1 making the most of mathematical tasks john mason overton jan 2011 the open university maths dept...
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Making the Most of Mathematical Making the Most of Mathematical TasksTasks
John MasonJohn Mason
OvertonOverton
Jan 2011Jan 2011The Open University
Maths Dept University of OxfordDept of Education
Promoting Mathematical Thinking
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AimsAims
To develop strategies for promoting To develop strategies for promoting learning from experiencelearning from experience
To develop questioning that To develop questioning that promotes extension, variation, and promotes extension, variation, and generalisationgeneralisation
To consider a variety of tasks which To consider a variety of tasks which can be used to stimulate reasoningcan be used to stimulate reasoning
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Teaching & LearningTeaching & Learning
Children are given mathematical tasks to doChildren are given mathematical tasks to do Tasks stimulate activityTasks stimulate activity Activity provides experienceActivity provides experience
– of the use of their powersof the use of their powers– of mathematical themesof mathematical themes– of mathematical topics, techniques, reasoning …of mathematical topics, techniques, reasoning …
Experience may contribute to learningExperience may contribute to learning– especially when learners are prompted to especially when learners are prompted to
withdraw from activity and reflect upon it withdraw from activity and reflect upon it
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What did What did you have to you have to
do to do to accomplish accomplish
this?this?
Make a copy of the following Make a copy of the following repeating patternrepeating pattern
ReproductionReproduction
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Patterned WheelsPatterned Wheels
…
An inked roller has made An inked roller has made at least two full at least two full revolutionsrevolutions
What colour is the 100What colour is the 100th th square?square?
Where is the 100Where is the 100thth red square? red square?
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Order! Order!Order! Order! A, B, C, D, and E are in a queueA, B, C, D, and E are in a queue
– B is in front of C B is in front of C – A is behind EA is behind E– There are two people between D and EThere are two people between D and E– There is one person between D and CThere is one person between D and C– There is one person between B and EThere is one person between B and E
BC
EA
BC EA
BC EA
BCEA D
What did you do?What did you do?
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Say What You SeeSay What You See
There are There are 16 canoes16 canoes5 asteroids5 asteroids4 wedges4 wedges4 peaks4 peaks
and these account for the total areaand these account for the total area
Also 6 arches; 6 Also 6 arches; 6 troughs; troughs;
What did you do?What did you do?
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Same & DifferentSame & Different
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15 6 What distinguishes it What distinguishes it from the others?from the others?
Pick an Pick an entry.entry.
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And AnotherAnd Another Write down two numbers that Write down two numbers that
differ by 3differ by 3– And another pairAnd another pair– And another pairAnd another pair
Write down two numbers that Write down two numbers that differ by 3 that you think no-one differ by 3 that you think no-one else will write downelse will write down
Write down two numbers that Write down two numbers that differ by 3 and that make that differ by 3 and that make that difference as obscure as possibledifference as obscure as possible
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Smallest UniqueSmallest Unique
Write down a positive number that you Write down a positive number that you think no-one else will write downthink no-one else will write down
The ‘winner’ is the person who writes The ‘winner’ is the person who writes down the smallest such number!down the smallest such number!
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WhatWhat’’s The Difference?s The Difference?
What could be varied?
– =
First, add one to eachFirst, add one to each
First, First, add one to the first and add one to the first and subtract one from the secondsubtract one from the second
What then would be
the difference?
What then would be
the difference?
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WhatWhat’’s The Ratio?s The Ratio?
What could be varied?
÷
=
First, multiply each by 3First, multiply each by 3
First, First, multiply the first by 2 and multiply the first by 2 and divide the second by 3divide the second by 3
What is the ratio?What is the ratio?
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Marbles (Bob Davis)Marbles (Bob Davis)
I have a bag of marblesI have a bag of marbles I take out 7, then put in 3, then I take out 7, then put in 3, then
take out 4. What is the state of my take out 4. What is the state of my bag now?bag now?– Variations?Variations?
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Speed ReasoningSpeed Reasoning
If I run 3 times as fast as you, how If I run 3 times as fast as you, how long will it take me compared to long will it take me compared to you to run a given distance?you to run a given distance?
