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Generalisation in Mathematics:who generalises what, when, how and why?

John MasonTrondheimApril 2009

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Some Sums

4 + 5 + 6 =9 + 10 + 11 + 1216

Generalise

Justify

Watch What You Do

Say What You See

1 + 2 =3

7 + 8= 13 + 14 + 15

17 + 18 + 19 + 20+ = 21 + 22 + 23 + 24

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Four Consecutives

Write down four consecutive numbers and add them up

and another and another Now be more

extreme! What is the same, and

what is different about your answers?

+ 1

+ 2

+ 3

+ 64

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One More

What numbers are one more than the product of four consecutive integers?

Let a and b be any two numbers, 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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CopperPlate Calculations

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Structured Variation Grids

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Extended Sequences

Someone has made a simple pattern of coloured squares, and then repeated it a total of at least two times

State in words what you think the original pattern was

Predict the colour of the 100th square and the position of the 100th white square…

…

Make up your own: a really simple

one a really hard

one

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Raise Your Hand When You Can See

Something which is 1/4 of something 1/5 of something 1/4-1/5 of something 1/4 of 1/5 of

something 1/5 of 1/4 of

something 1/n – 1/(n+1) of

something

What do you have to do with your attention?

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In how many different ways can you count them?

Gnomon Border

How many tiles are needed to surround the 137th gnomon?

The fifth is shown here

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Perforations

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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Honsberger’s Grid

17181920

2122

23

24

25

10111213141516

5

6

7

8

9

1

2

4

3

4

31

43 57 73 91 111

133 147

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Painted Cube

A cube of wood is dropped into a bucket of paint. When the paint dries it is cut into little cubes (cubelets). How many cubes are painted on how many faces?

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Attention

Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of

properties

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The Place of Generality

A lesson without the opportunity for learners to generalise mathematically, is not a mathematics lesson

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Text Books

Turn to a teaching page– What generality (generalities)

are present?– How might I get the learners to

experience and express them?– For the given tasks, what inner

tasks might learners encounter?New conceptsNew actionsMathematical themesUse of mathematical powersRehearsal of developing skills and

actions

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Roots of & Routes to Algebra

Expressing Generality– A lesson without the possibility of learners

generalising (mathematically) is not a mathematics lesson

Multiple Expressions– Purpose and evidence for the ‘rules’ of algebraic

manipulation Freedom & Constraint

– Every mathematical problem is a construction task, exploring the freedom available despite constraints

Generalised Arithmetic– Uncovering and expressing the rules of arithmetic

as the rules of algebra

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MGA & DTR

Doing – Talking – Recording

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DofPV & RofPCh

Dimensions of possible variation– What can be varied and still

something remains invariant Range of permissible change

– Over what range can the change take place and preserve the invariance

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Some Mathematical Powers

Imagining & Expressing Specialising & Generalising Conjecturing & Convincing Stressing & Ignoring Ordering & Characterising

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Some Mathematical Themes

Doing and Undoing Invariance in the midst of

Change Freedom & Constraint

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Consecutive Sums

Say What You See

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For More Details

Thinkers (ATM, Derby)Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby)Mathematics as a Constructive Activity (Erlbaum)Listening Counts (Trentham)

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

j.h.mason@open.ac.uk

Structured Variation Grids

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