Is it beneficial to practice derivations instead of solving problems

Question

During much of my time studying physics on my own time, I have prioritized practicing the derivations out of the book because I feel like I can learn more by trying to master the theoretical structure rather than applying the theory to solve problems. And I don't mean that I read over them, I mean that I will repetitively derive an equation anywhere from 1-10 times until I feel like I completely understand and have memorized it. However, I do not want to waste time. Is it beneficial to spend most of my study time trying to master the derivations already given, or should I mainly skim over the theory and dedicate most of my time-solving problems? I want to choose a route that will maximize the amount I learn and what will lead me to become a more efficient physicist. I don't know if it will make a difference but this question is aimed toward more advanced physics classes/students.

Answer 1

This is a great question. I think many theorists (and I am one) follow this path. For one thing that is part of our job, to derive things. Also, I think you are correct in that by deriving things we learn at a deeper level. However, sometimes solving specific problems allows us to apply mathematical techniques and reasoning in a different way compared to going through derivations. Also, as a theorists I was indoctrinated into the thinking that constants didn't matter. We tacitly set them to 1. Later in life I regretted that. There is great value in "knowing things", like the value of the Gravitational constant, the speed of light, etc. But it sounds to me like you are looking for an optimal learning path. One that takes up as little time as needed to maximize your skill set. I am not sure that there is a perfect ratio of theory to application. It is probably different for each person. But I would offer some advice. It is often said that we should play to our strengths while working on strengthening our weaknesses. So, be honest with yourself, are you really great at deriving and sloppy at solving? Or, is solving problems really easy and uninteresting for you? If you can do it well no need to bother practicing. But if you are better at one than the other you might want to spend more time on the thing you are bad at.

I often had the experience of reading something abstract on QFT or another topic, deriving the numbered equations and feeling like I had mastered it. Then when it came to doing a simple calculation I got stumped. In some cases the approach to solving a problem or calculating something involves deciphering the presentation of the problem and fitting it into the theoretical paradigm you learned. This is not always obvious and involves a different type of thinking. One does not beget the other. Also, we often find that to completely solve a problem we need to make simplifying assumptions that the theory does not necessarily teach us to do or we need some clever trick. This is always worth learning.

When it comes to deriving the basic results, it is helpful but applying them can be more useful. I'd say that if you want to be a theorist then you will want to learn the art of proving things, and deriving things. This will be part of your job so it is definitely worth the effort.