In a recent article in the Atlantic entitled The Math Revolution, Peg Tyre discusses the growth of extracurricular math programs. More and more students, Tyre reports, are able to advance to levels far beyond what their school math classes are taking them.
An article like this one presents an opportunity to critique the shortcomings of school math classes; unfortunately, Tyre, once again, misses the mark. Notice, for example, the subtle bias in this paragraph:
Broadly speaking, there have been two opposing camps. On one side are those who favor conceptual knowledge—understanding how math relates to the world—over rote memorization and what they call “drill and kill.” (Some well-respected math-instruction gurus say that memorizing anything in math is counterproductive and stifles the love of learning.) On the other side are those who say memorization of multiplication tables and the like is necessary for efficient computation. They say teaching students the rules and procedures that govern math forms the bedrock of good instruction and sophisticated mathematical thinking. They bristle at the phrase drill and kill and prefer to call it simply “practice.”Apparently there are no well-respected math gurus worthy of a second parenthetical who say that "memorization of multiplication tables and the like is necessary for efficient computation"--let alone for more advanced learning and for making math problems less tedious. For more on this issue, see Barry Garelick's excellent response to Tyre in Education News.
The extracurricular math programs Tyre focuses on include the Russian School, which she places squarely in the Constructivist camp:
The new outside-of-school math programs like the Russian School vary in their curricula and teaching methods, but they have key elements in common. Perhaps the most salient is the emphasis on teaching students to think about math conceptually and then use that conceptual knowledge as a tool to predict, explore, and explain the world around them. There is a dearth of rote learning and not much time spent applying a list of memorized formulas. Computational speed is not a virtue.
The pedagogical strategy at the heart of the classes is loosely referred to as “problem solving,” a pedestrian term that undersells just how different this approach to math can be. The problem-solving approach has long been a staple of math education in the countries of the former Soviet Union and at elite colleges such as MIT and Cal Tech. It works like this: Instructors present small clusters of students, usually grouped by ability, with a small number of open-ended, multifaceted situations that can be solved by using different approaches.It's all there, supposedly: "conceptual thinking," "problem solving," open-ended problems, multiple solutions, exploration, real-world application, and group activities. The only deviation from hard-core Constructivism is that the groups are "usually" homogenous in terms of ability.
Of course, there is plenty of conceptual thinking and problem solving in any good math curriculum, Russian Math included. But, here in America, the terms "conceptual thinking" and "problem solving," pedestrian though they may be, are by now so bastardized by Reform Math that anyone writing about them needs to be absolutely clear about what she is talking about. When it comes to non-American math (and pre-1970s American math), "conceptual thinking" and "problem solving" have very different meanings from those assumed by the mainstream American edworld. Tyre's article, failing to acknowledge this, will only further entrench current practices: particularly, the emphasis on group work; problems with high ratios of verbiage and "real-world" application to actual mathematical challenge; and requirements that students solve problems in multiple ways and explain their answers verbally and pictographically.
That is not what Russian Math is all about.
Tyre does cite one problem that she presents as representative of Russian Math and its ilk: a problem from the nascent math-and-science site Expii.com:
Imagine a rope that runs completely around the Earth’s equator, flat against the ground (assume the Earth is a perfect sphere, without any mountains or valleys). You cut the rope and tie in another piece of rope that is 710 inches long, or just under 60 feet. That increases the total length of the rope by a bit more than the length of a bus, or the height of a 5-story building. Now imagine that the rope is lifted at all points simultaneously, so that it floats above the Earth at the same height all along its length. What is the largest thing that could fit underneath the rope? The options given are bacteria, a ladybug, a dog, Einstein, a giraffe, or a space shuttle. The instructor then coaches all the students as they reason their way through. Unlike most math classes, where teachers struggle to impart knowledge to students—who must passively absorb it and then regurgitate it on a test—problem-solving classes demand that the pupils execute the cognitive bench press: investigating, conjecturing, predicting, analyzing, and finally verifying their own mathematical strategy. The point is not to accurately execute algorithms, although there is, of course, a right answer (Einstein, in the problem above). Truly thinking the problem through—creatively applying what you know about math and puzzling out possible solutions—is more important. Sitting in a regular ninth-grade algebra class versus observing a middle-school problem-solving class is like watching kids get lectured on the basics of musical notation versus hearing them sing an aria from Tosca.But this open-ended and verbiage-filled problem is not at all representative of the kinds of problems students encounter in Russian Math and the other high-level extracurricular math programs that Tyre is ostensibly writing about. Here's a link to some sample problems from the Russian School website. And here is one of the sample Russian Math problems given for 7-8th grade:
Solve: (x-4)(x-5)(x-6)(x-7) = 1680It is this kind of problem, not the Expii.com problem, that distinguishes Russian Math from contemporary U.S. math.
And one of the biggest reasons why U.S. math is underserving American kids (of all levels of talent) is because it has too many group-centered discovery problems of the open-ended, verbiage-intensive variety, and not enough problems of the sort that are truly representative of Russian math.