsclassrooms,onthegroundsthatthepossessionofthesecompetenciesrelatesstronglytoincreasedlevelsofmathematicalliteracy.Thepresenterarguesthatwidespreadunder-representationofthesecompetenciesamongthegeneralpopulacecontributestounacceptablylargemeasuresonthemathematics terror index.

TheargumentinsupportofthesecompetenciescomesoutoftheOECD’sProgrammeforInternationalStudentAssessment(PISA).ItisbasedontheresultsofresearchconductedbymembersofthePISAmathematicsexpertgroup.Thatresearchwillbedescribed,thecompetenciesunderdiscussionwillbedefined,andthecaseforgreateremphasisonthesecompetencieswillbemade.

Introduction

TheOECD’sProgrammeforInternationalStudentAssessment(PISA)aimstomeasurehoweffectively15-year-oldscanusetheiraccumulatedmathematicalknowledgetohandle‘real-worldchallenges’.Themeasureswederivefromthisprocessarereferredtoasmeasuresofmathematical literacy.TheliteracyideaseemstohavereallytakenholdamongthosecountriesthatparticipateinPISA.Itisgenerallyregardedasveryimportantthatpeoplecanmakeproductiveuseoftheirmathematicalknowledgeinappliedandpracticalsituations.

InthispresentationIwilldemonstratesomeillustrativePISAitemsasawayofintroducingasetofmathematicalcompetenciesthatarefundamentaltothepossessionanddevelopmentofmathematicalliteracy,andwillproposethatthesedeserveastrongerplaceinourmathematicsclasses.


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Illustrative PISA items

TwoitemsfromtheunittitledExportsinvolveinterpretingdatapresentedinabargraphandapiechart.Thefirstquestioncallsforthedirectinterpretationofafamiliargraphform:identifyingthatthebargraphcontainstherequiredinformation,locatingthebarfor1998andreadingtherequirednumberprintedabovethebar.

Thesecondquestionismoreinvolved,sinceitrequireslinkinginformationfromthetwographspresented:applyingthesamekindofreasoningrequiredinthefirstquestiontoeachofthetwographstolocatetherequireddata,thenperformingacalculationusingthetwofiguresfoundfromthegraphs(find9%of42.6million).

AfurtherquestionCarpenterispresented,whichrequiressomegeometricalknowledgeorreasoning.Familiaritywiththepropertiesofbasicgeometricshapesshouldbesufficient toestablishthatwhilethe‘horizontal’

componentsofthefourshapesareequivalent,theobliquesidesofDesignBarelongerthanthesumofthe‘vertical’componentsofeachoftheothershapes.

Whatdowefindwhenproblemssuchasthesearegiventorandomsamplesof15-year-oldsacrossover60countriesaroundtheworld?

Table1presentsthepercentcorrectdataforallstudentsinternationallyandallAustralianstudentswhoweregiventhelistedquestionsinthePISA2003survey.

ThechartinFigure1showswherethesepublicallyreleasedquestionsfitinthecontextofthewholePISA2003surveyinstrument.Theinternationalpercentcorrectfortheillustrativeitemsarelabelled,amidstthe84itemsusedinthesurvey(withabarforeachitem,orderedbytheirinternationalpercentcorrectvalue).ExportsQ1wasoneoftheeasieritemsinthetest,whileExportsQ2wasamoderatelydifficult

Table�1:PercentcorrectforthreeillustrativePISAmathematicsquestions

QuestionPer�cent�correct��

(all�students)Per�cent�correct��(Aus�students)

ExportsQ1 67.2 85.8

ExportsQ2 45.6 46.3

Carpenter 19.4 23.3

100.0

19.4

45.6

67.2

90.0

80.0

70.0

60.0

50.0

40.0

30.0

20.0

10.0

0.0

Figure�1:InternationalpercentcorrectofallPISA2003mathematicsquestions


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item.Carpenterwasoneofthemostdifficultitems.

Is there a problem?

WecouldspeculateaboutdifferencesinperformancelevelsbetweenAustralianandinternationalstudents,butformyimmediatepurpose,Imightsimplysuggestthatasamathematicsteacher,Iwouldhavehopedthatmost15-year-oldscouldanswerquestionslikethesecorrectly.Thisalsohasimplicationsforwhathappenstothose15-year-oldswhentheyleaveschool,sincethemathematicalcapabilitiesstudentsdemonstratebythetimetheyarenearingschoolleavingageforeshadowstheapproachthoseindividualswilltaketousingmathematicslaterinlife.

Istheproblemthatmanystudentsdon’tknowtherequiredmathematicalconcepts;thattheyhavenotlearnedtherequiredmathematicalskills?Orcoulditbethattoomany15-year-oldsaresimplyunabletoactivatetherequiredknowledgewhenitcouldbeuseful;thatthereisadisconnectbetweenthewayinwhichmanyofushavebeentaught,andtheopportunitiestousemathematicsinlifeoutsideschool?

Usuallytheopportunitiestousemathematicsthatwecomeacrossarenotpackagedinquitethewaytheywereinschool.There,youknewwhenyouweregoingtoamathematicsclass.Whenyouwenttothatclass,youdidsoexpectingthatyouwoulddothingsrelatedtomathematics.Youhadamathematicsteacherwhotaughtanddemonstratedmathematicalideasandskills,gaveyousomeexamples,andthenpointedyoutoasetofexercisesmoreorlesslikethoseusedtodemonstratetheideaorskillyouwerelearning.Youweregiveninstructionslike‘counttheseobjects’,or‘addthesenumbers’,or‘drawthisgraph’,

or‘factorisetheseexpressions’.Theobjectiveswereclearlymathematical.

Intherealworld,that’snotnormallyhowmathematicscomestous.Wehavetomakethejudgmentsanddecisionsaboutwhatmathematicalknowledgemightberelevant,andhowtoapplythatknowledge.Thatassumeswearemotivatedenoughinthefirstplacetoevennoticethatmathematicsmightberelevant.

ThisbringsusbacktooneofthemostimportantandinfluentialideasthatunderpinsthePISAproject:itsemphasisonwhatiscalledliteracy.PISAmeasuresandreportsthedegreetowhichthe15-year-oldsinparticipatingcountrieshavedevelopedtheirliteracyskillsinmathematicsandtheothersurveydomainssothattheycanapplytheirknowledgetosolvecontextualisedproblems–problemsthataremorelikethechallengesandopportunitieswemeetinourwork,leisure,andinourlifeascitizens.Butwhatarethecapabilitiesthatequipadultstomeetsuchchallenges?

Mathematical competencies – the research

TheframeworksthatgovernedthemathematicspartofthePISAsurveysconductedin2000,2003,2006and2009describeasetofeightmathematicalcompetencies.Forthepurposesofaresearchactivitywehavecarriedout,thesehavebeenconfiguredasasetofsixcompetenciesthatarefundamentaltotheconceptofmathematicalliteracythatPISAespouses,namelythecapacitytouseone’smathematicalknowledgetohandlechallengesthatcouldbeamenabletomathematicaltreatment.OurresearchhasshownthatthesecompetenciescanbeusedtoexplainaverylargeproportionofthevariabilityinthedifficultyofPISAmathematicstestitems,possiblyasmuchas70percentofthatvariability.Toidentify

factorsthatexplainsomuchofwhatmakesmathematicsitemsdifficultisanimportantfinding.

Thosecompetenciescanbethoughtofasasetofindividualcharacteristicsorqualitiespossessedtoagreaterorlesserextentbyindividuals.However,wecanalsothinkaboutthesecompetenciesfromthe‘perspective’ofamathematicsproblem,orasurveyquestion:towhatextentdoesthequestioncallfortheactivationofeachofthesecompetencies?Inthefollowingsectionthesixcompetenciesaredefined,andthetask–leveldemandforactivationofeachcompetencyatdifferentlevelsisdescribed.

Communication

Mathematicalliteracyinpracticeinvolvescommunication.Reading,decodingandinterpretingstatements,questions,tasksorobjectsenablestheindividualtoformamentalmodelofthesituation,animportantstepinunderstanding,clarifyingandformulatingaproblem.Duringthesolutionprocess,whichinvolvesanalysingtheproblemusingmathematics,informationmayneedtobefurtherinterpreted,andintermediateresultssummarisedandpresented.Lateron,onceasolutionhasbeenfound,theproblemsolvermayneedtopresentthesolution,andperhapsanexplanationorjustification,toothers.

Variousfactorsdeterminethelevelandextentofthecommunicationdemandofatask.Forthereceptiveaspectsofcommunication,thesefactorsincludethelengthandcomplexityofthetextorotherobjecttobereadandinterpreted,thefamiliarityoftheideasorinformationreferredtointhetextorobject,theextenttowhichtheinformationrequiredneedstobedisentangledfromotherinformation,theorderingofinformationandwhetherthismatchestheorderingofthethoughtprocessesrequiredto


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interpretandusetheinformation,andtheextenttowhichdifferentelements(suchastext,graphicelements,graphs,tables,charts)needtobeinterpretedinrelationtoeachother.Fortheexpressiveaspectsofcommunication,thelowestlevelofcomplexityisobservedintasksthatsimplydemandprovisionofanumericanswer.Astherequirementforamoreextensiveexpressionofasolutionisadded,forexamplewhenaverbalorwrittenexplanationorjustificationoftheresultisrequired,thecommunicationdemandincreases.

Mathematising

Mathematicalliteracyinpracticecaninvolvetransformingaproblemdefinedintherealworldtoastrictlymathematicalform(whichcanincludestructuring,conceptualising,makingassumptions,formulatingamodel),orinterpretingamathematicalsolutionoramathematicalmodelinrelationtotheoriginalproblem.

Thedemandformathematisationarisesinitsleastcomplexformwhentheproblemsolverneedstointerpretandinferdirectlyfromagivenmodel;ortotranslatedirectlyfromasituationintomathematics(forexample,tostructureandconceptualisethesituationinarelevantway,toidentifyandselectrelevantvariables,collectrelevantmeasurementsandmakediagrams).Themathematisationdemandincreaseswithadditionalrequirementstomodifyoruseagivenmodeltocapturechangedconditionsorinterpretinferredrelationships;tochooseafamiliarmodelwithinlimitedandclearlyarticulatedconstraints;ortocreateamodelforwhichtherequiredvariables,relationshipsandconstraintsareexplicitandclear.Atanevenhigherlevel,themathematisationdemandisassociatedwiththeneedtocreateorinterpretamodelinasituationinwhichmanyassumptions,variables,relationshipsandconstraintsaretobeidentifiedor

defined,andtocheckthatthemodelsatisfiestherequirementsofthetask;ortoevaluateorcomparemodels.

Representation

Thiscompetencycanentailselecting,devising,interpreting,translatingbetween,andusingavarietyofrepresentationstocaptureasituation,interactwithaproblem,ortopresentone’swork.Therepresentationsreferredtoincludeequations,formulas,graphs,tables,diagrams,pictures,textualdescriptionsandconcretematerials.

Thismathematicalabilityiscalledonatthelowestlevelwiththeneedtodirectlyhandleagivenfamiliarrepresentation,forexampletranslatingdirectlyfromtexttonumbers,orreadingavaluedirectlyfromagraphortable.Morecognitivelydemandingrepresentationtaskscallfortheselectionandinterpretationofonestandardorfamiliarrepresentationinrelationtoasituation,andatahigherlevelofdemandstillwhentheyrequiretranslatingbetweenorusingtwoormoredifferentrepresentationstogetherinrelationtoasituation,includingmodifyingarepresentation;orwhenthedemandistodevisearepresentationofasituation.Higherlevelcognitivedemandismarkedbytheneedtounderstandanduseanon-standardrepresentationthatrequiressubstantialdecodingandinterpretation;todevisearepresentationthatcapturesthekeyaspectsofacomplexsituation;ortocompareorevaluatedifferentrepresentations.

Reasoning and argument

Thisskillinvolveslogicallyrootedthoughtprocessesthatexploreandlinkproblemelementsinordertomakeinferencesfromthem,checkajustificationthatisgiven,orprovideajustificationofstatements.

Intasksofrelativelylowdemandforactivationofthisability,thereasoning

requiredinvolvessimplyfollowingdirectinstructions.Ataslightlyhigherlevelofdemand,itemsrequiresomereflectiontoconnectdifferentpiecesofinformationinordertomakeinferences(forexample,tolinkseparatecomponentspresentintheproblem,ortousedirectreasoningwithinoneaspectoftheproblem).Atahigherlevel,taskscallfortheanalysisofinformationinordertofolloworcreateamulti-stepargumentortoconnectseveralvariables;ortoreasonfromlinkedinformationsources.Atanevenhigherlevelofdemand,thereisaneedtosynthesiseandevaluateinformation,touseorcreatechainsofreasoningtojustifyinferences,ortomakegeneralisationsdrawingonandcombiningmultipleelementsofinformationinasustainedanddirectedway.

Devising strategies

Mathematicalliteracyinpracticefrequentlyrequiresdevising strategies for solving problems mathematically.Thisinvolvesasetofcriticalcontrolprocessesthatguideanindividualtoeffectivelyrecognise,formulateandsolveproblems.Thisskillischaracterisedasselectingordevisingaplanorstrategytousemathematicstosolveproblemsarisingfromataskorcontext,aswellasguidingitsimplementation.

Intaskswitharelativelylowdemandforthisability,itisoftensufficienttotakedirectactions,wherethestrategyneededisstatedorobvious.Ataslightlyhigherlevelofdemand,theremaybeaneedtodecideonasuitablestrategythatusestherelevantgiveninformationtoreachaconclusion.Cognitivedemandisfurtherheightenedwiththeneedtodeviseandconstructastrategytotransformgiveninformationtoreachaconclusion.Evenmoredemandingtaskscallfortheconstructionofanelaboratedstrategytofindanexhaustivesolutionora


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generalisedconclusion;ortoevaluateorcomparedifferentpossiblestrategies.

Using symbolic, formal and technical language and operations

Thisinvolvesunderstanding,manipulating,andmakinguseofsymbolicexpressionswithinamathematicalcontext(includingarithmeticexpressionsandoperations)governedbymathematicalconventionsandrules.Italsoinvolvesunderstandingandutilisingformalconstructsbasedondefinitions,rulesandformalsystemsandalsousingalgorithmswiththeseentities.Thesymbols,rulesandsystemsusedwillvaryaccordingtowhatparticularmathematicalcontentknowledgeisneededforaspecifictasktoformulate,solveorinterpretthemathematics.

Thedemandforactivationofthisabilityvariesenormouslyacrosstasks.Inthesimplesttasks,nomathematicalrulesorsymbolicexpressionsneedtobeactivatedbeyondfundamentalarithmeticcalculations,operatingwithsmalloreasilytractablenumbers.Moredemandingtasksmayinvolvedirectuseofasimplefunctionalrelationship,eitherimplicitorexplicit(forexample,familiarlinearrelationships);useofformalmathematicalsymbols(forexample,bydirectsubstitutionorsustainedarithmeticcalculationsinvolvingfractionsanddecimals);oranactivationanddirectuseofaformalmathematicaldefinition,conventionor

symbolicconcept.Increasedcognitivedemandischaracterisedbytheneedforexplicituseandmanipulationofsymbols(forexample,byalgebraicallyrearrangingaformula),orbyactivationanduseofmathematicalrules,definitions,conventions,proceduresorformulasusingacombinationofmultiplerelationshipsorsymbolicconcepts.Andayethigherlevelofdemandischaracterisedbytheneedformulti-stepapplicationofformalmathematicalprocedures;workingflexiblywithfunctionalorinvolvedalgebraicrelationships;orusingbothmathematicaltechniqueandknowledgetoproduceresults.

TheresearchonthesecompetenciessawagroupofexpertsassignratingstoPISAmathematicsitemsaccordingtothelevelofeachcompetencydemandedforsuccessfulcompletionofeachitem.Setsofitemswereratedbyseveralexperts,andtheratingswereanalysed:theaverageratingswereusedaspredictorsinaregressionontheempiricaldifficultyoftheitems.Thelevelofdemandforactivationofthesesixcompetenciesisanextremelygoodpredictorofthedifficultyofthetestitem.

InTable2thecompetencyratingsoftheillustrativeitemspresentedearlier,assignedbythreeexperts,arereported.

ForExportsQ1,arelativelyeasyitem,thecommunicationandrepresentationcompetenciesarethemoststrongly

demanded,withtheothersdemandedlittleornotatall.Thecommunicationdemandliesintheneedtointerpretreasonablyfamiliarneverthelessslightlycomplexstimulusmaterial,andtherepresentationdemandliesintheneedtohandletwographicalrepresentationsofthedata.ForQ2,therepresentationdemandisevenhigherbecauseoftheneedtoprocessthetwographsinmoredetail.Eachoftheothercompetenciesisalsocalledontosomedegree,withtheneedforreasoning,somestrategic thinking,andcallingonsomelow-levelprocedural knowledgetoperformtherequiredcalculation.

ForCarpenter,thereasoningrequiredcomprisesthemostsignificantdemand,buteachoftheothercompetenciesisdemandedtosomedegree.

The message?

Ofcoursethisresearchhasfurthertogo;nevertheless,theresultsofthisworkareencouragingenoughformetomakesomeconjecturesabouttheimportanceofthissetofcompetencies,andabouthowthisinformationmightbeusedinmathematicsclassrooms:

• Possessionofthesesixcompetenciesiscrucialtotheactivationofone’smathematicalknowledge.

• Themoreanindividualpossessesthesecompetencies,themoreableheorshewillbetomakeeffectiveuseofhisorhermathematical

Table�2:CompetencyratingsofthreeexpertsforthefourillustrativePISAitems

Rating��(from�raters�1/2/3)

Competency

ItemCommun-

icationMathematising

Repres-entation

Reasoning�and�argument

Devising�strategies

Symbols�and�formalism

ExportsQ1 1/1/2 1/0/0 1/1/1 0/1/0 0/0/0 0/1/0

ExportsQ2 1/1/2 1/0/1 2/2/2 1/1/1 2/0/1 0/1/1

Carpenter 2/2/1 1/0/1 1/1/1 2/3/2 2/1/1 1/1/1


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knowledgetosolvecontextualisedproblems.

• Thesecompetenciesshouldbedirectlytargetedandadvancedinourmathematicsclasses.

Ingeneral,notenoughtimeandeffortisdevotedinthemathematicsclassroomtofosteringthedevelopmentinourstudentsofthesefundamentalmathematicalcompetencies.Moreover,thecurriculumstructuresunderwhichmathematicsteachersoperatedonotprovideasufficientimpetusandincentiveforthemtofocusonthesecompetenciesascrucialoutcomes,alongsidethedevelopmentofthemathematicalconceptsandskillsthattypicallytakecentrestage.

What actions can be taken to improve this situation?

WemustrecognisetheimportanceofthefundamentalmathematicalcompetenciesthatIhavereferredto.Thesecompetenciesmustbegivenaconsciousfocusinourmathematicsclasses,throughteachingandlearningactivities,andthroughassessment.

Inmyview,akeyplacetostartiswiththenatureofdiscussionthatisfacilitatedinmathematicsclassrooms.Studentsneedtobegivenopportunitiestoarticulatetheirthinkingaboutmathematicstasksandaboutmathematicalconcepts.Obviouslyteachersplayacentralroleinorchestratingthatkindofdiscussioninclassandthisprovidesthebasisforencouragingstudentstotakethenextkeystep,writingdowntheirmathematicalarguments.Givingemphasistothecommunicationofmathematicalideasandthinking,bothinoralandwrittenforms,isessentialbothtoimprovingcommunicationskills,butalsotodevelopingthemathematicalideascommunicatedandthecapacitiestousethem.


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Robert�A.�ReeveUniversity of Melbourne

RobertReevecompletedhishonoursdegreeinpsychologyattheUniversityofSydneyin1976.AftercompletingaPhDatMacquarieUniversityin1984,underthesupervisionofJacquelineGoodnow,hemovedtotheUniversityofIllinoistotakeupapostdoctoralpositionwithAnnBrown.In1986hewasawardedaNationalAcademyofEducationfellowshiptostudytheoriginsofchildren’smathematicaldifficulties.Hewasalsooffered,andaccepted,afacultypositionattheUniversityofIllinoisinthesameyear.HemovedtotheUniversityofMelbourneintheearly1990swhereheiscurrentlyanassociateprofessorintheDepartmentofPsychologicalSciences,intheFacultyofMedicine,DentistryandHealthSciences.HerunstheDevelopmentalMathCognitiongroupinPsychologySciences,membersofwhichstudythenatureandoriginsofchildren’smathematicallearningdifficulties.IncollaborationwithBrianButterworthofUniversityCollegeLondon,hehasbeenawardedresearchgrantstoinvestigate(1)indigenousmathematics,and(2)thenatureofdevelopmentaldyscalculia.Since2003,hehasbeenworkingwithIndigenousgroupsintheNorthernTerritory,studyingethnomathematics.Hehasalsorecentlycompletedasix-yearlongitudinalstudydesignedtoidentifyearlymarkersofdyscalculiainyoungchildren.Heservesontheeditorialboardofseveralinternationalchilddevelopmentjournals.

AbstractHerewedescribethenatureanduseofspatialstrategiesinastandardnon-verbaladditiontaskintwogroupsofchildren,comparingchildrenwhospeakonlylanguagesinwhichcountingwordsarenotavailablewithchildrenwhowereraisedspeakingEnglish.WetestedspeakersofWarlpiriandAnindilyakwaagedbetween4and7yearsoldattworemotesitesintheNorthernTerritoryofAustralia.Thesechildrenusedspatialstrategiesextensively,andweresignificantlymoreaccuratewhentheydidso.English-speakingchildrenusedspatialstrategiesveryinfrequently,butreliedanenumerationstrategysupportedbycountingwordstodotheadditiontask.ThemainspatialstrategyexploitedtheknownvisualmemorystrengthsofIndigenousAustralians,andinvolvedmatchingthespatialpatternoftheaugendsetandtheaddend.Thesefindingssuggestthatcountingwords,farfrombeingnecessaryforexactarithmetic,offeronestrategyamongothers.Theyalsosuggestthatspatialmodelsfornumberdonotneedtobeone-dimensionalvectors,asinamentalnumberline,butcanbeatleasttwo-dimensional.

