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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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
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
28 + 15 = 43
20 + 21 = 41
43 – 41 = 2
4
7
3
4x7 + 5x3
5
4x5 + 7x3
4x(7–5) + (5–7)x3
= 4x(7–5) – (7–5)x3
= (4-3) x (7–5)
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
Powers
Themes
Are students being encouraged to use their own powers?
orare their powers being usurped by textbook, worksheets and …
with and in front of students?
Imagine the vertical and horizontal movements of the red point as it traverses the perimeter
Now imagine them being graphed against time
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