The other day, my 11-year-old daughter left a note on the scratch paper she was using to do her math homework. It read (caps hers):
I HATE THE MATH IN THIS UNIT! IT MAKES ME CRY, AND IT DEPRESSES ME. THIS MATH TORTORES [sic] ME. — JThis note depressed me. But quite frankly, it did not surprise me. The math program in our public school—a Canadianized version of the reform math so reviled in the US—continually frustrates and confuses my twin daughters, both of whom are A students in math. Both of them have declared, on many, many occasions during the past three or four years of struggling through this program, that they hate math. This really depresses me, because my husband and I have gone to great lengths to instill in our girls a love of math, a sense that it can be interesting and fun and challenging, and that, contrary to the message they may be receiving from the culture in general, it is something about which girls and boys should be equally enthusiastic.
Before I go any further, let me state a few facts about myself. Yes, I dislike reform math or "fuzzy" math or constructivist math, or whatever you want to call it. But . . . I am not an educational conservative, a back to basics advocate, or a nostalgic drill-and-kill enthusiast. On the contrary, I am a firm believer in progressive, child-friendly public schooling for all. I feel I have to say this because the "math wars" have been so politicized, both in the US and here in Canada (where in true Canadian style, the "war" was more of a minor skirmish followed by complete capitulation), that anyone who opposes the current math curriculum is branded as educationally retrograde. I think in order for an intellectually honest and productive discussion of math education to occur, this politicization and presumptive name-calling has to stop.
So why do I object to constructivist math? One reason is that it is, by-design, non-incremental or "spiral": its textbooks jump around from topic to topic, never staying on a subject long enough to allow for deep understanding or competence. I also dislike reform math because it frowns upon direct instruction. Since constructivist math teachers believe children can "construct" or "discover" mathematical truths and come up with their own algorithms to solve problems, they offer students minimal guidance, and are not averse to putting the cart before the horse: e.g., assigning algebra-type problems before teaching the tools of algebra, or asking kids to divide or multiply by decimals or fractions without having first taught them how decimals and fractions work.
All of this—the bouncing around from topic to topic, the "challenging" problems, the lack of direct teaching—constructivists defend in the name of what they call "conceptual" learning, which they oppose to both abstract instruction and their favourite straw man, "drill-and-kill" work. But there are two problems with this normative use of the term "conceptual." First of all, "conceptual" and "abstract" constitute a false binary opposition: a concept can be abstract, and an abstraction is not necessarily unconceptual. Take the standard algorithm for long division. Because this method of performing division—like all mathematical algorithms—can be separated from concrete or specific division problems, it is deemed to be abstract. Proponents of constructivist math argue that presenting it upfront would be tantamount to teaching division in a manner that does not allow kids to understand the concept behind it or why and how it works. But a mathematician (and it's interesting to me that most of the authors of constructivist math textbooks are not mathematicians) might counter that the algorithm embodies the concept—otherwise it would not work. So, let's say a teacher were to demonstrate the standard algorithm for long division at the outset of a lesson; he or she could, conceivably, set aside class time for practice and mastery, and then—with student participation—pick apart the algorithm to find out how and why it works. Would this be less conceptual than making kids stumble through division problems on their own, hoping they will discover an efficient algorithm, which most of them will never do?
Secondly, even if the terms conceptual and abstract were in fact polar opposites, why would we favour one over the other? There are some kids who love working in groups or with concrete materials (methods favoured by constructivists) but there are others, like both my daughters, who simply enjoy playing with symbols on a page, and who find all the illustrations, and colourful doodads in their current textbook patronizing and distracting. Why do we assume that math instruction must be a one size-fits-all proposition?
But my real opposition to the privileging of the conceptual in constructivist math is that it is misleading and even hypocritical: in my experience, constructivist textbooks do not encourage conceptual understanding at all. Indeed, my main problem with reform math is that it does not promote mathematical understanding, full stop.
The note from my daughter with which I started this post, in which she expresses her ongoing frustration with math, was sparked by a revealing instance of the true non-conceptual nature her constructivist math text. The problems my daughters were working on for their homework that night involved perimeter and area. In certain questions, they had to compare perimeters given in different metric units. To do that, they had to convert, for instance, metres to centimetres or vice versa in order to figure out which of two given perimeters was bigger. My daughters had no problem with this, but then they were confronted with a problem in which they had to compare the areas of two rectangles—one measuring 8400 centimetres squared and the other measuring .84 metres squared—and, again, indicate which was bigger. Their first instinct was simply to multiply .84 by 100 in order to carry out the comparison. This was my first instinct as well, but something (a residual spark of mathematical reasoning?) told me that in the case of area, it didn't quite work this way. Confused, I flipped back a page or two to see if any explanation of this type of problem had been given. I found no explanation, but I did find, in a coloured bubble in the margin of the previous page, these instructions:
When you convert an area in metres squared to centimtres squared, each dimension is multiplied by 100. So, the area is multiplied by 100 x 100, or 10,000.So there it was: a formula! No verbal or visual exposition, just an easily-missed bubble telling the kids what to do. You can't get any less "conceptual" than that. My daughters read the instructions and understood them, but they wanted to know why the formula worked. I asked them if the teacher had explained it, and they said he had not. I tried, unsuccessfully, to explain it. I then enlisted the help of my computer-scientist husband. He drew diagrams, and took my daughters, step-by-step, through the hows and whys of the formula given by the textbook; in doing so he was able to teach the girls how to carry out conversions from any metric unit squared to another—which the textbook formula, restricted as it was to conversions from metres squared to centimetres squared, was unable to do.
My point here is neither to ridicule my daughters' math textbook nor to blame the school for choosing it; it is, after all, one of a handful of textbooks approved and financially supported by the provincial government. My purpose, rather, is to demonstrate that this so-called constructivist, "conceptual" textbook is neither. It's just poorly-presented, pedagogically dubious, bad math. Which is why I concur with my daughter: THIS MATH DEPRESSES ME.