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Making Effective Use Making Effective Use of Examplesof Examples

John MasonJohn Mason

MSORMSOR

Sept 2010Sept 2010

The Open UniversityMaths Dept University of Oxford

Dept of EducationPromoting Mathematical Thinking

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OutlineOutline

Familiar PhenomenonFamiliar Phenomenon Mathematical & Pedagogical Mathematical & Pedagogical

ThemeTheme– Invariance in the midst of changeInvariance in the midst of change

Examples, Example Spaces & Examples, Example Spaces & VariationVariation

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PhenomenaPhenomenaStudents often ignore conditions Students often ignore conditions when applying theoremswhen applying theorems

Students often have a very limited Students often have a very limited notion of mathematical concepts ...notion of mathematical concepts ...

What is it about some examples that What is it about some examples that makes them useful for students?makes them useful for students?

What do students need to do with the What do students need to do with the examples they are given?examples they are given?

Students often ask for more Students often ask for more examples ...examples ...

But do they know what to do with the But do they know what to do with the examples they have?examples they have?

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What do students say?What do students say?

““I seek out worked examples and I seek out worked examples and model answers”model answers”

““I practice and copy in order to I practice and copy in order to memorise”memorise”

““I skip examples when short of time”I skip examples when short of time”

““I compare my own attempts with I compare my own attempts with model answers”model answers”

NO mention of mathematical objects NO mention of mathematical objects other than worked examples!other than worked examples!

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On worked examplesOn worked examples

Templating: changing numbers to Templating: changing numbers to matchmatch

What is important about worked What is important about worked examples?examples?

knowing the criteria by which each step knowing the criteria by which each step is chosenis chosen

knowing things that can go wrong, knowing things that can go wrong, conditions that need to be checkedconditions that need to be checked

having an overall sense of directionhaving an overall sense of direction

having recourse to conceptual having recourse to conceptual underpinning if something goes wrongunderpinning if something goes wrong

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Examples of Mathematical Examples of Mathematical ObjectsObjects

What do you do with examples?What do you do with examples?

What would you like students to What would you like students to do with examples?do with examples?

Unfortunately, students Unfortunately, students sometimes over-generalise, mis-sometimes over-generalise, mis-generalise or fail to appreciate generalise or fail to appreciate what is being exemplifiedwhat is being exemplified

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Really Simple Examples Really Simple Examples (sic!)(sic!)

To find 10%, you divide by 10;To find 10%, you divide by 10; so, to find 20%, you ... so, to find 20%, you ...

To differentiate xTo differentiate xnn you write nx you write nxn-1n-1

so, to differentiate xso, to differentiate xxx you write ... you write ...

(x-1)(x-1)22 + (y+1) + (y+1)22 = 4 has centre at (1, -1) and = 4 has centre at (1, -1) and radius 2radius 2

if f(3) = 5 then f(6) = 10 if f(3) = 5 then f(6) = 10 other appearances:sin(2x) = 2sin(x); ln(3x) = 3ln(x)sin(A+B) = sin(A)+sin(B) etc.

To find 10%, you divide by 10;To find 10%, you divide by 10; so, to find 20%, you ... so, to find 20%, you ...

To differentiate xTo differentiate xnn you write nx you write nxn-1n-1

so, to differentiate xso, to differentiate xxx you write ... you write ...

(x-1)(x-1)22 + (y+1) + (y+1)22 = 4 has centre at (1, -1) and = 4 has centre at (1, -1) and radius 2radius 2

if f(3) = 5 then f(6) = 10 if f(3) = 5 then f(6) = 10 other appearances:sin(2x) = 2sin(x); ln(3x) = 3ln(x)sin(A+B) = sin(A)+sin(B) etc.

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Mathematical ObjectsMathematical Objects

Who produces examples?Who produces examples?– Lecturer?Lecturer?– Students?Students?

What are they expected to do What are they expected to do with them?with them?

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CobwebsCobwebs

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Cobwebs ReviewedCobwebs Reviewed

What was being exemplified?What was being exemplified?– Cubic specified by 4 points, Quartic Cubic specified by 4 points, Quartic

by 5by 5– Any cubic; Any quarticAny cubic; Any quartic– Cobweb constructions and reading Cobweb constructions and reading

graphsgraphs– Way to imagine composite Way to imagine composite

functionsfunctions– Properties of composite functionsProperties of composite functions

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Linear TransformationsLinear Transformations What is being exemplified?What is being exemplified?

