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Asking QuestionsAsking Questionsin order to promotein order to promote

Mathematical ReasoningMathematical Reasoning

John MasonJohn Mason

East LondonEast London

June 2010June 2010

The Open UniversityMaths Dept University of Oxford

Dept of EducationPromoting Mathematical Thinking

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OutlineOutline

Some tasks to work onSome tasks to work on in a in a conjecturingconjecturing atmosphere atmosphere in order to experience in order to experience

different forms of reasoning, different forms of reasoning, and the questions and and the questions and prompts that may promote prompts that may promote themthem

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Reasoning TypesReasoning Types

Logical Deduction (cf arithmetic)Logical Deduction (cf arithmetic) Empirical (needs justification)Empirical (needs justification) Exhaustion of casesExhaustion of cases ContradictionContradiction (Induction)(Induction) Issue is often

what can I assume? what can I use? what do I know?

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Wason’s cardsWason’s cards Each card has a letter on one side and an Each card has a letter on one side and an

numerl on the other.numerl on the other. Which 2 cards Which 2 cards mustmust be turned over in order be turned over in order

to verify that to verify that ““on the back of a vowel on the back of a vowel

there is always an even number”?there is always an even number”?

A 2 B 3

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Revealing ShapesRevealing Shapes

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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

propertiesproperties

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Location, LocationLocation, Location

One letter has been chosen.One letter has been chosen. If you name a letter, you will be If you name a letter, you will be

toldtold– ““Hot” if the chosen letter is the same Hot” if the chosen letter is the same

as, or next to the letter you nameas, or next to the letter you name– ““Cold” otherwiseCold” otherwise

You can assert which letter has You can assert which letter has been chosen, but you have to be been chosen, but you have to be able to justify your choiceable to justify your choice

A

B

CD

E

Dimensions of possible variation?

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CopperPlate CopperPlate CalculationsCalculations

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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?

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Any colour-Any colour-symmetric symmetric

arrangement?arrangement?

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )

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Four ConsecutivesFour Consecutives

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

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

what is different about what is different about your answers?your answers?

+ 1

+ 2

+ 3

+ 64

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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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ClubbingClubbing

47 totaltotal

47–3147–3147–2947–29

31–(47–29)31–(47–29)29–(47–31)29–(47–31)

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poetspoets 29painterspainters

In a certain club there are 47 people altogether, of whom 31 are poets and 29 are painters. How many are both?

How many poets are not painters?

How many painters are not poets?

31 + 29 – 4731 + 29 – 47 Tracking Arithmetic

Tracking Arithmetic

14

14

11

15 musicians

poets painters

28 total23 musicians or painters

21 poets or musicians

22 poets or painters

28–23 28–21

28–22

14+15–21

11+15–23

(14+15-21) + (14+11-22) + (11+15-23) – (28– ((28-23) + (28-22) + (28-21)) 2

14+11–22

In a certain club there are 28 people. There are 14 poets, 11 painters and 15 musicians; there are 22 who are either poets or painters or both, 21 who are either painters or musicians or bothand 23 who are either musicians or poets or both.How many people are all three: poets, painters and musicians?

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Square DeductionsSquare Deductions

Each of the inner quadrilaterals is a square.Each of the inner quadrilaterals is a square.

Can the Can the outer outer quadrilateraquadrilateral be square?l be square?

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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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Geometrical ReasoningGeometrical Reasoning

What properties are agreed?What properties are agreed? What relationships are What relationships are

sought?sought? How are these connected?How are these connected?

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Eyeball ReasoningEyeball Reasoning

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Behold!Behold!

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DiscerningDiscerning

How many triangles?

What is the same, and what is different about them?

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VectenVecten

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Dimensions ofPossible Variation

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ReasoningReasoning

What makes it difficult?What makes it difficult?– not discerning (what others discern)not discerning (what others discern)– not seeing relationshipsnot seeing relationships– not perceiving propertiesnot perceiving properties

How can it be developed?How can it be developed?– Working explicitly onWorking explicitly ondiscerning; relating; property discerning; relating; property

perceiving;perceiving;reasoning on the basis of those reasoning on the basis of those

propertiesproperties