The Sweet Spot


Despite what many schools actually do, I think most of us can agree that learning is highly personal. What works for you might not work for me, and there’s nothing wrong with that. Thankfully, there’s more than one way to learn a subject.

However, what I fear happens early on in school is that, if a student struggles with the explanations and teaching style of a particular teacher, the student writes off the subject they are learning as unattainable. This can only be made worse when one sees the other students around them succeeding, while the ideas in the subject can’t seem to sink in to the student. It wouldn’t be surprising to me if this is a frustrating experience that also makes one have bad memories of a particular subject.

I’m pretty sure I’ve mentioned this before, but I’m eternally thankful to my parents for helping me learn arithmetic early on (as well as to my own, perhaps slightly better than average ability), which meant that the early subjects in mathematics during elementary school was easy for me. Without this boost, I feel like I would have been like many of my friends, where mathematics and science were subjects that one “survived”, but never really liked. From this small advantage, I was able to turn it into a whole education trajectory, where I’m now studying mathematics and physics in university.

The tragedy here is twofold. First, there’s the simple fact that mathematics is not linear. Despite what one may think after looking at a school curriculum, there’s not a required progression. Of course, it’s always nice to know arithmetic to speed up calculations within other fields of mathematics, and learning algebra is definitely a prerequisite to calculus, but should you take calculus first, or a class on proofs? Should you study graph theory before probability, or linear algebra before number theory? The answers to these questions depend heavily on the subjects, but the point I want to illustrate is that you don’t necessarily have a set progression in which you have to learn one subject before the next. Instead, you have a range of fields, and many can be studied independently of the others (though of course, one can always find interesting connections between fields).

Coming back to arithmetic, the problem with this is that arithmetic is now seen as the first “hurdle” to get over while learning mathematics at school. If you can’t get a grasp of arithmetic, then it’s as if you’ll be incapable of being good at other branches of mathematics (or, at least, you’ll have a lot of difficulty). But as I mentioned in the last paragraph, there are other fields of mathematics where the reliance on arithmetic may not be as great, so the student would be fine. However, the way mathematics progression is set up now, these small discrepancies between students now grow very large after only a few years, which begs the question, “Why arithmetic first?” If we started with a different subject, would that create a different set of students who seem to “excel” at mathematics? My feeling is that it would to some extent, which means we need to be careful in treating the beginning of mathematics education as a hurdle.

The second part of the tragedy is that, as I mentioned in the beginning, learning is personal. As such, there are different ways to teach subjects to a student. Different ways will work for different students, so the most critical thing to do when you don’t understand a subject is to try and learn it in a different way. If your class has a large emphasis on equations and solving abstract systems of equations, perhaps you need to try and transform this into a visual. It’s incredible how helpful it can be to have a visual when solving abstract equations. It might not make the manipulations easier, but it gives clarification as to why a particular strategy is used. And often, that can be the big difference.


I know that I’ve rambled on about both “higher” mathematics and more elementary mathematics, but my message is the same. If you’re having difficulty with a subject (which happens at every level), look for a different way to understand the material. You might look for a visual, or perhaps you rather work with only the abstract equations. Maybe diagrams help. The possibilities are numerous, so don’t feel like you have to learn things a certain way.

What I’m referring to here is your sweet spot, which is the amount of abstraction that you feel comfortable with while learning. I feel like I have a good grasp on calculus, but if you make it too mathematical and abstract, I get lost. It’s the same story with physics. I like to have some mathematical sophistication, but I’m not able to go off the deep end with it. Therefore, when learning a new subject, I try to find explanations and material that’s at my level. I know there’s no point trying to learn physics in the most mathematical and abstract way for now, because I’ll get lost. Instead, I start at my level, and I let myself become familiar with the subject. Then, when I start feeling more comfortable, I can extend my studies to the more sophisticated concepts. There’s no use to metaphorically bang your head against a brick wall when you simply don’t understand a concept. Use different or easier material as a stepping stone for what you want to learn.

This is exactly what I had to do when I worked in the physics department of my university this summer. I had to learn general relativity, so did I take the most mathematical and sophisticated book on general relativity? Of course not! I used the much more introductory book by Sean Carroll (which is great, mind you) to help me get through the initial hurdle of learning the subject. Without this book, it would have been extremely difficult to understand what was going on in my work. However, since this book was made for people like myself (who were just starting to learn), it made my journey into general relativity much easier.

Therefore, my advice is simple. Figure out what your sweet spot is when learning, and don’t feel afraid to seek out alternative explanations to what you get in class. Almost always, there’s another way to understand a concept, so use that alternative perspective to help you learn. Remember, you’re not necessarily like every other student in your class, so chances are high that you’ll get a class where the explanations the professor uses doesn’t mesh well with you. When this happens, seek the alternatives, and this will help you a lot as you advance your studies.

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