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Where is the RealityWhere is the Realityofof

Algebra & GeometryAlgebra & Geometry??

John MasonJohn Mason

Surrey HoDsSurrey HoDs

Feb 2009Feb 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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AssumptionsAssumptions

Learning is extending the Learning is extending the range of possible actions that range of possible actions that come to mind in a situationcome to mind in a situationDDiscerning detailsiscerning detailsRRecognising relationshipsecognising relationshipsPPerceiving propertieserceiving propertiesRReasoning on the basis of easoning on the basis of

propertiesproperties

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Do you know any students Do you know any students who who ……

DDon’t seem to participate in on’t seem to participate in mathematics lessonsmathematics lessons

SSay “I can’t ay “I can’t …”…” Ask “Why are we doing this?”Ask “Why are we doing this?” Appear to do the minimum to Appear to do the minimum to

get through a lesson?get through a lesson?

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TracksTracks

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MechanismMechanism

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Where Were You?Where Were You?

Where was your attention?Where was your attention?– In or on the picture?In or on the picture?– IImagining somethimagining somethingng moving? moving?– In or on recognising relationships?In or on recognising relationships?– In or on pIn or on perceiving instances of erceiving instances of

properties?properties?– In or on reasoning about those In or on reasoning about those

relationships?relationships?

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Imagine a Number-line Imagine a Number-line …… Imagine a copy sitting on top;Imagine a copy sitting on top; Rotate the copy through 180° about the Rotate the copy through 180° about the

originorigin Restore the copy. Restore the copy. Now rotate it through 180° about the Now rotate it through 180° about the

point 3 on the original fixed number-linepoint 3 on the original fixed number-line0 1 2 3 4 5

-5

-4

-3

-2

-1

Now rotate it through a further 180° Now rotate it through a further 180° about the point 1 on the original fixed about the point 1 on the original fixed number-linenumber-line

What is special about the points 3 and 1What is special about the points 3 and 1 How could you develop or extend this How could you develop or extend this

task?task?

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What ‘world’ were you What ‘world’ were you occupying?occupying?

TheThe world of the screen? world of the screen? TThe world of your imagination?he world of your imagination? AA mathematical world? mathematical world? TThe material world?he material world?

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KitesKites

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Where Were You?Where Were You?

Where was your attention?Where was your attention?– In or on the paper?In or on the paper?– In or on the folding?In or on the folding?– In or on relationships?In or on relationships?– Reasoning about those Reasoning about those

relationships?relationships?

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Square FormationSquare Formation

a b

a+b a+2b2a+b

a+3b3a+b

3b-3a

2 3

587

911

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3(3b-3a) = 3a+b

12a = 8b

So a/b = 3/2

For an overall square

4a + 4b = 2a + 5b

So 2a = b

For n squares upper left

n(3b - 3a) = 3a + b

So 3a(n + 1) = b(3n - 1)

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Where Were You?Where Were You?

Where was your attention?Where was your attention?– On material objects?On material objects?– In or on some diagrammatic In or on some diagrammatic

presentation?presentation?– What were you manipulating?What were you manipulating?

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Triangle CountTriangle Count

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Seven CirclesSeven Circles

How many different angles can you discern, using only the red points?How do you know you have them all?How many different quadrilaterals?

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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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Leibniz’s TriangleLeibniz’s Triangle

1

2

1

2

1

3

1

6

1

3

1

4

1

5

1

1

4

1

12

1

12

1

20

1

5

1

20

1

30

1

60

1

30

1

6

1

30

1

60

1

6

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Ages AgoAges Ago

Two persons, Two persons, AA, and , and BB were talking of their were talking of their Ages: Says Ages: Says AA to to BB, seven Years ago I was , seven Years ago I was thrice as old as you at that Time; and seven thrice as old as you at that Time; and seven Years hence I shall be just twice as old as Years hence I shall be just twice as old as you will be: I demand their present Ages? you will be: I demand their present Ages? – [Mole 1788 problem V p129][Mole 1788 problem V p129]

Make a guess: A is 14Make a guess: A is 14Now check: (14 Now check: (14 –– 7) = 3(9 7) = 3(9 –– 7) so B is 9? 7) so B is 9?And (14 + 7) ?=? 2(9 + 7) No And (14 + 7) ?=? 2(9 + 7) No … but… but(A – 7) = 3(B – 7) and (A+7) = 2(B + 7)(A – 7) = 3(B – 7) and (A+7) = 2(B + 7)

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SharingSharing

A hungry hunter came upon two A hungry hunter came upon two shepherds, one of whom had 3 shepherds, one of whom had 3 loaves and the other 5, all of loaves and the other 5, all of the same size. The loaves were the same size. The loaves were divided equally among the divided equally among the three. The hunter paid 8 cents three. The hunter paid 8 cents for his share. How should the for his share. How should the shepherds divide the money?shepherds divide the money?

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Debbie’s ApplesDebbie’s Apples

Debbie bought apples at 25p each. She Debbie bought apples at 25p each. She ate 2 and sold the rest for 32p each. Her ate 2 and sold the rest for 32p each. Her profit was £1.88. How many apples did profit was £1.88. How many apples did she buy?she buy?

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32

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Do you know any students Do you know any students who who ……

DDon’t seem to participate in on’t seem to participate in mathematics lessonsmathematics lessons

SSay “I can’t ay “I can’t …”…”– Convert can’t into didn’tConvert can’t into didn’t

Ask “Why are we doing this?”Ask “Why are we doing this?”– Customers want particular (price …)Customers want particular (price …)– Entrepreneurs & Managers need general Entrepreneurs & Managers need general

(policy, …)(policy, …) Appear to do the minimum to get Appear to do the minimum to get

through a lesson?through a lesson?

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Are they being encouraged Are they being encouraged to to ……

uuse their natural powers tose their natural powers to– IImagine & Expressmagine & Express– SSpecialise & Generalisepecialise & Generalise

mmake significant mathematical ake significant mathematical choiceschoices

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Reality is …Reality is … Being intrigued, surprised, engagedBeing intrigued, surprised, engaged Using your own powers toUsing your own powers to

– Imagine & ExpressImagine & Express– Specialise & GeneraliseSpecialise & Generalise– Conjecture & ConvinceConjecture & Convince– Extend & Restrict attentionExtend & Restrict attention

It is NOT restricted toIt is NOT restricted to– Utility, or even purposeUtility, or even purpose– Your own past experienceYour own past experience– Some imagined use in the futureSome imagined use in the future

Reality is where your focal attention Reality is where your focal attention isis

One of the corecontributions of schooling

is … … experiencing ‘realities’

you might not otherwiseencounter

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Becoming RealBecoming Real

Holding Wholes (gazing)Holding Wholes (gazing) Discerning detailsDiscerning details Recognising relationshipsRecognising relationships Perceiving properties (as Perceiving properties (as

being instantiated)being instantiated) Reasoning on the basis of Reasoning on the basis of

propertiesproperties

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WorldsWorlds

Material

World

World of

Symbols

Inner World

of Imager

y

enactive iconic symbolic

Worlds during Modelling and Problem-SolvingAspects of Conceptual Development

Modes of AttendingEpistemological Stances

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Follow-UpFollow-Up

These slides (and others) These slides (and others) available onavailable on– http://mcs.open.ac.uk/jhm3

Contact me atContact me at– j.h.mason@open.ac.uk