1 Mathematical Reasoning (Proofs & Refutations ) John Mason Oxford PGCE February 2010

2 Aims  To involve us in experiencing mathematical reasoning  To consider implications for teaching –Comprehending reasoning –Re-constructing reasoning –Reasoning for oneself

3 Outline  Silent Mathematics  Components of mathematical Reasoning (Deduction & Induction; Toulmin; Attention)  Comprehending reasoning  Re-constructing reasoning  Reasoning for oneself AM: algebra PM: geometry

4 Tasks  = 3  Carpet theorem  Abundant, perfect & deficient numbers  Homework tasks  Root 2 as comprehension & as re-construction  Geometric Construction  Revealing Shapes

5 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!)

6 Visual Carpet Theorem

7 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 with Empirical (scientific) Induction

8 Toulmin Toulmin, S. (1969). The Uses of Argument, Cambridge, England: Cambridge University Press

9 Jigsaw Proofs Does this generalise … … to √n ? … to n p/q ?

10 JigSaw Proofs Does this generalise … … to √n ? … to n p/q ?

11 Home (Reflections) Work  Not interested in actual reasoning, but in what you found yourself DOING in seeking proofs x x is square for x = x x is square for x =

12 Square Deduction a b a+ba+b a+2b 2a+b2a+b a+3b 3a+b3a+b 3b-3a (3b-3a) = 3a+b 12a = 8b So 3a=2b 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) Could these all be squares?

13 Abundant, Perfect & Deficient

14 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) 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 Mathematics, Australian Mathematics Teacher, 59 (4) p17-25

15 Magic Square Reasoning –= 0Sum( )Sum( ) Try to describe them in words What other configurations like this give one sum equal to another? 2 2 Any colour-symmetric arrangement?

16 More Magic Square Reasoning –= 0Sum( )Sum( )

17 Geometrical Reasoning Outline  Reprise on Reasoning  Tasks through which to refresh experience of geometrical reasoning –Angle reasoning –Length reasoning –Diagonal properties  familiar properties –Unfamiliar Problems

18 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

19 Tasks  Geometrical Construction (scrambled)  Angle Reasoning (angle at centre, at tangent)  Length Reasoning (alternating sums)  Diagonal properties  usual properties  Reflected Tangent

20 Geometric Construction 1

21 Geometric Construction 2

22 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 are these related?

23 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?

24 Reflected Tangent

25 Reflected Tangent (2) Allow the diagonal to be a chord

26 Reflected Tangent (3) Allow the tangent to be at some other angle to the radius

27 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.

28 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?

29 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?

30 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) 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 Mathematics, Australian Mathematics Teacher, 59 (4) p17-25

31 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

32 Further Reading  Logic problems – Reasoning-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  … see WebLearn site  mcs/open.ac.uk/jhm3