At *The Atlantic* today, I have an essay weighing in on the decades-long debate over memorization, trying to cut a middle path between two extremes:

1. “Memorization is the enemy. It’s the antithesis of critical thinking and conceptual learning. Memorization’s defenders are wilfully blind soldiers marching for an outdated tradition.”

2. “Memorization is an essential tool for students. It’s the surest path to retaining important facts. People who denounce it are letting liberal orthodoxy get in the way of our children’s achievement.”

I’d summarize my view along these lines:

3. “Memorization is a generally-not-great shortcut. It’s better than not knowing at all, but it’s not nearly as enduring, effective, and powerful as meaningful learning.”

In math, the classic example of a thing that “must” be memorized is the times tables. Amidst the din of debates about education, the one agreed-upon truth seems to be that all kids ought learn their times tables. It’s comparable to the moral rightness of *Brown v. Board of Education*–a universally accepted belief.

That makes the times tables a good test case for whether memorization is really necessary. So here’s my times-table story, in which a colleague succinctly captures the entire point of my *Atlantic* piece:

A friend who teaches Physics once told me that, growing up in Moscow, he’d never learned the times tables. “I don’t really get the American system, with the flashcards and the multiplication facts,” he told me.

“Really?” I said. “You don’t know 9 times 7?”

“63,” he said without hesitation. “I just picture ten 7s, which is 70, and then I take away one 7, which leaves 63.”

“You can’t be thinking it through like that,” I said. “You answered too fast.”

“Well, that’s how I learned it at first. But if something makes sense to you, and you use it enough, you don’t need to memorize it.” He shrugged. “You just know it.”

Anyway, go check out the piece!

EDIT 9/11/2013: Just wanted to address two criticisms of the piece, each quite valid, I think. First:

@benorlin May not have been intent, but opening of piece read a bit like you were proud to catch & fail student, which I found unfortunate.

— Patrick Honner (@MrHonner) September 11, 2013

I regret writing so flippantly about that case of cheating. It’s a complex, sad story (mine was far from the only test he cheated on), and I’m relieved to say it has a happy ending. The student learned his lessons, got into a good college, and I know he’ll do great things in the future. He won back my respect, and then some, to the point where I’m very proud to have known and taught him.

Second, the more common complaint:

**Sometimes, memorization is really useful. Even necessary.**

It depends on how you define memorization. If you accept my (admittedly debateable) definition, then “memorization” refers to the combination of rote learning and memory tricks (like mnemonics). These are almost never ideal. Memorization treats facts as arbitrary and interchangeable, which is silly. You’ll remember facts better, and use them more effectively, if you treat them as the interconnected web they are.

The exception–which nobody, from what I’ve seen, has really prosecuted–is vocabulary. By nature, words in a language bear only an arbitrary, symbolic relationship to the concepts they signify. For learning vocabulary (especially in a foreign language), memorization is a useful tool, even under my limited definition.

Your example is a good one. But, your Russian friend has indeed memorized the multiplication facts. He has memorized in a different, and better, way than most students in America, but without automaticity of basic facts, he wouldn’t today be a Physicist.

Great example. The difference is that he has a way to work it out when he doesn’t know the answer straight away. Students who memorise often don’t.

Definitely! That’s a major advantage of acquiring knowledge this way, instead of by rote: it’s much easier to reconstruct.

(Most memory works by reconstruction, anyway–our methods should probably embrace that fact.)

Agreed.

I think the only difference is how we define “memorization.” I define it as “learning a fact through deliberate effort, in isolation from other facts.” Under that definition, my friend didn’t memorize anything, and in general, memorization is rarely the best alternative.

If you use a broader definition–like “taking steps to remember something long-term”–then my friend’s strategy counts as memorization, and in general, memorization of that type is very valuable.

Memorization is where to start from and build upon- the template which serves as guide and reference for all later information…

I think that’s a fair stance–one of the reasons memorization isn’t totally evil.

Sometimes, the rote memorization comes first and then as we use the recalled facts, the connections and non-rote features of what we’ve memorized have a chance to surface. This was certainly true for me with the multiplication tables.

That’s true, and well-said.

For some students, these connections will inevitably surface, given enough time. For other students, it takes deliberate effort on the teacher’s part to bring them to light. It’s for the latter students that rote memorization poses a greater danger–once they’ve learned a fact, they feel that they know it, and have no need to inspect it further, which closes the door on understanding deeper features.

Whenever I’m teaching my math classes, I try as hard as I can to try and explain not only how to do each step, but WHY I am doing each step. I can’t speak for other subjects, but I feel like a) most math teachers don’t bother to explain the ‘why’, and b) most students don’t care about the ‘why’.

So how do we fix those issues? I haven’t fully figured it out yet.

I think both issues are not mutually exclusive (yay math terms!), either. The students’ lack of caring about deep understanding in mathematics leads to frustration and abandonment of its teaching. I think every math teacher has done it — exasperated, you exhale loudly and go “You do this and get the answer.” It’s pretty discouraging to do.

