Learning to Think and to Reason Algebraically and the Structure of Attention

1. Learning to Think and to ReasonAlgebraicallyand the Structure of Attention John MasonSMC 2007

2. Outline • Some assumptions • Some tasks • Some reflections

3. Some assumptions • A lesson without the opportunity for learners to generaliseis not a mathematics lesson • Learners come to lessons with natural powers to make sense • Our job is to direct their attentionappropriately and effectively

4. ? ? 7 Grid Sums To move to the right one cell you add 3. To move up one cell you add 2. In how many different ways can you work out a value for the square with a ‘?’ only using addition? Using exactly two subtractions?

5. Grid Movement ((7+3)x2)+3 is a path from 7 to ‘?’. What expression represents the reverse of this path? • What values can ‘?’ have: • if only + and x are used- if exactly one - and one ÷ are used, with as many + & x as necessary x2 ? ÷2 7 What about other cells?Does any cell have 0? -7? Does any other cell have 7? Characterise ALL the possible values that can appear in a cell -3 +3

6. Varying & Generalising • What are the dimensions of possible variation? • What is the range of permissible change within each dimension of variation? • You only understand more if you extend the example space or the scope of generality

7. Number Line Movements Imagine a number line • Rotate it about the point 5 through 180° • where does 3 end up? • … • Now rotate it again bit about the point -2. • Now where does the original 3 end up? • Generalise!

8. Number Spirals 43 45 46 47 48 49 50 44 49 42 21 22 23 24 25 26 25 41 20 10 27 9 40 19 5 4 3 7 8 2 6 1 1 9 11 28 39 18 12 29 4 38 17 16 15 14 13 30 16 35 34 33 36 37 36 32 31

9. 43 45 46 47 48 49 50 44 42 21 22 23 24 25 26 41 20 10 27 40 19 1 8 9 7 6 5 4 3 2 11 28 39 18 12 29 38 17 16 15 14 13 30 35 34 33 64 81 37 36 32 31

10. CopperPlate Multiplication

11. Four Odd Sums

12. With the Grain Across the Grain Tunja Sequences -1 x -1 – 1 = -2 x 0 0 x 0 – 1 = -1 x 1 1 x 1 – 1 = 0 x 2 2 x 2 – 1 = 1 x 3 3 x 3 – 1 = 2 x 4 3 x 5 4 x 4 – 1 =

13. … … Tunja Display (1) … (-1)x2 - (-1) - 2 = (-2)x3 - 1 (-1)x3 - (-1) - 3 = (-2)x2 - 1 … 0x3 - 0 - 3 = (-1)x2 - 1 0x2 - 0 - 2 = (-1)x3 - 1 … 1x2 - 1 - 2 = 0x3 - 1 1x3 - 1 - 3 = 0x2 - 1 … 2x3 - 2 - 3 = 1x2 - 1 2x2 - 2 - 2 = 1x1 - 1 … 3x3 - 3 - 3 = 2x2 - 1 3x2 - 3 - 2 = 2x1 - 1 … 4x3 - 4 - 3 = 3x2 - 1 4x2 - 4 - 2 = 3x1 - 1 … 5x3 - 5 - 3 = 4x2 - 1 5x2 - 5 - 2 = 4x1 - 1 … … … Run Backwards Generalise!

14. Tunja Display (2) … 4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2 4x0x2 - 2x0 - 4x2 = (-2)x3 - 2 4x1x2 - 2x1 - 4x2 = 0x3 - 2 4x2x2 - 2x2 - 4x2 = 2x3 - 2 4x3x2 - 2x3 - 4x2 = 4x3 - 2 4x3x3 - 2x3 - 4x3 = 4x5 - 2 4x4x2 - 2x4 - 4x2 = 6x3 - 2 4x4x3 - 2x4 - 4x3 = 6x5 - 2 4x5x2 - 2x5 - 4x2 = 8x3 - 2 4x5x3 - 2x5 - 4x3 = 8x5 - 2 4x6x2 - 2x6 - 4x2 = 10x3 - 2 4x6x3 - 2x6 - 4x3 = 10x5 - 2 … … Run Backwards Generalise!

15. Structured Variation Grids Generalisations in two dimensions Available free at http://mcs.open.ac.uk/jhm3

16. One More What numbers are one more than the product of four consecutive integers? Let a and b be any two numbers, at least one of them even. Then ab/2 more than the product of: any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared.

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

18. 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?

19. 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 • …

20. 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 • …

21. 2 6 7 2 1 5 9 2 8 3 4 Sum( – Sum( ) = 0 ) Magic Square Reasoning What other configurationslike thisgive one sumequal to another? Try to describethem in words

22. Sum( ) – Sum( ) = 0 More Magic Square Reasoning

23. Perforations If someone claimedthere were 228 perforationsin a sheet, how could you check? How many holes for a sheet ofr rows and c columns of stamps?

24. Gasket Sequences

25. Toughy 1 2 3 4 5 6 7 8

26. Powers • Specialising & Generalising • Conjecturing & Convincing • Imagining & Expressing • Ordering & Classifying • Distinguishing & Connecting • Assenting & Asserting

27. Themes • Doing & Undoing • Invariance Amidst Change • Freedom & Constraint • Extending & Restricting Meaning

28. Some Reflections • Notice the geometrical term: • It requires movement out of the current space into a space of one higher dimension in order to achieve it

29. Attention • Gazing at wholes • Discerning details • Recognising relationships • Perceiving properties • Reasoning on the basis of properties

30. John Mason • J.h.mason @ open.ac.uk • http://mcs.open.ac.uk/jhm3 Developing Thinking in Algebra (Sage 2005)