I was doing a bit of random lunch-hour web-surfing, and came across a blog post by a swimming instructor. It starts with the eye-catching line “before you can teach something, you have to realize it’s *hard*”

The blogger writes about their insights into how (and how not) to teach swimming, and then wonders “*how much this applies to other areas (teaching math in elementary school, for example?)*” Having read the post, I’d say an awful lot does. Here’s my take on it.

The first thing they mention is the difference between beginner and expert learners. If you teach quadratic equations by giving examples and solving them, a good math student will quickly learn. A beginner math student might never learn from your examples. The reason is, there are zillions of possible scribbles you could put on a board, but the good math student already knows the rules of the game. They know what kinds of scribbles make sense in math, and they can see the sense in your example. To the beginner, your quadratic equation looks no more or less sensible than a random string of symbols. Let me paraphrase a passage from the swimming blog post :

“In order to bring a math beginner to the point of solving quadratic equations, you have to teach them, one at a time, a long list of math skills that have to be learned well enough to come naturally before you can move on. You have to teach them arithmetic, and you can’t just tell them to the rules; you have to tell them, one at a time, which numbers to write for each problem and correct their mistakes, and then you can let go. You have to teach them how there can be rules mapping numbers to other numbers. You have to slowly introduce algebra – how symbols can represent numbers; and slowly shape in their minds that the rules of algebra are just the rules of numbers in abstract form – until these rules become something that will actually move them through a problem. And then you can teach them the steps for solving quadratic equations, which comes with its own pages-long list of common errors or special cases and ways to fix them”

Schoolkids spend years at school learning the details. Math ability is like a brick wall that must be built carefully from the ground up. Sometimes a detail slips through the cracks. If a student is struggling with quadratic equations, is it just quadratic equations they are missing? Or was there a brick that slipped out three years ago that’s stopping progress on the wall? The math that the student needs explained is often not the math that the student is struggling with now. If you can identify exactly which bricks are missing from the wall, and patch them in, you’ll sometimes find that “hopeless” student suddenly races ahead. Before, everything was confusing. With that foundational brick in place, it suddenly all makes sense.

And as in swimming, there’s no point in a math lesson saying “that was bad. Do it again” or repeating an explanation that made no sense last time. If the student was listening to your explanation, and heard it, but still doesn’t understand, then don;t repeat it. Find a new way to explain, or try again to discover exactly what the student is confused about.

The swimming teacher closes with the comment that metaphors are useful in teaching swimming. “Kick like a ballet dancer,” for example. This is an example of the age-old teaching principle “Connect what you’re teaching with what the student already knows”. If the students already know baseball, is it too hard to tie quadratic equations to the parabolic path of a ball? If they’re already good at linear equations, is it too hard to say that quadratic equations are like linear equations, with a couple of extra tricks?

If you identify what the students *actually* don’t know, and find ways to tie new topic to old ones, you’ve already won half the teaching game.