1. 1 Reasoning Reasonably in Mathematics John Mason Schools Network Warwick June 2012 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Outline  Some Tasks to engage in  On which to reflect  In a conjecturing atmosphere –Everything said must be tested in your experience  Make Contact with –Natural powers used to reason mathematically –Mathematical themes –Pedagogic choices  Straw Poll: please raise your hand if you –Teach proof or reasoning … …to previously high attaining students …to previously low attaining students

3. 3 Secret Places  One of the places around the table is a secret place.  If you click near a place, the colour will tell you whether you are hot or cold: –Hot means that the secret place is there or else one place either side –Cold means that it is at least two places away either side Homage to Tom O’Brien (1938 – 2010) What is your best strategy to locate the secret place?

4. 4 Reflexive Stance  What did you notice yourself doing?  What actions were effective?  What actions were not effective?

5. 5 Magic Square Reasoning 519 2 4 6 83 7 – = 0 Sum( ) Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2 Any colour-symmetric arrangement?

6. 6 More Magic Square Reasoning – = 0 Sum( ) Sum( )

7. 7 Reflexive Stance  What did you notice yourself doing?  What actions were effective?  What actions were ineffective?

8. 8 Convincing & Justifying  Justifying actions –Social aspect …convincing yourself …convincing a friend …convincing a sceptic –Learning to be sceptical of other’s reasoning –Learning to be sceptical of your own reasoning

9. 9 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?  What numbers can be expressed as the sum of four consecutive numbers? + 1 + 2 + 3 + 6 4 Express 424 as a sum of four consecutive numbers Doing & Undoing

10. 10 Carpet Theorem How are the red and blue areas related?

11. 11 Doug French’s Fractional Parts What’s the same and what different? What can be varied? What is the ‘whole’? Construct your own

12. 12 Square Sums Imagine a triangle … Imagine the circumcentre; Drop perpendiculars to the edges; On each half of each edge, put a square outwards, one yellow, one cyan. The sum of the Yellow areas = the sum of the cyan areas More generally … Does ???!!? ?

13. 13 Polygon Perimeter Projections  Imagine a square  Imagine a point traversing the perimeter of the square  Imagine projections of that point onto the x and y axes

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16. 16 Withdrawing from the Action  What actions did you carry out?  What mde you think of those?  Which ones were effective (for possible use in the future)?  Did I –Gaze (holding wholes)? –Discern Details –Recognise Relationships in the particular? –Perceive Properties as instantiated –Reason on the basis of properties?

17. 17 Eyeball Reasoning

18. 18 Square Deductions a b a+ba+ba+ba+b a+2b 2a+b2a+b2a+b2a+b a+3b 3a+b3a+b3a+b3a+b 3b-3a 2 3 5 8 7 9 11111111 3 3(3b-3a) = 3a+b 12a = 8b So 3a = 2b For an overall square 4a + 4b = 2a + 5b So 2a = b For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1) But not also 2a = b

19. 19 Square Deductions x y x+yx+2y x+3y 2x+y 3x+y To be consistent, 3x + y + x = x + 3y + y So 3x = 3y

20. 20 Rectangle Deduction x y x+y (x+3y)/2 (x+7y)/4 (3x+y)/2 (7x+y)/4 (x+y)/2 (x+3y)/4 (3x+y)/4 (x+7y)/8 + y = (7x + y)/4 + x 21x = 13y (x+7y)/8 Suppose the rectangles are two by one

21. 21 Wason’s cards  Each card has a letter on one side and a numeral 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 Which cards must be turned over to verify that “on the back of a red vowel there is a blue even number”

22. 22 Diamond Multiplication

23. 23 Seven Circles How many different angles can you discern, using only the red points? How do you know you have them all? How many different quadrilaterals?

24. 24 Bag Constructions (1)  Here there are three bags. If you compare any two of them, there is exactly one colour for which the difference in the numbers of that colour in the two bags is exactly 1. 17 objects 3 colours  For four bags, what is the least number of objects to meet the same constraint?  For four bags, what is the least number of colours to meet the same constraint?

25. 25 Reflective Stance  What did you notice yourself doing? –Checking? –Interpreting? –Specialising? –Being systematic?  What forms of attention did you notice? –Gazing (holding wholes)? –Discerning Details? –Recognising Relationships? –Perceiving Properties? –Reasoning on the basis of Properties?

26. 26 Bag Constructions (2)  For b bags, how few objects can you use so that each pair of bags has the property that there are exactly two colours for which the difference in the numbers of that colour in the two bags is exactly 1.  Construct four bags such that for each pair, there is just one colour for which the total number of that colour in the two bags is 3.

27. 27 Bag Constructions (3)  Here there are 3 bags and 2 objects.  [0,1,2;2]means that there are 0, 1 and 2 objects in three bags and 3 objects 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?  In how many different ways can you put k objects in b bags?

28. 28 Reasoning Types  Logical Deduction –Arithmetic: reasoning into the unknown –Algebra: reasoning from the unknown  Empirical (needs justification)  Exhaustion of cases  Contradiction  (Induction) Issue is often what can I assume? what can I use? what do I know?

29. 29 Attention Holding Wholes (gazing) Discerning Details (discriminating) Recognising Relationships (in a situation) Perceiving properties (as being instantiated) Reasoning On The Basis of Agreed Properties

30. 30 Powers  Imagining & Expressing  Specialisng & Generalising –Stressing & Ignoring  Conjecturing & Convincing (what is the status of what you say?)  Organising & Characterising

31. 31 Themes  Doing and Undoing  Invariance in the Midst of Change  Freedom & Constraint  Extending & Restricting

32. 32 Follow Up  mcs.open.ac.uk / jhm3 … presentations; applets;  j.h.mason @ open.ac.uk