Introduction

IndigenousAmazonians,whoselanguageslackourkindof‘count-list’,appearunabletoaccuratelycarryouttasksthatrequire‘thecapacitytorepresentintegers’(Gordon,2004;Pica,Lemer,Izard,&Dehaene,2004).TheAmazonianresearchers,therefore,claimthat‘Languagewouldplayanessentialroleinlinkingupthevariousnonverbalrepresentationstocreateaconceptoflargeexactnumber’(Picaetal.,p.499)andconclude‘Ourresultsthussupportthehypothesisthatlanguageplaysa

specialroleintheemergenceofexactarithmeticduringchilddevelopment’(Picaetal.,p.503).ThisisaWhorfianposition:conceptsofexactnumberareimpossiblewithoutcountingwords.Thatis,onecannotpossesstheconceptofexactlyfiveness,withouthavingawordcorrespondingtofive.

Thisviewisnotuniversal.GelmanandGallistel(1978)arguethatthechild’sdevelopmentofverbalcountingisaprocessofmappingastablyorderedsequenceofcountingwords(CW)ontoanorderedsequenceofmentalmarksfornumerositiestheycall‘numerons’.Thissystemissharedwithnon-verbalspeciessuchascrowsandrats,andisimplementedinan‘accumulator’systemthataccumulatesafixedamountofneuralenergyoractivityforeachitemenumerated.Eachnumeroncorrespondstoaleveloftheaccumulator.

Onecanthinkofthementalnumberline(MNL)asbeingascalethatiscalibratedagainsttheaccumulator.Similarly,onecanthinkofthecountlistasbeinglinedupagainstpointsorregionsontheMNL.Spatialmetaphorsofabstractconceptsandrelationsareextremelywidespreadinhumancognition:emotionsaredescribedashighorlow,personalrelationshipscanbecloseordistant,mostpeoplegoforwardintothefuture,backwardintothepast,etc.Itisnotthereforesurprisingthatcardinalnumbers,whichareabstractpropertiesofsets,shouldattractspatialmodels.Theunconsciousspatialrepresentationofnumbers,revealedinnumberbisectiontasks,isusuallythoughtofasone-dimensionalvectors–alinewithasingledirection.However,whereindividualshaveautomaticandconsciousrepresentationsofnumber–Galton’s

Usingmentalrepresentationsofspacewhenwordsareunavailable:StudiesofenumerationandarithmeticinIndigenousAustralia


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‘numberforms’(Galton,1880)–theseareindeedlines,butmorecomplex,intwooreventhreedimensions(Seron,Pesenti,Noël,Deloche,&Cornet,1992;Tang,Ward,&Butterworth,2008).

Hereweaskthequestion:whatwillindividualsdowhentheydonothavecountingwordsintasksthatrequireexactcalculation?TheWhorfianpositionwouldentailthatexactcalculationisimpossible.Ontheotherhand,thepositionespousedbyLocke(Locke,1690/1961)andWhitehead(Whitehead,1948),andsubsequentlybyGelmanandButterworth(2005),isthat‘Distinctnamesconducetoourwellreckoning’because,asWhiteheadnotes,‘Byrelievingthebrainofallunnecessarywork,agoodnotationsetsitfreetoconcentrateonmoreadvancedproblems,andineffectincreasesthementalpoweroftherace’(Whitehead,1948).

AreCWstheonly‘goodnotation’?HereweexaminetheabilityofIndigenousAustralianchildrenof4to7yearstocarryoutsimplenon-verbaladditionproblems.ThesechildrenlivedinremotesitesintheNorthernTerritory,andweremonolingualinoneoftwoAustralianlanguages,WarlpiriorAnindilyakwa.Theselanguageshaveverylimitednumbervocabularies.Althoughtheselanguagescontainquantifierssuchasfew,many,a lot,several,etc.,thesearenotrelevantnumberwords,sincetheydorefertoexactnumbers,andthetheoreticalclaimisaboutexactnumbers.OurcomparisongroupwasaschoolinMelbourne.

WehavealreadyshownthatthesechildrenperformaccuratelyasEnglish-speakingchildrenontasksthatrequiredrememberingthenumberofobjectsinanarrayandonmatchingthenumberofsoundswithanumberofobjects(Butterworth&Reeve,2008;Butterworth,Reeve,Reynolds,&Lloyd,

2008).Herewefocusonanon-verbalexactadditiontask.Additionistypicallyacquiredinstagesusingcountingprocedures.Wheretwonumbersortwodisjointsets,say3and5,aretobeaddedtogether,intheearlieststagethelearnercountsallmembersoftheunionofthetwosets–thatis,willcount1,2,3,andcontinue4,5,6,7,8,keepingthenumberofthesecondsetinmind.Inalaterstage,thelearnerwill‘count-on’fromthenumberofthefirstset,startingwith3andcountingjust4,5,6,7,8.Atastilllaterstage,thechildwillcountonfromthelargerofthetwonumbers,nowstartingat5,andcountingjust6,7,8.(Butterworth,2005).Itisprobablyatthisstagethatadditionfactsarelaiddowninlong-termmemory(Butterworth,Girelli,Zorzi,&Jonckheere,2001).Ifthelearnerdoesnothaveaccesstothesestrategies,becausehisorherlanguagelackstheCW,whatwilltheydo?(Note:Manylearnersduringthesestagesusetheirfingers–ahandyset–tohelpthemcount,especiallywhentheadditioninvolvesnumbersratherthansetsofobjects.Thatis,theywillrepresentthe3byraisingthreefingers,andthencountonusingthefivefingersoftheotherhand.Now,despitethefactthatmanycultureswithnospecialisednumberwordsusebody-partsandbody-partnamestocount,thisisnotwhathappensinAustralia.Althoughgesturalcommunicationsareverywidespreadthere(Kendon,1988),thereisnorecordofbody-partcountingorofshowingnumbersusingbody-parts.ThisseemstobeaconventionalformofcommunicationthatislackinginAustralia.Indeed,noneofourNorthernTerritorychildrenusedtheirfingerstohelpthemwiththesetasks.

Method

Wetested32childrenaged4to7years:13Warlpiri-speakingchildren,10Anindilyakwa-speakingchildren,

and9English-speakingchildrenfromMelbourne.ApproximatelyhalftheNorthernTerritorychildrenwere4to5yearsoldandhalfwere6to7yearsold.

InWillowraandAngurugu,bilingualIndigenousassistantsweretrainedbyaninterviewertoadministerthetasks,andallinstructionsweregivenbyanativespeakerofWarlpiriorAnindilyakwa.Toacquainthelperswithresearchpracticesandtofamiliarisechildrenwithtestmaterials(e.g.,counters),familiarisationsessionswereconducted.Childrenplayedmatchingandsharinggamesusingtestmaterials(countersandmats).Forthematchinggames,theinterviewerputseveralcountersonhermat,andchildrenwereaskedtomaketheirmatthesame.Childrenhadlittledifficultycopyingthenumberandlocationofcountersontheinterviewer’smat.

Inthebasicmemorytask,identical24-cm×35-cmmatsandbowlscontaining25counterswereplacedinfrontofachildandtheinterviewer.Theinterviewersatbesidethechild,asrecommendedinKearins(1981),ratherthanoppositeasistypicalintestingEuropeanchildren.Theinterviewertookcountersfromherbowlandplacedthemonhermat,oneatatime,inpre-assignedlocations.Foursecondsafterthelastitemwasplacedonthemat,allitemswerecoveredwithaclothandchildrenwereaskedbytheIndigenousassistantto‘makeyourmatlikehers’.FollowingthreepracticetrialsinwhichtheinterviewerandanIndigenousassistantmodelledrecallusingoneandtwocounters,childrencompleted14memorytrialscomprisingtwo,three,four,five,six,eight,orninerandomlyplacedcounters.Inmodellingrecall,counterswereplacedonthematwithoutreferencetotheirinitiallocation.Numberandlocationsofchildren’scounterrecallwererecorded.InearlieranalyseswefoundthatIndigenouschildrentendedto


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usespatialstrategiestoreconstructthenumerositiesofrandommemoryarrays(Butterworth&Reeve,2008).Ofinterestiswhethertheywouldusesimilarstrategiesinthenon-verbaladditiontask.

Thesamematerials(matsandcounters)wereusedinthenon-verbaladditiontask.Theinterviewerplacedonecounteronhermatand,after4seconds,coveredhermat.Next,theinterviewerplacedanothercounterbesidehermatand,whilethechildwatched,slidtheadditionalcounterunderthecoverandontohermat.ChildrenwereaskedbytheIndigenousassistantto‘makeyourmatlikehers’.Ninetrialscomprising2+1,3+1,4+1,1+2,1+3,1+4,3+3,4+2,and5+3wereused.Children’sanswerswererecorded.Wewereparticularlyinterestedinthewaysinwhichcomputedanswerstothenon-verbaladditionproblemswereapproached,andinwhetherIndigenouschildrenwouldusespatialstrategiesincomputinganswers.

Results

Thepatternsoffindingsarereasonablyclear.ComparedtotheirMelbournepeers,theyoungerNorthernTerritorychildrensolvedmarginallymorenon-verbaladditionproblemscorrectly(means=2.3and3.2problemscorrectrespectively,F(1,20)=3.27,p<.09).Further,theolderNorthernTerritorychildrensolvedmoreproblemscorrectlythantheyoungerNorthernTerritorychildren(means=3.2and4.5problemsrespectively,F(1,23)=10.15,p<.01).

Strategies

Ofinterestaredifferencesinthestrategiesusedtosolvethenon-verbaladditionproblemsbythedifferentgroupsofchildren(MelbournevsNorthernTerritory,andyoungervsolderNorthernTerritorychildren)

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1p<.01,2p<.05,3n.s.

Figure�1:Proportionofstrategyuseforcorrectnonverbaladditionresponsesasafunctionofchildren’slocationandage

andwhetherthesedifferences,iftheyexist,affectproblem-solvingsuccess.Thestrategyusedtosolveeachproblemwasclassifiedaseitheranenumerationorapatternstrategy.Foraproblem-solvingattempttobeclassifiedanenumerationstrategy,thetokensusedtoconveyanswerswereplacedbythechildonhisorhermatinarandomorlineararrangement(oftenwithaudibleenumeration).Foraproblem-solvingattempttobeclassifiedapatternstrategy,achildappearedtoconcatenatethetwopatterns(theoriginaltokenpattern,andthepatternofaddedtokens).Thepatternstrategyreflectsanattemptedreproductionofthespatiallayoutoftheinitialandaddedarrays.Inthiscase,noaudibleenumerationaccompaniedtokenplacement.Thesetwostrategiesappeartoreflecttwomeaningfullydifferentcomputationprocesses.

Whenproblemsweresolvedcorrectly,MelbournechildrenusedenumerationstrategiesmoreoftenthantheiryoungNorthernTerritorypeers,whoused

patternstrategiesmoreoften,χ2(1,N=56)=18.08,p<.001.Similarly,whencorrect,olderNorthernTerritorychildrenusedanenumerationstrategymoreoftenthanyoungerNTchildren,χ2(1,N=57)=4.30,p<.05.Forincorrectlysolvedproblems,theresultswerereversedforMelbourneandyoungNorthernTerritorychildren:youngNorthernTerritorychildrentendedtoerrwhentheyusedanenumerationstrategy,χ2(1,N=62)=14.91,p<.001.

Figures1and2showstrategyuseforcorrectandincorrectanswersasafunctionofageandtestlocation.Figure1showsthatMelbournechildrenaremorelikelytoobtainthecorrectansweriftheyusedanenumerationstrategy(p<.01),andthatthiseffectisreversedfortheyoungerNorthernTerritorychildren(p<.05).However,olderNorthernTerritorychildren’scorrectnon-verbaladditionproblem-solvingabilitydoesnotseemtodependonstrategyuse.However,Figure2showsthatolderNorthernTerritory


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participants.ItmaywellbethatnamingthenumberofobjectsinthearraytoberememberedisthepreferredstrategyfortheEnglish-speakingchildren,butnotfortheNorthernTerritorychildren.

Kearins(1986)considerstwopossibleexplanationsforthis.OneisagenetichypothesisproposedbyLockard(1971).Accordingtothis,thereisselectionofabilitiesaccordingtoniche,especiallywhereapopulationisrelativelyisolated.Desertdwellers,ofthesortthatKearinstested,arehunter-gathererswhoare‘possessorofunusualknowledgeandskillsinthenaturalworld.TheycanliveoffthelandwherealmostnoWesternerscandoso,findingwaterandfoodinapparentlyaridcountry.’PeoplebegantooccupyAustraliaatleast40000yearsago(Flood,1997)andhavebeenrelativelyisolatedfromotherpopulationsduringthattime.Thus,survivalinthishostileenvironmentmayhavefavouredthosewhocouldacquirethesespecialskills.Theabilitytoretainspatialandtopographicalinformationcouldmakethedifferencebetweenlifeanddeathinthedesert.Bycontrast,theinventionofagriculture10000yearsagoputanemphasisondifferentkindsofskills,andalsoresistancetoanimal-originateddiseasesthatarepandemicinEuropeandAsia,suchassmallpox,measlesetc.(Diamond,1997).ItisstrikingthereforethatinKearins’sstudy,bothsemi-traditionalparticipantswholivedinthedesertandnon-traditionalparticipantswholivedonthedesertfringeperformedequivalently,andbetteronalltasksthannon-indigenousparticipantsfromaforestryandfarmingarea.Theseresultsappeartosupportthegenetichypothesissinceitisnotwhereyoulivebutyourancestrythatiscritical.

However,Kearins(1986)raisesanotherpossibility:differencesinchild-rearingpractices.IndigenousAustralians,likeotherhunter-gatherers,

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Figure�2:Proportionofstrategyuseforincorrectnonverbaladditionresponsesasafunctionofchildren’slocationandage

childrenaremorelikelytoerriftheyusedanenumerationstrategy(p<.05).

Discussion

ItisclearthatEnglish-speakingchildreninMelbournealmostneverusethepatternstrategy,butperformthetaskusinganenumerationstrategy.Bycontrast,NorthernTerritorychildrenmatchedinagewiththeEnglish-speakers,usepatternstrategiesnearlytwiceoftenasenumeration.Whatisofparticularinterestisthefactthatthepatternstrategyismoreeffectiveforthem,andthatattemptingtoenumerateleadstoapreponderanceoferrors.Indeed,evenfortheEnglish-speakers,theonlyfourdocumentedusesofpatternwereallcorrect.TheolderNorthernTerritorychildrenhavebeguntousethepatternstrategymoreoften,nowmakingupabouthalfofallstrategiesused.However,themajorityoftheircorrectresponses(30vs24)andtheminorityoftheirincorrect

responses(5vs13)usedthepatternstrategy.

Theseresultssuggestthatapattern-matchingstrategyisaneffectivespatialheuristicwhenCWstosupportenumerationarenotavailable.Noticethatthepatternsusedherearetwo-dimensional,suggestingthataone-dimensionalorientednumberlineisnottheonlywayforchildrentorepresentnumbers.OnemightaskwhypatternmatchingisthepreferredstrategyfortheNorthernTerritorychildren.OnepossiblereasonisthatIndigenousAustraliansareverygoodatrememberingspatialpatterns.InaversionofKim’sgame,whereonehastorecallthelocationofavarietyobjectsonatray,Kearins(1981)showedthatIndigenousadolescentsandchildrenweresuperiortotheirnon-Indigenouscounterparts.Moreover,Kearinsfoundthatthenameabilityoftheobjectsinthearraytoberemembered,affectednon-indigenousparticipantsbutnotIndigenous


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rarelytransmitinformationorskillsbyverbalinstruction(‘Allthatnagging’).Ratherchildrenareencouragedtolearnbyobservation.Thismaymeanthatchildrenacquireskillsofrememberingwhattheyseeearlierorbetterthannon-indigenouschildren.ThisissupportedbyseveralstudiesthatKearinscites.Thus,parentsandthegenerallearningenvironmentofIndigenousAustralianchildrenencouragethoseskillsparticularlyusefulforthedesertniche,ofwhichgoodspatialmemoryandroutinedependenceonitareapart.Ofcourse,geneticfactorsandchild-rearingpracticesmaynotbeunrelated.

Wedonotdoubtthatagoodnotationishelpfulforcarryingoutmentalwork,inthiscase,carryingoutsimpleaddition.However,ourresultssuggestthatcountingwordsarenottheonlygoodnotation,andthatastrategyformappingitemstobeenumeratedontoaspatialrepresentationcouldalsobeeffectivewhencountingwordsarenotavailable.Therelationshipbetweenanaccumulatormechanismandatwo-orthree-dimensionalmentalspatialarrayisstilltobeelucidated.

ReferencesButterworth,B.(2005).The

developmentofarithmeticalabilities.Journal of Child Psychology & Psychiatry, 46(1),3–18.

Butterworth,B.,Girelli,L.,Zorzi,M.,&Jonckheere,A.R.(2001).Organisationofadditionfactsinmemory.Quarterly Journal of Experimental Psychology, 54A,1005–1029.

Butterworth,B.,&Reeve,R.(2008).Verbalcountingandspatialstrategiesinnumericaltasks:EvidencefromIndigenousAustralia.Philosophical Psychology, 21,443–457.

Butterworth,B.,Reeve,R.,Reynolds,F.,&Lloyd,D.(2008).Numerical thought with and without words: Evidence

from Indigenous Australian children.ProceedingsoftheNationalAcademyofSciencesoftheUSA,105,13179–13184.

Diamond,J.(1997).Guns, germs and steel: The fates of human societies.London:JonathanCape.

Flood,J.(1997).Rock art of the Dreamtime: Images of ancient Australia.Sydney,Australia:HarperCollinsPublishers.

Galton,F.(1880).Visualisednumerals.Nature, 21,252–256.

Gelman,R.,&Butterworth,B.(2005).Numberandlanguage:Howaretheyrelated?Trends in Cognitive Sciences, 9(1),6–10.

Gelman,R.,&Gallistel,C.R.(1978).The child’s understanding of number.(1986ed.).Cambridge,MA:HarvardUniversityPress.

Gordon,P.(2004).Numericalcognitionwithoutwords:EvidencefromAmazonia.Science, 306,496–499.

Kearins,J.(1981).VisualspatialmemoryofAustralianAboriginalchildrenofdesertregions.Cognitive Psychology, 13,434–460.

Kearins,J.(1986).VisualspatialmemoryinAboriginalandWhiteAustralianChildren.Australian Journal of Psychology, 38(3),203–214.

Kendon,A.(1988).Sign languages of Aboriginal Australia: Cultural, semiotic and communicative perspectives.Cambridge:CambridgeUniversityPress.

Lockard,R.B.(1971).Reflectionsonthefallofcomparativepsychology–Isthereamessageforusall?American Psychologist, 26,168–179.

Locke,J.(1690/1961).An essay concerning human understanding(BasedonFifthEdition,J.W.Yolton(Ed.).London:J.M.Dent.

Pica,P.,Lemer,C.,Izard,V.,&Dehaene,S.(2004).ExactandapproximatecalculationinanAmazonianindigenegroupwithareducednumberlexicon.Science, 306,499–503.

Seron,X.,Pesenti,M.,Noël,M.-P.,Deloche,G.,&Cornet,J.-A.(1992).Imagesofnumbers,or‘When98isupperleftand6skyblue’.Cognition, 44,159–196.

Tang,J.,Ward,J.,&Butterworth,B.(2008).Numberformsinthebrain.Journal of Cognitive Neuroscience, 20(9),1547–1556.

Whitehead,A.N.(1948).An introduction to mathematics((Originallypublishedin1911)ed.).London:OxfordUniversityPress.


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Merrilyn�GoosThe University of Queensland

MerrilynGoosisDirectoroftheTeachingandEducationalDevelopmentInstituteatTheUniversityofQueensland.From1998–2007ProfessorGoosco-ordinatedpre-serviceandpostgraduatecoursesinmathematicseducationatUQ.Herresearchinmathematicseducationhasinvestigatedsecondaryschoolstudents’learning,teachingapproachesthatpromotehigherorderthinking,mathematicsteachers’learninganddevelopment,andtheprofessionallearningofmathematicsteachereducators.ThisworkhasbeensupportedbytwoARCLargeGrantsandtwoARCDiscoveryGrants.ProfessorGooshasalsoledlarge-scale,cross-institutionalresearchprojectscommissionedbytheAustralianandQueenslandGovernmentsinnumeracyeducationandschoolreform.In2004shewonanAustralianAwardforUniversityTeaching,followedin2006byanAssociateFellowshipoftheCarrickInstituteforLearningandTeachinginHigherEducation(nowtheAustralianLearningandTeachingCouncil).ProfessorGoosiscurrentlyPresidentoftheMathematicsEducationResearchGroupofAustralasia.

AbstractWhatcountswhenitcomestousingdigitaltechnologiesinschoolmathematics?Istechnologytheretohelpstudentsget‘theanswer’morequicklyandaccurately,ortoimprovethewaytheylearnmathematics?Thewaypeopleanswerthisquestionisilluminatingandcanrevealdeeplyheldbeliefsaboutthenatureofmathematicsandhowitisbesttaughtandlearned.Thispresentationconsiderstheextenttowhichtechnology-relatedresearch,policyandpracticemightusefullyinformeachotherinsupportingeffectivemathematicsteachingandlearninginAustralianschools.Thefirstpartofthepresentationconsiderskeymessagesfromresearchonlearningandteachingmathematicswithdigitaltechnologies.Thesecondpartofferssomesnapshotsofpracticetoillustratewhateffectiveclassroompracticecanlooklikewhentechnologiesareusedincreativewaystoenrichstudents’mathematicslearning.ThethirdpartanalysesthetechnologymessagescontainedinthedraftAustralian curriculum – Mathematicsandthechallengesofaligningcurriculumpolicywithresearchandpractice.

