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

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

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

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

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

6. Principle 5
Learning is dependent on context, representation and tools

7. 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

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

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

10. 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?

11. 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)

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

13. … 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

14. 2,4,6,8 …
5,7,9,11 …
9,11,13,15 …

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

16. 2, 4, 6, 8 …
2, 5, 8, 11 …
2, 23, 44, 65 …

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

18. 2,4,6,8 …
3,6,9,12 …
4,8,12,16 …

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

20. 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

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

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

23. 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?

24. Can you see any fractions?

25. Can you see 1 ½ of something?

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

29. 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