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1
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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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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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
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