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Expressing Generalityand the role of

Attentionin Mathematics

John MasonSEMAT

Køge DenmarkMarch 2014The Open University

Maths Dept University of OxfordDept of Education

Promoting Mathematical Thinking

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Copy & Extend

…

…

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Reproduction

Copy this shape What are you doing with your

attention?Now extend it

To this shape

Explain your choice

to someone else

Now extend it

further according to same

principle

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Notice what you do with your attention to make sense of

these ways

Count How many white squares

are there?

Notice the effect on your attention between these two

task types

In how many different ways can you count the white squares?

How did you do it?

How were you attending?

Resolución de Problemas de Final Abierto Chile 2013 p67

Count blacks and subtract Notice 6 rows of 3 blacks & 3

rows of 1 black Notice 6 columns of 3 black & 3

columns of 1 black Notice diagonal stripes of blacks Columns: 3 lots of (3 + 3 + 1)

blacks Similar for whites

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What’s The Difference?

What could be varied?

– =

First, add one to each

First, add one to the larger and subtract one from the smaller

What then would be

the difference?

What then would be

the difference?

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Triangle Count

In how many different ways might you count the triangles?

15 x 4 + 1

What are you attending to?

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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

Now choose different numbers so that the answer is again 2

Choose numbers so that the answer is …

45 3

7

28 + 15 = 43

20 + 21 = 41

43 – 41 = 2

Action!

Undoing the Action!

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Differing Sums of Products

Tracking Arithmetic

4x7 + 5x3

4x5 + 7x3

4x(7–5) + (5–7)x3

= (4-3)x (7–5)= 4x(7–5) – (7-5)x3

So how can an answer of 2 be guaranteed?

In how many essentially different ways can 2 be the difference?

What is special about 2?

45 3

7

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Think Of A Number (ThOAN)

Think of a number Add 2 Multiply by 3 Subtract 4 Multiply by 2 Add 2 Divide by 6 Subtract the number

you first thought of Your answer is 1

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+ 2

3x + 6

3x + 2

6x + 4

6x + 6

+ 1

1

7

7

7

7

7

7

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Doing & Undoing

What operation undoes ‘adding 3’? What operation undoes ‘subtracting 4’? What operation undoes ‘subtracting from

7’?

What are the analogues for multiplication? What undoes ‘multiplying by 3’? What undoes ‘dividing by 4’? What undoes ‘multiplying by 3/4’?

Two different expressions! Dividing by 3/4

or Multiplying by 4 and dividing by 3 What operation undoes dividing into 12?

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Composite Doing & Undoing

I add 8 and the answer is 13.I add 8 and then multiply by 2;the answer is 26.

I add 8; multiply by 2; subtract 5;the answer is 21.I add 8; multiply by 2; subtract 5; divide by 3;the answer is 7.

What’s my number?

What’s my number?What’s my number?

What’s my number?

HOW do you turn +8, x2, -5, ÷3 answer into a solution?

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I am thinking of a number …

Generalise!

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Tabled Variations

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Raise your hand when you can see …

Something that is 3/5 of something else Something that is 2/5 of something else Something that is 2/3 of something else Something that is 5/3 of something else What other fraction-actions can you see?

How did your attention shift?

Flexibility in choice of

unit

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Raise your hand when you can see …

Something that is 1/4 – 1/5of something else

What did you have to do with your attention?

What do you do with your

attention in order to

generalise?

Did you look for something that is 1/4 of something else

and forsomething that is 1/5 of the same thing?

Common Measure

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Stepping Stones

…

…R

R+1

What needs to change so as to ‘see’ that

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SWYS

Find something that is , , , , , of something else

Find something that is of of something else

Find something that is of of something else

What is the same, and what is different?

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Consecutive Sums

Say What You See

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Diamond Multiplication

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16 27 38 49 60 71 82

13 22 31 40 49 58 67

10 17 24 31 38 45 52

7 12 17 22 27 32 37

4 7 10 13 16 19 22

Sundaram’s Grid

What number will appear in the Rth row and the Cth column?

Claim: N will appear in the table iff 2N + 1 is composite

The grid extends up and to the right by arithmetic progressions

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Two Journeys Which journey over the same distance at two

different speeds takes longer:– One in which both halves of the distance are done at

the specified speeds?– One in which both halves of the time taken are done

at the specified speeds?

distance time

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Counting Out

In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37). Which object will you end on?

A B C D E

1 2 3 4 5

9 8 7 6

…

If that object is elimated, you start again from the ‘next’. Which object is the last one left?

10

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Set Ratios In how many different ways can you place 17

objects so that there are equal numbers of objects in each of two possibly overlapping sets?

