Pedagogical Mathematics

1. Pedagogical Mathematics In this session I intend to raise and address some issues in teaching and learning mathematics: mathematical pedagogy In the next session I intend to show how mathematical questions arising from asking pedagogical questions led to mathematical explorations. In this session I intend to raise and address some issues in teaching and learning mathematics: mathematical pedagogy In the next session I intend to show how mathematical questions arising from asking pedagogical questions led to mathematical explorations.

2. Definitions Mathematical Pedagogy Strategies for teaching maths; useful constructs Mathematical Didactics Tactics for teaching specific topics or concepts Pedagogical Mathematics Mathematical explorations useful for, and arising from, pedagogical considerations This session is really an indulgence for me. Often I become aware of a problem. I then explore it for myself, paying attention to what I struggle with, and what I notice myself doing along the way. Often the questions arise for me as part of trying to contact the essence of a topic or a concept, and the epistemological and pedagogical obstacles that learners often encounter. Here is an example. I knew from experience that many people find generalising two things at once, or even in succession, a big challenge. I wanted to find a setting which would offer that opportunity. Pedagogical purpose: opportunity to work on multiple parameters; in context where examples are easy to come by, so Watch What You Do is available as a strategy, but also Say What You See with structural generalisation.This session is really an indulgence for me. Often I become aware of a problem. I then explore it for myself, paying attention to what I struggle with, and what I notice myself doing along the way. Often the questions arise for me as part of trying to contact the essence of a topic or a concept, and the epistemological and pedagogical obstacles that learners often encounter. Here is an example. I knew from experience that many people find generalising two things at once, or even in succession, a big challenge. I wanted to find a setting which would offer that opportunity. Pedagogical purpose: opportunity to work on multiple parameters; in context where examples are easy to come by, so Watch What You Do is available as a strategy, but also Say What You See with structural generalisation.

3. Perforations I discovered that inviting people to generalise two features at once could be a significant if not daunting challenge. I was looking for another version and came up with Perforations.I discovered that inviting people to generalise two features at once could be a significant if not daunting challenge. I was looking for another version and came up with Perforations.

4. Possible Strategies Watch What You Do Specialise but attend to what your body does as way of seeing & as source of generalisation Say What You See Reveal/locate distinctions, relationships, properties, structure Conjecture: these are the sorts of things you were doing, with or without these two strategies.Conjecture: these are the sorts of things you were doing, with or without these two strategies.

5. Perforations Generalised Dimensions of possible variation: Let go of practical matters! H,V and M for corners Get HC(R+1)+VR(C+1)+(R+1)(C+1)M which assumes there is at least one row and at least one column! This gives N = (H+V+M)(RC) + (V+M)R + (H+M)C + M as the number of perforations. 12RC+7R+8C+4 To characterise these numbers, note that (H+V+M)N + HV = ((H+V+M)R + (H+M)) ((H+V+M)C + (V+M) )Let go of practical matters! H,V and M for corners Get HC(R+1)+VR(C+1)+(R+1)(C+1)M which assumes there is at least one row and at least one column! This gives N = (H+V+M)(RC) + (V+M)R + (H+M)C + M as the number of perforations. 12RC+7R+8C+4 To characterise these numbers, note that (H+V+M)N + HV = ((H+V+M)R + (H+M)) ((H+V+M)C + (V+M) )

6. Structured Variation Grids Pedagogical Offshoot

7. Vecten

8. Chords Locus of midpoints of chords of a quartic? Locus of midpoints of fixed-width chords? Locus for quintic seems to be bounded by portions of a rational polynomial of degree 5 over 1. Locus for quintic seems to be bounded by portions of a rational polynomial of degree 5 over 1.

9. Cubic Construction Construct a cubic for which the root-tangents are alternately perpendicular It seems a reasonable task, except that it is impossible! Why is it impossible?What sorts of constraints are acting? What about quartics? In the last session I invited you to construct a cubic subject to certain constraints. Here is another such task.In the last session I invited you to construct a cubic subject to certain constraints. Here is another such task.

