1. 1 On the Structure of Attention & its Role in Engagement & the Assessment of Progress John Mason Oxford PGCE April 2012 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Attention  Macro –Locus, Focus, Scope  Micro –To be experienced  Meso –Student focus & disposition

3. 3 Present or Absent?

4. 4 Micro Attention  Holding Wholes (Gazing)  Discerning Details (making distinctions)  Recognising Relationships (in the particular)  Perceiving Properties (being instantiated)  Reasoning on the basis of agreed properties

5. 5 79645 6478964789 30 2420 361635 54242840 4230423245 28634836 497254 5681 63 5160119905 Find the error! 79645 64789 30 2420 361635 54242840 4230423245 28634836 497254 5681 63 5160119905 How did your attention shift?

6. 6 Movements of Attention in Geometry a b c d A B F E D yy C xx GG

7. 7 Rectangular Room with 2 Carpets How are the red and blue areas related?

8. 8 Tracking Arithmetic Becomes Algebra

9. 9 Differing Sums of Products  Write down four numbers in a 2 by 2 grid  Add together the products along the rows  Add together the products down the columns  Calculate the difference  What other grids will give the answer 2?  Choose positive numbers so that the difference is 7  That is the ‘doing’ What is an undoing? 45 37 28 + 15 = 43 20 + 21 = 41 43 – 41 = 2

10. 10 Differing Sums & Products  Tracking Arithmetic 45 37 4x7 + 5x3 4x5 + 7x3 4x(7–5) + (5–7)x3 = (4-3) x (7–5)  So in how many essentially different ways can 2 be the difference?  What about 7?  So in how many essentially different ways can n be the difference? = 4x(7–5) – (7–5)x3

11. 11 Think Of A Number (THOAN) How is it done? How can we learn to do it? Tracking Arithmetic!

12. 12 Club Memberships 31 47 total 29 47–3147–29 31–(47–29)29–(47–31) poets painters In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both?

13. 13 Club Memberships (3) 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?

14. 14 14 11 15 musicians poets painters 28 total 23 musicians or painters 21 poets or musicians 22 poets or painters 28–2328–21 28–22 14+15– 21 11+15– 23 (14+15-21) + (14+11-22) + (11+15-23) – (28– ((28-23) + (28-22) + (28-21)) 2 14+11– 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. 15 Tracking Arithmetic  Engage in some ‘calculation’ but don’t allow one (or more) number(s) to be absorbed into the arithmetic  Then replace those numbers by a symbol  Use in any task that calls for a generalisation or a method or a use of algebra

16. 16 Meso-Attention  What do you enjoy about thinking mathematically?  Could it be … –Getting an answer? –Knowing your answer is correct? –Using your natural powers? –Encountering increasingly familiar themes?

17. 17 Powers & Themes  Imagining & Expressing  Specialising & Generalising  Conjecturing & Convincing  Stressing & Ignoring  Doing & Undoing  Invariance in the midst of change  Freedom & Constraint Powers Themes Are students being encouraged to use their own powers? or are their powers being usurped by textbook, worksheets and …

18. 18 Teaching students to think mathematically … … involves developing a disposition to … think mathematically, to use powers mathematically, to be mathematical … to attend to situations mathematically  How often do you think mathematically with and in front of students?  What are they attending to? (and how?)  What are you attending to when interacting with students? (and how?)

19. 19 Meso Level of Attention  Discrete & Continuous –Integers -> fractions -> decimals  Additive & Multiplicative & Exponential Thinking  Arithmetic as the study of actions on objects  Finiteness & Infinity  Rules & Tools  Arbitrary (Convention) & Necessary  It looks right => It must be so because …  Procedures & Underlying Reasons  Adolescent concerns –self in relation to the social; sex

20. 20 Getting To Grips With Graphs  Imagine a square  Imaging a point on the edge of the square, traversing the perimeter at a constant speed  With your right hand, show the vertical movement of the point  With your left hand, show the horizontal movement of the point

21. 21 Perimeter Projections Imagine the vertical and horizontal movements of the red point as it traverses the perimeter Now imagine them being graphed against time

22. 22 Ride & Tie

23. 23 Put your hand up when you can see … … Something that is 3/5 of something else … Something that is 2/5 of something else … Something that is 2/3 of something else … Something that is 5/3 of something else … how many different fraction actions can be ‘seen’

24. 24 Put your hand up when you can see … … omething that is 1/4 – 1/5 of something else Construct your own example like this

25. 25 Elastic Multiplication  Imagine you have a piece of elastic.  You stretch it equally with both hands … what do you notice?  Hold one end fixed. Stretch the other so the elastic is four-thirds as long. Where is the midpoint? –Relative to the elastic –Relative to the starting position of the elastic

26. 26 Straight Line Constructions  Sketch the graph of a pair of straight lines such that –Their slopes differ by two –and their x-intercepts differ by two –and their y-intercepts differ by two –And the areas the triangles (origin, x-intercept, y- intercept) differ by 2.

27. 27 Tabled Variations

28. 28 Structured Variation Grids TunjaFactoringQuadratic Double Factors

29. 29 Sundaram Grids All rows and columns are arithmetic progressions How many entries do you need to fill out the grid?

30. 30 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 1 4 9 16 25 49 36Spiral

31. 31 12345678910111213181920212223242526272829303132 14151617333435 3637 38 394041 4243 44 454647484950 64 81Spiral

32. 32 Structure of the Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

33. 33 Structure of a Topic Language Patterns & prior Skills Imagery/Sense- of/Awareness; Connections Different Contexts in which likely to arise; dispositions Root Questions predispositions Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness Techniques & Incantations

34. 34 Attention  Macro –Locus, Focus, Scope  Micro –Holding wholes; discerning Details; Recognising Relationships; Perceiving Properties; reasoning on the basis of agreed properties  Meso –Student focus & disposition –Shifts in perception & conception

35. 35 To Follow Up  http://mcs.open.ac.uk/jhm3 –Presentations –Applets –Structured Variation Grids  j.h.mason@open.ac.uk