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Learning Learning to Think and to Reasonto Think and to Reason

AlgebraicallyAlgebraicallyand the and the

Structure of AttentionStructure of Attention

2007

John MasonSMC

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OutlineOutline

Some assumptionsSome assumptions Some tasksSome tasks Some reflectionsSome reflections

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Some assumptionsSome assumptions

A lesson without the opportunity A lesson without the opportunity

for learners to generalisefor learners to generalise

is is notnot a mathematics lesson a mathematics lesson

Learners come to lessons Learners come to lessons

with natural powers to make with natural powers to make

sensesense

Our job is to direct their Our job is to direct their

attentionattention

appropriately and effectivelyappropriately and effectively

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Grid SumsGrid Sums

In how many different ways can you work out a value for the square with a ‘?’ only using addition?

7

??To move to the right one cell you add 3.

To move up one cell you add 2.

Using exactly two subtractions?

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Grid MovementGrid Movement

7

?

+3

-3

x2÷2

((7+3)x2)+3

is a path from 7 to ‘?’.

What expression represents the reverse of this path?

What values can ‘?’ have:- if only + and x are used- if exactly one - and one ÷ are used, with as many + & x as necessaryWhat about other cells?Does any cell have 0? -7?Does any other cell have 7?

Characterise ALL the possible values that can appear in a cell

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Varying & GeneralisingVarying & Generalising

What are the dimensions of What are the dimensions of possible variation?possible variation?

What is the range of What is the range of permissible change within permissible change within each dimension of variation?each dimension of variation?

You only understand You only understand moremore if if you extend the example space you extend the example space or the scope of generalityor the scope of generality

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Number Line MovementsNumber Line Movements

Rotate it about the point 5 through 180°– where does 3 end up?where does 3 end up?– ……

Now rotate it again bit about the point -2. Now where does the original 3 end up? Generalise!

Imagine a number line

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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Number Spirals

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

10

aaa

30361635542428402863483642364232455681497254635160119905

24206478979654CopperPlate CopperPlate

MultiplicationMultiplication

aaa

30361635542428402863483642364232455681497254635160119905

79654242064789

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Four Odd SumsFour Odd Sums

a

a

a

a

a

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Tunja SequencesTunja Sequences

1 x 1 – 1 = 2 x 2 – 1 = 3 x 3 – 1 = 4 x 4 – 1 =

0 x 2 1 x 3 2 x 4 3 x 5

0 x 0 – 1 = -1 x 1

-1 x -1 – 1 = -2 x 0

Across the Grain

With the Grain

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Tunja Display (1)Tunja Display (1)

2x2 - 2 - 2 = 1x1 - 13x2 - 3 - 2 = 2x1 - 1

4x2 - 4 - 2 = 3x1 - 1

5x2 - 5 - 2 = 4x1 - 1

…

3x3 - 3 - 3 = 2x2 - 1

4x3 - 4 - 3 = 3x2 - 1

5x3 - 5 - 3 = 4x2 - 1

…

Generalise!

Run Backwards

1x2 - 1 - 2 = 0x3 - 1

0x2 - 0 - 2 = (-1)x3 - 1

(-1)x2 - (-1) - 2 = (-2)x3 - 1

2x3 - 2 - 3 = 1x2 - 1

1x3 - 1 - 3 = 0x2 - 1

0x3 - 0 - 3 = (-1)x2 - 1

……

…

(-1)x3 - (-1) - 3 = (-2)x2 - 1

…

…

…

…

…

……

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Tunja Display (2)Tunja Display (2)

4x3x2 - 2x3 - 4x2 = 4x3 - 24x4x2 - 2x4 - 4x2 = 6x3 - 2

4x5x2 - 2x5 - 4x2 = 8x3 - 2

4x6x2 - 2x6 - 4x2 = 10x3 - 2…

4x3x3 - 2x3 - 4x3 = 4x5 - 24x4x3 - 2x4 - 4x3 = 6x5 - 2

4x5x3 - 2x5 - 4x3 = 8x5 - 2

4x6x3 - 2x6 - 4x3 = 10x5 - 2…

Generalise!

Run Backwards

4x2x2 - 2x2 - 4x2 = 2x3 - 2

4x1x2 - 2x1 - 4x2 = 0x3 - 2

4x0x2 - 2x0 - 4x2 = (-2)x3 - 2

4x(-1)x2 - 2x(-1) - 4x2 = (-4)x3 - 2

…

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Structured Variation GridsStructured Variation Grids

Generalisations in two dimensions

Available free at

http://mcs.open.ac.uk/jhm3

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One MoreOne More What numbers are one more than the What numbers are one more than the

product of four consecutive product of four consecutive integers?integers?

Let a and b be any two numbers, at least one of them even. Then ab/2 more than the product of: any number, a more than it, b more than it and a+b more than it, is a perfect square, of the number squared plus a+b times the number plus ab/2 squared.

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Remainders of the Day (1)Remainders of the Day (1)

Write down a number which Write down a number which when you subtract 1 is divisible when you subtract 1 is divisible by 5by 5

and anotherand another and anotherand another Write down one which you think Write down one which you think

no-one else here will write down.no-one else here will write down.

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Remainders of the Day (2)Remainders of the Day (2)

Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2

and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33

and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4

Why must any such number be Why must any such number be divisible by 3? divisible by 3?

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Remainders of the Day (3)Remainders of the Day (3)

Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2

and which is 2 more than a and which is 2 more than a multiple of 3multiple of 3

and which is 3 more than a and which is 3 more than a multiple of 4multiple of 4

… …

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Remainders of the Day (4)Remainders of the Day (4)

Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2

and 1 more than a multiple of and 1 more than a multiple of 33

and 1 more than a multiple of and 1 more than a multiple of 44

… …

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

7

– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?2

2

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More Magic Square More Magic Square ReasoningReasoning

– = 0Sum( ) Sum( )

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PerforationsPerforations

a

How many holes for a sheet of

r rows and c columnsof stamps?

If someone claimedthere were 228 perforations

in a sheet, how could you check?

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Gasket SequencesGasket Sequences

a

a

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ToughToughyy

12345678

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PowersPowers

Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Imagining & ExpressingImagining & Expressing Ordering & ClassifyingOrdering & Classifying Distinguishing & ConnectingDistinguishing & Connecting Assenting & AssertingAssenting & Asserting

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ThemesThemes

Doing & UndoingDoing & Undoing Invariance Amidst ChangeInvariance Amidst Change Freedom & ConstraintFreedom & Constraint Extending & Restricting Extending & Restricting

MeaningMeaning

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Some ReflectionsSome Reflections

Notice the geometrical term:Notice the geometrical term:– It requires movement out of the It requires movement out of the

current space into a space of one current space into a space of one higher dimension in order to higher dimension in order to achieve itachieve it

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AttentionAttention

Gazing at wholesGazing at wholes Discerning detailsDiscerning details Recognising relationshipsRecognising relationships Perceiving propertiesPerceiving properties Reasoning on the basis of Reasoning on the basis of

propertiesproperties

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John MasonJohn Mason J.h.mason @ open.ac.ukJ.h.mason @ open.ac.uk http://mcs.open.ac.uk/jhm3http://mcs.open.ac.uk/jhm3

Developing Thinking in Algebra (Sage 2005)