Usingideasof'resourcesystem'toanalysetheworkofmathematicsteaching

Kenneth Ruthven University of Cambridge, UK

Basicideasof‘resource’and‘system’

•  Inestablishedeverydayusage,aresourceisanasset–typicallyamonetary,materialorhumanasset–thatiscapableofprovidingsomeformofsupport.

•  Thecentralideaofa'system'isoneoforganisation:–  Somestructureresultingfrommultipleentitiesbeingorganisedtoformafunctioningwhole;

–  Someschemeormethodwhichprovidesabasisforthistypeoforganisation.

•  Traditionallyeducationhasmadeuseoftwoideas/typesof'resourcesystem’tosupportteachingandlearning:–  Thetextbook–asystematiccurriculumschemecombiningmanyunitsofresourceintoacoherentprogramme;

–  Thelibrary–arepositoryofresourcesorganisedsystematicallytomakeitscontentsreadilysearchableandusable.

Euclid'sElementsasasystematiclogicalorganisationofresources

•  TheElementswasacrucialtextinthedevelopmentofWesternmathematics.

•  ItwascreatedbyEuclidaround300BCEandwidelystudieduntiltheearlyyearsofthetwentiethcentury.

•  ThankstotheremarkableLibraryofAlexandria,EuclidwasabletodrawondisparatemathematicaltextsfromacrosstheancientworldincompilingtheElements.

•  WhatmakestheElementssoremarkable?•  First,Euclidcombinedandadaptedmathematicalmaterial

fromanextensiverangeofexisting(re)sourcestoproduceacomprehensivetextpresentingacoreofclassicalmathematics.

•  Foremost,however,itestablishedasystematicapproachtoorganisingthismaterialinalogicallycoherentway.

GlobalandlocalstructureintheElements

•  Aglobalstructureforthetextisestablishedasfollows:–  Itstartswith'definitions’ofentitiesandaxioms,takentobeself-evident,eitherintheformof'postulates'aboutgeometricalentitiesor'commonnotions'aboutmagnitudes.

–  Fromthese,itderivesasuccessionof'propositions’whicharenumberedsequentiallyandgroupedthematicallyinto'books’.

•  Aconsistentlocaltemplateisusedtotreateachproposition:–  An'enunciation'statesthesituationgivenandtheresultsought.–  Ifrequired,a'setting-out’providesalabelledfigureexemplifyingthesituation;anda'construction'or'mechanism'elaboratesthefiguretosupportreasoningtoproducethedesiredresult.

–  Astep-by-step'proof’showsthewarrantforeachstepbyindexingtherelevantdefinition,axiomorpriorproposition.

–  A'conclusion'relatesthedemonstrationbacktotheoriginalenunciation.

Euclid'sElementsasatextforstudy

•  Asatextforstudy,theElementsprovidesaccessnotjusttomathematicalcontentbuttothelogicalmethodemployedtoorganiseit.

•  Inthefieldofmathematics,theElementscametofulfilanimportantsociocognitivefunction,establishingasharedargumentativeframeworkforthediffusionofknowledge.

•  Inthefieldofeducation,theElementswasembracedbyaninfluentialapproachwhichsoughttoexposestudentstotheclassicalmodelsofthoughtdisplayedin'greatbooks'.

•  Indeed,theElementscametobestudiedprimarilyforthehabitsofmindthatsuchstudywasthoughttoinculcate.

•  InductiontotheEuclideansystemthrough‘exposuretotheElements’wasintendedtoteachstudentstoreasoninanabstractrealmremovedfromsensoryperception.

From‘greatbook’to‘schoolbook’

•  Yetactualeducationalpracticecouldbeverydifferent.•  Forexample,inEnglandintheearly20thcentury,reformers

criticisedoneoftherequirementstograduateatOxfordUniversity:tomemorisetwobooksofEuclid,evendowntotheletteringoffigures,withnooriginalexercisesrequired.

•  SuchexamplesshowhowtheElementshadoftenbecomeassociatedwithareductivemnemonicpedagogy.

•  Asreformersgainedtheupperhand,then,the'greatbook'gavewaytothe'schoolbook’.

