Flash cards don't always work for multiplication facts

What makes math different and why that changes how we memorize

Say that you want to memorize the capitals of the United States. Most of us could probably pull this off. If I was a different kind of nerd as a kid — more interested in facts than I’ve ever been — I might have ended up committing them to memory. As is, I know very few. But if I wanted to learn them now, I’d just make a bunch of flashcards and start practicing.

I think it’s important to understand why this doesn’t necessarily work for math. Or at least why it’s not so simple.

When I practice state capitals, there is only one way to get the question right: I have to remember the answer. I guess I could also come up with little invented mnemonics that would help — this being maybe a kind of procedural knowledge rather than declarative knowledge— but mnemonics are in general not very useful procedures and I’ll mostly have to rely on retrieving the answer from memory.

Math, especially single-digit multiplication, isn’t like that. If I see 4 x 7 on a card, I don’t have to remember that the answer is 28. I have options. I can also figure it out.

I can figure it out slowly, by drawing four groups of 7s and counting them one-by-one. I can figure it out efficiently by remembering that 3 x 7 is 21, and adding another 7. Either way, my memory of “28” need not be involved.

OK and this is important! It’s maybe the most important fact about memorizing things in math. Because it means that there are a lot of fake-memorization activities in mathematics. There are a lot of things that look like activities that lead to memorization in subjects like history, science, geography, etc., but they don’t work for math. Because math is special. Math can be derived.

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Ted Hasselbring is a researcher in Special Education at Vanderbilt.

(Incidentally, and this is a total aside, I brought my kids to a Columbia University Women’s Basketball game on Sunday against Vandy. Columbia went up quickly in the first quarter but collapsed in the second half. Final score, 74-63. It was the first my son experienced the team he was rooting for lose. So, yeah, f*** you, Vanderbilt!)

In one study he conducted, Hasselbring gathered a bunch of kids with learning disabilities who didn’t know their addition facts. He put them in front of a drill-and-practice game on the computer. (This is in 1988, by the way.) And, then, basically, he watched. What was interesting was that kids who solved the problems by counting never ended up memorizing anything. They just kept on counting:

In a recent study examing the effect of arcade-type drill and practice software on the automatization of addition facts, we found that this form of drill and practice had no effect on developing automaticity in learning handicapped youngsters unless they had established declarative knowledge networks between problems and answers before engaging in the drill and practice. Even after as many as 70 sessions on the computer, children who came to the activity using counting strategies to solve basic facts left the activity using the same counting strategies.

I have seen the same thing in my own classrooms, especially for multiplication. (Addition may be different in important ways, but we’ll have to talk about that some other time.¹) Kids who derive multiplication facts often never end up moving to memorization. They just keep on deriving, and deriving, and deriving. Unless the teacher designs the activity in a way that makes it harder to use procedural approaches, memorization will often never happen.

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I have written about this before, for example here and here. But this week I feel like I finally figured out how to make this work. Here’s the routine:

1. “Hey 4th Graders, take out your card decks. Now flip through and pick three cards that you don’t yet know by heart. Or if you know them all by heart, the three that are hardest to remember. OK fine yes if you know them all perfectly than just pick some cards you want to know better.”

2. “Now everybody, just practice those three until you know them by heart. Try them backwards and forwards — look at 56 and try to remember what’s on the other side. Do this until it’s easy. I need quiet.”

3. “What I want everyone to do is pick 10 more cards from your deck that you already know pretty well. Mix these up so that you have 13 cards now. And I want you to practice these until they’re all easy. Why are you sharpening your pencil. Please sit down."”

4. “By the way! If you get a card wrong, or if you don’t know it by heart, no big deal! Look at it, and then don’t put it in the back of the deck. Put it just a card or two away. And if you get it wrong again? No big deal! Just review it and put it back into the deck, a card or two back. No you can’t go to the bathroom right now.”

5. “You can put all those cards into the mix then!”

This routine sticks very closely to what Hasselbring recommends in that paper. His list for making flashcard activities work for memorization:

1. Determine learner's level of automaticity.

2. Build on existing declarative knowledge.

3. Instruct on a small set of target facts.

4. Use controlled response times.

5. Intersperse automatized with targeted nonautomatized facts during instruction.

I’m asking students to determine their level of automaticity. I’m asking them to instruct themselves on a small set of target facts. I’m encouraging them to not spend too long thinking about each card, so that they’re focusing on remembering and not deriving. And I’m asking them to mix up those 3 cards with the 10. This isn’t super-complicated, most fact apps do something similar, but I hadn’t figured out how to do it with my whole group at once until this week.

I’m trying to do this for ~5 minutes a few times a week. I think it’s helping, though I still walk around and find kids feeling frustrated, staring at a card and trying to derive it quickly. “Let’s flip it over,” I tell them, and then put the card right back in their deck. And I watch until they get it right.

Math facts are one of those topics that people like to yell about. And it’s fun to write about things when you think pretty much everybody is wrong, that’s always a good time. But part of what’s going on here is that lots of people say that math fact instruction stresses them out, and as a math teacher and a person who just wants people to feel good, this makes me sad.

And as I guess you’d expect, I think the problem is not math fact instruction but bad instruction. I think a lot of math fact practice puts students in a difficult position, putting them in a position where they are clearly expected to retrieve facts by memory when they are only able to derive them. Imagine you’re taking a state capital quiz and you’ve got a list of the capitals but it’s written in a code that takes three-steps to translate. And you’re sitting there carefully decoding the answers while time runs out, in a totally silent room, where your neighbors are scribbling away. What a nightmare.

But the key is to understand that math is different, that even things that we just know we can also derive, and that a teacher has to engineer situations where kids get to focus on just one or the other.

1. Let’s put it like this: Fuchs et al got great results from an intervention that taught kids explicitly to count up to solve addition and subtraction problems. It did as well as memorization drill did. All this is wrapped up somewhat in a confusing controversy about whether competent adults actually do some sort of ultra-efficient derivation for addition also, see for example this. I don’t know where that controversy ends up, probably with addition being memorized, but the fact that it’s at all controversial I think sort of goes to show that addition procedures can be pretty great! Multiplication, less so.