teaching 3d geometry in a dynamic geometric environment

Teaching3Dgeometryinadynamicgeometricenvironment(DGE):Rethinkingtheroleofdidacticalperformanceinteacher’sorchestrationShaoMingyu,ENSdeLyonandECNUShanghai

ThedidacticalchallengeofDynamicGeometryEnvironmentENSdeLyon,7thNovember2019

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Researchproblemandbackground Students'thinkingwithgraphicalrepresentationsTeacher’schallenges

Thinkingprocesseswithgraphicalrepresentationsof3Dobjects—anoverview 3Dgeometricobjects;InternalandexternalGraphicalrepresentationsStudents'thinkingprocessesintheFrenchandChinesetasks

Discussion

Instrumentalorchestration:aframeworktoanalyzeteacher’sperformanceinaDGEequippedclass

Whatshouldbetherolesplayedbytheteacher

Relevanceofadditionaltheoreticalframeworks?

AnalyzingthetwocaseswithTheoryofSemioticMediationSemioticpotentialofartifacts;Interventionofteacherasamediator

Necessityofbalancingintuitionandtheoretic-deductivereasoninginDGE

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ProblematicofmythesisBackground

HowateacherintegratesDGEinclasstofacilitatestudents’thinkingwithgraphicalrepresentationsin3Dgeometry?‒  Moreprecisely,whatcanwelearnfromdeepeningdidacticalperformanceininstrumental

orchestrationframework

Student’sthinkingwithgraphicalrepresentations

•  Importancebutdifficultytointerpretgraphicalrepresentations;

•  Increasedinterestofintuitionin3DGeometry(Chinesecurriculum,2017;CREM,2004)

Teacher’schallengesinDGE,inspiteofitsadvantages

•  Resourcepreparation

•  Artifacts/resourcesorganizing

(Ruthven,2009;Drijvers,2010)

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Discussion





3DGeometricobject:afiguralconceptwithadoublenature(Fischbein,1993;Mithalal&

Balacheff,2019)

•  Atheoreticalreferentdefinedbytheconjunctionofgeometricalproperties

•  asetofrepresentations,eithergraphical,textualorsymbolic

Internal(mental)graphicalrepresentation

•  Mentalimage:amentalrepresentationoftheobjectwhenitisnotpresent(Marchand,2006)

•  Conceptimage:acognitiveconstructionthatcrystalizesallthepropertiesofthegeometrical

objectintooneimage,completelycontrolledbyadefinition(Accascina&Rogora,2006)

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Thinkingprocesseswithgraphicalrepresentationsof3Dobjects


External(concrete)graphicalrepresentation

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HOME

RealisticModels Drawings DigitalModels/Diagrams

•  Intuitivesupporttowardsinternalimages

•  Easytoconstruct/deconstruct?

•  Transformable(rotate,translate…)

+++Veryclosetotheobjectrepresented;Noninfinity;

+++

+Constraintofmaterials;Somecanbecut,butnoteasytorecover;

+Lossofinformation;Perceptualobviation;Noninfinity

+++NoConstraintofmaterials;Easytoeraseorrecover;Principlesofparallelprojection;+

++Dragging,changingpointofview;Verycomplexconfigurations;Noninfinity;++NoConstraintofmaterials;Easytohide/redisplay/cutGeometricknowledgeneededforrobustconstruction++instrumentaltechniquesneeded

adheringtosomeconventions;Prototypeeffect;unnecessarylinks;

Drawingsintextbook/curricula

(Accascina&Rogora,2006;Fujita,2012;Bridoux&Nihoul,2015;Parsysz,1988;)


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Students’thinkingprocesseswithgraphicalrepresentations

Twoabilitiesinvisualization(Bishop,1983):

•  visualprocessing:emerging,transformation,manipulationandextrapolationofvisualimages;

•  interpretationoffiguralinformation:‘reading’and‘interpreting’ofvisualimages,eithermentalor

external,togetfromthemanyrelevantinformationStudents’interpretationofdrawingsof3Dobjects(Bridoux&Nihoul2015):

possiblyinfluencedbyperceptualintuition,conventionsintextbooks,discoursessituatingthe

representations,andthetheoretic-deductivereasoningprocess


ConceptImage Mental

Images

Otherinformationincludingrelationship/properties.

initiate

Addconstructions

Externalgraphical

representations

RealisticModels

Digitalmodels

FrenchtaskConstructtheintersectionofline(EF)andplane(ABC)onthedigitalmodelsinGeoGebra

Wheredigitalmodelscanfacilitate:Conflictingperceptualeffectfromvaryingviewpoints—stimulatedeductive-theoreticreasoninginordertobetterunderstandthephenomenononthescreen

theoretic-deductivereasoning/conjecturebasedonperception

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Analysisaprioriofstudents’thinkingintheFrenchtask

Definitionofgeometricobject

Control

Interpretation Visualprocessing

Analysisaprioriofstudents’thinkingintheChinesetask

MentalImages

Otherinformationincludingrelationship/properties.