•  Textswrittenspecificallytointroduceschoolpupilstogeometrygaveaplacetopracticalexperimentationandtookalessrestrictiveapproachtomodesofreasoning–inlinewiththedidacticalpreceptsofthereformmovement.

Durell'sANewGeometryasasystematicdidacticalorganisationofresources

•  Tocharacterisethe'schoolbooks'whichtooktheeducationalplaceofEuclid's'greatbook',IwillusetheexampleofDurell'sANewGeometryforSchools.

•  InhisreviewofEnglishmathematicstextbooksofthe20thcentury,QuadlingdescribesDurellas"themostprolificauthorofthecentury”sothathis"namewasformanypupilsalmostsynonymouswithmathematics”.

•  Intheprefacetothetext,Durellsetsouthisdidacticallyinspiredorganisationalschemeforthesystematicsequencingofactivitywithineachtopicalunitofthetext.

•  Thisschemeinvolvesfourstages,witheachstagelinkedtoparticulartypesoflearninggoalandacorrespondingformofclassroomactivity.

Durell’s4-stagedidacticalscheme

•  Theearlierstagesprovideaninformalintroduction:–  Stage(i)Examplesfororaldiscussiontogivepupilsaclearunderstandingoffacts,familiarisethemwithargumentsusedlaterinformalproofs,andtraintheminmethodsforsolvingrelatedproblems(‘riders’).

–  Stage(ii)Anexerciseofnumericalexamplestogivepracticeinapplyingfactsandensureafirmgraspofthem.

•  Thelaterstagesdevelopamoreformaltreatment:–  Stage(iii)Formalproofsofthecorrespondingtheorems,focusingonthekeytheoremineachgroup,togivepracticeinwritingoutproofs.

–  Stage(iv)Anexerciseofriders,inwhichearlyexamplesaresimpleapplicationsandlaterexamplesincludeestablishingrelatedtheoremsinthegroup.

Durell’sGeometryasatextforstudy

•  Consistentuseofthisschemethroughoutthetextbookaccustomsteacherandpupilstothestagedapproach,allowingthemtofocusonthemathematicaltasksandlearninggoalsinplay.

•  ThethirdstageprovidesadegreeofcontinuitywiththeapproachoftheElements,butismoreeconomicalinthetheoremschosentoreceiveformalproof.

•  Organisingmaterialconceptuallyaroundakeytheoremincorporatesapowerfullearningprinciple.

•  Equallythetwofinalstagesproduceabalancebetweenexposuretoformalproofandexperienceofsolvingriders(includingthoserelatingtoothertheoremsinthegroup).

Durell'sANewGeometryasasystematicdidacticalorganisationofresources•  ThecoreofANewGeometryisthesequenceoftopic-specific

resourceunitsformingtheindividualchaptersofthebook,eachemployingthestagedorganisationjustoutlined.

•  Thiscorrespondstothefirstsenseofresourcesystemasasystematiccurriculumsequencecombiningresourcesintoacoherentprogramme.

•  Ancillaryfeaturesofthetext–indexesandappendices-correspondtothesecondsenseofresourcesystemasarepositoryofreadilysearchableandusableresources.

•  Suchatextisdesignedasacompact,comprehensiveall-in-oneresourcesystem,meetingthevariousneedsofteachersandpupilsoverthecourseasawhole.

•  ItisthisexplicitandsystematicdidacticalorganisationwhichmakesDurell'stextidentifiablyatextbook.

Beyondthe‘teacher-proof’text

•  Wehaveseenhowboththesetextsweredesignedsoastoprovideaparticulartypeoforganisedresourcesystem.

•  Butwehavealsoseenhowtheusersofsuchatextmayfailtohonourtheorganisingsystem,asillustratedbythesometimesdebasedtreatmentoftheElements.

•  Equally,amongstmorethan1000editionsoftheElements,manysoughtto‘improve’itsorganisingsystem,oftenwithdidacticalandpedagogicalconsiderationsinmind.

•  Indeed,beyondacertainpoint,whilestillclearlyindebtedtotheElements,reformedtextsmadesuchfundamentalchangestotheorganisingsystemastoclaimdistinctidentities,asDurell’sGeometryillustrates.

