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Thinking MathematicallyThinking Mathematicallyas as

Developing Students’ PowersDeveloping Students’ Powers

John MasonJohn Mason

OsloOslo

Jan 2009Jan 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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AssumptionsAssumptions What you get from this session will be What you get from this session will be largely what you notice happening for youlargely what you notice happening for you

If you do not participate, I guarantee you If you do not participate, I guarantee you will get nothing!will get nothing!

I assume a conjecturing atmosphereI assume a conjecturing atmosphere– Everything said has to be tested in experienceEverything said has to be tested in experience– If you know and are certain, then think and If you know and are certain, then think and listen;listen;

– If you are not sure, then take opportunities to If you are not sure, then take opportunities to try to express your thinkingtry to express your thinking

Learning is a maturation process, and so Learning is a maturation process, and so invisibleinvisible– It can be promoted by pausing and withdrawing It can be promoted by pausing and withdrawing from the immediate action in order to get an from the immediate action in order to get an overviewoverview

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OutlineOutline

Some tasks to work on togetherSome tasks to work on together Some remarks about what might Some remarks about what might have been noticedhave been noticed

Each task indicates: a domain of similar tasks a style or structure of tasks

More important than particular tasks: ways of working with learners ON tasks

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Imagining & ExpressingImagining & Expressing

Where can the centre get to?

… … a fixed point P and a fixed point P and a circle passing through Pa circle passing through P

… … two distinct fixed points two distinct fixed points P and Q and a circle P and Q and a circle passing through both pointspassing through both points

… … three distinct points three distinct points P, Q & R and a circle P, Q & R and a circle passing through all three pointspassing through all three points

Where can the centre get to?

Where can the centre get to?

Where can the centre get to?

Imagine a mathematical plane, Imagine a mathematical plane, and lying in it, a …and lying in it, a …

… … circlecircle

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Only then Check!

One SumOne Sum

I have two numbers which sum to 1I have two numbers which sum to 1

Which will be larger:Which will be larger:

The square of The square of the larger the larger added to the added to the smaller?smaller?

The square of The square of the smaller the smaller

added added to the to the larger?larger?

Make aConjecture!

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One Sum DiagramsOne Sum Diagrams

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1

(1- )2

Anticipating,not waiting

1-2

2 (1- )+

(1- )2+=

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Reading a DiagramReading a Diagram

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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Triangle CountTriangle Count

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VariationVariation

Dimensions-of-possible-Dimensions-of-possible-variationvariation

Range-of-permissible-changeRange-of-permissible-change Invariance in the midst of Invariance in the midst of changechange

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Structured Variation GridsStructured Variation Grids

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Up & Down SumsUp & Down Sums

1 + 3 + 5 + 3 + 1

3 x 4 + 122 + 32

1 + 3 + … + (2n–1) + … + 3 + 1

==

n (2n–2) + 1 (n–1)2 +

n2

==

Generalise!See

generalitythrough aparticular

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Reading GraphsReading Graphs Imagine the graph of a cubic polynomialImagine the graph of a cubic polynomial Imagine also the graph of a quarticImagine also the graph of a quartic Imaging also the graph of y = xImaging also the graph of y = x Now, imagine a point x on the x-axis;Now, imagine a point x on the x-axis;

– proceed vertically up proceed vertically up (or down) to the cubic;(or down) to the cubic;

– proceed horizontally proceed horizontally to the line y=xto the line y=x

– proceed vertically upproceed vertically up(or down) to the quartic(or down) to the quartic

– proceed horizontally untilproceed horizontally untilyou are directly in verticalyou are directly in verticalline with the x you started line with the x you started withwith

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Cubical PropertyCubical Property

Imagine a cubicImagine a cubic Imagine a chord, extended to a Imagine a chord, extended to a line;line;Find the midpoint of your chordFind the midpoint of your chord

Imagine a second chord with the Imagine a second chord with the same midpoint; extend it to a same midpoint; extend it to a lineline

What do you imagine will happen? What do you imagine will happen?

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Chord-slopesChord-slopes

Imagine a quartic polynomialImagine a quartic polynomial Imagine an interval of fixed Imagine an interval of fixed width on the x-axiswidth on the x-axis

The interval determines a The interval determines a chord. The mid-point of the chord. The mid-point of the chord is markedchord is marked

The slope of the chord is shownThe slope of the chord is shown

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KitesKites

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PowersPowers

Am I stimulating learners to use Am I stimulating learners to use their own powers, or am I abusing their own powers, or am I abusing their powers by trying to do their powers by trying to do things for them?things for them?– 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 extend & to restrictTo extend & to restrict

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ReflectionReflection

What did you notice happening What did you notice happening for you mathematically?for you mathematically?

What might you be able to use What might you be able to use in an upcoming lesson?in an upcoming lesson?

Imagine yourself in the Imagine yourself in the future, using or developing future, using or developing or exploring something you or exploring something you have experienced today!have experienced today!

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More ResourcesMore Resources

Questions & Prompts for Mathematical Questions & Prompts for Mathematical Thinking Thinking ((ATM Derby: primary & secondary ATM Derby: primary & secondary versions)versions)Thinkers (Thinkers (ATM Derby)ATM Derby)Mathematics as a Constructive Mathematics as a Constructive Activity Activity (Erlbaum)(Erlbaum)Designing & Using Mathematical Tasks Designing & Using Mathematical Tasks (Tarquin)(Tarquin)http: //http: //mcs.open.ac.uk/jhm3mcs.open.ac.uk/jhm3j.h.mason @ open.ac.ukj.h.mason @ open.ac.uk