My primary and secondary school teachers are very lucky that Paul Lockhart didn’t write his Mathematician’s Lament when I was still in school. I can only imagine the hell I would have raised with his article in-hand. Lockhart is a mathematician at Saint Ann’s School, where he teaches K-12. Recently, a blistering critique of mathematics education he wrote in 2002 was re-published. I would say that much of his criticism could be applied to a varying degree to ALL classes, but math may indeed be the most egregious offender. (I really hope that in 100 years we look at our current educational system the way we look back at the practice of medicine from 100+ years ago.) Below are my favorite passages; believe it or not, I tried to parse them, but there’s a lot there. I’d suggest reading the original, as the whole thing is good. For example, I have ommitted the part where Lockhart theorizes what art class would look like if taught like math, as well as the type of mathematicians the current system creates (Lockhart’s not a fan).
Lockhart believes mathematical ideas to be “beautiful little poems of thought … sonnets of pure reason.”
This is why it is so heartbreaking to see what is being done to mathematics in school. This rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be memorized and procedures to be followed.
In place of a simple and natural question about shapes, and a creative and rewarding process of invention and discovery, students are treated to this:
“Triangle Area Formula: A = 1/2 b h h b. The area of a triangle is equal to one-half its base times its height.”
Students are asked to memorize this formula and then “apply” it over and over in the “exercises.” Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time— there is nothing left for the student to do.
Now let me be clear about what I’m objecting to. It’s not about formulas, or memorizing interesting facts. That’s fine in context, and has its place just as learning a vocabulary does— it helps you to create richer, more nuanced works of art.
But it’s not the fact that triangles take up half their box that matters. What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you.
By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this— at least it’s understood that there is an art of sculpture that I am being prevented from appreciating).
By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant.
Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.
And I haven’t even mentioned the lack of mathematical criticism in school. At no time are students let in on the secret that mathematics, like any literature, is created by human beings for their own amusement; that works of mathematics are subject to critical appraisal; that one can have and develop mathematical taste. A piece of mathematics is like a poem, and we can ask if it satisfies our aesthetic criteria: Is this argument sound? Does it make sense? Is it simple and elegant? Does it get me closer to the heart of the matter? Of course there’s no criticism going on in school— there’s no art being done to criticize!
Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them.
It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.
A similar problem occurs when teachers or textbooks succumb to “cutesyness.” This is where, in an attempt to combat so-called “math anxiety” (one of the panoply of diseases which are actually caused by school), math is made to seem “friendly.” To help your students memorize formulas for the area and circumference of a circle, for example, you might invent this whole story about “Mr. C,” who drives around “Mrs. A” and tells her how nice his “two pies are” (C = 2πr) and how her “pies are square” (A = πr2) or some such nonsense.
But what about the real story? The one about mankind’s struggle with the problem of measuring curves; about Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which is more interesting— measuring the rough dimensions of a circular piece of graph paper, using a formula that someone handed you without explanation (and made you memorize and practice over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles for god’s sake!
What other subject is routinely taught without any mention of its history, philosophy, thematic development, aesthetic criteria, and current status? What other subject shuns its primary sources— beautiful works of art by some of the most creative minds in history— in favor of third-rate textbook bastardizations?
The main problem with school mathematics is that there are no problems. Oh, I know what passes for problems in math classes, these insipid “exercises.” “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee.
But a problem, a genuine honest-to-goodness natural human question— that’s another thing. How long is the diagonal of a cube? Do prime numbers keep going on forever? Is infinity a number? How many ways can I symmetrically tile a surface? The history of mathematics is the history of mankind’s engagement with questions like these, not the mindless regurgitation of formulas and algorithms (together with contrived exercises designed to make use of them). A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions.
A triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way? I can understand the idea of training students to master certain techniques— I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process— that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.
Now I’m not saying that math teachers need to be professional mathematicians— far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it? If teaching is reduced to mere data transmission, if there is no sharing of excitement and wonder, if teachers themselves are passive recipients of information and not creators of new ideas, what hope is there for their students? If adding fractions is to the teacher an arbitrary set of rules, and not the outcome of a creative process and the result of aesthetic choices and desires, then of course it will feel that way to the poor students. Teaching is not about information. It’s about having an honest intellectual relationship with your students.
Also, as I said before, just because a subject happens to have some mundane practical use does not mean that we have to make that use the focus of our teaching and learning. It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way— to force third graders to fill out purchase orders and tax forms— it wouldn’t work!
We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.
SIMPLICIO: Then what should we do with young children in math class?
SALVIATI: Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
SIMPLICIO: It seems like we’d be taking an awful risk. What if we de-emphasize arithmetic so much that our students end up not being able to add and subtract?
SALVIATI: I think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate them on standardized tests—“Preparing tomorrow’s workforce today!”
SIMPLICIO: But surely there is some body of mathematical facts of which an educated person should be cognizant.
SALVIATI: Yes, the most important of which is that mathematics is an art form done by human beings for pleasure! Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises.
Real mathematics doesn’t come in a can— there is no such thing as an Algebra II idea. Problems lead you to where they take you. Art is not a race. The ladder myth is a false image of the subject, and a teacher’s own path through the standard curriculum reinforces this myth and prevents him or her from seeing mathematics as an organic whole. As a result, we have a math curriculum with no historical perspective or thematic coherence, a fragmented collection of assorted topics and techniques, united only by the ease in which they can be reduced to step-bystep procedures.
SIMPLICIO: So we’re supposed to just set off on some free-form mathematical excursion, and the students will learn whatever they happen to learn?
SALVIATI: Precisely. Problems will lead to other problems, technique will be developed as it becomes necessary, and new topics will arise naturally. And if some issue never happens to come up in thirteen years of schooling, how interesting or important could it be?