•  Equally,recognisingthatteachersoftenseekatextalignedwiththeirpreferredteachingapproach,Durellproducedseveralothergeometrytexts,organisedinalternativeways.

Summaryofpart1–thecomprehensiveresourcesystem

•  Ineverydaylanguage,a'resource'issomeformofsupportiveasset.

•  Likewisea'system'issomeformofmethodicalorganisation.•  Viewedinsuchterms,textssuchasEuclid'sElements(orThe

NineChaptersontheMathematicalArt九章算术)canbeseenasaformofmathematical'resourcesystem’.

•  Thesefoundationalmathematicaltextsadaptedandorganisedunitsofmaterialfrommanysources.

•  LatertheirplacewastakenbyschooltextbookssuchasDurell'sGeometry.

•  Throughanoverarchingdidacticalorganisation,textbooksaimtoprovideacomprehensivecurricular'resourcesystem'adaptedtothevariousneedsofteachersandstudents.

Thegenesisofthecurricular'resource'

•  Inthefieldofeducation,aspecialisedideaof'resource'cameintouseduringthe1960s,referringspecificallytomaterialssupportingcurricularactivity.

•  Theemergenceofthiseraofcurricular'resources’wasfosteredbytechnologicalchanges–notablytheincreasingavailabilityofaudio-visualandreprographicfacilities.

•  Thesechangesbroadenedtherangeofmediainwhichcurricularmaterialscouldbecreated,andfacilitatedtheirlocalproductionandreproduction.

•  Morerecentlyfurthertechnologicalchangeshaveenabledashifttowardsdigitalisingresourcesandaccessingthemonline.

•  Thisshifthasfacilitatedtheexchangeofcurricularmaterialsonamuchwiderscale,accompaniedbythedevelopmentoforganisationsaimingtofostersuchexchanges.

‘Resource-based’learningandteaching

•  InWesterncountries,therehasbeenincreasinginterestin'resource-based'formsoflearningandteaching,involving,respectively,moreindependentstudybypupilsandmorelocalisedcurriculumdesignbyteachers.

•  Theinstitutionalisationofsuchtrendshasbeenmarkedbytherenamingofthetraditionallibraryasthemodern'resourcecentre’.

•  Assuchapproacheshavebecomeincreasinglyinfluential,therehasbeenashiftawayfromusingacomprehensivecoursetextbooktowardscombiningmanysmallerresourceunitsfromdifferentsources.

•  Nevertheless,someskeleton'resourcesystem’remainsnecessary,providingaframeworkfororganisingresourcessoastoprovideacomprehensiveandcoherentcurriculum.

SMILEasasystematiccurricularorganisationofmulti-sourcedresources

•  Akeyfeatureofearlyresource-basedinitiativesinschoolmathematics,suchasSMILEinEngland,wasdevelopmentofacurriculummapintowhichcarefullychosen(orspeciallydevised)‘tasks’fromdifferentsourcescouldbeinserted.

•  InSMILE,thebasisforthismapwasagridcombiningthedimensionsoftopicareaandofconceptualleveltocreateasystemofcellsintowhicheachtaskresourcewasmapped.

•  Withinthisgrid,furtherdependenciesbetweencellsandbetweentaskscouldthenberecordedingreaterdetail.

•  Teacherandstudentusedthisgridtoplan,takingaccountoftheattainmentofthestudentandtasksalreadyundertaken.

•  Guidedbythegrid,theychoseappropriatetasksforthestudenttoundertaketosupportprogressioninlearning.

GAIMasasystematiccurricularandcognitiveorganisationofassessmenttasks

•  Thiscurriculummapcametobeparalleledbyagradedassessmentsystem,GAIM.

•  GAIMcreatedalevelledsystemofcriteriadescribingspecificcognitively-basedstrategieswhichrepresentsignificantstepsinmathematicaldevelopment.

•  Atthistime,whatwasparticularlynotableaboutthisdevelopmentwasthecreationofasystemwhichcombinedacurricularmodelofdomainsofmathematicalknowledgewithacognitivemodelofprogressioninmathematicalthinking.

