Shallow Knowledge, Deep Knowledge

And the curse of knowledge. Knowledge knowledge knowledge.

Lots of math teachers make conceptual understanding their priority. The argument is that we need to move past "rote memorization" to "deep understanding" etc etc. I've been in this camp in the past but I've moved away from it over the last few years.

One Big Idea

Here is one big idea that has informed my take on conceptual understanding:

Shallow understanding and deep understanding are not opposites. Shallow understanding is a necessary step on the way to deep understanding.

I’ve always wanted students to develop a deep understanding of math, but I used to make shallow understanding the enemy along the way. Let's say I was teaching students to solve equations. I wanted students to understand equations deeply. I wanted them to look at equations as statements about two things that are equal, and to try to figure out what value or values of a variable make that statement true. I wanted them to think in terms of inverse operations and maintaining equality. I didn't want students to say "ok to solve the equation x + 12 = 20 I subtract" because that doesn't reflect deep understanding, it won't help them solve different types of equations in the future. I would tell them not to think that way, and try to help them see equations from a deeper perspective.

And sure, a student who only thinks about that equation in terms of subtraction is missing a lot. I want that student to learn more in the future. But in the past I looked at that shallow knowledge as wrong, as unhelpful, and as something to be avoided. But that knowledge isn't something to avoid — it's where students have to begin. It's something to build on. Hopefully it's one piece of knowledge that we will add more and more to in the future. If I leave it there, or never create a need for students to build on that foundation, then that's the problem. Knowledge is a complex web. It has to start somewhere. It's my job to find somewhere to start, and then to keep building and building until that web connects all the little pieces in a way that helps students apply what they know in lots of different places. If I try to attack every piece of shallow knowledge along the way I kneecap students when they’re just getting started.

A pedantic note: I’m using the words “knowledge” and “understanding” interchangeably here. I don’t think there’s much categorical difference between the two. We typically use knowledge to describe things on the shallow side and understanding to describe things on the deeper side, but they’re two sides of the same coin. I would define shallow knowledge as knowledge that students have trouble applying in a new context, and deep knowledge as knowledge that’s flexible and transfers to lots of new situations. I find the distinction between shallow and deep much more useful than the distinction between knowledge and understanding.

The Curse of Knowledge

One reason I've fallen into the trap of thinking I can go straight to deep understanding is the "curse of knowledge." I know a lot about equations, but most of my knowledge is invisible to me. It's automatic, so deep in my long-term memory that I don't realize it's there. Since it's invisible to me it's easy for me to believe that students can get by without it. I look at my own experiences, and I think "well when I was a student I only understood the procedures, I was a math robot. Now I truly understand equations, and I want my students to have that level of understanding." I can't see my starting point as a step on the way to where I am now because of the curse of knowledge. I end up thinking I can short-circuit the system. And sure, I was a bit of a procedural robot as a student. I want my students to move beyond that. The question is how get them there.

When I thought I could get students straight to conceptual understanding it wasn't because I was being dumb or ignorant. It's a quirk of human cognition that the things we know best exist outside of our consciousness, simply because we know them so well. That quirk led me to believe that present me doesn’t have any shallow knowledge and only has deep understanding. That gets me to the thesis of this blog post. I'm arguing that shallow knowledge is a prerequisite for deep knowledge, but that's not the whole point. The point is that when humans try to teach other humans things, we are hardwired to underestimate the importance of shallow knowledge. It's an inevitable truth of education. It's something I forget all the time. I'm writing this post to help me remember a little more often. Shallow knowledge isn’t the enemy, it’s just a step along the way.