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Progress Progress in in

Mathematical ThinkingMathematical Thinking

John MasonJohn Mason

BMCEBMCEManchesterManchester

April 2010April 2010

The Open UniversityMaths Dept University of Oxford

Dept of EducationPromoting Mathematical Thinking

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OutlineOutline

What is progress in mathematical What is progress in mathematical thinking?thinking?

Progress in what aspect?Progress in what aspect?– Performance (Performance (behaviourbehaviour))– Conceptual appreciation and understanding; Conceptual appreciation and understanding;

connectedness; articulacy (connectedness; articulacy (cognitioncognition))– Independence & Initiative (Independence & Initiative (affectaffect))– Ways of working individually and Ways of working individually and

collectively(collectively(milieumilieu)) Need for a sufficiently precise vocabularyNeed for a sufficiently precise vocabulary

– to make thinking, discussion and negotiation to make thinking, discussion and negotiation possiblepossible

Tasks that reveal progress

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In BetweenIn Between How many circles could there be between the two shown?How many circles could there be between the two shown?

How many numbers could there be betweenHow many numbers could there be between

1.50 and 1.591.50 and 1.591.500 and 1.59871.500 and 1.5987

Range of permissi

ble change

Discrete&

Continuous

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Difference of 2Difference of 2

write down 2 numbers with a difference of 2

Sketch the two lines with slopes differing by 2

write down an integral over two different intervals whose values differ by 2

And another And another And another

And another And another And another

Primary Secondary Upper Secondary

Progression is visible in the range of choices exhibited; in the richness of the example space being sampled

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Shifts ConjectureShifts Conjecture

Every technical term indicates a shift in Every technical term indicates a shift in perspective, in ways of perceiving; perspective, in ways of perceiving;

The name or label serves as a reminder of The name or label serves as a reminder of trigger for that shift;trigger for that shift;

in order to use the term effectively, learners in order to use the term effectively, learners need to experience a similar shiftneed to experience a similar shift

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Seeing AsSeeing As

✎ Raise your hand when you can Raise your hand when you can see something that issee something that is1/3 of something; 1/3 of something;

again differentlyagain differently

A ratio of 1 : 2A ratio of 1 : 2

Range of permissi

ble change

Dimensions of

possible variation

Threshold Concept:

Clarifying the unit✎ What else can you ‘see as’?What else can you ‘see as’?

✎ What assumptions are you making?What assumptions are you making?

4/3 of 4/3 of somethingsomething

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1

n−

1n+1

=1

n n+1( ) 1

a−

1b

=b−aab

Seeing through the particular to a generality

Hands up when you can see something that is:One fifth of somethingOne fourth of somethingOne fourth of something take away one fifth of the same thing

Now Generali

se!

8Dimensions-of-Possible-Variation

RegionalRegional

Which is the smallest and Which is the smallest and which the largest shaded which the largest shaded area?area?

Generalise!

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Doug French Fractional PartsDoug French Fractional Parts

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Making Mathematical SenseMaking Mathematical Sense

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Which way did the bicycle go?Which way did the bicycle go?

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Triangle CountTriangle Count

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Reading a Diagram: Seeing As …Reading a Diagram: Seeing As …

a

a

x3 + x(1–x) + (1-x)3

x2 + (1-x)2

x2z + x(1-x) + (1-x)2(1-z)

xz + (1-x)(1-z)xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-

z)

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Length-Angle ShiftsLength-Angle Shifts

What 2D shapes have the property that What 2D shapes have the property that there is a straight line that cuts them there is a straight line that cuts them into two pieces each mathematically into two pieces each mathematically similar to the original?similar to the original?

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TangentialTangential

At what point of y=eAt what point of y=exx does the tangent go does the tangent go through the origin?through the origin?

What about y = eWhat about y = e2x2x?? What about y = eWhat about y = e3x3x?? What about y = eWhat about y = eλxλx?? What about y = μf(λx)?What about y = μf(λx)?

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Progress in What?Progress in What? Use of their own powersUse of their own powers

– To imagine & to expressTo imagine & to express– To specialise & to generaliseTo specialise & to generalise– To conjecture & to convinceTo conjecture & to convince– To stress & to ignoreTo stress & to ignore– To persist and to let goTo persist and to let go

Enrichment of their accessible example spacesEnrichment of their accessible example spaces Awareness of the pervasiveness of Awareness of the pervasiveness of

mathematical themes:mathematical themes:– Doing & Undoing (inverses)Doing & Undoing (inverses)– Invariance in the midst of changeInvariance in the midst of change– Freedom & ConstraintFreedom & Constraint– Extending & Restricting MeaningExtending & Restricting Meaning

and of opportunities to think mathematically and of opportunities to think mathematically outside of classroomsoutside of classrooms

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Conjecture: Progression can be seen in Conjecture: Progression can be seen in terms ofterms of

Dimensions-of-Possible-Variation &Dimensions-of-Possible-Variation &Range-of-Permissible-ChangeRange-of-Permissible-Change

Use of powers on own initiativeUse of powers on own initiative– E.g. Specialising in order to re-GeneraliseE.g. Specialising in order to re-Generalise

Construction tasks to reveal richness of Construction tasks to reveal richness of accessible example spacesaccessible example spaces

Self-Constructed TasksSelf-Constructed Tasks Using Natural Powers toUsing Natural Powers to

– Make sense of mathematicsMake sense of mathematics– Make mathematical senseMake mathematical sense

Manifesting results of shifts in Manifesting results of shifts in perspectiveperspective– Discrete & ContinuousDiscrete & Continuous– It just is – I was told it – It must be becauseIt just is – I was told it – It must be because– Seeing-As; Seeing-As;

BehaviourDisposition (affect)CognitionAssenting &

AssertingReacting & Responding

Shifts Conjecture

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Progress & The PsycheProgress & The Psyche

Only behaviour is trainableOnly behaviour is trainable Only Awareness is educableOnly Awareness is educable Only emotion is harnessableOnly emotion is harnessable So progress in mathematical thinking So progress in mathematical thinking

includes coordination of progress in all three includes coordination of progress in all three aspects;aspects;

All classroom actions involve an element of All classroom actions involve an element of each aspecteach aspect

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My Website & Further ReadingMy Website & Further Reading

j.h.mason @ open.ac.ukj.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 go to mcs.open.ac.uk/jhm3 go to

PresentationsPresentations New Edition of Thinking Mathematically due New Edition of Thinking Mathematically due end of Aprilend of April77 new problems related to the curriculum77 new problems related to the curriculum

Special conference price of £20 regularly Special conference price of £20 regularly £25£25

Designing Mathematical Tasks (Tarquin)Designing Mathematical Tasks (Tarquin) Questions & Prompts (ATM)Questions & Prompts (ATM) Thinkers (ATM)Thinkers (ATM) Fundamental Constructs in Maths Edn (Sage)Fundamental Constructs in Maths Edn (Sage) Researching Your Own Practice (Routledge)Researching Your Own Practice (Routledge)