Math-Mapperasasystematicdevelopmentalorganisationofcurriculum/assessmenttasks

•  Amoderncounterpartofthistypeofsubject-specific'shell’–formanagingcurricularresourcesandintegratingassessment–istheMath-Mapperdigitallearningsystemorganisedaroundlearningtrajectories.

•  Toguidethelearningeffortsofstudents(andteachersupportfortheseefforts)itsetsappropriatelearningtargetsandidentifiescorrespondinglearningopportunities(throughdigitalcurriculumresourcesmappedintothe'shell’).

•  Thenitmakesdiagnosticassessmentstoprovidefeedbackonthesuccessoftheseefforts,andanalysesprogresstoinformappropriatenextstepsininstructionandlearning.

Fromamulti-sourcedcollectionofresourcestoanorganisedsystem•  Thelocalinsertionofresourcesintosuch'shells'makes

considerabledemandsonteachers.•  ThusthereareversionsofbothSMILEandMath-Mapper

prepopulatedwithsuitablecurricularresources,sotakingonaformclosertothetraditionaltextbookormoderndigitalcurriculumprogramme.

•  IndeedSMILEdevelopedintoacomprehensivecurriculumprogramme,distributedwellbeyondthecontributingschoolsandteachers,andsustainedbyagroupofcoreparticipantsresponsiblefor‘mindingthesystem’.

•  Suchtrendsindicatethecontinuingimportanceofcoherent,comprehensive,externallydevelopedsystemsofresourcesinsupportingmathematicseducationinschools.

Summaryofpart2:themulti-sourcedresourcesystem

•  InWesterncountries,duringthe1960s,aspecialisededucationalideaof'resource'emerged,referringtoanytypeofmaterialcapableofsupportingcurricularactivity.

•  Interesthassubsequentlygrownin'resource-based'formsoflearningandteaching.

•  Theseinvolve,respectively,moreindependentstudybypupilsandmorelocalisedcurriculumdesignbyteachers.

•  Often,too,thisinvolvesashiftawayfromusingasinglecoursetextbooktowardscombiningmanysmallerresourceunitstakenfromdifferentsources.

•  Nevertheless,somekindofskeleton'resourcesystem’isstillneededtoorganisemanydisparateresourcessoastoprovideacomprehensiveandcoherentcurriculum.

Overviewofpart3–recentideasfromresearch

•  Suchdevelopmentshaveledresearcherstogiverenewedattentiontoteachers’useofcurricularresources.

•  Thishasproducedfurtherideasof'resourcesystem'attunedtotheparticularfocusofresearch.

•  Ruthven(2009)focusesonthepracticesandrationalesthroughwhichmathematicsteachersseektocreateafunctional'resourcesystem’,particularlyoneincorporatingdigitalresourcesalongside(orinplaceof)classicalones.

•  Gueudet&Trouche(2009)focusontheorganisationandevolutionofthe'resourcesystem'consistingofthestructuredcollectionofresourcesusedbyateacherinplanninganddeliveringlessons,andontheschemesguidingtheseprocesses.

‘Resource’and‘system’inRuthven(2009)

•  ForRuthven(2009),‘resources’comprisethemathematicaltoolsandcurriculummaterialsinuse,andthe‘system’relatestotheorganisationoftheirmodesofuse.

•  Thefocusisonthefunctionalityoftheresourcesystem,particularlyinasmuchasitincorporatesdigitalresourcesalongside(orinplaceof)classicalresources.

•  Thepracticesandrationalesreportedbyteachersprovidetheevidentialbaseonwhichthefunctioningoftheresourcesystem(asperceivedbythem)isreconstructed.

The‘resourcesystem’forageometryunit

•  Asanexample,considerhowtheresourcesystemusedbyateacherforanexistinggeometryunitforafirst-yearsecondary-schoolclasswasevolving.

•  Asinthepast,theunitstartedwithpaper-and-pencilworkonconstructionswithclassicaltools(straightedge/compass).

•  Ithadnowbeenextendedtoincludeanewcomputer-basedinvestigation,makinguseofdynamicgeometrysoftware.

•  Theresourcesystemforthiscurriculumunit,then,wasadualone,involvingsequentialuseofclassicalthendigitaltools.

•  Theanalysisexaminesthepracticesandrationalesthroughwhichtheteachersoughttomakethisresourcesystemfunctional.

