Download - 1 Wherein lies the Richness of Mathematical Tasks? John Mason Windsor & Datchett Feb 2008
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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
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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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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