## Sunday, February 19, 2006

### Math

Everyone is outraged! An incredibly stupid columnist insists that algebra doesn't matter. After pondering the tale of a high school drop-out who left high school diploma-less thanks to LA's new requirement that students graduate with a passing grade in algebra and geometry, he writes:

Here's the thing, Gabriela: You will never need to know algebra. I have never once used it and never once even rued that I could not use it. You will never need to know -- never mind want to know -- how many boys it will take to mow a lawn if one of them quits halfway and two more show up later -- or something like that. Most of math can now be done by a computer or a calculator.
This is a lie. PZ Myers explains:
Algebra is not about calculating the answer to basic word problems: it's about symbolic reasoning, the ability to manipulate values by a set of logical rules. It's basic stuff—I know many students struggle with it, but it's a minimal foundation for understanding mathematics and everything in science.
But even this gives Richard Cohen far too much credit. Algebra is taking childish calculations (what's 2+2?) and acknowledging that sometimes the world doesn't serve up a really simple problem. When you go to dinner with 5 other people and you agree to split the check between the two couples and the person who went stag, you're doing algebra (5 * \$X=total cost of meal, X=total cost/5, couples pay 2X, the stag pays X). Hell, you're doing simple algebra when you check that the bill was calculated correctly, or that the gas pump didn't screw you. And we haven't even gotten to calculating tips.

Algebra is basic math. Being able to even compute the exact answer is often less important than being able to set up the problem right. If you have a calculator in your pocket you still have to be able to do the algebraic manipulation. And 90% of the time, some quick algebraic manipulation gets you something you can estimate, the precise answer isn't necessary. You look at the price of gas (\$2.129) and you know your gas tank holds about 15 gallons. To fill it up will cost about \$30, maybe a little more. If you get charged \$40, something's wrong. If the pump stops at \$20, maybe you want to give it another squeeze. Algebra+estimation=life in the modern world.

Yes, algebra teaches abstract reasoning, but that's really the argument for mandatory calculus in high school, not the argument for algebra.

Calculus is the core of all scientific calculations. In honor of Cohen's praise of typing class as superior to algebra, let's work that analogy. Algebra is the equivalent of recognizing the symbols on the keyboard and knowing how to assemble them, hunt and peck, into words, perhaps even sentences. Calculus is touch-typing. It involves not just manipulating symbols, but getting at what the symbols mean, and working with them in big, integrated blocks, or deriving some deeper meaning from them.

Matt Yglesias says algebra is necessary to understand the budget. Sure, but only in combination with calculus. You don't necessarily have to take the derivative or compute an integral, but when the President says "We've now cut the rate of growth in non-security discretionary spending each year I've been in office," someone who's taken calc. thinks "Aha, changes in the rate of growth are a second derivative. That means that the curve of spending since 2001 is concave downward. It doesn't mean spending has actually declined. That would require a negative first derivative." If you care about fiscal discipline, you realize that he didn't tell you about the only statistic that matters, the first derivative of spending. He told you the second derivative of 15% of the budget. Woooo!

In statistics, I know that p-values are the equivalent of an integral of the probability distribution function. I haven't taken an integral in a long time, but knowing the underlying principle lets me understand something important about how p-values work, and gives me a way to check my thinking.

When I think about how small mutations add up to major evolutionary changes, I'm thinking that I've got an integral, and that evolutionary change is the integral of all the infinitesimal changes in each generation. Calculus is the study of how infinitely small things can add up to really big, even infinitely big, things. There are tons of processes in the world where small changes add up to huge effects, and that's not algebra, it's calculus. People who don't understand how that works are incredibly abundant, and are running on fumes in many discussions about the world. Do such people deserve a high school diploma? Kevin Drum isn't sure, but I don't feel ashamed of thinking they don't.

Can a columnist accurately assess the significance of economic policy changes if he doesn't understand algebra (let alone calculus)? I say no. I say Richard Cohen is untrustworthy on any topic that isn't purely emotional, and the Washington Post is doing its readers a disservice by letting him talk about policy that involves math (ie, policy). I assume he just accepts whatever BS numbers a think tank feeds him, and that's a problem.

It's said that there are three kinds of lies: lies, damn lies, and statistics. By statistics, people usually just mean "numbers." But the numbers don't lie, and statistics don't lie. People lie about what the numbers mean. And a numerically literate society is one where citizens can make an assessment of the meaning of a number independent of the text someone typed about it. If I can use some algebra and estimation, I can check that the numbers match up with what I know of reality, and I'll avoid stupid errors. I won't parrot liberal numbers that support my position, because I'll see they're dishonest, and I can pick apart conservative lies, or acknowledge valid mathematical claims from either side. I can decide for myself. If I'm mathematically illiterate, I'm stuck trusting someone else. It's as debilitating as the inability to read, but harder to detect. I don't think the illiterate have an inherent right to a high school diploma.

Black Math ” by The White Stripes from the album Elephant (2003, 3:03).

Listen master, can you answer a question?
Is it the fingers, or the brain
That you're teaching a lesson?
I can't tell you how proud I am
I'm writing down things that I don't understand

Complicated ” by Avril Lavigne from the album Let go (4:04).

Life's like this
That's the way it is

Update: I thought I posted this days ago, but blogger seems to have eaten it.