Algorithms are not the answer

Oh, look! It’s another article about discovery vs rote math! Here we go again….

I thought I had solved this back in 2011 (twice!), but apparently a couple of people in the world didn’t listen, so now here I am, back at it again. It seems like every five years when the latest round of test scores shows that the sky is falling we’ll be doomed to have the same arguments over and over about how to teach, and in particular, how to teach math. It’s tiresome. I’ve only been a teacher for four years, and I’m already tired of it.

Let me qualify that statement of fatigue: I never tire of talking about how to teach. Education is serious business, and having a healthy debate over the best and most effective strategies is important. I am not out to label anyone in any of the camps as wrong or extremist in their views, and I believe that most of the participants in these discussions genuinely want what is best for our children. When I say that I’m “tired” of this discussion, I’m not here to say, “I’m right” (even though I am) and, therefore, we should shut down the whole thing and get on with it. That would be terrible.

No, what I want to rant against is the way this discussion keeps rearing its head in media—almost, as I said above, like clockwork. And it’s always the same talking heads from either side and the same few chestnuts being pulled out and passed off as brand new truths. That’s why I’m tired: it’s boring, and it’s counter-productive. By all means, let’s have these discussions. But let’s actually come up with new things to discuss about how to teach math, instead of making the same tired arguments again and again. After all, we aren’t politicians here.

More Rote Math is Not the Answer

My position from 2011 hasn’t greatly changed: I very much support the so-called “discovery math” that is supposed to be prevalent in our classrooms and that some fear is ruining our test scores. Note that this isn’t the same as supporting the Americanized version as enshrined in the Common Core—where the problem is not so much the Common Core, but the insane pressure on teachers to somehow cover the Common Core and prepare students for bogus standardized tests. But I digress….

I’m not going to question the contention of those among old-school or the rote math camp that knowing basic arithmetic is a good idea. It is totally important that people have basic numeracy and number sense. Where I break from them is in their narrow view of how students can acquire that number sense.

In the CBC article I linked to at the top of this post, the CBC quotes Marian Small arguing that students would be better off using a calculator instead of learning how to multiply 2- or 3-digit numbers by hand. Then, in their attempt to offer “balance,” they counter it with this nonsense:

But Craigen rejects Small's assertion.

That sort of skill "is an important benchmark in the development of an overall conception of how numbers and arithmetic work, in preparation for algebra, which is the gateway in high school to the higher mathematical disciplines they will see in college," he says.

"It's long been settled that the establishment of basic facts, in memory, and the development of automatic skills for the most basic tasks is really of fundamental importance in developing long-term skills."​

At face value this is a very reasonable statement, and I totally agree that you need basic skills in order to develop long-term skills. I have two issues with the idea that this somehow discredits discovery math and supports the necessity of rote math. Firstly, Craigen and I seem to be disagreeing about what, exactly, constitutes a “basic skill.” Secondly, the beloved algorithms that are at the heart of rote math are beautiful if you understand how our number system works, yes—but I don’t agree that these algorithms by themselves encourage understanding of our numbers.

What, Exactly, is a “Basic Skill”?

Mathematics is not about numbers.

I’m going to repeat that, because it’s a really important idea but it might seem strange to you: mathematics is not about numbers.

Mathematics is about solving problems involving relationships between quantifiable entities.

If that sounds a little complex, then good. Mathematics, like a great many other things in life, is a complex field (pun intended). And the beauty of mathematics, at least in my opinion, is that once you scratch the number-infested surface and dig down beneath it, you discover that numbers themselves are entirely made up! Numbers are just a very common way of representing what’s happening deeper down the mathematics rabbithole. They also happen to be useful, I suppose, in real-world applications. But that’s what computers and … engineers … are for. I don’t worry about that stuff. Give me a spherical cow any day.

Really, you can a great deal of difficult mathematics with very few numbers. Don’t believe me? Just look at, oh, most of Western history. The Greeks got along fine with only the positive integers and a bunch of geometrical shapes. We only got negative numbers, and then zero, in the fifteenth and sixteenth centuries, and Arabic numerals were themselves only a couple of hundred years old at that point. So the vast majority of the numbers that exist today—that is, the rest of the reals and the complex numbers—were invented in the past three hundred or so years. And, mathematically speaking, those numbers dwarf all the numbers that came before, so I could make the argument that practically no numbers existed before the 1700s. (Welcome to calculus class!)

So when it comes to “basic skills” in mathematics, there are a few things that are important: counting, obvs; ordering (also obvs); and logic. We spend a lot of time on the first two in elementary school and a fair bit of time on the last one, though perhaps not as explicitly. When it comes to counting and ordering, numbers are useful but not necessary. What matters more is that you can take these three skills and apply them to successfully solve problems—that’s what math is about! Math is not computation and is not the ability to perform a few narrowly-defined algorithms.

I don’t want to get bogged down in the minutiae of which skills are truly basic, though. You are free to disagree with my list above, just as I disagree with the idea that long multiplication or division are basic skills. Should kids learn their times tables? Certainly. It’s one of the first things I did with a class of 15-year-olds who were struggling with math: times tables are important. Memorizing basic facts is important. Memorizing algorithms? Not so much.

Long multiplication and long division are just means to an end. Calculators can do it faster and better than we can. And while these two algorithms exploit some beautiful underlying properties of our positional, decimal number system, those properties are not necessarily exposed by using the algorithm, unless a skilled teacher points them out. Furthermore, there are plenty of practical alternatives to these algorithms that often work faster and more reliably depending on, ironically, one’s grasp of properties of arithmetic and algebra.

Which brings me to my second point.

