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

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

Task types  Example giving  … and another

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

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)

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

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

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

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

Task affordances  …… ?????

Task = = = …… = ??? = = …….

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

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

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

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

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

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