“The first thing to understand is that mathematics is an art,” writes Paul Lockhart at the beginning of A Mathematician’s Lament, his brief but powerful book on the failures of contemporary mathematics education. Elsewhere in the book he enriches this definition, describing math as poetic, radical, subversive, psychedelic, and “nothing less than the distilled essence of who we are.” But as a society this is not how we understand math, and in our schools this is certainly not how we teach it. Instead, math is presented as something boring but necessary, a subject one learns for practical purposes, not for fun. Lockhart despises this conception of math, and in this book he attempts to discredit it.
A Mathematician’s Lament (which could just as aptly be titled A Mathematician’s Manifesto) is divided into two parts: “Lamentation” and “Exultation.” In the first part, Lockhart complains about the way math education is mishandled in our schools and attacks teachers and institutions for their damaging inadequacy. He expresses sympathy for students, claiming that they are right to consider math class stupid and boring. “If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making,” he writes, “I couldn’t possibly do as good a job as is currently being done.” In language that is acerbic and provocative, he then proceeds to tear down nearly every part of the math establishment.
In the second part, Lockhart tries to counterbalance his complaints and criticisms with a more positive, constructive attitude. He describes why he thinks math is so wonderful and offers an alternative way for one to experience it. Not content to discuss math in general terms, he presents specific problems and illustrates how one might go about solving them. The solutions, however, are not the most important things, for Lockhart does not believe that the value of mathematics lies in the “truths” it establishes. He is interested in why things are the way they are and in how we can come to understand them. This process of discovery involves much more than the manipulation of numbers and symbols; there is an essential aesthetic value to it as well: a mathematical proof should be more than a series of facts leading to an answer—it should be a thing of beauty.
“It will be said that the joy of mental adventure must be rare, that there are few who can appreciate it, and that ordinary education can take no account of so aristocratic a good,” wrote Bertrand Russell in Why Men Fight. “I do not believe this. The joy of mental adventure is far commoner in the young than in grown men and women. Among children it is very common, and grows naturally out of the period of make-believe and fancy. It is rare in later life because everything is done to kill it during education.”
This “joy of mental adventure” is the very thing that Lockhart wants to encourage and protect. Like Russell, he believes that our educational institutions are destroying the imaginative and creative impulses of our children. Nowhere is this more evident, according to Lockhart, than in our math classes. And who is to blame for this? Lockhart accuses nearly everyone involved, but he is especially hard on his fellow teachers. Because he sees math as an art, he thinks that it should be taught by artists, or at least by people who appreciate its artistic nature. But in our current system, math is forced upon students by people who have never produced original work, are ignorant of the history and philosophy of their subject, and are not up to date on recent developments. The problem is not that these people are teaching math poorly, it is that they are not teaching math at all. Instead, they are mindlessly implementing a clunky curriculum, or what Lockhart describes as a “confused heap of disinformation.”
A Mathematician’s Lament is a short book, but it should be even shorter. The first part is succinct, insightful, impassioned, thought-provoking, and occasionally eloquent; the second part is somewhat excessive. It is clear that Lockhart cares deeply about math, that he enjoys sharing his passion with others, and that he wants to save students from the stale and ill-conceived curriculum to which they are subjected. It is admirable that Lockhart has devoted himself to the cause of improving math education, but in his eagerness to rebel against the establishment he has sacrificed certain subtleties. His definition of math is too narrow, and his dismissal of the school system is at times rather indiscriminate. (“It’s no shock to learn that math is ruined in school,” he writes in one particularly embarrassing sentence, “everything is ruined in school!”) His overuse of exclamation points becomes wearisome, and there are moments when he seems inspired by a conspiracy theorist’s enthusiasm. He also undermines his own argument regarding the essential nature of math: at first he tries to establish it as a fine art, but then later in the book he wants us to believe that it is a childish activity that we do simply in order to have a good time. Surely there is more to art than that.
Despite these flaws, A Mathematician’s Lament remains a compelling and inspiring book. Much of Lockhart’s prose possesses an aphoristic quality that makes it especially memorable—and quotable. It is clear that Lockhart is a devoted and passionate teacher who genuinely cares about his students. Although he is vehemently critical of the present state of math education, he is not driven by his negativity, but rather by the overwhelming love for his subject.
Unfortunately, it is improbable that much progress will be made in the argument between Lockhart and his adversaries, for the two sides have chosen to define mathematics in entirely different ways. The disagreement is not only about conclusions, but about premises as well. Before any solutions can be found, the right problems must be identified. Lockhart has clearly identified what he thinks the problems are, but his definition of math is a peculiar, unconventional one, and therefore his criticisms of teachers, administrators, and textbook publishers are likely to strike them as irrelevant. (Ford Madox Ford’s remark comes to mind: “It is an easy job to say that an elephant, however good, is not a good warthog.”) But perhaps his criticisms will cause them to reflect on their assumptions and to honestly consider if certain changes are necessary. The students deserve this much, at least.
I will end by saying what might have been better said at the beginning of this essay: There is almost no one to whom I would not recommend this book. It is not only about the importance of mathematics, but about the importance of ideas in general, and it should be of interest to anyone who values imagination, creativity, and intellectual curiosity.
A. M. Kaempf is the founding editor of The Northwest Review of Books. His work has also appeared in The Threepenny Review, The Los Angeles Review of Books, The Millions, and Full Stop.