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Mathematical thinking

Mathematical thinking. is of course. very special. Why should we care about whether it’s special?. Because we’re asking society to fund us to teach it. Because we want to be able to recognise mathematical thinking when we see it.

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Mathematical thinking

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  1. Mathematical thinking

  2. is of course

  3. very special

  4. Why should we care about whether it’s special?

  5. Because we’re asking society to fund us to teach it. Because we want to be able to recognise mathematical thinking when we see it. Because somebody might ask us – at a party or in a classroom.

  6. Because a teacher’s political position + her general educational philosophy + her views about nature of mathematics and numeracy = (sort of) her approaches to teaching, and to curriculum and accreditation issues Based on Ernest, P., 1991, The Philosophy of Mathematics Education, Basingstoke, Flamer Press

  7. What do you really hope or believe about the “specialness” of maths? Hopes and beliefs exercise

  8. What makes maths special? • Content? • Style of thinking? • Style and standards of proof?

  9. Maths is ABOUT something It’s about numbers or shapes or symbols or mental objects or........

  10. Bain, I. 1986. Celtic Knotwork. London: Constable

  11. Bain, I. 1986. Celtic Knotwork. London: Constable

  12. Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

  13. Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

  14. Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

  15. Zaslavsky, C. 1973 Africa Counts. Westport. Lawrence Hill & Co.

  16. Deal or No Deal. Any mathematical thinking going on there?

  17. OK...... it’s not about things...... it’s about FACTS about the things. Maths is really a set of facts about the world... like 1 + 1 = 2

  18. Or....... “for every line, L, and point, P, which is not on that line, there exists a unique line, M, through P that is parallel to L.” Is that a fact? A mathematical fact?

  19. Ok, forget content, forget facts. Maths isn’t a noun, it’s a verb. It’s about a style of thinking.

  20. style of thinking..... logical objective challenging integrated   stuck but happy knitting ideas together deductive consistent compartmentalised creative questioning step-by-step disciplined rule-generating   speculating generalising enquiring   practical abstract well-organised rule-following proof refutation algorithmic  structured by leaps and bounds intuitive

  21. How about proof? If you prove something, you’ve been doing mathematical thinking........? And if you haven’t proved something, you haven’t .......?

  22. When is a proof really a proof?

  23. Formal ? Algebraic? Computer-generated? Visual? Intuition? Consensus? Proof-building by “incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations” Lakatos, I. (1976). Proofs and Refutations. Cambridge: Cambridge University Press.

  24. Mathematicians as enquirers • Visual - thinking in pictures, often dynamic • Analytic - thinking symbolically, often formalistically • Conceptual • thinking in ideas, classifying Burton, L. (2004). Mathematicians as Enquirers - Learning about Learning Mathematics.Dordecht: Kluwer Academic Publishers.

  25. And finally....... 0 1 9 2 8 7 3 4 6 5

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