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What Makes An Example Exemplary?

Promoting Active Learning Through Seeing Mathematics As

A Constructive ActivityJohn Mason

Birmingham Sept 2003

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Functions on R

Sketch a function on R and another and another What makes them ‘typical’?

What about them is exemplary?

Example-Spaces

Thinking of Students …

Dimensions-of-possible-variationRanges-of-permissible-change

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Write down a function on R …

which is continuous and differentiable everywhere

except at one point

What is exemplary about your example?

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Exemplary-ness

What can change and it still be an example?

Dimensions-of-possible-variation

Range-of-permissible-change

Seeing the general through the particular

Seeing the particular in the general

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Variations

Write down a function twice differentiable everywhere except at one point

Write down a function differentiable everywhere except at two points

Dim-of-Poss-Var?

Dim-of-Poss-Var?

What sets can be the points of non-differentiability of a function on R?

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Imagine a vector space

of dimension 5

What happened inside you?

What dimensions-of-possible-variationare you aware of?

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Sketch a function on R …

with a discontinuity at 1 and with a different type of discontinuity

at 0 and with a different type of discontinuity

at –1How many different typesof discontinuity at a point

are there?

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Sketch a function on R

with a discontinuity of the same type at 1/2n for all positive integers n

and with a discontinuity of a different type at 1/(2n –1) for all positive integers n

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Sketch a typical cubic

which has a local maximum and a local minimum

and which has three distinct real roots and which has an inflection tangent with

positive slope

Now go back and make sure that each example is NOT an example for the succeeding stage

Surprised? Need to re-think?

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Active Learning

Increasingly taking initiative Assenting –> Asserting, Anticipating Conjecturing; Justifying–Contradicting

– Specialising & Generalising – Imagining & Expressing – Constructing objects

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Assumptions

You don’t fully appreciate-understand a theorem or concept … unless you have access

to a range of familiar examples Mathematics starts from identifying

phenomena: material, electronic-screen, mental-screen,

and trying to explain, characterise, generalise

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Sketch graph of xy = yx (x≥ 0, y ≥ 0)

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xy =p yx

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Doing & Undoing

Typical calculation for a specified differentiable function:

x –> 2

f(x) – 5x– 2

find lim

So what can you tell me about f if the answer is given as 3?

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Double Limit

Lim

f(p) – f(q)p – q

f(p) – f(r)p – r–

q – rq –> pr –> p

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Rolle Points

Given a function f and an interval [a, b], where in the interval would you expect to find the Rolle points?

The point x = c is a Rolle Point for f on [a, b] if …

Did you form a mental image? Draw a diagram? Try some simple functions? Which ones?

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Perpendicular Root-Slopes Find a quadratic whose root-slopes are

perpendicular Find a cubic whose root-slopes are

consecutively perpendicular Find a quartic whose root-slopes are

consecutively perpendicular

For what angles can the root slopesbe consecutively equally-angled?

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Limits of properties

Write down a property which is not preserved under taking limits

Write down another And another

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Bury The Bone Construct a function which requires three

integrations by partsShow how to generalise

Construct a pair of numbers which require four steps of the Euclidean algorithm to find the gcdShow how to generalise

Construct a limit which requires 3 uses of l’Hôpital’s ruleShow how to generalise

Get learners to construct ‘as complicated’

& ‘as general’‘problems’ as they can

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Object Construction

Recall familiar object Adjust details of familiar object Glue or join familiar objects Compound familiar objects Impose algebraic constraints on general

object Bury The Bone

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Active Learners … Experience lecturers actively engaging with

mathematics Develop confidence as they discover that

they too can construct new objects Learn how to learn mathematics

which they come to see as a constructive & creative enterprise

Dimensions-of-possible-variation and ranges-of-permissible-change to extendlearners’ example-spaces so that examples are actually exemplary