1. 1 Digging at the Foundations of Mathematics Education (Part 1) John Mason PTAN Karachi August 2008

2. 2 Outline of the Day Part 1: Using our mathematical powers Part 2: Probing mathematical awareness Part 3: Using these in the classroom

3. 3 Remainders of the Day (1)  Write down a number which when you subtract 1 is divisible by 7  and another  Write down one which you think no-one else here will write down.

4. 4 Remainders of the Day (2)  Write down a number which is 1 more than a multiple of 2  and 1 more than a multiple of 3  and 1 more than a multiple of 4  … … … …

5. 5 Remainders of the Day (3)  Write down a number which is 1 more than a multiple of 2  and which is 2 more than a multiple of 3  and which is 3 more than a multiple of 4  … … … …

6. 6 Remainders of the Day (4)  Write down a number which when you subtract 1 is divisible by 2  and when you subtract 1 from the quotient, the result is divisible by 3  and when you subtract 1 from that quotient the result is divisible by 4  Why must any such number be divisible by 3?

7. 7 One Sum  Write down two numbers whose sum is 1  And another pair  I have a pair; I calculate –A: The square of the larger plus the smaller –B: The square of the smaller plus the larger –Which of these answers will be the larger?

8. 8 One Sum Diagrams 1 1 (1- ) 2 Anticipating, not waiting 1- 2

9. 9 Reading a Diagram x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

10. 10 Reasoning from Diagrams … … has a long tradition!

11. 11 Some Sums 4 + 5 + 6 = 9 + 10 + 11 + 12 16 Generalise Justify Watch What You Do Say What You See 1 + 2 =3 7 + 8 = 13 + 14 + 15 17 + 18 + 19 + 20+ = 21 + 22 + 23 + 24

12. 12 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.  For four bags, what is the least number of objects, and the least number of colours to meet the same constraint? 15 objects 3 colours

13. 13 Bag Constructions  Here there are 3 bags and two objects.  There are [0, 1, 2;2] objects in the bags and 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?  In how many ways can you put k objects in b bags?

14. 14 Another & Another  Write down a pair of numbers whose product is 12  and another pair

15. 15 Another & Another  Write down a pair of numbers whose product is 13  and another pair  and a pair that you think no-one else in the room will write down  and a pair that perhaps no human being has ever written down

16. 16 Example Spaces  The examples that come to mind when you hear a word or see symbols  Dimensions of possible variation  Ranges of permissible change

17. 17 Fractional Difference  Write down two fractions that differ by 3/4  and another pair  and a pair that make it as obscure as possible

18. 18 Constrained Decimal  Write down a decimal number between 2 and 3  and which does not use the digit 5  and which does use the digit 7  and which is as close to 5/2 as possible

19. 19 Constrained Quadrilateral  Draw a quadrilateral  with a pair of equal edges  and with a pair of perpendicular edges  and with a pair of parallel edges  How many different ones can you find?

20. 20 Perpendicularity  Draw a quadrilateral which has both pairs of opposite sides perpendicular  Trouble? –Try just one pair of opposite sides perpendicular

21. 21 Sentenced 37 + – 37 = 49 Make up your own like this 3 ÷ 4 = 15 ÷ Make up your own like this What is the ‘like this’ of your example?

22. 22 Distribution  Write down five numbers whose arithmetic mean is 5 –What are the dimensions of possible variation: how much freedom?  and whose median is 6 –how much freedom now?  and whose mode is 7 –how much freedom now?

23. 23 Iteration  Write down a number between 0 and 1  add 1 and divide by 2  repeat this over and over …  What happens in the long run?

24. 24 Iteration Too  Write down a number between 0 and 1  Add 1 and multiply by your starting number  Repeat this over and over until your calculator no longer changes  Take the reciprocal of this; add 1; take the reciprocal of this

25. 25 Iteration As Well  Write down a positive number  Take its square root  Keep taking the square root of the result … what happens?  Did you try a number between 0 and 1?

26. 26 Digging at the Foundations of Mathematics Education Part 2 John Mason PTAN Karachi August 2008

27. 27 Outline  Some observations  Some tasks  Some reflection  Some planning

28. 28 Some Observations (pre-paring)  Students often make mistakes when carrying out procedures  Procedures are ‘actions’ based on concepts  Students often develop a limited appreciation of mathematical concepts  ‘Awareness’ is the basis for action  ‘Digging at the Foundations’ means exploring core awarenesses

29. 29 Reflection  Expressing generality as the core awareness behind algebra  Reading Diagrams as core awareness for changing (re)presentations  Construction tasks used to reveal awarenesses

30. 30

31. 31

32. 32 Square Count

33. 33 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties

34. 34 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?

