Exploring example spaces: what are they like and how do we move around them? Anne Watson Jasper, October 2006

2,4,6,8 … 5,7,9,11 … 9,11,13,15 …

► Make up a similar sequence of your own for which your neighbour will find the sum of the first five terms.

2, 4, 6, 8 … 2, 5, 8, 11 … 2, 23, 44, 65 …

► Make up a similar sequence of your own for which your neighbour will find the sum of the first five terms.

2,4,6,8 … 3,6,9,12 … 4,8,12,16 …

► Make up a similar sequence of your own for which your neighbour will find the sum of the first five terms.

Principle 1 ► All learners have a natural propensity to  see patterns,  to seek structure,  classify,  generalise ….

2,4,6,8 … 5,7,9,11 … 9,11,13,15 …

2, 4, 6, 8 … 2, 5, 8, 11 … 2, 23, 44, 65 …

2,4,6,8 … 3,6,9,12 … 4,8,12,16 …

Principle 2 ► Example spaces can be characterised by their dimensions of variation and ranges of of change

The largest … ► Sketch a quadrilateral whose sides are all equal in length. Area? ► Sketch a quadrilateral for which two pairs of sides are equal in length, and which has the largest possible area. ► Sketch a quadrilateral for which three lines are equal in length, and which has the largest possible area. ► … same for no lines equal

Principle 3 ► Constraints make the problem more interesting/ harder/ more conceptual

► Write down a pair of numbers which have a difference of 2 ► ….. and another pair

Principle 4 ► Example spaces are individual, and learners can be prompted to extend their example spaces

► Write down a pair of numbers which have a difference of 9 ► ….. and another pair

► On a nine-pin geoboard, create a triangle which has a height of two units.  and another ► Using dynamic geometry software, find the class of triangles which have a height of two. ► Construct a triangle which has a height of two and a height of one.  and another

Principle 5 Example spaces are dependent on context and tools Example spaces are dependent on context and tools

Example of what?

Principle 6 ► Examples have to be examples of something:  classes of objects  concepts  techniques  problems and questions  appropriate objects which satisfy certain conditions  ways of answering questions  ways to construct proofs  …. so on

Sorting examples ► Think of a number ► Add 3 to it and also subtract 3 from it; also multiply it by 3 and divide it by 3 ► Now put your four answers in increasing order, and label then with their operations ► If you change the 3 to something else, is the order always the same for your starting number? ► If you change your starting number, but preserve 3, what different orders can you achieve? ► What if you change both the starting number and the 3?

Principle 7 ► You can explore and extend your example spaces by:  sorting  comparing  combining  … what else?

Can you see any fractions?

Can you see 1 ½ of something?

Principle 8 ► The process of creating examples is dependent on the way it is prompted

Examples of methods ► Think of as many ways as you can to enlarge a rectangle by a scale factor of 2

► Sequences:  what does “like this” mean?  we all look for patterns ► Quadrilateral  start from what we know and make it harder by adding constraints ► Difference of 2 ..and another – push beyond the obvious ► Triangle with height 2  fix properties to encourage play with concepts ► Grid - of what?  similarity as a tool, or as a muddle? ► Use 3 to +, -, ×, ÷  Using learners’ own example spaces to sort, compare, relate … ► Seeing fractions  open/closed questions ► Enlarging rectangles  shifting to more powerful methods

Principle 1 ► All learners have a natural propensity to  see patterns,  to seek structure,  classify,  generalise ….

Principle 2 ► Example spaces can be characterised by their dimensions of variation and ranges of of change

Principle 3 ► Constraints make the problem more interesting/ harder/ more conceptual

Principle 4 ► Example spaces are individual, and learners can be prompted to extend their example spaces

Principle 5 Example spaces are dependent on context and tools Example spaces are dependent on context and tools

Principle 6 ► Examples have to be examples of something:  classes of objects  concepts  techniques  problems and questions  appropriate objects which satisfy certain conditions  ways of answering questions  ways to construct proofs  …. so on

Principle 7 ► You can explore and extend your example spaces by:  sorting  comparing  combining  … what else?

Principle 8 ► The process of creating examples is dependent on the way it is prompted

If I had to describe my conclusion [as to a method of studying] in one word, I’d say examples. They are to me of paramount importance. Every time I learn a new concept I look for examples … (Halmos 1985).

On arriving at any new rule or process, the student should work a number of examples sufficient to prove to himself that he understands and can apply the rule or process in question…. He may choose an example for himself, and his previous knowledge will suggest some method of proving whether his result is true or not. (De Morgan 1831).