Algorithms Make Our Students Lazy; Calculators Make Them Smart

Teach a kid long multiplication and long division. Go ahead. I dare you.

Did you do it? Good. Did you notice how their jaw kind of slacked, and their eyes glazed over? And after you tested them a hundred times and reported the scores to the province, which gave you a thumbs-up, because hey, scores are the most important part of this whole endeavour, am I right—did you notice how the kid kind of hates math now, and has no idea how they figured out that 1352*412 = 557024? (And yes, I totally googled that instead of working it out by hand.)

Learning algorithms is a cop-out. You know what uses algorithms? A computer, that’s what. And a computer is dumb. It has no understanding of what “numbers” are or the quantities and relationships inherent in the problems we ask it to solve. It just takes input from humans, runs it through a series of algorithms, and reports back on the output. It has no awareness of mathematics.

And that is exactly what we do if we try to educate children by force-feeding them algorithms. We rob them of their awareness of numbers, and we bore them and teach them that math is not a creative or interesting experience. This seems like the opposite effect intended by the well-meaning supporters of rote mathematics.

Long multiplication is really cool, but unless your teacher is super-cool with math (and, all respect to them, a lot of elementary teachers are not comfortable with it), then chances are they don’t actually explain what’s going on when you use long multiplication. In fact, it wasn’t until I had to teach myself “grid multiplication” when I was in England (because that’s how must of my students there were multiplying 2- and 3-digit numbers) that I had the epiphany about what that placeholder zero is doing when you long-mutiply with columns. When I was a kid, I was just taught that if I want to multiply, say, 32*15, then I would multiply 32 by 5 on the first line, put a “placeholder zero” in the right column on the second line, and then multiply 32 by 1. Actually, what I’m doing is multiplying 32 by 5, and then multiplying 32 by 10, and then adding the result—exactly as if I were using the grid method.

The grid method and column method are exactly the same, minus layout changes. Mind. Blown.

Maybe everyone else in the world already understood this and I’m just really late to the party. But it’s precisely because I was told, “do it this way to multiply big numbers” that I never made the connection, never really dug into what was happening with the numbers. I treated it as an operation on digits when I should have treated it as an operation on place values. Similarly, long division actually works using powers of 10 (but try teaching that to a Grade 4 student).

My point is simply that it’s unrealistic to claim that students gain an awareness and understanding of the properties of our number system by learning rote computational algorithms. If anything, these algorithms have the opposite effect. Take a more complicated algorithm, for example: the quadratic formula. Does knowing the quadratic formula give you any insight into how quadratic equations and parabolas work? Absolutely not. Deriving it might, but that’s a different story altogether.

If you teach a student an algorithm, and they forget the algorithm, they are screwed. They have no way to recover from that. The response to this idea is usually, “well, they need to memorize and practise more.” But that only goes so far, and if students can’t see the value in being asked to do it, the returns won’t be very high.

When I teach compound interest, I don’t just say, “this is the compound interest formula; here’s how you use it.” We do eventually talk about the formula, and eventually I say, “Just use the formula and your calculator.” (Because, otherwise, trying to calculate the interest on an investment after 20 years of monthly compounding would be torture!) However, it is important for students to investigate, with me, where that formula comes from and why it works. Then they practice the hell out of it.

If you teach a student how to use a calculator properly, then they suddenly have an entire world of mathematics open to them that was previously inaccessible. I say “properly” because calculators are not crutches. They can’t solve your problem for you, and if you make a mistake, they will happily compound it. We need to educate our students about how to use calculators, and in particular, help them distinguish common errors. You don’t need to be able to multiply 1352*412 by hand—but you should be able to tell me what the answer will be within an order of magnitude. Having conversations about topics like this—orders of magnitude, estimation, geometrical and spatial reasoning—is so important. Those are the basic skills I want to see in my classroom.

When Small is arguing in favour of calculators, she is just being practical. We are so lucky to live in a time and place with access to limitless cheap computation. Face it: the amount of time that students have available to learn math has not, largely, changed over the past few generations. The amount that we need to cover has changed, thanks to more pressure to stream students into places like university. Fortunately, technology has kept apace—or even outstripped that pace. We can cover a lot more, because we are able to investigate it, explore it, and, yes, discover it, using technology.

Room for Improvement

I don’t think there is much debate that we should be striving to improve math education. This is a difference of opinion on what needs to be done, not whether anything needs to be done.

Discovery math versus rote math is a red herring. Instead, we need to talk about the attitudes of mathematical thinking we instill in our students, and whether our techniques actually improve numeracy and mathematical literacy. Ensuring that kids have a good grounding in math while in the youngest grades is crucial, yet we lack specialists in primary math. I have the utmost respect for primary teachers—I just couldn’t do it. However, I know that a lot of primary teachers don’t receive the training they need to teach mathematical concepts, and in many cases, to undo the math anxiety that set in while they were in school. That anxiety comes across to students, and that, to me, is much more concerning than whether the teacher is using “discovery” or “rote” methods.

Also, at the end of the day, test scores are only ever going to tell you so much. There are some concepts that just don’t test well. And, of course, rote memorization of algorithms and formulas will tend to test well—but is that really preparing our students for life outside the classroom? Are we really doing this because we want students to be better mathematicians, or are we doing this because we want some arbitrary scores to nudge higher?

Mostly, though, I’m irate not at those promulgating one type of mathematical education over another. I’m disappointed in the cursory, “balanced,” and uncritical news articles that appear at regular cycles in our media. It’s always nice when I see a good one that digs deeper into the issues. But most of them, like the CBC article that prompted this post, simply dredge up the same, tired and dogmatic stances that we’ve seen before. We need to have real conversations about this, and our media should step up in that regard.