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

2 Functions on R Sketch a function on R / and another / What makes them ‘typical’? What about them is exemplary? Example-Spaces Thinking of Students … 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? What sets can be the points of non-differentiability of a function on R?

6 Sketch a function on R … / with a single discontinuity at 0 / and another

7 Imagine a vector space  of dimension 5 What happened inside you? What dimensions-of-possible-variation are you aware of?

8 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 types of discontinuity at a point are there?

9 Sketch a function on R / with a discontinuity of the same type at 1/2 n for all positive integers n / and with a discontinuity of a different type at 1/(2 n –1) for all positive integers n

10 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?

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

12 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

13 Sketch graph of x y = y x ( x ≥ 0, y ≥ 0)

14 x y =p y x

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

17 Double Limit Lim f ( p) – f( q ) p – q f ( p) – f( r ) p – r – q – r q –> p r –> p

18 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?

19 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 slopes be consecutively equally-angled?

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21 Limits of properties / Write down a property which is not preserved under taking limits / Write down another / And another

22 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 Get learners to construct ‘as complicated’ & ‘as general’ ‘problems’ as they can

23 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

24 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 extend learners’ example-spaces so that examples are actually exemplary