Versatile Mathematical Thinking in the Secondary Classroom

1. Versatile Mathematical Thinking in the Secondary Classroom Mike Thomas The University of Auckland

2. Overview • A current problem • Versatile thinking in mathematics • Some examples from algebra and calculus • Possible roles for technology

3. What can happen? • Why do we need to think about what we are teaching? • Assessment encourages: Emphasis on procedures, algorithms, skills Creates a lack of versatility in approach

4. Possible problems • Consider But, the LHS of the original is clearly one half!!

5. Concept not understood Which two are equivalent? Can you find another equivalent expression? A student wrote… …but he factorised C!

6. Procedural focus

7. Procedure versus concept • Let • For what values of x is f(x) increasing? • Some could answer this using algebra and but…

8. Procedure versus concept For what values of x is this function increasing?

9. Versatile thinking in mathematics First… process/object versatility—the ability to switch at will in any given representational system between a perception of symbols as a process or an object

10. Examples of procepts

11. Lack of process-object versatility (Thomas, 1988; 2008)

12. Procept example

13. Procept example

14. Effect of context on meaning for

15. Process/object versatility for • Seeing solely as a process causes a problem interpreting and relating it to

16. Student: that does imply the second derivative…it is the derived function of the second derived function

17. Visuo/analytic versatility Visuo/analytic versatility—the ability to exploit the power of visual schemas by linking them to relevant logico/analytic schemas

18. A Model of Cognitive Integration conscious unconscious

19. Surface (iconic) v deep (symbolic) observation • “” Moving from seeing a drawing (icon) to seeing a figure (symbol) requires interpretation; use of an overlay of an appropriate mathematical schema to ascertain properties

20. External world Interact with/act on interpret external sign ‘appropriate’ schema Internal world

21. Schema use Booth & Thomas, 2000 We found  e

22. Example This may be an icon, a ‘hill’, say We may look ‘deeper’ and see a parabola using a quadratic function schema This schema may allow us to convert to algebra

23. Algebraic symbols: Equals schema • Pick out those statements that are equations from the following list and write down why you think the statement is an equation: • a) k = 5 • b) 7w – w • c) 5t – t = 4t • d) 5r – 1 = –11 • e) 3w = 7w – 4w

24. Surface: only needs an = sign All except b) are equations since:

25. Equation schema: only needs an operation Perform an operation and get a result:

26. The blocks problem

27. Solution

28. Reasoning

29. Solve ex=x50

30. Solve ex=x50 • Check with two graphs, LHS and RHS

31. Find the intersection

32. How could we reason on this solution?

33. Antiderivative? What does the antiderivative look like?

34. Task: What does the graph of the derivative look like?

35. Method easy

36. But what does the antiderivative look like? How would you approach this? Versatile thinking is required.

37. Maybe some technology would help • Geogebra Geogebra

38. Representational Versatility Thirdly… representational versatility—the ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with representations

39. "…much of the actual work of mathematics is to determine exactly what structure is preserved in that representation.” J. Kaput Representation dependant ideas... Is 12 even or odd? Numbers ending in a multiple of 2 are even. True or False? • 123? • 123, 345, 569 are all odd numbers • 113, 346, 537, 469 are all even numbers

41. Representations can lead to other conflicts… The length is 2, since we travel across 1 and up 1 What if we let the number of steps n increase? What if n tends to ? Is the length √2 or 2?

42. Representational versatility • Ruhama Even gives a nice example: • If you substitute 1 for x in ax2 + bx + c, where a, b, and c are real numbers, you get a positive number. Substituting 6 gives a negative number. How many real solutions does the equationax2 + bx + c = 0 have? Explain.

43. 1 6

44. Treatment and conversion (Duval, 2006, p. 3)

45. Treatment or conversion? 25

46. Integration by substitution

47. Integration by substitution

48. Integration by substitution

49. Integration by substitution

50. Linking of representation systems • (x, 2x), where x is a real number Ordered pairs to graph to algebra