Transcript Document

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