Introduction

Digitaltechnologieshavebeenavailableinschoolmathematicsclassroomssincetheintroductionofsimplefour-functioncalculatorsinthe1970s.Sincethen,computersequippedwithincreasinglysophisticatedsoftware,graphicscalculatorsthathavemorphedinto‘all-purpose’hand-helddevicesintegratinggraphical,symbolicmanipulation,statisticalanddynamicgeometrypackages,andweb-basedapplicationsofferingvirtuallearningenvironmentshavechangedthemathematicsteachingandlearningterrain.Orhavethey?Thispresentationconsiderstheextenttowhichtechnology-relatedresearch,

policyandpracticemightusefullyinformeachotherinsupportingeffectivemathematicsteachingandlearninginAustralianschools.

Thefirstpartofthepresentationconsiderskeymessagesfromresearchonlearningandteachingmathematicswithdigitaltechnologies.Thesecondpartofferssomesnapshotsofpracticetoillustratewhateffectiveclassroompracticecanlooklikewhentechnologiesareusedincreativewaystoenrichstudents’mathematicslearning.ThethirdpartanalysesthetechnologymessagescontainedinthedraftAustralian curriculum – Mathematicsandthechallengesofaligningcurriculumpolicywithresearchandpractice.

Key messages from research on learning and teaching mathematics with digital technologies

Fearsaresometimesexpressedthattheuseoftechnology,especiallyhand-heldcalculators,willhaveanegativeeffectonstudents’mathematicsachievement.However,meta-analysesofpublishedresearchstudieshaveconsistentlyfoundthatcalculatoruse,comparedwithnon-calculatoruse,haseitherpositiveorneutraleffectsonstudents’operational,computational,conceptualandproblem-solvingskills(Ellington,2003;Hembree&Dessart,1986;Penglase&Arnold,1996).Adifficultywiththesemeta-analyses,however,isthattheyselectstudiesthatcomparetreatment(calculator)andcontrol(non-calculator)groupsofstudents,withtheassumptionthatthetwogroupsexperienceotherwiseidenticallearningconditions.Experimentaldesignssuchasthisdonottakeintoaccountthepossibilitythattechnologyfundamentallychangesstudents’mathematicalpracticesandeventhenatureofthemathematicalknowledgetheylearnatschool.

Usingtechnologytosupporteffectivemathematicsteachingandlearning:Whatcounts?


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Technology and mathematical knowledge

Intheircontributiontothe17th ICMI Study on Mathematics Education and Technology,OliveandMakar(2010)analysedtheinfluenceoftechnologyonthenatureofmathematicalknowledgeasexperiencedbyschoolstudents.Theyarguedasfollows:

Ifoneconsidersmathematicstobeafixedbodyofknowledgetobelearned,thentheroleoftechnologyinthisprocesswouldbeprimarilythatofanefficiencytool,i.e.helpingthelearnertodothemathematicsmoreefficiently.However,ifweconsiderthetechnologicaltoolsasprovidingaccesstonewunderstandingsofrelations,processes,andpurposes,thentheroleoftechnologyrelatestoaconceptualconstructionkit.(p.138)

Theirwordsencapsulatethecontrastingpurposesoftechnologythatwereforeshadowedintheopeningparagraphofthispaper.Forlearners,mathematicalknowledgeisnotfixedbutfluid,constantlybeingcreatedasthelearnersinteractwithideas,peopleandtheirenvironment.Whentechnologyispartofthisenvironment,itbecomesmorethanasubstituteformathematicalworkdonewithpencilandpaper.Consider,forexample,thewayinwhichdynamicgeometrysoftwareallowsstudentstotransformageometricobjectby‘dragging’anyofitsconstituentpartstoinvestigateitsinvariantproperties.Throughthisexperimentalapproach,studentsmakepredictionsandtestconjecturesintheprocessofgeneratingmathematicalknowledgethatisnewforthem.

Technology and Mathematical Practices

Learningmathematicsisasmuchaboutdoingasitisaboutknowing.How

knowinganddoingcometogetherisevidentinthemathematicalpracticesoftheclassroom.Forexample,schoolmathematicalpracticesthat,inthepast,wererestrictedtomemorisingandreproducinglearnedprocedurescanbecontrastedwithmathematicalpracticesendorsedbymostmoderncurriculumdocuments,suchasconjecturing,justifyingandgeneralising.Technologycanchangethenatureofschoolmathematicsbyengagingstudentsinmoreactivemathematicalpracticessuchasexperimenting,investigatingandproblemsolvingthatbringdepthtotheirlearningandencouragethemtoaskquestionsratherthanonlylookingforanswers(Farrell,1996;Makar&Confrey,2006).

OliveandMakar(2010)arguethatmathematicalknowledgeandmathematicalpracticesareinextricablylinked,andthatthisconnectioncanbestrengthenedbytheuseoftechnologies.TheydevelopedanadaptationofSteinbring’s(2005)‘didactictriangle’thatinitsoriginalformrepresentsthelearningecologyasinteractionsbetweenstudent,teacherandmathematicalknowledge.Introducingtechnologyintothissystemtransformsthelearningecologysothatthetrianglebecomesatetrahedron,withthefourverticesofstudent,teacher,taskandtechnologycreating‘aspacewithinwhichnewmathematicalknowledgeandpracticesmayemerge’(p.168).

Withinthisspace,studentsandteachersmayimaginetheirrelationshipwithtechnologiesindifferentways.Goos,Galbraith,RenshawandGeiger(2003)developedfourmetaphorstodescribehowtechnologiescantransformteachingandlearningroles.Technologycanbeamasterifstudents’andteachers’knowledgeandcompetencearelimitedtoanarrowrangeofoperations.Studentsmaybecomedependentonthetechnologyiftheyareunabletoevaluatetheaccuracyoftheoutputitgenerates.Technologyisa

servantifusedbystudentsorteachersonlyasafast,reliablereplacementforpenandpapercalculationswithoutchangingthenatureofclassroomactivities.Technologyisapartnerwhenitprovidesaccesstonewkindsoftasksornewwaysofapproachingexistingtaskstodevelopunderstanding,exploredifferentperspectives,ormediatemathematicaldiscussion.Technologybecomesanextension of self whenseamlesslyintegratedintothepracticesofthemathematicsclassroom.

PierceandStacey(2010)offeranalternativerepresentationofthewaysinwhichtechnologycantransformmathematicalpractices.Theirpedagogical mapclassifiestentypesofpedagogicalopportunitiesaffordedbyawiderangeofmathematicalanalysissoftware.Opportunitiesariseatthreelevelsthatrepresenttheteacher’sthinkingabout:

• thetaskstheywillsettheirstudents(usingtechnologytoimprovespeed,accuracy,accesstoavarietyofmathematicalrepresentations)

• classroom interactions(usingtechnologytoimprovethedisplayofmathematicalsolutionprocessesandsupportstudents’collaborativework)

• thesubject(usingtechnologytosupportnewgoalsorteachingmethodsforamathematicscourse).

Snapshots of classroom mathematical practice

Twosnapshotsarepresentedheretoillustratehowtechnologycanbeusedcreativelytosupportnewmathematicalpractices.

Changing�tasks�and�classroom�interactions

Geiger(2009)usedthemaster-servant-partner-extension-of-selfframeworktoanalyseaclassroomepisodeinwhichheaskedhisYear11studentstousethedynamicgeometryfacilityontheir


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syntaxwascorrect,butsaidtheyshouldthinkharderabouttheirassumptions.

Eventually,theteacherdirectedtheproblemtothewholeclassandonestudentspottedtheproblem:‘Youcan’thaveanexponentialequaltozero’.Thisresultedinawholeclassdiscussionoftheassumptionthatextinctionmeantapopulationofzero,whichtheydecidedwasinappropriate.Theclassthenagreedonthepositionthatextinctionwas‘anynumberlessthanone’.StudentsusedCAStosolvethisnewequationandobtainasolution.

Inthisepisodetheteacherexploitedthe‘confrontation’createdbytheCASoutputtopromoteproductiveinteractionamongtheclass(technologyaspartner).Usingthispedagogicalopportunityallowedtheteachertorefocuscourse goals and teaching methodsonpromotingthinkingaboutthemathematicalmodellingprocessratherthanonpracticeofskills.

Aligning curriculum with research and practice?

Thebriefresearchsummaryandclassroomsnapshotspresentedaboveshowhowdigitaltechnologiesprovidea‘conceptualconstructionkit’(Olive&Makar,2010,p.138)thatcantransformstudents’mathematicalknowledgeandpractices.TowhatextentdoestheAustralian curriculum – Mathematicssupportthistransformativeviewoftechnology?

TheshapepaperthatprovidedtheinitialoutlineoftheK–12mathematicscurriculum(NationalCurriculumBoard,2009)madeitclearthattechnologiesshouldbeembeddedinthecurriculum‘sothattheyarenotseenasoptionaltools’(p.12).Digitaltechnologieswereseenasofferingnewwaystolearnandteachmathematicsthathelpeddeepenstudents’mathematicalunderstanding.Itwasalsoacknowledgedthatstudentsshouldlearntochooseintelligently

Table�1:Drawaline√—45unitslong

Classroom�interaction Role�of�technology

Studentsfindthesquarerootsofvariousnumbers. Servant

Studentspasscalculatorsbackandforthtoshareandcritiqueeachother’sthinking.

Partner

Teacherinvitesstudenttopresentcalculatorworktowholeclass.Audienceidentifiesmisconceptionsabouthowcalculatorsdisplaydecimalversionsofirrationalnumbers.

Master(priorgroupwork)thenpartner(wholeclassdisplayanddiscussion)

Teacherhint:thinkabouttriangles.StudentssearchforPythagoreanformulationwithoutgeometricrepresentation.

Servant

Teacherredirectsstudentstoconsidergeometry,notjustnumbers.Studentinterruptsgroupdiscussiontoproposegeometricsolution;passeshiscalculatoraroundgrouptoshareanddefendhissolution.

Partner

CAScalculatorstodrawaline√—45

unitslong.Hisaimwastoencouragestudentstothinkaboutthegeometricrepresentationofirrationalnumbers.TheanticipatedsolutioninvolvedusingthePythagoreanrelationship62+32=(√—

45)2toconstructaright-angledtrianglewithsides6and3unitslongandhypotenuse√—

45unitslong.Figure1summarisestheflowoftheepisodeandhowtechnologywasused.

Inthisepisode,technologywasinitiallyusedasaservanttoperformnumericalcalculationsthatdidnotleadtothedesiredgeometricsolution.Itbecameapartnerwhenstudentspassedtheircalculatorsaroundthegroupordisplayedtheirworktothewholeclasstoofferideasforcommentandcritique.Asapartneritgavethestudentwhofoundthesolutiontheconfidenceheneededtointroducehisconjecturedsolutionintoaheatedsmallgroupdebate.IntermsofPierceandStacey’s(2010)pedagogicalmap,thisepisodeillustratesopportunitiesprovidedbyataskthatlinknumericalandgeometricrepresentationstosupportclassroom interactionswherestudentsshareanddiscusstheirthinking.

Changing course goals and teaching methods

Geiger,FaragherandGoos(inpress)investigatedhowCAStechnologiessupportstudents’learningandsocialinteractionswhentheyareengagedinmathematicalmodellingtasks.Inthissnapshot,Year12studentsworkedonthefollowingquestion:

Whenwillapopulationof50,000bacteriabecomeextinctifthedecayrateis4%perday?

Onepairofstudentsdevelopedaninitialexponentialmodelforthepopulationyatanytimex,y=50000x(0.96)x.Theythenequatedthemodeltozeroinordertorepresentthepointatwhichthebacteriawouldbeextinct,withtheintentionofusingCAStosolvethisequation.WhentheyenteredtheequationintotheirCAScalculator,however,itunexpectedlyrespondedwithafalsemessage.Thestudentsthoughtthisresponsewasaresultofamistakewiththesyntaxoftheircommand.Whentheyaskedtheirteacherforhelp,heconfirmedtheir


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betweentechnology,mental,andpencilandpapermethods.

Thedraftconsultationversion1.0oftheK–10mathematicscurriculumexpected‘thatmathematicsclassroomswillmakeuseofallavailableICTinteachingandlearningsituations’.TheintentionisthatuseofICTistobereferredtoincontentdescriptionsandachievementstandards.Yetthisisdonesuperficiallyandinconsistentlythroughoutthecurriculum,withtechnologyoftenbeingtreatedasanadd-onthatreplicatesby-handmethods.Thisisseen,forexample,inthefollowingcontentdescriptionfromtheYear8NumberandAlgebrastrand:‘Plotgraphsoflinearfunctionsandusethesetofindsolutionsofequationsincluding use of ICT’(emphasisadded).

Inthecorrespondingconsultationversionsofthefourseniorsecondarymathematicscourses,theaimsforallcoursesrefertostudentschoosingandusingarangeoftechnologies.Nevertheless,eachcoursecontainsacommontechnologystatement–‘Technologycanaidindevelopingskillsandallaythetediumofrepeatedcalculations’–thatbetraysalimitedviewofitsrole.Acrossthecourses,variablemessagesabouttheuseoftechnologyareconveyedinwordslike‘assumed’and‘vital’inEssentialandGeneralMathematicsto‘shouldbewidelyusedinthistopic’,‘canbeusedtoillustratepracticallyeveryaspectofthistopic’,ornomentionatallforsometopicsinMathematicalMethodsandSpecialistMathematics.

InboththeK–10andseniorsecondarymathematicscurricula,usesoftechnology,wheremadeexplicit,aremostlyconsistentwiththeservantmetaphorofGoosetal.(2003),despitethemoretransformativeintentionsevidentintheinitialshapingpaper.PedagogicalopportunitiesaffordedbythecurriculumarerestrictedtotheleveloftasksinPierceandStacey’s

(2010)taxonomy,inthattechnologymaybeusedtomakecomputationandgraphingquickerandmoreaccurateandpossiblytolinkrepresentations.

AlthoughthetechnologymessagescontainedintheAustralian curriculum – Mathematicsdonotdojusticetowhatresearchtellsusabouteffectiveteachingandlearningofmathematics,itisalmostinevitablethattherearegapsbetweenanintendedcurriculumandthecurriculumenactedbyteachersandstudentsintheclassroom.Manyteachersarealreadyusingtechnologyeffectivelytoenhancestudents’understandingandenjoymentofmathematics.Intheirhandsliesthetaskofenactingatrulyfutures-orientedcurriculumthatwillpreparestudentsforintelligent,adaptiveandcriticalcitizenshipinatechnology-richworld.

ReferencesEllington,A.(2003).Ameta-analysisof

theeffectsofcalculatorsonstudents’achievementandattitudelevelsinprecollegemathematicsclasses.Journal for Research in Mathematics Education, 34,433–463.

Farrell,A.M.(1996).Rolesandbehaviorsintechnology-integratedprecalculusclassrooms.Journal of Mathematical Behavior, 15,35–53.

Geiger,V.(2009).Learning mathematics with technology from a social perspective: A study of secondary students’ individual and collaborative practices in a technologically rich mathematics classroom.Unpublisheddoctoraldissertation,TheUniversityofQueensland,Brisbane,Australia.

Geiger,V.,Faragher,R.,&Goos,M.(inpress).CAS-enabledtechnologiesas‘agentsprovocateurs’inteachingandlearningmathematicalmodellinginsecondaryschoolclassrooms.Mathematics Education Research Journal.

Goos,M.,Galbraith,P.,Renshaw,P.,&Geiger,V.(2003)Perspectives

ontechnologymediatedlearninginsecondaryschoolmathematicsclassrooms. Journal of Mathematical Behavior, 22,73–89.

Hembree,R.,&Dessart,D.(1986).Effectsofhand-heldcalculatorsinpre-collegemathematicseducation:Ameta-analysis. Journal for Research in Mathematics Education, 17,83–99.

Makar,K.,&Confrey,J.(2006).Dynamicstatisticalsoftware:Howarelearnersusingittoconductdata-basedinvestigations?InC.Hoyles,J.Lagrange,L.H.Son,&N.Sinclair(Eds.),Proceedings of the 17th Study Conference of the International Commission on Mathematical Instruction.HanoiInstituteofTechnologyandDidiremUniversitéParis7.

NationalCurriculumBoard(2009).Shape of the Australian curriculum: Mathematics.RetrievedMay29,2010fromhttp://www.acara.edu.au/verve/_resources/Australian_Curriculum_-_Maths.pdf

Olive,J.,&Makar,K.,withV.Hoyos,L.K.Kor,O.Kosheleva,&R.Straesser(2010).Mathematicalknowledgeandpracticesresultingfromaccesstodigitaltechnologies.InC.Hoyles&J.Lagrange(Eds.),Mathematics education and technology – Rethinking the terrain. The 17th ICMI Study(pp.133–177).NewYork:Springer.

Penglase,M.,&Arnold,S.(1996).Thegraphicscalculatorinmathematicseducation:Acriticalreviewofrecentresearch.Mathematics Education Research Journal, 8,58–90.

Pierce,R.,&Stacey,K.(2010).Mappingpedagogicalopportunitiesprovidedbymathematicsanalysissoftware.International Journal of Computers for Mathematical Learning, 15(1),1–20.

Steinbring,H.(2005).The construction of new mathematical knowledge in classroom interaction: An epistemological perspective.NewYork:Springer.


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Shelley�DoleThe University of Queensland

ShelleyDoleisaseniorlecturerinmathematicseducationatTheUniversityofQueensland.Dr.DoleisDirectorofthePrimaryandMiddleYearsTeacherEducationProgramsandteachesinBachelorandMasterofEducationcourses.Dr.Doleisanexperiencedclassroomteacher,havingtaughtinprimaryandsecondaryschoolsinVictoria,NorthernTerritoryandQueensland.ShehasalsobeenatertiaryeducatorinuniversitiesinQueensland,TasmaniaandVictoria.Herresearchinterestsincludestudents’mathematicallearningdifficulties,misconceptionsandconceptualchange;assessmentinmathematics;middleyearsmathematicscurriculum;mentalcomputation;thedevelopmentofproportionalreasoningandmultiplicativethinkingwithinthestudyofrationalnumber,andmentalcomputationandnumeracy.Dr.Dole’sresearchinterestsfocusparticularlyonpromotingstudents’conceptualunderstandingofmathematicstoencouragesuccessandenjoymentofmathematicalinvestigationsinschool.

AbstractThefocusofthispaperisonproportionalreasoning,emphasisingitspervasivenessthroughoutthemathematicscurriculum,butalsohighlightingitselusiveness.Proportionalreasoningisrequiredforstudentstooperatesuccessfullyinmanyrationalnumbertopics(fractions,decimals,percentages),butalsoothertopics(scaledrawing,probability,trigonometry).Proportionalreasoningisalsorequiredinmanyotherschoolcurriculumtopics(forexample,drawingtimelinesinhistory;interpretingdensity,molarity,speedcalculationsinscience).Inthispaper,anoverviewofmathematicseducationresearchonproportionalreasoningwillbepresented,highlightingthecomplexnatureofthedevelopmentofproportionalreasoningandimplicationsforlearningandinstruction.Throughpresentationofresultsofacurrentresearchprojectonproportionalreasoninginthemiddleyears,teachingapproachesthathavecapturedandengagedstudents’interestinexploringproportion-relatedsituationswillbeshared.

Background

Proportionalreasoningisafundamentalcornerstoneofmathematicsknowledge(Lesh,Post,&Behr,1988).Proportionalreasoningistheabilitytounderstandsituationsofcomparison.Examplesofeverydaytasksthatrequireproportionalreasoningincludeestimatingthebetterbuy,interpretingscalesandmaps,determiningchancesassociatedwithgamblingandrisk-taking.Proportionalreasoninghasbeendescribedasoneofthemostcommonlyappliedmathematicsconceptsintherealworld(Lanius&Williams,2003).Underdevelopedproportionalreasoningpotentiallyimpactsreal-worldsituations,

sometimeswithlife-threateningordisastrousconsequences,forexample,incorrectdosesinmedicine(Preston,2004).Proportionalreasoningthereforeisamajoraspectofnumeracy,yetitisimplicitinschoolcurriculaandoftenlimitedtothestudyofrateandratioinmathematicsonly.

Thedevelopmentofproportionalreasoningisacomplexoperation,and

...[it]requiresfirmgraspofvariousrationalnumberconceptssuchasorderandequivalence,therelationshipbetweentheunitanditsparts,themeaningandinterpretationofratio,andissuesdealingwithdivision,especiallyasthisrelatestodividingsmallernumbersbylargerones.Aproportionalreasonerhasthementalflexibilitytoapproachproblemsfrommultipleperspectivesandatthesametimehasunderstandingsthatarestableenoughnottoberadicallyaffectedbylargeor‘awkward’numbers,orthecontextwithinwhichaproblemisposed.(Post,Behr&Lesh,1988,p.80)

Proportionalreasoningisintertwinedwithmanymathematicalconcepts.Forexample,EnglishandHalford(1995)statedthat:‘Fractionsarethebuildingblocksofproportion’(p.254).Similarly,Behretal.(1992)statedthat‘theconceptoffractionorderandequivalenceandproportionalityareonecomponentofthisverysignificantandglobalmathematicalconcept’(p.316).Also,Streefland(1985)suggestedthat‘Learningtoviewsomething‘inproportion’,or‘inproportionwith...’precedestheacquisitionoftheproperconceptofratio’(p.83).Developingstudents’understandingofratioandproportionisdifficultbecausetheconceptsofmultiplication,division,fractionsanddecimalsarethebuilding

Makingconnectionstothebigideasinmathematics:Promotingproportionalreasoning


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blocksofproportionalreasoning,andstudents’knowledgeofsuchtopicsisgenerallypoor(Lo&Watanabe,1997).

