Wednesday, March 10, 2010

American Math Education? "Scrap it!"

A friend recently forwarded me this rather marvelous 2002 paper by mathematician/high school teacher Paul Lockhart about the sad state of math education in this country. His thesis: math, far from being the useful tool we were all constantly told it was, bears more of a resemblance to our conception of an art than it does to our idea of a science. So when we force kids to repetitively apply set-in-stone formulas to "solve" contrived non-problems, rather than give them real problems to creatively explore, we're choking one of humanity's great sources of fun and expression while teaching little more than a hatred of "math."

The mathematics curriculum doesn’t need to be reformed, it needs to be scrapped. All this fussing and primping about which “topics” should be taught in what order, or the use of this notation instead of that notation, or which make and model of calculator to use, for god’s sake— it’s like rearranging the deck chairs on the Titanic!


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.

In the midst of section headings like "High School Geometry: Instrument of the Devil" and examples like

A) pharmaceutical companies : doctors
B) record companies : disk jockeys
C) corporations : congressmen
D) all of the above

Lockhart proposes a harsh reevaluation of our national teaching style and a radical departure into techniques that most curricula reserve for art classes. Which is to say, replace terminology and an emphasis on usefulness with interesting unsolved problems and the encouragement of intelligent creative expression. Worried that our adults will be woefully ignorant of mathematics? We've already done that, Lockhart retorts. At least we could let the kids have some fun while they do it, particularly if they have a chance of learning more anyway.

Do I think he's right? Lord knows - I'm no math teacher. But I do certainly agree with this: "How sad that fifth-graders are taught to say “quadrilateral” instead of “four-sided shape,” but are never given a reason to use words like “conjecture,” and “counterexample.”" Just speaking for myself, I would probably have turned in more than the small fraction of math assignments I did in high school if they'd involved exploration or research rather than repetition of something I already understood. I'd even say that being forced to do a larger number of easy problems poorly prepared me for college, where math homework has fewer problems but does sometimes require a fair bit of thought and research.

I also deeply agree with one of Lockhart's premises about knowledge. As a culture, we tend to think of "knowledge" as set, known, impartable truths rather than as intellectual choices we've made. I wouldn't want to deny that there are facts about the world; but there are also a hell of a lot of facts. So many that our choice of the ones we investigate is almost always a more important feature of our knowledge than what the facts themselves happen to be. Teaching taste in facts - now there's an educational aspiration.

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?

A risk, maybe; but education already is.


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  8. Hey guys it's been a while. I'll take a look at the whole article today, but I figured I'd comment now since I am a math teacher.
    I agree that math curriculums need more problem-solving, but what Lockhart fails to talk about is that real problems cannot be solved/engaged with if the kids don't have the basic skills first. Otherwise you waste too much mental energy on the things that are supposed to be easy and can't focus on the difficulty or nuances of the task at hand. I agree that kids aren't really taught to think, but from my experience, very few of them have a mastery of the basic skills needed to free your mind to think about challenging problems. It's hard to think about the diagonals of shapes or the infinity of prime numbers if you don't know the names of the shapes or what a prime number is. It's hard to solve complicated problems without a mastery of the multiplication tables or the ability to add and multiply fractions without a calculator.
    As much as it sucks, drill exercises are needed to master the basics before you think about hard, real-world problems. As a sports analogy: it doesn't matter how complicated and sophisticated the plays are that your team runs, you can't score on anybody if you can't dribble. I guess my problem with the curriculum is that there is no attempt at true mastery, and they don't give you enough time as a teacher to explore real-world problems consistently. It has nothing to do with throwing out drill exercises; they are needed, too.

  9. I was hoping you'd comment on this.

    Yeah, it does seem like Lockhart's approach entails a catch 22: you need kids that are good at math to make kids good at math. A student already has to be a bit of a nerd to get pumped about something like figuring out areas of strange shapes for herself.

    As you'll probably see when you finish the paper, though, I think he's a little more sensitive to the need for drills than his polemical style (and my summary) makes him appear. He does talk about how the problem isn't the formulas we've come up with – they're great – so much as that we rarely teach anything BUT the formulas. Anyway, I'd love to hear more when you get through the whole thing.

  10. @6.54 I could not disagree more. I do not think kids need to be good at math to teach them. And, I don't think you need to get kids excited about using their imagination. Math just needs to be taught with that in mind.

    @LSouth I do not think Lockhart's message was about removing drills. To me it was more about questioning the use of drills if kids never get to play the game; whether the game is math or basketball. How do you get kids to do drills in basketball? They want to get better so eventually they get to play basketball. Right? even if they suck at it. But where is the math equivalent to that end game? When after all of your times tables and formulas and terminology do you get to play and have fun with math? In my experience it was never. I would learn math facts, the bell would ring and I would go onto the next class. Why can't math be fun like basketball? Couldn't you even use basketball to teach kids about math?

  11. Oh, I didn't mean to imply I thought that you can only teach math to math geniuses – just that it could be a solid critique of Lockhart's proposed approach, since creativity can be difficult and expecting kids to just jump right in and have success might be too extreme.

    Word to the second part of your comment.

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