The Unapologetic Mathematician

I’ve posted my notes from Zuckerman’s second lecture. Again, there’s a lot to unpack here, so it goes after the jump.
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From over at Mathematics Weblog (a paragon of naming wit), I find a few videos on YouTube dealing with math education. The first one shows me why Washington State has to cripple its graduation requirement exams. Math Education: An Inconvenient Truth illustrates some of the “reform math” texts for fourth and fifth graders, and shows the incredibly intricate — yet somehow called simpler — algorithms for basic place-value multiplication and division. The New Math overshot students’ capacity to learn abstract concepts, but these methods undercut their ability to learn at all, and at every step they come back to their all-powerful god: calculator.

I think it’s well summed up in this quote from the teachers’ reference for one such program:

The authors of Everyday Mathematics do not believe it is worth students’ time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole-number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator.

Okay, here’s why this is a complete load: the extensibility of the algorithm. The standard long division algorithm leads to polynomial division, and even power series division without any modification at all! If a student has mastered place-value long division, they are prepared for these later necessary algorithms. And students today simply are not. In fact, I just demonstrated power series division to my second-semester calculus classes the other day, and I’m pretty sure that I smelled some of the students wetting themselves when I said, “It just goes like long division.” These are ivy-league college students reduced to gibbering terror over a basic method that I remember being taught in elementary school less than twenty years ago. I can only imagine what it’s like at other universities.

Fine, I’ll grant you that most people will never have to long-divide even a polynomial in their “everyday” lives. But without this ability, and the comfort with mathematical algorithms it acts as a marker of, students will never achieve in mathematics, science, or engineering. What the authors are really saying in the above quote is, “we don’t think most of your kids are ever going to do much of anything in science or engineering, so we’re going to cripple all of them — even those who might otherwise have done.”

And here’s another point: you don’t always have a calculator. To indulge a bit of hyperbole, there’s a reason that a naval submarine carries paper copies of its own schematics rather than a CD-ROM of them. Jim Lovell didn’t have a calculator when things went wrong on Apollo 13. Pencils and paper never get shorted out, never run out of batteries, never have hardware glitches. In a mission-critical situation, having the ability to do without a calculator or computer is essential. Yes, most people will never be slingshotted around the dark side of the moon in a tin can, but the authors of these curricula are saying they have no problem with preventing that being your child on the first manned spaceflight to Mars.

The second video, Math Education: A University View, shows the ultimate effect of these “reform” math curricula on the quality of incoming undergraduates, particularly at the University of Washington. It also illustrates the general stylistic trends of such curricula, like the emphasis on students “discovering” mathematical facts on their own rather than being instructed by a teacher.

I knot that what I’m about to say is horribly politically incorrect (”counter-revolutionary”?), but the most basic mathematical methods must be taught — they cannot be discovered without mastery of the basic algorithms. I know I’m more interested in “why” than in “how” in higher mathematics, but this is simply not how people lay the foundations of familiarity with mathematics. It’s just not how human beings learn to think mathematically.

You want to know how people learn mathematics? There’s actually a movie about it, though you might not have recognized it as such at the time: The Karate Kid. Mr. Miyagi starts by making Daniel wax his car, paint his fence, sand his deck, and so on. Daniel doesn’t understand why he’s supposed to go through these motions, but Mr. Miyagi knows that the understanding will come only after mastery of the basic techniques, not before.

I have not met anyone who understands why the place-value algorithm for multiplication works without having learned how to do it first. After the student is forced to master the formal manipulations, authoritarian as it might seem, the underlying reasons may be taught or discovered. Besides, there’s simply too much depth to any of these manipulations for any student to understand why before how. I only just recently found out (via Michi’s blog) that place-value addition “really” arises from group cohomology. Did my not knowing this impede my doing addition problems? Of course not. And neither did my not understanding the ring-theoretical underpinnings of place-value algorithms before I learned them, nor my not understanding the set-theoretical underpinnings of natural number arithmetic before I learned them.

The algorithm comes first, and understanding comes later. Mathematics simply is. It cannot be negotiated. Mathematics education as realized in the NCTM standards has been taken over by sociologists, or even Critical Theorists. They are vehemently opposed to the seemingly-authoritarian rote method and saying “just do it like this and don’t ask why”. Never mind the fact that in this case “why” comes naturally after “how”. And it’s about time for mathematicians to come down and start kicking some ass over this, or we’ll be left with nobody capable of replacing us.