Is it normal to feel lost while following a mathematical / physical reasoning?

Question

I re-uploaded this question here, since a user from Physics Stack Exchange told me it might fit better.

TL;DR: I feel like I could not come up with most reasonings myself, so I end up feeling like I'm a phony, a bad student, or not good enough, and I can't focus on the actual problem.

I have been meaning to post this question for a long time. I am aware that this might be a little too personal for this forum, but at the same time I think I might not be alone here. And whether this is a common phase to go through during one's mathematical education or just something people have to learn to deal with, I think it could be of use to others. So here it is.

I'm a third year physics student in university. I am not American, but Spanish, and university/college here does not work like most Americans are used to, which is why I want to clarify this first. (You can skip this if you know how college works in most countries outside the USA). People can't choose what they want to study with as much freedom as people do in the USA: you just choose a degree, like Physics, or Mathematics, and then follow the curriculum with very little deviation. You can choose a bunch of optional classes to start leaning to a certain branch of your discipline, but that's it. No history, language, arts, cinema or biology classes if you're a Physics student. Therefore, a physics student is expected to have passed all of their mathematical training in their second year of university: linear algebra, real and complex calculus, differential equations, etc., and all of the major branches of Physics after their fourth year: Mechanics, Thermodynamics, Electromagnetism, Quantum Mechanics, and the rest of them. There is very little actual research, so students are expected to spend four years studying to get "the basics" down before they can help or research for themselves.

I wanted to tell you this to give you some kind of idea of the type of content that I might struggle with. Now, my problem is the following:

Every time I have to follow some kind of mathematical reasoning (even if it is applied to a certain physical problem), I get nervous and lost if I can't grip it the first time I read it. Sometimes because I don't remember the mathematics too well, sometimes because I just don't understand why they chose to complete a step in the way they did, kind of like when a professor suggests a change of variable that works for solving an integral and you think you could have never come up with that. Even after seeing the complete reasoning, I just don't know why they knew they had to do what they did to get there. And the most common one for me: sometimes I can follow the reasoning, but I have the feeling that I could have never thought about that myself, and therefore I'm just being a phony. When I feel like that, I just feel like giving up and checking the final result: "I am going to forget this anyway, since I could not come up with that line of reasoning".

Is it normal to feel like this at this point of one's education? This might be impostor's syndrome, or just the fact that maybe I am not very well prepared, or that I might need to find some kind of mental peace in the fact that this is common before I can actually sit down and do things.

Thank you all. You are a great community. I could talk to my therapist about this, but first I wanted to talk to people who might actually have encountered this problem in their lives.

Answer 1

What you are experiencing is very normal for a student. It takes a lot of practice solving simpler problems before you can, on your own, solve the harder ones. You have also discovered the truth that it is harder to produce a proof than it is to follow it. But it is also easer to read and follow a fictional story than it is to create one. That also takes a lot of practice.

In fact, it takes a lot of failed attempts before you learn to see the successful path. Even then it is hard.

Moreover, if you are reading the professional literature (published papers), note that most mathematicians will leave out all but the most essential steps. This is both because of page limits, but also because they are writing for people just like themselves, and not for novices, including students.

So, while it is discouraging, it isn't something to think of as an indication that you are a failure. It is just genuinely difficult to follow sparsely written proofs and even harder to produce the steps between what you already know and what you want to prove. And the more practice you get, the easier it will become, partly because that practice helps you recall what you once knew but don't remember very well, having not used it actively for a while.

Press on.

Answer 2

Every time I have to follow some kind of mathematical reasoning (even if it is applied to a certain physical problem), I get nervous and lost if I can't grip it the first time I read it.

Why do you think that it's unusual to have to go through a line of mathematical reasoning several times the first time that you've seen it in order to fully absorb it? I do that all the time when I come across something that I haven't seen before.

As for not understanding why someone chose to perform a certain step in a mathematical procedure (like a change of variable in an integral) or a physical argument and then feeling inadequate because you feel that you wouldn't have been unable to think of the step yourself, realize that when you're reading a book describing some mathematical proof you're looking at a final, polished product. You don't see all the dead-ends and speed bumps and wrong turns that the person took in the course of delivering that final product.

With time, experience should make you feel more comfortable and confident. You'll notice that oftentimes a mathematical proof is making use of a trick or concept or procedure that you remember seeing before. You'll start to see familiar, recognizable patterns in the mathematical proofs and procedures more and more often as your experience grows.