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Thinking MathematicallyThinking Mathematicallyand and

Learning MathematicsLearning MathematicsMathematicallyMathematically

John MasonJohn Mason

GreenwichGreenwich

Oct 2008Oct 2008

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Conjecturing AtmosphereConjecturing Atmosphere

Everything said is said in Everything said is said in order to consider order to consider modifications that may be modifications that may be neededneeded

Those who ‘know’ support Those who ‘know’ support those who are unsure by those who are unsure by holding back or by asking holding back or by asking revealing questionsrevealing questions

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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

==

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Remainders of the DayRemainders of the Day Write down a number that leaves a Write down a number that leaves a reminder of 1 when divided by 3reminder of 1 when divided by 3

and anotherand another and anotherand another Choose two simple numbers of this type Choose two simple numbers of this type and multiply them together: and multiply them together: what remainder does it leave when what remainder does it leave when divided by 3?divided by 3?

Why?Why? What is special What is special about the ‘3’? about the ‘3’?

What is special about the ‘1’?What is special about the ‘1’?

What is special about the ‘1’?

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PrimalityPrimality

What is the second positive What is the second positive non-prime after 1 in the system non-prime after 1 in the system of numbers of the form 1+3of numbers of the form 1+3nn??

100 = 10 x 10 = 4 x 25100 = 10 x 10 = 4 x 25 What does this say about primes What does this say about primes in the multiplicative system of in the multiplicative system of numbers of the form 1 +3numbers of the form 1 +3nn??

What is special about the ‘3’?What is special about the ‘3’?

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Inter-Rootal DistancesInter-Rootal Distances

Sketch a quadratic for which the Sketch a quadratic for which the inter-rootal distance is 2.inter-rootal distance is 2.

and anotherand another and anotherand another How much freedom do you have?How much freedom do you have? What are the dimensions of possible What are the dimensions of possible variation and the ranges of variation and the ranges of permissible change?permissible change?

If it is claimed that [1, 2, 3, 3, If it is claimed that [1, 2, 3, 3, 4, 6] are the inter-rootal 4, 6] are the inter-rootal distances of a quartic, how would distances of a quartic, how would you check?you check?

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Bag Constructions (1)Bag Constructions (1) Here there are three Here there are three bags. If you compare bags. If you compare any two of them, there any two of them, there is exactly one colour is exactly one colour for which the difference for which the difference in the numbers of that in the numbers of that colour in the two bags colour in the two bags is exactly 1.is exactly 1.

17 objects

3 colours

For four bags, what is For four bags, what is the least number of the least number of objects to meet the same objects to meet the same constraint?constraint? For four bags, what is For four bags, what is the least number of the least number of colours to meet the same colours to meet the same constraint?constraint?

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Bag Constructions (2)Bag Constructions (2) Here there are 3 bags and Here there are 3 bags and two objects.two objects.

There are [0,1,2;2] There are [0,1,2;2] objects in the bags with 2 objects in the bags with 2 altogetheraltogether

Given a sequence like Given a sequence like [2,4,5,5;6] or [1,1,3,3;6] [2,4,5,5;6] or [1,1,3,3;6] how can you tell if there how can you tell if there is a corresponding set of is a corresponding set of bags?bags?

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StatisticalityStatisticality

write down five numbers whose write down five numbers whose mean is 5mean is 5

and whose mode is 6and whose mode is 6 and whose median is 4and whose median is 4

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ZigZagsZigZags

Sketch the graph of Sketch the graph of yy = | = |x x – 1|– 1| Sketch the graph of Sketch the graph of yy = | |x - 1| - = | |x - 1| - 2|2|

Sketch the graph of Sketch the graph of y = | | |y = | | |xx – 1| – 2| – 3| – 1| – 2| – 3|

What sorts of zigzags can you make, What sorts of zigzags can you make, and not make?and not make?

Characterise all the zigzags you can Characterise all the zigzags you can make using sequences of absolute make using sequences of absolute values like this.values like this.