Thedevelopmentofproportionalreasoningisagradualprocess,underpinnedbyincreasinglysophisticatedmultiplicativethinkingandtheabilitytocomparetwoquantitiesinrelative(multiplicative),ratherthanabsolute(additive)terms(Lamon,2005).Theessenceofproportionalreasoningisonunderstandingthemultiplicativestructuresinherentinproportionsituations(Behr,Harel,Post&Lesh,1992).Children’sintuitivestrategiesforsolvingproportionproblemsaretypicallyadditive(Hart,1981).Theteacher’srole,therefore,istobuildonstudents’intuitiveadditivestrategiesandguidethemtowardsbuildingmultiplicativestructures.Strongmultiplicativestructuresdevelopasearlyasthesecondgradeforsomechildren,butarealsoseentotaketimetodeveloptoalevelofconceptualstability,oftenbeyondfifthgrade(Clark&Kamii,1996).Behretal.(1992)suggestedthatexploringchangewillhelpstudentsdevelopmultiplicativeunderstanding.Forexample,studentscanbeencouragedtodiscussthechangeto4whichwillresultin8.Fromanadditiveview,4canchangeto8byadding4.Fromamultiplicativeview,4canchangeto8bymultiplyingby2.Thedifferencebetweentheadditiveandmultiplicativeviewcanbeseenbylookingatothernumbers.Theadditiveruleholdsfor13changingto17,butnotthemultiplicativerule.AccordingtoBehretal.(1992),‘theabilitytorepresentchange(ordifference)inbothadditiveandmultiplicativetermsandtounderstandtheirbehaviourundertransformationisfundamentaltounderstandingfractionandratioequivalence’(p.316).MovingstudentstowardsformalratioandproportionprinciplesandproceduresistermedbyStreefland(1985)as‘anticipatingratio’,wheretheteachercapitaliseson

students’informalintuitiveproblemsolvingprocedures,guidingstudentsto‘formulaeandalgorithmisation’(p.84).SuchanapproachwastakeninateachingexperimentconductedbyLoandWatanabe(1997)whereaYear5childwasexposedtoproportionalreasoningtaskstopromoteintuitivemultiplicativereasoningskillsandhencedevelopproportionalreasoning.

Researchhasindicatedthatstudents’(andteachers’)understandingofproportionisgenerallypoor(e.g.,Behretal.,1992;Fisher,1988;Hart,1981).Streefland(1985)statedthat‘Ratioisintroducedtoolatetobeconnectedwithmathematicallyrelatedideassuchasequivalenceoffractions,scale,percentage’(p.78).EnglishandHalford(1995)suggestedthatproportionalreasoningistaughtinisolationandthusremainsunrelatedtoothertopics.Behretal.(1992)stated,‘Webelievethattheelementaryschoolcurriculumisdeficientbyfailingtoincludethebasicconceptsandprinciplesrelatingtomultiplicativestructuresnecessaryforlaterlearninginintermediategrades’(p.300).Behretal.alsoadded,‘Thereisagreatdealofagreementthatlearningrationalnumberconceptsremainsaseriousobstacleinthemathematicaldevelopmentofchildren...Incontrastthereisnoclearargumentabouthowtofacilitatelearningofrationalnumberconcepts’(p.300).

Astheproportionconceptisintertwinedwithmanymathematicalconcepts,thishasimplicationsforinstruction.Thedevelopmentofarichconceptofrationalnumber,andthusproportionalrelationships,takesalongtime(Streefland,1985).Theproportionalnatureofvariousrationalnumbertopicsmustbethefocusofinstructionasthesetopicsarerevisitedcontinuallythroughoutthecurriculum,inordertobuildandlinkstudents’proportionalunderstanding(Behretal.,1992).Buildingproportionalreasoningmustbethroughmultipleperspectives

(Postetal.,1988).Theliteratureprovidesvarioussuggestionsforactivitiesandstrategiesforpromotingtheproportionconcept.Theuseofratiotableshasbeensuggestedasonemeansforbuildingstudents’ratiounderstanding(English&Halford,1995;Middleton&VandenHeuvel-Panhuizen,1994;Robinson,1981;Streefland,1985).EnglishandHalford(1995)providedthefollowingexampleofaratiotable,whichassistsinthecomparisonofthenumberofsoupcubesperperson:

soupcubes 2468

people 481216

EnglishandHalfordstated,‘Atableofthisnatureprovidesaneffectivemeansoforganisingtheproblemdataandenableschildrentodetectmorereadilyalltherelationsdisplayed,bothwithinandbetweentheseries...itservesasapermanentrecordofproportionasanequivalencerelation’(p.254).

The MC SAM project

PromotingproportionalreasoninghasbeenthefocusofalargeresearchprojectundertakenbyTheUniversityofQueensland(2007–2010).Notonlydidthisprojecttargetproportionalreasoninginmathematicsbutinscienceaswell,asproportionalreasoningisfundamentaltomanytopicsinbothmathematicsandscience(Lamon,2005).TheMCSAMproject,anacronymforMakingConnections:ScienceandMathematics,broughttogethermiddleyears’mathematicsandscienceteachersaroundthisimportanttopic,providinganopportunityforteacherstoexploretheproportionalreasoninglinkagesbetweentopicsinbothmathematicsandscience,andtocreate,implementandevaluateinnovativeandengaginglearningexperiencestoassiststudentstopromoteandconnectessentialmathematicsandscienceknowledge.Theprojecthadtwomajoraims.First,


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itaimedtodevelopaninstrumenttoassessmiddleyearsstudents’proportionalreasoningknowledge.Second,itaimedtousethisdatatodevelopandtrialspecificlearningexperiencesinbothmathematicsandsciencethatmaysupportstudents’accesstoparticulartopicsinthosesubjectsandpromoteproportionalreasoningskills.

Thereisalargecorpusofexistingresearchthathasprovidedanalysisofstrategiesappliedbystudentstovariousproportionalreasoningtasks(e.g.,Misailidou&Williams,2003;Hart,1981),Suchresearchhashighlightedissuesassociatedwiththeimpactof‘awkward’numbers(thatis,commonfractionsanddecimalsasopposedtowholenumbers),thecommonapplicationofanincorrectadditivestrategy,andtheblindapplicationofrulesandformulaetoproportionproblems.Priorresearchhasalsoemphasisedthecomplexityofthedevelopmentofproportionalreasoningandtheneedforfurtherandcontinuedworkinthefieldtosupportstudents’developmentofproportionalreasoning.Infact,itisestimatedthatapproximatelyonly50percentadultscanreasonproportionately(Lamon,2005).Inourstudy,wewantedtotakeasnapshotofalargegroupofstudents’proportionalreasoningontasksthatrelatetomathematicsandsciencecurriculuminthemiddleyearsofschooling.Thiscomponentoftheprojectwasconcernedwiththedevelopmentofaninstrumentthatwouldprovidea‘broadbrush’measureofstudents’proportionalreasoningandtheirthinkingstrategies,andthatwouldhavesomedegreeofdiagnosticpower.Thischallengewasundertakenwithfullawarenessofboththepervasivenessandtheelusivenessofproportionalreasoningthroughoutthecurriculumandthatitsdevelopmentisdependentuponmanyotherknowledgefoundationsinmathematicsandscience.

DevelopingtheinstrumentwasguidedbyliteratureandespeciallytheAmericanAssociationfortheAdvancementofScience(AAAS)(2001)AtlasofScienceLiteracy.TheAtlasidentifiestwokeycomponentsofproportionalreasoning:RatiosandProportion(partsandwholes,descriptionsandcomparisonsandcomputation)andDescribingChange(relatedchanges,kindsofchange,andinvariance).TheAAASprovidedtheframeworkforthedevelopmentoftheproportionalreasoningassessmentinstrument.Thetestincludeditemsondirectproportion(wholenumberandfractionalratios),rateandinverseproportionitems,aswellasfractions,probability,speedanddensityitems.GuidedbythewordsofLamon(2005),whosuggestedthatstudentsmustbeprovidedwithmanydifferentcontexts,‘toanalysequantitativerelationshipsincontext,andtorepresentthoserelationshipsinsymbols,tables,andgraphs’(p.3),theitemsincludedcontextsofshopping,cooking,mixingcordial,paintingfences,graphingstories,savingmoney,schoolexcursionsanddualmeasurementscales.Foreachitemonthetest,studentswererequiredtoprovidetheanswerandexplainthethinkingtheyappliedtosolvetheproblem.

Approximately700studentsinthemiddleyearsofschooling(Years4–9)participatedinthisassessment.Initially,projectteachershadmixedfeelingsaboutthetest’scapacitytoassesstheirstudents’proportionalreasoning.Theninthgradeteachersstatedthattheythoughtthetestwouldbetooeasyfortheirstudents;thefourthgradeteachersstatedthatthetestwastoohard.Thehighestaveragescorehowever,fortheninth-gradersononeitemwasjust75percent,withthefourth-gradersaveraging15percentforthatitem.Onseveralotheritems,theeighthandninthgradersscoredlessthan50percent.Ononeparticularitem,the

ninthgradersaveragedjust21percentandthefourthgradersaveraged5percentforthesameitem.Theresultswereawake-upcalltoallteachersintheproject:thefourthandfifthgradeteachersrealisedthatthereweresomeverygoodproportionalreasonersintheirgrades,andtheeighthandninthgradeteachersrealisedthattheyweretakingforgrantedtheproportionalreasoningskillsoftheirstudents.Itemanalysisandstudents’resultsprovideddirectionfortargetedteaching.Collectively,resultsofthewholetestsuggestedthatamuchgreaterfocusonproportionalreasoningmustoccurinallclassesateveryopportunity.

Throughouttheproject,aseriesofintegratedmathematicsandsciencetaskshasbeendeveloped,sharedandadaptedbytheteachers.Oneofthesimplest,andonethathasbeentakenupmostwidelybyallfourthgradetoninthgradeteachers,isanexplorationintowhypenguinshuddle,incorporatingthesurfaceareatovolumeratio.Byusingthree2-cmcubicblocks,penguinscanbecreated.Focusingononepenguin,thesurfaceareaofthepenguincanbefoundbycountingthefacesofthecubes(14)andthevolumecanbecountedbycountingthenumberofcubes(3).Ahuddleisformedbyputting9penguinsintoacubicarrangement.Adatatableisconstructedandstudentscananalysetheresultstoconsiderhowthesurfaceareatovolumeratiochangesasthehuddlegetsbigger.

Oneofthecapstoneelementsoftheprojecthasbeenthedevelopmentofaunitofworkondensity.Althoughdensityistypicallyregardedasatopicwithinthemiddleyearssciencecurriculum,conceptualunderstandingofdensityrequiresunderstandingofmathematicstopicsincludingmassandvolume,aswellasnumbersenseandmentalcomputation.Italsorequiresdatagathering,dataanalysis,interpretationofdata,graphing,


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measuring,usingmeasuringinstruments,problemsolving,problemposing,conductingexperimentsandcontrollingvariables,whicharecomponentsofbothmathematicsandsciencecurricula.Theintegratedunitondensitywasdevelopedandtrialledinanumberofmiddleyearsmathematicsand/orscienceclassrooms.Itwasimplementedtovaryingdegreesinmostclassesbyprojectteachers,butwasspecificallyimplementedbytheprojectteaminafifthandseventhgradeclassroom.Atthebeginningoftheunit,thestudents’hadlimitedknowledgeofdensity,withdevelopingunderstandingofmassandvolume.Attheendoftheunit,studentscoulddescribehowanobjectmightsinkorfloatinwaterbysimultaneouslyconsideringbothitsvolumeandmass.Allstudentscouldverbalisetheconceptofdensityandshowedgreaterconceptualisationofunitsofmeasureforvolume.Resultsofthisstudyprovideevidenceofthecapacityoftargeted,integratedmathematicsandscienceunitsforthedevelopmentofconnectedmathematicsandscienceknowledgeandpromotionofproportionalreasoningskills.

Concluding comments

Thedevelopmentofproportionalreasoningisaslowprocessexacerbatedbyitsnebulousnatureandlackofspecificprominenceinschoolsyllabusdocuments.Ourprojectteachershaverevisitedtheirtraditionalworkprogramanditstwo-weekmathematicsunitonratioandproportion.Theyhaveputgreateremphasisonproportionalreasoningandmultiplicativethinkinginthestudyofscaledrawing,linearequations,trigonometry,percentages,numberstudy,mapping,ratioandratesituations.Scienceteachersintheprojectagreaterawarenessofthemathematicalfoundationsofproportionalreasoningandhowsciencetopicsandpresentationsofequations(e.g.,densityequationand

forceequation)maybebasedonassumptionsofstudents’proportionalreasoningthatarenotstable.Thesignificanceofthisprojecthasbeenthatitbroughttogethermathematicsandscienceteacherstoexplorethesynergiesbetweenmathematicsandsciencecurriculumthroughproportionalreasoning.

ReferencesAmericanAssociationforthe

AdvancementofScience(AAAS).(2001).Atlas of Science Literacy:Project 2061.AAAS.

Behr,M.,Harel,G.,Post,T.,&Lesh,R.(1992).Rationalnumber,ratioandproportion.InD.Grouws(Ed.),Handbook on research of teaching and learning(pp.296–333).NewYork:McMillan.

Clark,F.&Kamii,C.(1996).IdentificationofmultiplicativethinkinginchildreninGrades1–5.Journal for Research in Mathematics Education, 27(1),41–51.

English,L.,&Halford,G.(1995)Mathematics education: Models and processes.Mahwah,NJ:Erlbaum.

Fisher,L.(1988).Strategiesusedbysecondarymathematicsteacherstosolveproportionproblems. Journal for Research in Mathematics Education, 19(2),157–168.

Hart.K.(1981).(Ed.).Children’s understanding of mathematics 11–16.London:JohnMurray.

Lamon,S.(2005).Teaching fractions and ratios for understanding(2nded.).Mahwah:Erlbaum.

Lanius,C.S.,&Williams,S.E.(2003).Proportionality:Aunifyingthemeforthemiddlegrades.Mathematics Teaching in the Middle School, 8(8),392–396.

Lesh,R.,Post,T.,&Behr,M.(1988).Proportionalreasoning.InJ.Hiebert&

M.Behr(Eds.),Number concepts and operations in the middle grades(pp.93–118).Hillsdale,NJ:Erlbaum.

Lo,J-J.,&Watanabe,T.(1997).Developingratioandproportionschemes:Astoryofafifthgrader.Journal for Research in Mathematics Education, 28(2),216–236.

Middleton,J.,&VandenHeuvel-Panhuizen,M.(1995).Theratiotable.Mathematics Teaching in the Middle School, 1(4),282–288.

Misailidou,G.,&Williams,J.(2003).Diagnosticassessmentofchildren’sproportionalreasoning.Journal of Mathematical Behaviour, 22,335–368.

Post,T.,Behr,M.,&Lesh,R.(1988).Proportionalityandthedevelopmentofprealgebraunderstandings.InA.F.Coxford&A.P.Shulte(Eds.),The Ideas of Algebra, K–12(pp.78–90).Reston,VA:NCTM.

Preston,R.(2004).Drugerrors&patientsafety:Theneedforachangeinpractice.British Journal of Nursing, 13(2),72–78.

Robinson,F.(1981).Rate and ratio: Classroom tested curriculum materials for teachers at elementary level.TheOntarioInstituteforStudiesinEducation,Ontario:OISEPress.

Streefland,L.(1985).Searchingfortherootsofratio:Somethoughtsonthelongtermlearningprocess(towards...atheory).Educational Studies in Mathematics, 16,75–94.


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Sue�ThomsonAustralian Council for Educational Research

SueThomsonisaPrincipalResearchFellowattheAustralianCouncilforEducationalResearchintheNationalandInternationalSurveysresearchprogram.

DrThomsonistheNationalResearchCoordinatorforAustraliaintheTrendsinInternationalMathematicsandScienceStudy(TIMSS),whichmeasuresachievementinmathematicsandscienceforstudentsingrades4and8,theProgressinInternationalReadingLiteracyStudy(PIRLS),whichmeasuresreadingliteracyofgrade4students,andtheNationalProjectManagerforAustraliafortheOECDProgrammeforInternationalStudentAssessment(PISA),whichexaminesreading,mathematicalandscientificliteracyof15-year-oldstudents.

DrThomson’sresearchatACERhasinvolvedextensiveanalysisoflarge-scalenationalandinternationaldatasets–theLongitudinalSurveysofAustralianYouth(LSAY),aswellasTIMSSandPISA.

DrThomsonwasengagedasanexpertwriterontheNationalNumeracyReview,andhasconsultedwithDEEWR,FaHCSIAandtheVictorianandACTDepartmentsofEducationonavarietyofdataanalysisprojectsrelatedtoTIMSSandPISA.

BeforejoiningACER,DrThomsonlecturedatanumberofuniversitiesinStatisticsandResearchMethodologywhileshecompletedherPhDfocusingonstudents’attributionsandengagementinmathematicsoverthetransitionfromprimarytosecondaryschool.

DrThomsonhaspublishedavarietyofarticlesandresearchreportsbasedonherworkatACER(seealsounderSueFullarton),andhaspresentedfindingsatconferencesinternationallyandnationally.

AbstractTeachersandschoolleaderswillbefamiliarwithNAPLAN–asacensusofstudentsinYears3,5,7and9itinvolvesalleducators.However,aspartoftheNationalAssessmentProgram,Australiaalsoparticipatesintwointernationalassessments,PISAandTIMSS,whichare,bydesign,lightsampleassessmentsandinvolveonlyasmallproportionofschools.Thestudentsweareeducatingtodaywillcompeteinaglobalmarket,andwehavetobesurethattheeducationweareprovidingthemwithisonethatwillprovidethemwithastrongbase,bothinknowledgeandskillsandintheabilitytoapplythoseskillstoreal-worldproblems.Inadditiontotheassessments,PISAandTIMSScollectaricharrayofcontextualinformationfromstudents,teachersandschools–includingbackgroundfactors,andattitudesandbeliefsaboutlearningmathematics.Whatshouldbeparticularlyinterestingforeducatorsisnotjusthowwellstudentsperformontheinternationalassessments,buthowmuchtheotherinformationwegathercantellthemaboutwhatAustralianstudentscanandcan’tdo.

Introduction

In1999,theMinistersresponsibleforschooleducation,theMinisterialCouncilonEducation,Employment,TrainingandYouthAffairs,agreedtoanewsetofNational Goals for Schooling in the Twenty-first Century(MCEETYA,1999).TheaimofthesegoalswastoprovideAustralianstudentswithhigh-qualityschoolingtoprovidethemwiththenecessaryknowledge,understanding,skillsandvaluesforaproductiveandrewardinglife.MCEETYAalsosetintrainaprocesstoenablenationallycomparablereportingofprogressagainsttheseNational

Goals.TheMeasurement Framework for National Key Performance Measures(MCEETYA,2008)setsouttheNational Assessment ProgramasabasisforreportingongoingprogresstowardsthegoalsbydrawingonagreeddefinitionsofKeyPerformanceMeasures.TheFrameworkisdesignedtobealivingdocument,inthatitwillbeupdatedtoreportonthemostrecentgoalsasdefinedintheMelbourne Declaration on Educational Goals for Young Australians,allowingittorespondtonewgoalsandchallenges.

TheNationalAssessmentProgramencompassesalltestsendorsedbyMCEETYA,suchasthenationalliteracyandnumeracytests(NAPLAN),three-yearlysampleassessmentsinscienceliteracy,civicsandcitizenship,andICTliteracy,andAustralia’sparticipationintheinternationalassessmentsPISAandTIMSS.

TeachersandschoolleadersarefamiliarwithNAPLAN–asacensusofstudentsinYears3,5,7and9itinvolvesalleducators.However,manymaynotbeawareofPISAandTIMSS,astheyarelightsampleassessmentswhich,bydesign,involveonlyaproportionofschools.Inadditiontotheassessments,PISAandTIMSScollectaricharrayofcontextualinformationfromstudents,teachersandschools–includingbackgroundfactors,andattitudesandbeliefsaboutlearningmathematics.Whatshouldbeparticularlyinterestingforeducatorsisnotjusthowwellstudentsperformontheinternationalassessments,buthowmuchtheotherinformationwegathercantellthemaboutwhatAustralianstudentscanandcan’tdo.

Thepresentationwillbestructuredaroundthequestionsteachersoftenask:

• WhatarePISAandTIMSS?Whoparticipates?

Mathematicslearning:WhatTIMSSandPISAcantellusaboutwhatcountsforallAustralianstudents


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• WhydoweneedtheseassessmentsaswellasNAPLAN?

• Whatcanthesestudiestellmeaboutwhatourstudentslearncomparedtoothercountries?

• Whatcantheytellmeaboutourstudents’motivation,engagementandself-efficacy–andhowthiscomparestoothercountries?

• Whatcanthesestudiestellusaboutequity–bothwithinAustraliaandinternationally?AresomestudentsdisadvantagedinAustralia,andisthiscommoninternationally?

TIMSS and PISA – some details

TheTrendsinInternationalMathematicsandScienceStudy(TIMSS)isalong-runningstudyofachievementinmathematicsandscience,managedbytheInternationalAssociationfortheEvaluationofEducationalAchievement(IEA).TheassessmentsoccureveryfouryearsatYears4and8,andAustralia’sparticipationinTIMSS2011willbeourfifthsincethecombinedmathematicsandscienceassessmentevolvedfromseparateinternationalassessmentsin1985.UnderpinningTIMSSisaresearchmodelinwhichthecurriculum,broadlydefined,isusedasthemajororganisationalconceptinconsideringhoweducationalopportunitiesareprovidedtostudents,andthefactorsthatinfluencehowstudentsusetheseopportunities.TheTIMSScurriculummodelhasthreeaspects:theintendedcurriculum(whatsocietyexpectsstudentstolearnandhowthesystemshouldbeorganisedtofacilitatethis),theimplementedcurriculum(whatisactuallytaughtinclassrooms,whoteachesitandhowitistaught)andtheachievedcurriculum(whichiswhatthestudentshavelearned,andwhattheythinkaboutthesesubjects).

TheProgrammeforInternationalStudentAssessment(PISA)istheothermajorinternationalassessmentincludedintheNational Assessment Program,andAustraliabeenaparticipantsincethestudybeganin2000.PISAismanagedbytheOrganisationforEconomicCo-operationandDevelopment(OECD);ittestscompetenciesinreading,mathematicsandscientificliteracy,andoccurseverythreeyears.TheunderlyingPISAmodelaimstomeasurehowwell15-year-olds,approachingtheendoftheircompulsoryschooling,arepreparedformeetingthechallengestheywillfaceintheirlivesbeyondschool.Withitsgoalofmeasuringcompetencies,thePISAassessmentfocusesonyoungpeople’sabilitytoapplytheknowledgeandskillstheyhavelearnedthroughouttheirschoollivestoreal-lifeproblemsandsituations.

