Sunday, July 29, 2012

Consider the Gebra

Today's Times features an oped which is sure to excite a swarm of comment. The author, a retired political scientist at one of the senior CUNY schools named Andrew Hacker, asks "Is algebra necessary", in the sense of "is the sine qua non status of algebra in our elementary-to-collegiate education tracking justified". His response is decidedly negative.

Now, normally I'd sarcastically begin my response with "There's only one thing wrong with this..." and then proceed to let the author know why he should leave and go feed penguins or something. Normally, this kind of attack begins from some deeply flawed assumptions, and it just gets worse from there. This piece, though, isn't that. So instead of sarcastically pointing out the one little thing the author got wrong, I think it's useful to mention the only thing the author got right: there is a deep and low-level problem with how we teach and learn school mathematics, and in particular with what role we expect mathematics instruction to play in a student's larger academic trajectory.

My aim is not to spare students from a difficult subject, but to call attention to the real problems we are causing by misdirecting precious resources. The toll mathematics takes begins early. To our nation’s shame, one in four ninth graders fail to finish high school. In South Carolina, 34 percent fell away in 2008-9, according to national data released last year; for Nevada, it was 45 percent. Most of the educators I’ve talked with cite algebra as the major academic reason.

The problem pointed to here is real: even at elite universities like my own, math courses required for graduation are serious GPA drains, and many students drop out midsemester, derailing progress toward their intended degrees temporarily if not permanently. Our courses are never below calculus-level, and one needs considerable algebraic dexterity to do well in them. At Hacker's institution and many others, courses begin below the calculus level; these courses too see high rates of attrition and failure.

But from there, Hacker's thesis is off the rails. After correctly identifying that this failure is a symptom of a deep problem, he proceeds to make a completely nonsensical diagnosis:
I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame... Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship.
Hacker goes on to repeatedly set up an opposition between the ability to parse quantitative information and the skills taught in algebra classrooms, toward the end of doing away with the latter, at least for the general population of students. (Recognizing that modern life is predicated on lots and lots of math, he explicitly carves out exceptions for those learning the trades where such procedures belong to the everyday toolbox.)

His argument, however, is incoherent: there is nothing separating the skills he doesn't want taught to the typical student, from those he does. And while it's certainly true that most students' mathematical education don't produce a lick of number sense, numeracy, or logic skills, there doesn't seem to be any connection between a proposal to scrap algebra and doing better at any of these things.

Algebra, you see, isn't any harder than any of the mathematical topics that Hacker thinks we all should become good at in school. Just teaching number sense is tremendously difficult, even if a student has good symbol-manipulation skills; teaching someone to be a sophisticated consumer and critic of quantitative data and arguments is even harder than teaching them to be a sophisticated consumer and critic of media in general -- the struggle of a lifetime, a constant battle against the forces of bullshit.

But as I've said, Hacker's pointing out symptoms of a real disease, even if he's misdiagnosed it. And I've alluded to one of these above: the tendency for school math to reduce to mindless symbol-manipulation. (I'm probably more sensitive than is Hacker to the argument that mindless rote [1] can be a legitimate goal for an educator; very briefly, there is a strong case to be made that correct and efficient algorithms, especially those which "trivialize" problems which we once thought to be bafflingly difficult, are among the most significant artifacts of our civilization.) Because of course, you and I know that math isn't just meaningless manipulation of symbols -- why, we know that it powers all of our society! But even that fact isn't at all obvious to a student in an algebra course.

(Never mind that many of the students we encounter in math courses have a real block on even mindlessly following the instructions in the algorithms; sometimes it seems that their minds are rebelling against being left out of the process, but other times it just seems that they're bad at following directions.)

Of course, the mere fact that our modern world would not run without lots of math, both that taught in high school algebra and otherwise, isn't much of an argument for requiring everyone to know something about it. I don't know much thermodynamics past the Three Laws, and that stuff is what propels me about the landscape whenever I take a car or plane -- and I teach math, for FSM's sake. I certainly wouldn't complain if more thermodynamics had been part of the high school or college core, but I would have been at a distinct disadvantage if, say, it had held the place it did in my high schooling, but my ignorance had dealt a fatal blow to my chances of attending a good university.

