In an effort to Continuously Increase my Effectiveness, I have spent quite a bit of time this summer reading various books and articles about teaching. I recognize that I am by no means an expert teacher; part of the reason I wanted to move to a charter school was that I knew that they would help me develop in my profession in a meaningful way. I can’t help close the achievement gap if I just stay as effective as I am now. As a teacher, there’s always more you can do, and so here I am, spending my summer working. (And I love it, by the way!)
I was sent A Mathematician’s Lament through a TFA-Bay Area listserv that I am a member of. Since I’m only going to be teaching math to second graders next year, I figured it was highly relevant to me. What I read humbled me.
The author, Paul Lockhart, starts with a hypothetical situation about a musician trapped in a world without music. Children are being taught to write music in sheet form, but they are never allowed to hear music or taught to play it. The beauty and art has been sucked out of music. This, Lockhart claims, is exactly what has happened to math in our public schools.
Sadly, our present system of mathematics education is precisely this kind of nightmare. 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— I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.
Everyone knows that something is wrong. The politicians say, “we need higher standards.” The schools say, “we need more money and equipment.” Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, “math class is stupid and boring,” and they are right.
Well, I certainly agree that most math classes are boring. In fact, I’m pretty sure I slept through much of my high school math education. I remember often feeling like my teachers were kidding me when they tried to convince me that “I would need to know this later.” Yeah, right. I’m pretty sure that I use the quadratic equation every single day- thank goodness I spent all that time learning it!
To be fair, though, I can’t remember much of my lower elementary math education. I remember doing multiplication tables in third grade and hating every second of it. Is it possible I enjoyed first and second grade math?
Lockhart continues by explaining to us that the reason nobody sees math as an art is because nobody understands what mathematicians do. He claims, “[M]athematicians sit around making patterns of ideas. [...] If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”
O…k. So, I’m supposed to teach my students about imaginary things? Right. It’s hard enough to get some of them to stop daydreaming about fairies and princesses as it is. But, maybe I’m missing some key piece of information about math. It’s not, after all, my favorite subject. I’ll give him the benefit of the doubt for now and accept that math is about imaginary things.
Lockhart takes us through a discussion of how to derive the formula for discovering the area of a triangle: A=1/2BH. (Yep, I remember that from my geometry days.) He has chopped a triangle inscribed in a rectangle in half, thereby discovering that the triangle fills exactly half of the rectangle. He says, ” But it’s not the factthat 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.”
Ok, fair enough, I see his point. He’s arguing for more of a self-discovery process of math- of appreciating its beauty and using your own natural curiosity to learn about the mathematical world. It’s similar to making a painting- children should be given artistic freedom to create something beautiful on their own. I can appreciate this point of view.
But, here’s my problem with it. When I place myself back in my younger self’s shoes, sitting in a desk in a classroom in front of a math teacher, I remember that I never once cared what the answer to a math problem was. I would do the work and find the answer, but I never thought that it was fun or beautiful. Maybe this is a result of the fact that I was simply handed all the formulas I ever needed and told to use them. There was no self-discovery process for me. Is it possible that if I had had a teacher who showed me a triangle inscribed in a rectangle and asked me, “How much of the box does the triangle take up?” that I would have found that interesting? I’d have to say probably not. I (and I’d be willing to bet, most of my classmates) would have responded, “Who cares?” I simply fail to see the beauty in this problem.
Lockhart uses the rest of his article to give a scathing report on how horrible everyone involved with mathematics education is, from the top of the government right down to the classroom teacher. Nobody gets math, he claims, so of course we are failing to teach it to our kids. The very establishment designed to impart math knowledge has, in essence, killed it. “There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it.”
Now there is a statement I can get on board with! I may not have ever seen the fun or beauty in math, but I have always loved reading. I think this statement applies to all subject areas- we have become so focused on the results of some meaningless test that we just force our students to test prep all day long. In the end, what have we taught them? How to fill in bubbles? Awesome. I’m sure that will help them in life.
Anyway, Lockhart does go on to suggest improvements to math instruction.
So how do we teach our students to do mathematics? By choosing engaging and natural
problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.
Certainly, that’s a great ideal to strive towards. But I think it sort of makes the assumption that, as the math teacher, you see math as beautiful and artistic. Since the education establishment is so horrible, how many teachers out there really see it this way? Everyone in my life who actually does see math this way absolutely did not go into teaching. They went into engineering. Who is left to do the teaching? People who were brought up by and buy into the very system that Lockhart condemns.
So, Mr. Lockhart, I will do my best to see math as an art from now on, to push my students to reach their own mathematical conclusions, and to discover the beauty of it on their own. But I don’t think that any of us should be surprised if I’m not successful in every single unit. Tell me, where is the beauty in subtraction with borrowing? I can see the beauty in 3-D shapes or in multiplication. But subtraction? It’s pretty ugly, if you ask me.
It would be awesome if I could get my students to see math in this way. How much fun would they have, and how much more would they actually learn? I am sure the possibilities are limitless. But if there’s one thing I’ve learned during the past two years, it’s that you have to believe what you are teaching. If I lacked the artistic mathematical instruction from my education, how am I going to bring that to my students?
Frankly, articles like these make me feel hopeless. If seeing math as art is truly the way to go (though I’m not convinced it is) then how are we supposed to get there? The problem is so systemic that unless all those mathematicians who do see math that way come out of their high-paying jobs to teach, the problem will persist.
There must be another way.