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Discerning Discerning In and Between In and Between

Theories Theories in Mathematics in Mathematics

EducationEducation John MasonJohn Mason

OxfordOxford

Nov 2010Nov 2010The Open UniversityMaths Dept University of Oxford

Dept of Education

Promoting Mathematical Thinking

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OutlineOutline

‘‘Theory’ as ‘perceiving’ Theory’ as ‘perceiving’ (discernment)(discernment)

Notable Theorists in Maths EdnNotable Theorists in Maths Edn AttentionAttention Structuralist View of TheoriesStructuralist View of Theories Appreciating TheoriesAppreciating Theories

Specific to Mathematics Education

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Theory as PerceivingTheory as Perceiving

‘‘Theory’Theory’– From Greek: a way of seeing … From Greek: a way of seeing …

perceiving … hence ‘perspective’perceiving … hence ‘perspective’ We see what we expectWe see what we expect

‘seeing’ ‘seeing’ isis believing believing ““I see”: I see”:

– comprehend, understand, comprehend, understand, appreciate, make sense of … appreciate, make sense of …

– ““I see what you are saying”I see what you are saying”

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Say What You SeeSay What You See

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More on Theory as More on Theory as PerceivingPerceiving

‘‘Theoretical perspective’ Theoretical perspective’ predisposes us to attend to predisposes us to attend to certain ‘things’ and not others …certain ‘things’ and not others … … … with lots of consequenceswith lots of consequences

On multiple levelsOn multiple levels– Macro, Micro & Meso TheoriesMacro, Micro & Meso Theories

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ExamplesExamples

Skemp: Intelligence, Skemp: Intelligence, Learning & ActionLearning & Action

Dubinsky: APOSDubinsky: APOS Sfard: Sfard:

CommognitionCommognition

MacroMacro MicroMicro Pirie & Kieren: Pirie & Kieren:

Growth of Growth of UnderstandingUnderstanding

Tall et al: proceptsTall et al: proceptsConcept imageConcept image

Bruner: enactive-Bruner: enactive-iconic-symboliciconic-symbolicMesoMeso

Van Hiele levelsVan Hiele levels Vertical & Horizontal Vertical & Horizontal

MathematizationMathematization RBC + CRBC + C Background

ForegroundGrain Size

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Notable Grand TheoristsNotable Grand Theorists

JesuitsPlato (Socrates) 500 BCE

Comenius (1592-1671)Froebel (1782-1852)

Montessori (1870-1952)

Dewey (1859-1952)

Freudenthal (1905-1990)

Skemp (1919-1995)Brousseau (1933)Chevallard

Gattegno (1911-1988)

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Different WorldsDifferent Worlds

World of External FactsWorld of External Facts– Repeatable experiments; sciences, Repeatable experiments; sciences,

psychologypsychology World of Opinion and BeliefWorld of Opinion and Belief

– Surveys & Questionnaires; sociologySurveys & Questionnaires; sociology World of Others’ ExperienceWorld of Others’ Experience

– Participant Observation; ethnographicParticipant Observation; ethnographic World of Involvement in ActionWorld of Involvement in Action

– Change & prediction; action researchChange & prediction; action research World of Personal ExperienceWorld of Personal Experience

– Sensitising to notice; phenomeno-logic & -Sensitising to notice; phenomeno-logic & -graphicgraphic

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Features of WorldsFeatures of Worlds

EpistemologyEpistemology Ontological commitmentOntological commitment Value system: what is researchableValue system: what is researchable Psychology of researcher & Psychology of researcher &

subjectssubjects How environment influences actorsHow environment influences actors

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Say What You SeeSay What You See

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AttentionAttention

Holding Wholes (Gazing)Holding Wholes (Gazing) Discerning Details (Hyparchic Discerning Details (Hyparchic

acts)acts) Recognising RelationshipsRecognising Relationships Perceiving PropertiesPerceiving Properties Reasoning on the basis of Reasoning on the basis of

agreed propertiesagreed properties

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ClassroomClassroom

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DevelopmentDevelopment Manipulating familiar Manipulating familiar

confidence inspiring confidence inspiring objects (specialising, objects (specialising, particularising)particularising)

