1. 1 Thinking Mathematically as Developing Students’ Powers John Mason Oslo Jan 2009 The Open University Maths Dept University of Oxford Dept of Education

2. 2 Assumptions  What you get from this session will be largely what you notice happening for you  If you do not participate, I guarantee you will get nothing!  I assume a conjecturing atmosphere –Everything said has to be tested in experience –If you know and are certain, then think and listen; –If you are not sure, then take opportunities to try to express your thinking  Learning is a maturation process, and so invisible –It can be promoted by pausing and withdrawing from the immediate action in order to get an overview

3. 3 Outline  Some tasks to work on together  Some remarks about what might have been noticed Each task indicates: a domain of similar tasks a style or structure of tasks More important than particular tasks: ways of working with learners ON tasks

4. 4 Imagining & Expressing Where can the centre get to? … a fixed point P and a circle passing through P … two distinct fixed points P and Q and a circle passing through both points … three distinct points P, Q & R and a circle passing through all three points Where can the centre get to? Imagine a mathematical plane, and lying in it, a … … circle

5. 5 Only then Check! One Sum I have two numbers which sum to 1 Which will be larger: The square of the larger added to the smaller? The square of the smaller added to the larger? Make a Conjecture!

6. 6 One Sum Diagrams 1 1 (1- ) 2 Anticipating, not waiting 1- 2 2 (1- )+ (1- ) 2 + =

7. 7 Reading a Diagram x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

8. 8 Triangle Count

9. 9 Variation  Dimensions-of-possible-variation  Range-of-permissible-change  Invariance in the midst of change

10. 10 Structured Variation Grids

11. 11 Up & Down Sums 1 + 3 + 5 + 3 + 13 x 4 + 12 2 + 3 2 1 + 3 + … + (2n–1) + … + 3 + 1 == n (2n–2) + 1 (n–1) 2 + n 2 = = Generalise! See generality through a particular

12. 12 Reading Graphs  Imagine the graph of a cubic polynomial  Imagine also the graph of a quartic  Imaging also the graph of y = x  Now, imagine a point x on the x-axis; –proceed vertically up (or down) to the cubic; –proceed horizontally to the line y=x –proceed vertically up (or down) to the quartic –proceed horizontally until you are directly in vertical line with the x you started with

13. 13 Cubical Property  Imagine a cubic  Imagine a chord, extended to a line; Find the midpoint of your chord  Imagine a second chord with the same midpoint; extend it to a line  What do you imagine will happen?

14. 14 Chord-slopes  Imagine a quartic polynomial  Imagine an interval of fixed width on the x-axis  The interval determines a chord. The mid-point of the chord is marked  The slope of the chord is shown

15. 15 Kites

16. 16 Powers  Am I stimulating learners to use their own powers, or am I abusing their powers by trying to do things for them? –To imagine & to express –To specialise & to generalise –To conjecture & to convince –To stress & to ignore –To extend & to restrict

17. 17 Reflection  What did you notice happening for you mathematically?  What might you be able to use in an upcoming lesson?  Imagine yourself in the future, using or developing or exploring something you have experienced today!

18. 18 More Resources  Questions & Prompts for Mathematical Thinking (ATM Derby: primary & secondary versions)  Thinkers (ATM Derby)  Mathematics as a Constructive Activity (Erlbaum)  Designing & Using Mathematical Tasks (Tarquin)  http: //mcs.open.ac.uk/jhm3  j.h.mason @ open.ac.uk