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Ben Babcock

I read, write, code, and knit.

More on math from Margaret: Arithmetic should be boring

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:

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

Or, you could do it this way:

  • Recognize that 110 = 100 + 10.
  • Divide 100 by 5 to get 20.
  • Divide 10 by 5 to get 2.
  • 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.