Blog: Articles from December 2011

The year is almost over, and unless I finish a book tomorrow, it looks like I will end 2011 with 115 books read. Not too shabby, I suppose. Far cry from my goal, which was to tie with 2009’s best of 156 books. But still pretty good, all things considered. Indeed, from time to time people exclaim their awe at how much I read. I don’t like to draw too much attention to the quantity, which is after all no indicator of quality, because it feels too much like bragging. But today someone on Goodreads asked me how I manage to read so much, and as I was composing my reply, I realized it was getting too lengthy. Lengthy enough for a blog post, in fact.

It’s quite simple. I have a time machine, you see, and that allows me to go back in time and spend more time reading throughout the day….

Well, I wish that weren’t so much science fiction!

Last year, which was a very good year for me, I averaged 2.6 days per book; this year I have been slightly busier, so I took 3.1 days per book. Considering that most people, i.e., people who do not bother joining a social networking site about books, are probably lucky if they read 10 or 15 books in a year, I suppose I do read quite a bit. However, I’m far from abnormal—some of my friends here are up to the 200s when it comes to books, and I suspect they must be speed-readers.

I am nothing of the sort. I probably do skim quite a bit, by which I mean that my reading comprehension has developed to a point where I don’t have to focus my eyes on every single word in order to get the gist of a passage. The way that the human brain and the human eyes interact is really quite amazing and not very much like a camera. I suspect (because I Am Not a Neurologist) that my practice reading means that my brain can predict what words will be before they have fully registered. Indeed, when I encounter an unfamiliar word I do tend to “stumble” and slow down (while I pull out my dictionary!). I know I’m not a speed reader because I still need to focus carefully when I read technical, academic, or legal documents where rigorous attention to the word choice is more important.

My “secret” is a patent-pending formula discovered through years of careful, painstaking research, including an ill-fated expedition to a Tibetan monastery long thought lost to the ravages of time and war. And, for the low payment of $99.95, or three easy payments of $39.95, you can have it too….

I don’t have much of a secret. When people ask me how I manage to read so much, the answer is always the same: I make reading a priority. I allocate a great deal of my free time to reading, more so than almost anything else. And this has been true for a long time. I read a lot when I was a kid, and I’ve continued this habit my entire life. That doesn’t mean you can’t start reading voraciously now—but like any skill, reading becomes easier with experience.

Also, keeping track of your reading helps too. Goodreads has been really good for me in that regard; I’m a lot more aware of which books I read in a year and which ones I want to read next. I’m not saying you need to review every book like I do; you don’t even have to join a site like Goodreads (though I certainly recommend it!). But even just keeping a list of which books you’ve read each year, and looking it over every few months so you can see your progress, might help. You could even develop a goal. You might choose to try to read a certain number of books in a year. One of my friends is working her way through the BBC’s list of top 100 novels as voted by readers.

Of course, I also have to admit that I probably program my life in such a way that I have more free time to read. I am lucky enough to be financially stable right now (I still live with my dad). I don’t find my schoolwork particularly challenging, and with the possible exception of this year, it has never felt time-consuming either. Most people seem to engage in a dazzling array of extracurricular activities, including sports, music, and volunteering. I don’t do many of those things, and while I feel that has sheltered me in many respects, I also recognize that my vast experience reading has opened my eyes to new worlds. So while I don’t have the same experiences as my peers, I wouldn’t necessarily say mine have been of inferior quality. But it’s definitely the case that I make time to read, because for me, reading is a priority.

This year, my final year of my undergrad, has given me a taste of what I might suspect once I get a full-time job. In the five weeks of my practicum I only read three books. Terrifying! And one of my instructors mentioned that most of the teachers he knows only have time to read a few books while they are on break during the summer. I certainly hope my own personal drive to read shields me from such misfortune!

The person on Goodreads whose question prompted this post also mentioned that he reads audiobooks and probably couldn’t listen fast enough to match my pace, even at double speed. When people ruminate on how they can read more, I do tend to suggest audiobooks as a part of the solution. Audiobooks are awesome: you can listen to them “on the go” in the car, while you’re exercising, or while you’re cleaning or cooking. They are excellent for people who just don’t have the time to sit with a book for an hour (or even half an hour) a day. Even so, I tend not to listen to many myself. Even with the ability to alter the playback speed, audiobooks are a little too much like a movie or television show: you go at their pace, not the other way around. Books, among all our entertainment devices, have a marvellous and singular capability to take as long as you desire. You might choose to devour a good book in an afternoon, or draw out the pleasure for a few days. This is one of my favourite things about reading, and it’s the one aspect that audiobooks, for all their advantages, do not replicate.

