1 Asking Questions Asking Questions in order to promote in order to promote Mathematical Reasoning Mathematical Reasoning John Mason John Mason East London East London June 2010 June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking
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1 Asking Questions in order to promote Mathematical Reasoning John Mason East London June 2010 The Open University Maths Dept University of Oxford Dept.
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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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2
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?
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