Yesterday I wrote about my frustration with the book The Pleasures of Counting and its failure to communicate to me the answers to many of the problems it presented. But I didn't mention the ironic part: I teach math for a living. Sort of. What I actually teach is test prep. I've taught the law school entrance exam to elderly women and I've helped junior high kids get into Stuyvesant. I've taught the children of bajillionaire celebrities and I've taught in slums like Bushwick and East New York. On the SAT front I've had students for whom it was Cornell or bust and students who were just trying to get into a college without the word "Community" in its name. But over the last couple of months it's mostly been math, and on a level I haven't had to teach before. See, Massachusetts recently started requiring high school students to pass a couple of exams in order to receive a high school diploma. One's in English, one's in math. The math one is ludicrously easy. It requires the student to be able to solve for x, find the area of a circle, read a chart, and that's pretty much it. That in and of itself would seem to indicate that the state is not setting its standards very high when it comes to considering a person educated — but there's more. The test is scaled from 200 to 280, and passing only takes a 220. And kids get five chances to pass this thing. Five chances to get a quarter of the possible points on a test that should be a breeze to a kid in middle school. My assignment was to deal with the kids who found this too much to ask.

Let me put it another way. Take your typical junior high. All the kids who manage to make it to high school? I wasn't dealing with those. I was dealing with the kids who couldn't hack it, over at the vocational school. Then, from that group, set aside the ones who managed to pass the MCAS (for that is what the test is called) on one of the first two tries — I didn't have them either. Get the idea? Grad school conditioned me to think of myself as kinda dumb because I was usually not one of the brighter people in my classes; sure, simply being accepted to the schools I went to meant that I was judged to be among the top 50 or so people in my field among recent graduates, but being #50 in the country marks you as a serious lightweight when you're talking to numbers 1 through 20. Similarly, reading books like The Pleasures of Counting and spending big chunks of my time around professional computer programmers get me thinking that I'm a drooling idiot when it comes to quantitative reasoning. But if ever you find yourself thinking you're stupid because your percentile rank in a given subject is merely somewhere in the 90s instead of the 99.99th, spend some time hanging with people who're smack in the middle of percentile zero.

There is no way I can exaggerate how limited these kids' math skills were. I received more than one test book where addition and subtraction of single-digit numbers had been done with tally marks. It'd be sad if the kids were remotely sympathetic, but they weren't: most showed up less than half the time (often far less), and of those who did attend on a given day, most would either put their heads down on the table and sleep or else talk to their friends as if there weren't a lesson going on. Those who did participate did so mostly to mouth off. All in all it was a pretty dire experience. But I did manage to get some value out of it. See, they say that great basketball players make terrible coaches because they expect all their players to just naturally have the same sorts of skills they had back in their playing days. Similarly, brilliant people aren't necessarily at all good at teaching because they expect their students to make the same sort of leaps they do — "Can't you see the answer? You can't? Then I don't know what to tell you. It's right there in front of you!" But as noted, my math ability is hardly prodigious. Having struggled with The Pleasures of Counting and the like, I know the sorts of things that tend to help: breaking complex methods up into tiny steps; fleshing out abstractions with a slew of concrete examples; offering very simple versions of the problem at hand and gradually building back up to the real problem's level of difficulty; that sort of thing. But these techniques weren't working with the MCAS kids. So partly in hopes of being able to actually teach them something, and partly just out of interest, I started keeping track of exactly what mental deficits they exhibited. It was my first extended contact with the far left end of the bell curve, and I think I learned a fair amount.

"Intelligence" comes from inter- + legere: "to read between." IE, the ability to read between the lines. That's probably the #1 thing these kids couldn't do. Not just in solving math problems, but in understanding the purpose of the problems. I'd say things like, "Now, what are you supposed to do with these two numbers here?" and they'd reply, "Add... subtract... multiply... divide...", just rattling off answers and waiting for me to say, "Yeah, that's right," and continue, as if it were good enough to get the answer right via wild guess on the fourth try. Or I'd get into one of these exchanges:

Me:What's the answer to multiple choice question #5?
Kid:I put C.
Me:Okay, why?
Kid:Because it had to be A or B or C or D and I picked C.
Me:All right, let's put this one up on the board. The first step is—
Kid:Wait, is C right?
Me:Well, let's find out. First you—
Kid:C'mon, is it right?
Me:Yes, because—
Kid:Well if it's right then what do you need to explain it for?
Me:Because you only got it right by guessing, and you might not be so lucky on the real test.
Kid:(confused look)

This sort of thing came up over and over again. The kids could not grasp that when the test asked, "What's the greatest common factor of 30 and 72?", it was really asking — reading between the lines, as it were — "Do you know how to determine a greatest common factor?" Perhaps they thought that the government of the Commonwealth of Massachusetts really needed to know the answer to this particular question and was hitting up the students of small-town vocational schools for help — but, more likely, they hadn't put any thought at all into why these tests might be given. And I mean even to the extent of "the state wants to know how much math I know," let alone, "the state wants to know how much math I know because it's a good thing for the citizenry to know, for example, how percentages work." Most seemed to think, in all seriousness, that mathematics was a meaningless fabrication designed for the express purpose of tormenting them personally. I started collecting examples of when I used simple algebra in real life to figure stuff out, but it was hopeless:

