A Grade 7 Math Question

The other day my daughters were assigned a perplexing math question for homework. It was a question straight out of their Grade 7 math textbook, which is the French (immersion) version of Math Makes Sense 7. Math Makes Sense is a Trillium-approved, "constructivist-lite" math textbook series published by Pearson Education Canada, and widely used across the province of Ontario. Here is the question:

Use a place value chart. Explain why you add one or more zeros to the end of a number that you multiply by 10, by 100, or by 1000. [translation mine]

The girls thought about it for a while. They understood that adding the zeros had something to do with the fact that you move the decimal place to the right when you multiply by 10, 100, or 1000,  but they got stuck on that word "why." Why do you move the decimal, thereby adding the zeros?

Now, this type of question is not uncommon in the Math Make Sense series. Proponents of what is variously called "discovery," "constructivist" or "reform" math would say it exemplifies the kind of challenging question that leads children into authentic mathematical "discovery." But does it?

The problem with this question, and others of its ilk that we have encountered over the years with this series (and with Nelson Mathematics —the "competition" to Math Makes Sense), is that the type of analytical reasoning needed  to answer it adequately is not commonly taught in the contemporary math classroom. What the writers of the question are looking for is a kind of conceptual grasping, written in English. For instance, here is the answer provided in the back of the book:

For example: When I multiply a number by 10, it becomes 10 times bigger. In a place value chart, each digit of the number moves one position to the left. The digit 0 occupies the last position.  [translation mine]

For a series that prides itself on furnishing teachers and students alike with a conceptual approach to mathematics, this answer is quite curious. It substitutes one mechanical trick—adding zeros—for another: moving the decimal place. But both tricks are answers to a "how" question, and not to the "why" question  posed.

The inconvenient fact of the matter is that it is nearly impossible to answer the question in a way that is mathematically precise using English alone. A mathematically correct answer requires a mixture of notation (with which kids at this level are mostly unfamiliar) and English. In fact, it requires a proof like this one:

A decimal number is written as \(a_k \ldots a_3 a_2 a_1 a_0\) (for some \(k\))
and represents the value \[\sum_{i\ge 0}^k a_i 10^i.\]

So \(10^d\) is represented by a 1 followed by \(d\) 0's.

Given a decimal number \(x\) represented by \(a_k \ldots a_3 a_2 a_1 a_0\),
what does the representation of \(10^dx\) look like?

\[10^d x = 10^d (\sum_{i \ge 0}^k a_i 10^i) = \sum_{i \ge 0}^k a_i
  10^{i+d} = (\sum_{i \ge d}^{k+d} a_{i-d}10^i) + \sum_{0 \le i < d} 0 \cdot 10^i\]

So \(10^dx\) has the representation \(a_k \ldots a_3 a_2 a_1 a_0\)
  followed by \(d\) 0's,
  as we were required to show.*

Show me the Grade 7 student who can "discover" that.

(See also THIS MATH DEPRESSES ME)

*Proof courtesy of Prabhakar Ragde, professor of Computer Science at the University of Waterloo.

Is It A Boy or a Girl?

Today's guest blogger, Prabhakar Ragde, is a professor of Computer Science at the University of Waterloo in Ontario. In the 1990s, he and his wife, quietly and without fanfare, made the decision not to reveal the sexes of their two children. In this post, he reflects upon that decision and its repercussions in light of the unprecedented media frenzy surrounding the so-called "genderless baby."

Is It A Boy or a Girl?
by Prabhakar Ragde

Twitter is my link to the zeitgeist. It's where I learned of the Japanese earthquake and the death of Osama bin Laden. But I also learn about many less momentous events and situations, such as the one described in an article in the Toronto Star about a Toronto couple who weren't announcing the sex of their third child.

The article went "viral", exploding on both the Web and in traditional media, eliciting much ignorant reaction from anonymous readers and only slightly more nuanced expressions of concern from so-called experts. Back in my Twitterverse, some of my tweeps offered their own 140 characters of acerbic comment. I argued back, more confidently than usual, because I had something they didn't: empirical evidence. My wife and I had done the same thing, with the birth of our first child Arju in 1992, and again in 1995 with Zazuki (Zuki), and anyone who knows our teenagers knows how well they have turned out.

Why would we want a "genderless baby"? Well, we didn't, and neither did the Toronto couple. There are three related notions of gender here. The first is biological sex, for which people often use the word "gender" as a euphemism. The second is psychological gender, or gender identity — the sex with which a person self-identifies. The third is the social role assigned to a man or woman, leading to the quote, "Gender is a social construct". The correct answer to the question "What is the baby's gender?" is probably "No one knows yet," for all babies. But what the question really is asking is "What is the baby's biological sex?"

The asker probably wants to know in order to fit the baby into a social role, and in doing so, change the nature of interaction. Although the asker will probably steadfastly deny that they would treat a boy baby and a girl baby differently, it's not hard to turn up peer-reviewed studies demonstrating otherwise. We cited a few of these in our birth announcement (we couldn't resist the conceit of including a bibliography).

