Expressing Generality and the role of Attention in Mathematics

1. The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking Expressing Generalityand the role of Attentionin Mathematics John Mason SEMAT Køge Denmark March 2014

2. Copy & Extend … …

3. Reproduction What are you doing with your attention? Copy this shape Now extend it To this shape Now extend it further according to same principle Explain your choice to someone else

4. Count How did you do it? • How many white squares are there? How were you attending? • In how many different ways can you count the white squares? • Count blacks and subtract • Notice 6 rows of 3 blacks & 3 rows of 1 black • Notice 6 columns of 3 black & 3 columns of 1 black • Notice diagonal stripes of blacks • Columns: 3 lots of (3 + 3 + 1) blacks • Similar for whites Notice the effect on your attention between these two task types Notice what you do with your attention to make sense of these ways Resolución de Problemas de Final Abierto Chile 2013 p67

5. – = What’s The Difference? What then would be the difference? What then would be the difference? First, add one to each First, add one to the larger and subtract one from the smaller What could be varied?

6. Triangle Count • In how many different ways might you count the triangles? 15 x 4 + 1 What are you attending to?

7. Differing Sums of Products 4 7 3 5 • Write down four numbers in a 2 by 2 grid • Add together the products along the rows 28 + 15 = 43 • Add together the products down the columns 20 + 21 = 41 43 – 41 = 2 • Calculate the difference Action! • Now choose different numbers so that the answer is again 2 • Choose numbers so that the answer is … Undoing the Action!

8. Differing Sums of Products 4 7 3 5 • Tracking Arithmetic 4x7 + 5x3 4x5 + 7x3 4x(7–5) + (5–7)x3 So how can an answer of 2 be guaranteed? = 4x(7–5) – (7-5)x3 = (4-3)x (7–5) In how many essentially different ways can 2 be the difference? What is special about 2?

9. Think Of A Number (ThOAN) 7 + 2 7 • Think of a number • Add 2 • Multiply by 3 • Subtract 4 • Multiply by 2 • Add 2 • Divide by 6 • Subtract the number you first thought of • Your answer is 1 3x + 6 7 7 3x + 2 6x + 4 7 6x + 6 7 + 1 7 1

10. Doing & Undoing • What operation undoes ‘adding 3’? • What operation undoes ‘subtracting 4’? • What operation undoes ‘subtracting from 7’? • What are the analogues for multiplication? • What undoes ‘multiplying by 3’? • What undoes ‘dividing by 4’? • What undoes ‘multiplying by 3/4’? • Two different expressions! • Dividing by 3/4 or Multiplying by 4 and dividing by 3 • What operation undoes dividing into 12?

11. Composite Doing & Undoing What’s my number? I am thinking of a number … I add 8 and the answer is 13. What’s my number? I add 8 and then multiply by 2;the answer is 26. I add 8; multiply by 2; subtract 5;the answer is 21. What’s my number? I add 8; multiply by 2; subtract 5; divide by 3;the answer is 7. What’s my number? HOW do you turn +8, x2, -5, ÷3 answer into a solution? 7 Generalise!

12. Tabled Variations

13. Raise your hand when you can see … • Something that is 3/5 of something else • Something that is 2/5 of something else • Something that is 2/3 of something else • Something that is 5/3 of something else • What other fraction-actions can you see? How did your attention shift? Flexibility in choice of unit

14. Raise your hand when you can see … Something that is 1/4 – 1/5of something else Did you look for something that is 1/4 of something else and for something that is 1/5 of the same thing? Common Measure What did you have to do with your attention? What do you do with your attention in order to generalise?

15. Stepping Stones … R What needs to change so as to ‘see’ that … R+1

16. SWYS , , , , , of something else Find something that is What is the same, and what is different? Find something that is of of something else Find something that is of of something else

17. Consecutive Sums Say What You See

18. Diamond Multiplication

19. Sundaram’s Grid The grid extends up and to the right by arithmetic progressions Claim: N will appear in the table iff 2N + 1 is composite What number will appear in the Rth row and the Cth column?

20. Two Journeys • Which journey over the same distance at two different speeds takes longer: • One in which both halves of the distance are done at the specified speeds? • One in which both halves of the time taken are done at the specified speeds? time distance

21. Counting Out • In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37). Which object will you end on? A B C D E 1 2 3 4 5 9 8 7 6 10 … If that object is elimated, you start again from the ‘next’. Which object is the last one left?