In2010/2011morethan60educationalsystems,fromcountriesasdiverseasGhana,SaudiArabia,England,Honduras,UnitedStatesofAmericaandGermanywillparticipateinTIMSS.Inthefollowingyear,67countrieswillparticipateinPISA,includingallOECDcountriesplusagrowingnumberofnon-OECDorpartnercountries,againfromlocationsasdiverseasShanghai,QatarandAzerbaijan.Thegrowingnumberofcountriesparticipatinginoneorbothstudiesreflectsthevaluethatgovernmentsplaceonobtaininginternationalcomparativedata.

NAPLAN, PISA and TIMSS

SowhydoweneedNAPLANandPISAandTIMSS?Theanswerslieinwhoareassessed,howtheassessmentsareconstructed,andtheadditionalinformationgainedfromtheinternationalassessments.

InNAPLANallstudentsaretested,andthedataprovideresultsatthestudentlevel.NAPLANisintendedtoprovidediagnosticinformationabout

astudent’sindividualprogressagainstnationalstandards.Incontrast,alightsample(about5%ofallAustralianstudentsateachyearoragelevel)ofstudentsistestedintheinternationalassessments.Thissampleisanationallyrepresentativerandomsample,stratifiedtoensureaccuratedataforeachstate,eachschoolsector(government,Catholicandindependent)andeachgeographiclocationband(metropolitan,regional,rural).Thesedataenableustoexamineoureducationalsystemagainstinternationalstandards.

Intermsofwhatisassessed,theNAPLANtestsareinformedbytheNationalStatementsofLearninginEnglishandMathematicsthatunderpinthecurrentstateandterritorylearningframeworks;incontrasttheTIMSSandPISAassessmentsaredevelopedagainstframeworksdevelopedataninternationallevel.TheTIMSSframeworkisdevelopedafterextensiveconsultationbetweenrepresentativesofallcountriesinvolvedandanexpertpanelofmathematicseducators,andrepresentsthosegoalsofmathematicseducationthatareregardedasimportantinasignificantnumberofcountries.MathematicsintheTIMSSassessmentisreadilyrecognisableasthemathematicsinmostcurricula–thecontentdomainsofnumber, algebra, measurement, geometry and data (data display, geometric shapes and measures and numberatYear4),andthecognitivedomainsknowing, using concepts, applying and reasoningarefamiliarterritorytoteachers.

ThePISAmathematicalliteracyframeworkrevolvesaroundwiderusesandapplicationsofmathematicsinpeople’slives,andhasthreemaindimensions:mathematicalcontent,mathematicalprocessesandthesituations or contextsinwhichmathematicsisused.MathematicalcontentisdefinedintermsofSteen’s(1990)deepmathematicalideas,adaptedasoverarching ideas.These


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overarchingideasarequantity, space and shape, change and relationships, and uncertainty.ThePISAframeworkalsoidentifiesanumberofcompetencies–labelledasthereproductioncluster(relativelyfamiliaritemsthatrequireessentiallythereproductionofknowledgealreadyacquired),theconnectionscluster(problemsthatextendordevelopfromfamiliarsettingstoaminordegree)andthereflectioncluster(buildsfurtherontheconnectionscluster–itemsrequiresomeinsightorcreativityinidentifyingsolutions).

Soallthreestudiesareembeddedindifferentmodels–NAPLANandTIMSSincurriculummodels,butonenationalandtheotherinternational,andPISAasayieldstudy,lookingatwhetherstudentshaveinfactlearnedwhatweexpectthemtohavelearnedoverthecumulativeyearsofeducation.

Theinternationalassessmentsalsoprovideuswithawealthofcontextualinformation–becausethefocusisnotjustonwhataparticularstudentisabletodo,andbecauseforsuchstudiesthecontextoflearningisconsideredasimportantasthelearningitself.BothTIMSSandPISAcollectbackgrounddataonstudents–theeducationalresourcestowhichtheyhaveaccess,theeducationalexperienceoftheirparents,andtheirattitudestowardsandbeliefsaboutschoolingandthemselvesaslearners,inparticularinrelationtomathematics.TIMSScollectsdatafrommathematicsteachersaswell,asTIMSSissampledonintactclasses,whereasPISAsamples15-year-oldstudentsrandomlyacrossclasseswithinaschool.

What can we learn from PISA and TIMSS?

IfyouhaveheardofPISAandTIMSSinAustralia,itismostlikelythatyouwillhaveheardwherewerank,orwhichcountriesscorehigherthanus,orhowourscorescomparetothose

inNewZealand(orKazakhstan1).Thereis,ofcourse,alotmorethatispublishedinournationalreports,andthispaperwillpresentsomeoftheseresults.Largely,thispaperwillreportresultintermsofproficiencylevelsforPISAandbenchmarksforTIMSS.InPISA,sixproficiencylevelshavebeendescribed,representingacontinuumofmathematicsachievement.MCEETYAhavesetproficiencylevel3astheminimumstandardforAustralianstudents.InTIMSS,therearefourbenchmarksrangingfromlowtohigh,alsorepresentingacontinuumofmathematicsachievement.WhilenobaselevelshavebeensetbyMCEETYAforTIMSS,studentsperformingatthelowbenchmarkornotachievingthelowbenchmarkmustbethoughtoftobeatrisk,particularlyatYear8.

Content

It’simportantthatanyassessmentofmathematicsshouldreflectthemathsthatitismostimportantforstudentstolearn.WhatdoPISAandTIMSStellusthatourstudentsknowwell,andinwhatareasaretheylaggingbehindinternationally?

PISAresultsfrom2003,whichwasthelastfullassessmentofmathematicalliteracy(enablingustoreportonsubscales),showthatAustralian15-year-oldstudentshaveagenerallyhighlevelofoverallmathematicalliteracy,significantlyhigherthantheOECDaverage.AustralianstudentsoverallalsoscoredatalevelsignificantlyhigherthantheOECDaverageoneachofthesubscales–notquiteaswellinquantitybutbetterinuncertainty.Butintermsofproficiencylevels,one-thirdofAustralianstudentsdidnotachieveproficiencylevel3ontheoverall

1 Manyoftheheadlinereports(eveninbroadsheetssuchasTheAustralian)forthelastreleaseoftheTIMSS2007resultswerealongthelinesof“Borat’skidsbeatAussiekidsinmathsandscience”

mathematicalliteracyscale.WhilethisisclearlybetterthantheOECDaverageof42percentofstudents,wecanaimtodobetter.InHongKong,forexample,oneofthehighestperformingcountries,only25percentofstudentsdidnotachieveproficiencylevel3.

AtYear8,inTIMSS2007,Australianstudentsperformedataroundtheinternationalaverageinmathematicsoverall.Inthecontentdomainofdata and chance,Australianstudentsperformedatalevelsignificantlyhigherthantheinternationalaverage;however.inthecontentareasofalgebraandgeometry,Year8studentsinAustraliaperformedatalevelsignificantlylowerthantheinternationalaverage.Thirty-ninepercentofAustralianYear8studentswereeitheratthelowbenchmarkordidnotachievethelowbenchmarkinmathematicsoverall.

AustralianYear4studentsachievedatalevelsignificantlyhigherthantheinternationalaverageinTIMSS2007,withperformanceindata and chancesignificantlyhigherthantheinternationalaverage,andperformanceinnumberatalevelsignificantlylowerthantheinternationalaverage.Around30percentofAustralianstudentsachievedatorbelowthelowbenchmarkinmathematicsoverall.

Summingup,Australianstudentsperformbetterthantheinternationalaverageatalllevelsintopicsrelatedtodata and chance,whileachievementintheareasofnumberandalgebraarepotentiallyweakerthaninothercountries.However,thesedataindicatethatthereisasubstantialproportionofstudentsexhibitingpoorlevelsofmathematicalunderstandinginAustralianschoolsatallyearlevels.

Equity

Mathematicsisnolongerjustaprerequisitesubjectforscienceandengineeringstudents,butafundamentalliteracyrequirementforthe21st


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century.Equityimpliesthateverystudenthasanopportunitytolearnthemathematicsthatisassessed.CanPISAandTIMSShelpidentifysubgroupsofstudentswhoarenotachievingaswellaswewouldhope?Whatelsecanwefindoutaboutthesegroupsofstudentsthatmayprovidesomecluesastowhyachievementislowerthancouldbeexpected?

WhiletheAustralianPISAandTIMSSdataaregenerallyreportedbygender,Indigenousbackground,immigrantstatus,socio-economicbackgroundandgeographiclocationofschoolinthenationalandinternationalreports,thispaperwillfocusontwoimportantfactors.

Gender

InPISA2003,mathematicalliteracywasinmanycountriesamale-orientedsubject,withboysin28outofthe41countriessignificantlyoutperforminggirls.OnlyinIcelanddidgirlsoutperformboys.InAustralianosignificantgenderdifferenceswerefoundontheoverallmathematicalliteracyscale.Unpackingthisalittlefurther,however,itwasalsofoundthatwhiletherewerenodifferencesoverall,orinthesubscalesforquantityorchange and relationships,Australianboysperformedsignificantlybetterthangirlsonthesubscalesspace and shapeanduncertainty.Therewerenogenderdifferencesinthelowerproficiencylevels,with33percentofbothmaleandfemalestudentsnotachievingproficiencylevel3.Atthehigherlevelsofachievementslightlymoreboys(7%)thangirls(4%)achievedtheveryhighestproficiencylevel,butthesameproportionofmaleandfemalestudentsachievedatthenexttwohighestachievementlevels.

MathematicsinTIMSS2007wasgenerallynotasgenderedinternationally.AtYear4level,thereweresignificantgenderdifferencesin

20ofthe37participatingcountries.In12ofthosecountriesthegenderdifferenceswereinfavourofboysandtheremaining8,infavourofgirls.Australiawasoneofthe18countriesinwhichtherewerenosignificantgenderdifferencesinthecompositemathematicsscore.Withinthesubscales,however,boyssignificantlyoutperformedgirlsinnumber,whilegirlssignificantlyoutperformedboysindatadisplay.

In25ofthe49countriesparticipatinginTIMSS2007atYear8therewerenogenderdifferences.In16ofthecountriesthereweresignificantgenderdifferencesinfavourofgirls,andinonly8countries,ofwhichAustraliawasone(Algeria,Lebanon,Syria,ElSalvador,Tunisia,GhanaandColumbiaweretheothers),weretheresignificantdifferencesinfavourofboys.ThenationalTIMSS2007report(Thomson,Wernert,Underwood&Nicholas,2008)notedthatthiswasnotbecauseofanincreaseinthescoresofboys,butadeclineintheaveragescoreforgirls.Contrarytothefindingsinternationally,inwhichgirlsperformedsignificantlybetterthanboysinalldomainsotherthannumber,Australianboysoutscoredgirlsindata and chance,andnumber,whiletherewasnosignificantdifferenceintheotherdomains.Moreboysthangirlswereachievingatthehigherbenchmarksinbothyearlevels(Year4andYear8)inTIMSS2007.

Tosummarise,AustralianboysoutperformedgirlsinPISA2003intheareasofspace and shapeanduncertainty,inTIMSS2007atYear4innumber,andinYear8innumberanddata and chance.GirlsoutperformedboysinTIMSS2007atYear4indata display.Therewerenosignificantgenderdifferencesonanyothersubscale.Giventhesefewdifferences,itisinterestingtolookatstudents’attitudesandbeliefsaboutmathematics.

InPISA2003,15-year-oldAustraliangirlsreportedsignificantlylowerlevelsof instrumental motivation, self-concept in maths, self-efficacyandinterest in maths,andsignificantlyhigherlevelsofmaths anxiety.Thisfindingholdsevenwhenstudentsachievingatthesameproficiencylevelarecompared.Italsoheldinternationally–inallcountries(evenIceland)boyshadhigherlevelsofself-conceptandself-efficacy,andinthevastmajorityofcountries(therewereapproximatelytwoexceptions)interest in mathematicsandlowerlevelsofmathematics anxiety.

SimilarlyinTIMSS2007atYear4inAustralia,therewasasignificantlyhigherproportionofboysreportinghighlevelsofself-confidenceinmathematics(withnoassociateddifferenceinscorebetweenmaleandfemalestudents).AtYear8just39percentofgirlscomparedto51percentofboysreportedhighlevelsofself-confidence–andalmostone-quarterofgirls(24%)reportedlowlevels.Thiswasbroadlythecaseinmostparticipatingcountries2.Infurtheranalysis(seeThomson,Wernert,Underwood&Nicholas,2008),theeffectofgenderonachievementwasfoundtobesubstantiallyexplainedbythedifferencesinself-confidenceinlearningmathematics.Inotherwords,itisnotbeingagirlinandofitselfthatmakesthedifference,butthatbeingagirlmeansastudentislesslikelytohavehighlevelsofself-confidencethatcanleadtohigherlevelsofachievementinmathematics.

2 However,atYear8inanumberofMiddle-Easterncountries(Oman,Qatar,Palestine,Bahrain,SaudiArabiaandKuwait),girlssignificantlyoutperformedboysandingeneralhadhigherlevelsofself-confidencethanboys–significantlysoinQatar,BahrainandSaudiArabia.Therewereonlyfourcountriesinwhichasignificantlyhigherproportionofgirlsreportedhighlevelsofself-confidencethanboys,incontrasttothe26countriesinwhichtheoppositewasreported.


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Theseareimportantfindingsforteachersandresearchers.Whyisitthattherearestillgenderdifferencesinfavourofmalesinsomanycountriesinallareasofmathematicalliteracy,asshowninPISA,whileamorecurriculum-basedassessmentsuchasTIMSSfindsgenderdifferencesinfavourofboysinsomecountriesandgirlsinothers?Whyareboysmoreself-confidentandhavehigherlevelsofself-conceptandlowerlevelsofanxietyinmathematics,evenwhengirlsoutperformthem?Conversely,whydogirlsstilldoubttheirabilitiesevenwhentheyareclearlyachievingatahighlevel?Ifgirlsdonotseemathematicsasanareaofstrength,despitetheirachievementlevels,andsufferfromhigherlevelsofanxiety,thenitisunlikelythattheywillcontinuetheirstudiesthroughtouniversitylevel.

Indigenousstudents

AspecialfocusofbothPISAandTIMSSinAustraliahasbeentoensurethatthereisasufficientlylargesampleofIndigenousstudents,sothatvalidandreliablecomparisonscanbemade.Inbothstudies,therandomselectionofstudentsinPISAandclassesinTIMSSensuresthatsomeIndigenousstudentsarepartofthemainsample.Inadditiontothis,however,alleligibleIndigenousstudents(i.e.15-year-oldsinPISA,andYear4orYear8studentsinTIMSS)aresampledandaskedtoparticipate.TheNationalCentreandtheEducationMinisterscommunicatewithschoolprincipalstoexplainthepurposeofthisextrasampleandtoconveytothemtheimportanceofencouragingIndigenousstudentstoattendtheassessmentsession.

IthasbeenwidelyreportedthattheachievementlevelsofIndigenousstudentscontinuetolagwellbehindthoseofnon-Indigenousstudents.InmathematicalliteracyinPISA2003,Indigenousstudentsperformed86scorepointsloweronaveragethan

non-Indigenousstudents(DeBortoli&Thomson,2009).Thisrepresentsmorethanonefullproficiencyleveldifference.ThescoregapbetweenIndigenousandnon-Indigenouswassimilaracrossallsubscales.

Inaninternationalperspective,thisplacesourIndigenousstudentsatalevelsignificantlylowerthanstudentsin30othercountries,thesameasstudentsinGreeceandSerbia,andhigherthanstudentsinTurkey,Uruguay,Thailand,Mexico,Indonesia,TunisiaandBrazil.

Intermsofachievementatproficiencylevels,70percentofIndigenousstudents,comparedto32percentofnon-IndigenousstudentswerenotachievingattheMCEETYAstandardoflevel3orabove.Forty-threepercentofIndigenousstudentswerenotachievingatthebasicOECDacceptablestandardoflevel2orabove,thattheyargueisabaselinelevelofproficiencyatwhichstudentsbegintodemonstratethetypeofskillsthattheyneedtobeabletofullyparticipateinsocietybeyondschool.About5percentofIndigenousstudentswere,however,achievingatthehighesttwoproficiencylevels.

AtbothYear4andYear8inTIMSS2007,non-IndigenousstudentsscoredatasubstantiallyhigherlevelthanIndigenousstudents–91scorepointsatYear4and70scorepointsatYear8.AtYear4,Indigenousstudents’scoreswere,onaverage,almostonestandarddeviationlowerthanthoseofnon-Indigenousstudentsinnumber,andaroundthree-quartersofastandarddeviationlowerindata displayandgeometric shapes and measures.AtYear8also,Indigenousstudentsscoredatasignificantlylowerlevel(between54and67scorepoints)thannon-Indigenousstudentsineachofthesubscales.

However,intermsofattitudesandmotivationamongstIndigenous

students,thereweresomeinterestingfindings,recentlydescribedinDeBortoli&Thomson(2010).AmongstAustralian15-year-oldstudentsinPISA2003,aspreviouslydescribed,thereweresignificantgenderdifferencesininstrumental motivation, self-concept in maths, self-efficacyandinterest in maths,andmaths anxiety.AmongstIndigenousstudents,however,therewerenosignificantgenderdifferencesininterest,instrumental motivation or anxiety,althoughIndigenousgirlshadveryhighscoresonthislatterconstruct,reflectinglevelsofanxietyinmathematicsmuchhigherthantheOECDortheAustralianaverage.Inself-concept in maths,significantdifferenceswerefoundforIndigenousstudents,buttheyweresmallerinmagnitudethanthosefornon-Indigenousstudents.

InTIMSS2007,thereweresignificantlygreaterproportionsofAustralianboysthangirlsinthehighlevelsofbothself-confidenceandvaluing mathematics.However,amongsttheIndigenouspopulation,thiswasnotthecase,withsimilarproportionsofboysandgirlsreportinghighlevelsofboth.

Furtherinvestigationisneededtoexaminethesefindings–tofindoutwhethertheyreflectactualdifferencesinbeliefsamongstIndigenousboysandgirlsorwhetheritissimplyanartefactofthesamplesize,sincestandarderrorsarelargerfortheIndigenoussample.PISA2012will,wehope,providesomeoftheseanswers–thefocusisagainonmathematics,andAustraliaisimplementingadifferentsamplingmethodologywhichwehopewillresultinamuchbiggersampleofIndigenousstudentsthaneverbefore.

IntermsoffactorsinfluencingtheachievementofIndigenousstudents,theeffectofsocio-economicbackgroundissubstantial.However,theeffectofstrong,positiveattitudesandbeliefsisalsosignificant,andcanbeencouragedthroughschoolprograms.Also


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importantisattendanceatschool–Indigenousstudentswerefoundtobefarmorelikelythannon-Indigenousstudentstobelatetoschoolonaregularbasis,tomissconsecutivemonthsofschoolingandtochangeschoolsseveraltimes.Inadditiontolowerlevelsofhomeeducationalresourcesandparentaleducationexperience,thegapsthatappearatthebeginningofprimaryschoolwidenasaresultofpoorattendanceatschool.

Summary

ItissometimesdifficultforteachersandschoolleaderstoseethepurposeofPISAandTIMSS.However,thestudentsweareeducatingtodaywillcompeteinaglobalmarket,andwehavetobesurethattheeducationweareprovidingthemwithisonethatwillprovidethemwithastrongbase,bothinknowledgeandskillsandintheabilitytoapplythoseskillstoreal-worldproblems.PISAandTIMSSprovidesuswiththatinformation,andmuch,muchmore.

ReferencesMinisterialCouncilonEducation,Employment,TrainingandYouthAffairs(MCEETYA)(1999).TheAdelaidedeclarationonnationalgoalsforschoolinginthetwenty-firstcentury.Availablehttp://www.curriculum.edu.au/mceetya/nationalgoals/index.htmaccessedMay2010

MeasurementFrameworkforNationalKeyPerformanceMeasures(MCEETYA,2008)Availablehttp://www.mceecdya.edu.au/verve/_resources/PMRT_Measurement_Framework_National_KPMs.pdfaccessedMay2010

DeBortoli,L&Thomson,S.(2009).The achievement of Australia’s Indigenous students in PISA 2000- 2006.Camberwell:ACER.

Steen,L.A.(Ed).(1990).On the shoulders of giants: New approaches to

numeracy. WashingtonD.C.:NationalAcademyPress.

Thomson,S.,Wernert,N.,Underwood,C.&Nicholas,M.(2008).TIMSS2007:Taking a closer look at mathematics and science in Australia. Camberwell:ACER.

Poster presentations


83

1 Ken Lountain, Barbara Reinfeld, Phil Kimber and Vivienne McQuade

Department of Education and Children’s Services South Australia. Learning Inclusion Team

Maths�for�Learning�Inclusion�–�action�research�into�pedagogical�change

MathsforLearningInclusionisaninitiativefocussedonimprovingtheteachingandlearningofmathematicsin28primaryschoolsin6clustersservinglowsocio-economiccommunities.

Theaimsoftheprojectare:

• allstudentsachieving

• challengingandengagingcurriculum

• sustainableprofessionallearningcommunities

• improvementinformedbyevidenceandresearch

Professionallearningiscomposedofmathsknowledgeandpedagogy,learninginclusionprinciplesandpractices.TeachersaresupportedtoestablishandmaintainafocusonnarrowingtheachievementgapforAboriginallearnersandstudentsfromlowsocio-economicbackgroundsthroughdevelopinganactionresearchquestion.

Learningisshared,analysed,critiquedandsustainedasappropriateacrossschoolsandclustersbyteachers’andleaders’participationincommunities of practice.

Theprogramissupportedbyaconcurrentandrigorousevaluationcomposedofmultipledatasetsincludingteachernarrativesreflectingonpedagogicalchange.Thesenarrativeswillbepresentedattheconference.

