Variation as a Pedagogical Tool in Mathematics
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Transcript of Variation as a Pedagogical Tool in Mathematics
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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
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7 8 9 10
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12
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18
19
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21 22 23 24 25 26
27
28
29
30
3132
14151617
3334353637
38
39
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42
43 44 45 46 47 48 49 50
1
4
9
16
25
49
36
15
1
2 3 4
5
6789
101112
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18 19 20
21
22
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242526272829
303132
14 15 16 17
33
34
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36 37 38 39 40 41 42 43 44
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64
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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