mentalrotation

initiate

OtherMentalImages

Drawings

Digitalmodels

ChinesetaskAplaneαcuttingthoughacubeisatthesameangletoalltheedgesofthecube,thenwhatisthemaximumareaofthesectionsofthecube?

conjecturebasedonperception

Wheredigitalmodelscanfacilitate:Renderingvisiblethecomplexstructure:cube’srotationsandmanypossiblesections–mentalimagessupportingproductionofdrawings;intuitivesupportforconjectureA 3√�3 /4  B 2√�3 /3  c 3√�2 /4  D √�3 /2 

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cuttingsections

ConceptImage

Definitionofgeometricobject

Control

Externalgraphical

representations

theoretic-deductivereasoning

Interpretation Visualprocessing

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Discussion






Differentepistemologicalvalidationof(perceptual)argumentation,theoretic-deductivereasoningandmathematicalproof(Balacheff,2019).Frompersonalconvictiontowardscollectiveconviction

Teacher’sworkorchallenges:•  Avoidperceptualdeviation,promote“mathematical

conjectures”basedonempiricalevidences;•  Avoidstudentsfromenteringintocomplicated

deductivereasoningunlessnecessary.

Teacher’sworkorchallenges:•  highlighttheconflictingperceptualeffects,

provoketheintellectualneedforproof(Linetal,2012)

•  HelpstudentsmastertheinstrumentaltechniqueswithDGE,withmathmeaningattached

Intuitionsfromdifferentviewpointscauseconfusion

Intuitionsuggeststherightconclusion,theoretic-deductivereasoningonlyneededforstudyingthepropertiesofacertainsection

Frenchcase

Chinesecase


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Discussion






Instrumentalorchestration:aframeworktodescribeteacher’sperformanceinaDGEequippedclass

connectmathematicalmeaningtotheartifactsigns

(digitalones);Theoryofsemiotic

mediation(Mariotti,2009)

orchestratesmultiplevoices’toimprovethestudents’

formulations;Toulmin’sdiagramofargumentation(2003)

Usingthemousetohighlightpartsofthescreenduringher

oralexplanations;Multimedia-based

instructionaldesign(Mayer,2009)

DidacticalconfigurationExploitationmode(Trouche,2004)Frenchcase:technical-guide-and-explainChinesecase:explain-screen-board

Didacticalperformance(Drijvers,2010)—Ageneralrecordingofhowtheteachingactuallytakesplace,havingcoveredmanyaspects,noparticularfocus,notstructuredenoughtocaptureteachingprinciples

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anaspecttobedevelopedanddeepened

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Discussion






Semioticpotentialofanartifact:Signs,personalproductionoftheirmeaningintheuseofartifactforsolvingtasks;

French case Chinese case

Artifactsigns—mathematicalmeaning

—alineasgeometricobjectdefinedbytwopoints

—intersectionoftwosetsofpoints

—thecorrespondinggeometricobject

+teachers’gesturesoftranslatingthepenalongadirection

—symmetrypropertyofthecubewithrespecttoitsdiagonal.

—interactionbetweenthepropertiesofaplaneandacube

AnalyzingthetwocaseswithTheoryofSemioticMediation

—theideaofalinebeinginfiniteandcuttingintoaplane

Artifactsigns—mathematicalmeaning

—projectionofa3Dconfigurationontothe2Denvironment

Relativemathematicalmeanings,theeducationalgoal

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+

‒  Mathematicaltermsgivenatverybeginning;JOSE4:Toutd’abord,j’aiécritautableau…«positionsdedeuxdroites:soitellessontsécantes,soitellessontparallèles».‒  Reaffirmthetask;JOSE13： Exercice1,tracerlepointd'intersectionde(EF)avec(ABC)commeça(Fig.1).‒  Provideasynthesis,mentalprolongationinitiatedbypaper-

pencilartifact;theoretic-deductivereasoningJOSE13： •  J’aimontréunefeuillepourfaireleplan(ABC),etladroite(EF)

représentéeparunstylo,sijeprolongeladroiteetleplan,çavasecouperenunpoint....

•  C’estquesiladroiten’estpasparallèleauplan,niconfondue,alorsladroitevacouperleplan.

Interventionoftheteacherasamediator

Fig.1

AnalyzingthetwocaseswithTheoryofSemioticMediation—French

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JOSE4,13:excerptsfrominterview,sameforthefollowing

‒  Contextualizationandspecialization;JOSE13:pourtrouverlepointd’intersectionduplanavecladroite,•  premièreidée,prolonger[EF]•  Deuxièmeidée,prolonge[AB]et[EF],cesdeuxdroitesvontse

couperenunpoint…l’intersectionde(EF)avec(ABC).JOSE14：•  Aveclebouton,ilsontconstruitladroiteparEetF,et

(BA).•  Aveclebouton,ilsontconstruitl’intersectionde(EF)avec

(AB).‒  Fromacommonexperiencetowardsdivergentpersonal

trajectoriesJOSE18:«cequiontréussiàlefairepeuventpasseràl’exercicesuivant». Fig.1

