1

On the Structure of Attention

&its Role in Engagement& the Assessment of

ProgressJohn MasonOxford PGCEApril 2012The Open University

Maths Dept University of OxfordDept of Education

Promoting Mathematical Thinking

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

7964564789

302420

36163554242840

4230423245286348364972545681

635160119905

Find the error!7964564789

302420

36163554242840

4230423245286348364972545681

635160119905 How did your

attention shift?How did your

attention shift?

6

Movements of Attention in Geometry

a

b

c

d

A

BF

ED

y

C

xG

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Rectangular Room with 2 Carpets

How are the red and blue areas related?

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Tracking Arithmetic Becomes Algebra

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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 3

7

28 + 15 = 43

20 + 21 = 4143 – 41 = 2

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Differing Sums & Products Tracking Arithmetic 4

5 37

4x7 + 5x34x5 + 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

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Think Of A Number (THOAN)

How is it done?

How can we learn to do it?

Tracking Arithmetic!

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Club Memberships

31

47 total

2947–3147–29

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?

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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 bothand 23 who are either musicians or poets or both.How many people are all three: poets, painters and musicians?

14

14

11

15 musicians

poets painters

28 total

23 musicians or painters

21 poets or musicians

22 poets or painters

28–23 28–21

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 bothand 23 who are either musicians or poets or both.How many people are all three: poets, painters and musicians?

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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

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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?

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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?

orare their powers being usurped by

textbook, worksheets and …

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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 mathematicallywith and in front of students?

What are they attending to? (and how?)

What are you attending to when interacting with students? (and how?)

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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

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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

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Perimeter Projections

Imagine the vertical and horizontal movements of the red point as it traverses the

perimeterNow imagine them being graphed against time

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Ride & Tie

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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

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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.

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Tabled Variations

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Structured Variation Grids

Tunja Factoring

Quadratic DoubleFactors

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Sundaram Grids

All rows and columns are arithmetic progressionsHow many entries do you need to fill out the grid?

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1 2

345

6

7 8 9 10

11

12

13

18

19

20

21 22 23 24 25 26

27

28

29

30

3132

14151617

3334353637

38

39

40

41

42

43 44 45 46 47 48 49 50

1

4

9

16

25

49

36

Spiral

31

1

2 3 4

5

6789

101112

13

18 19 20

21

22

23

242526272829

303132

14 15 16 17

33

34

35

36 37 38 39 40 41 42 43 44

45

46

47

48

49

50

64

81

Spiral

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Structure of the PsycheImageryAwareness (cognition)

Will

Body (enaction)

Emotions (affect)

HabitsPractices

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Structure of a TopicLanguage Patterns& prior Skills

Imagery/Sense-of/Awareness; Connections

Different Contexts in which likely to arise;dispositions

Root Questionspredispositions

Standard Confusions

& Obstacles

Only Behaviour is TrainableOnly Emotion is Harnessable

Only Awareness is Educable

Behaviour

Emotion

Awareness

Techniques & Incantations

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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

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To Follow Up

http://mcs.open.ac.uk/jhm3– Presentations– Applets– Structured Variation Grids

j.h.mason@open.ac.uk