Some time ago while stumbling around online I came across a rather interesting essay about the state of mathematical education. Entitled A Mathematician’s Lament, it was written back in 2002 by Paul Lockhart who argues that there is next to no real math being taught at the K-12 level. His essay calls for a radical reform of the educational system to change the way the public perceives mathematics. Years later in 2009 Lockhart expanded the original 25 page essay into a 140 page book which is currently for sale. I find Lockhart’s essay to be thought provoking and though I agree with him on a few of his overall points, I take issue with a decent portion of his argument. In what’s to become a three part series I will discuss Lament and include my own objections to Lockhart’s argument.
A Mathematician’s Lament opens with the hypothetical world wherein which music is taught in schools much the way Lockhart envisions math to be. In this world before a student can ever sing, play an instrument, compose an original piece, or even listen to music, they must first complete numerous courses in music theory and notation. Topics such as listening or composing are considered too advanced for students and are put off until college where only the few students who study music ever get to apply what they’ve learned up until that point. Much like music, art class also suffers a similar fate in Lockhart’s mythical world. Instead of painting on blank canvas or sculpting works from clay, students simply learn the proper way to hold brushes and identify colors. Should they ever want to actually put paint on a canvas they have to enroll in the college prep course Paint-by-Numbers. Even the teachers in this world have never done anything as abstract or expressive as painting freely or writing any original music themselves; they’ve gone through college merely learning how to teach what they learned back in K-12 to new K-12 students. Everyone runs around wondering why kids aren’t showing real interest in these subjects and complain how boring they are. As an introduction to the rest of his essay, I think this portion of Lockhart’s essay serves its purpose and is pretty decent overall. It hints at the most of the major points he’s going to cover in later portions of his argument and sets the tone for the rest of the piece rather well.
The first section of Lament discusses Lockhart’s opinions on how mathematics is viewed in our culture and why it needs to change. Our society, he says, typically sees math as a subject shrouded in mystery and mathematicians as the ones who actually know what all the stuff people learned in high school is good for. Nobody understands what math actually is or how to go about doing any real math. As an engineer who oftentimes has trouble making people understand what I do as an applications engineer in the semiconductor industry, I sympathize with Lockhart here. Witnessing the misconceptions people have about your profession isn’t exactly fun and I’d love for the general populace to better understand what it is STEM professionals actually do. Lockhart’s answer to the public persona of math is treat mathematics as an art instead of science. In his opinion mathematics is art because it is not bounded by physical laws but rather governed by a unifying principle that simple is beautiful. Mathematicians deal with both the patterns and the imaginary and that is why they should be called artists in Lockhart’s eyes. They can be equated to artists in that as they both practice their respective crafts they can develop a distinctive style and taste.
While classifying math as an art form is all well and good, I suppose I have to ask what about the students who currently are bored with music and art class? Those who wonder when any of what they do in those classes will be used in the real world? Simply calling math an art instead of a science isn’t going to stop students from wondering when they’ll ever use it and why it’s important. Plenty of kids (myself included to be honest) simply coast through art and music in high school to fulfill a requirement for graduation and never develop a real appreciation for the subject matter. Why would math class be any different if it was considered an art? The rest of this essay doesn’t seem address this issue in too much detail. From my reading of A Mathematician’s Lament, Lockhart’s views on math as art can be summed up as “no one is learning anything the way things current are so why not try something new.” Perhaps making math a free form art class where students can creatively solve math problems will at least give them some appreciation for the subject. But more on that to come. (Side note: It’s possible that Lockhart’s book does discuss this point further. I don’t know as I haven’t read it yet.)
I would argue against calling math an art in Lockhart’s sense because it can’t affect people on a primal level the way in which music and art are able to do. Everyone can look at a painting or hear a piece of music and know almost immediately if they like it or not without any prior exposure to the subject matter. Music, film, painting, and sculpture can move people and play to their emotions. They’re just as much experiences as they are intellectual exercises. While throughout history plenty of people have been very passionate about mathematics, they’ve always been in the minority and in my opinion just because you’ve made something your passion doesn’t make it an art form to the world at large.
Lockhart also claims that there is currently too much rote memorization in mathematics education and having students being presented with lists of facts and formulas to memorize and regurgitate is detrimental to their enjoyment of math. He calls for more creative problems solving in the class room and for learning the why behind concepts as opposed to just the concepts themselves. On this high-level idea I completely agree with Lockhart. In his opinion, the best way to learn math is by doing what he considers to be real math and not standard textbook problems. From my own experiences as a student I definitely think that all STEM classes, not just math, could be improved by doing away with simply memorizing facts and theories just because someone said they’re important. Letting students get their hands dirty and apply what they learned instead of just grinding out drill problems is a huge step in the right direction towards making STEM fields more enticing to young people.
My main issue with A Mathematician’s Lament is only touched upon briefly in the first section so I will just quickly discuss it here and talk more about it in the upcoming parts of the series. What gets my blood boiling is his position that teaching the applications of math is essentially worthless and only purely theoretical math courses should be taught. In his eyes the public “are apparently under the gross misconception that mathematics is somehow useful to society.” No practitioner of mathematics has ever attempted to solve a down to earth problem using math, their goals are always more lofty than that. The fact that there just happens to be applications of their work is inconsequential. As someone who loves seeing the real world connection to theories I’ve learned I disagree with this notion whole heartedly. While reading Lament my mind immediately jumped to Netwon and Laplace. Both were excellent mathematicians who invented brand new fields of mathematics while trying to explain the motion of the planets. Clearly the application there can and should be easily ignored. If Lockhart had his way people wouldn’t be excited by seeing what a solid understanding of mathematics can do for them be it in science, engineering, accounting, whatever. They would instead be in awe over purely imaginary triangles with perfect sides that cannot exist in the real world because everything is just simply atoms in motion and thus is always changing shape. Excuse me while I wash the horrible taste out of my mouth with the biggest bottle of mouthwash I can find.