What People Get Wrong About Memorizing Math Facts

I haven’t done a survey or anything, but I’m pretty sure the most common way people try to teach math facts (like 3 x 7 or 9 + 6) is by giving kids a bunch of problems on a page and asking them to solve them. Sometimes the activity is timed, sometimes not, but that’s pretty much it. I think the lesson starts with putting, like, 10 problems on the board, and then sort of asking “ok so who knows how to do 3 x 7, Timmy do you know” until all the problems are solved, and then they move to the worksheet.

One of the most popular curriculums in the US for elementary school is EngageNY/Eureka, and they have a version of this activity that they call “Sprints.” A “sprint” is 44 problems that you’re supposed to solve as quickly as you can. It’s a bunch of problems on a page.

Some people love these activities, but a lot of people in math education hate them. Many of them are haunted by memories of “Mad Minute” drills in school, where you’re supposed to answer as many math fact questions as you can in a minute — that’s mad. These opponents prefer non-stressful practice that doesn’t have a time constraint. In their classrooms the facts are maybe presented one at a time on the board, and students are asked to derive the solution any way they like. Then they talk about the strategies people used. It’s not problems on a page, it’s a problem on the board, and we’re talking about it.

A good example of a proponent of this view is Jo Boaler, whose YouCubed has a position paper titled “Fluency Without Fear.” Besides for number talks (talking about a problem on the board) she also calls for practice that is fun and engaging, stuff like dice games and puzzles. As usual, the essay talks a lot about “brain research” that supposedly proves that this stuff works. Sure it does, whatever. This is the other perspective.

And what I’d argue is that actually both of these approaches make the same mistake, and it’s a big one. Because what both approaches assume is that if kids solve a problem a bunch of times, they will commit it to memory. And the thing is that this is not true, at least not true enough.

What we need is a theory — an explanation, really — for when people remember something. Let’s not make it complicated, we can put it very simply: people remember something when they’ve successfully remembered it a bunch of times. I don’t think this is saying anything that retrieval practice advocates haven’t already said — the best practice for remembering something is practicing remembering it.

Suppose that I ask you to find the product of 12 and 5. What is going on in your head as you answer the question? Maybe you start counting by 5s. Or maybe you remember that 12 x 4 is 48, and you do 48 + 12. Or you grab a piece of paper and start adding 12s. None of this is retrieval practice — none of this is practicing pulling the fact out of memory.

(To be fair, maybe you did practice pulling 12 x 4 out of memory. And maybe, when we’re talking about this problem after you solve it, you’ll end up having to remember that your answer to 12 x 5 was 60. But neither of these are sure things, though they might help if you keep doing them for enough time.)

There is a very simple question we can ask to see if math fact practice is likely to help in the most direct way: Are kids practicing remembering? Or are they practicing something else?

Let’s apply this test to EngageNY’s Sprints. Will kids be practicing remembering when working on them? Here’s a basic point: not unless they are successful. If a kid is unsuccessful at pulling a math fact out of memory, what are they going to do? They are going to try to derive it, using some sort of strategy. Maybe they’ll try using the most basic sort of strategy, something like skip-counting for multiplication or counting on fingers for addition. But of course they’ll derive — what else are they supposed to do?

The “Sprint” aspect of this is encouraging kids to move quickly, which is really only possible if they have very efficient strategies or have many facts in memory. True, if kids are successful then they’ll get some good practice. But if kids are not successful, they will be forced to derive solutions, which will take more time and is not the sort of thing we were trying to help kids get better at. They will complete much fewer problems, especially if they thoughtfully arrive at solutions to the questions. It’s a trap. Not hard to believe that this stresses some kids out.

Which means that “problems on a page” is lousy practice that helps strong kids get stronger but leaves students who know fewer facts spending time practicing something else, a.k.a. Not Very Good News.

OK, but we have precisely the same problem with the “Fluency Without Fear” activities. What are kids thinking about during a Number Talk? They are thinking about — the whole point is to think about — the various strategies that we’ve used to derive some fact. That’s great if you’re trying to study strategies. But it’s not giving anyone a chance to practice remembering stuff.

(And I really do think it’s good to teach strategies. For some facts, like 9 + 8, a lot of successful people just use a very efficient strategy and never end up memorizing it. Plus, you can’t memorize every useful fact, at some point mental strategies come into play. But also because it’s easier to memorize 9 x 5 if you know that it’s going to end in a 5 or a 0, easier to remember 7 x 9 if you know the digits are going to sum to 9, easier to remember 9 + 8 if you know it’ll be in the teens. Strategies are useful, but practicing with strategies isn’t retrieval practice.)

In sum: the vast majority of math fact instruction doesn’t focus students on the thing that it’s purporting to teach. Good news, though, it’s far from impossible to engineer practice that does focus on memorization. But how?

The basic answer here is “flash cards,” which has two big advantages. First, if you fail you can turn over the card and stick the fact back in your memory, then try again. You aren’t left to derive the fact (though you can). Second, you can repeat problems frequently to practice the problems that you didn’t answer correctly.

There are more complicated things to say. Brian Stockus just wrote a great post showing how he is doing one-on-one flashcard work with his daughter. I’ve written about some of the ways I’ve used flashcards in my 3rd and 4th Grade classrooms. And of course there are a million computer programs that promise to help teach kids math facts…they all are basically flashcards, each and every one of them, combined with some sort of gamey practice. You want research on how to help kids with difficulty learning facts how to learn facts? You’re going to find a lot of flashcards.

Nothing is a sure thing, and I don’t mean to make this sound easy. There are no guarantees in teaching, especially when you’re working with a whole class. Follow some of the links above and you’ll find lots of practical advice on how to manage the difficulties. Given our focus on retrieval practice, it goes without saying that you should only introduce a few new facts at a time, and the goal needs to be for students to be successful at remembering them by the end of the practice session.

There’s no point bemoaning the state of discourse in education, it’s bad, everyone knows it’s bad. Stop me if it sounds like I’m bemoaning, but it seems to me that pretty much every discussion about math facts misses the point, viz. everything I said above. People don’t ask the right question, which is “how do kids remember stuff?” Or rather they do, but answer the question in clearly insufficient ways.

People do not necessarily remember the things they derive. Repeatedly deriving something in a way is practice avoiding retrieval from memory, which is (I admit!) a very mathematical thing to do. Mathematicians love talking about formulas that they derive every time and can never seem to remember. These theorems or formulas aren’t anything but upper-level math facts.

So we should remember that this is a real phenomenon, and that it’s true for little kids as well. If you want people to remember something, it’s often not enough to get them to derive it, whether on a big page of problems or as part of a number talk. As usual in education, the people with the strongest opinions have missed the point, and apologies for just a bit of bemoaning.