Thursday, October 28, 2010


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):
This 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.

(See also THIS MATH DEPRESSES ME—Update and A Grade 7 Math Question)


  1. I agree completely. These reform math curricula sound great in theory, but our experience is that in practice they're generating math anxiety more than anything else.

    One of the frustrations we've had with our school's program, Everyday Math, is that its expectations often seem very unclear. In our house, at least, that seems to breed a strong feeling of insecurity and frustration around the subject of math, which I'm afraid is going to outweigh any benefit that might come from the program's attempts to take a more conceptual approach to the subject.

    Everyday Math is a "spiral" program, too, and it seems like it's often just trying to introduce certain concepts to the kids that will be revisited and expanded upon later, and that it's not really expecting the kids to master those new concepts, at least not right away. I can see some logic in that, but at the same time, if that's true, I wish someone would tell the kids that it works that way. It's only natural for them to think that they're supposed to be able to do whatever it is they're being asked to do, and to feel as if they have failed if they can't do it.

    Who knows, maybe it somehow raises math test scores. What difference does it make, if the kids end up hating math? That seems to embody the entire approach of schools today: raise scores at any cost.

  2. Chris, I think you're right that programs like Everyday Math don't expect kids fully to understand certain concepts the first time around. But I think proponents of these types of programs fail to take into consideration the frustration a child feels when he or she does not fully understand something. And how this frustration contributes to math anxiety.

    As for tests scores, I don't think these new reform curricula have actually raised math scores since they were introduced over a decade ago. On international tests--as opposed to state or provincial tests that may well be testing only how well kids are absorbing the new curriculum--the US and Canada don't perform particularly well. Year after year, they are outperformed by countries that don't use a constructivist math curriculum--countries such as Singapore and China. I'm not saying higher international test scores should be the goal of math instruction, but I also don't think test scores can be used to defend the current curriculum. I also find it interesting that National Council of Teachers of Mathematics (NCTM) has recently backed down on some of the more controversial reformist proposals set forth in their (in)famous 1989 report.

    In any case, the debate about math instruction, both in the elementary grades and in high school, continues, as this interesting article in Scientific American attests:

  3. northTOMom, this is fabulous! Can I repost it to Kid-Friendly Schools? Thanks!

  4. Thanks FedUpMom. Yes, you can re-post it to Kid-Friendly-Schools.

  5. We are fighting the same reform math programs here in the US. See our blog:

  6. Your blog is a great source of information on this issue. Thanks.

  7. I'm sorry to hear that the constructivist/reform math craze has spread to Canada. In the US, we've been fighting a losting battle with school boards/districts to have the various programs removed (Everyday Math, Investigations in Number, Data and Space; Connected Math Program; those are the chief culprits). But the publishers are good marketers and the politics is too much for the average community. On occasion they get tossed out, but even when bad texts are NOT used, the method of teaching math is to do it in groups, collaboratively, as the teachers are taught to do in ed schools.

    I have written about it extensively (see for an article that describes the state of US K-12 math education and how it got that way), and have decided to teach math when I retire in about five months. I've gone through the ed school hoops but there are still more hoops they make you jump through. Ridiculous hoops, and then they complain about a shortage of qualified math teachers. But that's another topic for another time.

  8. Thanks for your comment. Your article is very informative and insightful. I too was schooled in the old "new math" and it actually suited me quite well. In a comment on Max Ray's Math Forum blog (, I wrote a bit about how "reform" math seems to favor one type of math learner, to the near exclusion of all others. This is part of what I said:

    "I . . . was a kid who only began to enjoy math when it became more abstract, from about 8th grade on. Sure, I liked real-world problems, but the fun of math problems for me was in reducing them to formulas and numbers on the page. I think reform curricula does not encourage that kind of purely symbolic enjoyment of math—in fact it actively discourages it in the early grades. One of my daughters showed an interest in Algebra, so we taught her the basics, and she enjoys doing simple Algebra problems a lot more than anything she does in math at school. They are like puzzles to her, puzzles she can solve with the tools we’ve given her. Does she fully understand what she’s doing? I don’t know, but I don’t think doing this little bit of Algebra is hurting her mathematical understanding. (To keep up her interest in math, my husband is also planning to introduce her to very basic computer programming.)"

    I think what I was trying to point out in my comment was that "discovery" methods—group work, concrete materials, etc.—are not the only ways to interest kids in math.

    I also found your comments about the older math textbooks versus the new ones interesting. When my husband changed careers as an adult, and went back to university to get a degree in computer science, he needed a way to brush up on his math. He used the math textbook from my final year of high school. It was written by mathematicians, and covered every thing he needed to bring himself up to speed, including calculus. He actually found it quite difficult, but when he took his first math courses at university (his computer science program was part of the math program at the time), he was very well prepared, whereas the students coming directly from high school found themselves struggling with math, especially calculus.

  9. Regarding your comment that, "On international tests [...] the US and Canada don't perform particularly well. Year after year, they are outperformed by countries that don't use a constructivist math curriculum--countries such as Singapore and China." Don't disregard who is teaching our children - in other top-performing countries it's the top students, whereas in the US, anyway, it's far from that. See for one explanation of a recent report.

    I would argue that the curriculum isn't nearly as important as the person in charge of the classroom.

  10. Annie Fetter -- Good point. Here in the U.S. we pay our top students to be Wall Street bankers who wreck our economy. I'd agree that there's something wrong with that picture.

  11. Annie, I agree as well. I recently discovered that Finland--whose school system I greatly admire--uses a math program that is not much different from ours. (They call it a "problem-solving" approach, but it sounds a lot like "discovery" math to me). Yet their students still perform phenomenally well on international tests. I think it's because in Finland teachers all have masters degrees, and teaching is a well-respected profession. So you could be right, the problem could be that here in North American we don't have math specialists teaching elementary math, and our generalist teachers often don't like math or they just don't understand it. I do believe that constructivist math programs demand a lot more from the teacher, and that many (generalist) elementary teachers--through no real fault of their own--are simply not up to the task.