Math as Art

It is taken for granted by most parents that their children must learn mathematics to have a complete education and grow into competent adults. Moreover, outside of small circles, what comprises an education in mathematics has long been settled and formed into a rigid curriculum. We all know already what math is. And thus, when Paul Lockhart^[1] wrote A Mathematician’s Lament, he was well aware that he was up against a deep and strong prejudice. His lamentation springs precisely from this recognition, but it also yields the possibility of an exultation.

Lockhart does not use the philosophical language of “the good.” However, it is worth noting that as a culture we have elevated the learning of math as an essential good, as is evidenced by our emphasis on it as a prerequisite to high school graduation and a benchmark for suitable college applicants. At the same time most of us probably wondered why we had to learn what was often, if not mostly, mystifying material. Not only was its practical value questionable, but what was being taught was unclear. Many are familiar with the feeling that high school math classes were a tiring exercise in the memorization of formulae and of learning occult signs, something akin to witchcraft insofar as one must have memorized the correct spell to apply to the correct problem to, magically, produce the solution, which itself brought no enlightenment. A veteran schoolteacher, Lockhart knows this dismal scene well, and he brilliantly and entertainingly restores a sense of the interest and goodness of mathematics.

Lockhart begins by sketching two nightmarish scenarios in which a musician and a painter find that their respective arts have been mandated as essential school subjects and committees of educators (void of any working musicians or painters) have set out a course of study meant to generate competent students in music and painting. Music is reduced to musical notation, now conceived as the language without which a child could never express himself musically. A rigorous study of notation and music theory becomes prerequisite even to listening to music, let alone composing it. Meanwhile, in the painter’s nightmare, learning to paint has been reduced to learning colors, matching colors, color theory, and paintbrush composition; only at the highest level are students allowed to paint-by-number. Meanwhile, the students in these classes—never allowed real contact with the arts they study or their interesting history—appear bored, disengaged, uninspired, motivated primarily by the value these classes are given by college admission boards. Far from inspiring students, these classes have turned music and painting into drudgery, requiring effort hardly worthy of human beings. In Lockhart’s telling, both artists wake from their dreams relieved to know that their art forms have not actually been so degraded. The mathematician is less lucky. He writes: “In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern making, I couldn’t possibly do as good a job as is currently being done” in standard mathematics curricula.

This starting point does more than just make for entertaining writing; it reveals something of the good that is at stake here. Mathematics is an art, despite the implicit lesson in most math courses that math is a tedious slog of purely utilitarian value. However, unlike most art forms, the mathematician makes patterns with ideas. Mathematics relies entirely on the uniquely human capacity for abstract thought. A mathematician begins not with particularities, but with ideal and universal ideas. For example, a triangle conceived of by a mathematician will be an ideal and perfect triangle, which could never exist in material according to its idealized conception. The triangle might be perfectly symmetrical, with perfectly equal sides and angles, on a perfectly flat plane, or on a sphere, or on a wave. What is of interest to the mathematician is that once he has described the conditions of this ideal triangle, logical truths will follow. This creature of the triangle will behave according to its nature. The mathematician is the one who explores these patterns. He can imagine any pattern, and then can interrogate this pattern, but not with the instruments of science, but through further efforts of his imagination, through thought and logic. 

Throughout the text Lockhart gives several examples of what “doing math” looks like, in contrast to what teaching math has become. I offer two for your pleasure:

In the latter part of his book, Lockhart guides his reader through some mathematical problems. He points out a pattern that has to do with the behavior of odd numbers: when we add the first two odd numbers, 1 + 3, we get 4, which happens to be a square number. (A number that can be illustrated as a square ::). If to this result we add the next odd number, 5, we get another square number, 9. This seems to continue: 1+3+5+7+9+11 = 36, or 6 x 6, another square. The mathematician is the one who asks, and wants to prove, will this pattern hold for every odd number, that is, infinitely? Or does it fail at 15,968,352,147? Unlike in the scientific method, where we would test this pattern many times and then conclude that it will hold in all cases if it does in most, the mathematician searches for a golden image or law that reveals the truth of the observed pattern. 

