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What Makes An Example Exemplary?

What Makes An Example Exemplary?. Promoting Active Learning Through Seeing Mathematics As A Constructive Activity John Mason. Birmingham Sept 2003. Functions on R. Thinking of Students …. Sketch a function on R and another and another

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What Makes An Example Exemplary?

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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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