Lockhart begins with a terrifying metaphor -- what if classes in music were taught as exercises in rote manipulation of musical notation, or art classes were classes in Paint-by-Numbers? Yet this is the state of affairs in school mathematics education. Sometimes it astounds me that our schools manage to produce anybody who's actually any good at mathematics. (The best thing a lot of my teachers did was to just get out of my way.)
Lockhart writes (p. 3):
The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.... Part of the problem is that nobody has the faintest idea what it is that mathematicians do.
I will not explain what it is that mathematicians do -- I'm preaching to the choir here, I suspect -- but I agree with this. One of the things I find myself doing, when people find out I'm a mathematician, is to try to briefly explain that mathematics is not about numbers and formulas but about abstract ideas and patterns, and that yes, it is possible to create new mathematics! (People seem to think that mathematics is complete.) I hope this missionary zeal doesn't go away -- but it very well might as I age. A lot of older mathematicians I know have said that they used to do this sort of thing but now don't bother any more.
Later (p. 6):
Many a graduate student has come to grief when they discover, after a decade of being told they were “good at math,” that in fact they have no real mathematical talent and are just very good at following directions. Math is not about following directions, it’s about making new directions.
This reminds me of something that I overheard a friend of mine saying a few days ago. He was talking to one of the young faculty in our department, trying to get some insight about how the process of creating mathematics works. And he said that he didn't know what real mathematics was -- just graduate school mathematics! And this is a man who will quite likely be receiving a PhD in mathematics in three short years. (Let it be said that I am not saying I was not in the same situation at that point in my own graduate career.)
On sources (p. 9):
What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?
Indeed, friends of mine in the humanities are surprised to learn that we don't routinely learn from the writings of the masters. I've heard various explanations for this. First is the claim that one often doesn't want to see the original proof of some important theorem because it's a mess -- and yes, that might be true. So we write various explanations of it, and include those first in our research papers, sanding off a few rough edges here and there. Second is the fact that a lot of mathematicians just don't write that well, because the culture doesn't reward that.
Lockhart claims, in fact, that "there is no actual mathematics being done in our mathematics classes" -- I don't want to believe this, but he'd know better than I would. He's most annoyed at the fact that when school mathematics does teach the idea of "proof" -- which for traditional reasons is done in geometry classes -- the proofs that students produce are so different from real proofs as to be unrecognizable. For those of you who don't know, students in American schools are subjected to something called a "two-column proof" in which a sequence of statements is made in one column, and justifications for those statements is made in an adjacent column. I can imagine how this would be useful for very carefully checking if a proof is correct. But anybody who was actually reading such a proof in an attempt to learn something would probably attempt to translate it back to natural language first! I suspect that the real reason such "proofs" are so common in such courses is because they're much easier to grade than a proof actually written out in sentences and paragraphs. (This seems true of a lot of counterintuitive aspects of our educational system.) Lockhart advocates replacing the standard course with courses in which students actually struggle with trying to come up with their own mathematics. It sounds like it's working for him.
But Lockhart teaches at what appears to be a very good private school that has considerable latitude in choosing its students and gives considerable autonomy to its teachers. Would something like this work within the "mainstream" educational system? I don't know. I'd like to find out.