1. Example spaces: how to get one and what to do with it! Anne Watson Matematikbiennalen 2008

2. Task 1  Write down an example of an equation in which x = 5  ….. and another

3. Task types  Example giving  … and another

4. Task affordances  Working backwards  Pushing beyond the examples which first come to mind  …other?

5. Task 2  Write down two numbers that multiply to give 48  Write down two numbers that multiply to give 48, and one of them is odd  Write down two numbers that multiply to give 48, and one of them is not an integer  Write down two numbers that multiply to give 48, and ….. (make it harder)

6. Task types  Make an example  Construct another example with constraints which push you away from easy cases

7. Task affordances  Go beyond integers  Engage with multiplication beyond ‘times tables’  … other

8. Task 3  Construct a two-dimensional shape  … with four straight sides  … and with two opposite sides equal  … and with two pairs of adjacent angles equal  What other properties follow?  Change one of the constraints

9. Task types  Construct an example for which you are given minimum essential information  … then vary one of the constraints  … or add unusual constraints

10. Task affordances  …… ?????

11. Task 4 4 2 - 3 2 = 4 + 3 5 2 - 4 2 = 5 + 4 6 2 - 5 2 = 6 + 5 7 2 - ……. 4 2 - 2 2 = 4 + 2 ??? 5 2 - 3 2 = 6 2 - 4 2 = 7 2 - …….

12. A design principle for exploring example spaces Dimensions of possible variation Ranges of permissible change

13. Task types  Systematic example generation in a range of change (small positive integers here!)  Make an initial generalisation  Change a dimension of variation  Make a further generalisation  … and so on

14. Summary of task types  Give an example  Reaching for something obvious to you  Constructing something special  Construct with constraints which push you beyond obvious examples  Construct with minimum information: vary the information  Extend the range of change and vary a different dimension

15. Further task types  Give a hard example of …  Give an easy example of …  Give an example using …  Give an example using three different representations  Give an example of … using something new you learnt last week

16. Example spaces  Initially we reach for obvious examples (concept images; canonical examples; model examples)  Learning can be seen as a process of exploring, enriching, reorganising and extending example spaces  Constructing new objects can ‘force’ exploration, enrichment, reorganisation, extension of a personal available example space

17. A. Watson & J. Mason: Mathematics as a Constructive Activity: Learners generating examples published by Lawrence Erlbaum Associates  Colours chosen by George