1. 1 While you are waiting: Warm Up 1: in a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both? Warm Up 2: in a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or both and 23 who are either musicians or poets or both. How many people are all three: poets, painters and musicians?

2. 2 Resolving 31 47 total 29 47–3147–29 31–(47–29) 29–(47–31) poets painters In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both?

3. 3 14 11 15 musicians poets painters 28 total 23 musicians or painters 21 poets or musicians 22 poets or painters 28–2328–21 28–22 14+15– 21 11+15– 23 (14+15-21) + (14+11-22) + (11+15-23) – (28– ((28-23) + (28-22) + (28-21)) 2 14+11– 22 In a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or both and 23 who are either musicians or poets or both. How many people are all three: poets, painters and musicians?

4. 4 Learning & Teaching John Mason AIMSSEC Jan 2010 The Open University Maths Dept University of Oxford Dept of Education

5. 5 Reflections  Re-entering a situation re-constructing what you noticed preparing to act in the future  Doing some actions; Talking about those actions; Recording those actions; reflecting on the effects of those actions  Do–Talk–Record

6. 6 Reflections  Do you recall being invited to talk to a neighbour (talking in pairs)? –Getting you asserting (taking initiative, conjecturing) –Not simply assenting to what others say  Do you recall being asked to stop and reflect on what you had been doing? –Tasks ––> Activity ––> Experience –BUT withdrawing from the action and considering the effects of those actions, the way you use your powers, is vital in order to learn from that experience. Re-entering a situation re-constructing what you noticed preparing to act in the future

7. 7 Finding The Question  You are given cards 123456  Arrange the cards to make the calculation true: 413265 148 = – What might this do for learners? What could I change in the mathematics of the task so that through practice learners integrate that experience? What could I change in the format of the task? Why might I? What might this do for learners? What could I change in the mathematics of the task so that through practice learners integrate that experience? What could I change in the format of the task? Why might I? Re-entering a situation re-constructing what you noticed preparing to act in the future

8. 8 Worlds of Mathematics  Enactive: doing, manipulating  Iconic: imagining  Symbolic: presenting with symbols Doing Talking Recordin g Manipulating … Getting a sense of … Articulating … Relationships ––> Properties … what? why ?

9. 9 Revealing Shapes The coloured shapes on the right are somewhere in the white cells on the left. By clicking on a shape and then on a cell, you can test your conjectures as to which shape is where.

10. 10 Reflections  What actions were you called upon to perform?  What was the effect on you?  What does the task offer learners?  What can be changed in the content of the task?  What can be changed in the format of the task? Re-entering a situation re-constructing what you noticed preparing to act in the future

11. 11 Addition Table Completions totals How many entries do you need in order to fill out the table? 27 51 60-9 totals How many entries do you need in order to fill out a table with r rows and c columns altogether? What can you predict already? 69 -36 1845 15

12. 12 Reflections  What actions were you called upon to perform?  What was the effect on you?  What does the task offer learners?  What can be changed in the content of the task?  What can be changed in the format of the task? Re-entering a situation re-constructing what you noticed preparing to act in the future

13. 13 Reconstructing Multiplication Tables x 2 5 7 3 61521 5102535 6123042

14. 14 Generalising by Specialising x b/sc/s sa/b a bc/a s b c x 50/s60/s 12s /25 2412 5 s50 60 What about fractions? What about quadratics?

15. 15 Reflections  What actions were you called upon to perform?  What was the effect on you?  What does the task offer learners?  What can be changed in the content of the task?  What can be changed in the format of the task? Re-entering a situation re-constructing what you noticed preparing to act in the future

16. 16 Mill-Dienes-Bloor Animations

17. 17 Reflections  What actions were you called upon to perform?  What was the effect on you?  What does the task offer learners?  What can be changed in the content of the task?  What can be changed in the format of the task? Re-entering a situation re-constructing what you noticed preparing to act in the future

18. 18 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 Any colour-symmetric arrangement?

19. 19 More Magic Square Reasoning –= 0Sum( )Sum( )

20. 20 Reflections  What actions were you called upon to perform?  What was the effect on you?  What does the task offer learners?  What can be changed in the content of the task?  What can be changed in the format of the task? Re-entering a situation re-constructing what you noticed preparing to act in the future

21. 21 Worlds of Mathematics  Enactive: doing, manipulating  Iconic: imagining  Symbolic: presenting with symbols Doing Talking Recordin g Manipulating … Getting a sense of … Articulating … Relationships ––> Properties … what? why ?

22. 22 My best Wishes … … may you continue to enjoy your teaching more and more JohnM http://mcs.open.ac.uk/jhm3