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Nature of Mathematics Nature of Mathematics as a Human Endeavouras a Human Endeavour

Oxford N1AOxford N1A

§2 §2 20122012

The Open UniversityThe Open UniversityMaths DeptMaths Dept University of OxfordUniversity of Oxford

Dept of EducationDept of Education

Promoting Mathematical ThinkingPromoting Mathematical Thinking

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OutlineOutline

Phenomena provoking mathematical Phenomena provoking mathematical thinkingthinking– Material; imagined & virtual; symbolicMaterial; imagined & virtual; symbolic

Ways of Working on MathematicsWays of Working on Mathematics Use of Powers/ProcessesUse of Powers/Processes

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Conjecturing AtmosphereConjecturing Atmosphere

Everything said in this room is a Everything said in this room is a conjecture …conjecture …– To be tested out in your experienceTo be tested out in your experience– To be tested mathematicallyTo be tested mathematically

If you KNOW immediately take the If you KNOW immediately take the opportunity to focus on what you know, or opportunity to focus on what you know, or ask questions that might assist others to ask questions that might assist others to see what you seesee what you see

If you are uncertain or stuck, take the If you are uncertain or stuck, take the opportunity to try to express what you do opportunity to try to express what you do understand, or ask about what you do understand, or ask about what you do not!not!

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Square DifferenceSquare Difference

If the difference of two whole numbers If the difference of two whole numbers is is eveneven, then their product is the , then their product is the difference of two whole number squaresdifference of two whole number squares

If the difference of two whole numbers If the difference of two whole numbers is is oddodd, then the product of either with , then the product of either with one more than the other is a difference one more than the other is a difference of two whole number squaresof two whole number squares

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One More Than …One More Than …

What numbers can be expressed as one What numbers can be expressed as one more than the product of four more than the product of four consecutive integers?consecutive integers?

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ReminderReminder

Being Stuck is an honourable and Being Stuck is an honourable and valuable statevaluable state– It is then possible to learn something of It is then possible to learn something of

advantage for the futureadvantage for the future What matters is not so much ‘being What matters is not so much ‘being

stuck’ as what you do to get unstuckstuck’ as what you do to get unstuck You get the most from working on a You get the most from working on a

problem/task when you stop and think problem/task when you stop and think back over back over – What actions you undertook What actions you undertook – What actions were effective, What actions were effective, – What `ctions you might use in the future.What `ctions you might use in the future.

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Good Advice …Good Advice …

"The best way of overcoming a difficult "The best way of overcoming a difficult Probleme is to solve it in some Probleme is to solve it in some particular easy cases. This gives much particular easy cases. This gives much light into the general solution. By this light into the general solution. By this way Sir Isaac Newton says he overcame way Sir Isaac Newton says he overcame the most difficult things.the most difficult things.”” (David Gregory 1705 correspondence with Newton)(David Gregory 1705 correspondence with Newton)

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10891089

Write down a three-digit numberWrite down a three-digit number Reverse the digits then subtract smaller Reverse the digits then subtract smaller

from largerfrom larger Reverse digits of that, then add the two Reverse digits of that, then add the two

numbersnumbers

Extend/Generalise!Extend/Generalise!

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PhenomenonPhenomenon

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PhenomenonPhenomenon

Imagine a drinks Imagine a drinks cancan

Where is the centre of Where is the centre of gravity before it is opened?gravity before it is opened?

Where is the centre of Where is the centre of gravity when it is empty?gravity when it is empty?

Where is the centre of Where is the centre of gravity lowest?gravity lowest?

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Centres of GravityCentres of Gravity

Imagine a triangleImagine a triangle– Where is the centre of gravity?Where is the centre of gravity?

VerticesVertices EdgesEdges LaminaLamina

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Chords & CobwebsChords & Cobwebs

Cubic Chord

s

Cobwebs

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SundaramSundaram’’s Sieves Sieve

16 27 38 49 60 71 82

13 22 31 40 49 58 67

10 17 24 31 38 45 52

7 12 17 22 27 32 37

4 7 10 13 16 19 22

What number will appear in the Rth row and the Cth column?

Claim: N will appear in the table iff 2N + 1 is composite

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Max & MinMax & Min

For any rectangular array of numbers,For any rectangular array of numbers,– For each row, calculate the maximum in that rowFor each row, calculate the maximum in that row– For each column calculate the minimum in that For each column calculate the minimum in that

columncolumn– Calculate Min-Max: the minimum of the row Calculate Min-Max: the minimum of the row

maximamaxima– Calculate Max-Min: the maximum of the column Calculate Max-Min: the maximum of the column

minimaminima Is there any relationship between Max-Min Is there any relationship between Max-Min

and Min-Max true in all cases? and Min-Max true in all cases? In some precisely specified cases?In some precisely specified cases?

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ReflectionReflection

Mathematics as a human endeavourMathematics as a human endeavour– A way of making sense of phenomena often brought A way of making sense of phenomena often brought

to attention via some sort of disturbance or fractured to attention via some sort of disturbance or fractured expectation in a world of experience: expectation in a world of experience: material, mental-virtual, symbolic material, mental-virtual, symbolic

– Modelling the material world in a manipulably Modelling the material world in a manipulably expressive language;expressive language;

– Locating underlying structural relationshipsLocating underlying structural relationships Use of natural powers:Use of natural powers:

Imagining & ExpressingImagining & Expressing Specialising & GeneralisingSpecialising & Generalising

Conjecturing & ConvincingConjecturing & Convincing Organising & Organising & ClassifyingClassifying

Stressing & IgnoringStressing & Ignoring Extending & RestrictingExtending & Restricting

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Where is Mathematics?Where is Mathematics?

Do we discover/uncover pre-existant Do we discover/uncover pre-existant relationships (invariances in the midst relationships (invariances in the midst of change)?of change)?

Do we construct mathematical Do we construct mathematical relationships as part of our sense-relationships as part of our sense-making of our experience of material making of our experience of material and mental worlds?and mental worlds?

Does it ever make sense to ask for a Does it ever make sense to ask for a location for mathematics?location for mathematics?

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Follow-UpFollow-Up

Mcs.open.ac.uk / jhm3Mcs.open.ac.uk / jhm3 (go to applets or to presentations)(go to applets or to presentations)j.h.mason@open.ac.ukj.h.mason@open.ac.uk

Thinking MathematicallyThinking Mathematically– www.Pearson-books.com / www.Pearson-books.com /

ThinkingMathematicallyThinkingMathematically