1. 1 Making Use of Students’ Natural Powers to Think Mathematically John Mason Grahamstown May 2009 The Open University Maths Dept University of Oxford Dept of Education

2. 2 Some Sums 4 + 5 + 6 = 9 + 10 + 11 + 12 16 Generalise Justify Watch What You Do Say What You See 1 + 2 =3 7 + 8 = 13 + 14 + 15 17 + 18 + 19 + 20+ = 21 + 22 + 23 + 24

3. 3 Consecutive Sums Say What You See

4. 4 CopperPlate Calculations

5. 5 Difference Divisions 4 – 2 = 4 ÷ 2 4 – 3 = 4 ÷ 3 1 2 1 2 5 – 4 = 5 ÷ 4 1 3 1 3 6 – 5 = 6 ÷ 5 1 4 1 4 7 – 6 = 7 ÷ 6 1 5 1 5 3 – 2 = 3 ÷ 2 1 1 1 1 0 – (-1) = 0 ÷ (-1) 1 -2 1 2 1 oops 1 – 0 = 1 ÷ oops 1 1 How does this fit in? Going with the grain Going across the grain

6. 6 Leibniz’s Triangle 1

7. 7 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 5  and another  Write down one which you think no- one else here will write down.

8. 8 Remainders of the Day (2)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

9. 9 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4  … … … …

10. 10 Remainders of the Day (4)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4  … … … …

11. 11 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

12. 12 Triangle Count

13. 13 Max-Min 256832 417761 294689 589825 972198 371969

14. 14 Max-Min  In a rectangular array of numbers, calculate –The maximum value in each row, and then the minimum of these –The minimum in each column and then the maximum of these  How do these relate to each other?  What about interchanging rows and columns?  What about the mean of the maxima of each row, and the maximum of the means of each column?

15. 15 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

16. 16 Differences Anticipating Generalising Rehearsing Checking Organising

17. 17 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

18. 18 Reflections  Much of mathematics can be seen as studying actions on objects  Frequently it helps to ask yourself what actions leave some relationship invariant; often this is what is studied mathematically

19. 19 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

20. 20 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 1 4 9 16 25 49 36

21. 21 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 64 81

22. 22 Gasket Sequences

23. 23 Perforations How many holes for a sheet of r rows and c columns of stamps? If someone claimed there were 228 perforations in a sheet, how could you check?