Me:Just a few days ago Jennifer was deciding between a Toyota that cost $1800 and got 35 miles per gallon and a Geo that cost $2800 and got 48 miles per gallon. So we figured out which is the better buy by—
Guy:The Geo.
Me:Well, before jumping to conclusions we thought we'd better—
Guy:The car with better mileage always pays off.
Me:But that's only true if—
Guy:No. The Geo's better.
Me:But look. You need to make up a $1000 difference at the pump. Will she in fact save $1000 in gas? The amount she pays for gas is the number of gallons she buys times the price per gallon. The number of gallons she buys is the number of miles she goes divided by the miles per gallon, so plug that in and you get miles times price over mileage. And so if miles times price over 35 — that's the total price of the gas that goes into the Toyota over its lifetime — minus miles times price over 48 — same for the Geo — is greater than a thousand, she should get the Geo. Get a common denominator and you get 13 times miles times price over 1680 equals 1000. Multiply by the reciprocal — remember that? we did that last week — and you get miles times price is about 130,000. If the price of gas is about a buck fifty that means that if she drives a bit under 90,000 miles over the lifetime of the vehicle the Geo will then turn out to be the better deal overall. That was pretty fast, but the point here is that we actually did use this to figure which was truly the better deal, at least considering only these two factors.
Guy:I still say the Geo's better.
Girl:I'd just pick whatever car was cuter.

Sticking to arbitrary fixed ideas rather than doing the work to find out what the actual answers are, and then dismissing those answers when the work is done by someone else... that'll keep you crammed over at the left end, all right.

Or maybe he didn't see the work at all. I had a professor who liked to tell the following anecdote: when Europeans first landed in Australia and eventually managed to communicate with the aborigines, one sticking point was the question of how the Europeans had come to such a faraway land. The Europeans proudly pointed at the ships they'd come in, resplendent in their mighty sails and whatnot. We came from across the ocean in those ships, they said. The aborigines were confused. You swam here? Er, no, the Europeans said, like we just mentioned, we came here in those huge-ass ships we're pointing at. See, with the masts and the sails and all? The aborigines looked out at the ocean, scanning the horizon from left to right... you see, this professor said, they didn't understand the concept of ships, so they just mentally edited them out without even realizing they were doing so. Now, I of course assumed this story to be entirely apocryphal — until I saw it in action. I had a kid read a problem aloud. The problem had the term 3√¯7 in it (that is, three times the square root of seven.) She said, "The value of three seven lies between—" and I stopped her: "Read that again, please?" "The value of three seven lies—" "Aren't you leaving something out?" I asked (echoing the words of Dan Quayle at the spelling bee, but I digress.) "No," she said. Maybe her book has a misprint, I thought, so I went to her table to check it out. There was the radical sign, right where it was supposed to be. "See that sort of checkmark there?" I asked. "Any idea what that is?" She blinked. "Oh!" she said. "I didn't know what that was so I didn't notice it. Now I see it. What is it?"

And I told her, but chances are the answer knocked some other information out of her head. This was something that just astonished me — these kids seemed able to hold onto only one piece of information at a time. Take positive/negative rules. When the class started, I quickly discovered to my amazement that they didn't understand positive vs. negative, and persisted in Not Getting It even after my usual analogies: positive is money and negative is debt, positive is a pile of dirt and negative is a hole, etc. They persisted in thinking that the negative sign was like a hat that numbers wore more or less randomly and that predicting whether a number would wear that hat or not was an impossible task. I finally resorted to trying rote memorization. It seemed to work at first, but...

Me:Okay, here's Rule #1. A negative plus a negative is a negative. Repeat that please.
Kid:A negative plus a negative is a negative.
Me:Good. Again, what's Rule #1?
Kid:A negative plus a negative is a negative.
Me:One more time: a negative plus a negative is...
Kid:...a negative.
Me:And Rule #2 is: a negative times a negative is a positive. What's rule #2?
Kid:A negative times a negative is a positive.
Me:And what's Rule #1?
Kid:I forget.
Me:C'mon, we just went over it three times less than a minute ago. A negative plus a negative is...
Kid:I forget.

I quickly dropped the rote memorization drills, but the same problem kept rearing its head in other contexts: new information seemed to just bounce off these kids. They had space in their heads for one new thing, and that thing could be knocked out by any other new thing (including "look, out the window — a tree!") Perhaps this is why early on came the saddest exchange of all:

Me:Hey, uh, you might want to actually listen to this, since I'm explaining how to do #1 and I see here that you got this one wrong.
Kid:I yon' give a care. If I failed the first two times how'm I gonna pass the seventh or eighth time?
Me:Well, like I said, I'm going to be explaining how to do all these, so listen up and take notes and stuff and you should learn how to do them by the time you have to take it again.
Kid:Huh?
Me:Well, see, they just ask the same questions over and over with slightly different numbers. So you learn how to do #1 here, and then on the next test, find the problem that looks like this one and go through the same steps.
Kid:Huh?
Me:You asked how you're going to pass the next time. The answer is to listen and learn some math.
Kid:Huh?

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