# The futile proofs of high school

There are two math subjects taught in high school which involve the students doing a lot of proofs: geometry and trigonometry. Advocates say that it's important for students to do proofs, since they are the bulding blocks that are used to construct the whole edifice of mathematics. Though the latter is true, the former does not at all follow from it. The edifice has already been constructed; students can simply live in it, rather than being tasked with rebuilding it. And the proofs taught in those two classes are amazingly useless.

Geometry is the worse of the two. Euclid is said to have replied "There is no royal road to geometry" to a king who asked for an easy introduction to the subject. These days there is a royal road: analytic geometry, where all the points are given coordinates and geometric problems are reduced to algebraic and trigonometric ones. It's so much easier and more powerful that no working engineer uses Euclid's sorts of proofs; high school geometry is just something he was taught a long time ago and has thoroughly forgotten.

Trigonometry is better: the things that are proven can actually be useful results. But there is a much easier way to prove them. The proofs that students are asked to do involve showing that one trigonometric formula is equal to another. The easy way of doing such a proof is to convert both formulas into complex exponentials; then the proof becomes a completely mechanical exercise. But students of trigonometry are taught the hard way, where one memorizes a list of basic trigonometric relations and tentatively substitutes them in, manipulating one or the other formula to try to make the two identical. This can be quite a puzzle, which is why it takes half a school year to teach. The easy way requires a more advanced background (complex exponentials, usually a college rather than a high school subject), but then needs almost no teaching.

It might be thought that it's just a cruel trick of fate that both sorts of high school proofs are so futile. But part of what makes them accessible to high school students in the first place is that each is in a very small playground. In math as a whole, there's a very large arsenal of things that have been proven, and it's common for a single proof to draw on a wide variety of them. Even the set of techniques for proving things is much larger: there are basic techniques such as proof by induction that aren't mentioned at all in these high school classes, since in those small playgrounds there is no need for them. But with the much larger arsenal comes a much larger search space and much greater difficulty; the vast majority of the proofs that mathematics is built on aren't things you can ask the average high school student to do on an exam. (On occasion it's been done, and it can be a way of figuring out if you have a genius in your class, but it can hardly be standard procedure.) So it's only in small playgrounds that proofs can be assigned to high school students as a routine matter, and the world has a way of passing by small playgrounds.

Now, there is at least one useful sort of proof that can be taught at the high school level: the proofs of Boolean algebra. Computer programmers, for instance, have to know that the opposite of the proposition (A or B) is (not A and not B), so as to be able to manipulate "if" statements accordingly. That is the same sort of manipulation that one does in an abstract proof in Boolean logic. Digital hardware designers, likewise, often have to manipulate one Boolean expression into another that is more easily implemented or whose implementation is faster. So proofs in Boolean logic are a useful exercise.

But there should be no desire to teach proofs simply for the sake of teaching proofs. The edifice of mathematics doesn't need mass indoctrination to get people to respect it; living in the edifice and finding it completely functional is quite enough to convince most students that it was soundly built. Proofs are used because they defeat even the strongest challenges, not as items of worship. "You can go down that path of challenge, but there are the bones of the people who tried it before you" is more impressive than "Here, let me take you on a guided tour of the playground on the first part of the path".

Of course when students reach a higher level, teaching them to do proofs is proper: people who are actually extending mathematics need to prove things. That doesn't just apply to people who earn a salary as mathematicians and publish results in math journals; there is plenty of filling-in of details that isn't groundbreaking math but still is worth doing if you have a real use for it, and is easier to redo than to look up in some reference which might not even have it.

But these high school proofs are remarkably futile. For such wastes of time, one often hears the excuse that it's "good mental exercise". But there are more than enough good mental exercises that are useful that students shouldn't be forced to waste time on ones that aren't. It's still better than being forced to learn something that's actively wrong (as is too common in softer subjects), but that's not saying much.