Poetry, botany, arithmetic, memorization

August 30, 2017

It’s probably a coincidence, but last weeks NYTimes had two articles in the Sunday Review on the value of memorizing relatively arcane knowledge.

One encourages readers to commit poems to memory, and recite them to each other; the other encourages us to learn the names of the trees that surround us, and start paying attention to their active, wild lives.

Two things jump out at me as I read these articles. The first is that memorization, so often maligned in educational circles, has a place in learning. It serves a function: to focus attention, to help us notice the invisible. There is no question that the role memorization plays in education can be overstated and overplayed. In math education especially, entire subjects are reduced to incoherent laundry lists of isolated, irrelevant facts that we’re made to memorize for some reason. Still, I worry that saying memorization serves no role at all is to remove a meaningful learning tool.

The second thing I notice is my own state in reading these articles. My mind quiets and fills as I think of the poetry I know, and want to know. I want to organize a party where everyone shows up and recites a poem to the group. Even though my own knowledge of botany is laughably poor, the invitation to spend time with a tree is welcome, and while I may not seek out the opportunity to learn the names of trees, I’d certainly be happy to if the opportunity arose.

Memorization is one of the great bugaboos of mathematics learning. As someone who usually didn’t have a great memory—acquiring foreign languages, remembering biological and chemical structures, and so on was always tricky for me—math was always appealing because it didn’t require much memorization. Or if it did, it was a kind of memorization that fit my love of stories. I can remember lines in a play if I understand what the characters want, and what motivates them; in this case, their words make sense. Math is like this for me too: the unexplained pattern cries out for explanation, and the facts and formulas are short hands for the arguments that make the structure sensible. On the other hand, memorizing all the rivers in Europe was an empty exercise when I had to do that in seventh grade. Those river names are all gone from memory now. I bet if we had learned the stories of history at the same time, I would have been more interested, though; I still remember the Arno, in Italy, because in Arieti’s historical novel The Parnas the characters have to decide whether or not to flee to the river, which marks the dividing line between the Allied and Axis forces in Pisa. I care, so I remember.

The point of all this is that context matters. Unmotivated memorization is one of the great ways to kill interest. But even in an age where information is everywhere, I maintain that memorization serves a function in learning, as long as it acts as an invitation to see something new.

For example, I find being able to know and be fluent with some basic arithmetic helps you see new ideas and patterns in mathematics, just like knowing the names of trees helps you see the natural world alive around you. I once led a class where we interested in the areas of tilted squares on a grid.

What is the area of this square?

There’s no simple formula to calculate the area of this square (well, there is, in a way, but my students were young enough not to know it).  And memorizing formulas to calculate the areas of geometric shapes is usually a waste of time and energy. “Memorize arguments, not formulas,” as a professor of mine used to say. Which is to underline the point that most – probably 95% – of the memorization we ask students to do in math class is a waste of time, and counterproductive.

But there still is that 5% that’s worth it. And in this case, after we’d spent several days finding the areas of squares and organizing our lists in various ways to make sure we had them all, we had discovered this list of numbers, representing the areas of tilted squares:

2, 5, 8, 10, 13, 17, 18, 20, 25, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, …

What in the world is going on here? This is the kind of pattern that can leave us bewildered unless we actually have enough earlier patterns memorized to see structure in the noise. And this is precisely the kind of pattern that emerges in math sometimes, where there’s a deep, strange connection that reaches in from a completely different section of mathematics.

On the one hand, who cares. But on the other hand, exploring the interplay of patterns in the sequence above leads to a truly tremendous breakthrough known as quadratic reciprocity, which in its own right inspires vast new explorations in mathematical thought. But unless you have some previous patterns and arithmetic in your working memory, finding this kind of pattern is pretty much out of reach. This may seem like a peculiar example, but I think this is common in math: instant recall of certain objects, patterns, and algorithms helps us to see what can be invisible otherwise.

Forcing students to memorize something is like making them put something in their backpack before you hike up a mountain. It works best if they want to climb the mountain, and believe this tool in their backpack is going to be useful. And some tools can be picked up on the way! However, if they have to stop and go back to grab a missing tool every time they take a step, they never experience the beauty of the hike. That, to me, is the experience of the student who has to pull out a calculator when they are subtracting one and two digit numbers; they are constantly distracted from the beautiful experience of finding meaningful patterns.

So memorization has a place, but we have to be very selective. The moment the preparation goes too long, the experience becomes about being loaded up with a heavier and heavier backpack, filled with junk we don’t need, to climb a mountain we don’t want to go up anyway. Confusing memorization with learning is a huge error. Recognizing that memorizing a small body of selective knowledge at the appropriate time in order to aid learning and reveal beauty in the mathematical (or real) world is something else entirely.

Confusing memorization with learning is a huge error.

Still, I get a little nervous defending memorization at all. I recently had a back and forth on twitter with Eugenia Cheng, discussing if there’s any merit to memorizing in math at all. The pendulum has historically been so far in the direction of memorizing that the impulse to push it back is strong. If we had to memorize poems we didn’t care about (or that were in different languages we didn’t understand) for 12 years, we’d quickly come to say that memorizing poetry was pointless too. And yet, I can’t shake the sense that we are in danger of going too far; that teachers see the value in the selective use of memorization for students, and are afraid that it’s so strongly counter-indicated by experts that they should never do it.

So even as I emphasize the journey up the mountain, and all the reasons for taking it, I’ll keep reassuring teachers that they are allowed to help students memorize a select body of terms, ideas, and algorithms that will make the journey richer for them.

But keeping your eyes on the real point of learning – that’s wanting to take the trip – is always primary. As a sign off on that note, here’s a quote from Paul Lockhart’s new book Arithmetic: “There are a lot of people who hate arithmetic (far too many to count!), and it makes me sad. Usually it’s because they were made to do something they weren’t interested in doing. Let’s not have that be you.”


One thing I realized in looking this post over is that I didn’t really define what memorization means to me. I tend to find context-less memorization (mnemonic devices and the like) useless or worse than useless. For me, useful memorization is the act of going over facts and ideas that you understand to keep the details fresh, and which might slip away from you otherwise. It’s memorizing lines in a play where you have a sense of the motivations of the characters, as opposed to the useless memorizing lines in a language you don’t understand. Sometimes definitions and vocabulary need to be memorized, but I usually like those to come from a genuine need to name something as well. So I guess there’s much more I need to say on this topic to articulate my own thinking.

Ben Orlin has some nice articles on the evils and value of memorizing in learning. (Interestingly, I don’t agree entirely with his “memorize on Monday” idea, but both articles are still definitely worth reading.)

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