The New York Times ran a deeply contrarian editorial Saturday about math education in the United States. In it, political scientist Andrew Hacker argues that the youth of America is being crucified on a cross of higher math.
A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.
My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus. …There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong—unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)
Hacker argues that the math used in the typical American workplace—even a technical, highly quantitative workplace—does not much resemble math as it is taught in the American classroom. Engineers, doctors, and bankers rarely use algebra as such. What we probably should be doing, Hacker thinks, is to foster mathematical intuition amongst students who can’t master higher levels of abstraction.
Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.” This would not be a backdoor version of algebra, as in the Advanced Placement syllabus. Nor would it focus on equations used by scholars when they write for one another. Instead, it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.
It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted—and include discussion about which items should be included and what weights they should be given.
I’ve had some Canadians point out Hacker’s editorial to me in the spirit of “This guy’s crazy, right?” But his remarks have to be understood in an American context. There has been a strong push in the U.S. for pretty high universal national standards in mathematics, standards which have seen many states run afoul of what’s called “Algebra II” in curriculum circles.
American high school graduates are taking harder math and science classes, according to a recent report by the National Center for Education Statistics.
In September, Tom Luce, former CEO of the National Math and Science Initiative, said that the United States needs a “STEM-literate population” that starts by “convinc[ing] the entire country that every child must conquer Algebra II.” America has made steady progress toward that goal—in 1982, just 40 percent of high school graduates took Algebra II; in 2009, more than 75 percent did.
Keeping those figures in mind, here’s a short sample from an Algebra II exam: I note with some alarm that it features a question about complex numbers, which I don’t think I ever learned in the classroom despite having fought through Alberta’s Math 31 high-school course and a year of university calculus and statistics. This is pretty esoteric stuff to be expecting “every child” in a large, diverse country to conquer. (I think only physicists would ever actually use complex numbers at work, though I know electrical engineers are expected to master them as part of their theoretical education.) Insofar as Hacker is just pointing that out, his op-ed falls into the category of “man identifies patent, unaddressed insanity swirling around him” rather than “man quarrels with high educational expectations.”
Math has a special, awkward place in education. It is no wonder that it stirs passions and raises fears, for it is pure concentrated abstraction, and everybody senses on some level that how far you can go in math (speaking as someone who got the equivalent of a B in first-year calc) is a very precise, cruel measure of one’s cognitive separation from the cleverer beasts. America is pushing Algebra II because, of all high-school courses, it is “the leading predictor of college and work success”. But you don’t need Riemann curvature tensors to understand the logical flaw in the proposition “If Algebra II predicts success, making everyone pass Algebra II will make everyone successful.” Understanding the square root of minus one is no use to most of us, in itself; yet it is true that those who can be taught to understand it will, over time and as a group, earn and accomplish much more than those who don’t. This is true of any reasonably abstract concept, which is why there is always confusion over the actual value of learning to read a musical score or figure out a left fielder’s on-base percentage.
Andrew Hacker’s “algebra problem” is an interesting symbol of how rampant egalitarianism is in the American academy. Primordial America possessed an intellectual counterweight to the Jeffersonian faith in education; the minor Founding Father Fisher Ames is said to have responded to the notion that “All men are created equal” with the retort “…but differ greatly in the sequel”. Today’s American right, however, takes the tactical ground that no child must be “left behind”—that all can be educated for a STEM future, just as any goose can make foie gras in his liver if he is stuffed full enough. This happened because American public education became compromised by the teacher trust and its slovenly “easier-for-us” ideology: it became too tempting to whack the education industry over the head with standardized testing and with the excellent results of other countries’ education systems, as President Bush 2.0 did. It is probably not really realistic to expect the U.S. to compete in mass mathematics education with a small, homogenous Nordic country like Finland—but what American will admit to that? We built the Bomb and went to the Moon, man! (One supposes it would be unkind to note that the “we” in that sentence denotes, respectively, “a bunch of Europeans who fled the Nazis” and “a bunch of Nazis who fled the Russians”.)
Even Hacker, whose essential complaint seems to be that 25% would be a much better estimate of the number of American children capable of mastering Algebra II than 100%, won’t put it in such a direct, offensive way. But it doesn’t help that he conflates “algebra as taught in American schools” with algebra as such. Algebra is, above all, a single step up in abstraction. It’s a step that most children can take: once you give them the core idea of performing operations on “x” rather than on a particular number, the royal road is open.
And that much—the whole notion of a variable—is something ordinary people do require, if they hope to have the instincts, training, and mental safeguards Hacker agrees that they need. How the hell are you going to teach even the crudest statistics or concepts of probability to a student who doesn’t get what a variable is? How could an ordinary Cartesian graph be comprehensible? How, indeed, would one teach the difference between arithmetic and geometric growth—and hence the power of compound interest?
I suppose you do it, though Hacker specifically denies it, by sneaking the abstraction in through the back door: you give eleventy real-life examples of compound interest at work, until the student eventually realizes that the principle’s the same no matter what the actual principal and the particular interest rate are. He learns to identify a “variable” without being intimidated up front by xs and ys. There might be some merit in such an approach, for it is in fact the symbology that seems to frighten the math-averse. In this article about the rigours of Algebra II, a person claiming to be an accountant says “Most people I know who are lower income couldn’t solve 2x = 14 if their life depended on it.” And maybe he’s right. Yet there can’t literally be many people who couldn’t stumble into an answer to “Two times what is fourteen?” if it were presented that way and they were given a few minutes to wrestle with it.