1. 1 Making the Most of Mathematical Tasks John Mason Overton Jan 2011 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Aims  To develop strategies for promoting learning from experience  To develop questioning that promotes extension, variation, and generalisation  To consider a variety of tasks which can be used to stimulate reasoning

3. 3 Teaching & Learning  Children are given mathematical tasks to do  Tasks stimulate activity  Activity provides experience –of the use of their powers –of mathematical themes –of mathematical topics, techniques, reasoning …  Experience may contribute to learning –especially when learners are prompted to withdraw from activity and reflect upon it

4. 4 What did you have to do to accomplish this?  Make a copy of the following repeating pattern Reproduction

5. 5 Children’s Copied Patterns 4.1 yrs Marina Papic MERGA 30 2007 model

6. 6 Children’s Own Patterns Marina Papic MERGA 30 2007 5.4 yrs 5.0 yrs5.1 yrs

7. 7 Patterned Wheels … An inked roller has made at least two full revolutions What colour is the 100 th square? Where is the 100 th red square?

8. 8 Order! Order!  A, B, C, D, and E are in a queue –B is in front of C –A is behind E –There are two people between D and E –There is one person between D and C –There is one person between B and E BC EA BCEA BCEA BCEAD What did you do?

9. 9 Say What You See  There are 16 canoes 5 asteroids 4 wedges 4 peaks and these account for the total area Also 6 arches; 6 troughs; What did you do?

10. 10 Same & Different 1012 156 What distinguishes it from the others? Pick an entry.

11. 11 Revealing Shapes Applet

12. 12 And Another  Write down two numbers that differ by 3 –And another pair  Write down two numbers that differ by 3 that you think no-one else will write down  Write down two numbers that differ by 3 and that make that difference as obscure as possible

13. 13 Smallest Unique  Write down a positive number that you think no-one else will write down  The ‘winner’ is the person who writes down the smallest such number!

14. 14 What’s The Difference? What could be varied? –= First, add one to each First, add one to the first and subtract one from the second What then would be the difference?

15. 15 What’s The Ratio? What could be varied? ÷= First, multiply each by 3 First, multiply the first by 2 and divide the second by 3 What is the ratio?

16. 16 Marbles (Bob Davis)  I have a bag of marbles  I take out 7, then put in 3, then take out 4. What is the state of my bag now? –Variations?

17. 17 Speed Reasoning  If I run 3 times as fast as you, how long will it take me compared to you to run a given distance?  If I run 2/3 as fast as you, how long will it take me compared to you?

18. 18 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 ¾ ’? Two different expressions! Two different expressions!

19. 19 Additive & Multiplicative Perspectives  What is the relation between the numbers of squares of the two colours?  Difference of 2, one is 2 more: additive  Ratio of 3 to 5; one is five thirds the other etc.: multiplicative

20. 20 Raise your hand when you can see  Something which is 2/5 of something  Something which is 3/5 of something  Something which is 2/3 of something –What others can you see?  Something which is 1/3 of 3/5 of something  Something which is 3/5 of 1/3 of something  Something which is 2/5 of 5/2 of something  Something which is 1 ÷ 2/5 of something

21. 21 Why is (-1) x (-1) = 1?

22. 22 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

23. 23 More Magic Square Reasoning –= 0Sum( )Sum( )

24. 24 Teaching  Selecting tasks  Preparing Didactic Tactics and Pedagogic Strategies  Prompting extended or fresh actions  Being Aware of mathematical actions  Directing Attention Teaching takes place in time; Learning takes place over time

25. 25 The Place of Generality  A lesson without the opportunity for learners to generalise mathematically, is not a mathematics lesson

26. 26 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

27. 27 Some Mathematical Powers Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Organising & Characterising

28. 28 Rich tasks, Rich Use of tasks  It may not be the task that is rich  But the way the task is used

29. 29 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint

30. 30 For More Details Thinkers (ATM, Derby) Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby) Mathematics as a Constructive Activity (Erlbaum) Thinking Mathematically (new edition) mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk Structured Variation Grids Revealing Shapes Other Publications This and other presentations