10. Discovery Suppose a cubic has three distinct real roots. Then the sum of the cotangents of the root-angles is 0. More generally, for a polynomial of degree d, the sum of the products of the root-slopes taken d – 1 at a time is zero.

11. Extension Suppose a line cuts a polynomial of degree d > 1 in d distinct points. What is the sum of the cotangents of the angles the line makes with the polynomial at the intersection points?

12. Cutting-Angles (1) Let the line L(x) have slope m

13. Cutting-Angles (2) Put f(x) = p(x)-L(x) We know that Mathematical exploration arising from asking pedagogical questions and producing possible pedagogical tasks to stretch learner comprehension of slopesMathematical exploration arising from asking pedagogical questions and producing possible pedagogical tasks to stretch learner comprehension of slopes

14. Root-slope polynomial Coefficients of the root-slope polynomial are symmetric functions in the root-slopes, and so they are symmetric functions in the coefficients of pCoefficients of the root-slope polynomial are symmetric functions in the root-slopes, and so they are symmetric functions in the coefficients of p

15. Chordal Triangles Locus of centroids of chordal triangles? Locus of Circumcentre of chordal triangles with fixed chord widths? Locus of area of triangles with fixed chord widths? Limit of circumcentres? Limit of excentres of chordal triangle? Limit of centre of Bevan Circle? Limit of area/product of chord widths?

16. Mean Menger Curvature Given three points on a curve, the Menger curvature is the reciprocal of the radius of the circle through the three points Given three points on a function, they determine an interval on the x-axis.Is there a point in the interior of that interval, at which the curvature of the function is the Menger curvature of the three points? Try something easier first.

17. Rolle-Lagrange Mean-Parabola Given three points on a function but not on a straight line, there is a unique quadratic function through them. Is there a point in the interval spanned, to which some point on the parabola can be translated so as to match the function in value, slope and second derivative at that point? The mean value, or Rolle-Lagrange mean Value theorem says that you can translate a chord to be tangential. A lot of fiddling using the R-L Mean Value Theorem three times gives the result.The mean value, or Rolle-Lagrange mean Value theorem says that you can translate a chord to be tangential. A lot of fiddling using the R-L Mean Value Theorem three times gives the result.

18. Why ‘Mean Value’?

19. Mean Menger Curvature Given a circle through three points (Menger circle), is there a point on the spanned interval with the same curvature? Actually shows that if f and g agree at three distinct points then somewhere in that interval they have the same curvature!Actually shows that if f and g agree at three distinct points then somewhere in that interval they have the same curvature!

20. Cauchy Mean Value Theorem Let [f(t), g(t)] trace a differentiable curve in the plane in each interval [a, b] there exists a point s at which [g(b) – g(a)] f’(s) = [f(b) – f(a)] g’(s)

21. Cauchy Mean Menger Curvature?

22. Procedural-InstrumentalConceptual-Relational Human psyche is an interweaving of behaviour (enaction)emotion (affect)awareness (cognition) Behaviour is what is observable Teaching: Expert awareness is transposed into instruction in behaviour The more clearly the teacher indicates the behaviour expected, the easier it is for learners to display it without generating it from and for themselves

23. Tasks & Teaching Tasks are only a vehicle for engaging in mathematical thinking Learners need to be guided, directed, prompted, and stimulated to make sense of their activity: to reflect To manifest a reflection geometrically as a rotation, you need to move into a higher dimension! The same applies to mathematical thinking!

24. Pedagogical Mathematics ‘Problems’ to explore probe learner awareness and comprehension, and afford opportunity to use their own powers to experience mathematical thinking These may not be at the cutting edge; Often they may be hidden in the undergrowth of mathematics of the past but they can be intriguing! Effective teaching of mathematics results in learners with a disposition to explore for themselves.

25. Further Reading