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Developing Mathematical Thinking

John MasonJohn Mason

FlFlötur, ötur, SelfossSelfoss

Sept 2008Sept 2008

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Some Throat ClearingSome Throat Clearing

What you get from this session What you get from this session will be what you notice will be what you notice happening inside youhappening inside you

Everything said is to be Everything said is to be treated as a conjecture, and treated as a conjecture, and tested in your experiencetested in your experience

If you don’t engage in my If you don’t engage in my tasks, you will get nothing!tasks, you will get nothing!

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How often do you arrange for your students

to use this power for themselves?

Getting GoingGetting Going

Specialisingin order to

(re)generalise

If the difference of two numbers is even, then their product is the difference of two squares

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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 is any two of them, there is exactly one colour for exactly one colour for which the difference in which the difference in the numbers of that the numbers of that colour in the two bags is colour in the two bags is exactly 1.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) For For bb bags, how few bags, how few

objects can you use so objects can you use so that each pair of bags that each pair of bags has the property that has the property that there are exactly two there are exactly two colours for which the colours for which the difference in the numbers difference in the numbers of that colour in the two of that colour in the two bags is exactly 1.bags is exactly 1.

Construct four bags such Construct four bags such that for each pair, there is that for each pair, there is just one colour for which just one colour for which the total number of that the total number of that colour in the two bags is colour in the two bags is 3.3.

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Bag Constructions (3)Bag Constructions (3) Here there are 3 bags and Here there are 3 bags and

two objects.two objects. There are [0,1,2;2] objects There are [0,1,2;2] objects

in the bags and 2 in the bags and 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 is how can you tell if there is a corresponding set of a corresponding set of bags?bags?

In how many different In how many different ways can you put ways can you put kk objects objects in in bb bags? bags?

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QuickTime™ and aAnimation decompressor

are needed to see this picture.

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Triangle CountTriangle Count

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AttentionAttention

Holding Wholes (gazing)Holding Wholes (gazing)

Discerning DetailsDiscerning Details

Recognising RelationshipsRecognising Relationships

Perceiving PropertiesPerceiving Properties

Reasoning on the basis of Reasoning on the basis of agreed propertiesagreed properties

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Doing & UndoingDoing & Undoing

What operation undoes ‘adding 3’?What operation undoes ‘adding 3’?What operation undoes ‘subtracting What operation undoes ‘subtracting 4’?4’?What operation undoes What operation undoes ‘subtracting from 7’? ‘subtracting from 7’?What are the analogues for What are the analogues for multiplication?multiplication?

What undoes multiplying by 3?What undoes multiplying by 3?What undoes dividing by 2?What undoes dividing by 2?What undoes multiplying by 3/2?What undoes multiplying by 3/2?What undoes dividing by 3/2?What undoes dividing by 3/2?

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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

Across the Grain

With the Grain

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

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– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?2

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )

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Map Drawing ProblemMap Drawing Problem

Two people both have a copy of the same map Two people both have a copy of the same map of Iceland.of Iceland.

One uses Reykjavik as the centre for a scaling One uses Reykjavik as the centre for a scaling by a factor of 1/3by a factor of 1/3

One uses Akureyri as the centre for a scaling by One uses Akureyri as the centre for a scaling by a factor of 1/3a factor of 1/3

What is the same, and what is different about What is the same, and what is different about the maps they draw?the maps they draw?

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Some Mathematical PowersSome Mathematical Powers

Imagining & ExpressingImagining & Expressing Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Stressing & IgnoringStressing & Ignoring Ordering & CharacterisingOrdering & Characterising Seeing Sameness & Seeing Seeing Sameness & Seeing

DifferenceDifference Assenting & AssertingAssenting & Asserting

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Some Mathematical ThemesSome Mathematical Themes

Doing and UndoingDoing and Undoing Invariance in the midst of Invariance in the midst of

ChangeChange Freedom & ConstraintFreedom & Constraint

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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 Harnessable

Only Awareness is EducableOnly Awareness is Educable

Behaviour

Behaviour

Behaviour

Behaviour

EmotionEmotionEmotionEmotion

Awar

enes

s

Awar

enes

s

Awar

enes

s

Awar

enes

s

Imagery/Sense-of/Awareness; Connections

Standard Confusions

& Obstacles

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For More DetailsFor More Details

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

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

Structured Variation GridsStudies in Algebraic ThinkingOther PublicationsThis and other presentations