Some people spend so much time gaming they turn it into a lifestyle, even a career. Others become master speedcubers, or Olympic-class athletes. We all have our talents and our interests. Reading is mine. And at the rate my to-read list has been growing in these years since I joined Goodreads, I wish I could read even faster! No matter how many books you read in a year, however, the fact that you are reading is pretty amazing. Keep it up.

Once again Margaret Wente, my favourite Globe and Mail columnist, has delved into the gritty underworld of math education to expose the truth. This time she is concerned that we’re not teaching basic arithmetic in schools any more. She takes issue with recent trends in math education, which emphasize discovery-based learning over drill or rote-based learning. As a consequence of this shift, the standard algorithms for addition, subtraction, multiplication, and division are no longer a core part of the curriculum. Wente, as well as some parents and teachers, thinks this is a bad idea. And while I agree with her on one point—it’s essential for students to know basic arithmetic as they go on to high school—once again I have to protest how she has chosen to argue that point.

Before I discuss Wente’s arguments, I think it’s important to mention one thing that Wente does not make explicit. Education falls under the mandate of the provincial governments. Hence, every province and territory in Canada has different math curricula. There are similarities, but we still have to be careful when we are talking about math education across the entire country as if it were some uniform curriculum.

Canada is “Behind the Times”

One of Wente’s more absurd reasons for rejecting the current curriculum is that “this approach to math education has been repudiated” in the United States, and this apparently makes us “behind the times”. Heaven forfend that we don’t copy the United States in every respect! You would think we’re a sovereign country or something crazy like that. We are allowed to structure our curriculum differently from our neighbour to the south. And without being too indelicate, let’s just say that the American education system in all its forms does not instil much confidence, at least for me personally. I’m not sure it’s something we should be striving to emulate.

This Is a Plot by Private Tutoring Firms (And Thus Puts the Poor at a Disadvantage)

Since kids no longer learn arithmetic in school, parents are forced to turn to private tutoring companies—Wente names Kumon as one example—for these skills. At the conclusion of her article, Wente decides to deploy the heavy Scare Tactic weapon:

The biggest losers aren’t your kids, of course. The biggest losers are the kids of parents who can’t afford tutoring, or don’t have the time to teach them times tables, or don’t even know their kids need help. It’s called two-tier education. And it’s here.

I love the way Wente phrases this: the biggest losers aren’t your kids; they’re the kids of those poor people. Those two sentences tell you all you need to know about who Wente assumes is reading The Globe and Mail.

I’m not sure how to refute this argument simply because it’s a conspiracy theory, and Wente doesn’t even try to disguise that fact. I suspect that if our curriculum magically amended itself to reflect Wente’s visions, then Kumon and its ilk would find other ways to get clients. They are a business and target their marketing accordingly. It just so happens they’ve found a niche here.

But two-tiered education has always been around. That second tier is called private school.

The Standard Algorithms are Better Because They are Efficient

Wente calls the standard algorithms “efficient and foolproof”. And they are. She blasts the alternative methods, “such as breaking numbers into units of thousands, hundreds, tens and ones” for not being efficient. I would disagree, but first we probably need to decide what we mean by efficient.

If one’s goal is to add two numbers efficiently, then my suggestion would be to use a calculator. Savants aside, computers are just better than humans at adding and subtracting. And now that calculators are commonplace, not just in schools but on our computers and even in our phones, there is no reason not to encourage their use. Should you be able to add basic numbers without a calculator? Absolutely! There will be instances where a calculator isn’t in reach, for whatever bizarre reason, and you will be glad you can do arithmetic. But those instances are becoming increasingly rarer. And so when they come, are we really worried about efficiency?

An algorithm is a series of steps that one repeats until one reaches a pre-determined stopping point. The first algorithm that most of us learned (and one that is apparently no longer taught, to Wente’s chagrin), is the long division algorithm. In this case, you repeat the same step over (dividing a digit of the dividend) until the remainder is less than the divisor (your pre-determined stopping point). As Wente points out, the nice thing about algorithms is that they are foolproof. Whether you are dividing 30 by 12 or 3000 by 1250, assuming you recall the steps correctly and don’t make a mistake while implementing them, you will always come up with the correct answer. This is comforting.

But algorithms are also cumbersome for humans. Unlike computers, which thrive on algorithms because that’s the way we built them, our brains do not always think linearly. We make intuitive leaps, and we often think spatially. Thus, training ourselves to use algorithms to do arithmetic might be a waste of our brains’ potential. The method that Wente disparages, which we can call the “place-value method” is ingenious: it short-circuits arithmetic by allowing us to take advantage of our base 10 number system.