Clarifyinghowdigitaltoolscomplementclassicaltools

•  Theteachernotedhowthesoftwaresupportedexplorationofdifferentcasesandovercamepracticaldifficultiesofstudentsattemptingsuchaninvestigationbyhand:

o  Youcanmoveitaroundandseethatit’salwaysthecaseandnotjustthatoneoffexample.ButIalsothinktheygetboggeddownwiththetechnicalitiesofdrawingthethingsandgettingtheircompassesright,and[dealingwith]theirpencilsbroken.

•  Buttheteachersawthecontributionofthesoftwareasgoingbeyondeaseandaccuracy;usingitrequiredpropertiestobeformulatedpreciselyingeometricalterms:

o  Andit’stheprecisionofrealisingthatthecompassconstruction...isaboutthedefinitionofwhattheperpendicularbisectoris...AndGeometer’sSketchpadforcesyoutousethegeometryandknowtheactualpropertiesthatyoucanexplore.

Reconcilingincongruenceoftoolsasdistinctivefunctions

•  Atthesametime,theteacherfeltthattherewasadegreeofincongruencebetweentoolsbecausecertainclassicaltechniquesappearedtolackdigitalcounterparts.

o  Whenyoudocompasses,youusecirclesandarcs,andyoukeepyourcompassesthesame.AndIsaytothem:“Nevermoveyourcompassesonceyou’vestarteddrawing.”...WellGeometer’sSketchpaddoesn’tusethatnotionatall.

•  Accordingly,digitalandclassicalwereseenasinvolvingdifferentmethodsandhavingdistinctfunctions:

o  Soit’sadifferentmethod.Whethertheythencantranslatethatintocompassesandpencilconstruction,Idon’tthinkthere’sagreatdealofconnection.Idon’tthinkit’sawayofteachingconstructions,it’sawayofexploringthegeometry.

Justifyingfamiliarisingstudentswithsoftwareoperation

•  Theteacherwaswillingtoinvesttimeinfamiliarisingstudentswithusingthesoftwarebecausehesawthisasengagingthemindisciplinedinteractionwithageometricsystem:

o  IalwaysintroduceGeometer’sSketchpadbysaying‘‘It’saveryspecific,you’vegottotellit.It’snotjustdrawing,it’sdrawingusingmathematicalrules.’’

•  Thisalsocapitalisedonearlierinvestmentinusingclassicaltools,aneconomyofthejointresourcesystem:

o  Andalsothey’redoingtheconstructionsbyhandfirst,tosee,gettingallthewords,thekeywords,outoftheway.

•  Likewise,hewaswillingtospendtimemakingstudentsawareoftheunderlyingconstructionprocessoffigures:

o  Ialwaysliketostartwithablankpageandactuallyputittogetherinfrontofthestudentssotheycanseewhereit’scomingfrom.

Developingstudentuseofdistinctivedigitaltechniques

•  Theteachersupportedstudentsexperiencingdifficultiesinmanipulatingthesoftware,andturnedthesedifficultiestoadvantageinreinforcingthemathematicalissueathand:

o  Understandingtheideaofperpendicularbisector...youselectthelineandthe[mid]point...There’safewpeoplethatmissedthatanddrewrandomlines...Soyouhavetoclickawayandde-selectthings,andthatcausedabitofconfusion…But...quiteafewdiscussionsIhadwiththememphasisedwhichlineisperpendiculartothatedge.

•  Likewise,hepromptedstudentstousethemostdistinctivefunctionalityofthesoftware,draggingadynamicfigure:

o  Theydidn’tspotthattheyallmetatapointaseasily...Idon’tthinkanybodygotthatwithoutsomesortofprompting.It’snotthattheydidn’tnoticeit,buttheydidn’tseeitasasignificantthingtolookfor...WhenItalkedaboutmeetingatapoint,theywereabletomoveitaround.

