Constructing Versatile Mathematical Conceptions

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 versatility—the 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 versatility—the 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 versatility—the 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. External world Interact with/act on interpret external sign ‘appropriate’ schema Internal world Icon to symbol requires interpretation through appropriate mathematical schema to ascertain properties 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 dependson 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=x2is 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 x1 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 system—algebra

34. Work within the representation system—algebra Same process

35. Conversion u v v 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 Semiotic representations systems — Semiotic registers Processing — transformation within a register Duval, 1999, p. 4

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. "…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

43. Representational interactions We can interact with a representation by: • Observation—surface or deep (property) • Performing an action—procedural or conceptual Thomas, 2001

44. Representational interactions We can interact with a representation by: • Observation—surface or deep (property) • Performing an action—procedural or conceptual Thomas, 2001

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

46. Procedure versus concept 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. A B D C Icon, index, symbol ABCD — symbol