If it's Tuesday, this must be trigonometry, or why HS math sucks

I've been reading Steven Hawking's God Created the Integers lately. This is a book that is likely to end up decorating the bookshelves of a lot of people who expect it to be a popularization, like "A Brief History of Time." In fact, it's Hawking's commentary on some of the mathematicians he considers most important in history, from Euclid through Turing, with the majority of the text being simply the text of the mathematician's work.

It's slow going; I'm not even through the Euclid yet. I admit to skimming some too, especially in the part about ratios. It's really hard to slog through all the unfamiliar old Euclidean terms for things that we have better words for today, or to decipher the difficult verbal descriptions of things that can be described much simpler using algebraic notation. Which makes more sense to you, this:

If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitude equal in multitude, then whatever multiple one of the magnitudes is of one that multiple will be of all.

Or this:

ma + mb + mc... = m(a + b + c...)

Of course, as a math teacher I think it is also good for me to spend time seeing familiar concepts described in unfamiliar ways; it helps me to remember what it's like for many of my kids, for whom the algebraic description of the first theorem of ratios would be as confusing as the verbal version.

And then there are the really beautiful parts, like re-reading Euclid's proof of theorem i.47, more commonly known as the Pythagorean Theorem. I've read the Elements a few times before, including that proof. But I had forgotten the exact elements of it, just remembering that it involved congruent triangles using the Side Angle Side postulate. But I had forgotten some of the most beautiful steps in the reasoning. For example these congruent triangles are of the same measure as the triangles based on the diagonal of the rectangles composing the square of the hypotenuse, being on a parallel base. I was trying in my geeky way to explain to my wife, who is a poet and author of a this amazing book, that it was like reading a particularly sublime poem, like an Emily Dickenson.

To a lot of people, math is entirely pragmatic. Not just people at a cash register punching in sums of christmas shopping. Engineers might have to learn Differential Equations and epidemiologists might have to learn incredibly complex statistical analyses, often far above anything I can do. But there is no need for them to appreciate the beauty of these procedures in order to use them. Not to say that many of them don't, just that even at that level math is often taught for the purpose of getting some real-world result.

Let me point out now that I am not by any definition a mathematician. Some people, knowing my job, have mistakenly called me that; in response I wave my hands and say, "No, no, no! I'm a high school Algebra teacher. Believe me, there is a difference." I am a mathematician in the same way that the neighborhood softball coach who practices swinging his bat like Mark McGwire is a major leaguer, or the family piano teacher who tinkles along in the style of Glenn Gould is a concert pianist.

But I do love math, not just for its astounding practical applications (and really is there any other branch of human thinking that has produced such amazing results?), but because it is one of the great towers of human achievement. The graph of a quadratic equation, which I have always been tempted to have tattooed on my arm, is as deep a penetration into the nature of reality as a Shakespeare tragedy or a Beethoven sonata. It took millenia of proofs and deductions, from the conics of Appolonius to the Principia of Newton, to understand how that one shape explains so many changes in the world. Which of course is only a drop in the unfathomable ocean of human reasoning that math is today. Standing at the edge of it, looking out over the horizon, one can be so overwhelmed with the beauty of it that it's difficult to speak.

And the real agony is that in my teaching I do not know how to get across the slightest bit of the beauty in math to my students. Actually, that is not entirely true. I have an idea how it might be done. But given the constraints I am under, it simply seems impossible.

It is not, let me emphasize, because of the condition or weaknesses of my students. I teach in the South Bronx, and when I say that people will chuckle and ask me if I wear a bulletproof vest to school, etc, etc. In fact, my students are nearly all good kids with high moral standards who come from families that want them to succeed and do well like most families do. I teach a larger number of kids with serious problems, from foster care to poverty to homelessness to being teen parents. Drug use, contrary to what people would expect, is not a top problem. I teach a few stoners, but I taught a lot more in the upper-middle class suburbs of Albuquerque where kids could afford the drugs. The biggest problem I struggle with is a sort of learned helplessness, the tendency of a minority, but a significant minority, of my kids to give up on anything hard before they even try. "Mister, I don't get it," they'll say. "What part don't you get?" I ask. "I don't get all of it," they reply, not even having looked yet at what they are supposed to do.

If I taught in Suffolk County on Long Island, or in a Manhattan top prep school, that would not be such a problem. Many more of my kids would be able to solve the problems I gave them, and be able to pass the dreaded Math Regents that tails so many urban kids through their high-school years. More would do their homework, and less would be tempted to hurl their calculators' plastic covers across the room at each other.

But they would nevertheless see math as a means for graduating high school and getting into college. It would be an entirely practical necessity to be taken care of and gotten out of the way. Beyond a few units in Geometry, they'd never have to deal with a mathematical proof, or put more than a moment's thought about why a^2 + b^2 = c^2, or what that means.

To do that, one would have to slow down, and spend some time on a topic, dig deeply into it. Imagine a class that, in talking about the Pythag. Theorem, explored how the Greeks from day one had doubts about the parallel postulate (at least that's what Hawking thinks), but had to use it because they couldn't prove the P.T. or much of anything else without it. And how the undecidability of that theorem led eventually to non-Euclidean geometry, Reimannian geometry, which led to Einstein's Theorem of General Relativity, and less directly to the nuclear bomb. They wouldn't have to understand all of the math (lord knows I don't) to see the beauty of the fact that from early on people sensed that hidden in the struggle to settle on the provability of that postulate was, quite literally, the key to the shape of the entire universe.

But slowing down is most definitely something a high school math teacher can't do today. The New York State Math curriculum, which is based primarily on the Regents exam, is so crammed with topics that one must as quickly as reasonably possible get students to where they seem likely able to solve the kind of problems they're likely to be confronted with, then speed on. In the freshman year alone they cover algebra from linear through quadratic, including inequalities, as well as probability, statistics, a little bit of logic and basic trig.

But why do you have to make such a big deal of covering everything in the curriculum? you might ask. It's a nice thing to think that coverage isn't so important until the kids get to the regents and don't even recognize half the problems on it. But wouldn't the kids do better if they had a deep understanding? Absolutely, but deep understanding takes time and is far more unpredictable and harder to measure.

I don't ever want to say it can't be done. If Jaime Escalante was doing my job perhaps he could have them all ready for pre-calc by the end of the year and appreciating every bit of it. And if he walks in my classroom and wants to take over, I'll hand him the chalk and sit in the front row and listen.

Instead I'm just another urban teacher struggling to get my kids to where they can graduate, and I pray go on to some kind of post-secondary and succeed. I'm just someone who loves math, and loves teaching, but hates the way I have to teach math. And I know something has to change, but I just don't know how or what yet.