For example, I was teaching my Algebra II students the other day how to work with fractions in equations (I have a saying — ‘fractions are your friend’ — oh, no? well, at least be their acquaintance). My method is getting them to find a common denominator for each term, put each term in that common denominator and finally rewrite the equation with only numerators, essentially making the denominators — and fractions — ‘disappear.’ At first I introduce it as a trick, like a magician, but I make sure to explain many times exactly what mathematical procedure I am doing (multiplying each term by its now common denominator). I explain that it comes into play in other topics — like literal equations and finding a single variable — when multiplying by a denominator is important. I’d like to think that this explanation helped their understanding more than just saying “hey guys, remember the trick I taught you!” but I really can’t be sure.

The way I try to assess deep understanding is a written or verbal explanation of the problem. (Oh my gosh, writing and speaking in math?! NEVER!) Sometimes, I’ll have the kids get up and ‘teach’ the class and make them explain exactly what they did. I’m there for support — moral and mathematical — but hearing a student be able to explain it like I do is, to me, the final level of understanding.

I also hate memorization of formulas. I am getting ready to teach factorization of the difference of cubes, and, as I write this, I don’t remember the formula. This is the first time I’m teaching it. I’m a math teacher…so what’s the point of making a bunch of juniors memorize it? Knowing when to use it is the most important aspect.

Anyway, it’s sometimes difficult to translate abstract concepts to their usefulness in everyday life — which, largely, is problem-solving skills in the real world. When will you need to be able to factor the difference of cubes? Likely never, but knowing how to problem solve will be extremely useful later in life.

Agreed. I certainly didn’t know the formula for factorizing a difference of cubes the first time I taught it–which is fine, because it’s easy enough to determine if you just divide the original expression by the linear factor.

I know that resigned feeling of, “Ugh, fine, here’s how you do it.” I try to avoid it when I’m teaching a class, but when you’re tutoring another class, or preparing a kid for a test, it’s sometimes hard to escape.

One reason to seek computational proficiency in our students–not a very highminded reason, but a real one–is to prepare them for later math classes. You need strong algebra skills to thrive in college math, and while you might not need college math to thrive in the real world, it sure opens a lot of doors.

Ben

Loved The Atlantic piece. Shared it with my department and was just discussing some of the issues with my physics teacher colleague. I have to admit that after 27 years I have not come to peace with the balance between valuing facts and equation recall with believing (REALLY believing) that testing thinking needs to be my primary goal. You seem to advocate the formula sheet in the article – do you allow it/recommend it in practice? I do agree with some of the commenters there that a well put together formula sheet rarely gets referenced due to its helpful organizing power. How do you handle this in your day to day class life?

Glad you enjoyed the piece!

The irony of my essay is that, even after the cheating experience I describe, I still didn’t allow a formula sheet. In trigonometry, I make a few concessions, giving them the Law of Cosines and sometimes the formula for cos(a-b) (but none of the related formulas).

I expect them to deduce the rest of their formulas. All the trigonometric identities are variations on a few key ones (sin^2 + cos^2 = 1, cos(pi/2 – x) = sin(x), cos(a-b) = cos(a)cos(b) + sin(a)sin(b)). Having spent several weeks in class seeing how these formulas connect, and using them nightly on the homework, students should know them cold for quizzes and tests.

Sometimes I’ll assign a formula sheet for homework, because I do agree that making the sheet can be a helpful exercise. But I don’t allow them to refer to it during assessments.

I have to confess that some students probably memorize those formulas in precisely the way I counsel against in the piece. But I urge them NOT to memorize by rote or by mnemonic trick, and most seem to heed that advice. They mostly retain them by what I consider better methods–repeated use, and meaningful connections to other knowledge.

Even though I remember memorizing my times tables in the 4th grade, there are very few Products that I quote from memory. (7×8=56 is one example of strict memorization. I was having trouble remembering the answer until I realized that the four digits in the equation are a “straight” – 5678.) For just about every other combination of multiplier/multiplicand I feel like I don’t trust my memorization, such that even if I do remember the answer, I check it before proceeding… and I use tricks like your Russian friend.

In the case of 9 * x = 10x – x… the two operations on the right side of the equation are a lot easier and quicker to do in my head than the one on the left. BONUS: once you learn a trick like this with small numbers, you can apply it to tougher situations. For example, I can start with 45 x 9 and arrive at 405 very quickly by turning it into 450 – 45; and certainly a lot faster than “9×5=45, hold the 5, carry the 4, 9×4=36, plus 4=40, wait, what was the first digit I was holding?”. LESSON: It is better to memorize a single method (which you retain by usage) than several individual answers.

That’s a great example of how limited memorization’s power is compared with actual understanding. A memorized fact sits in place, isolated. But a deep understanding you can carry with you, and use it to acquire other facts.

This is why I don’t make my chemistry students memorize the periodic table. They already know a few elements (hydrogen and oxygen, from H2O), but probably haven’t grokked the association of O = oxygen before. They will learn a few more elemental symbols by the end of the first semester. But I really care about how they use the information. Just *knowing* that CO2 has carbon and oxygen in it tell you nothing about its behavior. Because of the elements’ positions in the periodic table, CO2 is likely to (and does indeed have) strong covalent bonds, is very non-polar, and therefore will interact well only with other non-polar molecules… now that’s more useful.

