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
= ??? = =
72 - …….
= 4 + 3 = =
72 - …….

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