But the largest influence on our children in the early years was, of course, their parents. We knew their biological sex. We'd grown up in an era when it was unusual for married women to work outside the home, or to keep their last names on being married. We're not self-aware or iron-willed enough to avoid our own gender biases, even if we wanted to completely eliminate them (which it's not clear we should, considering that our children have to live in a gender-biased world). So this was never about affecting the children, except indirectly in the examples we set as parents as they grew up. It was a minor bit of consciousness-raising among our immediate circle of family, friends, and acquaintances. Minor in the grand scheme of things, that is; it loomed fairly large for us at the time, despite the lack of media attention.

Fortunately, we have no traditions in this culture of routinely displaying the genitals of newborns, so the only things we needed to do were to avoid dressing our children in all pink or all blue, and avoid using words such as "he" or "she". It's not hard to do this in writing, especially if one is willing to adopt the singular "they" (for which there is historical precedent). Extemporaneous speech is another matter. I managed it by dint of furious concentration and much stammering, and it got easier with practice. If someone asked directly, we would briefly explain our stance, but otherwise we simply never corrected anyone's assumptions, except to spare them embarrassment. And people would make assumptions based on the flimsiest of evidence (even on whether they thought the children's names, which we made up, sounded like they referred to one sex or the other). I can remember four occasions when I slipped up and used a sex-specific pronoun to refer to Arju, but it was because the person I was talking to was using them. Twice I used "he", and twice I used "she".

My wife's parents, thousands of miles away, accepted our decision; mine, closer by, did not. Siblings and other family members we were close to were supportive. Among our friends, some were enthusiastic about the idea, and some were dubious, and it was sometimes surprising to us who took which stance. My wife and I are both professors in the same department, and our secretary reported mostly puzzlement among the academic staff. One woman was concerned that Arju would turn out gay (hard to see the logic in that one), and a few said that because they didn't know Arju's sex, they couldn't buy gifts (which we'd asked them not to do in the birth announcement, anyway). When we were out in public, we didn't make a point of bringing the topic up, but sometimes people would ask, and we'd gently explain. We never encountered any hostility; at worst, the subject would be abruptly dropped. More often, we got some clarifying questions, and maybe a nice expression of support.

We tried to choose sensible, attractive clothing in a range of colours, which meant choosing from both racks in the store, as pinks and pastels were on the girls' side, and other bright colours were on the boys' side. We gravitated towards toys that were not only fun but stimulated creativity and imagination; that meant a wide array, including both dolls and trucks, building blocks, and miniatures for role play both domestic and "on the job". (It probably helped that I did the cooking, while my wife mowed the lawn.)

Our university, at the time, topped up salary during maternity leave, but only for thirteen weeks. We had a particular daycare in mind, but we hadn't put in an application before Arju was conceived, as we would have had to do to get a spot at three months. It was nearly a year later when a slot opened up. We'd visited several times in between, to keep up our visibility, and the director of the daycare had been one of those making an incorrect assumption about Arju's sex. So when we handed in the completed registration forms and the first cheque, we had to gently explain why we hadn't corrected her.

We didn't ask for special treatment, but the director clearly took the lesson to heart, and discussed it with the staff. Years later, my friend L attended a party where she overheard a conversation in which my children came up. One of the participants had been a worker at our daycare; she didn't know that L knew us and would report back. She said that attitudes had changed among the staff as a result of the situation; they thought about possible biases in their actions, and went about their jobs in a more thoughtful, introspective fashion. How long-lasting that effect was, it's impossible to say. But this is one way that progress occurs, through small, local changes.

When Zazuki was born in 1995, no one blinked an eye when we said we were doing it again (and this time, we had put in a daycare application before conception!). Arju was by then a delightful, talkative creature, and any fears they might have had, had long since been put to rest. I kept the birth announcements up on my Web page for a while, though as the kids grew up, they seemed like old news, and I took them off. But history has a way of resurfacing.

Back in 2011, my "been there, done that" tweets must have been noticed, because a reporter from Postmedia (which owns the National Post) e-mailed me requesting an interview. I refused phone contact but gave a short statement by e-mail. He wanted more detail, and I agreed to answer by e-mail the questions he would have asked over the phone. As a consequence of my being able to compose my responses, the article had fewer distortions than usual. What I hadn't expected were the phone calls from TV networks. I let them e-mail me, and turned them down. Apparently, the Toronto couple who started the latest furor did the same, as the woman explained in a rational and intelligent article written entirely in her own words. And with that, the attention of the world turned elsewhere.

I wrote my answers to the reporter using gender-neutral language to refer to my children (as I have done here) to make a point, even though he had done his research on the Web (looking, perhaps, at their Facebook profile photos, in which it's fairly obvious) and figured out which pronouns he needed to use. His article highlights their sexes in the lede, so you can click through, if you wish, and find out for yourself. But before you do, ask yourself why that particular bit of information is so important. Really, it isn't. The desire to know, on the other hand, that is worth thinking about.