1 Asking Questions in order to promote Mathematical Reasoning John Mason East London June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Outline  Some tasks to work on  in a conjecturing atmosphere  in order to experience different forms of reasoning, and the questions and prompts that may promote them

3 Reasoning Types  Logical Deduction (cf arithmetic)  Empirical (needs justification)  Exhaustion of cases  Contradiction  (Induction) Issue is often what can I assume? what can I use? what do I know?

4 Wason’s cards  Each card has a letter on one side and an numerl on the other.  Which 2 cards must be turned over in order to verify that “on the back of a vowel there is always an even number”? A2B3

5 Revealing Shapes

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

7 Location, Location  One letter has been chosen.  If you name a letter, you will be told –“Hot” if the chosen letter is the same as, or next to the letter you name –“Cold” otherwise  You can assert which letter has been chosen, but you have to be able to justify your choice A B C D E Dimensions of possible variation?

8 CopperPlate Calculations

9 Magic Square Reasoning –= 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?

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

11 Four Consecutives  Write down four consecutive numbers and add them up  and another  Now be more extreme!  What is the same, and what is different about your answers?

12 Tunja Sequences 1 x 1 – 1 = 2 x 2 – 1 = 3 x 3 – 1 = 4 x 4 – 1 = 0 x 2 1 x 3 2 x 4 3 x 5 0 x 0 – 1 =-1 x 1 -1 x -1 – 1 =-2 x 0 Across the Grain With the Grain

13 Clubbing total 47 total 47–3147–29 31–(47–29)29–(47–31) 31poets 29 painters In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both? How many poets are not painters? How many painters are not poets? – 47 Tracking Arithmetic

musicians poets painters 28 total 23 musicians or painters 21 poets or musicians 22 poets or painters 28–2328–21 28– – – 23 ( ) + ( ) + ( ) – (28– ((28-23) + (28-22) + (28-21)) – 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?

15 Square Deductions  Each of the inner quadrilaterals is a square.  Can the outer quadrilateral be square?

16 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 2?  What undoes multiplying by 3/2?  What undoes dividing by 3/2?

17 Geometrical Reasoning  What properties are agreed?  What relationships are sought?  How are these connected?

18 Eyeball Reasoning

19 Behold!

20 Discerning How many triangles? What is the same, and what is different about them?

21 Vecten

22 Dimensions of Possible Variation

23

24 Reasoning  What makes it difficult? –not discerning (what others discern) –not seeing relationships –not perceiving properties  How can it be developed? –Working explicitly on discerning; relating; property perceiving; reasoning on the basis of those properties