2 Paul Waddell, Patrick Murray and Stephen Murray

Mathematics.com.au NSW

Online�Maths�Resources�–�Creating�deep�mathematical�thinking�or�lazy�teachers�dispensing�‘busy�work’?

Withaplethoraofonlinemathsprogramsavailabletoteachers,studentsandparents,howdoweaseducatorsdistinguishbetweenthosethatwerecreatedtoentertainandoccupystudentsfromthosethatencourageanddevelopdeepmathematicalthinking?Aneffectivedigitalmathematicsresourcewillbedesignedwithstudentlearningasthekeygoal.Itshouldclearlydemonstratestrategiestodevelopthebuildingblocksofnumeracy,provideopportunitiestodiscoverbetterandvariedwaysofsolvingproblems,andfocusonthestepsonthejourneyofdiscoveryaswellasthedestinationofimprovedstudentlearning.

Thisposterpresentationwillprovideadviceonstrategiestoevaluatethepurposeandplaceofdigitalresourcesintheteachingandlearningofmathematics.Insightsdrawnfromover12yearsofpracticeintheevaluationanduseofdigitalresourcestosupporteffectivestudentlearningwillinformthisposterpresentation.

3 Alex NeillNew Zealand Council for Educational Research

Processes�surpass�products:�Mapping�multiplicative�strategies�to�student�ability

Whenmakingjudgementsaboutstudentunderstanding,thestrategiesthattheyusearefarmorerevealingoftheirlevelofthinkingthantheanswerstheyproduce.Theposterwilldisplayarangeofstudentresponsestosomemultiplicationproblems,andexploretherelationshipbetweenstudents’overallabilityandthestrategiesthattheyemploy.


84

4 Cathryn MorrisAustralian Association of Mathematics Teachers

Make�it�count�–�Numeracy,�mathematics�and�Indigenous�learners

TheAustralianAssociationofMathematicsTeachers(AAMT)inc.hasestablishedthisnationalfouryearprojecttodevelopanevidencebaseofpracticesthatimproveIndigenousstudents’learninginmathematicsandnumeracy.Theposterwillprovide:

• Informationabouttheprojectanditseightclustersofschoolsframeworksforintersectingcommunitywithclassroomandthedevelopmentofculturallyresponsivemathematicseducation

• Storiesfromtheclustersinvolveddirectlyintheproject

• Professionaldevelopment,communicationandcollaborationthroughanonlinelearningcommunity(networkring)

• Examplesofresearch/inquiryanddatacollection

• Partnerships/friendshipsbetweencommunity,schoolanduniversitiesthatsupportimprovedlearningoutcomesofIndigenousstudents

• AresourceforotherswantingtohelptheirIndigenousstudentsbetterreachtheirpotentialinmathematicsandnumeracy

ThisprojectisfundedbytheAustralianGovernmentundertheClosingtheGapInitiative.

5 Sonia White and Dénes Sz cs

The Queensland University of Technology and The University of Cambridge

Number�line�estimation�behaviours:�Influence�of�strategy?

ThepurposeofthisstudywastoinvestigatenumberlineestimationbehavioursofchildreninYears1-3andexplorethepotentialinfluenceofstrategyduringsuchtasks.Childrenwereaskedtopositiontargetdigitsonaseriesof0-20numberlinesandtheirresponseswereanalysed.Existingcognitiveresearchhastypicallymodelledthedevelopmentofnumberestimationasbeingaprogressionfromlogarithmictolinearrepresentations.ThistrendwasconfirmedinthisstudywithchildreninYears2and3demonstratingasignificantpreferenceforalinearmodel;aresultnotevidentintheYear1participants.Thismodellingapproachhadlimitationswhenattemptingtounderstandtheinfluenceofstrategyinnumberlineestimation.Toascertainstrategy,weanalysedestimationaccuracyforindividualtargetdigits.Thesefindingspointtoalinkbetweendevelopmentalprogressionandstrategyapplicationforcertaintargetdigits.Itwasconcludedthatfurtherexplorationsintothetypesofstrategieschildrenemploywhenperformingnumberestimationtaskswouldbeofgreatvalue,particularlywhenreferencedtoclassroompracticeandtheovertteachingofstrategyinmathematicseducation.

6 Michael JenningsThe University of Queensland

First-year�university�students’�mathematical�understanding

Inrecentyearstherehasbeenanoticeableincreaseinthediversityofbackgrounds,abilitiesandaspirationsofstudentsenteringbridgingandfirst-yearmathematicscoursesatTheUniversityofQueensland.Muchresearchhasbeenundertakenintoprimaryandsecondarymathematicseducationbutlittleincomparisonhasbeendoneintotertiarymathematicsandstudents’transitionfromsecondarytotertiarymathematics.WiththenumberofstudentsenteringAustralianuniversitiesincreasing,itisimportanttoknowwhatlevelofmathematicalunderstandingtheybringwiththem.

Diagnostictestingoffirst-yearengineeringandsciencestudentsatTheUniversityofQueenslandhasbeenconductedatthebeginningoffirstsemesterforthepastfouryears.Thedatafromthecompetencytestswasanalysedtodecidethebestwaytoimprovestudents’mathematicalknowledgeandunderstanding.Resultsfromthetestsandsubsequentoutcomeswillbepresented.

Conference program


87

Sunday 15 August6.00-7.30PM Cocktails�with�the�Presenters–CrownConferenceCentre–Entertainment by Fly Right Trio

Monday 16 August7.30AM Conference�Registration Level2–CrownConferenceCentreHall

8.30AM Welcome�to�Country IanHunter

8.45AM Conference�Opening ProfessorGeoffMasters,ChiefExecutiveOfficer,ACER

9.00AM Keynote�Address�1 Speaking in and about mathematics classrooms internationally: The technical vocabulary of students and teachers.ProfessorDavidClarke,UniversityofMelbourne

Crown Conference Centre Hall Chair : Dr. John Ainley, ACER

10.15AM Morning�tea�and�poster�presentations

10.45AM Concurrent�Sessions�Block�1

Session�A

Issues of social equity in access and success in mathematics learning for Indigenous students

ProfessorRobynJorgenson,GriffithUniversity

M 12 &13 Chair : Kerry-Anne Hoad, ACER

Session�B

Primary students’ decoding mathematics tasks: The role of spatial reasoning

ProfessorTomLowrie,CharlesSturtUniversity

M11 Chair : Cath Pearn, ACER

Session�C

Promoting the acquisition of higher order skills and understandings in primary and secondary mathematics

ProfessorJohnPegg,UniversityofNewEngland

Crown Conference Centre Hall 1 Chair : Dr Lawrence Ingvarson ACER

Session�D

Mathematics assessment in primary classrooms: Making it count

AssociateProfessorRosemaryCallingham,UniversityofTasmania

Crown Conference Centre Hall 2&3 Chair : Dr Hilary Hollingsworth, ACER

Session�E

Conversation with a Keynote

ProfessorPaulErnest,UniversityofExeter

Restricted to designated delegates only.

M14

12.00PM Lunch�and�poster�presentations

12.15PM Lunchtime�talkback Mathematics or Numeracy – what are we actually talking about here? Does it matter?TalkbackledbyMrWillMorony,ExecutiveOfficer,AAMT.Opentoalldelegates–bringyourlunchandyourviews.

M 15 &16

1.00PM Keynote�Address�2 Standards, what’s the difference?: A view from inside the development of the Common Core State Standards in the occasionally United StatesMrPhilDaro,UniversityofCalifornia


2.15PM Afternoon�tea�and�poster�presentations

2.45PM Concurrent�Sessions�Block�2

Session�F

The case of technology in senior secondary mathematics: Curriculum and assessment congruence?

DrDavidLeigh-Lancaster,VictorianCurriculumandAssessmentAuthority

M 12 & 13 Chair : Ray Peck, ACER

Session�G

Reconceptualising early mathematics learning

AssociateProfessorJoanneMulligan,MacquarieUniversity

Crown Conference Centre Hall 2 &3 Chair : Kerry-Anne Hoad, ACER

Session�H

Learning about selecting classroom tasks and structuring mathematics lessons from students

ProfessorPeterSullivan,MonashUniversity

Crown Conference Centre Hall 1 Chair : Dr. Lawrence Ingvarson, ACER

Session�I

Identifying cognitive processes important to mathematics learning but often overlooked

Mr.RossTurner,ACER

M11 Chair : Marion Meiers, ACER

Session�J


ProfessorKayeStacey,UniversityofMelbourne


M14

4.00PM Close�of�Day�1

6.45PM Pre�dinner�drinks� CrownConferenceCentreHallEntertainment by Regent Strings

7.00PM Conference�dinner CrownConferenceCentreHallEntertainment by Pot Pourri


88

Tuesday 17 August9.00AM Keynote�Address�3 Mathematics teaching and learning to reach beyond the basics

ProfessorKayeStacey,UniversityofMelbourne


10.15AM Morning�tea�and�poster�presentations

10.45AM Concurrent�Sessions�Block�3

Session�K

Using mental representations of space when words are unavailable: Studies of enumeration and arithmetic in Indigenous Australia

AssociateProfessorRobertReeve,UniversityofMelbourne

M 12 & 13 Chair : Cath Pearn, ACER

Session�L

Using technology to support effective mathematics teaching and learning: What counts?

ProfessorMerrilynGoos,UniversityofQueensland

Crown Conference Centre Hall 2 & 3 Chair : Kerry-Anne Hoad, ACER

Session�M

Making connections to the big ideas in mathematics: Promoting proportional reasoning

DrShelleyDole,UniversityofQueensland

M 11 Chair : Marion Meiers, ACER

Session�N

Mathematics learning: What TIMSS and PISA can tell us about what counts for all Australian students

DrSueThomson,ACER

Crown Conference Centre Hall 1 Chair : Dr. Hilary Hollingsworth, ACER

Session�O


MrPhilDaro,UniversityofCalifornia,Berkley


M 14

12.00PM Lunch�and�poster�presentations

12.15PM Lunchtime�talkback Mathematics or Numeracy – what are we actually talking about here? Does it matter? (repeat)TalkbackledbyMrWillMorony,ExecutiveOfficer,AAMT.Opentoalldelegates–bringyourlunchandyourviews.

M 15 &16

1.00PM Keynote�Address�4 The social outcomes of school mathematics: Standard, unintended or visionary? ProfessorPaulErnest,UniversityofExeter


