The Right Time
Do you hate mathematics? Have you found that the rules you have to follow seem to make no sense? No matter what you do, it never feels obvious, like the teacher says it should. Perhaps you can follow your teacher through a problem, but you know that if you ever had to do the problem on your own, there’s no way you could do it.
For someone that identifies with the above paragraph, I want to let you in on a secret. I feel the same way at least once a week, in terms of things not making sense. And I’m in the midst of completing a degree in mathematics.
The reason is simple: learning mathematics isn’t easy, and it comes in different levels. The latter is a crucial distinction to understand, because it can be the difference between finding mathematics engaging and creative versus being just a bunch of symbols.
There’s a reason that students have so many mathematics classes during elementary and secondary school. It’s because there’s a lot of material to cover, and one has to have a certain level of mathematical maturity in order to make sense of it all. Imagine if you had to learn about trigonometry when you were in the fifth grade. Sure, maybe you could learn how to manipulate a few symbols, but I doubt most would understand how trigonometry works. This is because students in the fifth grade aren’t ready to tackle those kinds of topics yet. As such, it would be a mistake to teach them trigonometry at that time^{1}.
This example generalizes to any topic within mathematics. Mathematics is a process of layering new abstraction and tools to understand objects. Expecting students to understand topics without grasping the layers underneath is a recipe for confusion. Not only that, but it frustrates students. If you ask many of them, I know they want to learn. It’s just that some concepts make no sense to them, and they don’t have the time to go back and master the previous layers during school. This leads to poor results and lack of desire to do mathematics. Should we be surprised by their low enthusiasm?
After my first year in the physics program, I did research for the university in the summer. The field I worked in was alternative theories of gravity. These are theories that modify the usual recipe of general relativity in order to explain different features of our universe that we observe (particularly on the largescale).
To understand this area of research, it should come as no surprise that a knowledge of general relativity is a must. To understand general relativity, you need to be comfortable with tensor equations, which are an abstraction of the idea of vectors and matrices. Notice that these don’t give you any insight into how these alternative theories of gravity work. They’re just the prerequisites that allow one to comprehend what’s going on.
Coming into the summer, I hadn’t taken any course that helped with this knowledge. I didn’t take a course in special relativity, electromagnetism, Lagrangian mechanics, or any mathematics course that introduced tensors. This meant I had to learn everything on my own, from scratch.
There are some who are able to learn things on their own with ease. I’m not one of those people. It was a struggle to wade through the concepts. Not only was I learning new physics, I was also learning new mathematics. It was difficult, because I had no prior tools to work from. By the end of the summer, I would say that I had an “operational” understanding of the subject. I could do some computations, but I didn’t have a holistic understanding of the subject by any means.
I was in the same scenario as the one I described at the beginning. Things were confusing, and it was difficult to see the “whole” picture. This made it frustrating because I could more or less follow an argument, but I couldn’t see why it began as it did. In essence, I wasn’t approaching the subject at the right time.
Fast forward a year later, and things make a lot more sense. I’ve taken courses in special relativity and Lagrangian mechanics, which has made my understanding of general relativity much better. Just going over the same notes I wrote last year is so much easier. It’s remarkable how much difference a year makes, once you’ve taken the right courses.
My recommendation is this. If you find yourself struggling with a subject, ask yourself if you’re comfortable with the material that has come before. Chances are, you will find that it’s the surrounding details that make the current discussion difficult.
Everyone is capable of learning something new. However, you need to keep in mind the timing. If you’re trying to understand concepts that are more advanced than you are ready for, you’re bound to feel frustrated and confused. That’s not a sign that mathematics is not your thing^{2}. It’s a sign that you’re not prepared.
What can you do about it? The first step is to be honest that you’re in over your head. This can be difficult, particularly if you’re in school and are supposed to know the material that was covered in previous years. However, the most important thing is understanding. There’s no shame in admitting that you don’t quite grasp a subject that you’ve already “finished”. In fact, it will help you out in the long term.
The second step is to go back within the subject until you can explain the concepts to someone else. That’s your new “foundation point”. You will build up your knowledge from there. It doesn’t matter how far back you go. Keep going until the concepts are obvious to you. Once you’ve found that point, revisit the next concepts one by one, until each one seems evident to you. This will take some time, but it will ensure that you don’t have a shaky foundation.
I can’t deny this is timeconsuming. Worse, you will have to juggle doing this while also working through your present subject in school. But, the idea here is to get back to a place where the learning feels like the “next logical step”. If learning is frustrating, we are unlikely to continue. That frustration is often a symptom of learning new things without mastering previous material, so a great way to proceed is to go back back and strengthen the fundamentals.
It’s all about tackling new subjects at the right time. If you do that, learning will be much more rewarding.

Note that I’m not saying that young students can’t earn advance material. Instead, we should be aware of the prior knowledge that students need in order to tackle certain topics. ↩

I’ve concentrated on mathematics and physics here, but this concept applies in general. You want to make sure that you don’t try to take something on without the proper preparation. ↩