How would you divide 110 by 5? Margaret Wente would like you to use long division, in which case you would follow these steps:

1. Recognize that 5 goes into 11 evenly 2 times.
2. Multiply 2 by 5 to get 10.
3. Subtract 10 from 11 to get 1.
4. Bring down the 0 to get a new divisor of 10.
5. Recognize that 5 goes into 10 evenly 2 times.
6. Multiply 2 by 5 to get 10.
7. Subtract 10 from 10 to get a remainder of 0.

Or, you could do it this way:

1. Recognize that 110 = 100 + 10.
2. Divide 100 by 5 to get 20.
3. Divide 10 by 5 to get 2.
4. Add 20 and 2 to get 22.

I suspect that the second method is probably closer to what most people do in their heads, whether they were taught that way or not. You can do this for multiplication too. It’s all thanks to the nifty distributive property. And hey, look, suddenly instead of memorizing two algorithms, you only need to know one strategy. Furthermore, the long division algorithm is exclusive to, well, long division; it’s very difficult to use it as a template for solving different types of problems. In contrast, knowing how to exploit the place values of our number system will leave you in good shape for a variety of problems. Generalized knowledge!

I’m sure some people prefer the long division algorithm instead, and that’s fine. In fact, that brings me to Wente’s next argument.

Discovery-Based Learning Sucks Because Students Have to Start from Scratch on Every Problem

Wente dislikes the alternatives to the standard algorithms because, instead of just giving other methods to students, teachers instead encourage students to find those methods themselves. In addition to her lament that this is not efficient enough for her tastes, it also means

every time a student sees a new problem, he has to start from scratch—and pick his “strategy”. It’s like playing the piano without ever learning scales, or hockey without basic drills.

Those are quite evocative analogies; it’s a shame they’re false. Solving math problems bears little resemblance to playing hockey and even less to playing the piano. When playing piano, the goal is to reproduce a series of sounds by triggering the correct sequence of keys. For a given composition, that sequence is always the same—and you know the sequence beforehand (unless you’re playing some kind of weird piano game where you reproduce a sonata by ear). A new math problem, by definition, is one a student has not seen before.

Let me tell you from personal experience on my practicum: if you put a problem on a test that is identical, except for the numbers themselves, to one on the review, the majority of students will not recognize this fact and will instead approach it as a novel problem. Encouraging students to make connections is one of the most difficult tasks a math teacher faces. Those moments when a student goes to ask a question and then says instead, “Wait, it’s like what we did yesterday, right?” are golden—and far too few.

So let us suppose students do have trouble making such connections, that they do approach each problem from scratch even if it is of a type they have seen before. What can we do to help them solve the problem anyway? Wente would have the student, like a good computer, apply one of the standard algorithms and arrive at the solution. No need to find a new strategy! Yet this assumes the student recognizes which operations are necessary to find the solution. And therein lies the crux of the problem. Incidentally, this is also why computers suck at solving word problems.

Discovery-based learning actually works better in this case. By encouraging students to look at what they know and what they need to find out, then develop their own strategy to get there, we are building general-purpose skills that will work whether they recognize the type of problem or not. This is the beauty of mathematics: there is one correct solution but not one correct method. Wente would rob students of this beauty.

Failing to Learn the Standard Algorithms Makes It More Difficult to Learn “Higher” Math

I left this argument for last because it’s the one with the most validity. One of the reasons I am so passionate about teaching high school mathematics is because I have seen why my peers are struggling with their university math, and it’s usually not because the university concepts are too hard. No, most university students just suck at fractions. And I saw this while on my practicum too: fractions and basic algebra are concepts that students fail to master in grades 7, 8, and 9, and it haunts them for the rest of their schooling.

So Wente has a point here: students do need basic skills in order to go on to higher-order thinking. And it’s not clear-cut, despite what either side might have you believe, whether drill-based or discovery-based learning is superior in teaching these skills. I can’t really evaluate them properly, because despite my passion for this subject, I’m a fledgling teacher with very little experience in the field. I can tell you what I have reasoned a priori, but experienced teachers are expressing frustration, so there must be something else going on.

While on my practicum, I had the opportunity to sit in on a meeting between Grade 9 math teachers at my school and Grade 7 and 8 teachers from the “feeder” schools. This very dilemma came up during our discussion: the push from the Ministry of Education and curriculum experts is to have students discover their own strategies and take ownership of their learning. At the same time, however, these teachers feel a responsibility to ensure that students are prepared for high school and for their EQAO tests, and sometimes discovery-based learning makes this difficult—for one thing, it can take more time. So I can see why there is frustration among teachers who are trying to work with this new curriculum but seeing less-than-stellar results.

