Monday, September 07, 2009

More laments about Math

Lockhart's Lament - The Sequel

A while back, I posted a link called Lockhart's lament-all about the dismal state of affairs in how math is taught in schools. The best way to teach math, according to mathematician Paul Lockhart, is by 'doing' and by that he means 'owning' one's mathematics-mathematics that has meaning for the learner. It's an unschooling approach-one that aims at drawing upon the interest and motivation of the child. A way that is not divorced from life!

Well it seems that this strange and radical idea sparked a controversy and you can read about it at

I thought Lockhart's response is worth quoting here:

I would like to begin by reminding readers that what I have written is a Lament, not a Proposal. I am not advocating any particular plan of action; I am merely describing the extremely sad and painful (and probably hopeless) state of affairs as I see it: mathematicians are not interested in teaching children, and teachers are not interested in doing mathematics.

If I am advocating anything, it is only the obvious (and time-tested) idea of "learning by doing." If I have a method, it is only to convey my love for my subject honestly, and to help inspire my students to engage in a delightful and fascinating adventure - to actually do mathematics, and to thereby gain an appreciation for the depth, subtlety, and yes, utility, of this quintessentially human activity. Is that really such a strange and radical idea? Have we really reached a point where one has to argue for teaching that "awakens and stimulates students' natural curiosity?" As opposed to what? I thought that was the definition of teaching!

I find it a bit frustrating that I am put in the position of having to defend such a simple and natural idea as having students engage in the actual practice of mathematics. Shouldn't it rather be the proponents of the current regime who should have to defend their bizarre system, and explain why they have chosen to eliminate from the classroom the actual ideas of the subject? You say I take a hedonistic approach to mathematics education? I call it a mathematical approach to mathematics education!

What I find so pathetic about our math education system is that it reduces a lively, creative, and messy human art form to a sterile set of notations and procedures, then attempts to train students to master them and become "technically skilled." Of course it fails even on its own terms because there is no coherent narrative - the teacher doesn't know where the natural logarithm came from, what its problem history is, what it means within the context of modern mathematics, only that it's on the test and the students need to "know" it. So the students cram some formulas into their heads for a day or two, pass a test, and promptly forget them. Of course most people can't retain dry, meaningless hieroglyphic information that they had no role in creating or contextualizing, so they get classified by the teacher (and by themselves) as "bad at math." (I worry that the most talented mathematician of our time may be a waitress in Tulsa, Oklahoma who considers herself bad at math.)

What are the goals of K-12 mathematics education?
One theme that seems to recur in discussions of my essay is this idea of training the 21st century workforce to be responsive to the needs of industry and to be "competitive in the global economy." I am no economist, but this seems to be more a matter concerning college and graduate level education, not the K-12 setting with which my essay is nominally concerned. Of course (as you may easily imagine) I have quite a bit to say about the disastrous state of affairs at the university level, but perhaps this deserves a separate discussion. (I have, however, received numerous emails from graduate students and researchers in mathematics and the physical sciences who feel that my essay hit the nail on the head for them as well.) So let's save the economic discussion for another time.

So the question is, what should be the goals of K-12 mathematics education? Or, to put it in somewhat more inflammatory terms, what whole categories of human experience do you want hidden from your child? Any other "enjoyable and challenging intellectual pursuits" you wish to prevent your youngster from engaging in? Painting and music certainly don't seem very practical, and neither does all this literature and poetry. Why should society expend resources to impart knowledge of any form of beauty? My god, there's so much unprofitable, non-industrial fluff our young economic units are being wastefully exposed to!

But seriously, are we really saying that introducing children to mathematics and helping them to develop a mathematical aesthetic is a bad thing? Inspiration, wonder and excitement can only lead to positive results. And it is especially valuable to have this kind of energy and enthusiasm when learning to master a new technical skill. Practicing a new scale is a lot easier when it occurs as part of an interesting, challenging, and beautiful piece of music.

Look. A child will have only one real teacher in her life: herself! I see my role as not to train, but to inspire and to expose my students to a wide range of ideas and possibilities; to open up new windows. It is up to each of us to be students - to have zeal and interest, to practice, and to set and reach our own personal artistic and scientific goals. Children already know how to learn: you play around and have fun and struggle and figure it out for yourself. Grownups don't need to hold infants up and move their legs for them to teach them to walk; kids walk when there is something interesting in the room that they want to get to. So a good teacher is someone who "puts interesting things in the room," so to speak.

No? Alright, fine. I propose a curriculum for reading which has students first learn all the words that begin with the letter 'A' and then proceeds through the alphabet. The course of study would be divided into 26 Units, and naturally one could not 'skip' to the advanced 'Q' class without having taken the 'P' prerequisite. (Reading actual books would come much, much later of course.) I wonder why we don't currently do this? Could it be because parents and teachers actually do read from time to time, so they know what matters and what does not? But the only source of information about what mathematics actually is comes from school itself: the 37th-generation photocopy of the same blinkered misconceptions, the perpetual feedback loop of School Math.

Suppose the devil were to offer you this deal: your child will get a perfect score on the English section of the SAT, but will never again read a book for pleasure. I would like to believe that no parent would make that deal. But how many would gladly shake the devil's other hand? Math is not something we want our children to enjoy, it is something we want them to get through.

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