– How a linear mapping is How a linear mapping is specifiedspecified

– Relation between vector and Relation between vector and imageimage

– Action of transformation on a Action of transformation on a setset

– Eigenvectors etcEigenvectors etc– Two dimensions ‘typical’ of Two dimensions ‘typical’ of

higher dimensionshigher dimensions– How ICT can be used to How ICT can be used to

support mental imagerysupport mental imagery– How ICT can be used to How ICT can be used to

prompt questionsprompt questions What would you want learners What would you want learners

to do with this experience?to do with this experience?

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Essence of These ‘Examples’Essence of These ‘Examples’

Stressing what can be changed Stressing what can be changed and what remains invariant and what remains invariant (dimensions of possible (dimensions of possible variation)variation)(range of permissible change)(range of permissible change)

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What would you use as an What would you use as an example of ... example of ... a continuous function on R differentiable everywhereeverywhere except at a single point?

a continuous function on R differentiable everywhereeverywhere except at a single point?

sin(x)0

2π∫

sin(x)0

2π∫

|x| |x|

a function whose integral requires integration by parts?

a function whose integral requires integration by parts?

a function whose integral on a finite subinterval of R is 0?

a function whose integral on a finite subinterval of R is 0?

xsin(x),xex ,ln(x)

0dx

a

b

∫

x1

3

x sin1

x⎛⎝⎜

⎞⎠⎟,x≠0

0,x=0

⎧

⎨⎪

⎩⎪

1−x( )dx0

2

∫

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ProposalProposal

The first time you give an example of a The first time you give an example of a mathematical object (not a worked mathematical object (not a worked example)example)

ask students to write down on a slip of ask students to write down on a slip of paper what they expect to do with the paper what they expect to do with the exampleexample

Near the end of the course, when you give Near the end of the course, when you give (or get them to construct) an example(or get them to construct) an example

ask students to write down on a slip of ask students to write down on a slip of paper what they expect to do with the paper what they expect to do with the exampleexample

get them to put their initials or some other get them to put their initials or some other identifying mark on the papers!identifying mark on the papers!

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Example ConstructionExample Construction

What features need to be salient?What features need to be salient?contrasting several examplescontrasting several examples

What can be changed (What can be changed (dimensions of dimensions of possible variationpossible variation))and over what range (and over what range (range of range of permissible changepermissible change))

Learners Constructing ExamplesLearners Constructing Examples

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Imposing ConstraintsImposing ConstraintsSketch a cubicSketch a cubicwhich does not go through the originwhich does not go through the originand which has a local maximumand which has a local maximumand for which the inflection slope is positiveand for which the inflection slope is positive

Is there a quadratic whose root slopes are Is there a quadratic whose root slopes are perpendicular?perpendicular?Is there a cubic whose root slopes are Is there a cubic whose root slopes are consecutively perpendicular?consecutively perpendicular?A quartic ... ?; A quartic ... ?; for what angles is it possible? for what angles is it possible?

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ConjectureConjecture

When learners construct their When learners construct their own examples of mathematical own examples of mathematical objects they:objects they:

extend and enrich their accessible extend and enrich their accessible example spacesexample spaces

become more engaged with and become more engaged with and confident about their studiesconfident about their studies

make use of their own make use of their own mathematical powersmathematical powers

experience mathematics as a experience mathematics as a constructive and creative constructive and creative enterpriseenterprise

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Bringing Possibilities to Bringing Possibilities to AttentionAttention

Construct an injective function from Construct an injective function from [-1, 1] to R for which [-1, 1] to R for which

the limit as x approaches ±1 is 1, the limit as x approaches ±1 is 1,

and f(0) = -1and f(0) = -1

PossibilitiesPossibilitiesf(x) = x if first decimal digit is odd, f(x) = x if first decimal digit is odd, else f(x) = -x else f(x) = -x

use rationals and irrationals differently for use rationals and irrationals differently for x>0 and x<0x>0 and x<0

Is there a non-constant periodic Is there a non-constant periodic function on R with no positive minimal function on R with no positive minimal period?period?

