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How, Why and What of Algebra

The Future of the . Teaching. and. Learning. of Algebra. How, Why and What of Algebra. Kaye Stacey University of Melbourne. The Future of the . Teaching. and. Learning. of Algebra. AIMS. Continue international survey (Australia, Italy, Norway, USA, UK) Give further exemplification

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How, Why and What of Algebra

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  1. The Future of the Teaching and Learning ofAlgebra How, Why and What of Algebra Kaye Stacey University of Melbourne

  2. The Future of the Teaching and Learning ofAlgebra AIMS Continue international survey (Australia, Italy, Norway, USA, UK) Give further exemplification Think about putting research into a form that can affect practice

  3. Bushwalking with Kim.Kim wants to walk from A to B through varying terrain. What route should she take? B C A Faster here Slower here P Q R

  4. Yr 12 assessment version 2 km/hr in bush, 5 km/hr in clearing (45 deg) |AB| = 14 km, |CR| = 7 km, C midpoint B C A Faster here Slower here P Q R

  5. Formulation of the mathematical problem Total time = sum of times on three segments = distance 1 /speed 1 + distance 2 /speed 2 + distance 3 /speed 3 AIM: minimise total time

  6. Setting up the variables and expressions….

  7. |AP| found by cosine rule: d(x) = long expression in x Define function t(x, b, c) = 2x/c + 2d(x)/b d(x) = sqrt (2x2 + 72 -2.sqrt2.7.cos 45)

  8. Time calculated from “raw” distances

  9. Minimum time for b=2, c=5

  10. ..or find a minimum graphically

  11. With CAS, can find minimum as function of k = c/b …and then explore what this answer means… what if k=1?, domain of validity of solution… what if distances change?

  12. Bushwalking with Kim.Kim wants to walk from A to B through varying terrain. What route should she take? • Solution 1: Algebraic formulation, with calculus for optimisation • Solution 2: Algebraic formulation, with graphical optimisation • Solution 3: Dynamic geometry formulation, with “dragging” optimisation

  13. Orienteering with Kim: a numerical “drag” solution can deal with a clearing in any position Faster here Slower here B A Time = dist 1 /speed 1 + dist 2 /speed 2 + dist 3 /speed 3

  14. Some observations . . . • Our move to reduce symbolic manipulation has been good - but we need to develop algebraic expectation and symbol sense with only minimal practice. • Emphasis on formulation and interpretation essential; we still need more in schools (and more research). • What would an intellectually strong numerical algebra be like? Can enough cognitive obstacles be avoided to justify big change ? (Note need for reification in both) • Basic algebra can no longer be justified by pragmatic reasons - so we need to clarify the epistemic reasons.

  15. A “black box” is OK, but a doubly black box is not!

  16. Thank you

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