In order to get a sense of In order to get a sense of underlying structural underlying structural relationships (modelling, relationships (modelling, axiomatising, justifying, axiomatising, justifying, proving …proving …

Bringing this experience to Bringing this experience to articulation, which over articulation, which over time, becomes more time, becomes more succinct and useable succinct and useable (manipulable)(manipulable)

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Purposes & UsesPurposes & Uses

DescriptiveDescriptive IllustrativeIllustrative ExplanatoryExplanatory InformativeInformative PredictivePredictive EvaluativeEvaluative

Grain Size

Using theories to make sense of experience …Experiencing the sense made as a result of theories

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An Overtly Structural View of An Overtly Structural View of TheoriesTheories

Neo-Pythagorean (qualities of Neo-Pythagorean (qualities of number)number)– Provides a justification for listsProvides a justification for lists

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Describing

Illustrating; Probing

Informing

Explaining

Predicting

Evaluating

Six Modes of Interaction with Six Modes of Interaction with TheoryTheory

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Enquirer

Theory

SituationDescribing

Situation

Enquirer

Theory

Informing

Enquirer

TheoryExplaining

Situation

Enquirer

Theory

SituationPredicting

Enquirer

Theory

Situation

Evaluating

Enquirer

TheoryIllustratingProbing

Situation

Six Modes of Interaction with Six Modes of Interaction with TheoryTheory

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Teacher

Student

ContentExpounding

Student

Content

Teacher

Exploring

Content

TeacherExamining

Student

Student

Content

TeacherExercising

Student

Content

Teacher

Expressing

Teacher

StudentExplaining

Content

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ActivityActivity

Goal

Current State

Resources TasksDistinctionsRelationshipsPropertiesActions

Having come to mind

Choosing (?)Exploiting

Unfamiliar Situation

InsightInfluenceWant

Know

Introduce, Try

DrawUpon

BackgroundForeground

Distinguish: -‘placing in scheme’ - gaining insight from

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Essence: theories as websEssence: theories as webs

Desire to know, to understand to act

Vocabulary

Basis ofperception

InsightAction

Format for findingsView of

validation

Formulate enquiry

Coherent view of objects

Methods:view of data; analysis

BackgroundForeground

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Possibilities in the MomentPossibilities in the Moment

Conservationof practices

Informative insight

Mechanicalapplication

InspirationIdeals

CreativeResponses

RestrictionsBlinkers

BackgroundForeground

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Appreciating a TheoryAppreciating a Theory

BricolageBricolage ImmersionImmersion

Combining Theories: completeness consistency

Combining Theories: completeness consistency

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Structure of TheoriesStructure of TheoriesAssociated ActionsAssociated Actions

DistinctionsDistinctions Labelled ––> constructsLabelled ––> constructs Assembled into collections ––> Assembled into collections ––>

Frame(works)Frame(works) Supporting recognition of relationshipsSupporting recognition of relationships Articulated and taken-as propertiesArticulated and taken-as properties

Resulting in:Phenomena (Hyparxis & Ontology)Leading to actions

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Metaphor?Metaphor?

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Follow UpFollow Up

mcs.open.ac.uk/jhm3 mcs.open.ac.uk/jhm3 (Presentations)(Presentations)

SourcesSources The Role of Theory in Mathematics The Role of Theory in Mathematics

Education and Research (1996) in Bishop Education and Research (1996) in Bishop et aet a

Theories of Mathematics EducationTheories of Mathematics Education (2010) (2010)

Sriraman & English Sriraman & English PME research group on Theory in Maths PME research group on Theory in Maths

EducationEducation Systematics (J. G. Bennett 1960s)Systematics (J. G. Bennett 1960s)