1. 1 Mathematics: with good reason John Mason Exeter April 2010 The Open University Maths Dept University of Oxford Dept of Education

2. 2 Aims  To experience shifts from ”It just is” to “It must be because …”  To consider a variety of tasks which can be used to stimulate reasoning

3. 3 Revealing Shapes

4. 4 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

5. 5 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;

6. 6 Bag Re-Constructions  Here there are 3 bags and two objects.  There are [0,1,2;2] objects in the bags with 2 altogether  Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there is a corresponding set of bags?

7. 7 Why is (-1) x (-1) = 1?

8. 8 Fractional Increase and Decrease (1 + ) 1 2 (1 – ) 1 3 = (1 + ) 2 5 (1 – ) 2 7 = = 1 (1 – ) (1 + ) a b By how much do I have to decrease in order to undo an increase by one-half? By how much do I have to increase in order to undo a decrease by two- sevenths? (1 + ) 3 8 (1 – ) 31 = = 1(1 – ) (1 + ) a b Make up your own! b a+ba+b 1 1 1

9. 9 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?

10. 10 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?

11. 11 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?

12. 12 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?

13. 13 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!

14. 14 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

15. 15 More Magic Square Reasoning –= 0Sum( )Sum( )

16. 16 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

17. 17 The Place of Generality  A lesson without the opportunity for learners to generalise mathematically, is not a mathematics lesson

18. 18 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

19. 19 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Ordering & Characterising  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Ordering & Characterising

20. 20 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint  Extending & Restricting Meaning

21. 21 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 out any day) mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk Structured Variation Grids Revealing Shapes Studies in Algebraic Thinking Other Publications This and other presentations