Establishingnormsforappropriatedigitaltooluse

•  Somefeaturesofthesoftwarewerenotwhollywelcome.•  Forexample,studentscouldbedeflectedfroma

mathematicalfocusbyanoverconcernwithpresentation.o  Theyspend… threequartersofthelessonmakingthefontlooknice

andmakingitalllookpretty[but]gettingawayfromthemaths.•  Theteacherhadtriedoutanewtechniqueformanagingthis,

byprojectinganexampletoillustrateappropriateuse:o  ItshowedthatIdidwantthemtothinkaboutthepresentation,Idid

wantthemtoslightlyadjustthefontandchangethecoloursalittlebit,toemphasisethemaths,nottomakeitjustlookpretty.

•  Heretheteacherisestablishingsociomathematicalnormsforusingthetooltocreateabetterfunctioningresourcesystem.

Replacingoldtoolsandtechniquesbybetternewones

•  Theteacherwasdevelopingideasabouthowusingdigitaltext-boxescouldmakestudentsmorewillingtoconsiderrevising(andsorethinking)theirwrittenobservations:

o  Theyhadtowritein[atextbox],andtheycouldchangeitandeditit.Theycould… thenthinkaboutwhattheywerewriting,howtheydescribe… Withhandwritten,ifsomeonewritesawholesentencenexttoaneatdiagram,andyousay,“Whataboutthatword?Canyouaddthisin?”,you’vejustruinedtheirwork.Butwithtechnologyyoucanjustchangeit… addonanextrabit,andtheydon’tmind.

•  Thiswashelpinghimtodevelopstudents’capacitytoexpressthemselvesclearlyingeometricalterms:

o  Iwasfocusingongettingthemtowritearuleclearly…Therewerealotwriting“Theyallmeet”...Soweweretryingtodiscusswhat“all”meant,andagirlatthebackhad“Theperpendicularbisectorsmeet”….“Meetatapoint”:havingthatsortofsentencethere.

‘Resource’and‘system’inGueudet&Trouche(2009)

•  ForGueudet&Trouche(2009),‘resources’comprisenotonlycurricularmaterialsbutanyothersourcesonwhichteachersdrawinplanningandconductinglessons.

•  Thustheir‘resources’canbenon-material,suchasdiscussionsbetweenteachersorstudentreactionstolessons.

•  Usersconvertsuchresourcesintotoolsbyconstructingcognitive‘utilizationschemes’:thecombinationofresourceandschemeistermeda‘document’.

•  Thefocusinthisapproachisontheschematicstructureswhichorganiseteachers’‘documentationsystems’.

•  Inlaterwork,however,Gueudet&Trouchedouse‘resourcesystem’torefertothesetofresourceswithinateacher’sdocumentationsystem(i.e.withouttheschemes).

The‘resourcesystem’ofanexpertteacher

•  Acentralhypothesisofthisapproachisthatteachers’resourcesystemsreflecttheirworkingpracticesandprofessionalexpertise,andco-evolvewiththese,andsocanprovideawindowintothem.

•  Thisapproachwastakeninaresearchstudy(Pepin,Xu,Trouche&Wang,2017)oftheresourcesystemsofChinesemathematicsteachers,regardedasexpertsbytheeducationauthorities,leadingteacherresearchgroupsintheirschool.

•  Herewewillfocusononeteacher’sresourcesystem.•  Asapreliminarytointerviewingtheteacherabouthisuseof

resources,theresearchersaskedhimtodrawamapofhisresourceswithrespecttothedifferentassociatedactivities.

Theteacher’sSchematicRepresentationofhisResourceSystem(SRRS)•  Thebasicpartsof

theresourcesystemare:

•  homecomputer(foraccessingandstoringresourcesfromonlinegroups,forumsandsites);

•  officecomputer(forpreparinglessonsandtests);

•  papermaterial(forexternalresources,andownlistingsofstudenterrors).

Exploitationandevolutionoftheteacher’sresourcesystem•  Forlessonpreparationtheteacherreportedthathewould

firstdevelopabasiclessonplanbyhimself,knowingwhatthecurriculumstandardswere.

•  Hewouldthenturntootherresourcessuchasteachingguidelines,curriculumstandardsandselectedteaching-aiddocuments.

•  Herehisfirstsourcewasthe“lessonpreparationgroup”(thegrade-specificpartoftheteacherresearchgroup);andhissecondsource,mathematicsresourcewebsites.

