Jeremy Côté

Knowledge From Repetition

When I’m sitting in class and listening to the professor, I get frustrated when they go over a concept once and then continue on as if everything is clear. I think to myself, “Do they really expect us to understand a concept from just one example?”

And yet, I realize that I do the same thing when I’m working with a student.

This is a bit disconcerting, and it reminds me that teaching is a tricky business. You never want to make things confusing or too brief for your students, but you also can’t read their minds. You don’t know know how well they are absorbing the material. Furthermore, the most difficult part is when a topic seems straightforward to you, but actually isn’t that clear for newcomers.

I’ve studied mathematics and physics for many years now, so a lot of ideas seem clear to me. For example, I can deal with algebraic expressions without batting an eye (as can any student with enough years of experience). I’m not stymied if I see fractions on top of fractions, and I can factor or expand expressions as I see fit.

At this point, there’s nothing “special” about this ability. It’s what I would call basic knowledge (for someone with the years of working with the concepts as I have), but the crucial point is that this isn’t basic to many people. If you’re used to doing multivariable calculus and dealing with derivatives and integrals until the cows come home, it can be difficult to remember that some people don’t have this skill. Even if it’s obvious to you, there’s a good chance it isn’t obvious to others.

I’m not trying to admonish anyone here. Rather, I’m writing this to remind myself that the students I tutor aren’t necessarily able to “soak up” the crux of a concept through one example. I know that I’ve felt like I’m wasting time when I do multiple examples with a student. After all, we went through the computation once. What’s the use in doing basically the same thing over again with slightly different numbers?

It doesn’t seem like a big difference, but I know from experience that it’s what helps you make sense of new concept. Multiple examples are worth the time, even if they seem superfluous for you, the teacher. For the student, it’s exactly what they need.

I don’t know why it took me so long to figure this out. It has been right in front of me for years. That’s one of the good things about being a tutor while I’m still a student. Even though I might be a few years removed from the classes that the students I work with are taking, I still am in their position as a student myself. The class content isn’t the same, but the experience I have as a student is still similar.

I now try to incorporate more examples and problems into the tutoring sessions I hold with students. Instead of assuming that they have a firm grasp after one example, I do multiple with them in order to let them get comfortable with the idea. Is it as efficient as zipping through many concepts with only one example each? No, but I think the extra time is worth it.

When I think back to my own experience, the times I became really comfortable with concepts are when I did many problems. It was at this point that I understood all of the features of a given idea. After that, I didn’t need to go through more examples to understand at a deeper level, but it took a lot of initial work to get to that point.

I like to think of it a bit like building any other skill. Suppose you want to get better at basketball. You can work on your ball handling, your shooting, your passing, and your movement. If you want to improve any of these areas, there’s not much to it in terms of seeing and understanding what’s happening. However, there’s a big difference between seeing what it means to have good shooting skills and actually having a decent shot. For the latter, you need to go through hours of repetition. It’s not that there’s anything “new” that you couldn’t see by watching someone else shoot, but by going through the repetition you get to feel what it’s like to take a good shot. That only comes with repetition, and I think it’s a similar story with learning mathematics.

As educators, I think we need to be mindful of this when working with students. It might seem like one example is enough to get the point across to a student, but I suspect this often isn’t the case. Instead, we need to be better at offering multiple examples to students. If we don’t, we can expect only a vague understanding. Getting a concept across is a strong function of the amount of repetition which is dedicated to it. Do only a few examples and students won’t come away from the experience with a deep understanding. This is true no matter how basic the idea is. Therefore, I would suggest to forget about how “easy” it seems to you. That’s probably a bad barometer. If you get to the point where you feel like doing another example would be a waste of time, do it. That’s probably a good start in correcting our biased perspective.

In some sense, learning is repetition. It’s as simple (and as difficult) as that. Repetition often feels like a waste of time, but I think we need to shift our thinking on this point. The more I think about it, the more I like the idea of repetition and a greater number of examples.

Don’t be afraid of wasting time. Going through a bunch of content but only having students understand the ideas at a shallow level isn’t really better.