Proof: Why bother?
Ida Ah Chee Mok
Department of Curriculum Studies, the University
of Hong Kong
At the end
of a lecture, a student asked, "Why do we need to learn proofs?"
"What is proof?
Why do we need it?" These questions have been discussed on different occasions.
For example, Plumpton et al. (1984) say,
The mathematician has an intuitive, instinctive feeling
that some proposition may be true. The essence of proof is to establish
whether the result is, indeed, true or whether he has been deceived by
such a feeling. .... Fundamentally, mathematical proof is based on logical
argument, that is, to establish from a hypothesis ‘p is true’ a conclusion
‘q is true’. (p.1).
The importance
of proof in mathematics is evident but why students still raise the question.
It may be likely that proof in the curriculum is presented as "an obituary
of mathematics" which contains the important facts but not the living sensations
that the mathematician felt (Austin, 1991). To capture a mathematician’s
living sensation may sound ambitious. However, effort should at least be
made to help students appreciate the need for a proof and how proof helps
them understand the heart of the matter. In what follows, I would like
to demonstrate how an ordinary exercise in A-level mathematics can be extended
to provide an arena for further learning of the nature of mathematics.
The exercise (*)
To show that there are an infinite number of prime
numbers.
A proof by contradiction:
Suppose that there are finitely many primes, say 
.
Now consider the number 
.
None of the existing prime numbers, 
,
is a factor of 
.
Thus
must also be a prime.
This is absurd. Therefore, the initial assumption, that
there are finitely many primes, must be wrong.
Q.E.D.
Referring to the
above argument, the following question can be posed:
Is
always a prime?
After trialling
of the first few cases, the guess appears to be more plausible. Nevertheless,
verifications cannot be sufficient (see Table 1).
Table 1
| The prime numbers |
|
Is
a prime? |
| 2 |
2+1=3 |
Yes
|
| 2, 3 |
2x3+1=7 |
Yes
|
| 2, 3, 5 |
2x3x5+1=31 |
Yes
|
| 2. 3, 5, 7 |
2x3x5x7+1=211 |
Yes
|
| 2, 3, 5, 7, 11 |
2x3x5x7x11+1=2311 |
?
|
| 2, 3, 5, 7, 11, 13 |
2x3x5x7x11x13+1=30031 |
?
|
The next hurdle
will probably be how to check whether a large number "n" (such as 2311,
30031) is a prime. After investigating the factorization of numbers, students
may find that they need only to test the divisibility with prime numbers
not greater than ? n. This result will probably
resume their interest in Table 1 and they will be happy to find "30031=59x509."
Discussion
needs not stop at this stage. It may be proceeded to enhance students'
understanding in the following directions:
-
How does the hypothesis, "Is
always a prime?", arise?
-
If
does not always give a prime, will the proof still be valid? Why?
-
What results have we proved?
-
What techniques of proof have been used?
-
How observations help in making conjectures or formulating
a proof?
-
When do we need a proof?
I hope that
the above can illustrate that, besides producing a deductive argument to
convince, proof serves other purposes in classroom teaching. As Davis and
Hersh (1980) suggest,
Proof, in its best instances, increases understanding
by revealing the heart of the matter. Proof suggests new mathematics. The
novice who studies proofs gets closer to the creation of new mathematics.
(p.151)
References:
Austin, K. (1991). See the butterfly in flight. Theta,
5(2).
Davis, P. J., & Hersh, R. (1981). The Mathematical
Experience. England: Penguin
Books Ltd.
Plumpton, C., Shipton, E., & Perry, R.L. (1984). Proof.
London: Macmillan
Education Ltd.
(*) The writer used the problem while teaching
in a secondary school many years ago. The proof in fact originated from
Euclid (Book IX, Proposition 20).