What about requiring that there be twice as many in the left set as in the right set?

What about requiring that the ratio of the numbers in the left set to the right set is 3 : 2?

What is the largest number of objects that CANNOT be placed in the two sets in any way so that the ratio is 5 : 2?

What can be varied?

S1 S2

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Bag Reasoning Axiom: Every bean is in some bag. Axiom: There are finitely many bags. Definition: a bean b is loose in a bag B means

that although b is in B, b is not in any bag that is in B.

Must every bean be loose in some bag? How do you know?

Definition: the number of beans in bag B is the total number of beans that would be loose in bag B if all the bags within B disintegrated.

How few beans are needed in order that for k = 1, 2, …, 5, there is exactly one bag with k beans?

What is the most beans there could be? Can it be done with any number of beans

between the least and the most?

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What Do I know?

√2 (√2)2 = 2

√3 (√3)2 = 3

x2 = 2 & x > 0 => x = √2

x2 = 3 & x > 0 => x = √3

√3 – √2

X4 – 10x2 + 1 = 0 & 1 > x > 0 => x = √3 – √2√3 + √2

X4 – 10x2 + 1 = 0 & x > 1 => x = √3 + √2

(X2 – 5)2 = 24

(X2 – 5)2 = 24

?

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Covered Up Sums

Cover up one entry from each row and each column. Add up the remaining numbers.

The answer is (always) the same!

Why?

    

    

    

 

 

 

      

0 -2 2 -4

6 4 8 2

3 1 5 -1

1 -1 3 -3

Example of seeking invariant relationshipsExample of focusing on actions

preserving an invarianceOpportunity to generalise

Stuck? Speciali

se!

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Covered Up Sums

Opportunity to generalise

    

    

    

 

 

 

      

Opportunity to quantify freedom of

choice

How much freedom of choice do you have when making up your own?

a

b

c

d

e

f

ge-(a-b)

a b

e ?

a b c d

e

f

g

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Marbles 1

If Anne gives one of her marbles to John, they will have the same number of marbles.– What can you say about the number of marbles they

started with? Buses

– If one person got off the first bus and got onto the second, there would be the same number of people in both buses.

Ages– If A was one year older and B was one year younger,

then A and B would be the same age.

1/4

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Marbles 2

If Anne gives one of her marbles to John, they will have the same number of marbles;if John now gives two of his marbles to Kathy, they will have the same number.– What can you say about the relation between Anne’s

and Kathy’s marbles to start with. Buses

– If one person were to get off the first bus and onto the second, then the buses would have the same number of passengers;if two people then got off the second bus and got onto the third bus, those two buses would have the same number.

2/4

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Marbles 3

If Anne gives John one of her marbles, she will then have one more than twice as many marbles as John then has.

If John started with 12 marbles, how many did Anne start with?

Buses– If one passenger gets off the first bus and onto the

second bus, then there will be one more than twice as many passengers on the first bus as on the second.

– If the second bus started with 12 passengers, how many did the first bus start with?

3/4

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Marbles 4

If Anne gives John one of her marbles, she will then have one more than twice as many marbles as John then has.

However, if instead, John gives Anne one of his marbles, he will have one more than a third as many marbles as Anne then has.

How many marbles have they each currently? Buses!

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FrameworksDoing – Talking – Recording

(DTR)

Enactive – Iconic – Symbolic

(EIS)

See – Experience – Master(SEM)

(MGA)

Specialise … in order to locate structural

relationships …then re-Generalise for

yourself

What do I know?What do I want?

Stuck?

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Mathematical Thinking

How describe the mathematical thinking you have done so far today?

How could you incorporate that into students’ learning?

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Possibilities for Action

Trying small things and making small progress; telling colleagues

Pedagogic strategies used today Provoking mathematical thinking as happened

today Question & Prompts for Mathematical Thinking

(ATM) Group work and Individual work

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Human Psyche

ImageryAwareness (cognition)

Will

Body (enaction)

Emotions (affect)

HabitsPractices

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Three Only’s

Language Patterns& prior Skills

Imagery/Sense-of/Awareness; Connections

Different Contexts in which likely to arise;

dispositions

Techniques & Incantations

Root Questionspredispositions

Standard Confusions

& Obstacles

Only Behaviour is TrainableOnly Emotion is Harnessable

Only Awareness is Educable

Behaviour

Emotion

Awar

enes

s

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Follow Up

j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Presentations Questions & Prompts (ATM) Key ideas in Mathematics (OUP) Learning & Doing Mathematics (Tarquin) Thinking Mathematically (Pearson) Developing Thinking in Algebra (Sage)