1 learner generated examples in the teaching of mathematics john mason grahamstown may 2009 the open...
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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