I was good at math in high school, but I used to spend hours trying to figure stuff out in my mind on my own time. I would sit and listen in class, and then I would go home and try to make sense of what I had heard. It was like working out a giant puzzle. I wanted to know how today’s lesson fit in with all of the math I had learned previously. Part of the impetus for this was my poor memory. If I didn’t understand something and make it all connect, I’d never be able to remember it.
Learning happens when we experience something, and then go out and make sense of it. We make sense of an experience by constructing and integrating it into mental models—internal theories we build to understand the world. Students who are successful at math in school don’t actually learn math simply by sitting in the classroom. They have to make sense of what they learned by actively constructing mental models on their own.
When I became a math teacher (mostly middle school), I realized that there is a ton of stuff that you need to know in order to understand math, but it’s never taught in class. Successful students figure it out on their own. Students who pay attention dutifully in class but who don’t go out of their way to puzzle out what they learned on their own time end up being utterly lost.
An example of this is order of operations. For many students, order of operations is PEMDAS. I learn it and I’m done, what’s next? But our understanding of order operations needs to scale well beyond PEMDAS. We use order of operations to chunk expressions. That’s never taught, but some students figure that out on their own. And being able to glance at an expression and chunk it in your mind with no conscious effort is critically important in understanding algebra.