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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 - PowerPoint PPT Presentation
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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?
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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
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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
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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
34
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
35
To Follow Up
http://mcs.open.ac.uk/jhm3– Presentations– Applets– Structured Variation Grids