Mathematical thinking and task design Anne Watson Singapore, 2012

Toolkit from experience control variables vary one thing systematically superimpose patterns combine aspects put constraints within combinations turn questions round

Create a need to: bring in new areas of mathematics seek relations express generality compare representations see old ideas from a new direction use pattern as a purposeful tool .....

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

Principle 4 Learning is dependent on context, representation and tools

Principle 5 Constraints make mathematics more interesting/ harder/ more conceptual

Principle 6 The way a task is done is dependent on the way it is prompted and the norms of the classroom

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

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

Principle 8 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

Conceptual development Tasks, and the ways they are presented, mediate formal mathematical ideas for learners Multiple examples: given or constructed Natural/scientific concepts: how introduced? Intuitive/formal understanding: how shifted? Further experience to embed new ideas 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)