Last year, Professor Steven Strogatz of Cornell University wrote a series of op-eds for the New York Times that discussed the presence of mathematics in unlikely places. I discussed one of these columns here.  Now, either those articles were well-received, or Professor Strogatz is well-connected, because this year he’s back in the Times with a much more ambitious series of articles.  This time around, Strogatz is attempting to “[write] about the elements of mathematics, from preschool to grad school, for anyone out there who’d like to have a second chance at the subject.”

Preschool to grad school is a significant amount of ground to cover, but thus far Strogatz has used his articles to assault this goal with gusto.  To date, he has tackled counting, patterns in addition, negative numbers, division, and basic high school algebra.  This doesn’t really do justice to his content, though.  Along the way he gives the reader some Sesame Street, and discusses a number of tangential topics, including the inability of Verizon employees to do math, the half-your-age-plus-seven rule, and pre-WWI European history.  The latter comes about in a discussion of that old adage which is familiar to anyone who saw the first Alien vs. Predator movie: the enemy of my enemy is my friend.

Predators must be awesome at math.

While some of Professor Strogatz’s explanations are a bit hand wavy (in particular, his explanation of why (-1) x (-1) = 1 is a lacking), on the whole they are quite good.  In particular, he offers a nice explanation of what it is for a mathematical argument to be “elegant.”  But even more impressive than his writing is its location – to have a discussion of mathematics with as wide an audience as the New York Times readership is commendable.  Even if people are not inspired to learn more mathematics after reading these pieces, hopefully they will have at least learned something.  As with exercise, a little mathematics is better than no mathematics at all.

Moreover, these articles highlight aspects of math not usually seen in popular discourse.  Much like Paul Lockhart’s A Mathematician’s Lament (which Strogatz references), these snack-size essays are focused on simple mathematical ideas, and the beautiful (and sometimes unexpected) results that follow.  Nowhere here does Professor Strogatz multiply two really big numbers together; in fact, he’s quite sympathetic to the fact that for many people, there is nothing more tedious than calculation.  By leading the conversation in this way, he’s hopefully able to give a taste of what makes math beautiful to an audience for whom such a statement might otherwise be labeled heresy.

I don’t know where this series of articles is headed, but I look forward to finding out, and hope you do to.  Professor Strogatz’s articles are grouped together here.

(Hat tip to dad for sending me a few of these articles.)

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.

• 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.