AnalyzingthetwocaseswithTheoryofSemioticMediation—Chinese

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Interventionoftheteacherasamediator

‒  FromthecommonexperiencewiththeGeoGebra,focusonperceptualeffectsconsistentwithcube’sgeometricproperty,withhiswords,gesturesandinductiveprinciples

Interventionoftheteacherasamediator:


Fig.2

Fig.3 18

HE10:•  rotatethecubearoundthediagonal(Figure2)…thisedge

willberotatedtothatedge,thesamefortheotheredges,•  i.e.,thepositionsoftheedgebeforeandaftertherotation

areequivalentHE11:•  Idragthisplanealongthediagonal…eithertheplanemoves

upwardordownward,thesection’sareawillbecomesmaller(Figure3)

•  thenthemaximumareamustexistsomewhereinthemiddle…itisenoughtoguessouttherightanswer.

Interventionoftheteacherasamediator:


‒  Provideasynthesis•  Inordertocalculatetheconcretemeasureofarea,the

teacherwillintroducetheoreticdeductivereasoningwiththecube’sproperties;

•  Detachthesectionfromthedigitalmodeltotheboard;‒  Personalproductionofpapersignsbasedonthe

mentalimagesinitiatedbydigitalmodels•  Studentsconstructdrawingsofcubeonthepaper,and

addtheirowndraftstofiguratethesections.

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Discussion





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whatcanwelearnfromthiscomparison

DidacticperformanceundertheperspectiveofTSMBothprovideasynthesisfromacommonexperiencewithGeoGebratothemathematicalmeaning.Differencesinthefollowingaspects

Instrumentalorchestration:Explain-screen-board:teacherisinthewholecontrolofGeoGebra

Discussion

Technical-guide-and-explain:students’actionswithGeoGebraaremoreatstake

Frenchcase Chinesecase

•  Atrajectoryofartifactsigns,mental/conceptimages,andmathematicalmeaning•  howintuition/theoretic-deductivereasoningarebalanced

Presenceofdigitalmodels—perceptualeffectsconsistenttothecube’sproperties—mathmeaning；mentalimageswithdigitalmodels--drawingsonthepaper

Improperintuitionwithdigitalmodels—mentalprolongationwithpaper-pencil—conceptimage—mathmeaning--digitalmodelsandinstrumentaltechniques

theoretic-deductivereasoning+paper-pencilartifact

intuitionbasedconjecture(inductiveprinciples+gestures,words);alittledeductivereasoning

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Perspectivesoffutureinvestigation

Potentialsoftheothertwoapproachesdeepeningthedidacticalperformance•  Toulmin’sdiagramofargumentation(2003)

Data,warrant,backing,qualifier-claimsemerginginstudents’argumentation,teacher’sroleinestablishingnormsinacommunitythatallowstovalidateanargumentation—somethingrelatedtodeductive-theoreticreasoning,orproof.

•  Multimedia-basedinstructionaldesign(Mayer,2009)Coordinateteacher’sgestures,oralexplanationstothemultimediainformationdisplayedtodirectstudents’attentioninasystematicandorderedway.

Takingintoconsiderationteacher’sresourcepreparation,togetherwiththeirknowledgeandorientations.

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Accascina,G.,&Rogora,E.(2006).UsingCabri3DDiagramsforTeachingGeometry.InternationalJournalforTechnologyinMathematicsEducation,13(1),11–22.

Balacheff,N.(2019).L’argumentationmathématique:précurseurproblématiquedeladémonstration.PresentationofCORFEM(33).Strasbourg.

Bishop,A.J.(1983).Spatialabilitiesandmathematicalthinking.InProceedingsofthefourthinternationalcongressonmathematicaleducation(pp.176–178).

Bridoux,S.,&Nihoul,C.(2015).Difficultésdesélèvesàinterpréterdesconstructionsdansl’espace.Uneétudedecas.Petitx,(98),53–76.

CREM:Ballieu,M.,&Guissard,M.-F.(2004).Pouruneculturemathématiqueaccessibleatous.Élaborationd’outilspédagogiquespourdevelopperdescompétencescitoyennes.Nivelle,Belgique.

Drijvers,P.,Doorman,M.,Boon,P.,Reed,H.,&Gravemeijer,K.(2010).Theteacherandthetool:Instrumentalorchestrationsinthetechnology-richmathematicsclassroom.EducationalStudiesinMathematics,75(2),213–234.

Fischbein,E.(1993).Thetheoryoffiguralconcepts.EducationalStudiesinMathematics,24(2),139–162.Fujita,T.(2012).Learners’levelofunderstandingoftheinclusionrelationsofquadrilateralsandprototypephenomenon.TheJournalofMathematicalBehavior,31(1),60–72.

Fujita,T.(2012).Learners’levelofunderstandingoftheinclusionrelationsofquadrilateralsandprototypephenomenon.TheJournalofMathematicalBehavior,31(1),60–72.

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References

teaching 3d geometry in a dynamic geometric environment

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