2.15PM Closing�Address ProfessorGeoffMasters,ChiefExecutiveOfficer,ACER

Crown Conference Centre map and

floorplan


91


92

Conference delegates


95

Dinner table no. Delegate Name Delegate Organisation

13 MrRossAbbottHead of Mathematics

MaristCollege,Canberra,ACT

MsBelindaAdamsDeputy Principal

LockleysNorthPrimarySchool,SA

2 DrJohnAinleyDeputy CEO (Research) and Research Director

ACER,VIC

MrRonaldAldermanNumeracy Coach

DECS,SA

MsJulesAldous ShelfordGirls’Grammar,VICMsRosannaAlgeriMaths Teacher

CasimirCatholicCollege,NSW

MsMariaAliceProject Officer : Primary and Numeracy

CEO,InnerWesternRegion,NSW

15 MsJudithAllenPrincipal

BrightonPrimarySchool,SA

MrNicholasAmbrozyMaths HOD

StAnthony’sCatholicCollege,QLD

8 DrJudyAndersonAssoc. Prof. Mathematics Education

TheUniversityofSydney,NSW

MrsKayAndersonMaths Teacher

TheGlennieSchool,QLD

MrLorneAndersonMaths Coordinator

TaylorsLakesSecondaryCollege,VIC

MrsNoxiaAngelidesCurriculum Director

CaulfieldJuniorCampus,VIC

MsJanineAngoveManager Content Development

HOTmaths,NSW

MrsTaniaAngroveMaths Coordinator

CatholicCollegeBendigo,VIC

9 MsGayleApplebyAdministration Coordinator

ACERINSTITUTE,VIC

MrsSueanneAquilinaTeacher

StAndrew’sPrimarySchool,NSW

MrsRebeccaArmisteadTeacher

KillaraPrimarySchool,VIC

MrsMaryAsikasPrincipal

Seaford6-12School,SA

MsCynthiaAthaydeMaths Coordinator

StJohnBoscoCollegeEngadine,NSW

MrsCatherineAttardLecturer

UniversityofWesternSydney,NSW

MrBrianAulsebrookPrincipal

SacredHeartSchool,NSW

MsVivienneAwadDeputy Principal

LoretoKirribilli,NSW

MrsJessieAzizSales Coordinator, VIC

Jacaranda,JohnWiley&SonsAustraliaLtd,

MissVeronicaAzzopardiClassroom Teacher



96


MrsMarijaBaggioDeputy Principal

LefevrePrimarySchool,SA

14 MsJillBainTeacher Mathematics

WildernessSchool,SA

MrAndrewBakerTeacher

StJerome’sPrimarySchool,NSW

MsJulieBakerCoordinator

StMary’s,Toukley,NSW

MrsRuthBakogianisTeacher

StMaryoftheAngelsSec.College,VIC

19 MsMariaBallHead of Maths

AllHallows’School,QLD

MrMichaelBarraEducation Officer Mathematics

BrisbaneCatholicEducationOffice,QLD

MsSueBarringtonAssistant Principal

StTherese’sPrimary,NSW

MrTravisBartlettDeputy Principal

AllenbyGardensPrimarySchool,SA

MrsKimBastockMaths Coordinator

PresbyterianLadiesCollege,NSW

MrMarkBatemanPrincipal

OLGCCatholicSchool,NSW

MsJaneBattrickLeading Teacher - Numeracy

MiddleParkPrimarySchool,VIC

MrKevinBauerPrincipal

HolyFamilyCatholicPrimarySchool,NSW

17 MsGeraldinaBaxterTeacher

IrympleSecondarySchool,VIC

MrsDonnaBeauchamp-WhylieTeacher

CarwathaCollegeP-12,VIC

MsNaomiBelgradeHead of Mathematics

WoodcroftCollege,SA

10 MsAnneBellertAdditional Needs Officer

CatholicEducationOffice,NSW

MrRichardBennettsPrincipal

MalvernPrimarySchool,VIC

MrSteveBentleyTeacher

TheFriends’School,TAS

7 MsDagmarBevanRegional Curriculum Consultant

DECS,SA

11 MsSuzanneBevanPrincipal

StPhilipNeri,Northbridge,NSW

MrChrisBiefeldAssistant Principal

StMartin’sSchool,NSW

3 MsMargaretBigelowSPO Mathematics

ACARA,NSW

20 MrsMichelleBinneyTeacher

WhitsundayAnglicanSchool,QLD

20 MrGrahamBishopAssistant Coordinator - Mathematics

UWSCollegePtyLtd,NSW


97


MrAndrewBlackwoodTeacher

ClaremontCollege,TAS

MrJohnBlecklyNumeracy Coach

DECS,SA

8 MrChristopherBloodHead of Mathematics

BrisbaneBoys’College,QLD

11 MsJanetBohanDeputy Principal

StMary’sPrimarySchool,VIC

MrsElizabethBortolotRegional Numeracy Coach

WesternMetropolitanRegion,VIC

10 MsTrishBoschettiMaths for Learning Inclusion Co-ordinator

PrimaryMathematicsAssociation,SA

MrsCarolineBoulisMaths Coordinator

StJoseph’sPrimary,Belmore,NSW

MsMaryBoutrosTeacher

WoorannaParkPrimarySchool,VIC

20 MrRobertBowdenDeputy Principal

WestBeachPrimarySchool,SA

MsBenitaBowlesHead of Learning Support

OurLadyofMercyCollege,VIC

MrTonyBoyd OurLadyofFatimaPrimary,NSWMrRussellBoyleDean of Mathematics

RuytonGirls’School,VIC

19 MsDeborahBrassingtonPrincipal

TorrensvillePrimarySchool,SA

MrsNatalieBratbyTeacher

HolyFamilySchool,NSW

MrsKarenBredenhannMaths Coordinator

HeightsCollege,QLD

MrBernardBreePrincipal

StuartParkPrimarySchool,NT

MrChristopherBrennanMaths Teacher

StAidan’sAnglicanGirls’School,QLD

MissKellieBrennanTeacher

KingstonStateSchool,QLD

MrsJulieBridgenTeacher

MaryMacKillop,NSW

7 MrsFionaBrimmerPPO - Mathematics

EducationQueensland,QLD

MrPhilBrockbankHead of Mathematics

AllSaints’College,WA

MrDavidBrooksMaths Teacher


MrsCarolineBrownTeacher

SacreCoeur,VIC

10 MrGarryBrownDeputy Principal

QldAcademyforHealthSciences,QLD

MrGregBrownTeacher

SeafordRisePrimarySchool,SA


98


MrsJulieBrownHead of Mathematics

CatherineMcAuley,Westmead,NSW

15 MrsSafiaBrownTeacher

StClaresCatholicCollege,ACT

MsJulieBrozConsultant

StepsProfessionalDevelopment,WA

MrStevenBruceTeacher


MrStuartBrunsdonTeacher

MaryMacKillopforGirls,NSW

MrsEmilyBuckleyTeacher

CanterburyPrimarySchool,VIC

MrsSuzanneBuddNumeracy Leader

AllSaintsPrimarySchool,SA

MsFionaBuiningTeacher

OranaSteinerSchool,ACT

MsJoanBurfittConsultant

CatholicEducationOffice,WA

MsToniBurfordCoordinator, Maths

LittlehamptonPrimarySchool,SA

MrPaulBurkeTeacher


21 MrsMicheleBurnsCurriculum Leader Mathematics

GenazzanoFCJCollege,VIC

MissFionaBylsmaAssistant Principal

ChristtheKingPrimarySchool,NSW

MrsDaleCainLiteracy/Numeracy Consultant

CatholicSchoolsOffice,NSW

MrsJacquelineCainNumeracy Coach

DECS,SA

MrsKateCalleaNumeracy Coordinator

StMartinofToursPrimarySchool,VIC

3 ProfRosemaryCallingham UniversityofTasmania,TASMsHilaryCameronAssistant Principal

StGerard’sPrimarySchool,NSW

20 MsAnneCannizzaroPrincipal

WestLakesShoreSchools,SA

MrDavidCareyMathematics Coordinator

StAndrew’sCollege,NSW

MrPeterCarmichaelProject Officer - Mathematics


MsBeverleyCarrTeacher


MrsBethCarrollMaths Domain Leader

StJoseph’sCollege,VIC

MrsCristiCarrollMaths Coordinator

StFrancisCollege,NSW

MrShaunCarrollMaths Facilitator

ConcordiaInternationalSchool,CHINA


99


MsAmandaCarterHead of Mathematics

DamascusCollege,VIC

MrsLouiseCaruana StMary’sPrimarySchool,NSWMrGregCashmanTeacher, Mathematics

MonteSant’AngeloMercyCollege,NSW

MrDarylCastellinoMaths Coordinator

PatricianBrothers’CollegeFairfield,NSW

13 MrsMarianneCastor StDominic’sCollege,NSWMrSteveCauchiCoordinator

MaryMacKillop,NSW

MsMelissaChabranProgram Officer

BillandMelindaGatesFoundation,USA

MsCateCharles-EdwardsDirector of Maths

WestbourneGrammarSchool,VIC

6 MrGraemeCharltonPrincipal

WoodvillePrimarySchoolandCHI,SA

MrSengChongMathematics Coordinator

InternationalEducationServicesLtd,QLD

6 MsMeredithChristie-LingAssistant Principal

WoodvillePrimarySchoolandCHI,SA

ProfDavidClarkeDirector

TheUniversityofMelbourne,VIC

MissRuthClarkeActing Head of Mathematics

WycliffeChristianSchool,NSW

MsNicoleClaxtonNumeracy Coach


MsKathrynCleary StPeterChanelPrimary,NSWMrGrantCliftonHead of Mathematics

AitkenCollege,VIC

MrLanceCoadTeacher

StMichael’sCollegiate,TAS

MrFrankCohenPrincipal

StJohntheBaptistCatholicPrimary,NSW

MrIanColemanHead of Department

StAugustine’sCollege,QLD

MrsLeeCollieDirector

MacmillanProfessionalLearning,VIC

MsCarolCollinsTeacher

BraybrookSecondaryCollege,VIC

MrsPatConheadyPrimary Maths Specialist

NorthShorePrimarySchool,VIC

10 MrVinceConnorSchools Consultant


MsMelanieCookMaths Coordinator

GoodSamaritanCatholicCollege,NSW

MrsBiancaCookeTeacher

GoodShepherdSchool,NSW

MrsMerilynCostaMaths Coordinator

MalvernPrimarySchool,VIC


100


MsJulieCostelloeTeacher

OLRTheEntrance,NSW

MrsSandraCottamCurriculum Officer - Numeracy

DepartmentofEducationofWA

12 MrNoelCovillHead of Mathematics

StJosephsCollege,QLD

MrIanCowanTeacher

TerraSanctaCollege,NSW

MrsMelissaCowanReading Recovery Teacher


MrPeterCranneyAssistant Principal

StJoseph’sPrimarySchool,NSW

MrDavidCreesHead of Mathematics

FlindersChristianCom.College,VIC

MrsKimberleyCromptonLeslieAcademic Enrichment Coordinator

BarkerCollege,NSW

21 MrsShelleyCrossTeacher Mentor Maths

A.B.PatersonCollege,QLD

MsSusanCrouchMaths Teacher

BrownsPlainsSHS,QLD

MrsJacintaCrowePrincipal

OurLadyofRosarySchool,NSW

8 MrsKarenCrowleyHead of Maths

TrinityLutheranCollege,QLD

MrTomCrowleyMaths Coordinator

StMichael’sPrimarySchool,NSW

MrGregCummingDeputy Principal

StBrendan’sSchool,NSW

MrsNicoleCummingPrincipal

StPatrick’sPrimarySchool,NSW

MrsDeborahCurkpatrickDirector, Student Learning Support & Extension


MrsRobinCurleyTeacher

LandsdalePrimarySchool,WA

MrChrisDalyTeacher

MacGregorPrimarySchool,QLD

MrsAngelaD’AngeloAdviser

CatholicEducationOfficeSydney,NSW

MrMichaelDarcyHead of Mathematics

AssisiCatholicCollege,QLD

1 MrPhilDaro UniversityofCalifornia,USAMsAndreaDartHead of Curriculum

OvernewtonCollege,VIC

MsMaureenDavidsonNumeracy Coach

DECS,SA

MrsBeverleyDaviesPrimary Teacher

WycliffeChristianSchool,NSW

15 MrGaryDaviesHead of Mathematics

NewingtonCollege,NSW


101


MrsHelenElizabethDaviesPrincipal

GinGinStateSchool,QLD

MsTraceyDaviesDeputy Principal

KidmanParkPrimarySchool,SA

MsPatriciaDavisHead of Maths

WenonaSchool,NSW

12 DrAlexandreDavyskibSenior Teacher

St.AloysiusCollege,NSW

MissSusanDawsonHead Teacher Aboriginal Education

CampbelltownP.A.HighSchool,NSW

MsFionadeStGermainYear 5 Teacher

StRoseCatholicSchool,NSW

MsEvaDeVriesPrincipal Project Officer

AustralianCatholicEducation,Qld

MsSandyDeamAssistant Principal

KilkennyPrimarySchool,SA

15 MrMichaelDeleanAssistant Principal

BrightonPrimarySchool,SA

MrsTheaDelfosTeacher

StJohn’sRegionalCollege,VIC

4 MrDeanDell’oroHead of Mathematics

GeelongGrammarSchool,VIC

MrsTraceyD’eltonCoordinator

LowtherHallAGS,VIC

16 MsJoDenton DaramalanCollege,ACT10 MrChrisDerwin

Schools ConsultantCatholicEducationOffice,NSW

5 MrLanceDevesonLibrary and Information Manager

ACER,VIC

MrsElizabethDevlinAssistant Principal

StOliver’sPrimarySchool,NSW

MrsElizabethDevlinTeacher

MaryMacKillop,NSW

MrsMargaretDevlinTeacher

StuartholmeSchool,QLD

MsJennieDewMaths Coordinator

LloydStreetSchool,VIC

15 MrsTinaDiSanoTeacher

SaintIgnatiusCollege,SA

MsLouiseDickMaths Teacher

AschamSchool,NSW

MissAlisonDicksonMaths Coordinator

StThomastheApostle,VIC

MrsSueDietrichPrincipal

MacKillopCatholicCollege,NSW

17 MissClaireDillmannMathematics Teacher

KingstonCollege,QLD

MrRichardDipaneHead of Mathematics

GeorgianaMolloyAnglicanSchool,WA


102


MrsLouisaDohertyHead of Mathematics

CalvinSecondarySchool,TAS

3 DrShelleyDole UniversityofQueensland,QLDMsLynDonaghueNumeracy Coordinator

LearningServicesNorth-West,TAS

MrPhilipDonatoDeputy Principal

OurLadyoftheSacredHeartCollege,SA

MrPaulDooleyTeacher

StUrsula’sCollege,QLD

MrMichaelDoonerMaths Coordinator

ClancyCatholicCollege,NSW

14 MsHelenDouvartzidisHead of Mathematics

WildernessSchool,SA

MrJohnDoveyHead of Mathematics

MelbourneHighSchool,VIC

MsAmandaDowdellTeacher

StPeterChanelPrimary,NSW

15 MrGraemeDownwardTeacher

NewingtonCollege,NSW

MsMelanieDoyleNumeracy Coach

DECS,SA

17 MrGlennDudleyHead of Mathematics

PymbleLadies’College,NSW

MsJeanneDudleyMaths Coordinator

AllSaintsCatholicGirlsCollege,NSW

9 MrsMary-AnnDudleyMaths Teacher/Pastoral Coordinator

MtStBenedictCollege,NSW

MissAnneDuncanPrincipal

StJohntheApostlePrimarySchool,NSW

MrBruceDuncanNumeracy Coordinator

WoodbridgeSchool,TAS

MissKerryDundas ShelfordGirls’Grammar,VICMrDavidDunstanNumeracy Consultant

AISWA,WA

11 MissDominiqueDybalaTeacher


MrsTrishDykesTeacher


MrsMariaDyneMaths Coordinator

QueenofPeacePrimarySchool,VIC

MsSylviaEadieNumeracy Support Teacher

LearningServicesNorth-West,TAS

MrsCherylEatherAdministration Coordinator

LoyolaSeniorHighSchool,NSW

MrsJoEdwardsHOC

BerserkerStreetStateSchool,QLD

MrGavinEdwardsSenior Project Officer

DEECD,VIC

MrsHeatherEfraimsenPrincipal

DECS,SA


103


MrDebEldridgeMaths Coordinator

BallaratGrammarSchool,VIC

MsHelenElliottAssistant Principal

StMichael’sPrimary,NSW

7 MsAnn-MarieEllisMaths for All Facilitator

DECS,SA

MsSueEllisTeacher


MsCateElshaugAssistant Principal

LLoydStreetSchool,VIC

MrAndrewEmanuelAssistant Principal

ChisholmCatholicPrimarySchool,NSW

MrsNatalieEmbertonTeacher

AllSaintsPrimarySchool,SA

1 ProfPaulErnest TheUniversityofExeter,UKMsGailErskineTeacher Educator

StJerome’sPrimary,NSW

19 MrsSueEvansMaths DBA Leader

OberonHighSchool,VIC

9 MsFrancesEveleighResearch Fellow

ACER,NSW

MrsCaitlinkFaimanHead of Gifted

BialikCollege,VIC

MrsMarilynFaithfullSenior Mathematics Administrator

KoonungSecondaryCollege,VIC

MrsWendyFalconerNumeracy Adviser

UniversityofWaikato,NZ

MrsRobynFarnellAssistant Principal

HamptonPrimarySchool,VIC

MsSallyFarrellHOD

PalmBeachHighSchool,QLD

MrAntonioFazziniHead of Mathematics

SaintIgnatius’College,SA

MrLukeFenslingNumeracy Coordinator

McKinnonPrimarySchool,VIC

15 MsCandiceFereyCoordinator Learning Enrichment

SantaSabinaCollege,NSW

MrsMargaretFergusonTeacher/Leadership Team

HolyFamilyPrimarySchool,NSW

MrBruceFerringtonTeacher

RadfordCollege,ACT

MrsAnitaFewsterTeacher


18 MsJocelynFieldTeacher

PenrhosCollege,WA

16 MrsJoanneFindlayTeacher

BundabergS.H.S.,QLD

MsAnneFinlayMathematics Coordinator

DeLaSalleCollegeAshfield,NSW


104


MrSimonFinniecomeAdmin/Teacher

DomremyCollege,NSW

13 MrsMeganFinniganTeacher

MaristCollege,Canberra,ACT

MrsLaurenFitzhenryAssistant Principal

StKevin’sCatholicPrimarySchool,NSW

MrsLanaFleiszigMaths Coordinator

MtScopusMemorialCollege,VIC

MrsKrishnaFlemingCoordinator

AquinasCollege,VIC

MrsSharonFlemingTeacher

LoretoCollege,SA

MrKenFletcherYear 1 Teacher

EmmanualCollege,QLD

MsJackyFoleyMaths Coordinator

NagleCollege,NSW

MrsMargaretFordTeacher


MrsRobynFordTeacher

BarkerCollege,NSW

MsMichelleFothergillEducation Sales Coordinator

CambridgeUniversityPress,VIC

MrsJoFoxPrincipal


MsKathrynFoxHead of Teaching & Learning Services


MrsElizabethFragopoulosTeacher Educator

StJoseph’sPrimary,Belmore,NSW

MrDavidFrancisHead of Mathematics

CitipointeChristianCollege,QLD

MrsBeaulahFranksonTeacher


MissKylaFrazerTeacher

CarwathaCollegeP-12,VIC

MsDanielleFreeman EvertonParkStateSchool,QLDMrPhilFreemanHOD - Mathematics

CraigsleaSeniorHighSchool,QLD

MrsDanielleGagliardiTeacher


MsAmandaGahanTeacher


MrsSusanGahanStage One Co-ordinator

OlsosPrimarySchool,NSW

15 MrsDonielleGaleTeacher

StIgnatiusCollege,NSW

MrToddGallacherSenior School Maths

CareyBaptistGrammar,VIC

MrMichaelGallagherAssistant Principal

StJoseph’sCatholicPrimarySchool,NSW


105


MsGinaGalluzzoCurriculum Officer

CatholicEducationOffice,ACT

MrCraigGannonDeputy Principal

ClarksonCommunityHighSchool,WA

MsNicoleGardnerTeacher


MsMarthaGarkelHead of Mathematics

SacredHeartGirls’College,VIC

MsRobynGarnettTeacher


MrsJudyGastinPrincipal

StMichael’sPrimary,NSW

MrsElizabethGauldMathematics Coordinator

StMargaretMary’sCollege,QLD

MissMichelleGawronskiPrimary Maths Specialist

NorthShorePrimarySchool,VIC

MrAndrewGearLeading Teacher

CedarsChristianCollege,NSW

MrsKatherineGeePrincipal

MariaReginaCatholicPrimarySchool,NSW

5 MsKatieGeeringsTeacher

LorneAirey’sInletP12College,VIC

MsLindaGelatiNumeracy Consultant

CatholicEducationOffice,SA

MrGregGeorgiouAssistant Maths Coordinator

GoodSamaritanCatholicCollege,NSW

MrsDeborahGibbsMathematics Adviser

MasseyUniversityCollegeofEducation,NZ

MissMelissaGibbsTeacher

MountGambierHighSchool,SA

6 MrsBernadetteGibsonEducation Officer


14 MsRhiannonGilesMathematics Teacher

WildernessSchool,SA

MsKarenGillespieAssistant Principal

CraigburnPrimarySchool,SA

MrsTrishGleesonEducation Officer

CSOMaitland-Newcastle,NSW

2 ProfMerrilynGoosDirector

TheUniversityofQueensland,QLD

MrsJohannaGordonTeacher

BrisbaneGrammarSchool,QLD

MsHaleyGrahamCo Head of Middle School Maths

BallaratClarendonCollege,VIC

MrRichardGrechAssistant Principal

DelanyCollege,NSW

MrDavidGreenTeacher

SydneyGrammarSchool,NSW

19 MrJamesGreenHead of Mathematics

TrinityCatholicCollege,NSW


106


MrsDeniseGreenbergAssistant Head of Mathematics

WenonaSchool,NSW

MrMartinGregoryTeacher

XavierCollege,VIC

MrWilliamGrieveHead of Maths

StBrendansCollege,QLD

MsLindaGrofInstructional Practice Coach

StKildaPrimarySchool,VIC

MsJackyGruszkaMaths Teacher

TaylorsCollegeWaterloo,NSW

MrsSusanGuilfoylePrincipal


MrPeterHackettHola

CorpusChristiCollege,WA

MrsRobynHadfieldMaths Teacher


MrsSueHageTeacher


MsBelindaHaleyTeacher

LockleysNorthPrimarySchool,SA

MrMichaelHallAssistant Principal

StAndrew’sCollege,NSW

MrsLynHamilton DECS,SAMrsJulieHancock CatholicEducationOffice,SAMsJudithHankeManager, Secretariat

DEECD,VIC

11 MrsCynthiaHarborTeacher


4 MsChristineHardieTeam Leader

UnivofAuckland,FoEd,TeamSolutions,NZ

MissMarinaHardyAssistant Principal

MaryMacKillop,NSW

MrMattHardyTeacher

PaduaCollege,QLD

MsJoannaHarrissonTeacher

AustralindSeniorHighSchool,WA

14 MrBedeHartPrincipal

StAnne’sPrimarySchool,NSW

MsJanHarteCurriculum Adviser


12 MrDaveHartleyNumeracy Coach

MerrimacStateSchool,QLD

MsJodieHartmann ToorminaHighSchool,NSW5 MsJudyHartnett

LecturerQueenslandUniversityofTechnology,QLD

MsLibertyHatzidimitriouTeacher

LowtherHallAGS,VIC

MrsKerrinHazardNumeracy Project Officer

CSOBrokenBay,NSW


107


MsCarmelHealeyPrincipal

SacredHeartCatholicSchool,NSW

MsTracyHealyCoordinator

LowtherHallAGS,VIC

14 MrsChristineHeathHead of Middle School Mathematics

PembrokeSchool,SA

MsJayneHeathAssistant Principal

Aust.Science&MathSchool,SA

MissKarleyErinHefferanNumeracy Coach

DECS,SA

MsTracyHerft StrathconaBGGS,VIC7 MrsJenniHewett

Maths Facilitator/Numeracy CoordinatorDECS,SA

MsAnnHewittTeacher

GympieSHS,QLD

14 MrIanHilditchHead of Mathematics

PembrokeSchool,SA

MsJackyHiscockTeacher


3 MsKerry-AnneHoadDirector

ACERINSTITUTE,VIC

MrsGianninaHoffmanAssessor Trainer

SACEBoardofSA,SA

5 MrJohnHogan RedgumConsulting,WA

MrsBirgitHolleyTeacher

StuartholmeSchool,QLD

7 DrHilaryHollingsworthTeaching Fellow

ACERINSTITUTE,VIC

10 MrsJanetteHolmesQuality Teaching Consultant

DeptofEducation,NSW

11 MrsMaryHorAssistant Principal

StPhilipNeri,Northbridge,NSW

7 MsRhondaHornePrincipal Education Officer

DET,QLD

MrNicholasHoughtonTeacher

StAnthony’sPrimarySchool,NSW

MrRodneyHowardAssistant Principal

BedePoldingCollege,NSW

MrsRebeccaHuddyCurriculum Coordinator

DECS-WesternAdelaideRegion,SA

MrCameronHudsonHead of Mathematics

TheHutchinsSchool,TAS

MsJudithHuntNumeracy Coordinator

DECS,SA

MsJanetHunterTeacher

AschamSchool,NSW


108


MrsKylieHydeTeacher

HolyFamilyCatholicPrimary,NSW

9 MrMalcolmHylandManager

MinistryofEducation,NZ

MrsDiannHynesSchools Consultant


ProfLawrenceIngvarsonPrincipal Research Fellow

ACER,VIC

6 MrsBernadetteIrvinEducation Officer


MrsJaneIrvinHead of Department Mathematics

MorayfieldStateHighSchool,QLD

MissKimIrvineCoordinator

ItalianBilingualSchool,NSW

MrsTerryJackaHead of Faculty - Mathematics

StHilda’sSchools,QLD

MrsAnnJacksonExecutive Team


MsDeirdreJacksonDirector, Assessment Services

ACER,VIC

10 MrsLorraineJacobSenior Lecturer

MurdochUniversity,WA

MsKylieJagoTeacher

QueechyHighSchool,TAS

MsJacintaJamesTeacher

SimondsCatholicCollege,VIC

18 MissLaurenJamesHigh School Teacher

NorthSydneyGirlsHighSchool,NSW

11 MrsSherylJamiesonCoordinator

NuriootpaPrimarySchool,SA

MrMichaelJenningsLecturer

TheUniversityofQueensland,QLD

21 MrPaulJohansenHead of Department

StPaul’sSchool,QLD

MsJanetJohnsonTeacher

OceanViewCollege,SA

18 MrsNicoleJohnsonTeacher

PenrhosCollege,WA

MrKevinJonesPrincipal


MsMaureenJonesPrincipal

ChristtheKingPrimary,NSW

MissBriannaJordanNumeracy Coach

DECS,SA

1 ProfRobynJorgenson GriffithUniversity,QLDMsFranKaneAssistant Principal

OLRTheEntrance,NSW


109


MissPaulineKaszubowskiTeacher


MrsClareKavanaghHead of Mathematics

StPatrick’sCollege,VIC

MrsRobynKayDeputy Principal


10 MrAlexanderKeechClassroom Teacher

Dept.ofEducation,QLD

MsJoKellawayCoordinator

Aust.Science&MathSchool,SA

DennisKellyTeacher


MsMaryKellyAssistant Principal


MrPaulKellyHead of Mathematics

CatholicLadies’College,VIC

MrTimKellyEducation Officer

LismoreC.E.O.QLD

SrBrendaKennedyPrincipal


MsJenniferKerbyMaths Teacher

OurLadyofSionCollege,VIC

MissSuzanneKhatib McKinnonPrimarySchool,VICDrSiekToonKhooResearch Director

ACER,VIC

14 MrsDianeKibbleMathematics Co-ordinator

StCatherine’sCatholicCollege,NSW

MsKatherineKilburnTeacher

ShoreSchool,NSW

MissLindaKloedenTeacher

NorthHavenPrimarySchool,SA

MrsJacquiKlowssHOD Maths

MaristCollegeAshgrove,QLD

15 MsMargaretKnightAssistant Head of Primary

StColumbaCollege,SA

MrMichaelKnightTeacher

TerraSanctaCollege,NSW

MsPatKnightSenior Librarian

ACER,VIC

MrsRebeccaKnightNumeracy Coach

DECS,SA

MsCarolKnoxMaths Director

LindisfarneAnglicanGrammarSchool,NSW

MsKarenKnoxNumeracy Coach

DECS,SA

MrKimonKousparisMaths Coordinator

CasimirCatholicCollege,NSW

20 MsMiriamKrakovskaAcademic Teacher



110


MrAndreKristovskisTeacher

TheRiverinaAnglicanCollege,NSW

MsJanLadhamsMathematics Consultant


18 MrGregLadnerHead Maths

PresbyterianLadiesCollege,WA

MsAnniLahdesluomaRetired Teacher

18 MsTaniaLambleTeacher


16 MrsSiobhanLanskeyTeacher

BundabergS.H.S.,QLD

MsFelisaLapuzHead of Mathematics

MarianCollege,VIC

MrsJennyLawrenceTeacher


MrsMaryLeaskPrincipal

NagleCollege,NSW

3 DrDavidLeigh-LancasterCurriculum Manager Mathematics

VictorianCurriculum&AssessAuthority,VIC

MsElisabethLendersDeputy Principal


17 MsDianneLeyTeacher

GilroyCatholicCollege,NSW

MrJohnLeyActing Principal

XavierCollege,NSW

MrCameronLievorePrincipal

OurLadyoftheNativitySchool,NSW

MrsDeborahLillyTeacher

LowtherHallAGS,VIC

MrJulianLindsayHead of Department - Mathematics

RuncornStateHighSchool,QLD

MrsHeatherLinesHead of Mathematics

WestminsterSchool,SA

MissCharlotteLipnickiYear 2 Teacher


MrsJeanetteLittleHead of Mathematics

LoretoCollege,QLD

MrsCaroleLiveseyEducation Officer

CatholicEducationOffice,VIC

MrsSharynLivyProfessional Officer

MAV,VIC

MsShayneLlandaTeacher

StMonica’sCollege,VIC

MrPeterLorenti ReservoirDistrictSec.College,VICDrIanLoweProfessional Officer

MAV,VIC

2 ProfTomLowrie CharlesSturtUniversity,NSW11 MsDonnaLudvigsen

Network ImprovementGrampiansDEECD,VIC


111


MrChrisLynaghTeacher

StLuke’sAnglicanSchool,QLD

MrsCarolLynchTeacher


MrDesLyristisMaths Department

HuntingTowerSchool,VIC

MrsAnnMacMillanCoordinator Maths For Learning inclusion

DECS,SA

MrMichaelMacNeillLearning Development

St.JosephsCollege,VIC

MsRobynMacready-BryanHead of Maths/IT-Senior School


MsChristineMaeCoordinator

StAloysius’Primary,NSW

DrBryanMaherAssistant Principal

StJoseph’sHighSchool,NSW

11 MissDanielleMahonyTeacher


8 MrChicriMaksoudCoordinator Mathematics

BrisbaneBoys’College,QLD

MrChrisMalbergAssistant Principal


MsNitaMaloneyNumeracy Coach

DECS,SA

MissAmandaMamoTeacher

DomremyCollege,NSW

18 MissAliceManningTeacher

PenrhosCollege,WA

MrsKatrinaMansfieldTeacher

CraigsleaSeniorHighSchool,QLD

MrPaulMansfieldHead of Curriculum - Mathematics

PaduaCollege,QLD

21 MrGarethMansonClassroom Teacher

ABPatersonCollege,QLD

MsJuvyMarcellanoMaths Teacher

NagleCollege,NSW

4 MrsAnneMartinMaths Teacher

GeelongGrammarSchool,VIC

MrDavidMartinTeacher

StPeter’sCollege,SA

1 ProfGeoffMastersCEO

ACER,VIC

MsStamatikiMatheosTeacher

NorthHavenPrimarySchool,SA

MsCatherineMathewsTeacher


MrLukasMatysekDean

CedarsChristianCollege,NSW

MrRichardMaynardProgram Manager


MrsCarolineMazurkiewiczTeaching and Learning Coach WMR

DEECD,VIC


112


19 MsFionaMcAlisterMathematics Teacher

AquinasCollege,WA

MsCaraMcCarthyProject Officer

DEECD,VIC

MrsSheilaMcCarthyMaths Specialist

NorlaneWestPrimarySchool,VIC

16 MsMargaretMcCaskieTeacher

DaramalanCollege,ACT

11 MrTerenceMcClellandHead of Department

MareebaStateHighSchool,QLD

MsCatherineMcCluskeyNumeracy Consultant

CatholicEducationOffice,SA

2 DrBarryMcCraePrincipal Research Fellow

ACER,VIC

MsKimMcDonaldAssistant Principal


MsMicheleMcDonaldTeaching & Learning Devl. Consultant


MrsYvonneMcGarryTeacher

CanberraGirls’Grammar,ACT

MsBernadetteMcGillMaths Domain Leader

OurLadyoftheSacredHeartCollege,VIC

MrsPatriciaMcGregorTeacher

StPaul’sManly,NSW

MrsKimMcHughNumeracy Consultant


MrJesseMcInnesTeacher

WesleyCollege,VIC

17 MsNarelleMcKay JamisonHighSchool,NSW

MrsJenniferMcKeownPrincipal

StThomasSchool,NSW

MsNicolaMcKinnonResearch Fellow

ACER,VIC

MrsEllenMcLaganTeacher

OurLadyofLourdesCatholicSchool,TAS

MrsLorraineMcLarenMaths Coordinator

ReservoirDistrictSec.College,VIC

12 MsJillianMcNamaraTeacher


MrColinMcNeilPublisher

MacmillanEducationAustralia,VIC

MrsFrancesMcPheeAssistant Principal

CaulfieldJuniorCampus,VIC

MsVivienneMcQuadeCurriculum Manager

DECS,SA

19 MrPeterMeeHead of Mathematics

MercedesCollege,WA

MrsAnitaMeehanAdministration Coordinator



113


MrsMargaretMeehanTeacher

MaryMacKillop,NSW

MsJennyMeibuschTeacher

CanberraGirls’Grammar,ACT

8 MrsMarionMeiersSenior Research Fellow

ACER,VIC

MrsSilvaMekerdichianMathematics Teacher

CovenantChristianSchool,NSW

MrPaulMendayHead of School Services


6 MrsCareyMenz-DowlingEducation Officer


7 MrsJennyMerrettHead of Mathematics

YarraValleyGrammar,VIC

MrsChrisMiethkeMaths & Science Facilitator

DECS,SA

MrChristopherMillsHead Teacher

RichmondRiverHighSchool,NSW

5 MrsDianneMillsPartnership Broker

SchoolsIndustryPartnership,NSW

MrsLeonieMitchellTeacher

MaryMacKillop,NSW

MrBrettMolloyManager

QldStudiesAuthority,QLD

9 MrNickMoloneyLearning Coordinator

MarcellinCollege,VIC

MsSamanthaMonteiroSenior Teacher

EducationQLD

9 MrDavidMoranTeacher

MarcellinCollege,VIC

3 MrWillMoronyExecutive Officer

Aust.Assoc.ofMathematicsTeachers,SA

MrsCatyMorrisNational Manger: Indigenous Programs

AustAssocofMathematicTeachers,SA

MrAndrewMorrisonMaths Leader

MossfielPrimarySchool,VIC

19 MrRodneyMorrisonAssistant Head of Mathematics

AquinasCollege,WA

19 MrsSallyMorseMaths Domain Leader

BelmontHighSchool,VIC

MsRachaelMoweTeacher

QueenwoodSchoolforGirls,NSW

3 AssocProfJoanneMulligan CRiMSEMacquarieUniversity,NSWMsKerryMulvogue OurLadyofMercyCollege,VICMsCatherineMurrayEducation Officer


