“Eureka! Do you see why mathematicians jump out of bathtubs and run naked through the streets?”
This is Paul Lockhart’s comment after he has shown us the beautiful and shockingly simple explanation of why odd numbers always add up to a perfect square.
1+3+5 = 9
1+3+5+7+9 = 25
1+3+5+7+9+11 = 36
This is mathematics, he argues in A Mathematician’s Lament (2009), in all its elegance and wonder, not the mind-crushingly dull exercises and memorization of rules and vocabulary that pass for math curriculum in our schools.
If you want to know why the odd numbers add up to a perfect square, or if you know why, and are delighted by it, read this short, funny, and tremendously readable book—a math page-turner, believe it or not. It is subtitled “How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.” And indeed Lockhart does explain, in passionate and convincing prose, how little of the math we learn (and teach) in schools has any purpose or beauty to it, and how little we learn (and teach) of the intrinsic elegance and wonder of mathematics.
Among the various conventional wisdoms Lockhart skewers:
The “ladder myth.” We have become convinced that we must learn math in the sequence that we do, because if not, we won’t be able to understand the next topic in the sequence. Lockhart points out that children learn to walk because they see something across the room that they want, not because we exercise their legs. He advocates for putting real problems (like the one above, not exercises) in front of students and letting them creatively wonder their way through them, with help and guidance.
The “real world” rationale. Aside from basic arithmetic, Lockhart contends that very little of the math we learn in school (especially high school) is ever needed in daily life. Anyone used the quadratic equation lately? While Lockhart espouses the student-centered, creative teaching approaches we all aspire to, he is relentless in his insistence that the elegance and joy of theoretical math is an end in itself. He rejects the notion of “making math real” that is dear to the hearts of many project-based educators.
Math as preparation for engineering, science, and “international competitiveness.” Lockhart’s discussion of the reduction of math to its utilitarian and commercial purposes is perhaps the most poignant aspect of the book. For him, mathematics is an exhilarating and awe-inspiring practice. He believes humans (including all children) naturally want to do math for the same reasons we want to read great literature, or create art, or listen to music—for the sheer joy of it. He finds it sad and demeaning that math must justify itself through its purported “usefulness.”
Math as a set of rules and truths. Maybe so, Lockhart argues, but it is a set that has evolved and changed in a fascinating way. What other discipline is ever taught completely stripped of its history and context?
Lockhart reserves his most venomous attack for high school geometry, that “wolf in sheep’s clothing” that offers a glimpse of the beauty and elegance of mathematical proof, but in the end “stuns and paralyzes” its “student-victims” and bores them to death. Instead of allowing students to come up with their own creative and individual proofs, or ”poems of reason,” they are instead forced to memorize hundreds of theorems and trot them out in a rigid, formal, stultifying manner. Q.E.D.
In case we are not convinced—or some of us actually liked geometry—Lockhart shows an example of a formal proof (why is a triangle inside a semicircle always a right triangle?) that would be standard fare in high school geometry, using postulates and theorems. It is hardly readable. On the next page, voila! We see a seventh grader’s “poem of reason”—not perfect, but nevertheless an awe-inspiring explanation in words and pictures of why this is always true.
The back-story on Lockhart and his lamentation is almost as interesting as the book itself. He worked as a research mathematician and university math professor for many years before deciding to devote himself to K-12 math education. I first read a version of “Lament” several years ago when it was passed on as a 25-page paper to my mother, who passed it to me, from a Bank Street School math teacher. Apparently someone passed it along to Keith Devlin of the Mathematical Association of America, who published it in his on-line column. It was met with an outpouring of responses, both positive and negative, from around the world. A sample of those, with Lockhart’s response, makes interesting additional reading. It is reprinted in the journal AntiMatters, vol. 2, no. 4 (2008).
Apparently, this dialogue led to the publication of his recent book, a greatly expanded version of the original paper. In the book, Lockhart has not only improved his original “Lamentation,” but also added a part two: “Exultation.” I, for one, am very glad of it, because after reading the so very true evisceration of our current math curriculum, I needed a little exultation.
Lockhart conjures up an image of a math class that resembles a studio art class. Each student is working on a problem that holds intrinsic interest for him/her. They are trying various approaches, their creative wonderings encouraged and guided by helpful suggestions from their math teacher as she walks from easel to easel (well, maybe not easels). They understand the historical context in which humans first grappled with their unsolvable problem. They are introduced, at an early age, to the ideas and practices of proof, conjecture, and counterexample. Critique by peers is an important part of this picture as well. Is this possible? I’m not sure, and neither is Lockhart. It certainly would require a revolution in the education of math teachers, not to mention state curriculum standards, the textbook industry, and indeed our whole society’s concept of math.
But until that revolution comes, there is certainly much we can learn from Lockhart’s Lament.
Lockhart, Paul. (2009). A Mathematician’s Lament. New York: Bellevue Literary Press.