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Discerning In and Between Theories in Mathematics Education

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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

- ‘Theory’ as ‘perceiving’ (discernment)
- Notable Theorists in Maths Edn
- Attention
- Structuralist View of Theories
- Appreciating Theories

Specific to Mathematics Education

- ‘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”

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

- On multiple levels
- Macro, Micro & Meso Theories

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

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)

- 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

- Epistemology
- Ontological commitment
- Value system: what is researchable
- Psychology of researcher & subjects
- How environment influences actors

- Holding Wholes (Gazing)
- Discerning Details (Hyparchic acts)
- Recognising Relationships
- Perceiving Properties
- Reasoning on the basis of agreed properties

- 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)

- Descriptive
- Illustrative
- Explanatory
- Informative
- Predictive
- Evaluative

Grain Size

Using theories to make sense of experience …

Experiencing the sense made as a result of theories

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

Describing

Illustrating; Probing

Explaining

Informing

Predicting

Evaluating

Situation

Theory

Situation

Enquirer

Theory

Enquirer

Evaluating

Explaining

Theory

Situation

Theory

Enquirer

Situation

Enquirer

Predicting

Informing

Enquirer

Theory

Situation

Describing

Enquirer

Situation

Theory

Illustrating

Probing

Student

Content

Teacher

Content

Teacher

Student

Expressing

Examining

Content

Student

Teacher

Student

Teacher

Content

Exercising

Exploring

Teacher

Student

Content

Expounding

Teacher

Content

Student

Explaining

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

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

Mechanicalapplication

Informative insight

Conservationof practices

InspirationIdeals

CreativeResponses

RestrictionsBlinkers

Background

Foreground

Bricolage

Immersion

Combining Theories:

completeness

consistency

- 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

- 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)