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)

Sunday, October 17, 2010

Separate Schools for LGBT Kids?

To its credit, the Toronto District School Board, has a long history of supporting alternative schools. In fact, it boasts North America's first public alternative secondary school, SEED, which stands for Shared Experience Exploration and Discovery, and which was originally a "free" school. There are currently 41 alternative schools operating within the board, as well as multiple special schools and programs for arts, athletics, science, etc. A few of the most recently inaugurated alternative schools have been controversial. The elementary "Africentric Alternative School" caused quite a stir when it was proposed in 2007; the typical arguments for and against separate schooling for minority children were trotted out, but in the end the school was approved and opened its doors to 90 children in 2009.

In light of the disturbing spate of suicides among gay and lesbian students in the US, one has to wonder if separate schooling should be considered in this case as well. In fact, there is a program, housed in one of Toronto's alternative schools, called Triangle. Here's how the school board's website describes it:

Unique in Canada, we offer academic and applied level programs for lesbian, gay, bisexual, transgender, and queer (lgbtq) students who are able to work independently with some guidance. Our program covers lgbtq history, literature, and issues as well as a lunch program, class field trips, access to lgbtq community events, and co-op education.

This program, then, provides the safe space for LGBT youth that is so sorely lacking in regular middle and secondary schools. Which is wonderful, but I find it sad that such a school should be necessary. The optimistic side of me believes that if anti-bullying education were taken seriously enough, started early enough, and were specific enough—if it included explicit discussion of words like "fag" and "queer" and "gay," and explanations of how and why they are used as slurs—then separate LGBT schools would not be needed. But I'm enough of a realist to know that this is not likely to happen anytime soon. In the meantime, programs like Triangle—in fact, whole schools—should be set up across North America as options for LGBT kids. If they prevent even one teen suicide, they will have been worth it.

Wednesday, October 6, 2010

Breeding Tolerance: Is it Possible?

In 2008 I published an article in a local newspaper about my daughters' emerging understanding of the word “gay.” The article was inspired in part by Ellen Degeneres’ emotional plea for tolerance in the wake of the murder of Lawrence King, a teenager from Oxnard, California killed by a classmate simply for being gay. Now, two years later, King's murder trial is underway, yet the bullying of gay teenagers continues unabated. Gay teens have been driven to suicide in Indiana, Texas and California. Tyler Clementi, a Rutgers University freshman, jumped off the George Washington Bridge to his death after his roommate posted a video online of him kissing a man in his dorm room. In light of these deeply disturbing incidents, I have decided to post an earlier, less-edited version of my 2008 article here; I believe it is—unfortunately—still relevant and timely.

Twin sisters question the meaning of “gay”

In February 2008, Lawrence King, a gay teenager from Oxnard California was killed by a classmate for openly expressing his sexuality. After reading the chilling details of the story on the Internet, I was left with a perturbing question: How does a child grow up to believe that hatred and murder are acceptable responses to difference?

My eight-year-old twin daughters have been brought up in a fairly typical, liberal heterosexual family. Yet the word “gay,” with all its ambiguous cultural freight, entered their lives at a young age. One fall day in Grade 1, they asked offhandedly, “What's ‘gay’”? My face must have registered surprise because E added by way of explanation, “Connor was talking about it.” Connor was a boy in their class who had older siblings and was clearly in a different league of worldliness.

I hesitated. How could I respond to such a question in a way that a 6-year-old could understand? “Well,” I ventured, “when you’re a man, and you want to spend most of your time or life with another man, you could be gay, or if you’re a woman and you’re more interested in living with a woman than a man, you might be gay.”

“Can two girls or two boys get married?” the other twin—J—asked.

Canada had recently passed legislation legalizing gay marriage, so I answered truthfully, “In this country, yes.”

E piped up: “When I grow up I’m going to be gay; I’m gay.”

“I’m not,” J said, “I don’t want to be gay.”

OK then, I thought, I’ll have one of each. Just give me grandchildren. Aloud, I said, “You don’t have to worry about this stuff for a long time.”

