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With and Across the Grain:With and Across the Grain:making use of learners’ powersmaking use of learners’ powersto detect and express generalityto detect and express generality

London Mathematics

CentreJune 2006

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OutlineOutline Orienting QuestionsOrienting Questions Tasks through which to experience Tasks through which to experience

and in which to noticeand in which to notice ReflectionsReflections

a lesson without the opportunitya lesson without the opportunityfor learners to generalisefor learners to generalise

is is notnot a mathematics lesson! a mathematics lesson!

Conjecture

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Orienting QuestionsOrienting Questions What can be changed?What can be changed? In what way can it be changed?In what way can it be changed? What is the What is the innerinner task? task? What pedagogical and didactic choices What pedagogical and didactic choices

are being made? are being made?

What you can get from this sessionWhat you can get from this session:

what you notice inside yourself;what you notice inside yourself;ways of thinking and acting;ways of thinking and acting;useful distinctions.useful distinctions.

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Seeing the general through the particularSeeing the general

through the particularSeeing the general

through the particular

Up and Down SumsUp and Down Sums

1 + 2 + 3 + 4 + 3 + 2 1 + 2 + 3 + 4 + 3 + 2 + 1 =+ 1 =

1 + 2 + 3 + 2 + 1 + 2 + 3 + 2 + 1 =1 =

1 + 2 + 1 + 2 + 1 =1 =

1 1 ==

1122

2222

3322

4422

……

In how many different In how many different ways can you see WHY ways can you see WHY itit works? works?

What is the What is the itit??

1 + 2 + 3 + 4 1 + 2 + 3 + 4 + + 3 + 2 + 1 3 + 2 + 1

= 4 x 4 = 4 x 4 = 4= 422

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Seeing the general through the particularSeeing the general

through the particularSeeing the general

through the particular

More Up and Down SumsMore Up and Down Sums

1 + 3 + 5 + 7 + 5 + 3 1 + 3 + 5 + 7 + 5 + 3 + 1 =+ 1 =

1 + 3 + 5 + 3 + 1 + 3 + 5 + 3 + 1 =1 =

1 + 3 + 1 + 3 + 1 =1 =

1 1 ==

1x0 + 11x0 + 1

2x2 + 1 2x2 + 1

3x4 + 13x4 + 1

4x6 + 14x6 + 1

……

In how In how many many different different ways ways can you see can you see WHY WHY itit works?works?

What is the What is the itit??

1 + 3 + 5 + 7 1 + 3 + 5 + 7 + + 5 + 3 + 15 + 3 + 1

= 4x6 + 1= 4x6 + 1

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Yet More Up and Down SumsYet More Up and Down Sums

1 + 4 + 7 + 10 + 7 + 4 1 + 4 + 7 + 10 + 7 + 4 + 1 =+ 1 =

1 + 4 + 7 + 4 + 1 + 4 + 7 + 4 + 1 =1 =

1 + 4 + 1 + 4 + 1 =1 =

1 1 ==

1x(-1) + 21x(-1) + 2

2x2 + 2 2x2 + 2

3x5 + 23x5 + 2

4x8 + 24x8 + 2

……

In how many In how many

different different ways ways can you see can you see WHY WHY itit works?works?

What is the What is the itit??

1 + 4 + 7 + 10 + 13 + 10 + 7 + 1 + 4 + 7 + 10 + 13 + 10 + 7 + 4 + 1 =4 + 1 =

5x11 + 25x11 + 2

1 + 4 + 7 + 10 + 1 + 4 + 7 + 10 + 13 + 13 + 10 + 7 + 4 + 110 + 7 + 4 + 1= 5x11 + 2= 5x11 + 2

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Same and DifferentSame and Different

What is the same and what is different What is the same and what is different about the three Up and Down Sum tasks?about the three Up and Down Sum tasks?

What dimensions of possible variation What dimensions of possible variation are there? (what could be changed?)are there? (what could be changed?)

What is the range of permissible change What is the range of permissible change of each dimension (what values could be of each dimension (what values could be taken?)taken?)

1 + 4 + 7 + 4 + 1 + 4 + 7 + 4 + 1 =1 =

3x5 + 23x5 + 2

1 + 3 + 5 + 3 + 1 + 3 + 5 + 3 + 1 =1 =

3x4 + 13x4 + 1

1 + 2 + 3 + 2 + 1 + 2 + 3 + 2 + 1 =1 =

3322

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ReflectionsReflections What inner tasks might be associated What inner tasks might be associated

with Up and Down Sums?with Up and Down Sums? What pedagogic and didactic choices What pedagogic and didactic choices

did you notice being made?did you notice being made?

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Some SumsSome Sums1 + 2 =1 + 2 =4 + 5 + 6 =4 + 5 + 6 =

9 + 10 + 11 + 12 9 + 10 + 11 + 12 ==

Generalise

Justify

Watch What You Do

Say What You See

7 + 87 + 813 + 14 + 1513 + 14 + 15

1616 ++17 + 18 + 19 + 2017 + 18 + 19 + 20==21 + 22 + 23 + 2421 + 22 + 23 + 24

33

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What’s The Difference?What’s The Difference?

What could be varied?

– =

First, add one to eachFirst, 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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a

a b

a+b a+2b2a+b

a+3b3a+b

3b-3a

a

2 3

587

911

3

3(3b-3a) = 3a+b

8b = 12a

So a : b = 2 : 3

For an overall square

4a + 4b = 2a + 5b

So 2a = b

Oops!

