If the question is not put front and center, we enter a rhetorical world where schools are well supplied with qualified emissaries of a more coherent view of mathematics, and the only thing holding them back is devotion to outdated educational orthodoxy. Lockhart seems to exist in that world— unaware that he might not be representative of the norm, and unwilling to make the obvious inference about the practical value of his lamentations.

My understanding of Lockhart’s essay is that it was originally intended mainly for reading by the mathematical or educational communities (however you want to define that) and that the more recent attention it has gotten grew out of various postings on the web (e.g. in Keith Devlin’s MAA column), then the book, and, by now, everywhere (e.g. its recent appearance in Steven Strogatz’s NYT blog). The shift in audience, from professional to public, makes me uncomfortable.

I do not think the public is able to distinguish an earnest gripe by someone who has strong subject knowledge (as Lockhart does) from empty demagoguery from someone with no subject knowledge. It is ironic: Lockhart’s rhetoric of lamentation has a long history of use by politicians moonlighting as educators (or vice versa) in support of pet projects— modifications of standards, curricula, textbook lists, you name it— that made education in America exactly what Lockhart finds so troubling.

Pick your least favorite attempt to “change mathematics education in America”. Among its proponents, do you not find someone offering the lamentation narrative? This is what’s wrong with education, this is how it should be, and isn’t it tragic that the ignorant and hidebound just don’t get it? The hallmark of this style is that no attention is paid to legitimate differences of opinion, interests and constraints that genuinely compete with one another and cannot be reconciled, or indeed anything that does not fit into the lamentation narrative.

Reading comments on various web forums, Lockhart’s essay seems to have gone over as a sort of Rohrschach test for what’s wrong with education. Anybody can find something they agree with. Teaching means openness and honesty and loving learning, teaching means talking to students and listening to them… who could disagree with that?

This is unfortunate. Reforming educational practice is necessarily controversial. Even assuming good faith on all sides (so rare in policy disputes), reasonable people will disagree about what should be done and compromises will have to be made. It is great when math PhDs take the time to participate in these kinds of discussions, but they have much more to offer than this.

To me, Lockhart’s lament is shop talk among professionals that should have stayed that way. It is as if someone found a rant around the office cooler so compelling that they printed it in a book and sent it to all employees of a company. Maybe it doesn’t harm anything, but it doesn’t help, either.

The “laboratory mathematics” style of teaching is almost never done well, and done poorly students grasp the point and the beauty of mathematics even less than they do in traditional classes.

He’s well-meaning, but mistaken.

There are two difficult question. First, how much maths should everyone have to know? I would probably draw the line well *below* what is taught as standard in most schools. Second, how should we teach it them? I agree with Lockhart to an extent that giving them more of an idea about what the math they’re learning is *for* would help, but I’m not sure I agree with his methods for doing this (they are excellent methods for training mathematicians, but most people are not going to be mathematicians).

If you’re going to teach elementary calculus to everyone, why not do it in a physics, or an economics course? Then you at least get to see that the “scales” that you’re practising can be put together to make some sort of tune.

(Disclaimer, I’m still not entirely sure what I think about this issue, and I’ve been thinking about it (at least) since I first read Lockhart’s essay a good few years ago – I’m sure I’ll sort my thoughts out in the end, and discussing it here has possibly helped me to do so).

Formal math education exists to teach people enough of the subject so they can use it when they need it.

Need it? That may be true for Arithmetic, but certainly not for the subjects deceptively taught under the names of “Algebra”, “Geometry” and “Calculus”? Most folks never need to solve a quadratic equation outside of school, let alone calculate derivatives and the like.

Math is important precisely because it is useful. The fact that it is often times beautiful is just gravy. But like all things that are useful, learning and doing math is a hard, arduous task. I think there should be more emphasis on actual problem solving and discovery (like Lockhart suggests students discover the rule that the area of a triangle is 1/2*bh), but if we require each generation to discover all things ALL READY known how much intellectual progress will be made? I hear people generally denigrate drill and memorization without really thinking the matter through. I know the definition of a limit. You know why? I memorized it. You know how? I had it drilled into me.

Lockhart disparages the idea of mimicry when solving math problems saying something like students are taught certain proofs, then taught to prove other things mimicking these proofs (I don’t remember exactly what he says) as if this is a bad thing. My immediate thought was: ALMOST ALL KNOWLEDGE IS GAINED THROUGH MIMICRY. Disparaging mimicry is disparaging the human experience.

In summary, Lockhart correctly diagnosis the idiocy of math education and makes a nice analogy through music. But then goes on to recommend an even greater idiocy by totally mischaracterizing what math is, then recommending a totally feel good, no accountability education style, which will leave us all worse off.

Similarly, almost none of the English literature, history, economics or biology we teach them will be useful: mathematics unfortunately has the baggage that a lot of what we *could* teach in schools as maths happens to be useful to a few specialists, whereas the other arts are (almost consciously) useless.

There may be a strong argument that 10+ years of compulsory education is just too much because it’s impossible to teach people useful stuff for 10 years. I can’t see any strong argument that if we *are* going to teach people useless stuff for 10+ years (because we think it makes them better people? because it keeps them off the streets?), we might as well teach them some useless maths, and make it interesting on the way.