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The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Discerning In and Between Theories in Mathematics Education. John Mason Oxford Nov 2010. Outline. ‘Theory’ as ‘perceiving’ (discernment) Notable Theorists in Maths Edn Attention

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Discerning In and Between Theories in Mathematics Education

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Discerning in and between theories in mathematics education l.jpg

The Open University

Maths Dept

University of Oxford

Dept of Education

Promoting Mathematical Thinking

Discerning In and Between Theories in Mathematics Education

John Mason

Oxford

Nov 2010


Outline l.jpg

Outline

  • ‘Theory’ as ‘perceiving’ (discernment)

  • Notable Theorists in Maths Edn

  • Attention

  • Structuralist View of Theories

  • Appreciating Theories

Specific to Mathematics Education


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Theory as Perceiving

  • ‘Theory’

    • From Greek: a way of seeing … perceiving … hence ‘perspective’

  • We see what we expect‘seeing’ is believing

  • “I see”:

    • comprehend, understand, appreciate, make sense of …

    • “I see what you are saying”


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Say What You See


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More on Theory as Perceiving

  • ‘Theoretical perspective’ predisposes us to attend to certain ‘things’ and not others …

    … with lots of consequences

  • On multiple levels

    • Macro, Micro & Meso Theories


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Examples

Macro

Micro

  • Skemp: Intelligence, Learning & Action

  • Dubinsky: APOS

  • Sfard: Commognition

  • Pirie & Kieren: Growth of Understanding

  • Tall et al: proceptsConcept image

  • Bruner: enactive-iconic-symbolic

Meso

  • Van Hiele levels

  • Vertical & Horizontal Mathematization

  • RBC + C

Grain Size

Background

Foreground


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Notable Grand Theorists

Plato (Socrates) 500 BCE

Dewey (1859-1952)

Jesuits

Gattegno (1911-1988)

Comenius (1592-1671)

Skemp (1919-1995)

Froebel (1782-1852)

Brousseau (1933)

Montessori (1870-1952)

Chevallard

Freudenthal (1905-1990)


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Different Worlds

  • World of External Facts

    • Repeatable experiments; sciences, psychology

  • World of Opinion and Belief

    • Surveys & Questionnaires; sociology

  • World of Others’ Experience

    • Participant Observation; ethnographic

  • World of Involvement in Action

    • Change & prediction; action research

  • World of Personal Experience

    • Sensitising to notice; phenomeno-logic & -graphic


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Features of Worlds

  • Epistemology

  • Ontological commitment

  • Value system: what is researchable

  • Psychology of researcher & subjects

  • How environment influences actors


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Say What You See


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Attention

  • Holding Wholes (Gazing)

  • Discerning Details (Hyparchic acts)

  • Recognising Relationships

  • Perceiving Properties

  • Reasoning on the basis of agreed properties


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Classroom


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Development

  • Manipulating familiar confidence inspiring objects (specialising, particularising)

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

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


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Purposes & Uses

  • Descriptive

  • Illustrative

  • Explanatory

  • Informative

  • Predictive

  • Evaluative

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 Theories

  • Neo-Pythagorean (qualities of number)

    • Provides a justification for lists


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Six Modes of Interaction with Theory

Describing

Illustrating; Probing

Explaining

Informing

Predicting

Evaluating


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Situation

Theory

Situation

Enquirer

Theory

Enquirer

Evaluating

Explaining

Theory

Situation

Theory

Enquirer

Situation

Enquirer

Predicting

Informing

Six Modes of Interaction with Theory

Enquirer

Theory

Situation

Describing

Enquirer

Situation

Theory

Illustrating

Probing


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Student

Content

Teacher

Content

Teacher

Student

Expressing

Examining

Content

Student

Teacher

Student

Teacher

Content

Exercising

Exploring

Teacher

Student

Content

Expounding

Teacher

Content

Student

Explaining


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Activity

Goal

Insight

Influence

Want

Tasks

Resources

Current State

Distinctions

Relationships

Properties

Actions

Having come to mind

Choosing (?)

Exploiting

Draw

Upon

Introduce, Try

Know

Unfamiliar Situation

Background

Distinguish:

-‘placing in scheme’

- gaining insight from

Foreground


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Essence: theories as webs

Format for findings

Action

View of validation

Insight

Methods:

view of data; analysis

Basis ofperception

Coherent view of objects

Vocabulary

Formulate enquiry

Desire

to know, to understand

to act

Background

Foreground


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Possibilities in the Moment

Mechanicalapplication

Informative insight

Conservationof practices

InspirationIdeals

CreativeResponses

RestrictionsBlinkers

Background

Foreground


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Appreciating a Theory

Bricolage

Immersion

Combining Theories:

completeness

consistency


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Structure of TheoriesAssociated Actions

  • Distinctions

  • Labelled ––> constructs

  • Assembled into collections ––> Frame(works)

  • Supporting recognition of relationships

  • Articulated and taken-as properties

Resulting in:

Phenomena (Hyparxis & Ontology)

Leading to actions


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Metaphor?


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Follow Up

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

  • Sources

    • The Role of Theory in Mathematics Education and Research (1996) in Bishop et a

    • Theories of Mathematics Education (2010) Sriraman & English

    • PME research group on Theory in Maths Education

    • Systematics (J. G. Bennett 1960s)


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