Of course, the curriculum will always need refinement. Continual revision and renewal of the curriculum at regular intervals is a hallmark, at least in Ontario, of the high quality of our education system. It’s never going to be perfect, and as our society and our needs change, so too will the curriculum. Right now, I think a lot of what we are seeing is simply growing pains—teachers who are used to the previous curriculum are still finding the their way with this new curriculum. Moreover, this is clearly a complex issue, one with a plurality of perspectives that should be considered.

And that’s why I take issue with Wente’s column: I agree that arithmetic is important, but once again I wonder why she feels the need to create a dichotomy where none need exist. She would have us return to the methods that turn kids off math and lend credence to their cries that “math is boring”. Picture me going to my knees and pleading as I say this: it doesn’t have to be that way. Math can be fun and full of wonder. Please, parents, don’t make math boring. Computers do their math in binary, but there is no reason our math education has to be an either/or scenario. And I wish The Globe and Mail would talk about that instead of choosing to be sensational and blame it all on the corporate interests of Big Tutoring.

My practicum is over.

But you might not have known it had even started. I kept meaning to blog about my experiences in my “professional year”, and then when my practicum began, about that. Yet I never got around to it. This has been my busiest year in long memory, and my practicum kept me busier than ever. So hopefully a short recap will suffice.

First, professional year—the first nine weeks. I enjoyed most of my classes. There was a lot more reading and many more assignments than I was used to in my previous years, which mostly consisted of weekly math assignments and the occasional essay. But my classes have raised important issues I need to consider as a teacher, and they have prepared me well for teaching. (I still hate group work.)

Now, the practicum. I was lucky with my assignment. I went to a local high school, to the math department. In fact, my associate teacher was the same teacher in whose classroom my group had taught a “mini-lesson” for my math instruction course. So I had already met her, and she had already seen me teach (sort of). This reduced my trepidation as I went into the placement.

My associate teaches two classes of Grade 12 University Preparation Advanced Functions, and one Grade 9 Applied mathematics class. I took over one of the Grade 12 classes at the end of my first week—they were beginning a new unit, and I felt ready. The other Grade 12 class followed in the third week, and I began teaching the Grade 9 class at the end of last week. The subject matter of the Grade 12 classes is definitely closer to my heart; I got to teach both rational functions and trig functions! Both grades presented challenges, though—Grade 12s are not necessarily as mature and independent as I had remembered from my own time in high school, so you can imagine the handful that a Grade 9 Applied classroom might be.

That being said, I had an excellent time. There were no serious issues in my placement, no moments when I said, “What the hell am I doing here?” It was difficult at times, just because of the amount of work involved—planning is so exhausting—but the good far outweighed any of the bad. Perhaps the most challenging aspect of my placement was simply confronting hard truths about being a teacher—the fact that there are times when you do everything possible to help a student succeed, and it’s still not enough. I’m not necessarily a brash idealist, but I’m still young enough and inexperienced enough to have a certain kind of naive optimism that will no doubt temper itself with time.

And of course, there were times when I felt like there is no other place I can be but in a classroom. There were students who were wonderful, who were enthusiastic or simply so earnest in their attempts to learn and expand their knowledge. It’s neither realistic nor productive to require one’s students to love math … I merely need them willing to be there and to listen; we will work from that. And I can’t say I reached everyone, or that I even reached most of them—for all I know, most of them are glad I’m gone! But I’d like to think I made many positive contributions.

Oh, and one student even made me a card! Before I show it to you, however, you need a little context. At the end of a lesson on graphing the trigonometric functions, I put this image on the homework slide:

This is a riff on the “Not sure if…” meme using Fry, from Futurama. I made it myself! Those of you familiar with your trigonometric functions will recognize the humour, of course. For those who don’t, I’ll ruin it with an explanation: owing to its symmetry, cos(x) = cos(-x).

Anyway, now you’ll understand why the student’s card looks like this:

That’s the outside, and here is the inside:

I’m keeping it.

I can’t believe the time went so fast. I enjoyed it, and I learned so much, both from what I did and from observing my associate and listening to her feedback. I’ve wanted to be a teacher for a long time—almost as long as I can remember—and after my first student teaching practicum, I’m finally starting to feel like one. Or maybe it’s just the tea talking….

And now I get three weeks off. One week to relax and do Christmas shopping, two weeks to work on those papers I have due in January! Good times.