On the Structure of Attention & its Role in Engagement & the Assessment of Progress

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

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

3. Present or Absent?

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

6. Movements of Attention in Geometry A c E a x G D d y b B C F

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

8. Tracking Arithmetic Becomes Algebra

9. Differing Sums of Products 4 7 3 5 • Write down four numbers in a 2 by 2 grid • Add together the products along the rows 28 + 15 = 43 • Add together the products down the columns 20 + 21 = 41 43 – 41 = 2 • Calculate the difference • That is the ‘doing’What is an undoing? • What other grids will give the answer 2? • Choose positive numbers so that the difference is 7

10. Differing Sums & Products 4 7 3 4x7 + 5x3 5 • Tracking Arithmetic 4x5 + 7x3 4x(7–5) + (5–7)x3 = 4x(7–5) – (7–5)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?

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

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

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. 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? 28 total 15 musicians 21 poets or musicians 23 musicians or painters 28–22 poets 14 painters 11 28–23 28–21 11+15–23 22 poets or painters 14+15–21 14+11–22 (14+15-21) + (14+11-22) + (11+15-23) – (28– ((28-23) + (28-22) + (28-21)) 2

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

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. 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. 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. 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. Ride & Tie

23. 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

24. 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.

25. Tabled Variations

26. Structured Variation Grids Tunja Factoring Quadratic Double Factors

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

28. Spiral 43 45 46 47 48 49 50 44 49 42 21 22 23 24 25 26 25 41 20 10 27 9 40 19 5 4 3 7 8 2 6 1 1 9 11 28 39 18 12 29 4 38 17 16 15 14 13 30 16 35 34 33 36 37 36 32 31

29. Spiral 43 45 46 47 48 49 50 44 42 21 22 23 24 25 26 41 20 10 27 40 19 9 8 7 6 5 4 3 2 1 11 28 39 18 12 29 38 17 16 15 14 13 30 35 34 33 81 64 37 36 32 31

30. Imagery Awareness (cognition) Will Emotions (affect) Body (enaction) HabitsPractices Structure of the Psyche

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

32. 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

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