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Wherein lies the RichnessWherein lies the Richnessof Mathematical Tasks?of Mathematical Tasks?

John MasonJohn Mason

Windsor & DatchettWindsor & Datchett

Feb 2008Feb 2008

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ConjecturesConjectures

The richness of mathematical The richness of mathematical tasks does NOT lie in the task tasks does NOT lie in the task itselfitself

NOR does it lie in the format of NOR does it lie in the format of interactionsinteractions

It DOES lie in the teacher’s It DOES lie in the teacher’s ‘being’, manifested in ‘being’, manifested in – teacher-learners relationshipsteacher-learners relationships– Teacher’s mathematical awarenessTeacher’s mathematical awareness

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More ConjecturesMore Conjectures

The richness of learners’ The richness of learners’ mathematical experience mathematical experience depends ondepends on– Opportunities to use and develop Opportunities to use and develop

their their ownown powers powers– Opportunities to make significant Opportunities to make significant

mathematical choicesmathematical choices– Being in the presence of Being in the presence of

mathematical awarenessmathematical awareness

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Conjecturing AtmosphereConjecturing Atmosphere

Everything said is said in order Everything said is said in order to consider modifications that to consider modifications that may be neededmay be needed

Those who ‘know’ support Those who ‘know’ support those who are unsure by those who are unsure by holding back or by asking holding back or by asking informative questionsinformative questions

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What is changingand what is invariant?

Some Galileo Sum RatiosSome Galileo Sum Ratios

1

3

1 + 3 + 5

7 + 9 + 11

1 + 3 + 5 + 7

9 + 11 + 13 + 15

1 + 3

5 + 7, , , , …

What is the sameand what is different?

A single task is of little interest!What variations & extensions

are possible?

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DifferencesDifferences

17=16−142

AnticipatingGeneralising

Rehearsing

Checking

Organising

18=17−156

=16−124

=14−18

13=12−16

14=13−112

=12−14

15=14−120

16=15−130

=12−13=13−16=14− 112

12=11−12

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Some SumsSome Sums

4 + 5 + 6 =9 + 10 + 11 + 1216

Generalise

Justify

Watch What You Do

Say What You See

1 + 2 =3

7 + 8= 13 + 14 + 15

17 + 18 + 19 + 20+ = 21 + 22 + 23 + 24

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Remainders of the Day (1)Remainders of the Day (1)

Write down a number which Write down a number which when you subtract 1 is divisible when you subtract 1 is divisible by 5by 5

and anotherand another and anotherand another Write down one which you Write down one which you

think no-one else here will think no-one else here will write down.write down.

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Remainders of the Day (2)Remainders of the Day (2)

Write down a number which when Write down a number which when you subtract 1 is divisible by 2you subtract 1 is divisible by 2

and when you subtract 1 from the and when you subtract 1 from the quotient, the result is divisible by quotient, the result is divisible by 33

and when you subtract 1 from that and when you subtract 1 from that quotient the result is divisible by 4quotient the result is divisible by 4

Why must any such number be Why must any such number be divisible by 3? divisible by 3?

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Remainders of the Day (3)Remainders of the Day (3)

Write down a number which is 1 Write down a number which is 1 more than a multiple of 2more than a multiple of 2

and which is 2 more than a and which is 2 more than a multiple of 3multiple of 3

and which is 3 more than a and which is 3 more than a multiple of 4multiple of 4

……

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Remainders of the Day (4)Remainders of the Day (4)

Write down a number which is Write down a number which is 1 more than a multiple of 21 more than a multiple of 2

and 1 more than a multiple of and 1 more than a multiple of 33

and 1 more than a multiple of and 1 more than a multiple of 44

……

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Consecutive SumsConsecutive Sums

a a

a

Say What You See

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More Or Less Percent & More Or Less Percent & ValueValue

50% of something is 20

more

same

less

moresameless

% of

Value

50% of 40 is 20

50% of 60 is 3040% of 60 is 24

60% of 60 is 36

40% of 30 is 12

60% of 30 is 20

40% of 50 is 20

40% of 40 is 16

50% of 30 is 15

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More Or Less Rectangles & More Or Less Rectangles & AreaArea

more

same

less

moresamefewer

area

Perimeter

same perimmore area

more perimsame area

more perimmore area

less perimmore area

less perimless area

more perimless area

same perimless area

less perimsame area

Draw a rectilinear figure which requires at least 4 rectangles in any decomposition into rectangles

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More Or Less Whole & PartMore Or Less Whole & Part

? of 35 is 21

more

same

less

moresameless

WholePart

3/5 of 35 is 21

3/4 of 28 is 21

6/7 of 35 is 30

3/5 of 40 is 24

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Algebra ReadingsAlgebra Readings

a

aa

a

Say What You See

Say What You See

Expresssymbolically

Expresssymbolically

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What Teachers Can DoWhat Teachers Can Do aim to be mathematical with and in front aim to be mathematical with and in front

of learnersof learners aim to do for learners only what they aim to do for learners only what they

cannot yet do for themselvescannot yet do for themselves focus on provoking learners tofocus on provoking learners to

– use and develop their (mathematical) powersuse and develop their (mathematical) powers– encounter (mathematical) themes & encounter (mathematical) themes &

heuristicsheuristics– learn about themselves (inner & outer tasks)learn about themselves (inner & outer tasks)– make mathematically significant choicesmake mathematically significant choices

direct attention, guide energiesdirect attention, guide energies

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Worlds of ExperienceWorlds of Experience

Material

World

World of

Symbols

Inner World

of imager

y

enactive iconic symbolic

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Principal FociPrincipal Foci

core awarenesses underlying topicscore awarenesses underlying topics familiar actions which need familiar actions which need

challenging, developing, extendingchallenging, developing, extending generating reflection through generating reflection through

drawing out of immersion in activitydrawing out of immersion in activity getting learners to make significant getting learners to make significant

choiceschoices prompting learners to use and prompting learners to use and

develop their natural powersdevelop their natural powers

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Task DomainsTask Domains

Dimensions-of-possible-variation Dimensions-of-possible-variation (what can change without method (what can change without method or approach changing)or approach changing)

Ranges-of-permissible-changeRanges-of-permissible-change(over what range can things (over what range can things change)change)

Ways of presenting tasksWays of presenting tasks Ways of interacting during activityWays of interacting during activity Ways of concluding activityWays of concluding activity

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Some Mathematical PowersSome Mathematical Powers

Imagining & ExpressingImagining & Expressing Specialising & GeneralisingSpecialising & Generalising Conjecturing & ConvincingConjecturing & Convincing Stressing & IgnoringStressing & Ignoring Organising & CharacterisingOrganising & Characterising

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Some Mathematical ThemesSome Mathematical Themes

Doing and UndoingDoing and Undoing Invariance in the midst of Invariance in the midst of

ChangeChange Freedom & ConstraintFreedom & Constraint