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Asking QuestionsAsking Questionsin order to promotein order to promote

Mathematical ReasoningMathematical Reasoning

John MasonJohn Mason

HampshireHampshire

Nov 2009Nov 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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Elastic Mathematics (1/3)Elastic Mathematics (1/3) With marks towards you, fingernails at With marks towards you, fingernails at

the end marks, stretch the elastic. the end marks, stretch the elastic. What do you notice?What do you notice?

Stretch and release the elastic so that Stretch and release the elastic so that the green mark stays still (in terms of the green mark stays still (in terms of left-right movement; you can’t stop it left-right movement; you can’t stop it thickening and thinning!).thickening and thinning!).

Hold one hand still and stretch the Hold one hand still and stretch the elastic. What is the same and what is elastic. What is the same and what is different?different?

What is the same, and what is What is the same, and what is proportionateproportionate??

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Elastic Moments (2/3)Elastic Moments (2/3) Now stretch the elastic so that the red Now stretch the elastic so that the red

mark stays still. What are you doing with mark stays still. What are you doing with your hands?your hands?

Using the marked line on the handout, Using the marked line on the handout, align the end marks with marks; then align the end marks with marks; then stretch the band until the green mark stretch the band until the green mark reaches the original end mark. What is the reaches the original end mark. What is the scale factor?scale factor?– What is the ratio of the new (stretched) green-What is the ratio of the new (stretched) green-

red segment to the original?red segment to the original? Start again and stretch the band so that Start again and stretch the band so that

the green mark aligns with where the red the green mark aligns with where the red mark was when at rest. Where is the red mark was when at rest. Where is the red mark now, in relation to the original marks mark now, in relation to the original marks when at rest?when at rest?

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Scaling a DrawingScaling a Drawing Place a point P to the left of and Place a point P to the left of and

level with the roof of the house.level with the roof of the house. Now for each point Q on the Now for each point Q on the

corners of the house and corners of the house and windows, go from P beyond Q by windows, go from P beyond Q by half the distance PQ;half the distance PQ;add in the linesadd in the lines

Now place a point R to the left Now place a point R to the left of and below your new house. of and below your new house. Mark points one third of the way Mark points one third of the way from points on the house to Rfrom points on the house to R

What do you notice? Why must What do you notice? Why must this be the case?this be the case?

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Revealing ShapesRevealing Shapes

The coloured shapes on the right are somewhere in the white cells on the left.By clicking on a colour and then on a cell, you can test your conjectures as to which shape is where

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Magic Square ReasoningMagic Square Reasoning

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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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Questions Used (1/2)Questions Used (1/2) Say What You See (SWYS)Say What You See (SWYS) What do you predict? What do you expect?What do you predict? What do you expect? What stays the same?What stays the same? What can you fix (hold constant)?What can you fix (hold constant)? Can you (stretch it) so that …?Can you (stretch it) so that …? What actions are available on what objects?What actions are available on what objects? What does it feel like in your body to … ?What does it feel like in your body to … ? Holding … fixed, perform action on …Holding … fixed, perform action on … By how much is the elastic scaled to perform By how much is the elastic scaled to perform

that action?that action? What have you found?What have you found? Those who have not spoken much say what you Those who have not spoken much say what you

think has been said about the situation by think has been said about the situation by others or what you seeothers or what you see

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Questions used (2/2)Questions used (2/2) What’s going on when … (action performed)?What’s going on when … (action performed)? When I’ve done this action, where does (x) end up?When I’ve done this action, where does (x) end up? Where is (x) in relation to (y) now? Where is (x) in relation to (y) now?

– positions; gaps between positionspositions; gaps between positions What can be said about the relation between (x) What can be said about the relation between (x)

and (y) before and after the action?and (y) before and after the action?– Note the build up to this relational questionNote the build up to this relational question

Make a record so that someone else can Make a record so that someone else can reconstructreconstruct– The relationshipThe relationship– What happens when action is performedWhat happens when action is performed

Do-Talk-RecordDo-Talk-Record– Each both arises from and informs the othersEach both arises from and informs the others

Catch yourself reasoningCatch yourself reasoning