When we learn new concepts in class, I think we tend to focus on what we’re taught, confining ourselves to the scenarios that were introduced in class. To be fair, that’s not a bad strategy, since most professors are only going to test the material that was explicitly seen in class. As such, there’s an implicit sort of agreement that students are not going to see any “surprises” on the test (not the euphemism).
But even though that’s true, I think there’s a huge benefit to looking at more than just the scenarios and cases that are outlined in class. In particular, I think it’s a useful practice to look at the edge cases of a concept, as well as alterations on the regular scenarios you’ve seen in class.
Just to give a brief example, consider the regular kind of differential equations you see in an introductory course. During this course, one learns about the various techniques that can be used to solve differential equations. One usually learns about integrating using the integrating factor and integrating via separable equations before they start learning about the characteristic equation.
If you were boring and didn’t like exploring, you’d cleanly separate each use case out and find ways to identify when one method should be used versus another. Or, you could be like me and try to skip the first techniques that I learned and try to apply the characteristic equation to first order differential equations, for example. It was here that I was able to try and see how the different methods could work and produce the same answer for a given equation. It also helped me gain an intuition about which methods are better for a problem.
Being good at a certain subject in mathematics or science can seem to others as though you have a magical ability to see things they don’t. One way to hone this ability is to explore different ways you can go about answering the same question. This may seem boring and repetitive at first, but the real reward is in the potential application of this experience to future problems. When you encounter a new problem that has given others pause, you might be able to solve it because you’ve already explored different ways to tackle a problem, and one of those ways can you help you out.
If you want to be good in mathematics and science (at least, on the theory side), it’s always a good idea to work on answering questions with multiple methods.