Writing Proofs

I will be teaching a course in graph theory to some secondary school kids in about three weeks and as I prepared the course materials, one key point stuck out: Students need to write proofs.

Writing proofs is inevitable in dealing with most mathematics. Many theorems come with their most commonly used proof or proofs. The Pythagoras' Theorem, for instance, can be proven over a hundred ways.

In comparison, some problem solving questions require a proof as a solution but there are unfortunately no standard proofs that the instructor can rely on or turn to.

Particularly in graph theory, many questions require solutions that are highly demanding. It is not enough to say that the answer is "n". Often, you have to show why all other answers other than "n" are not acceptable, even though they satisfy the general conditions of the question.

In this regard, constructing a logically water-tight proof is difficult because many of us do not use contradictions and exhaustion as strategies, unless we are involved in certain types of work. It would not be suitable for me to exclaim the beauty and prowess of such proof techniques as I don't want to be guilty of playing up my areas of interest and "semi-expertise" but having a good grasp of the techniques can help solve a lot of problems quite elegantly.

The proof by contradiction is often elusive and hard to figure out. One needs to know which condition is likely to undergo "reductio ad absurdum". One common claim that relies on the proof by contradiction is "square root 2 cannot be expressed as a fraction with whole number numerators and denominators." Intuitively, this looks simple and the claim seems more like an axiom than something to be proven. However, it is not axiomatic and how do we construct a proof to validate our intuition? Interested readers can click on the word proof to see a proof by contradiction of that claim.

Proof by exhaustion is something elementary learners of mathematics used to resort to when they first come across claims such as "there is no largest integer." Frequently, a child who has learnt concepts such as a million, a billion, a trillion, and so on will attempt things like "a trillion trillion trillion" and when you tell them, "how about you add one?" they will quickly come to terms that exhaustion is plain exhausting. And frequently, exhaustion does not prove anything. Before long, proof by exhaustion becomes an antiquated technique to these learners of mathematics, until they are forced to face their inner demons again.

In graph theory, in particular, the need to establish uniqueness in the solution often means that the other competing answers must be rendered inadmissible. While it is probably not sensible to show a full graph theory question to elicit thoughts on how to construct answers that rely on the proof by exhaustion, folks who have a propensity for factorization might delight in seeing how the proof by exhaustion is daintly applied in this example: If n is an integer, then (n^7 - n) is a multiple of 7.

The study of mathematics can often lead its learners to develop logical thought processes. Writing proofs is one of the tools that play an important role in developing a person's logical acuity. As with all pursuits, there is no gain until one gets their hands dirty and begin practicing.

I hope my students will appreciate the package that I'm putting together for them. After all, whoever said learning was always easy?