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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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ChildrenChildren’’s Copied Patternss Copied Patterns

4.1 yrs

Marina Papic MERGA 30 2007

model

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ChildrenChildren’’s Own Patternss Own Patterns

Marina Papic MERGA 30 2007

5.4 yrs

5.0 yrs 5.1 yrs

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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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Revealing ShapesRevealing Shapes

AppletApplet

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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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Why is (-1) x (-1) = 1?Why is (-1) x (-1) = 1?

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Magic Square ReasoningMagic Square Reasoning

51 9

2

4

6

8 3

7

– = 0Sum( ) Sum( )

Try to describethem in words

What other configurations

like thisgive one sum

equal to another?

2

2

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More Magic Square ReasoningMore Magic Square Reasoning

– = 0Sum( ) Sum( )

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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

j.h.mason@open.ac.uk

Structured Variation GridsRevealing ShapesOther PublicationsThis and other presentations