Jeremy Côté


Starting Simple

If you want to get a concept across, the best thing to do is to start simple. Learning can be difficult and many parts initially might not make sense, so it’s important to make the “jump” to that knowledge in a way that we can follow. If not, it will simply be too difficult to make that conceptual leap.

This is incredibly important when being introduced to a formula of some kind. The thing about formulas is that they can be used with little understanding of what is actually going on. It’s definitely possible to do well in a physics or chemistry or even mathematics class by only knowing formulas without actually knowing what is happening within the formulas.

For example, I was recently learning about the concept of curl and divergence in multivariable calculus. Without actually knowing the idea behind curl and divergence, I could simply used the formulas \(2dcurl(\textbf{v})=\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}\) and \(div(\textbf{v})=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\) respectively without any problems. Since I know how to compute partial derivatives, finding the value for curl and divergence is relatively trivial.

However, the deeper understanding comes from when a formula is explained. I hate having formulas simply presented to me. What do they mean? How did certain terms come into play for calculating this certain thing? These are questions that interest me, because they give a reason for a formula. Without a reason, I’m just blindly computing.

This is why I was pleasantly surprised to view these videos from Khan Academy, because the author (Grant Sanderson) always gave intuitions before presenting the formulas. This way, there wasn’t befuddlement as to why a certain formula was supposed to represent curl or divergence. Instead, it made sense because of the intuitions and explanations beforehand.

How did he do this? By constructing the simplest case possible, and building the formula up from there. Because he approached the formula like this, it made sense as to why a certain operation was being done. Consequently, the explanation was easy to follow, and the formula wasn’t surprising once it was presented.

Let’s face it: mathematics can be difficult and abstract at times, making it difficult to make sense of formulas. Often, it can seem like formulas pop out from nowhere, which make them look mysterious and exotic. However, by exploring the simplest cases possible for the situation and slowly working up the intuitions, it makes it much easier to understand the origin of a formula.

This idea can just as well be extended for ideas in physics or other sciences, and I firmly believe it should be. I’ve been in classes in which time didn’t allow for us to derive various formulas, so they were simply presented to us as correct. While efficient, this made it difficult to understand how each component of a formula worked to calculate the thing we were searching for.

Therefore, whenever you get a chance to understand how a formula came about, take it. Do not simply absorb the final answer. In the long run, it will be more productive to figure out how a formula works instead of just knowing it.