1. Mathematical thinking and task design
Anne Watson
Singapore, 2012

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

3. 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
.....

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

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

6. Principle 3
Learner responses are individual, and learners can be prompted to extend their responses beyond the obvious

7. Principle 4
Learning is dependent on context, representation and tools

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

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

10. 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?

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

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

13. 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

14. 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)