If I run 2/3 as fast as you, how long If I run 2/3 as fast as you, how long will it take me compared to you?will it take me compared to you?
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Doing & UndoingDoing & Undoing
What operation undoes What operation undoes ‘‘adding 3adding 3’’??What operation undoes What operation undoes ‘‘subtracting 4subtracting 4’’??What operation undoes What operation undoes ‘‘subtracting from 7subtracting from 7’’??What are the analogues for multiplication?What are the analogues for multiplication?
What undoes What undoes ‘‘multiplying by 3multiplying by 3’’??What undoes What undoes ‘‘dividing by 4dividing by 4’’??What undoes What undoes ‘‘multiplying by multiplying by ¾¾ ’’??
Two different expressions!Two different expressions!
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Additive & Multiplicative Additive & Multiplicative PerspectivesPerspectives
What is the relation between the What is the relation between the numbers of squares of the two colours?numbers of squares of the two colours?
Difference of 2, one is 2 more: Difference of 2, one is 2 more: additiveadditive
Ratio of 3 to 5; one is five thirds the Ratio of 3 to 5; one is five thirds the other etc.:other etc.: multiplicativemultiplicative
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Raise your hand when you can Raise your hand when you can seesee
Something which is 2/5 of somethingSomething which is 2/5 of something Something which is 3/5 of somethingSomething which is 3/5 of something Something which is 2/3 of somethingSomething which is 2/3 of something
– What others can you see?What others can you see? Something which is 1/3 of 3/5 of something Something which is 1/3 of 3/5 of something Something which is 3/5 of 1/3 of somethingSomething which is 3/5 of 1/3 of something Something which is 2/5 of 5/2 of somethingSomething which is 2/5 of 5/2 of something Something which is 1 ÷ 2/5 of somethingSomething which is 1 ÷ 2/5 of something
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Magic Square ReasoningMagic Square Reasoning
51 9
2
4
6
8 3
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– = 0Sum( ) Sum( )
Try to describethem in words
What other configurations
like thisgive one sum
equal to another?
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2
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TeachingTeaching
SSelecting taskselecting tasks PPreparing reparing Didactic Tactics Didactic Tactics
and and Pedagogic StrategiesPedagogic Strategies Prompting extended or fresh actionsPrompting extended or fresh actions Being Aware of mathematical actionsBeing Aware of mathematical actions Directing AttentionDirecting Attention
Teaching takes place in time;Learning takes place over time
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The Place of GeneralityThe Place of Generality
A lesson without the opportunity A lesson without the opportunity for learners to generalise for learners to generalise mathematically, is not a mathematically, is not a mathematics lessonmathematics lesson
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AttentionAttention
Holding Wholes (gazing)Holding Wholes (gazing)
Discerning DetailsDiscerning Details
Recognising RelationshipsRecognising Relationships
Perceiving PropertiesPerceiving Properties
Reasoning on the basis of agreed Reasoning on the basis of agreed propertiesproperties
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Some Mathematical PowersSome Mathematical Powers
Imagining & ExpressingImagining & ExpressingSpecialising & GeneralisingSpecialising & GeneralisingConjecturing & ConvincingConjecturing & ConvincingStressing & IgnoringStressing & IgnoringOrganising & CharacterisingOrganising & Characterising
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Rich tasks, Rich Use of tasksRich tasks, Rich Use of tasks
It may not be the task that is richIt may not be the task that is rich But the way the task is usedBut the way the task is used
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Some Mathematical ThemesSome Mathematical Themes
Doing and UndoingDoing and Undoing Invariance in the midst of ChangeInvariance in the midst of Change Freedom & ConstraintFreedom & Constraint
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For More DetailsFor More Details
Thinkers (ATM, Derby)Questions & Prompts for Mathematical Thinking Secondary & Primary versions (ATM, Derby)Mathematics as a Constructive Activity (Erlbaum)Thinking Mathematically (new edition)
mcs.open.ac.uk/jhm3
Structured Variation GridsRevealing ShapesOther PublicationsThis and other presentations