Rote Memorization in Mathematics

The number one murderer of truth is ideology. Ideology: we expect it in politics and religion, government and propaganda. We do not expect ideology in mathematics. Mathematics is supposed to be the one last refuge for absolute truth in a world battered by storms of confusion and uncertainty, right? At least that’s the standard soundbite. But there is one particular ideological weed which has taken root deep in the mathematics classroom. “Rote memorization is evil.” Time and again, math lecturers inure their students to the idea that memorization is to be avoided at all costs, that it is somehow mutually exclusive with deeper understanding.

As a math PhD student, teaching business calculus to pay my way, I’ve recently been infuriated by this anti-rote dogma. The week before a midterm which would cover multivariable implicit differentiation (among other things), I showed my students a formula which reduces the process from a 10-minute algebra-hell into a 30-second breeze. Then the weekend before the midterm, with almost no more time to talk to my students, the department sends me an email stating unambiguously that the students MUST do the work the hard way. Seeing red, I shot off an email demanding why they insist on this ridiculous policy. The response: “…memorizing a formula does not make sense to me.”

The situation really stirs up righteous fury in me. When I don my math educator hat, my job is to teach students how to do mathematics. The official talking point is, “you don’t use rote memorization in math”. However, this is a lie. And I refuse to go before students and teach them a lie. I’m certainly not going to lie just to cover my ass.

In case you don’t believe that the talking point is a lie, this very same calculus course, where the department said “memorizing a formula does not make sense”, the students are required to memorize the formula for the sum of the first n integers (1+2+…+n) and for the sum of the first n squares (1+4+9+…+n^2). I’m not calling the talking point a lie in some vague philosophical “what is memory… pass the bong, man” way. I’m saying that it’s a bald-faced, impeach-him-for-breach-of-oath, 1+1=3 whopper. It is fantasy. And it’s being fed to students as gospel, students who are paying good money to eat it, students who are placing faith and trust in their professors to teach them what is true.

MEMORIZATION VS. UNDERSTANDING

When I learned calculus, it was self-taught out of a calculus book my parents bought me for my 14th birthday. That gave me the freedom to go at my own pace and understand everything as I went. In the process, I memorized certain things like derivative rules semi-naturally, without really trying. Some things, like the quotient rule, I eventually had to memorize. (Math dork trivia: the quotient rule is actually unnecessary, since you can pull the fraction upstairs and give it a -1 exponent, then use the product rule and chain rule) Therefore, to the best of my knowledge, learning math and memorizing things are interwoven processes.

“But that’s obviously not the memorization they’re talking about,” comes the rejoinder. The memorization which we’re really talking about here is the type using flashcards and repetition. Conventional wisdom says that this type of memorization should never be used in the classroom. Conventional wisdom also said that a black man with middle name “Hussein” could never be elected President of the United States. Sometimes conventional wisdom is wrong.

The pro-rote argument has two main prongs. First, students are going to use it anyway. To assume otherwise is not noble or principled, it is delusional. Pretending that this isn’t the case, and teaching the class as if it isn’t the case, is an exercise in delusion. If I stand there and tell my students “you shouldn’t try to memorize any of this”, it paints me as totally out-of-touch and makes me look like I don’t know what I’m talking about.

The other prong is, in the standard university freshman calculus course, the material is simply covered too fast for the students to gain the type of deep wonderful understanding which makes rote memorization unnecessary. I would have trouble keeping up with it if I didn’t know it all already, and I’m some kind of bulging-cranium math-genius freak! If there’s any true ideology in math, it’s the “snowball” ideology, that math builds on itself and if you fall behind, it starts snowballing out of control until you go to lecture and it sounds like the professor’s speaking Ugaritic. This kind of makes it really important not to fall behind. I would even say that (GASP!) keeping up with the material is more important than having some sort of pure godlike rote-free understanding of it. If a freshman devotes all the time it takes to deeply understand Riemannian integration (something many mathematicians don’t really accomplish until grad school), then they’re going to fail the course because they didn’t have any time left over to learn anything else.

The preceding argument is a kind of “lesser of two evils” argument, but I’ll go even further and say there’s a place for rote memorization even in an ideal self-paced program. Look at language, for example. Trying to learn a language with nothing but deep understanding would take a really freakin’ long time.

SO DOES THIS MEAN XAMUEL WILL QUIT MATH?

I’ve stepped across a pretty serious line in this article. Taking a pro-rote stance in math education is like nailing Martin Luther’s manifesto to the doors of the catholic church. So, I’m a math heretic. Hey, that sounds pretty cool, actually. Does that mean I’m quitting math?

Quite the opposite. The passion I’ve been feeling lately about math education, means that it still inspires some kind of growth and change in me. The day I stop caring and just put in the time, that’s the day I’ll quit teaching math. If I don’t get fired first for being a mathemaverick.

FURTHER READING

This isn’t the only bone I have to pick with math education. Read my article “Problems” In Mathematics to learn why freshmen shouldn’t encounter many “problems” at all. They should encounter exercises and questions.

I’m pretty rare among pure math majors as someone who really likes teaching. I’m passionate about it. Look, I even wrote an article called, How To Be A Better Teacher.

If you sympathized with the article you just read, you might also sympathize with “How To Be Better At Math“. I promise it doesn’t contain any of the usual b.s. talking points you’d expect.