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Discerning In and Between Theories in Mathematics Education

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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  1. 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

  2. Outline • ‘Theory’ as ‘perceiving’ (discernment) • Notable Theorists in Maths Edn • Attention • Structuralist View of Theories • Appreciating Theories Specific to Mathematics Education

  3. 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”

  4. Say What You See

  5. 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

  6. 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

  7. 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)

  8. 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

  9. Features of Worlds • Epistemology • Ontological commitment • Value system: what is researchable • Psychology of researcher & subjects • How environment influences actors

  10. Say What You See

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

  12. Classroom

  13. 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)

  14. 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

  15. An Overtly Structural View of Theories • Neo-Pythagorean (qualities of number) • Provides a justification for lists

  16. Six Modes of Interaction with Theory Describing Illustrating; Probing Explaining Informing Predicting Evaluating

  17. 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

  18. Student Content Teacher Content Teacher Student Expressing Examining Content Student Teacher Student Teacher Content Exercising Exploring Teacher Student Content Expounding Teacher Content Student Explaining

  19. 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

  20. 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

  21. Possibilities in the Moment Mechanicalapplication Informative insight Conservationof practices InspirationIdeals CreativeResponses RestrictionsBlinkers Background Foreground

  22. Appreciating a Theory Bricolage Immersion Combining Theories: completeness consistency

  23. 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

  24. Metaphor?

  25. 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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