MrsVanessaMurrayTeacher

HolyFamilyCatholicPrimarySchool,NSW

MrBruceMurrieTeacher

DECS,SA


114


MrRobertMuscatelloEducation Officer


20 MrsAnneMyhillAssistant Head of Mathematics

WilliamCareyChristianSchool,NSW

MrsDebraNeedhamAssistant Hola

CorpusChristiCollege,WA

4 MrAlexNeillResearcher

NZCER,NZ

MrMichaelNekvapilTeacher

OranaSteinerSchool,ACT

MrMarkNewhouseManager of Curriculum

AssociationofIndependentSchoolsofWA

9 MrsKathyNolanProject Officer Maths


MsOliviaNorrisNumeracy Coordinator

StJerome’sPrimary,NSW

MsRosalieNottAssistant Director

CatholicEducationCommission,NSW

MrsDebbieOatesMaths Coordinator

SydneyGrammarSchool,NSW

8 MsGaylO’ConnorAssessment Advisor

EducationServicesAustralia,VIC

10 MsLisa-JaneO’ConnorEducational Consultant

PrimaryMathematicsAssociation,SA

MrsWendyOgilvieNumeracy Coach

DECS,SA

MrMichaelO’HalloranKLAC

AquinasCollege,VIC

MsDelwynOliverHead of Maths

BallaratHighSchool,VIC

18 MrsJenniferOlmaMathematics Co-ordinator

PerthCollege,WA

MsPatriciaOlsenTeacher

ChisholmInstitute,VIC

16 MrsSharonOlsenTeacher

BundabergS.H.S.,QLD

MsJoanneO’MalleyActing Assistant Principal

StKildaPrimarySchool,VIC

MrFrankO’MaraTeacher

DownlandsCollege,QLD

MsEffieOrlandoAssistant Maths Coordinator

MaryMacKillopCollege,NSW

17 MrsCarolOsborneHead of Mathematics

LoretoNormanhurst,NSW

MrPeterOslandMaths Inspector

BoardofStudies,NSW

MrsYvetteOwensAssistant Principal

StJohntheBaptistCatholicPrimary,NSW

MissAttiliaPaganoTeacher Educator



115


13 MrChrisPageMaths Teacher

Marist,Eastwood,NSW

MrMichaelPalmeHead of Mathematics

BrigidineCollege,NSW

MrsDeborahPalmer CEO,InnerWesternRegion,NSWMrsKathrynPalmerRegional Coach

WesternMetro.Region,VIC

14 MsKaterinaPapetrosMaths Teacher

SeymourCollege,SA

21 MrsLarraParonMathematics Teacher


MrsHeatherParringtonSenior Curriculum Coordinator

SACEBoardofSA,SA

MsSheilaParsonsTeacher


20 DrAnnePatersonTeacher

WesleyCollege,WA

MrsCarolPattersonHead of Mathematics

Haileybury,VIC

MrJacobPearceResearch Officer

ACER,VIC

7 MrsCathPearnTeaching Fellow

ACERINSTITUTE,VIC

MsMelindaPearsonProject Officer

AustralianAssocofMathTeachers,SA

MrsSuzannePearsonSenior Curriculum Officer

DET,WA

MrRayPeckSenior Research Fellow

ACER,VIC

2 ProfJohnPegg UniversityofNewEngland,NSWMrGeoffPellPrincipal


MsTeresaPelusoMaths Coordinator

CheltenhamSecondaryCollege,VIC

MrBrettPerkinsClassroom Teacher

St.Cecilia’sCatholicSchool,NSW

MsMichellePerryAssistant Principal

StPatrick’sCatholicPrimarySchool,NSW

MrGregoryPetherickAssistant Regional Director

DECS-WesternAdelaideRegion,SA

MrJoemonPhilipCoordinator

MountAnnanChristianCollege,NSW

MrRayPhilpotResearch Fellow

ACER,VIC

MsSuePickupCoordinator

OurLadyofMtCarmel,NSW

MrsSamanthaPinkertonTeacher

GuilfordYoungCollege,TAS

MsMeredithPlaistedHead of Maths/IT Senior School



116


MrsPaulinePollockTeacher

StThomasCatholicSchool,NSW

MrsKarenPostNumeracy Coach

DECS,SA

MsMaureenPricePrincipal

MossfielPrimarySchool,VIC

MrRobProffitt-WhiteNumeracy Coach


MsSusanneProsenicaTeacher

CopperfieldCollege,VIC

MsYiannaPullenAssistant Principal


MsRobynPurcellMaths Coordinator

MaristSister’sCollege,NSW

4 MrBrendanPyeProject Officer

ACERINSTITUTE,VIC

MsMaryQuillMathematics Coordinator

HolySpiritCollege,NSW

MrsKylieQuinTeacher


MsMaryQuinanePrimary Numeracy Officer

CatholicEducationOffice,ACT

MrJeremyRackhamTeacher


MsJaneRalston-PalmerSenior Teacher


MsChristineRatcliffNumeracy Coach

DECS,SA

MsDympnaReaveyLeader of Teaching & Learning

NagleCollege,NSW

MrMarkRedingtonTeacher


13 MrMaxRedmayneMaths Coordinator

Marist,Eastwood,NSW

3 AssocProfRobertReeve TheUniversityofMelbourne,VIC10 MissDeborahReeves

Numeracy AdviserWaikatoUniversity,NZ

6 MsGlenysReidPrincipal Consultant

DepartmentofEducation,WA

MrsJennyRendallPrincipal


MsAnnaRerakisProject Officer

DEECD,VIC

MrsFrancesReynoldsSchools Consultant


MsLouiseReynoldsCorporate Publicity & Comm. Manager

ACER,VIC

MsMaryReynoldsNumeracy Leader

ElthamCollegeofEducation,VIC


117


MrsPenelopeReynoldsCurriculum Officer - Numeracy


MsElisabethRhodesDeputy Principal

LowtherHallAGS,VIC

MrJoshuaRichmondMaths Teacher

BallaratGrammar,VIC

MsJoanneRiddellMathematics Adviser (Primary)


MrsJanetRidleyTeacher

LandsdalePrimarySchool,WA

MrPaulRijkenPrincipal

CardijnCollege,SA

MsNicoleRilesHead of Mathematics

StLaurence’sCollege,QLD

MsSueRiquelmeCoordinator

LowtherHallAGS,VIC

MissKarenRobertsLead Teacher

SandringhamEastPrimarySchool,VIC

12 MsTrishRobertsSupport Teacher


MrAndrewRobertsonHead Faculty

KingswoodCollege,VIC

8 MsLeanneRobertsonSenior Project Manager

EducationServicesAustralia,VIC

MrGregRobinsonProject Officer


MsKarenRobsonAssistant Principal

StPeter’sPrimarySchool,NSW

MrsKathleenRoffeyMathematics Coordinator

TrinityCatholicCollege,NSW

5 MsHonorRonowiczNumeracy Adviser


MrsSarahRosenwegHead of Faculty

ShelfordGirls’Grammar,VIC

4 MsLyndaRosmanManager Programs and Projects

ACERINSTITUTE,VIC

MrsJenniferRowlandNumeracy Coach

DECS,SA

MrPeterRundleHead of Mathematics

BarkerCollege,NSW

MrsIreneRuscignoNumeracy Coordinator

EppingViewsPrimarySchool,VIC

12 MrBradleyRyallMaths Coordinator

StJohn’sCollege,NSW

MsSophieRyanHead of School Service


MrsNicoleSadlerYear 6 Teacher


MrJohnSagnerHead of Department Mathematics

BrownsPlainsHighSchool,QLD


118


19 MrDariusSamojlowiczHead of Stage Two

TheHillsGrammarSchool,NSW

MrJaredSandersTeacher

CanterburyPrimarySchool,VIC

MrPeterSandersLecturer

LaTrobeUniversity,VIC

MrsSusanSandersHead of Maths

OurLadyofMercyCollege,VIC

MissAliciaSandersanTeacher


MsEmilySangsterActing Manager

QueenslandStudiesAuthority,QLD

MrsRosaSantopietroMaths Coordinator

OurLadyoftheSacredHeartCollege,SA

6 MrRalphSaubernGeneral Manager, Schools Program

ACER,VIC

MrKeatSawTeacher

AustralindSeniorHighSchool,WA

MrsFionaScannellHOD

PalmBeachHighSchool,QLD

MrsRonelleScheepersLearning Coordinator - Maths

StTeresa’sCollege,QLD

11 MrBruceSchmidtProject Officer

GrampiansDEECD,VIC

MsCathyScottPrincipal

ChisholmCatholicPrimarySchool,NSW

MrsLyndaSecombeAdviser

Assoc.ofIndependentSchoolsofSA

MsJudithSelbyHT Mathematics

CowraHighSchool,NSW

MrsEmmaSellarsCoordinator


MrMarkSellenHOD

ShoreSchool,NSW

MrsYvetteSemlerTeacher


MrsKatherineSerbinMaths Teacher

NagleCollege,NSW

21 MrFerruccioServelloMathematics Teacher


12 MsMichelleSextonTeacher


MrBarryShanleyPrincipal

StJohnFisherSchool,NSW

MsLindaShardlowHead of Mathematics

MethodistLadiesCollege,VIC

16 MrsAmyShawTeacher

BunburyCathedralGrammar,WA

MrsMargaretSheahanCoordinator

StOliver’sPrimarySchool,NSW


119


12 MrJamesSheedyPrincipal


MrsDebraSheehanTeacher


MrsKylieSheltonTeacher

BerserkerStreetStateSchool,QLD

MsDebraShephardTeacher

KillaraPrimarySchool,VIC

MrIanSheppardHead of Mathematics

WesleyCollege,WA

MrsJoyShortHead of School Service

CatholicEducationOffice,Parramatta,NSW

MissJodieSibbaldTeacher/Leadership Team


MrMichaelSicilianoAssistant Principal

StMichael’sPrimarySchool,NSW

MrsWendySilvestriNumeracy Coach

DECS,SA

MissVanessaSimieleTeacher


MissMeganSkinnerMaths Specialist


MissAmySkuthorpPrep Teacher


MsChristineSlatteryConsultant

CEO,SA

MrsJudySlatteryPrincipal

StJohntheBaptist,NSW

20 MrRoySmalleyTeacher

ChisholmInstitute,VIC

6 MrsBarbaraSmithSales Manager

ACER,VIC

MsCatherineSmithMaths Teacher

MaristSister’sCollege,NSW

4 MsDeniseSmithTeam Leader

UnivofAuckland,FoEd,TeamSolutions,NZ

21 MrGlenSmithHead of Studies, Senior School

StPaul’sSchool,QLD

MsJacquiSmithNumeracy Co-ordinator

WesternPortSecondaryCollege,VIC

16 MsJulieSmithTeacher

BunburyCathedralGrammar,WA

6 MrsMichelleSmithTeacher/Leadership Team


MissMichelleSmithSchools Consultant


8 MrVaughanSmithHead of Research

CaulfieldGrammarSchool,VIC

MsGabriellaSpadaroSpecial Needs Teacher

MarymountInternationalSchool,ITALY


120


4 MrKenSpanksTeacher

GinGinStateHighSchool,QLD

MrsSusanSpencerSpecial Education Consultant

SpencerEducation,VIC

MissDominiqueSpindlerExhibitions Administrator

Routledge,UK

18 MrPeterSprentTeacher


MrsLoisStaatzPrincipal

GattonStateSchool,QLD

1 ProfKayeStacey TheUniversityofMelbourne,VICMissEllieStanfordTeacher

AschamSchool,NSW

MrMitchellStaplesTeacher

CanterburyCollege,QLD

MissLizStarlingExecutive Team


MrDavidSteeleDept. Head of Campus

WesleyCollege,VIC

MrGregSteeleMaths Specialist

NorlaneWestPrimarySchool,VIC

MsMarieStenningTeacher


DrAndrewStephanouSenior Research Fellow

ACER,VIC

MrsRobynStephensMaths Coach

CroydonPrimarySchool,VIC

21 MrDavidStephensonMaths Coordinator

GrovedaleCollege,VIC

16 MrMichaelStjepcevicHOD Junior Maths

IpswichGrammarSchool,QLD

12 MsMelindaStockwellTeacher

TrinityAnglicanCollege,QLD

MrPeterStoneMaths Coordinator

UnleyHighSchool,SA

MrMaxStoweMaths Teacher

BallaratGrammarSchool,VIC

MrDirkStrasserPublishing Manager

PearsonAustralia,VIC

MrsSusannahStredwickLearning Enrichment Coordinator

RavenswoodSchoolforGirls,NSW

2 ProfPeterSullivan MonashUniversity,VICMrsMicheleSunnucksAssistant Principal

OLMCPrimarySchool,NSW

MsNancySuraceTeacher

MacKillopCollege,VIC

MsJennySuttonClassroom Teacher


MissCarlaSweetingTeacher

AschamSchool,NSW


121


MsCarmelTapleyEducation Officer


MrsBernadetteTaylorClassroom Teacher


MsChristineTaylorInspector, Primary

BoardofStudies,NSW

MsDebbieTaylorTeacher


9 MsMargaretTaylorAdministration Officer

ACERINSTITUTE,VIC

5 MrsGaynorTerrillNumeracy Adviser


18 MrKenTerryTeacher


20 MrGregThackerayHead Teacher

WilliamCareyChristianSchool,NSW

MrGregThomasDeputy Principal

StMartinofToursPrimarySchool,VIC

MsJulieThompsonAssistant Principal

StFrancisXavier’sPrimary,NSW

MrLincolnThompsonTeacher


2 DrSueThomsonPrincipal Research Fellow

ACER,VIC

4 MrGregoryThruppTeacher

GinGinStateHighSchool,QLD

MrGregoryTierTeacher

BrisbaneGrammarSchool,QLD

MrsLornaTobinAssistant Principal

StJohntheApostlePrimarySchool,NSW

MissMelissaTomaszewskiTeacher Educator


MsDianneTomazosPrincipal Curriculum Officer - Numeracy


MrLeighToomeyTeacher

AquinasCollege,VIC

MrPaulToomeyPrincipal


9 MrGeorgeTothSenior Project Officer Maths


MsDianeTouzellTeacher


16 MrIanTranentTeacher

BundabergS.H.S.,QLD

13 MrsTanyaTraversR.E.C.


7 MrBruceTrenerryPrincipal Education Officer

DET,QLD

MsJenniferTrevittLibrary Information Dissemination

ACER,VIC


122


MsLiesaTrinderMaths Coordinator

MaryMacKillopCollege,NSW

13 MrsGailTullSpecial Needs


MrGeoffTunnecliffeMaths Teacher

ElthamHighSchool,VIC

MsSusanTurnbullHead of Mathematics

AustralianIntern.School,HONGKONG

MrMarkTurkingtonRegional Director


16 MrsAnnMarieTurnerHOD Senior Maths

IpswichGrammarSchool,QLD

MrJohnTurnerTeacher

KingstonStateSchool,QLD

6 MrKevinTurnerPrincipal

OLHCPrimarySchool,NSW

1 MrRossTurnerPrincipal Research Fellow

ACER,VIC

MsStaceyVanderVeldersNumeracy Coordinator


MsAnjavanHooydonkTeacher

StMary’sCatholicCollege,QLD

MsChristineVanRyswykAssistant Mathematics Coordinator

CatherineMcAuley,Westmead,NSW

18 MissNikkyVanderhoutHead Teacher


MrsSapnaVatsTeacher


MsJackieVellaMathematics Adviser (Primary)


MsRosemaryVellarHead Educational Measurement

CEOSydney,NSW

17 MrsVeronicaVerdiTeacher

GilroyCatholicCollege,NSW

MrsJosieVescioPrincipal

StRoseCatholicSchool,NSW

MrsLorraineVincentAssistant Principal

OLGCCatholicSchool,NSW

MissJacintaVistoliYear 7 Teacher

LoretoCollege,SA

MsCatherineVolpeTeacher

KewHighSchool,VIC

9 MrPaulWaddellDirector & General Manager

Mathematics.com.auNSW

MrGregWagnerTeacher

MoriahCollege,NSW

MrJeffWaitPrincipal

CraigburnPrimarySchool,SA

MrsAngelaWaiteTeacher

LoretoCollege,QLD


123


MsJulieWalkerMathematics Coordinator

LoretoKirribilli,NSW

MrDougWallaceCurriculum Coordinator

WesleyCollege,VIC

MrsSueWalpoleHead of House


13 MrsLynWalshTeacher


MrsSueWalshHead of School Service


MrGraemeWaltersHead of Mathematics

KinrossWolaroiSchool,NSW

MrsRenaeWanTeacher

ItalianBilingualSchool,NSW

13 MsKerriWardGrade 5 Teacher


MrsRosemaryWardTeacher

XavierCollege,VIC

MsDoraWarlondProject Officer

DEECD,VIC

17 MrPaulWatersHOD Maths

MackayNorthSHS,QLD

13 MrRobertWattTeacher


5 MrJohnWattersExecutive Officer

SchoolsIndustryPartnership,NSW

8 DrJenniferWayAssociate Dean Undergraduate Programs

TheUniversityofSydney,NSW

MrsNeridaWayNumeracy Coach

HelensvaleStateSchool,QLD

MsMignonWeckertRegional Manager

Intern.BaccalaureateOrg.,SINGAPORE

MrRobertWellhamHead Teacher Mathematics

ErinaHighSchool,NSW

MrPieterWepenerMaths Teacher

CitipointeChristianCollege,QLD

MsHelenWestonInstructional Coach

SandringhamEastPrimarySchool,VIC

19 MrsValerieWestwellMaths Teacher (yrs 3 to 7)

BridgewaterPrimarySchool,SA

MrJonathanWeverFaculty Head

MentoneGrammar,VIC

14 MrGlenWhiffenAsst. Head of Mathematics

PembrokeSchool,SA

5 DrSoniaWhiteLecturer

QldUniversityofTechnology,QLD

MsAnnWhitmoreTeacher

Marist,Burnie,TAS

17 MrsRosemaryWiffenHead of Department - Mathematics

KingstonCollege,QLD


124


20 MsDamithWijeratneAcademic Teacher


17 MrsKylieWilesTeacher

LoretoNormanhurst,NSW

MrsDesleyWilliamsHOD - Mathematics

TaraAnglicanSchool,NSW

14 MrJohnWilliams StAloysius’College,NSWMrKevinWilliamsPrincipal


MrDavidWillmottMaths Coordinator

LaSalleCatholicCollegeBankstown,NSW

MsRobinWillmottTeacher


15 MsDebraWilsonHead of Mathematics

RosevilleCollege,NSW

MrsJennyWoodTeacher


MrsKathrynWoodTeacher


MsRosemaryWoodTeacher

BalaklavaPrimarySchool,SA

MrsJenniferWoodsTeacher

MLCSchool,NSW

MrPeterWoolfeMathematics Coordinator

WaverleyChristianCollege,VIC

MrGeoffWrightMaths Coordinator

DeLaSalleCollegeCronulla,NSW

MrLachieWrightHead of Junior School

ScotchOakburnCollege,TAS

MrsNoeleneWrightAssistant Principal

LindisfarneAnglicanGrammarSchool,NSW

MrJasonYatesNumeracy Coordinator

DECS,SA

MsJoannaYaxleyHead of Learning (Maths & IT)

JohnPaulCollege,QLD

21 MrAlecYoungCEO

ITE,TAS

7 MsCamilleYoungTeacher

TrinityCollegeSenior,SA

MrJohnYoungPrincipal

ClarksonCommunityHighSchool,WA

MrsMirellaZalakosTeacher


MrGregoryZerounianMaths Coordinator

StLeo’sCollege,NSW

MsStavroulaZoumboulisResearch Fellow

ACER,VIC

MrsDihnaZuvelaTeacher-in Charge for Mathematics

EducationDepartment,WA

767delegateslistedasofFriday,16July2010.

acer research conference proceedings (2010)

Documents