1. 1 Designing and Using Tasks Effectively for Conceptual Development Anne Watson John Mason Agder College Kristiansand Norway September 2006

2. 2 Outline  Exercise as object  Using structured variation  How many different ways …?  Construct a ….

3. 3 Exercise as object 17 – 9 = 27 – 9 = 37 – 9 = 47 – 9 = …

4. 4 Learning from experience  Patterns in layout  Patterns of digits  Familiarity  Generality  Going beyond mere answers

5. 5 Conceptual development  Tasks, and the ways they are presented, mediate formal mathematical ideas for learners –Multiple examples –Personal images –Natural/scientific concepts –Intuitive/formal understanding –Further experience

6. 6 Exercise as object ( x – 2 ) ( x + 1 ) = x 2 - x - 2 ( x – 3 ) ( x + 1 ) = x 2 - 2x - 3 ( x – 4 ) ( x + 1 ) = x 2 - 3x - 4 … …

7. 7 Reflections  Going beyond mere answers  Reflecting across the grain  Quasi-physical and visual repetition  How do learners know what to focus on?

8. 8 Stars

9. 9 Structured variation  Dimensions of possible variation  Range of permissible change

10. 10 Find the gradient between each pair of points  (4, 3) and (8, 12)(4, 3) and (4, 12)  (4, 3) and (7, 12)(4, 3) and (3, 12)  (4, 3) and (6, 12)(4, 3) and (2, 12)  (4, 3) and (5, 12)(4, 3) and (1, 12)

11. 11 Task elements  Quasi-physical, visual, notational patterns  Dimensions of possible variation (DofPV)  Range of permissible change (RofPCh)

12. 12 How many different ways … … can a unit fraction be written as the difference of unit fractions? e.g.

13. 13 Unit fraction differences Anticipating Generalising Rehearsing Checking Organising 111 32 =– 6 111 43 =– 12 11 2 =– 3 111 54 =– 20 111 65 =– 30 11 2 =– 4 11 3 =– 6 111 21 =– 2 11 4 =– 12 111 76 =– 42 111 87 =– 56 11 6 =– 24 11 4 =– 8

14. 14 Construct a … … pentagon of area 20 by moving only blue pegs to other peg positions

15. 15 Sources of data; Opportunities for learning Who creates the data What does the learner do? Data: from teacher DofPV: constrained by task RofCh: suggested by teacher Data: from learners DofPV: constrained by task RofCh: comes from learners How prompt learners to go beyond mere construction? How prompt learners to go beyond mere answers? Scope and freedom of what learner might think about 27 – 9 = (x – 2)(x + 1) = … Tasks: Unit fractions Pentagon construction Tasks:

16. 16 Follow-up

17. 17 Some Tools for task design and use  Structured Variation  How many different ways …?  Construct a …  Treating an exercise as a mathematical object DofPV, who decides? RofPC, who decides? Openness of input Openness of learning