Lots of teachers disagree about whether students need to know their times tables. I’m teaching 7th grade for the first time and I’m thinking a lot about times tables, because some of my students aren’t fluent and I want to understand whether or not that matters.

Context: I’m teaching proportions right now. Here’s a problem I recently gave my students:

I think a reasonable person does this with a calculator. Sure you could do it by hand, but why? Real humans who want to avoid making a mistake definitely use calculators.

Here’s another problem:

Sure, this one is totally reasonable with mental math. But if it’s ok to allow a calculator for the problem above, why not for this one?

Here’s my take:

Times tables aren’t useful for solving problems, they’re useful for learning to solve problems.

Let’s look at the two problems again. The first problem I wouldn’t want to use when introducing a new concept. There’s too much going on. But the second problem is a great introduction to proportions. If students can see that each number on the right is three times the number on the left, they will have a better understanding of the patterns in the table and the idea of a proportional relationship. They could use a calculator to see that pattern — but all the effort involved in using a calculator saps valuable working memory that could focus on making other connections.

Ok so takeaway one: times tables maybe aren’t useful for solving problems, they’re useful for learning to solve problems. Here’s takeaway two: the traditional 12×12 times table is silly and arbitrary. What’s so special about 12? After teaching 7th grade for the last six weeks, here are the times tables I wish my students knew:

Ok the tens look a little awkward, but the idea is simple. I can’t use the same numbers in every problem, so I would love students to be fluent in a decent variety of multiplication facts. But I don’t care about 9×7, or 12×8, or 6×11. It’s been rare in the last few weeks of my class that those facts come up.

Final point. It’s not only about multiplication. If a students knows that 4×6=24, that’s awesome. But it is just as important that they know that 24 / 6 = 4. And one piece of knowledge that keeps coming up, and that I think is another distinct skill, is knowing that if 4 x something = 24, that something is 6. Each of these facts creates opportunities to make connections and better understand new ideas because of the working memory they make available while looking at a new problem.

To summarize: times tables are important, but it’s all about quality over quantity. I don’t need students to know every fact, I need them to know a smaller subset forwards, backwards, and sideways to free up space in working memory. And the point of freeing up that space isn’t to solve problems, it’s to notice other patterns while solving problems that help students learn new ideas.