If you ask a group of young children to tell you the biggest number, there will be no shortage of suggestions - a million, nine hundred and ninety-nine billion, infinity, and so on.
Children understand easily the mathematical idea of a counter-example, for they will readily suggest examples which are bigger than those already suggested. Eventually some more thoughtful individual will say "There isn't one!" A good teacher will respond with "Why not?" and that challenge will provoke something like "because whatever number you choose, you can always find one bigger". Here we have the essence of a mathematical proof by contradiction - any suggested number cannot be the biggest because you can always construct a bigger number by simply adding on one.
Mathematics at all levels is full of interesting and surprising results, which can be used to provoke children's curiosity and develop their ability to think clearly. The subject is so often just presented as a set of procedures to be learned and applied, and many children do not become aware of the power of mathematics to explain and make sense of ideas and relationships. Learning about proof is an essential part of this.
Once children have learned to measure angles they are commonly asked to measure the angles of some triangles and add them up to establish that the sum of the angles is 180 degrees. This experimental activity is valuable in making the idea plausible and familiar, but it is not sufficient to show that the sum is exactly 180 degrees, because the nature of measurement, however accurate, is that it is always an approximation.
Likewise, tearing the corners off a paper triangle and placing the angles together suggest that their angle sum is 180 degrees, but the eye cannot discern if they lie exactly on a straight line. The first example below should serve as a warning of the danger of such plausible pictures, besides providing an interesting curiosity in itself that merits explanation. Returning to the angle sum of a triangle, the proof in example 2 provides a good illustration of a deductive proof based on properties of parallel lines.
Learning to appreciate and produce reasoned arguments is a vital part of children's mathematical education and should be encouraged from an early age by constantly asking "Why?", and by providing frequent examples of proof in the context of standard topics of the curriculum. During a GCSE course, for instance, we should expect students to encounter ways of proving Pythagoras' Theorem, the tests for divisibility by three and nine and some of the angle properties of a circle. For A-level, the ability to understand proofs is a clear, if often neglected requirement of all courses.
The purpose of emphasising proof is not so much that students should remember the details of particular proofs, but that they should understand the nature of mathematical proof, and develop the art of applying their knowledge. The Mathematical Association's book Are You Sure? Learning About Proof responds to recent concerns about the lack of emphasis given to proof in the school curriculum. Some of the examples here are from that book.
One of the unfortunate effects of the wide use of investigations in school mathematics has been to reinforce in students' minds the idea that a pattern or property observed in a few examples gives sufficient evidence to say that it always applies, rather than providing a conjecture requiring proof or disproof.
If students are asked what they notice about the product of three consecutive numbers they will produce a set of examples like those in example 3 and assert that they are all divisible by 6. This is not sufficient to be sure that 93 x 94 x 95, or any other such product, is always divisible by 6. A short argument proves the statement and shows, moreover, that proofs do not necessarily either need to be lengthy or to require lots of complicated algebra. None the less, algebra is a valuable and vital tool. Example 4 shows another divisibility proposition with two alternative proofs. Looking at alternatives is often instructive and here we have a contrast between a neat approach which will only work in a few cases and a powerful general method based on the factor theorem.
Learning about proof is important both because it provides students with explanations of mathematical propositions and because it develops their powers of understanding and reasoning. Above all, it is a stimulus to their curiosity, imagination and enjoyment, for mathematics is a subject to be liked as well as learned.
Doug French is a lecturer in education at the University of Hull and chairman of the teaching committee of the Mathematical Association. He chaired the group which wrote 'Are You Sure? Learning about Proof', pound;7.50 from The Mathematical Association, 259 London Road, Leicester LE2 3BE
* Example 1
Try cutting up an 8 by 8 square and rearranging the pieces to make a 5 by 13 rectangle as shown. The area of the square is 64, but the area of the rectangle is 65, and yet both are made from the same set of four pieces. Clearly something is wrong. In fact the pieces do not fit together to make an exact rectangle, although your eye will not detect the discrepancy and some calculation is needed to show that the fit is not exact.
* Example 2
A line through one vertex of a triangle and parallel to the opposite side leads to a simple proof that the sum of the angles of a triangle is 180 degrees. Using angle properties of parallel lines, we have: f = b alternate angles
g = c alternate angles
f + a + g = 180 angles on a straight line
It follows that a + b + c = 180, and so the angle sum of a triangle is 180 degrees.
* Example 3
A few examples suggest that the product of three consecutive numbers is divisible by 6.
I x 2 x 3 = 6
2 x 3 x 4 = 24
3 x 4 x 5 = 60
4 x 5 x 6 = 120
5 x 6 x 7 = 210
Any set of three consecutive numbers must include both an even number and a multiple of 3. The presence of both these in the product proves that it is divisible by 6.
* Example 4
Some examples provide evidence for the conjecture that 9n - 1 is divisible by 8 for all positive integers n.
91 - 1 = 8
92 - 1 = 80
93 - 1 = 728
94 - 1 = 6560
The conjecture can be proved by using the difference of two squares: 9n - 1 = 32n - 1 = (3n - 1)(3n + 1).
Since 3n is odd, 3n - 1 and 3n + 1 are consecutive even numbers and therefore one is divisible by 2 and the other by 4. It then follows that 9n - 1 is divisible by 8 for any positive integer n.
This is a neat argument for the particular example, but it only extends to a few other cases involving square numbers. An alternative more general strategy is to show algebraically that xn - 1 is divisible by x - 1, a special case of the factor theorem. Then, putting x = 9, we see that 9n - 1 is divisible by 9 - 1 = 8. Indeed, with this simple algebraic result, we can make with certainty all sorts of impressive looking assertions like the one shown below, which would be difficult to check by long division or even with a powerful calculator!
15937 - 1 is divisible by 158