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Exemplary Mathematics: Promoting Active Learning Through Constructive Examples

This document explores the concept of exemplary mathematical examples and their role in promoting active learning. It discusses the characteristics that make examples 'typical' or 'exemplary' and emphasizes the importance of identifying variations and permissible changes in mathematical functions. Through various exercises, including sketching functions with specific properties and analyzing continuity, the content encourages learners to engage deeply with mathematical concepts, fostering creativity and constructing a rich understanding of mathematics as an active process.

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Exemplary Mathematics: Promoting Active Learning Through Constructive Examples

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  1. What Makes An Example Exemplary? Promoting Active Learning Through Seeing Mathematics As A Constructive ActivityJohn Mason Birmingham Sept 2003

  2. Functions on R Thinking of Students … Sketch a function on R • and another • and another • What makes them ‘typical’?What about them is exemplary? Example-Spaces Dimensions-of-possible-variation Ranges-of-permissible-change

  3. Write down a function on R … • which is continuous • and differentiable everywhere except at one point What is exemplary about your example?

  4. 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

  5. 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?

  6. Imagine a vector space • of dimension 5 What happened inside you? What dimensions-of-possible-variationare you aware of?

  7. 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 –1 How many different typesof discontinuity at a pointare there?

  8. 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

  9. 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 Surprised? Need to re-think? Now go back and make sure that each example is NOT an example for the succeeding stage

  10. Active Learning • Increasingly taking initiative • Assenting –> Asserting, Anticipating • Conjecturing; Justifying–Contradicting • Specialising & Generalising • Imagining & Expressing • Constructing objects

  11. 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

  12. Sketch graph of xy = yx(x≥ 0, y ≥ 0)

  13. xy =p yx

  14. f(x) – 5 find lim x– 2 x –> 2 Doing & Undoing • Typical calculation for a specified differentiable function: • So what can you tell me about fif the answer is given as 3?

  15. Double Limit f(p) – f(q) f(p) – f(r) – p – q p – r Lim q – r q –> p r –> p

  16. Rolle Points The point x = c is a Rolle Point for f on [a,b] if … • Given a function f and an interval [a, b], where in the interval would you expect to find the Rolle points? Did you form a mental image? Draw a diagram? Try some simple functions? Which ones?

  17. 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?

  18. Limits of properties • Write down a property which is not preserved under taking limits • Write down another • And another

  19. Get learners to construct ‘as complicated’ & ‘as general’‘problems’ as they can Bury The Bone • Construct a function which requires three integrations by parts Show how to generalise • Construct a pair of numbers which require four steps of the Euclidean algorithm to find the gcd Show how to generalise • Construct a limit which requires 3 uses of l’Hôpital’s rule Show how to generalise

  20. 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

  21. 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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