Math for math’s sake is hard, satisfying, and deeply human work. Lockhart also calls it a useless, childish and compelling activity. That is, if it isn’t play.

Lockhart is eager to point out why he delights in this question. Here is pure math, a question of the logical truth of a pattern created by odd numbers. The idea of number is an abstraction from any physical thing, and yet these abstractions obey rules and behave in ways that can be logically described. And yet there is no seemingly practical application for this pattern.

Lockhart referred to his students as “apprentices.” Having laid out this pattern, he would have let them explore it and attempt to find if they could claim with certainty that this pattern would hold. When I first read his book, I did just that. My children played about me, the dinner waited to be cooked, and I pondered the nature of square numbers. Suddenly, light illuminated darkness, and I knew that, yes, this would continue infinitely. I have yet to forget the delight of that moment. The image that dawned in my mind I painted for this article (I refer you to the image accompanying this review). 

I then read on. Concerning mathematical proofs, Lockhart writes “we are trying to craft a ‘poem of reason’ that explains fully and clearly and satisfies the pickiest demands of logic, while at the same time giving us goosebumps.” A mathematical proof must have airtight logic and it must be elegant, simple and clear. To suddenly see that a square number is a collection “of nested L-shapes, and the L-shapes contain precisely the odd numbers” feels like a moment of revelation. To suddenly see the validity of the pattern, the literal shape of the proof, offers one simple good: delight to the mind in an idea that is more than the mind. 

The second example I will share is actually the first example Lockhart offers the reader in his book. It has to do with a triangle inside of a rectangle. The triangle shares one side with the rectangle, and its third angle touches the opposite side of the rectangle (again, I refer you to the illustration accompanying this review).

After conceiving of this shape, the mathematician might wonder how much of the area of the rectangle the triangle occupies. Do you know? Is it obvious to you? Precisely here is the work of the mathematician, in using his imagination and thought to attempt to solve this question. If you were a student of Lockhart, he would set you loose at this moment to see what you could come up with. Most teachers of math never even show students this picture; they simply offer the formula for determining the area of a triangle. Sadly, the premature solution destroys any chance for the problem taking root in the imagination as a real question. In other words, the chance to become a person who can work with a pattern, play with it and delight in it is never allowed the space to develop in most students. Along with that stolen opportunity, the implicit lesson taught is that formula memorization and application are what matter, not the creative engagement of one’s thought with an idea.

Anyone who wishes to reclaim some of the dignity stolen from them in high school geometry should stop here and find a solution to this. We might not find the best or most elegant solution, but it would be our solution, and we would have also welded ourselves to the world in this creative act, becoming more ourselves in the doing. And then, when we see Lockhart’s solution, we would actually be in dialogue with him. It will be a delight. The thrill of it is worth the time, the earned insight into what it means to be a human thinker is invaluable.

If you look at the triangle in the bottom right-hand corner above, you will see where Lockhart’s solution begins: he divides the rectangle into two smaller rectangles at precisely the point where the top of the triangle meets the rectangle. With this simple move, we immediately see a new relationship: each side of the triangle occupies exactly half of each new rectangle. The area of a rectangle is one side multiplied by the perpendicular side. Thus, the area of the triangle will be half of the area of the rectangle. This discovery can be expressed in mathematical notation: Area of Triangle = 1/2(h x b), where h is the height and b is the base. Such notation is good and useful, allowing for clear and concise expression of this pattern that we have thought through with spatial ideas. But it is more satisfying and meaningful after one has wrestled with this question and seen the beauty of this added line. When we skip over this, we lose the inspiration inherent in math, and we reduce it to the mechanical application of formulas and algorithms. Students are asked to be calculators, not mathematicians. The mathematician is one who encounters problems within this set of thought creations and brings his creativity to bear upon them. This is precisely what is not being done in math curricula.

In other words, the good of mathematics does not exist as mathematical information, but as a human activity. And this good can only be known when engaged in through mathematical acts. This will not primarily be applied math. Rather it will be the kind of work illustrated above, math for math’s sake. This is hard, satisfying, and deeply human work. Lockhart also calls it a useless, childish and compelling activity. That is, if it isn’t play.