I didn’t hear anything further about “gay” for a few months. Then one day while the girls were playing dolls with a friend in our kitchen, I overheard an interesting conversation. The friend, a good-natured North Toronto girl, said about her favourite doll: “When Taryn grows up she’s going to marry a girl, so she never has to kiss a boy.” The twins nodded approvingly. I tiptoed out to the living room stifling a laugh.

It wasn’t until the end of Grade 2 that I began to notice the twins’ neutrality towards all things gay and lesbian starting to erode. E no longer wanted to be gay because, as she explained—after I was forced to answer very pointed questions about where babies came from—she did not want to have to “to borrow a seed” from a sperm bank (a solution my husband had helpfully proposed). J asserted that being gay wasn’t the best option because it wasn’t “the tradition,” at least not in our family. Curious, I quizzed them on their attitudes, wondering if they had heard negative talk at school.

“Well, we don’t know anyone who’s gay,” E said, accusingly. I pointed out that there was a lesbian couple living a few houses away on our street. The girls were surprised; my husband had taken them trick-or-treating to that house, but neither they nor we knew the couple well.

J looked thoughtful. “Hmm,” she said, “maybe it would be nice to be a lesbian because you could have lots of nice teas on the verandah with your wife.” She proceeded to launch into a make-believe dialogue, playing both parts herself in a bad English accent: “Ella come and have tea with me on the verandah. In a minute dear. OK dear,” and so on.

“Is that how you think lesbians live?” I asked, laughing.


J's favourable—albeit Victorian—view of lesbianism seemed to persist. In the middle of Grade 3 she told me that her friend Sarah had taken to telling her almost daily that she loved her. “But,” J explained, “Sarah always adds, ‘as a friend,’ because otherwise she says we’d be gay.”

“What do you say?” I asked.

“Nothing. Except once when she said, ‘or we’d be gay,’ I said ‘well, we could be,’ and Sarah said ‘eeww,’ and ran away.” I considered telling J that I was proud of her for uttering that little phrase “well, we could be,” for daring to acknowledge, in her own childish way, that gayness exists, but I let it go. It seemed I’d said enough.

Except it's never enough. Time and again, I'm jostled out my doze of complacency—by a conversation, a word—into an awareness that as parents we can never do enough to inculcate acceptance of difference in all its incarnations.

Not long ago, the girls asked about another culturally freighted word, “queer.” A socially savvy friend had informed them there was another meaning besides “odd” or “strange,” but she wouldn't tell them what it was. I launched into a complicated, politically-correct explanation of how some people use “queer” as a not-so-nice way to say “gay,” but that some gays and lesbians had “taken back” the word and now used it to refer to themselves, which is OK because . . . Two pairs of 8-year-old eyes glazed over in unison. I left it at that.

Then, just last week, the girls rushed in the door after school, bubbling with excitement. “Mom,” E said, “Lauren finally told us what the other meaning of ‘queer’ is and it’s not what you said at all.”

“What is it then?”

“She said it means stupid.”

We can never do enough indeed.

Friday, October 1, 2010

Junk Lunch

I've had this post sitting in my drafts file since last spring. I hadn't worked on it much, but I'd thought about it often, and I had a lot to say. I was planning to make some pointed observations about school lunches. I also hoped to raise a few seldom-asked questions about the importance schools place—or do not place—on children's health and nutrition. More specifically, I wanted to write about: why lunch periods in elementary schools have been reduced in length from an hour and a half, when I was a kid, to one hour or less today; how and why kids are rushed out of unappealing lunch rooms (overseen by authoritarian "Lunch Ladies") after 10-15 minutes; why the one lunch program that is a given in almost all Toronto District School Board schools—the Pizza Lunch fundraiser—is a junk lunch. But then I was alerted to this report from CBS on school lunches in France, and I realized that not only does it tie into my previous post on the French Paradox, but that it speaks for itself, and for me. There's nothing more to say. (Thanks to Corey Mintz for the link.)