1. 1 Discerning In and Between Theories in Mathematics Education Discerning In and Between Theories in Mathematics Education John Mason Oxford Nov 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

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

3. 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. 4 Say What You See

5. 5 More on Theory as Perceiving  ‘Theoretical perspective’ predisposes us to attend to certain ‘things’ and not others … … with lots of consequences … with lots of consequences  On multiple levels –Macro, Micro & Meso Theories

6. 6 Examples  Skemp: Intelligence, Learning & Action  Dubinsky: APOS  Sfard: Commognition Macro Micro  Pirie & Kieren: Growth of Understanding  Tall et al: procepts Concept image  Bruner: enactive-iconic- symbolic Meso  Van Hiele levels  Vertical & Horizontal Mathematization  RBC + C Background Foreground Grain Size

7. 7 Notable Grand Theorists Jesuits Plato (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)

8. 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. 9 Features of Worlds  Epistemology  Ontological commitment  Value system: what is researchable  Psychology of researcher & subjects  How environment influences actors

10. 10 Say What You See

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

12. 12 Classroom

13. 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. 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. 15 An Overtly Structural View of Theories  Neo-Pythagorean (qualities of number) –Provides a justification for lists

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

17. 17 Enquirer Theory Situation Describing Situation Enquirer Theory Informing Enquirer Theory Explaining Situation Enquirer Theory Situation Predicting Enquirer Theory Situation Evaluating Enquirer Theory Illustrating Probing Situation Six Modes of Interaction with Theory

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

19. 19 Activity Goal Current State Resources Tasks Distinctions Relationships Properties Actions Having come to mind Choosing (?) Exploiting Unfamiliar Situation Insight Influence Want Know Introduce, Try Draw Upon Background Foreground Distinguish: -‘placing in scheme’ - gaining insight from

20. 20 Essence: theories as webs Desire to know, to understand to act Vocabulary Basis of perception Insight ActionFormat for findings View of validation Formulat e enquiry Coherent view of objects Methods: view of data; analysis Background Foreground

21. 21 Possibilities in the Moment Conservation of practices Informative insight Mechanical application Inspiration Ideals Creative Responses Restrictions Blinkers Background Foreground

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

23. 23 Structure of Theories Associated 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. 24 Metaphor?

25. 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)