The immediate implication of Lockhart’s understanding of mathematics is this: real math requires real teachers. Lockhart writes in a thrilling and terrifying passage:

Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children. 

Lockhart takes his students seriously in the way that only great teachers do. Instead of seeing them as empty receptacles to be filled with the collected data generated by mathematicians throughout history, he sees them as intellectual creatures able to engage in the real problems of math. These problems are not the same as the exercises assigned as homework. Rather a real problem is a “natural human question…. How long is the diagonal of a cube? Do prime numbers go on forever? Is infinity a number?” A real teacher adventures with his students into these questions. Mathematical technique arises and makes itself useful along the way.

Such an approach to teaching makes mathematical critique an intrinsic process of teaching. Just as an art teacher will critique a student’s work, introducing techniques that could improve it, a good mathematician will critique his students’ proofs, helping them to become more clear in their arguments, more elegant in their work, more skilled in the techniques that help them to discover more about the patterns they are engaging. And mathematical talent will make itself known in those students that seem to have a native inspiration and ability to see relationships.

However, it is precisely here that the weight of Lockhart’s critique bears down upon us. Teaching, like all truly human activity, cannot be a mechanical process. Textbooks are sold on the idea that they can deliver real knowledge to students almost without the involvement of a teacher. However, this relationship between a teacher and student cannot be automated or captured in a book. The teacher’s task is to communicate to the student the good of any subject. This cannot be done without engaging the good of the subject on its own terms. In the teacher the student has contact with a humanity that has become enlivened by the good that is that subject. Lockhart’s advice to math teachers is to do real math. If the questions and problems of mathematics mean nothing to the math teacher, the students can only hope to accidentally encounter these deeply human and interesting questions. What is left is to make a utilitarian argument for the pursuit of math. We have all learned this by rote. And yet it hasn’t produced adept students of math.

Lockhart drives home his critique by examining the typical and universal mathematics curriculum that every reader no doubt knows. In place of offering students a real experience of mathematics, a rigid, unvarying system of arbitrarily arranged mathematical ideas is forced upon students irrespective of their own interests or questions. The math curriculum is wholly instrumentalized, a ladder that leads to mathematical competence that will unlock future possibilities. However, Lockhart is certain that the ladder leads nowhere, least of all to “a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in mathematics.” A real mathematics classroom would be focused on problems, with students and teachers involved in the difficult and painful process of “having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work.” Only then would the specific techniques, methods, and nomenclature of mathematics organically arise, as it did historically, and show its value. Reduced to its techniques, methods, and nomenclature, mathematics will be suffered by teachers and students alike as an absurdity.

In his critique, Lockhart well conveys the world of mathematics as he knows it. Many readers will recognize in his description their own weary experience and even feel a sense of vindication, if not tragedy. If he convinces us that we were robbed of a real mathematical education, he is equally persuasive that the pursuit of math could be wonderous. And in the final chapter of this short book, he offers just that. In his “Exultation” he tries to share his love of math by observing patterns in numbers and setting mathematical problems before the reader. 

Lockhart went on to write three more books on mathematics: Measurement, Arithmetic, and Mending the Broken Bones: A Modern Guide to Classical Algebra. These are great, delightful and challenging reads, but they are a far cry from the classroom Lockhart describes. A book can simply never recreate what could take place in a mathematics classroom with “real” teachers.

It wouldn’t be surprising if Lockhart’s lament triggers a personal lament in his readers for the waste of their high school math classes. But it also affirms that the experience of boredom was actually the knowledge that was being offered did not correspond to our humanity. Lockhart reminds us that all education at its heart is a relationship to real things that requires us to become “real” people. In his own work as a teacher, which this book can only echo, he definitively answers the question: “Why do we have to do mathematics?” Because math is beautiful and a good in itself. That is to say, it is worthy of human love and effort. And those who learn it and love it will become more human for their effort—as will their teachers.