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On the Structure of Attention & its Role in Engagement & the Assessment of ProgressPowerPoint Presentation

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

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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

- Macro
- Locus, Focus, Scope

- Micro
- To be experienced

- Meso
- Student focus & disposition

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

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

A

c

E

a

x

G

D

d

y

b

B

C

F

How are the red and blue areas related?

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

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?

How is it done?

How can we learn to do it?

Tracking Arithmetic!

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

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?

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

- 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

- 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?

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 …

- 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?)

- 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

- 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

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

Now imagine them being graphed against time

- 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

- 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.

Tunja

Factoring

Quadratic Double

Factors

All rows and columns are arithmetic progressions

How many entries do you need to fill out the grid?

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

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

Imagery

Awareness (cognition)

Will

Emotions (affect)

Body (enaction)

HabitsPractices

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

- 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

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

- j.h.mason@open.ac.uk