22. Set Ratios • In how many different ways can you place 17 objects so that there are equal numbers of objects in each of two possibly overlapping sets? • What about requiring that there be twice as many in the left set as in the right set? • What about requiring that the ratio of the numbers in the left set to the right set is 3 : 2? • What is the largest number of objects that CANNOT be placed in the two sets in any way so that the ratio is 5 : 2? S1 S2 What can be varied?

23. Bag Reasoning • Axiom: Every bean is in some bag. • Axiom: There are finitely many bags. • Definition: a bean b is loose in a bag B means that although b is in B, b is not in any bag that is in B. • Must every bean be loose in some bag? How do you know? • Definition: the number of beans in bag B is the total number of beans that would be loose in bag B if all the bags within B disintegrated. • How few beans are needed in order that for k = 1, 2, …, 5, there is exactly one bag with k beans? • What is the most beans there could be? • Can it be done with any number of beans between the least and the most?

24. What Do I know? √2 (√2)2 = 2 x2 = 2 & x > 0 => x = √2 x2 = 3 & x > 0 => x = √3 √3 (√3)2 = 3 √3 – √2 (X2 – 5)2 = 24 X4 – 10x2 + 1 = 0 & 1 > x > 0 => x = √3 – √2 √3 + √2 (X2 – 5)2 = 24 X4 – 10x2 + 1 = 0 & x > 1 => x = √3 + √2 ?

25. -2 2 0 -4 Covered Up Sums 4 8 6 2 -1 1 5 3 • Cover up one entry from each row and each column. Add up the remaining numbers. • The answer is (always) the same! • Why? -1 3 1 -3 Stuck? Specialise! Example of seeking invariant relationships Example of focusing on actions preserving an invariance Opportunity to generalise

26. Covered Up Sums • How much freedom of choice do you have when making up your own? b c a d e a f a b b e ? g e-(a-b) e c f d g Opportunity to generalise Opportunity to quantify freedom of choice

27. Marbles 1 • If Anne gives one of her marbles to John, they will have the same number of marbles. • What can you say about the number of marbles they started with? • Buses • If one person got off the first bus and got onto the second, there would be the same number of people in both buses. • Ages • If A was one year older and B was one year younger, then A and B would be the same age. 1/4

28. Marbles 2 • If Anne gives one of her marbles to John, they will have the same number of marbles;if John now gives two of his marbles to Kathy, they will have the same number. • What can you say about the relation between Anne’s and Kathy’s marbles to start with. • Buses • If one person were to get off the first bus and onto the second, then the buses would have the same number of passengers;if two people then got off the second bus and got onto the third bus, those two buses would have the same number. 2/4

29. Marbles 3 • If Anne gives John one of her marbles, she will then have one more than twice as many marbles as John then has. • If John started with 12 marbles, how many did Anne start with? • Buses • If one passenger gets off the first bus and onto the second bus, then there will be one more than twice as many passengers on the first bus as on the second. • If the second bus started with 12 passengers, how many did the first bus start with? 3/4

30. Marbles 4 • If Anne gives John one of her marbles, she will then have one more than twice as many marbles as John then has. • However, if instead, John gives Anne one of his marbles, he will have one more than a third as many marbles as Anne then has. • How many marbles have they each currently? • Buses!

31. Frameworks Doing – Talking – Recording(DTR) (MGA) See – Experience – Master(SEM) Enactive – Iconic – Symbolic(EIS) Specialise … in order to locate structural relationships … then re-Generalise for yourself Stuck? What do I know? What do I want?

32. Mathematical Thinking • How describe the mathematical thinking you have done so far today? • How could you incorporate that into students’ learning?

33. Possibilities for Action • Trying small things and making small progress; telling colleagues • Pedagogic strategies used today • Provoking mathematical thinking as happened today • Question & Prompts for Mathematical Thinking (ATM) • Group work and Individual work

34. Imagery Awareness (cognition) Will Emotions (affect) Body (enaction) HabitsPractices Human Psyche

35. Three Only’s Language Patterns& prior Skills Imagery/Sense-of/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise;dispositions Techniques & Incantations Standard Confusions & Obstacles Emotion Behaviour Awareness Only Emotion is Harnessable Only Awareness is Educable Only Behaviour is Trainable

36. Follow Up • j.h.mason @ open.ac.uk • mcs.open.ac.uk/jhm3  Presentations • Questions & Prompts (ATM) • Key ideas in Mathematics (OUP) • Learning & Doing Mathematics (Tarquin) • Thinking Mathematically (Pearson) • Developing Thinking in Algebra (Sage)