Teaching is one of those things that’s always controversial.

There seems to be a constant tug of war between two approaches. In one, thought of as old-fashioned and practical, students are expected to work hard, study to memorize facts and formulas, and end up with an impressive ability to reproduce the knowledge of the past. In the other, presented as more modern or more permissive, students aren’t supposed to memorize, but to understand, to get intuition for how things work, and are expected to end up more creative and analytical, able to come up with new ideas and understand things in ways their predecessors could not. This whole thing then gets muddled further with discussions of which skills actually matter in the modern day, with the technology of the hour standing in. If adults can use calculators, why should students be able to do arithmetic? If adults can use AI, why should students be able to draw, or write, or reason?

I’ve taught a little in my day, though likely less than I should. More frequently, I’ve learned. And, with apologies to the teachers and education experts who read this blog, I’ve got my own opinion.

I don’t think anyone in the old-fashioned/new-fashioned tug of war is thinking about education right.

People talk about memorization, when they should be talking about practice.

We want kids to be able to multiply and divide numbers. That’s not because they won’t have calculators. It’s because we want to teach them things that build on top of multiplying and dividing numbers. We want some of them to learn how to multiply and divide polynomials, and if you don’t know how to multiply and divide numbers, then learning to multiply and divide polynomials is almost impossible. We want some of them to learn abstract generalizations, groups and rings and fields, and if you’re not comfortable with the basics, then learning these is almost impossible. And for everyone, we want them to get used to making a logical argument why something is true, in a context where we can easily judge whether the argument works.

This doesn’t mean that we need students to memorize their times tables, though. It helps, sure. But we don’t actually care whether students can recite 5 times 7 equals 35, that’s not our end goal. Instead, we want to make sure that students can do these operations, and that they find them easy to do. And ultimately, that doesn’t come from memorization, but from practice. It comes from using the ideas, again and again, until it’s obvious how to step ahead to the results. You can’t replicate that with pure understanding, like some more modern approaches try to. You need the “muscle memory”, and that takes real practice. But you also can’t get there by memorizing isolated facts for an exam. You need to use them.

Understanding is important too, though. We need students to know the limits of their knowledge, not just what they’ve been taught but why it’s true. It’s the only way to get adults who can generalize, who can accept that maybe there is a type of math with numbers that square to zero without dismissing it as a plot to corrupt the youth. It’s the only way to get students who can go to the next level, and the next, and then generate new knowledge on their own.

But that understanding often gets left by the wayside, when teachers forget what it’s for. If you try to teach the Pythagorean theorem by showing a few examples, or tell students stories where different types of energy are different “stuff”, you’re trying to convey an intuitive understanding, but not the useful kind. What you’re trying to give the students is stories about how things work. But the kind of understanding we need students to have isn’t of stories. It’s of justifications, and arguments. Students should understand why what they are taught is true, and understanding why doesn’t mean having a feeling in their hearts about it: it means they can convince a skeptic.

It’s easier, for a world full of overworked teachers from a variety of backgrounds, to teach the simpler versions of these. It’s easy for a traditionalist teacher to drill their students on memorization, and test them on memorization. It’s easier for a sympathetic teacher to tell students stories, based on stories the teacher thinks they understand.

But if you want the traditionalist approach to work, you have to actually do things, to practice using ideas rather than merely know them, to have that experience down as reflexively as those times tables. And if you want the modern approach to work, you have to actually understand why what you’re teaching is true, the way you would convince a skeptic that it is true, and then convey those justifications to the students.

And if you, instead, are a student:

Don’t worry about memorizing facts, you’ll drill too hard and stress yourself out. Don’t worry about finding a comfortable story, because no story is true. Use the ideas you’re learning. Use them to convince yourself, and to convince others. Use them again and again, until you reach for them as easily as breathing. When you can use what you’re learning, and know why it holds, then you’re ready to move forward.