For n squares upper left

n(3b - 3a) = 3a + b

So 3a(n + 1) = b(3n - 1)

a : b = 3n – 1 : 3(n + 1)

Square ReasoningSquare Reasoning

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Seeing the general through the particular;Expressing Generality

ReflectionsReflections What might the What might the innerinner tasks have been? tasks have been? What pedagogic and didactic What pedagogic and didactic choiceschoices

did you notice being made?did you notice being made?

In every lesson!

Say What You SeeCan You See?

With and Across the grain

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With & Across The Grain (part 2):With & Across The Grain (part 2):Sequences and GridsSequences and Grids

NumberGrid

What sorts of orienting questions What sorts of orienting questions are you ready to use in this are you ready to use in this session?session?

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Experiencing GeneralisationExperiencing Generalisation

Qu

ickTim

e™

an

d a

TIF

F (

Un

co

mp

re

sse

d) d

eco

mp

re

sso

ra

re n

eed

ed

to

se

e t

his

pic

tu

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.

Pleasure in use of powers; Pleasure in use of powers; disposition: affective generalisationdisposition: affective generalisation(Helen Drury)(Helen Drury)

Going across the grain: Going across the grain: cognitive generalisationcognitive generalisation

Going with the grain: Going with the grain: enactive generalisationenactive generalisation

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Use of Powers on Inappropriate DataUse of Powers on Inappropriate Data If 10% of 23 is 2.3If 10% of 23 is 2.3

– What is 20% of 2.3? What is 20% of 2.3? – .23 ?!?.23 ?!?

If to find 10% you divide by 10If to find 10% you divide by 10– To find 20% you divide by To find 20% you divide by – 20 ?!?20 ?!? Perhaps some ‘wrong answers’ arise Perhaps some ‘wrong answers’ arise

from inappropriate use of powers … from inappropriate use of powers …

so you can praise the use of powers so you can praise the use of powers but treat the response as a but treat the response as a conjecture … conjecture …

which needs checking and modifying!which needs checking and modifying!

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ReflectionsReflections Arithmetic has an underlying Arithmetic has an underlying

structure/logicstructure/logic What pedagogic choices are available?What pedagogic choices are available? What didactic choices are available?What didactic choices are available? What might you do if some learners ‘work What might you do if some learners ‘work

it all out’ quickly?it all out’ quickly? What might you do if learners are What might you do if learners are

reluctant to conjecture?reluctant to conjecture?

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Imagine …Imagine … A teaching page of a textbook, or a A teaching page of a textbook, or a

work card or other handout to learners work card or other handout to learners used or encountered recentlyused or encountered recently– What are its principal features?What are its principal features?– What are learners supposed to What are learners supposed to

get from ‘doing it’?get from ‘doing it’?

Conjecture: it contains

Examples of …

A lesson without the opportunity for learners

to generalise …

is NOT a maths lesson!

Problem typeTechniqueMethodUseConcept

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Descent to the Particular & the SimpleDescent to the Particular & the Simple Research on problem solving, task Research on problem solving, task

setting, textbooks and use of ICT setting, textbooks and use of ICT suggests thatsuggests that– tasks & problems get simplified tasks & problems get simplified

‘so learners know what to do’‘so learners know what to do’– learners’ powers are often learners’ powers are often

bypassedbypassed Counteract this by trying toCounteract this by trying to

– do only for learners what they do only for learners what they cannot yet do for themselvescannot yet do for themselves (even if it takes longer)(even if it takes longer)

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Promoting GeneralisationPromoting Generalisation Dimensions of Possible VariationDimensions of Possible Variation

Range of Permissible ChangeRange of Permissible Change– What can be changed, and over What can be changed, and over

what range?what range?

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ToughyToughy

12345678

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SummarySummary Expressing GeneralityExpressing Generality

– Enactive; Cognitive; AffectiveEnactive; Cognitive; Affective– with the grain; across the grain; with the grain; across the grain;

dispositiondisposition Dimensions of Possible VariationDimensions of Possible Variation

Range of Permissible ChangeRange of Permissible Change Say What You See & Watch What You DoSay What You See & Watch What You Do What pedagogic and didactic choices could be What pedagogic and didactic choices could be

made?made? What is the inner task? What is the inner task?

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Inner TaskInner Task What mathematical powers might be used?What mathematical powers might be used?

– Imagining & ExpressingImagining & Expressing– Specialising & GeneralisingSpecialising & Generalising– Stressing & IgnoringStressing & Ignoring– Conjecturing & ConvincingConjecturing & Convincing

What mathematical themes might arise?What mathematical themes might arise?– Invariance in the midst of changeInvariance in the midst of change– Doing and UndoingDoing and Undoing

What personal dispositions might emerge to be What personal dispositions might emerge to be worked on?worked on?– Tendency to dive inTendency to dive in– Tendency to give up or seek helpTendency to give up or seek help

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Consecutive SumsConsecutive Sums

a a

a

Say What You SeeCan You See …?

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For Further ExplorationFor Further Exploration

Mason, J. with Johnston-Wilder, S. & Graham, A. (2005). Developing Thinking in Algebra. London: Sage.Johnston-Wilder, S. & Mason, J. (Eds.) (2005). Developing Thinking in Geometry. London: Sage.

Open University Courses on teachingAlgebra, Geometry, Statistics and Mathematical Thinking

Structured variation Grids available (free) on

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

j.h.mason@open.ac.uk