A couple of “kids these days are bad at math” stories crossed my feed reader last week, first a New York Times blog post about remedial math, then a Cocktail Party Physics post on confusion about equals signs. The first was brought to my attention via a locked LiveJournal post taking the obligatory “Who cares if kids know how to factor polynomials, anyway?” tack, which was obvious bait for me, given that I have in the past held forth on the importance of algebra for science students (both of these are, at some level, about algebra).
Of course, these articles aren’t about science students, so you might want to say that the same arguments don’t carry over. I would tend to claim that algebra is still an important part of education. The example that comes to mind about this is always an incident from the house I lived in in grad school, involving circuit breakers.
When I was in grad school, I did my lab work at NIST, which is about half an hour from the UMD campus on a good day, and considerably more if traffic is bad. Accordingly, I moved closer to Gaithersburg once I was more or less done with my class work, and ended up renting a room in a crackerbox Cape Cod house on the edge of a bad neighborhood in Rockville, MD. The rent was nice and cheap, but since the house was so far from the local academic institutions, the other rooms were occupied not by other grad students, but by a rotating cast of people in low-wage jobs, or no job at all. This led to a number of colorful stories that I used to amuse folks on Usenet with back in the day, but the relevant one here is pretty simple.
The kitchen of this house had some slightly dodgy wiring, I believe in the hanging lamp over the table, which would occasionally trip the circuit breaker controlling the lights, the microwave, and the stove (which was gas, but had a clock and an electric starter). One morning, I woke up to find a note under my door from one of the other tenants, who we’ll call W, saying that he had gotten up in the middle of the night to get a snack, and found the power out in the kitchen. So he lit the stove with a match, cooked what he wanted, then wrote me a note and went back to bed.
This was, as you might imagine, a little boggling. When I asked him about it, he explained that he didn’t know where the fuse box was, but he knew I did know, and would fix it eventually. At least he hadn’t woken me up at 3am, right?
Even once I had explained to him where the box was, though, we still had problems. The next couple of times, he went down there, and couldn’t figure out which breaker was tripped, so he just turned each one off and back on in turn, resetting all the clocks in the house in the process. On another occasion, he tripped a breaker in his own room, and turned off the breaker for my room while he was trying to figure out which one was his. I ended up taping a sign with detailed instructions to the outside of the box, and putting big arrows pointing to the two breakers that most frequently went out.
What’s this got to do with algebra? The fundamental problem here is a lack of systematic thinking. The first time I had had a problem with the power in the kitchen, it wasn’t too hard to figure out by running through a sort of mental checklist. Since the power was on in the other rooms, I figured it had to be a breaker. I didn’t know where the box was, but I knew it wasn’t likely to be in one of the rented rooms, which meant it pretty much had to be in the basement. Since it wasn’t in the laundry room, that meant it had to be in the storage room, and hey, there it was. I found the breaker that was off, turned it back on, and there you go.
W. was, apparently, incapable of figuring this out. Neither was the other housemate he enlisted to help him one day when I was at work. They needed me to tell them where the box was, and even then couldn’t work out how to get the right circuit. Confronted with a problem and nobody to point them to the right answer, they just gave up.
That’s one of the things you’re supposed to get out of taking math classes like algebra in school. There’s nothing all that alien and intrinsically difficult about algebra, it just requires you to systematically apply a few very simple rules. When you’re asked to solve an equation like
x2 – 2x – 8 = 0
you shouldn’t need to have the quadratic formula programmed into your graphing calculator, or even committed to memory. You know you’re looking for something like:
(x ± a)(x ± b) = 0
Since the sign of the final term in the original equation is negative, that means the two factors need to have opposite signs, so we have
(x + a)(x – b) = 0
We also know that a times b has to be eight, which means one is two and the other four. The negative sign on the middle term of the original equation means that the negative term must be the larger of the two, so we have:
(x+2)(x-4) = 0
This is satisfied if x=-2 or x=4, so those are your solutions.
There’s nothing sneaky or arcane about this. You don’t need to know how to do anything other than basic arithmetic to solve this problem, you just need to approach it systematically.
The breakdown that leads most students to give up on this type of problem– usually saying “I don’t remember the quadratic formula”– is just a breakdown of systematic thinking. It’s the same failure that leads to nonsense like “I didn’t know where the fuse box was, so I left you a note.” It’s a failure to systematically attack a problem beyond the most superficial level. Hell, I’d be happier to see them start plugging integers in for x in hopes of finding a solution that way.
And that’s one of the benefits people ought to be getting from taking math classes. There’s a lot of repetitive drill in math classes, specifically to establish the habit of thinking systematically about problems. If you can learn to work through simple algebra problems, you ought to be able to apply that same sort of reasoning to real-life problems, even ones like “Why doesn’t the kitchen light come on?”
The problem we have is two-fold: 1) students compartmentalize to a remarkable degree, and often don’t even connect math to science, let alone apply the systematic thinking of math and science to other areas of their life, and 2) as a society, we enable this sort of compartmentalization by shrugging off poor math performance as just one of those things. I bang on this a lot, but it’s incredibly annoying that a student who doesn’t like to read represents a major crisis, but a student who doesn’t like to do math is steered to special classes for people who don’t like to do math. If the Times were to write about a remedial class for students who can barely read, I doubt it would get quite as flip a treatment as the math class did.
That’s a problem, because the skills you learn in math go well beyond learning to plug numbers into the quadratic formula. Math is about systematic thinking, and systematic thinking is what built human civilization. Without the ability to think systematically, we’d all be stuck huddling in caves, hoping the lions didn’t eat too many of us tonight.
Or at the very least, sitting in a dark kitchen, waiting for somebody else to fix the lights.