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Preparing the GroundPreparing the Ground

Construct a function F : [a, b] Construct a function F : [a, b] --> R --> R

which is continuous and which is continuous and differentiable on (a, b) differentiable on (a, b)

for which f(a) = f(b) for which f(a) = f(b)

but nowhere on (a, b) is f’(x) = 0but nowhere on (a, b) is f’(x) = 0

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Construction TaskConstruction TaskWrite down a pair of distinct functions Write down a pair of distinct functions for which the integral of the difference for which the integral of the difference over a specified interval is zero.over a specified interval is zero.

and anotherand another

and anotherand another

How did your attention shift so that How did your attention shift so that you could come up with those you could come up with those examples?examples?

Construct an example of a pair of Construct an example of a pair of distinct functions for which the distinct functions for which the integral of the difference is 0 over integral of the difference is 0 over finitely many finite intervals; over finitely many finite intervals; over infinitely many.infinitely many.

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Write down another integral like this one Write down another integral like this one which is also zerowhich is also zero

and anotherand another

and anotherand another

and another which is different in some and another which is different in some wayway

What is the same and what is different about What is the same and what is different about your examples and the original?your examples and the original? What features can you change and still it What features can you change and still it has integral zero?has integral zero?

Write down the most general integral you Write down the most general integral you can,can, like this, which has answer zero.like this, which has answer zero.

1−x( )dx=0

0

2

∫

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Exemplifying Mathematical Exemplifying Mathematical ConceptsConceptsSuppose you were about to introduce Suppose you were about to introduce the notion of relative extrema for the notion of relative extrema for functions from R to R.functions from R to R.

what examples might you choose, and what examples might you choose, and why?why?

would you use non-examples? why or would you use non-examples? why or why not?why not?

How might you present them?How might you present them?

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Sorting TaskSorting Task

Sort the function graphs Sort the function graphs according to relative extremaaccording to relative extrema

What is the same and what is What is the same and what is different about ...different about ...

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The Exemplification ParadoxThe Exemplification ParadoxIn order to appreciate a generality, it helps to have In order to appreciate a generality, it helps to have examples;examples;

In order to appreciate something as an example, it In order to appreciate something as an example, it is necessary to know the generality being is necessary to know the generality being exemplified;exemplified;

so,so,I need to know what is exemplary about something I need to know what is exemplary about something in order to see it as an example of something!in order to see it as an example of something!What can change and

what must stay the same, to preserve examplehood?

Dimensions of possible variationRanges of permissible change

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Applying a Theorem as a Applying a Theorem as a TechniqueTechnique

Sketch a continuous function on the Sketch a continuous function on the open interval (-1, 1) which is open interval (-1, 1) which is unbounded below as x approaches ±1 unbounded below as x approaches ±1 and which takes the value 0 at x = 0.and which takes the value 0 at x = 0.

Sketch another one that is different in Sketch another one that is different in some way.some way.

and another; and anotherand another; and another

What is common to all of them? What is common to all of them?

Can you sketch one which does not Can you sketch one which does not have an upper bound on the interval?have an upper bound on the interval?

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Student ConstructionsStudent Constructions

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More Student ConstructionsMore Student ConstructionsWhat can we change in the conditions of the task and still have the same (or similar) phenomenon, i.e. make use of the same reasoning?

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Undoing a Familiar DoingUndoing a Familiar Doing

familiar familiar doing:doing:

unfamiliar unfamiliar undoing:undoing:

Given f(x), find Given f(x), find

What can you say about What can you say about f iff if

Limx→ 3

f x( )−5x−3

= 2= 2

Limx→ 3

f x( )− f(3)x−3

Limx→ 3

f x( )− f(3)x−3

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Bury The BoneBury The Bone

Construct an integral which requires Construct an integral which requires two integrations by parts in order to two integrations by parts in order to complete itcomplete it

Construct a limit which requires three Construct a limit which requires three uses of L’Hôpital’s rule to calculate it.uses of L’Hôpital’s rule to calculate it.

Construct an object whose symmetry Construct an object whose symmetry group is the direct product of four group is the direct product of four groupsgroups

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|x||x|

Of what is |x| an example?Of what is |x| an example?

