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Ped'l Maths - Pedagogical Mathematics John Mason PIMS...

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Unformatted text preview: Pedagogical Mathematics John Mason PIMS Celebration Lecture SFU 2007 1 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 Mathematical from, pedagogical considerations from, 2 Perforations How many holes for a sheet of r rows and c columns of stamps? 3 If someone claimed there were 228 perforations in a sheet, how could you check? Possible Strategies  Watch What You Do Watch – Specialise but attend to what your body does Specialise as way of seeing & as source of generalisation generalisation  Say What You See – Reveal/locate distinctions, relationships, Reveal/locate properties, structure properties, 4 Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the Basis of Properties Perforations Generalised  Dimensions of possible Dimensions variation: variation: For Each Stamp: – number of horizontal perforations top & bottom – number of vertical perforations, left & right – number of perforations in the corners 5 Structural Generalisation: write down the number of perforations for R rows and C columns and characterise the numbers which can arise Structured Variation Grids  6 Pedagogical Offshoot Pedagogical Vecten Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties 7Reasoning on the Basis of Properties Chords Locus of midpoints of chords of a quartic?  Locus of midpoints of fixed-width chords?  Chord Slopes family of chords with one end fixed: slope locus; what happens as the length goes to zero? fixed width chords: slope locus; what happens as the width goes to zero? ? chords of fixed length: slope locus; what happens as the width goes to zero? ? envelope of slopes of chords through pt 8 Cubic Construction  Construct a cubic for which the Construct root-tangents are alternately perpendicular root-tangents  It seems a reasonable task, except that it is It impossible! impossible!  Why is it impossible? What sorts of constraints are acting?  What about quartics? 9 Discovery  Suppose a cubic has three distinct real Suppose roots. roots.  Then the sum of the cotangents of the Then root-angles is 0. root-angles  More generally, for a polynomial of More degree d, the sum of the products of the root-slopes taken d – 1 at a time is zero. 10 Extension  Suppose a line cuts a polynomial of Suppose degree d > 1 in d distinct points. degree  What is the sum of the cotangents of the What angles the line makes with the polynomial at the intersection points? polynomial 11 Cutting-Angles (1) • Let the line L(x) have slope m 1 cot(α ) = ταν(θ − ϕ ) 1 + tan(q ) tan(j ) = tan(q ) - tan(j ) = 12 1 + p ' ( rj ) m p ' ( rj ) - m α =θ -ϕ α QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. ϕ θ Cutting-Angles (2) Put f(x) = p(x)-L(x) d We know that • ∑ • 1 δ 1 1 =0=∑ f '(rj ) (ϕ 1 π∋ ρ ) − µ • But the sum of the cots of the angles between line and function is δ 1 + µ π ∋ ρϕ) ( ∑ 1 13 µ +1 = µ ∑1 + ∑ = µδ π ∋ ρϕ) − µ ( (ϕ 1 1 π∋ ρ ) − µ δ δ 2 Root-slope polynomial δ Given a polynomial p( x ) = α∏ ( ξ − ρ ) κ κ=1 Define the root-slope polynomial of p to be δ pρ ( ξ ) = α δ ∏ ( ξ − π ∋( ρκ )) κ =1 The constant term is (-1)d times the discriminant which is the square of the product of the inter-rootal distances. 14 The coefficient of x is 0, which can be thought of as the sum of the reciprocals of the root-slopes, times the product of all the root-slopes. Chordal Triangles Locus of centroids of chordal triangles?  Locus of Circumcentre of chordal triangles with Locus fixed chord widths? fixed  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?  15 Mean Menger Curvature Mean  Given three points on a curve, the Menger Given curvature is the reciprocal of the radius of curvature the circle through the three points the  Given three points on a function, they Given determine an interval on the x-axis. determine Is there a point in the interior of that Is interval, at which the curvature of the function is the Menger curvature of the three points? three  Try something easier first. 16 Rolle-Lagrange Mean-Parabola  Given three points on a function but not Given on a straight line, there is a unique quadratic function through them.  Is there a point in the interval spanned, Is 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? that 17 Why ‘Mean Value’? b Let f be integrable on [a, b]. The average (mean) value of f on [a, b] is ∫f a β − α A sensible ‘Mean Value’ property (MVP): b There exists c in [a, b] for which ∫ f =f (c) (b - a) a  The usual ‘Mean Value’ Theorem(s) then The become become 18 – Derivatives (being continuous) satisfy the MVP Derivatives on every subinterval on – Integrals of continuous functions satisfy the Integrals MVP on every subinterval MVP Mean Menger Curvature Given a circle through three points (Menger circle), is there a point Given on the spanned interval with the same curvature? on Suppose f and g are both twice differentiable on [a, b], that k(f) <k(g) on [a, b], and that f(a) =g(a). If f Õa ) ² gÕa ) th e n ( ( f or a ll x in (a , b ], QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. f (x ) ­ g (x ): QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. If f Õa ) ³ gÕa ) th e n ( ( f or at most one x in (a , b ], f (x ) = g(x ). If g has constant curvature (is part of a circle) and k(f) ≠k(g) on [a, b] then f and g can intersect at most twice. 19 Therefore there m ust be a point s on [a, b] at which k(f(s)) =Menger Curvature of the points a, b, c. Cauchy Mean Value Theorem Augustin Cauchy 1789­1857)  Let [f(t), g(t)] trace a differentiable curve trace in the plane in – in each interval [a, b] there exists a point s at there which [g(b) – g(a)] f’(s) = [f(b) – f(a)] g’(s) )] f’ )] g’ 20 Cauchy Mean Menger Curvature? Is there a Cauchy version of curvature for curves in the plane? NO! Counter Example: QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. 2 +t 2 2 3/ 2 (1+t ) QuickTimeª and a TIFF (Uncompressed) decompressor are needed to see this picture. 21 Procedural-Instrumental Conceptual-Relational Human psyche is an interweaving of behaviour Human behaviour (enaction) (enaction) emotion (affect) awareness (cognition)  Behaviour is what is observable  Teaching: transposition  22 – Expert awareness is transposed Expert into instruction in behaviour into – The more clearly the teacher indicates The the behaviour expected, the easier it is for learners to display it without generating it from and for themselves from didactique didactic tension Tasks & Teaching  Tasks are only a vehicle for engaging in Tasks mathematical thinking mathematical  Learners need to be guided, directed, Learners prompted, and stimulated to make sense of their activity: to reflect their – To manifest a reflection geometrically as a To rotation, you need to move into a higher dimension! dimension! – The same applies to mathematical thinking! 23 Pedagogical Mathematics  ‘Problems’ to explore probe learner Problems’ 24 awareness and comprehension, and afford and opportunity to use their own powers to experience mathematical thinking experience  These may not be at the cutting edge;  Often they may be hidden in the Often undergrowth of mathematics of the past undergrowth  but they can be intriguing!  Effective teaching of mathematics results in Effective learners with a disposition to explore for themselves. themselves. Further Reading Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing. Mason J. (2002). Mathematics as a Constructive Activity: learners generating examples. Erlbaum. Watson A. & Mason J. (2005). Counter­Examples in Calculus. manuscript, Klymchuk, S. & Mason, J. 25 http://mcs.open.ac.uk/jhm3 ...
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