So here's a second place where Hacker is responding to a real problem: the way that successful incorporation of one's mathematical schooling becomes a filter for access to all kinds of opportunity.

This is again, a fact that every math educator has to grapple with. It is especially true in universities, because satisfactory passage of the first math course is the pre- or co-requisite to any other STEM coursework; but Hacker indicates how a student's SAT math subscore can be used to filter them out of more selective universities in the first place, even if they have no intention of pursuing STEM concentrations.

But the response to this (genuinely abhorrent to me as a mathematician) state of affairs is not to blame the particular mathematical content knowledge used to filter out candidates, or even to bitch that it is mathematics which is given the rump job of filtering in the first place; instead, my response is to point out that despite this filtration, most students outside of a mathematics major -- including those going off to teach math in elementary, middle, and high schools -- will attain their highest degree, with its concomitant math requirements met, without ever having encountered any mathematics.

Anyone who has spent any time talking about this with me knows that I can't respond to a proposal like Hacker's without bringing in Lockhart. Go read it, if you haven't. It's long, but worth your while. More than anything else, my reaction to Hacker's piece is that he's striking out at, but missing, the following fact about math education in today's academic environment: we spend our time teaching our students to answer questions they've never asked. In the specific case of algebra, we spend tremendous amounts of energy showing (or just telling) students how to solve polynomial equations and draw pictures of polynomial curves, but the typical student has never been in the position of asking a question whose answer requires such a procedure. In many cases, this is because the student has never found themself in the position of mathematically modeling some actual process they are interested in.

A month or two ago, Gowers wrote a pair of interesting posts about the math that the general public ought to be offered. His examples were heavily weighted towards the discrete side, and indeed Gowers had this to say:
But another (possibly related) reason I have not focused on real-world problems that require algebra and trigonometry is that I have found it very hard to come up with good examples. I can come up with examples of some kind, but not ones that are interesting and engaging.
Now, I think this is wrong, but it's wrong for a good reason. That reason is that algebra is the language through which we deal with basic physics, and more generally with the continuous processes that we encounter in everyday life. By contrast, the discrete processes we encounter daily give rise very quickly to exponential math and factorials. Gowers is a combinatorialist, and so he spends his days thinking first about discrete processes, and there's no denying that there's a wealth of good problems one can pose to school-age students there. But I think he's discounting the opportunities one has to pose good physics problems which require a student to model some process algebraically; and I find it very telling that students come to my calculus class and are stumped by the challenge to write a function modeling the growth of a colony of bacteria which reproduce asexually by mitosis every hour. This is in some sense a discrete situation (made into a continuous one by treating the bacteria as infinitely divisible). On the other hand, if you have a force field accelerating an object, and that force field is, say, constant, or linear, then modeling the position of an object moving freely under that force will get you a polynomial -- but that's not at all obvious to a student who hasn't been given some time to chew over the problem.

Hacker spends a lot of time arguing that teaching students algebra has poor ROI as job training.
[A] definitive analysis by the Georgetown Center on Education and the Workforce forecasts that in the decade ahead a mere 5 percent of entry-level workers will need to be proficient in algebra or above.
I can't say I'm either surprised or chagrined; what is more surprising is the implication that he'd be willing to put his own field up to the same standard, since if less than one student in twenty finds themselves needing random access to algebra, how few will need the content of the political science courses they took in sophomore year?

Instead, I submit to Hacker that, at its best, algebra provides the student with an experience, and often her first experience, of the toolbox of abstraction and modeling; and simultaneously the experience of an environment where it is possible to really determine whether a problem is well-posed or even sensical. Much of the criticism leveled against the social sciences is because of this sense that somehow, a sufficiently skilled rhetor could talk us all into accepting a non-solution of a non-problem, all because English allows such infinitely shaded nuance. If the student does not speak the language of precision, which in this sense allows us to verify the well-posedness of (some) problems and then allows us to verify that they are in fact solved (or show that they are not), I propose that the culprit may indeed be that their education may up until now have never featured an opportunity to investigate the difference. It would be a shame if the only arguments a young artist or political scientist knows how to make are vulnerable to the "well, that's just your opinion" critique.

[1] Full disclosure: Akin was an instructor of mine, and wrote me letters of recommendation for graduate study.

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