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Towards the Blanc Mange Towards the Blanc Mange functionfunction

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Reading GraphsReading Graphs

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ExamplesExamples

Of what is |Of what is |xx| an example?| an example? Of what is Of what is yy = = xx22 and and example?example?– y = by = b + ( + (xx – – aa))22 ? ?

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Functional ImaginingFunctional Imagining

Imagine a parabolaImagine a parabola Now imagine another Now imagine another one the other way up.one the other way up.

Now put them in two Now put them in two planes at right planes at right angles to each other.angles to each other.

Make the maximum of Make the maximum of the downward parabola the downward parabola be on the upward be on the upward parabolaparabola

Now sweep your downward Now sweep your downward parabola along the upward parabola along the upward parabola so that you get a parabola so that you get a surfacesurface

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MGAMGA

aa

Getting-a-sense-ofManipulatingGetting-a-sense-ofArticulatingManipulating

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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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Teaching Trap Teaching Trap Learning TrapLearning Trap

Doing for the learners Doing for the learners what they can already what they can already do for themselvesdo for themselves

Teacher Lust:Teacher Lust:– desire that the desire that the learner learnlearner learn

– desire that the desire that the learner appreciate learner appreciate and understand and understand

– Expectation that Expectation that learner will go learner will go beyond the tasks as beyond the tasks as setset

– allowing personal allowing personal excitement to drive excitement to drive behaviourbehaviour

Expecting the teacher to Expecting the teacher to do for you what you can do for you what you can already do for yourselfalready do for yourself

Learner Lust:Learner Lust:– desire that the teacher desire that the teacher teachteach

– desire that learning will desire that learning will be easybe easy

– expectation that ‘dong expectation that ‘dong the tasks’ will produce the tasks’ will produce learninglearning

– allowing personal allowing personal reluctance/uncertainty to reluctance/uncertainty to drive behaviourdrive behaviour

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Human PsycheHuman Psyche

Training BehaviourTraining Behaviour Educating AwarenessEducating Awareness Harnessing EmotionHarnessing Emotion Who does these?Who does these?

– Teacher?Teacher?– Teacher with learners?Teacher with learners?– Learners!Learners!

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Structure of the PsycheStructure of the PsycheImageryImageryAwareness (cognition)Awareness (cognition)

WillWill

Body (enaction)Body (enaction)

Emotions Emotions (affect)(affect)

HabitsHabitsPracticesPractices

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Structure of a TopicStructure of a TopicLanguage Patterns& prior Skills

Techniques & Incantations

Different Contexts in which likely to

arise;dispositions

Root Questionspredispositions

Only Behaviour is TrainableOnly Behaviour is Trainable

Only Emotion is HarnessableOnly Emotion is HarnessableOnly Awareness is Only Awareness is

EducableEducable

Behaviour

Behaviour

Behaviour

Behaviour

EmotionEmotionEmotionEmotion

Awareness

Awareness

Awareness

Awareness

Imagery/Sense-of/Awareness; Connections

Standard Confusions & Obstacles

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Didactic TensionDidactic Tension

The more clearly I indicate the behaviour sought from learners,

the less likely they are togenerate that behaviour for themselves

(Guy Brousseau)

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Didactic TranspositionDidactic Transposition

Expert awareness

is transposed/transformed into

instruction in behaviour(Yves Chevellard)

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More IdeasMore Ideas

(2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester.

(2008). Counter Examples in Calculus. College Press, London.

(1998) Learning & Doing Mathematics (Second revised edition), QED Books, York.

(1982). Thinking Mathematically, Addison Wesley, London

For Lecturers

For Students

http://mcs.open.ac.uk/jhm3j.h.mason@open.ac.uk

Modes of interaction

Expounding

Explaining

Exploring

Examining

ExercisingExpressing

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Student

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Exploring

StudentContentTeacherExamining

StudentContent

TeacherExercising

Student

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Expressing

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StudentExplaining

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