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Variation Variation as a as a

Pedagogical ToolPedagogical Toolin Mathematicsin Mathematics

John Mason & Anne WatsonJohn Mason & Anne WatsonWitsWits

May 2009May 2009

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Pedagogic DomainsPedagogic Domains ConceptsConcepts TopicsTopics

– Arithmetic Al rgeb a Techniques (Exercises)Techniques (Exercises) TasksTasks

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Topic: arithmetic Topic: arithmetic algebra algebra Expressing Generality for oneselfExpressing Generality for oneself Multiple Expressions for the same thingMultiple Expressions for the same thing

leads to algebraic manipulationleads to algebraic manipulation– BBoth of these arise from becoming aware of oth of these arise from becoming aware of

variationvariation– SSpecifically, of pecifically, of dimensions-of-possible-dimensions-of-possible-

variationvariation

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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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What’s The Ratio?What’s The Ratio?

What could be varied?

÷

=

First, multiply each by 3First, multiply the larger by 2 and divide the smaller by 3

What is the ratio?What is the ratio?

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Counting & ActionsCounting & Actions If I have 3 more things than you do, and If I have 3 more things than you do, and

you have 5 more things than she has, you have 5 more things than she has, how many more things do I have than how many more things do I have than she has?she has?– Variations?Variations?

If Anne gives me one of her marbles, she If Anne gives me one of her marbles, she will then have twice as many as I then will then have twice as many as I then have, but if I give her one of mine, she have, but if I give her one of mine, she will then be 1 short of three times as will then be 1 short of three times as many as I then have.many as I then have. Do your

expressions express what you mean them to express?

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Construction before ResolutionConstruction before Resolution I start with 12 and 8I start with 12 and 8

– 1212 88 1212 88– 1111 99 1313 77– 1010 1010 1414 44– 1515 55

So if Anne gives John 2, they will then have So if Anne gives John 2, they will then have the same number; if John gives Anne 3, she the same number; if John gives Anne 3, she will then have 3 times as many as John then will then have 3 times as many as John then hashas

Construct one of your ownConstruct one of your own– AAnd anothernd another– AAnd anothernd another

Working down and up, keeping sum invariant, looking for a multiplicative relationship

Translate into ‘sharing’ actions

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PrinciplePrinciple Before showing learners how to answer a Before showing learners how to answer a

typical problem or question, get them to typical problem or question, get them to make up questions like it so they can see make up questions like it so they can see how such questions arise.how such questions arise.– EEquations in one variablequations in one variable– EEquations in two variablesquations in two variables– WWord problems of a given typeord problems of a given type– ……

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Four ConsecutivesFour Consecutives

Write down four consecutive Write down four consecutive numbers and add them upnumbers and add them up

and anotherand another and anotherand another Now be more extreme!Now be more extreme! What is the same, and what is What is the same, and what is

different about your answers?different about your answers?

+ 1

+ 2

+ 3

+ 64

– 1

+ 1

+ 2

+ 24

Alternative:I have 4 consecutive numbers in mind.They add up to 42. What are they?

D of P V?R of P Ch?

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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 integers?product of four consecutive integers?

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

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ComparingComparing If you gave me 5 of your things then I would If you gave me 5 of your things then I would

have three times as a many as you then had, have three times as a many as you then had, whereas if I gave you 3 of mine then you would whereas if I gave you 3 of mine then you would have 1 more than 2 times as many as I then had. have 1 more than 2 times as many as I then had. How many do we each have?How many do we each have?

If B gives A $15, A will have 5 times as much If B gives A $15, A will have 5 times as much as B has left. If A gives B $5, B will have the as B has left. If A gives B $5, B will have the same as A. [Bridges 1826 p82]same as A. [Bridges 1826 p82]

If you take 5 from the father’s years and divide If you take 5 from the father’s years and divide the remainder by 8, the quotient is one third the remainder by 8, the quotient is one third the son’s age; if you add two to the son’s age, the son’s age; if you add two to the son’s age, multiply the whole by 3 and take 7 from the multiply the whole by 3 and take 7 from the product, you will have the father’s age. How product, you will have the father’s age. How old are they? [Hill 1745 p368]old are they? [Hill 1745 p368]

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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

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Lee Minor’s Mutual FactorsLee Minor’s Mutual Factorsx2 + 5x + 6 = (x + 3)(x + 2)x2 + 5x – 6 = (x + 6)(x – 1)x2 + 13x + 30 = (x + 10)(x + 3)x2 + 13x – 30 = (x + 15)(x – 2)x2 + 25x + 84 = (x + 21)(x + 4)x2 + 25x – 84 = (x + 28)(x – 3)x2 + 41x + 180 = (x + 36)(x + 5)x2 + 41x – 180 = (x + 45)(x – 4)

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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

36

15

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

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Triangle CountTriangle Count

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Up & Down SumsUp & Down Sums