35. 35 Reflection  What other actions have analogues? square-rootingreciprocalnegatingexponentiation…  Core awarenesses: –doing & undoing –self-undoing (involutions)

36. 36 Action and Awareness  Mathematical thinking involves undertaking actions  Awarenesses are what enable actions  Awarenesses trigger action depending on what is being attended to  What are the core awarenesses that lie at the heart of different mathematical topics?

37. 37 Composite Doing & Undoing  I am thinking of a number  HOW do you turn +8, x2, -5, ÷3 answer 7 into a solution? I add 8 and the answer is 13. I add 8 then multiply by 2; the answer is 26. I add 8, multiply by 2, subtract 5; the answer is 21. I add 8, multiply by 2, subtract 5, divide by 3. The answer is 7

38. 38 Word Problems In 26 years I shall be twice as old as I was 19 years ago. How old am I? + 26 40 ?=?2( - 19) ?26 ? 19 ? =

39. 39 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

40. 40 Multiplication  What image comes to mind when you see 3 x 4?  What about 0.3 x 0.4?  Why are these different?  What is the essence of multiplication? – ‘repeated addition’ – ‘scaling’

41. 41 Scaling P Q M Imagine a circle C. Imagine also a point P. Now join P to a point Q on C. Now let M be the mid point of PQ. What is the locus of M as Q moves around the circle?

42. 42 Map Drawing Problem  Two people both have a copy of the same map of Pakistan.  One uses Karachi as the centre for a scaling by a factor of 1/3  One uses Islamabad as the centre for a scaling by a factor of 1/3  What is the same, and what different about the maps they draw?

43. 43 Digging at the Foundations of Mathematics Education Part 3 John Mason PTAN Karachi August 2008

44. 44 Trigonometry  What is an angle? –Independence of orientation, length of arms  How can angles be compared?  Thales as invariance in the midst of change … ratios as measures

45. 45 Trig Construction  Draw an angle whose tangent is 3/4  Draw an angle whose tangent is 2/3  Draw an angle which is the sum of your two angles. What is its tangent?  Can you describe how to do this ‘in general’? Use this to write down a formula for the tangent of the sum of two angles whose tangents are known rationals On squared paper:

46. 46 Tangent Addition Tan(A) = 2/5 Tan(B) = 3/4 A B 4x5–3x2 3x5+4x2 Tan(A+B) = = 3 4 2 5 + 3 4 2 5 1 – x

47. 47 Preparing for the Future  Tasks  are for generating recent experience  as fodder for learning from that experience  through reflection  and  Imagining acting in the future

48. 48 Mathematics & Creativity  Creativity is a type of energy  It is experienced briefly  It can be used productively or thrown away  Every opportunity to make a significant choice is an opportunity for creative energy to flow  It also promotes engagement and interest  For example –Constructing an object subject to constraints –Constructing an example on which to look for or try out a conjecture –Constructing a counter-example to someone’s assertion

49. 49 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

50. 50 Structure of a Topic Language Patterns & prior Skills Techniques & Incantations Different Contexts in which likely to arise; dispositions Root Questions predispositions Imagery/Sense- of/Awareness; Connections Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness

51. 51 Some Mathematical Powers  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Ordering & Characterising  Distinguishing & Connecting  Assenting & Asserting

52. 52 Some Mathematical Themes  Doing and Undoing  Invariance in the midst of Change  Freedom & Constraint

53. 53 Worlds of Experience Material World World of Symbol s Inner World of imagery enactiveiconicsymbolic

54. 54 Cubelets Say What You See

55. 55 For More Information  These slides: PTAN website or http://mcs.open.ac.uk/jhm3 look for ‘presentations & workshops’  Other resources:  my website; http://www.atm.co.uk http://www.atm.co.uk  Mathematics as a Constructive Activity (2005 Erlbaum)  Developing Thinking in Algebra (2005 Sage)  Developing Thinking in geometry (2005 Sage)  Questions & Prompts for Mathematical Thinking (2002 ATM)  Thinkers (2003 ATM  j.h.mason@open.ac.uk