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John MasonJohn Mason

MathsfestMathsfestCorkCork

Oct 2012Oct 2012

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

Dept of EducationDept of Education

Promoting Mathematical ThinkingPromoting Mathematical Thinking

ReasoningReasoningMathematically Mathematically

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Specific Aims for Ordinary LevelSpecific Aims for Ordinary Level

an understanding of mathematical concepts and an understanding of mathematical concepts and of their relationshipsof their relationships

confidence and competence in basic skillsconfidence and competence in basic skills the ability to solve problemsthe ability to solve problems an introduction to the idea of logical argumentan introduction to the idea of logical argument appreciation both of the intrinsic interest of appreciation both of the intrinsic interest of

mathematics and of its usefulness and efficiency mathematics and of its usefulness and efficiency for formulating and solving problemsfor formulating and solving problems

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ConjecturesConjectures

Everything said here today is a conjecture … to Everything said here today is a conjecture … to be tested in your experiencebe tested in your experience

The best way to sensitise yourself to learnersThe best way to sensitise yourself to learners– is to experience parallel phenomena yourselfis to experience parallel phenomena yourself

So, what you get from this session is what you So, what you get from this session is what you notice happening inside you!notice happening inside you!

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TasksTasks Tasks promote activity; activity involves actions; Tasks promote activity; activity involves actions;

actions generate experience; actions generate experience; – but one thing we don’t learn from experience is but one thing we don’t learn from experience is

that we don’t often learn from experience alonethat we don’t often learn from experience alone Something more is requiredSomething more is required

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Secret PlacesSecret Places One of the places around the table One of the places around the table

is a secret place.is a secret place. If you click near a place, the colour If you click near a place, the colour

will tell you whether you are hot or will tell you whether you are hot or cold:cold:– Hot means that the secret place Hot means that the secret place

is within one place either wayis within one place either way– Cold means that it is at least two Cold means that it is at least two

places awayplaces away

Homage to Tom O’Brien (1938 – Homage to Tom O’Brien (1938 – 2010)2010)

What is your What is your best best strategy to strategy to locate the locate the secret secret place?place?

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Counting OutCounting Out

In a selection ‘game’ you start at the left and In a selection ‘game’ you start at the left and count forwards and backwards until you get to a count forwards and backwards until you get to a specified number (say 37 or 177). Which object specified number (say 37 or 177). Which object will you end on?will you end on?

A B C D E

11 22 33 44 55

99 88 77 66

……

If that object is eliminated, you start again from the If that object is eliminated, you start again from the ‘next’. Which object is the last one left?‘next’. Which object is the last one left?

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Alternating Square SumsAlternating Square Sums

Imagine a triangleImagine a triangle Imagine a point inside the triangleImagine a point inside the triangle Drop perpendiculars to the three Drop perpendiculars to the three

sides of the trianglesides of the triangle Each side of the triangle comprises Each side of the triangle comprises

two segmentstwo segments On each segment of each edge, On each segment of each edge,

construct a squareconstruct a square

Conjecture: Conjecture: the sum of the areas of the the sum of the areas of the yellowyellow squares is the sum of the areas of the squares is the sum of the areas of the cyancyan squares.squares. For what hexagons is this the case?For what hexagons is this the case?

Alternately colour the squares Alternately colour the squares yellowyellow and and cyancyan around the trianglearound the triangle

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Selective SumsSelective Sums

Add up any 4 entries, one Add up any 4 entries, one taken from each row and taken from each row and each column.each column.

The answer is (always) 6The answer is (always) 6 Why?Why?

Example of (use of) permutationsExample of seeking invariant

relationshipsExample of focusing on actions preserving an invariance

Opportunity to generalise

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Selective SumsSelective Sums

Add up any 4 entries, Add up any 4 entries, one taken from each one taken from each row and each column.row and each column.

Is the answer always Is the answer always the same?the same?

Why?Why?

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Chequered Selective SumsChequered Selective Sums

Choose one cell in each row Choose one cell in each row and column. and column.

Add the entries in the dark Add the entries in the dark shaded cells and subtract shaded cells and subtract the entries in the light the entries in the light shaded cells.shaded cells.

What properties makes the What properties makes the answer invariant?answer invariant?

What property is sufficient to What property is sufficient to make the answer invariant?make the answer invariant?