What is the same and what different What is the same and what different aboutaboutgraphs of functions of the formgraphs of functions of the form

f x( ) = x−a −b −cf (x)= x−a −bf (x)= x−a......

Characterise these graphs in Characterise these graphs in some waysome way

f (x)= x−a1 + x−a2 + ...+ x−an

oror

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Using |x| ...Using |x| ...

... write down a continuous function ... write down a continuous function which is differentiable everywhere on which is differentiable everywhere on R except at two points;R except at two points;

and anotherand another

and anotherand another

and another but at three pointsand another but at three points

What sets of points can be the set of What sets of points can be the set of points at which a continuous function points at which a continuous function on R is not differentiable?on R is not differentiable?

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Presenting ExamplesPresenting Examples

‘‘give’ or ‘work through’ examplesgive’ or ‘work through’ examples

use a sorting taskuse a sorting task

get students to construct some get students to construct some objects for themselvesobjects for themselves(another & another)(another & another)

tinker publicly, even tinker publicly, even interactively,interactively,with simple ones, to indicate with simple ones, to indicate other possibilitiesother possibilities

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Probing AwarenessProbing AwarenessAsking learners what aspects of an Asking learners what aspects of an example can be changed, and in example can be changed, and in what way.what way.

learners may have only some learners may have only some possibilities come to mindpossibilities come to mind

especially if they are unfamiliar with especially if they are unfamiliar with such a probesuch a probe

Asking learners to construct Asking learners to construct examplesexamples

another & another; adding constraintsanother & another; adding constraints

Asking learners what Asking learners what concepts/theorems an object concepts/theorems an object exemplifiesexemplifies

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What Makes an Example What Makes an Example ‘Exemplary’?‘Exemplary’?

Awareness of Awareness of ‘invariance in the midst of ‘invariance in the midst of change’change’

What can change and still the What can change and still the technique can be used or the technique can be used or the theorem applied?theorem applied?

Particular seen as a Particular seen as a representative of a space of representative of a space of examples.examples.

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Useful ConstructsUseful ConstructsAccessible Example Space Accessible Example Space (objects + constructions) (objects + constructions)

Dimensions of Possible VariationDimensions of Possible VariationAspects that can changeAspects that can change

Range of Permissible ChangeRange of Permissible ChangeThe range over which they can changeThe range over which they can change

Conjecture:Conjecture:If lecturer’s perceived DofPV ≠ If lecturer’s perceived DofPV ≠ student’s perceived DofPVstudent’s perceived DofPVthen there is likely to be confusionthen there is likely to be confusion

If the perceived RofPCh are different, the If the perceived RofPCh are different, the students’ experience is at best impoverishedstudents’ experience is at best impoverished

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Pedagogic StrategiesPedagogic Strategies

Another & AnotherAnother & Another

Bury the Bone Bury the Bone

Sequential constraints designed to Sequential constraints designed to contradict simple examplescontradict simple examples

Doing & UndoingDoing & Undoing

Request the impossible as prelim to Request the impossible as prelim to proofproof

These are useful as study These are useful as study techniques as well!techniques as well!

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To Investigate FurtherTo Investigate Further

Ask your students what they do Ask your students what they do with examples (and worked with examples (and worked examples, if any)examples, if any)

Compare responses between first Compare responses between first and later yearsand later years

When displaying an example, pay When displaying an example, pay attention to how you indicate the attention to how you indicate the DofPV & the RofPCh.DofPV & the RofPCh.

Consider what you could do to Consider what you could do to support them in making use of support them in making use of examples in their studyingexamples in their studying

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For MANY more tactics:For MANY more tactics: Mathematics Teaching Practice: a guide for Mathematics Teaching Practice: a guide for

university and college lecturersuniversity and college lecturers, , Horwood Publishing, Chichester, 2002.

Mathematics as a Constructive Activity: Mathematics as a Constructive Activity: learners constructing exampleslearners constructing examples.. Erlbaum 2005.

Using Counter-Examples in Calculus Using Counter-Examples in Calculus College Press 2009..

J.H.Mason @ open.ac.uk mcs.open.ac.uk/jhm3 [for these slides & [for these slides &

Applets … for contributions to appreciating Applets … for contributions to appreciating functions, derivatives, linear functions, derivatives, linear transformationstransformations

MathemaPedia (NCETM website)