As I’ve written about many times here before, I’m a big proponent of understanding why one is using a certain strategy or procedure during a problem. In my mind, understanding the essence of the process is a great way to learn. However, this comes with a huge caveat, which is rarely talked about. The piece that sparked my reflection is an article on Nautilus, where the author explains how she went from being a translator (and someone that wouldn’t even look at science and mathematics) to reinventing herself and becoming a professor of engineering.

This is obviously a radical transition. Think of someone you know that is in the arts and is definitely not a science person. Then, picture them becoming an expert in material that you’ve dealt with all your life. Undoubtedly, this would be a little unsettling. The mental model you’ve created for this person would essentially be broken, becoming a terrible approximation for that person.

The article is a great read (and I recommend you do read it in its entirety), but the crux of her explanation for how she was able to transform so well was through focusing on fluency rather than conceptual understanding. Repetition and examples were the elements of her training that were responsible for her improvement.

I’ll be the first to admit that my stance has been largely the opposite of hers. By knowing the concept well, I believed (and still somewhat do) that you can solve a wide variety of problems that you wouldn’t be prepared for if you only knew about the brute calculations. Therefore, repetition and examples were only useful insofar as the core idea is understood.

I think this is a good way to go, but I’ve accepted that repetition isn’t as bad of a teaching tool as it might seem. I shouldn’t be surprised, either. After all, the main thing that I preach when other runners ask me how I’ve gotten to be as fast as I am has been to say through many kilometres of training. That is repetition in action.

I suppose I come from a place that is more about getting an idea. If you can calculate something but you don’t know what it means, are you really learning? This is the kind of question I’d ask myself. And I’d always arrive to the same answer: conceptual understanding is always best.

However, I’m not so sure anymore. The best example I can think of is this: imagine you’re in class with an amazing lecturer. Everything he or she says is understood by you perfectly, and you’re hanging on to every word. At the end of the session, you’d be able to confidently say that you followed everything they said. Would this person be able to do the problems assigned?

As much as I want to say “yes”, I think I’d be deceiving myself. The truth is that this student would probably struggle on the problems. Not because they are incompetent, but because they aren’t fluent in the process. They weren’t lying when they said they followed every word the professor said. They simply couldn’t apply it to their actual work because they had never practiced it.

How many times do you listen to a teacher and think, “Oh yeah, I understand what they’re doing. This is easy.”? If you’re anything like myself, this happens a lot. However, the assignment will then come and I’ll be confused. It made so much sense when the teacher did it, but starting from scratch with no teacher is much more difficult.

The reason it’s difficult is that you’ve never done it before. Taking derivatives and integrals of most basic functions aren’t difficult, but they are if it’s the first time you’ve done them. Therefore, it is hardly surprising that practice and lots of repetition is the key to improving one’s skills in taking derivatives and integrals. You won’t necessarily gain conceptual understanding after the hundredth integral, but you’ll definitely get used to how they work and how you should approach solving them.

As the author points out, this applies more broadly then just mathematics and science. In general, one needs to practice over and over again to get better. It’s as simple as that. Understanding the concept is definitely an important part of one’s education, but it shouldn’t be the only feature of one’s curriculum. Instead, there should be a place for both conceptual understanding and lots of practice. Before this article, I’d perhaps be in favour of more conceptual understanding and less robotic procedures on how to calculate, but I think that’s doing a real disservice to students who are going to need those practical skills.

As usual, relying on one method of learning is recipe for failure. Mix both conceptual learning and practical learning in, and students will be better suited to understanding mathematics as well as knowing how to do the actual work.

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