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. Attention. Macro Locus, Focus, Scope Micro

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On the structure of attention its role in engagement the assessment of progress

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


Attention

Attention

  • Macro

    • Locus, Focus, Scope

  • Micro

    • To be experienced

  • Meso

    • Student focus & disposition


Present or absent

Present or Absent?


Micro attention

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


Find the error

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


Movements of attention in geometry

Movements of Attention in Geometry

A

c

E

a

x

G

D

d

y

b

B

C

F


Rectangular room with 2 carpets

Rectangular Room with 2 Carpets

How are the red and blue areas related?


Tracking arithmetic becomes algebra

Tracking Arithmetic Becomes Algebra


Differing sums of products

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


Differing sums products

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?


Think of a number thoan

Think Of A Number (THOAN)

How is it done?

How can we learn to do it?

Tracking Arithmetic!


Club memberships

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


Club memberships 3

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?


On the structure of attention its role in engagement the assessment of progress

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


Tracking arithmetic

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


Meso attention

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?


Powers themes

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 …


Teaching students to think mathematically

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?)


Meso level of attention

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


Getting to grips with graphs

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


Perimeter projections

Perimeter Projections

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

Now imagine them being graphed against time


Ride tie

Ride & Tie


Elastic multiplication

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


Straight line constructions

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.


Tabled variations

Tabled Variations


Structured variation grids

Structured Variation Grids

Tunja

Factoring

Quadratic Double

Factors


Sundaram grids

Sundaram Grids

All rows and columns are arithmetic progressions

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


Spiral

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


Spiral1

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


Structure of the psyche

Imagery

Awareness (cognition)

Will

Emotions (affect)

Body (enaction)

HabitsPractices

Structure of the Psyche


Structure of a topic

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


Attention1

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


To follow up

To Follow Up

  • http://mcs.open.ac.uk/jhm3

    • Presentations

    • Applets

    • Structured Variation Grids

  • [email protected]


  • Login