Wednesday, November 25, 2009

Lockhart's Lament

This article on K12 math education was written in 2002, but it finally found its way onto my class's Google group in the past couple days:

Before you dash off to read it, I should note that there are a couple big flaws with the argument. The main one is that he claims that math is useless and that this is a good thing. Math is not useless and it would not be a good thing if it were. Further arguments are below, but you can go read the article first.

--- Waiting. Go read the article. ---

Okay. You've read it? Here we go...


Lockhart's stance that math is art and thus useless (but capable of, you know, enlightening people...) does a disservice to both artists and mathematicians. Sure, art can be done for art's sake, but good art is done because people (even people who are not the artists themselves!) like it and it is thus useful.

Indeed, Lockhart's view of art reminds me a lot of List A pieces from the Royal Conservatory of Music. These are the classical, historic masterpieces that are critically acclaimed and all that [lack of] jazz, but are dead boring to a lot of modern students, such as myself. Music, art, and math do not exist in a vacuum. They are made better by being applied to the real world, to real situations, and to real problems.

Lockhart asserts that trigonometry is useless to most people's lives; this is clearly for lack of trying. Trigonometric functions provide a basis for the analysis of all periodic systems, from electrical circuits, to mechanical oscillators, many biological processes, and so forth. Can the beauty of the periodicity of a periodic function be appreciated without realizing this? Sure, maybe, for some people who would undoubtedly make fine mathematicians. There are, however, many cases where math is not developed for its beauty, but for its practical application. Take the Dirac delta function: a vertical spike at the origin of infinite height but with area one. Does that sound beautiful? (Okay, honestly, it does to me, but that's mainly because I've read ahead.) It's hard to imagine someone coming up with the impulse function for purely aesthetic reasons — as it goes against pretty much everything math has to say about functions — but it turns out to be extremely useful (I mean, man, you have know idea how important this one concept is, seriously, yow) for signal processing and the design of linear-time invariant systems (ie. not quite everything, but a large subset of everything).

The point that I want to make though is that the art metaphor is not flawed, but Lockhart annoyingly neglects the idea of pop art (stuff that people can actually relate to) for stuffy avant-garde postmodern cruft that can supposedly be admired for its intrinsic celestial beauty. Art is only art because it has context; math is only math because it is grounded in applications. To paraphrase Lady Gaga, pop culture will never be lowbrow.

Thanks for reading!

P.S. For those of you who have noticed that I drop a lot more Lady Gaga references now than I did before, it's because I need an excuse to link to this video which is awesome. I don't care what you think of dance pop, if you can play the piano with your foot that is damned impressive.


  1. I get the feeling that the article was written from the vantage point of a person that's outside of math... how did he hope to comment on something he's not bothered to understand?

    Thanks for sharing :P

  2. I believe the author (Paul Lockhart) is a teacher at Saint Ann's School in Brooklyn, NY. I thought he seemed reasonably well informed, but it's interesting that you don't feel the same.

    On the Google group where I first saw this, the reaction has been entirely positive; I don't think all the praise is justified, but if you tell a class of undergrad engineers that it's the system's fault that they're failing, they will love you for it.