Math is Beautiful

I am not a mathematician, and don’t use math in my day to day work. But I’m prompted to write this, right now!, due to my horror at a proposal by a New York Times’ columnist that math be taught using personal finance as a motivator. Ick. Ugh. Help me get this out of my system. Thank you.

Math is the language of nature and the universe, while personal finance is the language of capitalism and efficiency. Yes, you can teach arithmetic by asking how many more quarters you need if you have fifty cents and want to buy a brownie for $2.00. You can also teach English by asking people to read the ingredients list on a cereal package. Neither is likely to inspire much wonder or creativity.

I remember learning about irrational numbers a half century ago, in high school. It was kind of neat that they were crazy (“irrational”) numbers, but I never really got to the amazing part.

We’d all like to think that if we have a ruler, we can divide it in half (*) as many times as we need to in order to use it and its halves, and its halved halves, and so on, to precisely measure anything. The whole point of irrational numbers is: that doesn’t work. Now maybe that isn’t going to grab all high school students, but it ought to interest some of them, as a challenge if nothing else. By the time students have learned about triangles, you can even prove it for something as simple as a diagonal of a square, and there are short written proofs, but also geometric proofs that many high school students should be able to get.

(*) or thirds, fourths, fifths, or 1897ths

Where does that lead? I think it makes a student realize that the obvious isn’t always true. Maybe our society doesn’t really want students to learn that in math. The columnist thinks we should be teaching them net present value.

What about symmetry? A fun little puzzle that elementary students can understand is to ask them to add the numbers from 1 to 100, without using a calculator. There’s a creative solution that points out the power of recognizing a symmetry: write the numbers out on a tape, cut the tape in half after the first 50, and move the second half just below the first half, upside down. You get tapes that are ordered like this:

001 02 03 04 … 49 50

100 99 98 97 … 52 51

(You don’t have to do the whole tape, just enough to see what happens.)

Asked to see how this helps (and perhaps with a bit of encouragement to add), a classroom can see that by splitting the tape this way, they got fifty pairs of numbers and each of the pairs adds up to 101. We can see why: every time the top row increases by one, the bottom row decreases by one. From there, it just takes one multiplication to get the answer. This is solving puzzles — there’s no need for it to come off as drudgery. And it teaches a lesson that looking for patterns, as opposed to just doing it the standard way, might turn out to be worthwhile. Middle managers everywhere might find that terrifying.

Just one more. Euler’s identity. Assuming you didn’t study math or engineering, you’ve probably never heard of it, which is just sad. But you probably learned about the numbers found in it: 0, 1, pi, e, and i. What does pi, the ratio of the circumference of a circle to its diameter, have to do with i, the square root of -1? Well, the identity is that the number “e” (involved with things like growth and rate of change) raised to the power of pi (involved with cyclical motion, among other things) times i (the square root of negative one), plus one, equals zero. What you say!? How crazy is that?

This equation bridges multiple areas of math in an unexpected way. There are books which explain in detail, and I’m absolutely not going to try here. A recent one is David Stipp’s 2017 “A Most Elegant Equation” but it is more for college students or grad students, I think, than for high school students, though by all means give it a try.

What these three examples have in common is they have the potential, in the right hands, to astonish students and, for the more creative ones, raise their curiosity to learn.

What does x + 3 = 8 do? It convinces students that they must be tortured to move to the next grade. Has anyone ever not looked at that and said x is 5, and then complained when their teacher writes out a set of lines, complete with a “check,” because “this is how we do it.”

Don’t get me wrong; I do understand that foundations need to be built, but when the building’s foundation is day after day of x + 3 = 8, by the time there’s any chance of interest, it has often been drummed out.

The non-solution textbook publishers seem to hit on is to use pretty pictures and made up stories: “Juanita/Karen/Abe/Nichelle has three dresses/ice cream cones/blocks/angels…” Students aren’t stupid. They know it’s just x + 3 in hiding, and they know that, somehow, subtracting three from both sides of the double horizontal line makes the math teacher happy, when the answer is clearly five from the beginning, whether it’s about Juanita’s dresses or Abe’s blocks.

Look. I suppose we need accountants to double-check on the AIs that are doing the books. And there are probably some students who will be happy to go into accounting, and it’s an honorable field. And those students are going to be happy to be taught algorithms (“recipes”) that will stand them in good stead. But there are also students who want to be poets, and they should find math courses inspiring. They don’t.

I’m not sure if this is helpful to anyone reading it, but it’s helped me cool down from that columnist’s dreck. Thank you.