Mathematical thinking and task design Anne Watson Singapore, 2012

Principle 1 All learners have a natural propensity to see patterns to seek structure classify generalise compare describe

Principle 2 Tasks can be characterised by their dimensions of variation and ranges of change

Principle 3 Constraints make mathematics more interesting/ harder/ more conceptual

Principle 4 Learner responses are individual, and learners can be prompted to extend their responses beyond the obvious

Principle 5 Learning is dependent on context, representation and tools

Principle 6 The objects we work with in mathematics include: classes of objects concepts techniques problems and questions appropriate objects which satisfy certain conditions ways of answering questions ways to construct objects …. so on

Principle 7 People explore and extend their ideas by: sorting comparing combining … what else?

Principle 8 The way a task is done is dependent on the way it is prompted and the norms of the classroom

Learning from experience Patterns in layout Patterns of digits Familiarity Generality Going beyond mere answers Drawing from other experience of looking at patterns of layout and digits, something vaguely familiar, to identify a generality here Going beyond the mere generation of answers Davydov related? Across the Grain?

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 The teacher provides a range of particular examples of some general structure, method, class of mathematical objects etc. in a classroom context in which these can be discussed, named, played with etc. From these experiences, learners develop personal images of a concept, including the associated language, notations, examples, uses Classroom mathematical ideas are a mixture of natural and scientific concepts (Vygotsky) or intuitive and formal understandings (Fischbein)

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

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

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.

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

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

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

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?

Can you see any fractions?

Can you see 1 ½ of something?

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

Sequences: Quadrilateral Difference of 2 Triangle with height 2 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