1 + 3 + 5 + 3 + 1

3 x 4 + 122 + 32

1 + 3 + … + (2n–1) + … + 3 + 1

==

n (2n–2) + 1 (n–1)2 + n2 ==

Generalise!See

generalitythrough aparticular

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PerforationsPerforations

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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DifferencesDifferences

17=16−142

AnticipatingGeneralising

Rehearsing

Checking

Organising

18=17−156

=16−124

=14−18

13=12−16

14=13−112

=12−14

15=14−120

16=15−130

=12−13=13−16=14− 112

12=11−12

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Tracking ArithmeticTracking Arithmetic If you can check an answer, you can If you can check an answer, you can

write down the constraints (express the write down the constraints (express the structure) symbolicallystructure) symbolically

Check a conjectured answer BUT don’t Check a conjectured answer BUT don’t ever actually do any arithmetic ever actually do any arithmetic operations that involve that ‘answer’.operations that involve that ‘answer’.

PedDoms

THOANsTHOANsThink of a numberThink of a numberAdd 3Add 3Multiply by 2Multiply by 2Subtract your first Subtract your first numbernumberSubtract 6Subtract 6You have your starting You have your starting numbernumber

77 + 32x7 + 62x7 + 6 – 72x7 – 77

+ 32x + 62x + 6 – 2x –

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ConceptsConcepts Name some concepts that students struggle Name some concepts that students struggle

withwith– EEg perimeter & area; g perimeter & area; – slope-gradient; slope-gradient; – annuity (?)annuity (?)– MMultiplicative reasoningultiplicative reasoning– AAlgebraic reasoninglgebraic reasoning

Construct an exampleConstruct an example– NNow what can vary and still that remains an ow what can vary and still that remains an

example?example?DDimensions-of-possible-variation; Range-of-imensions-of-possible-variation; Range-of-permissible-changepermissible-change

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ComparisonsComparisons

Which is bigger?Which is bigger?– 83 x 27 or 84 x 2683 x 27 or 84 x 26– 8/0.4 or 8 x 0.48/0.4 or 8 x 0.4– 867/.736 or 867 x .736867/.736 or 867 x .736– 3/43/4 of 2/3 of something, or 2/3 of of 2/3 of something, or 2/3 of 3/43/4 of of

somethingsomething– 5/3 of something or the thing itself?5/3 of something or the thing itself?– 437 – (-232) or 437 + (-232)437 – (-232) or 437 + (-232)

What variations can you produce?What variations can you produce? What conjectured generalisations are being What conjectured generalisations are being

challenged?challenged? What generalisations (properties) are being What generalisations (properties) are being

instantiated?instantiated?

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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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Teaching TrapTeaching Trap Doing for the learners what they can Doing for the learners what they can

already do for themselvesalready do for themselves Teacher Lust:Teacher Lust:

– desire that the learner learndesire that the learner learn– allowing personal excitement to drive allowing personal excitement to drive

behaviourbehaviour

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Mathematical ThemesMathematical Themes Doing & UndoingDoing & Undoing Invariance Amidst ChangeInvariance Amidst Change Freedom & ConstraintFreedom & Constraint Extending & Restricting MeaningExtending & Restricting Meaning

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ProtasesProtases

Only awareness is educableOnly awareness is educableOnly behaviour is trainableOnly behaviour is trainable

Only emotion is harnessableOnly emotion is harnessable

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Didactic TensionDidactic Tension

TheThe more clearly I indicate the behaviour sought from learners,

the less likely they are togenerate that behaviour for themselves

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Pedagogic DomainsPedagogic Domains ConceptsConcepts

– WWhat do examples look like?hat do examples look like?WWhat in an example can be varied? (DofPV; RofPCh)hat in an example can be varied? (DofPV; RofPCh)

TopicsTopicsLearners constructing examples (Solving as Undoing Learners constructing examples (Solving as Undoing

of building)of building)Learners experiencing variation (DofPV, RofPCh)Learners experiencing variation (DofPV, RofPCh)Learners constructing variations (Doing & Undoing)Learners constructing variations (Doing & Undoing)

Techniques (Exercises)Techniques (Exercises)– SSee above!ee above!– Structured exercises exposing DofPV & RofPChStructured exercises exposing DofPV & RofPCh

TasksTasks– VVarying DofPV; exposing RofPCharying DofPV; exposing RofPCh

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VariationVariation Object(s) of LearningObject(s) of Learning

– KKey understandings; Awarenessesey understandings; Awarenesses– IIntended; Perceived-afforded; Enactedntended; Perceived-afforded; Enacted– EEncountering structured variationncountering structured variation

Varying to enrich Example SpacesVarying to enrich Example Spaces Actions performedActions performed

– TTasks asks activity activity experience experience Reconstruction & Reflection on Action Reconstruction & Reflection on Action

(efficiency, effectiveness)(efficiency, effectiveness) Use of powers & Use of powers &

Exposure to mathematical themesExposure to mathematical themes– Affective: dispositionAffective: disposition

PsychePsyche– awareness, emotion, behaviourawareness, emotion, behaviour

DofPV & RofPChDofPV & RofPCh