I was surprised when Samantha David, the principal of the school that my four students attend, asked me to teach math concepts. In New York's inner-city, remedial instruction usually focuses on drilling and chanting to memorize basic facts and procedures. So why were these four getting the luxury of slow-paced concept-building? The answer was simple: they started the year so far behind grade-level that they will not make it to sixth grade next year anyway.
For most inner-city students, though, concept-oriented math instruction just doesn't seem to be worth the time. As Ms. David puts it, "there isn't an alternative universe that these kids will get to go to if they learn the conceptual stuff, and they won't learn the material as fast if we teach it that way." Ms. David recognizes the intellectual value of conceptual understanding, but expedience is more important: if you take too long getting kids up to grade level, you risk losing them altogether. (For a detailed discussion of how the circumstances of inner-city schools and the pressures of high-stakes testing force educators to teach procedures over concepts and critical thinking, see my December 23rd post.)
I find that pragmatic logic compelling, yet I believe that the neglect of conceptual understanding does damage that is difficult to measure. It's rare for even the weakest math students to get the kind of conceptual remediation that I give those four 5th graders—I'm not tooting my own horn or anything; it's just that what I'm doing with them is only possible with a tiny student-teacher ratio, and I only get that ratio because I'm working on a volunteer basis. Few schools, presumably, have access to experienced teachers willing to work without pay—or, to put a finer point upon it, few schools would deem it cost-effective to pay a qualified teacher for the slow work of rooting out elementary conceptual deficits. I have evidence, though, that such deficits don't disappear of their own accord: last year, I asked two ninth-graders at a No-Excuses charter in Central Harlem what number comes before 100. Both were stumped.
This is more than just a remedial math issue, though. I believe that the deficits displayed by my four students—and by those two ninth-graders in Central Harlem—are merely the starkest form of a type of deficit that is prevalent in the majority of grade-school students. True, the No-Excuses movement has made great strides towards closing the black-white achievement gap in math relying heavily on procedural instruction; but, though their success is laudable, it is not enough: if our education leaders up the conceptual rigor of our state exams, as they have promised to do (see my December 12th post), students who have memorized procedures to the exclusion of concepts will see their scores drop. Then too, closing the black-white achievement gap is one thing; closing the achievement gap between the US and the rest of the industrialized world is another.
Most importantly, though, I think that conceptual instruction, done right, could expedite procedural learning, rather than slow it down. See below.
Opposed to procedures-based math instruction is the open-ended, student-driven, constructivist approach favored by progressive educators and Scandinavian child psychologists. This method focuses on allowing students to explore mathematical problems and discover mathematical procedures on their own, rather than having them dished out ready-made. Its efficacy is supported by a large body of experimental data, but it usually requires low student-teacher ratios, which are difficult to achieve in public school classrooms, and as Ms. David points out, it produces slower results.
My own experience is that constructivist methods work best with strong, self-motivated math students. Weaker students and those who are more teacher-dependent seem to get lost in conceptual lessons and are rarely able to connect the logic and patterns that I show them back to the procedures that that logic is meant to explain. Frustrated by such failures, I have increasingly embraced more procedures-based methods—but always with a sense that I was deadening the subject and short-changing my pupils. It seems obvious that memorizing procedures in the absence of comprehension cannot be the quickest route to high math scores—but in a subject as abstract and endlessly interconnected as mathematics, how do you teach concepts without getting your students lost in them?
What initially struck me about the conceptual deficits I encountered in these four students was how elementary they were. Constructivist math seeks to teach kids the logic behind mathematical procedures—to show them not merely how to do long multiplication but why that procedure works—but what my students lack is a basic understanding of what numbers are: their order, their size, and how they're written. Those two categories of knowledge differ not only in complexity, but in kind: one deals with the inner workings of a mathematical tool (long multiplication), the other with the nature of the objects (numbers) that mathematical tools (procedures) operate upon. In this distinction, I have begun to see a new way of thinking about math concepts.
To put it in the broadest terms, I believe that we progressive math teachers (a category I am at least halfway in) have failed to distinguish between different types of math concepts. In our quest to purify math education, to make it truly mathematical and to shuffle off the dross of memorized procedures, we have tried to teach concepts that are not only too difficult for most students but which are irrelevant to effective problem solving. In a perfect world—one without time constraints or resource constraints, standardized exams or college admissions; in short, one without any set expectations—a purified math curriculum might enrich the mind of every student. By insisting on such purity, however, we have marginalized conceptual math and thus impoverished mainstream math education.
I want to propose a new way of categorizing math knowledge, one which will do away with the simplistic divide between "conceptual" and "procedural" knowledge. This more nuanced taxonomy will, I hope, make it easier to distinguish those categories of knowledge that are necessary to a cogent practical understanding of mathematical methods from those that are not. It being Wednesday night, and my having not yet posted anything this week, I will present this new categorization neeeeext week!
 ^ We tied two other nations for 25th out of the 34 nations in the OECD. That lousy performance appears to be persistent across various demographic subgroups of the population.
 ^ No-Excuses schools take a lot of heat for their high rates of grade retention. Critics argue that grade retention increases a child's chance of dropping out and inflates the school's test-scores by allowing extra years to prepare the weakest kids for a given round of exams. The first argument is specious, because there's no way to make causal claims about grade-retention and drop-out rates; yes, the two are highly correlated, but that's got a lot to do with the fact that kids who get retained are weak students are thus already at risk of dropping out before they get retained. Even if there is a causal relationship, it probably only effects unsupportive schools that make retained students feel like failures. The second argument is undeniable, but when a student arrives three or four years behind grade-level, taking an extra year to catch them up makes a hell of a lot of sense. Otherwise, such students will spend the rest of their academic lives playing catch-up, cramming for exams that they lack the fundamentals to really understand.
 ^ I don't have all this data in front of me. Some of it, indeed, has not yet been released by the schools, making it difficult to analyze. Others can be found on school websites. The Democracy Prep Public Schools website, for example, boasts that, in 2009, their 7th grade class "outscore[d] their counterparts in Scarsdale and all of Westchester County by almost 10%." Cherry-picked data is always a little suspect, but this is impressive nonetheless.