1. 1 Learning to Think and to Reason Algebraically and the Structure of Attention 2007 John Mason SMC

2. 2 Outline  Some assumptions  Some tasks  Some reflections

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

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

5. 5 Grid Movement 7 ? +3-3 x2 ÷2 ((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 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

6. 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. 7 Number Line Movements   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! Imagine a number line

8. 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 1 4 9 16 25 49 36 Number Spirals

9. 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 64 81

10. 10 CopperPlate Multiplication

11. 11 Four Odd Sums

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

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

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

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

16. 16 One More What numbers are one more than the product of four consecutive integers? 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. 17 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.

18. 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. 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. 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. 21 Magic Square Reasoning 519 2 4 6 83 7 –= 0Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

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

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?

24. 24 Gasket Sequences

25. 25 Toughy 12345678

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

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

28. 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. 29 Attention  Gazing at wholes  Discerning details  Recognising relationships  Perceiving properties  Reasoning on the basis of properties

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