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Learner Generated ExamplesLearner Generated Examplesin thein the

Teaching of MathematicsTeaching of Mathematics

John MasonJohn Mason

GrahamstownGrahamstown

May 2009May 2009

The Open UniversityMaths Dept University of Oxford

Dept of Education

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Another & AnotherAnother & Another

Write down a pair of numbers Write down a pair of numbers whose difference is 2whose difference is 2

andand another pair another pair andand another pair another pair

What did you notice?

Write down a pairwhich obscure thedifference of 2 as much as possible

What did you notice?

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Decimal Construction 1Decimal Construction 1

Write down a decimal number Write down a decimal number between 2 and 3between 2 and 3

butbut which does not use the which does not use the digit 5digit 5

andand which does use the digit 7 which does use the digit 7

andand which is as close to 5/2 as which is as close to 5/2 as possiblepossible

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More Or Less Rectangles & More Or Less Rectangles & AreaArea

more

same

less

moresamefewer

area

No. of rectangles

same rectsmore area

more rectssame area

more rectsmore area

fewer rectsmore area

fewer rectsless area

more rectsless area

same rectsless area

fewer rectssame area

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

How many can have the same perimeter?

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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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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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Decimal Construction 2Decimal Construction 2

Write down a decimal number which Write down a decimal number which has the property that every finite has the property that every finite string of digits appears consecutively string of digits appears consecutively somewhere in the digits of your somewhere in the digits of your numbernumber

Write down a decimal number in Write down a decimal number in which the string of digits for each which the string of digits for each whole number appears somewhere whole number appears somewhere as a consecutive string in your as a consecutive string in your numbernumber

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Fraction ConstructionFraction Construction

Write down a fraction which Write down a fraction which uses all of the digits from 0 to 9uses all of the digits from 0 to 9

andand which lies between 3 and 4 which lies between 3 and 4

andand which is as close to 10/3 as which is as close to 10/3 as possiblepossible

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ExtremesExtremes

Write down a number which you Write down a number which you think think no-one else in the room is likely no-one else in the room is likely to write downto write down

… … which no-one is ever likely to which no-one is ever likely to have written down!have written down!

Write down a positive integer. Write down a positive integer. The person writing down the The person writing down the smallest positive integer that no-smallest positive integer that no-one else writes down gets a prize!one else writes down gets a prize!

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Interlude on CreativityInterlude on Creativity

often identified with person, or often identified with person, or productproduct

often associated with noveltyoften associated with novelty these divert attention from the these divert attention from the

essence of creativity:essence of creativity: a flow of a particular kind of energya flow of a particular kind of energy

– Aha! Insight; construction; completionAha! Insight; construction; completion IssueIssue: how to encourage its : how to encourage its

appearance , and how to exploit the appearance , and how to exploit the energy when it arisesenergy when it arises

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Quadrilateral Construction 1Quadrilateral Construction 1

Draw a quadrilateralDraw a quadrilateral

which has one pair of sides which has one pair of sides parallel,parallel,

andand one pair of sides equal, one pair of sides equal,

andand one pair of angles equal one pair of angles equal

How many different ones can How many different ones can you find?you find?

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Quadrilateral Construction 2Quadrilateral Construction 2

Draw a quadrilateralDraw a quadrilateral

which has one pair of opposite which has one pair of opposite sides equal,sides equal,

andand one pair of opposite sides one pair of opposite sides perpendicular,perpendicular,

andand a second pair of opposite a second pair of opposite sides perpendicular,sides perpendicular,

andand a second pair of sides equal a second pair of sides equal

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Learner ChoiceLearner Choice

The more choices I make, The more choices I make, the more likely I am to be engaged the more likely I am to be engaged

Choices of:Choices of:– special or particular cases, in order to special or particular cases, in order to

comprehendcomprehend– example (complexity, generality)example (complexity, generality)– example meeting constraintsexample meeting constraints– constraints to be metconstraints to be met– distribution of activitydistribution of activity

all contributing toall contributing to– Sense of possible variation; generality; Sense of possible variation; generality;

access to richer example spacesaccess to richer example spaces

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Lined UpLined Up

Write down the equations of Write down the equations of two straight lines whose two straight lines whose xx--intercepts differ by 2intercepts differ by 2

and whose and whose yy-intercepts differ -intercepts differ by 2by 2

and whose slopes differ by 2and whose slopes differ by 2 Now: find all such!Now: find all such!

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Example SpacesExample Spaces

asking learners to construct objects asking learners to construct objects – reveals something of their awareness reveals something of their awareness

of the scope of generalityof the scope of generality– promotes the extending and enriching promotes the extending and enriching

of the examples available to them: of the examples available to them: their example spacestheir example spaces

The examples which come to mind The examples which come to mind and are available in a given and are available in a given situation form an situation form an example spaceexample space (Watson & Mason 2002)(Watson & Mason 2002)

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SpinnersSpinners

Colour the spinner so that the Colour the spinner so that the probability of getting a red is probability of getting a red is 1/4 1/4 and of a yellow is 3/8 and of a yellow is 3/8

Colour the spinner so that a Colour the spinner so that a red is red is ¾ as likely as a yellow ¾ as likely as a yellow

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Seven CirclesSeven Circles

How many different size angles can you discern, using only the red points?How do you know you have them all?How many different quadrilaterals?

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Creativity as Energy FlowCreativity as Energy Flow

moment of insightmoment of insight– requires preparationrequires preparation– entails perspiration and entails perspiration and

performance!performance! satisfaction of constructionsatisfaction of construction

Feel creative when you go beyond habit/routine/expectation

Energy flow enables you to take initiative, to respond freshly, to feel good

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Grid SquaresGrid Squares Draw a square Draw a square

with vertices on with vertices on your grid; & Ayour grid; & A……

Now multiply the sum of their Now multiply the sum of their edge lengths by the difference edge lengths by the difference between their edge lengthsbetween their edge lengths

Draw one Draw one square inside square inside anotheranother

Calculate the Calculate the difference in difference in their areastheir areas

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PowersPowers

Am I getting students to make Am I getting students to make significant mathematical choices for significant mathematical choices for themselves?themselves?

Am I stimulating learners to use their 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 things for powers by trying to do things for them?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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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 Activity Mathematics as a Constructive 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