There are plenty of ways to enjoy mathematics. You can attend a classroom lecture, you can read a textbook, you can look at a news article, you can watch a video, or you could just play with some concepts yourself. There’s not one way in particular that is better than any other. Rather, each one has its own advantages and disadvantages. It depends what you’re looking to get out of your session.
Despite this, I want to highlight one method in particular that I think is crucial when learning mathematics. That is the method of discovering more mathematics on your own.
In terms of pure productivity, it makes little sense. After all, you’re likely going to hit a bunch of roadblocks when learning new material on your own. When you’re going through a textbook and trying to decipher the ideas without the help of a teacher, it’s easy to get stuck. As soon as you have a question, you don’t have an expert on hand to consult. Your choices are to skip ahead or struggle with the concept until it makes sense. Either way, you won’t be blazing through a textbook when learning on your own.
Obviously, this isn’t ideal. In a best-case scenario, you would go through material quickly and understand it. However, that’s not going to happen if you’re trying to push your boundaries. By definition, you’re trying to understand something that you didn’t know before. As a result, progress will be slow (at least at first).
I’m speaking about this from my own experience. I recently took an independent course on advanced topics in quantum mechanics, which means I learned on my own from a textbook. That’s not a bad thing, but it meant I had to learn the topics without a lot of outside help. I’ll admit that it was a challenge at times, but it went well overall.
If working through ideas and concepts on your own is so difficult, why do I recommend it?
The first reason is that you start learning how to learn on your own. This is an important skill, because once your education is over, you shouldn’t stop learning. If anything, it will be even more important to keep learning as new developments occur. Once you’re out of school, you won’t have a group of teachers waiting to help you out and lead you by the hand while you learn. It will be up to you to figure it out, which means the sooner you start learning how to learn on your own, the easier the transition will be.
That’s the practical reason why you should try to learn on your own. It will help you in the future and free yourself from needing a teacher to go through topics with you. By finding your own path through material, you will figure out what’s important and what isn’t.
However, the reason I think learning on your own is so important is that discoveries in mathematics stick with you longer when you come up with them on your own.
I’ll give you an example. This summer, I was doing research in gravitational theory. Like in any research project, there were papers I needed to consult. In particular, there was one paper that was very important. When I read it, I could have just accepted the results and gone on with my research. Instead, I spent multiple hours going through all the calculations, doing each of them one by one. I did this because we used a different sign convention than the author, but it also had the side effect of helping me understand what was happening. By the end of all the calculations, I understood where each term came from. It wasn’t just something I needed to accept on faith. Rather, I knew it was correct, since I spent a lot of time working it out.
This sounds like a minor thing, but it actually changed the way I saw the subject. It wasn’t something that I knew was probably correct but had no clue how to do. By doing those calculations, it made sense to me. That’s a powerful feeling in mathematics.
If we want to apply this to learning in a classroom, my suggestion is that you spend more time trying to make sense of everything your teacher says. When they say that a particular calculation yields this specific result, can you see that? Do you know how it comes about, or do are you trusting that your professor did their algebra right? The point isn’t to find mistakes. The point is to be comfortable with the results.
There’s a lot to be said about feeling comfortable in mathematics. When you’re comfortable, results are easy to accept because you know why they are true. You can look past the final answer and into the steps that led to the answer. The answer itself ceases to be important. It’s the knowledge that you know how to get from the beginning to the end of the problem that’s crucial.
This is where learning on your own comes in. Even if you’re in a class with a teacher who leads you through the material, it’s important to discover mathematical results on your own. When your teacher gives you a result, it’s easy to forget it. After all, what’s so special about that result? It’s just another thing to remember. However, when you spend hours working through a calculation to get to an answer, you remember it longer. At least, this has been my experience while working within physics. It’s well and good to read papers and textbooks and have people present results to you, but if you really want to internalize the ideas, nothing beats taking out a pencil and working through them yourself.
It’s a lesson I’m learning over and over as I go through my education. The lazy way to learn mathematics is to listen to someone tell you the answer, or read it in a text. The long and painful but ultimately useful way is to go at it on your own. It’s not easy, and I can say from experience that it will result in you getting mad many times, but it’s the only way I’ve found that ends with you remembering the results and not just knowing them.
And in the end, isn’t that part of what mathematics is about? Knowing that the square root of two is irrational is good, but understanding why this is true is the real fun. Mathematics is about the “why” behind the results, not the results themselves. As such, when you take the time to discover the mathematics on your own, it will have a larger impact than if you passively consumed it from someone else.