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Mathematical Reasoning (Proofs & Refutations) John Mason Oxford PGCE February 2010 1 Aims To involve us in experiencing mathematical reasoning To consider implications for teaching – Comprehending reasoning – Re-constructing reasoning – Reasoning for oneself 2 Carpet Theorem Imagine a room with two carpets (rugs) NOT overlapping. One of the carpets is moved so as to overlap the other. – What can be said about the area of overlap and the change of area of uncovered floor? Alter the amount of overlap … (in spare time: generalise to more carpets!) 5 Visual Carpet Theorem 6 Deduction & Induction Aristotelian Deduction – If A, and if A implies B, then B Peano Induction – If P(1) and if for all natural numbers k, P(k) implies P(k+1) then for all natural numbers n, P(n) Contrast 7 with Empirical (scientific) Induction Toulmin 8 Toulmin, S. (1969). The Uses of Argument, Cambridge, England: Cambridge University Press Jigsaw Proofs Does this generalise … … to √n ? … to np/q ? 9 JigSaw Proofs Does this generalise … … to √n ? … to np/q ? 10 Home (Reflections) Work Not interested in actual reasoning, but in what you found yourself DOING in seeking proofs -4890 x2 + 2220 x + 54289 is square for x = -3 .. 3 -420 x2 +420 x + 5329 is square for x = -3 .. 4 11 Square Deduction 3b-3a a+3b 3a+b a b 2a+b a+b a+2b 3 7 12 2 3 5 For an overall square 4a + 4b = 2a + 5b So 2a = b For n squares upper left n(3b - 3a) = 3a + b So 3a(n + 1) = b(3n - 1) 1 1 9 3(3b-3a) = 3a+b 12a =these 8b all be Could squares? So 3a=2b 8 Abundant, Perfect & Deficient 13 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties Burger W. & Shaunessy J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education. 17 (1) 31-48 van Hiele, P. (1986). Structure and Insight: a theory of mathematics education. Developmental Psychology Series. London: Academic Press (2003) On The Structure of Attention in the Learning of 14Mathematics, Australian Mathematics Teacher, 59 (4) p17-25 Magic Square Reasoning 2 2 6 7 2 1 5 9 8 3 What other configurations like this give one sum equal to another? Try to describe them in words 4 Any colour-symmetric arrangement? Sum( 15 ) – Sum( ) =0 More Magic Square Reasoning Sum( 16 ) – Sum( ) =0 Geometrical Reasoning Outline Reprise on Reasoning Tasks through which to refresh experience of geometrical reasoning – – – – 17 Angle reasoning Length reasoning Diagonal properties familiar properties Unfamiliar Problems Aims To involve us in experiencing mathematical reasoning To consider implications for teaching – Comprehending reasoning – Re-constructing reasoning – Reasoning for oneself Warrants, Back-up, and Counter-Examples (Toulmin) Movements of Attention 18 Geometric Construction 1 20 Geometric Construction 2 21 Subtended Angle Theorem 1 Imagine a circle – Imagine a chord of that circle – Imagine the angle subtended by the chord at the circumference – Imagine the angle subtended by the chord at the centre How 22 are these related? Subtended Angle Theorem 2 Imagine a circle – Imagine a chord – Imagine at one end of the chord a tangent to the circle – Imagine also an angle subtended by the chord at the circumference (away from the tangent) How are the angle between the tangent and the chord, and the angle subtended at the circumference, related? 23 Reflected Tangent 24 Reflected Tangent (2) Allow the diagonal to be a chord 25 Reflected Tangent (3) 26 Allow the tangent to be at some other angle to the radius Characterising For any quadrilateral whose diagonals intersect at right angles, the alternating sum of the squares of the edge lengths is zero. – Alternating sum: a – b + c – d in cyclic order For any quadrilateral whose alternating sum of squares of the edge lengths is zero, the diagonals intersecting at right angles Gluing such quadrilaterals together edge to edge preserves the alternating sum of squares of edge lengths as zero. Any planar polygon with an even number of sides with alternating sum of squares of edge lengths zero can be formed by gluing together quadrilaterals with this property. 27 Other Alternating Sums For any convex quadrilateral with an inscribed circle (a circle tangent to each of the edges at an interior point of that edge) the alternating sum of the edge lengths is zero – What if the quadrilateral is not convex (the points of tangency may be on the edges extended) For any convex quadrilateral inscribed in a circle the alternating sum of the interior angles is zero. What about the converse of these? 28 Characterising Quadrilaterals The diagonals of a kite intersect at right angles and one is bisected The diagonals of a parallelogram bisect each other The diagonals of a rhombus bisect each other at right angles What additional properties do the diagonals of a square, rectangle, parallelogram, trapezium have? Converses? Do these properties characterise these classes of quadrilaterals? 29 Attention Holding Wholes (gazing) Discerning Details Recognising Relationships Perceiving Properties Reasoning on the basis of agreed properties Burger W. & Shaunessy J. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education. 17 (1) 31-48 van Hiele, P. (1986). Structure and Insight: a theory of mathematics education. Developmental Psychology Series. London: Academic Press (2003) On The Structure of Attention in the Learning of 30Mathematics, Australian Mathematics Teacher, 59 (4) p17-25 Aspects of Proof Didactics Reconstructing (jigsaws) ≠ comprehending Comprehending reasoning ≠ constructing your own Constructing your own is an attempt to convince: – – – – Yourself A friend A sceptic That they can see what you can see (theorem) Developing ‘warrants’ for assertions by calling upon previously agreed properties 31 Further Reading 32 Logic problems – www.scribd.com/doc/193599/Challenging-Logic-andReasoning-Problems Hanna, G. (1995) Changes to the Importance of Proof, For the Learning of Mathematics, 15 (3), p42–50. Abramsky (Ed.) (2002). Reasoning, Explanation and Proof in School Mathematics and Their Place in the Intended Curriculum: proceedings QCA international seminar, QCA, London. ISBN 1 85838 510 5 … see WebLearn site mcs/open.ac.uk/jhm3