Read a Mathematician’s Lament
I recently had the pleasure of stumbling across Paul Lockhart's essay, A Mathematician's Lament. Lockhart, a former research mathematician in analytic number theory who received his Ph.D. from Columbia in 1990, decided to leave academia in 2000 in order to concentrate on K-12 math education, which he hass been doing at Saint Ann's School in Brooklyn.
Lockhart's article lambasts the current state of mathematics education in this country. Some of his main points are the following:
- Mathematics is an art form, but unlike other art forms like music or painting, is not understood as such by the general population. As a result, students are not exposed to the beauty of mathematics, and are instead taught through drill and memorization, which effectively kills any natural curiosity the student may have.
- The most important part of mathematics lies not in the facts or theorems that students memorize, but in the arguments that show why these facts must be true. By stripping away the beauty and elegance that lies behind many of these arguments, students don't develop an appreciation for (or a real ability to do) mathematics.
- The only class that does emphasize proof (high school geometry) sterilizes the process so much that all the beauty is drained from the arguments.
- Math education spends too much time trying to force artificial connections to the real world, rather than exposing the natural beauty that lies within mathematics. Most word problems don't actually reflect any type of problem that one would find in the real world.
There's much more, of course, but the article itself does a much better job of expanding on these points than I could. Lockhart takes an extreme position, to be sure, but in so doing he exposes much of what is horribly broken with our current system.
More than anything else I've posted, I recommend you read the article and percolate on it. Lockhart originally wrote this around 2002, but it wasn't published until last year - since then it's made the rounds in academic circles, I'm sure, but I hadn't heard of it until it was posted on Slashdot earlier this summer. This is all well and good, but for most people with technical backgrounds, Lockhart is preaching to the choir. Since this blog caters to a more general audience, I would particularly encourage those who don't work in the sciences to read through what Lockhart says - much of it will resonate with you, especially if you hated math as a student.
Lockhart certainly offers plenty for debate. Here are some questions I have after reading the article:
- Lockhart has no love for the endless drilling that goes on in current math classes (the type of drilling that continues all the way up through calculus). But to what extent are drills a necessary evil? If you want to become a concert pianist, you'd better practice your scales. Nobody will argue that drills are particularly taxing, but they do have their purpose in other arts - shouldn't they in mathematics as well?
- Many are quick to point out the one major problem with comparing mathematics to other art forms: mathematics has wide applicability to other fields, whereas other art forms do not. Lockhart argues that even though this is the case, the essence of mathematics isn't its practical consequences. This may reflect his own personal bias (after all, he was a researcher in analytic number theory), and while it's a bias I share to a certain extent, I doubt that this is a universal belief among mathematicians in general.
- I often find that students feed into the current system of teaching the facts rather than the ideas, because the facts are easier to check on standardized tests. Most students want to know a technique for solving a problem, and couldn't care less about why the technique works, where it came from, or most importantly, its limitations. In essence, I see a tremendous lack of curiosity. Much of this seems to stem from a desire to get a good grade (which may lead to a good job), rather than wanting to learn for learning's sake. However, this is a problem that goes beyond mathematics - to what extent are the problems Lockhart address indicative of broader problems in education?
Give it a read - if nothing else, it will give you something to think about.
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