## Why All the Formality?

3/22/2011

One of the biggest stumbling blocks for many math students is the requirement that they show their work and write in proper mathematical notation.

This is simultaneously the area in which other students thrive: If you can follow directions, you will succeed in math class.

Students are told that mathematics is a language like French or Spanish.  This mostly just frightens the students, or bores them if we're lucky.

What students are not often told is why we use a special language for math and why we need to follow special rules.

The fact is, math is much more confusing when we don't use mathematical language.  Think about word problems:

Jane is two years older than Kent who is twice as old as Sam.  Sam is four years old.  How old is Jane?

Isn't it much easier to solve j = 2s + 2 for j when s = 4?
j = 2*4 + 2
j = 10

Well, maybe you don't agree, but I think it's easier.  And that's the only reason we use the language of math--to make our lives easier.  If you pick up an ancient Greek geometry text, it barely makes sense, even if you can read ancient Greek!  For one thing, there aren't any pictures.  For another thing, they didn't use very helpful notation.

Here's a great example from Euclid's Elements:
"If two triangles have two angles equal to two angles respectively, and one side equal to one side, namely, either the side adjoining the equal angles, or that opposite one of the equal angles, then the remaining sides equal the remaining sides and the remaining angle equals the remaining angle."

To see a slightly more comprehensible version of this statement (which has pictures, at least!) visit this site.  You can read more of Euclid's Elements on this site to find more nonsensical language to irritate your friends with!

Okay, so we get it.  Mathematical notation makes things easier.  But what about the super rigorous proofs we have to do in high school.  Aren't those just to irritate us?

Yes and no.  I'm not a fan of strict regulations on notation for proofs.  I'm also not a fan of having high school students prove the heck out of obvious facts.

One example of an obvious fact we shouldn't have to prove in high school is the Well-Ordering Principle.

The Well-Ordering Principle states that in a non-empty set of positive integers, there will be a least element.  In other words, if we have a list of positive numbers, there will be a number in that list that's smaller than all the rest.  This is obvious, and for high school students to prove this on their way to bigger and better proofs is simply a waste of time.

This is not to say that proving this fact is a waste of time in general.  It is extremely important to the integrity of mathematics for mathematicians to rigorously prove everything.  We can't just make assumptions or generalizations because sometimes our instinct is wrong.  But it's the job of mathematicians to scrutinize these things and tell us the results.  It's the job of high school students to learn and gain an appreciation for their subject.  They don't gain appreciation through being asked to prove obvious facts.

Some proofs are appropriate for high school students, but the language doesn't have to be overly formal as long as the proof holds.  Here is an example of a proof that a high school student could probably write.  The result isn't obvious (which makes it more interesting), but the technique is simple.  I've written my comments in italics to clarify things for the math-phobic, so go ahead and check it out!