1 Variations in Variation Theory and the Mathematical Theme of Invariance in the Midst of Change John Mason Loughborough May 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2 Variation Theory  To learn something, the learner must discern what is to be learned (the object of learning). Discerning the object of learning amounts to discerning its critical aspects. To discern an aspect, the learner must experience potential alternatives, i.e. variation in a dimension corresponding to that aspect, against the background of invariance in other aspects of the same object of learning. [ Marton & Pang 2006 ]

3 Variation Theory Rephrased  Learning is discerning variation in critical features not previously discerned or in discerning an extension of the range of permissible change.  These critical features may involve finer detail or extended context.  dimensions of (possible) variation are discerned through variation juxtaposed in space and time: varying too many or too few dimensions at once makes discernment either difficult or tedious.  similar considerations apply to the range of permissible change in any dimension of variation.

4 Variation & Invariance  Slope of a line is invariant under choosing different chord widths  Trig ratios depend on invariance of ratios in similar triangles (Thales’ theorem)  Second order phenomenon: slope of smooth curve at a point

5 Tangency  In how many different ways can a straight line be tangent to a function?

6 Continuous Function with Imperfection  Sketch a continuous function that is differentiable everywhere except at one point.  And another  In how many different ways can a function display this property?

7 Slopes  Chord-slope function at a point  fixed-width chord-slope function

8 Cross Ratio

9 Area So Far

10 Continuous Function on a Closed Interval show p(x) attains its maximum value x 102 p(x) = -x ax bx 99 + cx + d Technique: find a closed interval contained in the domain for which you can control the values at the end points. find a closed interval contained in the domain for which you can control the values at the end points.

11 Encountering Variation  Sketch a continuous function on the open interval (-1, 1) which is unbounded below as x approaches ±1 and takes the value 0 at x = 0.  Sketch another one that is different in some way.  And another.  What is common to all of them? Can you sketch one which does not have an upper bound on the interval?  What can we change in the conditions of the task and still have the same (or very similar) phenomenon?

12 Emergent Example Space Sts: “You can get as many bumps as you want” T: “Why can’t you draw one with no maximum?” Sts: “because it’s continuous” “a positive value guarantees a maximum but not necessarily a minimum” “if it crosses the x-axis then there will be both a maximum and a minimum”