This post is by James.
About two months ago, I signed up for Quora. Its been interesting following the questions and answers there, whether I agree with the most popular answers or not. I feel that I get more information the questions the answers. What people are curious about, and willing to ask anonymously on the Internet, shows you things about people that I’m not likely to run into in my everyday life.
The most useful feed for me is Math Education. I range from “expert” to “generally competent” in the other topics I follow, but because I am not personally a teacher (that’s Jason, my brother), the Math Education forum on Quora is a great way to discover common ideas about what math is, what it is used for, how to learn, and so on. In particular, you manage to read stories which really bring home how much “math as many people learn it” and “math as STEM professionals use it” are different.
Which brings me to today’s topic. Recently I saw two different posts on Quora which brought up a common theme. The first was a question:
“Do good math students do anything besides purely memorizing formulas and theorems when it comes to tests?”
The second post was an answer to a different question. I don’t remember exactly what the question was, but it involved memorization. The answer was from a mathematician in India who told a truly terrible story about how she grew up being told by her family she was bad at math, and would never amount to anything in the world, because she had trouble memorizing the multiplication table up to 20 by 20. (The ending was less depressing; she found her own way at college and discovered that nobody else important cared at all. She became a mathematician and is more than happy to put that behind her.)
Both of these touch on the question: what is the relationship between mathematics and memorization? Both of the questions start with the assumption that memorization is an important part of mathematics, and continue from there. Large parts of the answers are teachers/other students/college professors saying that it really isn’t. What’s going on here?
Before we get deeper into this problem, it will be good to define a couple of things. First and most importantly, what is memorization? Like many words, it’s hard to pin down. For instance, the first definition that pops up when I Google “definition memorize” is “commit to memory; learn by heart.” This isn’t really what I’m talking about today. By this definition, if you’ve learned something well enough not have to look it up, then you’ve memorized it. Some of the synonyms I find online include “learn; understand; recall.” I doubt the people answering the Quora questions above would have answered the same way if the questioners had replaced “memorize” with “learn” or “recall.”
This means that in this post, we need to be more careful. Language is tricky and the above definitions just aren’t good enough to really answer the questions people were asking on Quora. This is because if you used the most general definitions of memorization I found online, then there are definitely things most people need to memorize in order to be fluent in math. Being able to quickly recall the addition and times tables up to about 10×10 or 12×12 is very helpful, even in advanced mathematics. How you become fast at arithmetic is in theory not important, but in practice, most people need to memorize something.
So what are math educators talking about when they condemn memorization? Well, some of them are actually talking about knowing the basic math facts. This is surprising to me, but it turns out this issue is a bit contentious. From the context of the answers, though, I think that far more of them are talking about a different meaning of memorization. I’m going to call this kind of memorization “foreign song memorization,” named after one of the most common places where this kind of memorization happens.
This name comes from personal experience. Despite my families constant disbelief, I have actually been in a couple of choirs which didn’t immediately kick me out after listening to me sing (to be fair, these are the kind of choirs which don’t kick anybody out.). In a few of these choirs, we’ve actually been kind of ambitious and sung things in other languages, like Italian or Latin. When we did this, we generally had to learn how to pronounce the words on the page. This means that we had to do a lot of memorizing in order to know how the song was supposed to be sung. We’d often spend a lot of time carefully practicing the sounds and making sure we had the whole sequence down and in the right order. The music helped, of course, because we could use the written words as triggers to remember what we were supposed to sing.
One thing we practically never learned was what the words meant. Even if the choir director knew what the words meant, they seldom taught us; we just didn’t have enough time. We did sometimes spend quite a while learning how to read the language phonetically, because that was very important in getting the sound right. When we were finished, though, none of us would ever claim that we had actually learned Latin. Unless we did extra work on our own time, none of us would claim that we had learned even a little bit of Latin. We had the sounds right (maybe), but we had no understanding. If you had dropped us somewhere where the original language was spoken, we would be helpless.
This is the kind of memorization that drives math teachers bonkers. When students treat learning math facts the way we treated our Latin texts, it can be very impressive—as long as you stick to the script. But as I’ve said elsewhere on this site, math is a language. Drop somebody who has learned math like I have “learned” Latin into a place where real ideas are being discussed, where there are real problems to solve, and they will instantly be in over their head. I suspect that many former students, faced with this, decide that they are simply bad at math, when the truth is that they have simply learned the wrong things.
Exactly why this type of memorization is bad is a topic for a different post. Today I’ll just point out the biggest problem: when you make a mistake in repeating something memorized this way, there is no way for you to tell. If you don’t know the language, then you could easily say “he” where you are supposed to say “she,” or say “tater-roll” where you were supposed to say “table.” You can easily spout total nonsense with no way of knowing.
So, what to do? Well, as I noted above, students still need to learn their math facts. Fluency requires it. It would be a very odd foreign language class which didn’t include vocabulary and grammar, and math facts are the vocabulary and grammar of arithmetic.
So lets extend the analogy. When a student learns a foreign language, they do need to spend some real time learning the vocabulary. Good teachers do not stop there, though. They also give the students lots of time to practice their vocabulary in “real” situations, either with themselves or other students. They force the students to use the words they have learned in situations which aren’t in the book and weren’t in the lesson. They expect the students to understand. And when they test, they test for understanding. One method I remember from my German classes was watching a video with two people talking in German. After watching it, we were forced to answer detailed questions about what the people were talking about and doing. These situations used words from the class, but the actual situation wasn’t in the book or in the class. We were expected to understand it anyway.
I think that this model is how we must understand math, and especially the changes in math. Many of the attempts to reform math are simply ways to make the questions we ask the students more like a foreign language class and less like singing Latin in a choir. We throw new situations at students, confident that if they are learning the language, they will be able to handle it. We ask questions that they have never seen before, and we provide more context for them to do it in. In practice, this means more word problems and more careful explanation than is required if we just want them to repeat the correct words.
As for memorizing the math facts—we have to do it. The good news is that we probably don’t have to do it all by simply reciting the facts. Practicing speaking a language with other students helps the student remember the vocabulary. Every time the student has to pull a word or a fact from deep in their mind, it is made stronger and easier to remember. Time that we lose in raw memorization is recovered during actual use.
As for actually doing it—well, that’s not really my specialty. It’s what many teachers and educators are trying to do, and it’s certainly what actually worked with me. All I’m saying with this blog post is: help them out. When it looks like math is being made more complex than it really is, ask yourself: will this question make my child more fluent in math? Is there an idea that this teacher is trying to get across which perhaps my teacher didn’t teach me? Even better, ask the teacher what they are trying to do, and if the goal is fluency, help them out. It is hard work. We need help.