1. Constructing Versatile Mathematical Conceptions Mike Thomas The University of Auckland

2. Overview Define versatile thinking in mathematics Consider some examples and problems

3. Versatile thinking in mathematics First… process/object versatilitythe ability to switch at will in any given representational system between a perception of symbols as a process or an object Not just procepts, which are arithmetic/algebraic

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

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

6. A Model of Cognitive Integration conscious unconscious

7. Representational Versatility Thirdly… representational versatilitythe ability to work seamlessly within and between representations, and to engage in procedural and conceptual interactions with representations

8. Treatment and conversion Duval, 2006, p. 3

9. Icon to symbol requires interpretation through appropriate mathematical schema to ascertain properties External world Internal world external sign appropriate schema interpret Interact with/act on translation or conversion

10. 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

11. A possible problem The opportunity to acquire knowledge in a variety of forms, and to establish connections between different forms of knowledge are apt to contribute to the flexibility of students thinking (Dreyfus and Eisenberg, 1996). The same variety, however, also tends to blur students appreciation of the difference in status which different means of establishing mathematical knowledge bestow upon that knowledge. Dreyfus, 1999

12. 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

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

14. Another possible problem Compartmentalization This phenomenon occurs when a person has two different, potentially conflicting schemes in his or her cognitive structure. Certain situations stimulate one scheme, and other situations stimulate the other…Sometimes, a given situation does not stimulate the scheme that is the most relevant to the situation. Instead, a less relevant scheme is activated Vinner & Dreyfus, 1989, p. 357

15. A formula

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

17. Abstraction in context We also pay careful attention to the multifaceted context in which processes of abstraction occur: A process of abstraction is influenced by the task(s) on which students work; it may capitalize on tools and other artifacts; it depends on the personal histories of students and teacher Hershkowitz, Schwarz, Dreyfus, 2001

18. Abstraction of meaning for

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

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

21. To see this relationship one needs to deal with the function g as an object that is operated on in two ways Dreyfus (1991, p. 29)

22. Proceptual versatile thinking If, then write down the value of

23. Versatile thinking–change of representation system nb The representation does not correspond; an exemplar y=x 2 is used

25. Newton-Raphson versatility Many students can use the formula below to calculate a better approximation of the root, but are unable to explain why it works

26. Newton-Raphson

27. Newton-Raphson When is x 1 a suitable first approximation for the root a of f(x) = 0?

28. Student V1: It is very important that the approximation is close enough the root and not on a turning point. Otherwise you might be finding the wrong root.

29. Student knowledge construction Learning may take place in a single representation system, so inter-representational links are not made Avoid activity comprising surface interactions with a representation, not leading to the concept The same representations may mean different things to students due to their contextual schema construction (abstraction) Use multiple contexts for representations

30. Reference Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67-87. From: moj.thomas@auckland.ac.nz

32. Proceptual versatility– eigenvectors Two different processes Need to see resulting object or effect as the same

33. Work within the representation systemalgebra

34. Same process

35. Conversion v v u u

36. Student knowledge construction Learning may take place in a single representation system, so inter-representational links are not made Avoid activity comprising surface interactions with a representation, not leading to the concept The same representations may mean different things to students due to their contextual schema construction (abstraction) Use multiple contexts for representations

38. Conversions Translation between registers Duval, 1999, p. 5

39. Representations Duval, 1999, p. 4 Semiotic representations systems Semiotic registers Processing transformation within a register

40. Epistemic actions are mental actions by means of which knowledge is used or constructed

41. Representations and mathematics Much of mathematics is about what we can learn about concepts through their representations (or signs) Examples include: natural language, algebras, graphs, diagrams, pictures, sets, ordered pairs, tables, presentations, matrices, etc. (nb icons, indices and symbols here) Some of the things we learn are representation dependant; others representation independent

42. Representation dependant ideas... "…much of the actual work of mathematics is to determine exactly what structure is preserved in that representation. J. Kaput Is 12 even or odd? Numbers ending in a multiple of 2 are even. True or False? 12 3 ? 12 3, 34 5, 56 9 are all odd numbers 11 3, 34 6, 53 7, 46 9 are all even numbers

43. Representational interactions We can interact with a representation by: Observationsurface or deep (property) Performing an actionprocedural or conceptual Thomas, 2001

44. Representational interactions We can interact with a representation by: Observationsurface or deep (property) Performing an actionprocedural or conceptual Thomas, 2001

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

46. For what values of x is this function increasing?

48. Why it may fail

49. We should not think that the three parts of versatile thinking are independent Neither should we think that a given sign has a single interpretation it is influenced by the context

50. Icon, index, symbol A DC B ABCD symbol

51. Icon to symbol 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

52. Function