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Circles in CirclesCircles in Circles

How are the How are the red and red and yellow areas yellow areas related?related?

redred

orangorangee

yelloyelloww

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Carpet TheoremsCarpet Theorems

In a room there are two carpets whose combined In a room there are two carpets whose combined area is the area of the room.area is the area of the room.– The area of overlap is the area of floor The area of overlap is the area of floor

uncovereduncovered In a room there are two carpets. They are moved In a room there are two carpets. They are moved

so as to change the amount of overlap.so as to change the amount of overlap.– The change in the area of overlap is the The change in the area of overlap is the

change in area of uncovered floorchange in area of uncovered floor

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Rectangular Room with 2 CarpetsRectangular Room with 2 Carpets

How are the red and blue areas related?How are the red and blue areas related?

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Perimeter ProjectionsPerimeter Projections

The red point traverses The red point traverses the quadrilateralthe quadrilateral

The vertical movement of The vertical movement of the red point is tracked.the red point is tracked.

What shape is the graph?What shape is the graph?

Given a graphical track of Given a graphical track of the vertical movement the vertical movement and the horizontal and the horizontal movement,movement,

What is the shape of the What is the shape of the polygon?polygon?

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Square DeductionSquare Deduction

Could these all be squares?

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Square Deduction: tracking arithmeticSquare Deduction: tracking arithmetic

33 44

3+43+4 3+2x3+2x44

2x3+2x3+44

3+3x3+3x443x3+43x3+4

Track the 3 and the 4:Replace the 3 by a and the 4 by b

(3x3+4)(3x3+4)/3/33x4-3x4-3x33x3

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Square Deduction: acknowledging Square Deduction: acknowledging ignoranceignorance

aa bb

aa++bb aa+2+2bb22aa++bb

aa+3+3bb33aa++bb

33aa++b = b = 3(33(3bb-3-3aa))

12a = 8b 12a = 8b

So 3So 3a = a = 22bb

For an overall squareFor an overall square

44aa + 4 + 4bb = 2 = 2aa + 5 + 5bb

So 2So 2aa = = bb

For For nn squares upper squares upper leftleft

nn(3(3bb - 3 - 3aa) = 3) = 3aa + + bb

So 3So 3aa((nn + 1) = + 1) = bb(3(3nn - - 1)1)

But not also 2But not also 2aa = = bb

(3a+b)/(3a+b)/3333bb-3-3aa

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558877

991111

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ReflectionReflection

What aspects of reasoning…What aspects of reasoning…– Stood out for you?Stood out for you?– Involved some struggleInvolved some struggle

What actions …What actions …– Did you undertake?Did you undertake?– Were ineffective (why?)Were ineffective (why?)– Were effective (why?)Were effective (why?)

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TasksTasks Tasks promote activity; activity involves actions; Tasks promote activity; activity involves actions;

actions generate experience; actions generate experience; – but one thing we don’t learn from experience is but one thing we don’t learn from experience is

that we don’t often learn from experience alonethat we don’t often learn from experience alone It is not the task that is rich …It is not the task that is rich …

– but whether it is used richlybut whether it is used richly What matters more than the particular answer is …What matters more than the particular answer is …

– how do you know?how do you know?– what can you vary and still the same approach what can you vary and still the same approach

works?works?

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ReminderReminder

Do–Talk–RecordDo–Talk–Record– Generating need to communicateGenerating need to communicate

Provoking EngagementProvoking Engagement– SurpriseSurprise– Challenge (trust)Challenge (trust)

Promoting Mathematical thinkingPromoting Mathematical thinking– How do you know? … How do you know? … – Why must …?Why must …?

Active StudentsActive Students– ConstructingConstructing– ExtendingExtending

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Follow UpFollow Up

mcs.open.ac.uk/jhm3mcs.open.ac.uk/jhm3j.h.mason @ open.ac.ukj.h.mason @ open.ac.ukmcs.open.ac.uk/jhm3mcs.open.ac.uk/jhm3j.h.mason @ open.ac.ukj.h.mason @ open.ac.uk

Thinking Mathematically Thinking Mathematically (new edition)(new edition)

Designing and Using Mathematical TasksDesigning and Using Mathematical TasksQuestions and Prompts … (Primary version from ATM)Questions and Prompts … (Primary version from ATM)

ThinkersThinkers

Thinking Mathematically Thinking Mathematically (new edition)(new edition)

Designing and Using Mathematical TasksDesigning and Using Mathematical TasksQuestions and Prompts … (Primary version from ATM)Questions and Prompts … (Primary version from ATM)

ThinkersThinkersInstitute of Mathematical

PedagogyAugust 6-9 2013

mcs.open.ac.uk/jhm3