That makes sense–the parallels between chemistry and math seem pretty strong.

I remember my own (totally brilliant) high school chemistry teacher made us memorize the first three periods. That one’s an instance of memorization I think worked beautifully–it helped to have some structure at our fingertips as we learned the deeper content. But it’s only worthwhile if you actually GET to the deeper content, and I can see the advantages of not asking kids to memorize that.

When you say “memorized the first three periods”, did you memorize just element and symbol, or positions, or properties or…? Most people only memorize element and symbol, which is just an abbreviation. It’s like memorizing all of the two-letter state codes for the post office (which I had to do in 6th grade), and isn’t super useful, especially now that digital communication is much more prevalent.

Devil’s Advocate:

Did memorizing the first three periods of the table help more than, say, just looking at a printed table while discussing deeper content?

What I memorized was a string of syllables (“H-He, LiBeB C NOFNe…”) that allowed me to reconstruct the positions of the first handful of elements. It wasn’t super-emphasized–just part of one night’s homework–and I don’t think it made a HUGE difference, though it lubricated the problem-solving a little to know off the top of my head that, say, Lithium and Sodium were both alkali metals. We always had access to a periodic table on tests, though.

Actually, I think that kinda proves my point. Your strings of syllables don’t tell you that Li and Na are alkali metals (because families run vertically, not horizontally). You remembered the alkali fact on its own.

Now, if we made student remember elements vertically, then it might be more helpful.

That’s fair–I think what we’re saying is compatible. Having facts at your fingertips can be valuable, especially early in learning a subject; but it’s a very small part of the picture, and absolutely shouldn’t preclude learning the important stuff.

Some thoughts on memorization:

Memorization is a stop-gap measure that allows you to use a fact while gaining enough experience to internalize it.

Rote memorization is important when you kind-of know something. For example, if you see sines and cosines repeatedly and know that sine/cosine takes things with {0,1,2,3,4,6,pi, /} to things with {0,1,2,3,/, sqrt}, then memorization helps you keep track of what exactly goes to what. For me, when I develop a general intuition about some fact, I’ve only really nailed down that fact in concept space to a small neighborhood of the fact I want to remember. Memorization helps distinguish the real fact from small deviations that might seem true based on my intuition.

Reconstruction doesn’t answer the question: if you can reconstruct 7*9 = 63 by a simple modification of 7*10 = 70, then you’re really just reducing the problem to rote memorization of another fact.

You can’t make connections between facts that aren’t in your head to begin with.

On this track, you have to start somewhere. When begins as an isolated fact, memorized purely for its own sake with no connections to other knowledge, later becomes a point of contact to which you attach new facts as you learn. When I learned European History, the year 1517 started out as a meaningless date that I needed to memorize and only later did I gather enough other facts to understand its importance. In fact, it became an anchor that secured my understanding of the Reformation.

That’s a good perspective. I think you’re quite right about the role of some memorization as an (often useful) intermediate step between ignorance and well-integrated conceptual knowledge. One might say that learning often begins with a small act arbitrary memorization, and by the end, the arbitrariness is gone, though the fact remains.

A rock-climbing analogy I use: A memorized fact is like a hand-hold. Memorize too little, and you can’t climb at all. Memorize too much, and you’re not really climbing.

Still, when possible (which isn’t always), it helps to elaborate on connections immediately, while the “arbitrary” new fact is still sitting in short-term memory.

For example: When I define “even function” for my students, I show them a verbal definition (“opposite inputs give the same output”), a symbolic definition (“f(x) = f(-x)”), and some sample functions, while asking them several quick questions (“If (2,1) is on the graph of an even function, what other point is?” and “Is f(x) = c even?”) By the time we’re done, 5-10 minutes later, they’ve internalized the definition by connecting the various representations of a function. There’s no particular need for memorization.

That said, your example (the values of sine and cosine for common angles) may be one where a little memorization really is the best course.

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I non potevo astenersi dal commentare . Perfectly scritto !

This post is worth everyone’s attention. How can I find out more?

Hi Ben, thank you for this blog. Truly. Have you seen the study on elementary students and arithmetic by Gray and Tall? http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1994a-gray-jrme.pdf

They found that students who were successful in math (as judged by their teachers), used a combination of memorized facts and derived facts whereas less successful students relied on counting. Memorization does have it’s place, but only alongside reasoned thinking.

its not it’s!

If you understand everything but still sometimes at the exam time you might forget a bit for that memorising is a handy tool. I will say learn everything and then memorise it all that way you know it and remember small details for longer time and even get better marks .

My daughter had a hard time learning the multiplication table.

So I wrote an android application to help her 🙂

Each exercise you solve helps you to feed the cute panda.

You can get daily reminders to feed the panda (solve exercises) and you get rewarded with trophies upon goals completion.

This application helped her to learn the multiplication table without noticing 🙂

* Currently the application is only for Android phones *

https://goo.gl/9QutZx