1

Attention to AttentionAttention to Attentionin the Teaching and Learningin the Teaching and Learning

of Mathematicsof Mathematics

John MasonJohn MasonOpen University & University of OxfordOpen University & University of Oxford

FlFlöötur Selfosstur Selfoss

Sept 2008Sept 2008

2

Say What You SawSay What You Saw

3

One Sum DiagramsOne Sum Diagrams

1

1

(1- )2

Anticipating,not waiting

1-2

4

Reading a Diagram: Seeing Reading a Diagram: Seeing As …As …

a

a

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

x2z + x(1-x) + (1-x)2(1-z)

xz + (1-x)(1-z)xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-

z)

5

Reasoning from Diagrams …Reasoning from Diagrams …

… … has a long tradition!has a long tradition!

6

aaa

30361635542428402863483642364232455681497254635160119905

24206478979654CopperPlate CopperPlate

CalculationsCalculations

aaa

30361635542428402863483642364232455681497254635160119905

79654242064789

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AttentionAttention

Holding Wholes (gazing)Holding Wholes (gazing)

Discerning DetailsDiscerning Details

Recognising RelationshipsRecognising Relationships

Perceiving PropertiesPerceiving Properties

Reasoning on the basis of Reasoning on the basis of agreed propertiesagreed properties

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The calculation comes from an Arabic manuscript Hindu Reckoning written by Kushyar ibn-Lebban about 1000 C.E. (quoted in NCTM 1969 p133)

aa

562548254795478683983983983948 79 86

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Treviso & Pacioli Treviso & Pacioli CalculationsCalculations

9 3 4

36

09

27

12

16

03

04

09

12

4

1

3

2 9 3

6

7

2

Treviso and Pacioli MultiplicationsHistorical Topics for the Mathematics Classroom, NCTM p134.

27

09

12

09

03

04

36

12

16 4

1

3

2 7 6

2

9

3

9 3 4

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Word ProblemsWord Problems

In 26 years I shall be twice as old as I was 19 years ago. How old am I?

+ 2640

40

?=? 2( - 19)

? 26?

19?

19

=

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Mid-PointMid-Point

Where can the Where can the midpoint of midpoint of the segment the segment joining two joining two points each points each on a separate on a separate circle, get to?circle, get to?

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ScalingScaling

P

QM

Imagine a circle C. Imagine a circle C. Imagine also a point P.Imagine also a point P.

Now join P to a point Q on C.Now join P to a point Q on C.

Now let M be the mid point of PQ.Now let M be the mid point of PQ.

What is the locus of M What is the locus of M as Q moves around the circle? as Q moves around the circle?

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Additive & Multiplicative Additive & Multiplicative PerspectivesPerspectives

What is the relation between the What is the relation between the numbers of squares of the two numbers of squares of the two colours?colours?

Difference of 2, one is 2 more: Difference of 2, one is 2 more: additiveadditive

Ratio of 3 to 5; one is five thirds Ratio of 3 to 5; one is five thirds the other etc.:the other etc.: multiplicativemultiplicative

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Raise your hand when you Raise your hand when you can seecan see

Something which is 2/5 of somethingSomething which is 2/5 of something Something which is 3/5 of somethingSomething which is 3/5 of something Something which is 2/3 of somethingSomething which is 2/3 of something

– What others can you see?What others can you see? Something which is 1/3 of 3/5 of something Something which is 1/3 of 3/5 of something Something which is 3/5 of 1/3 of somethingSomething which is 3/5 of 1/3 of something Something which is 2/5 of 5/2 of somethingSomething which is 2/5 of 5/2 of something Something which is 1 ÷ 2/5 of somethingSomething which is 1 ÷ 2/5 of something

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What fractions can you What fractions can you ‘see’?‘see’?

What relationships between What relationships between fractions can you see?fractions can you see?

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Two-bit PerimetersTwo-bit Perimeters

2a+2b

What perimeters are possible using only 2 bits of information?

a

b

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Two-bit PerimetersTwo-bit Perimeters

4a+2b

What perimeters are possible using only 2 bits of information?

a

b

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Two-bit PerimetersTwo-bit Perimeters

6a+2b

What perimeters are possible using only 2 bits of information?

a

b

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Two-bit PerimetersTwo-bit Perimeters

6a+4b

What perimeters are possible using only 2 bits of information?

a

b