•  Inpost-lessonreview,hewouldwritedownhisreflectionsaboutstudents,theirthinking,theirinteractions,andsoon.

•  Theteacherwasproudthathehadbeengoodatcollectingandorganizinghisresourcesovertheyears,andhadmodifiedandappropriatedthemtobecomehisown.

A‘utilizationscheme’composedof‘rulesofaction’and‘operationalinvariants’

•  Perhapsthemainimportanceofhighlightingthecognitiveaspectistoanalysethe‘utilizationschemes’thatenable‘resourcesets’toform‘documentationsystems’.

•  Althoughthisaspectisnotprominentinmanyofthestudiesespousingthe‘documentationalapproach’,anexampleofastudywhereitisisGueudet(2017).

•  Thetableonthenextslidesummarisesananalysis,fromthatstudy,ofschemesassociatedwithauniversitymathematicsteacher’suseofresources.

•  Inparticular,itidentifiesthe‘rulesofaction’(formsofuse)and‘operationalinvariants’(underlyingorganisingprinciples)comprisingthe‘utilizationschemes’associatedwiththeuseofparticularresourcesforparticularpurposes.

Analysisofschemesassociatedwithoneuniversitymathematicsteacher’suseofresources.

Concludingthoughts:theprotean‘resourcesystem’•  Ideasof'resourcesystem'differconsiderablyinthewaysin

whichtheydemarcate'resources'andformulate'system'.•  Equally,closerexaminationshowsthatdifferentperspectives

situate'resourcesystem'incontrastingways:–  asadheringtoaparticulartypeofagent–teacher,student,designer–orasinterveningbetweensuchagents;

–  asrelatedtoaspecificeducationalentity–especiallythetext,theclassroom,thecourseorthelesson–orasrangingacrossandbeyondthese;

–  asgoverningthestructuringand/orexploitationofresources.•  Professionalsandresearchershaveclearlyfoundeachof

thesevariationsusefulforsomepurpose.•  Couldwebenefitfromdevelopingacorrespondingly

overarchingnotionof'resourcesystem’?

Keyreferences•  Allslides

–  Ruthven,K.(2019).Theconstructof'resourcesystem'asananalytictoolinunderstandingtheworkofteaching.InL.Trouche,G.Gueudet&B.Pepin(Eds.)The'Resource'ApproachtoMathematicsEducation(pp.43-59).Springer.

•  Slides3-6–  Cajori,F.(1910).Attemptsmadeduringtheeighteenthand

nineteenthcenturiestoreformtheteachingofgeometry.AmericanMathematicalMonthly17(10),181-201.

–  Chemla,K.(Ed.)(2012).Thehistoryofmathematicalproofinancienttraditions.CambridgeUniversityPress.

–  Heath,T.L.(Ed.).(1908).ThethirteenbooksofEuclid'sElements.CambridgeUniversityPress.

•  Slides7-10–  Durell,C.V.(1939).ANewGeometryforSchools.Bell.–  Quadling,D.(1996).Acenturyoftextbooks.MathematicalGazette

80(487),119-126.

Keyreferences(continued)•  Slides21-28

–  Ruthven,K.(2009).Towardsanaturalisticconceptualisationoftechnologyintegrationinclassroompractice:theexampleofschoolmathematics.Education&Didactique3(1),131–149.

–  Ruthven,K.,Hennessy,S.,&Deaney,R.(2008).Constructionsofdynamicgeometry:astudyoftheinterpretativeflexibilityofeducationalsoftwareinclassroompractice.Computers&Education51(1),297-317.

•  Slides29-34–  Gueudet,G.(2017).Universityteachers’resourcessystemsanddocuments.

InternationalJournalofResearchinUndergraduateMathematicsEducation,3(1),198-224.

–  Gueudet,G.,&Trouche,L.(2009).Towardsnewdocumentationsystemsformathematicsteachers?EducationalStudiesinMathematics71(3),199–218.

–  Pepin,B.,Xu,B.,Trouche,L.,&Wang,C.(2017).Developingadeeperunderstandingofmathematicsteachingexpertise:anexaminationofthreeChinesemathematicsteachers’resourcesystemsaswindowsintotheirworkandexpertise.EducationalStudiesinMathematics,94(3),257-274.