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Mathematics:Mathematics:with good reasonwith good reason

John MasonJohn Mason

ExeterExeter

April 2010April 2010

The Open UniversityMaths Dept University of Oxford

Dept of Education

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AimsAims

To experience shifts from To experience shifts from ”It just is” ”It just is” to to “It must be because …” “It must be because …”

To consider a variety of tasks To consider a variety of tasks which can be used to stimulate which can be used to stimulate reasoningreasoning

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Revealing ShapesRevealing Shapes

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Order! Order!Order! Order! A, B, C, D, and E are in a queueA, B, C, D, and E are in a queue

– B is in front of C B is in front of C – A is behind EA is behind E– There are two people between D and EThere are two people between D and E– There is one person between D and CThere is one person between D and C– There is one person between B and EThere is one person between B and E

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Say What You SeeSay What You See

There are There are 16 canoes16 canoes5 asteroids5 asteroids4 wedges4 wedges4 peaks4 peaks

and these account for the total areaand these account for the total area

Also 6 arches; 6 troughs;

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Bag Re-ConstructionsBag Re-Constructions 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 with 2 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 is how can you tell if there is a corresponding set of a corresponding set of bags?bags?

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Why is (-1) x (-1) = 1?Why is (-1) x (-1) = 1?

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Fractional Increase and Fractional Increase and DecreaseDecrease

(1 + )

12

(1 – ) 13

=

(1 + )

25

(1 – ) 27

=

= 1(1 – ) (1 + )

ab

By how much do I have to decrease in order to undo an increase by one-half?By how much do I have to increase in order to undo a decrease by two-sevenths?

(1 + )

38

(1 – ) 311

=

= 1(1 – ) (1 + )

ab

Make up your own!

b

a+b

1

1

1

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Marbles (Bob Davis)Marbles (Bob Davis)

I have a bag of marblesI have a bag of marbles I take out 7, then put in 3, I take out 7, then put in 3,

then take out 4. What is the then take out 4. What is the state of my bag now?state of my bag now?– Variations?Variations?

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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 first and subtract one from the second

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 first by 2 and divide the second by 3

What is the ratio?What is the ratio?

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Speed ReasoningSpeed Reasoning

If I run 3 times as fast as you, how If I run 3 times as fast as you, how long will it take me compared to long will it take me compared to you to run a given distance?you to run a given distance?

If I run 2/3 as fast as you, how long If I run 2/3 as fast as you, how long will it take me compared to you?will it take me compared to you?

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Doing & UndoingDoing & Undoing

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

What undoes ‘multiplying by 3’?What undoes ‘multiplying by 3’?What undoes ‘dividing by 4’?What undoes ‘dividing by 4’?What undoes ‘multiplying by What undoes ‘multiplying by ¾¾’?’?

Two different expressions!Two different expressions!

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

7

– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?

2

2

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )

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TeachingTeaching

SSelecting taskselecting tasks PPreparing reparing Didactic Tactics Didactic Tactics

and and Pedagogic StrategiesPedagogic Strategies Prompting extended or fresh actionsPrompting extended or fresh actions Being Aware of mathematical actionsBeing Aware of mathematical actions Directing AttentionDirecting Attention

Teaching takes place in time;Learning takes place over time

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The Place of GeneralityThe Place of Generality

A lesson without the A lesson without the opportunity for learners to opportunity for learners to generalise mathematically, is generalise mathematically, is not a mathematics lessonnot a mathematics lesson

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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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Some Mathematical PowersSome Mathematical Powers

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

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

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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 Extending & Restricting Extending & Restricting

MeaningMeaning

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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)Thinking Mathematically (new edition out any day)

mcs.open.ac.uk/jhm3

j.h.mason@open.ac.uk

Structured Variation GridsRevealing ShapesStudies in Algebraic ThinkingOther PublicationsThis and other presentations