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

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 Revealing Shapes

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

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 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 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 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 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 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 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2

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

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

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

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

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 Structured Variation Grids Revealing Shapes Studies in Algebraic Thinking Other Publications This and other presentations