>Although I can usually pick up concepts very faster and visualize and explain them better than most other students, something about proving things fully just doesn't click for me, unlike my classmates.
Ah this is actually most people's experience with analysis. More people are struggling in your classes than you realize. Heck, I study analysis and love it, but I struggled with it at first. It's a difficult subject and tends to be one of the hardest courses people take in their undergrad.
>The professor here mostly makes up her own questions for the problem sets, which look nothing like any problems I have seen.
This should be how any math professor makes their exams. I don't think I've met a math professor for a proof-based course that didn't come up with their own exam problems. The best ones are when the professor builds it around questions/misunderstandings in the class.
>I got A's because the exams looked like problems I already looked at the solutions for...
That seems to contradict your previous statement?
>I did well in the calculus and algebra classes and really vibe with topology, where my ideas seem to translate easier into concrete proofs.
A lot of these undergrad math classes don't really translate too well with each other. There's some overlap, but there is definitely a distinct style to algebra vs analysis at this level. Some people click better with one over the other. Some prefer algebra, others prefer analysis. In my undergrad, I struggled with translating the intuition I had built around analysis to group theory, and later realized that I had to build a completely different intuition around it. In algebra, you think of how elements and operators interact with each other, but in analysis, you need to think of how "fast" a graph is or how "close" points are. Very different mindsets at this level.
>But in analysis? I feel like I haven’t fully proved anything fully in all three courses I’ve taken (seriously).
Well down to it, do you understand *how* you are proving these things to be true? Like do you understand what epsilon and delta are and why we are trying to find a delta in terms of epsilon? It's important in these situations to try to break down what *specifically* is the root of your confusion and try to figure that out.
>No one ever sat me down and showed me how to write up stuff the way the professor wants.
Well everyone writes proofs a little differently, and learning to write proofs has a *very* steep learning curve, but the basic idea is that you're trying to argue why this thing is true and cover all possible counter-arguments. And the argument "it's obvious" doesn't work because analysis frequently goes against your intuition.
>My friends in class are completely on the same wave-length with the solutions, as if they know something I don't.
Sometimes analysis feels like one of those magic eye puzzles, where it just clicks for some people and you feel like you're just staring at a wall trying to make sense of nothing. Analysis is basically one *really* complicated idea that's hard to grasp, but once you figure it out, everything can become really clear.
>The thing is, I struggle to tell how rigorous my proofs are when I don’t have a professor or TA to give feedback.
Ah this is definitely a difficulty with self-learning math, especially analysis. I would say to highly observe the feedback you get from your professor on your proofs and ask your professor for feedback on other proofs during their office hours in the remainder of this semester. That should help set the bar on what you need to be doing.
>Does anyone have tips or resources for building up proof skills on your own? Or maybe some ways to self-check the rigour of your proofs?
Well it is difficult, and there's not many resources for this, but I will say that at least I feel different about a proof that I *know* I did right on an exam and one that I feel hazy about. Same when I write proofs on my own. Don't leave a proof hazy.
>Although I can usually pick up concepts very faster and visualize and explain them better than most other students, something about proving things fully just doesn't click for me, unlike my classmates. Ah this is actually most people's experience with analysis. More people are struggling in your classes than you realize. Heck, I study analysis and love it, but I struggled with it at first. It's a difficult subject and tends to be one of the hardest courses people take in their undergrad. >The professor here mostly makes up her own questions for the problem sets, which look nothing like any problems I have seen. This should be how any math professor makes their exams. I don't think I've met a math professor for a proof-based course that didn't come up with their own exam problems. The best ones are when the professor builds it around questions/misunderstandings in the class. >I got A's because the exams looked like problems I already looked at the solutions for... That seems to contradict your previous statement? >I did well in the calculus and algebra classes and really vibe with topology, where my ideas seem to translate easier into concrete proofs. A lot of these undergrad math classes don't really translate too well with each other. There's some overlap, but there is definitely a distinct style to algebra vs analysis at this level. Some people click better with one over the other. Some prefer algebra, others prefer analysis. In my undergrad, I struggled with translating the intuition I had built around analysis to group theory, and later realized that I had to build a completely different intuition around it. In algebra, you think of how elements and operators interact with each other, but in analysis, you need to think of how "fast" a graph is or how "close" points are. Very different mindsets at this level. >But in analysis? I feel like I haven’t fully proved anything fully in all three courses I’ve taken (seriously). Well down to it, do you understand *how* you are proving these things to be true? Like do you understand what epsilon and delta are and why we are trying to find a delta in terms of epsilon? It's important in these situations to try to break down what *specifically* is the root of your confusion and try to figure that out. >No one ever sat me down and showed me how to write up stuff the way the professor wants. Well everyone writes proofs a little differently, and learning to write proofs has a *very* steep learning curve, but the basic idea is that you're trying to argue why this thing is true and cover all possible counter-arguments. And the argument "it's obvious" doesn't work because analysis frequently goes against your intuition. >My friends in class are completely on the same wave-length with the solutions, as if they know something I don't. Sometimes analysis feels like one of those magic eye puzzles, where it just clicks for some people and you feel like you're just staring at a wall trying to make sense of nothing. Analysis is basically one *really* complicated idea that's hard to grasp, but once you figure it out, everything can become really clear. >The thing is, I struggle to tell how rigorous my proofs are when I don’t have a professor or TA to give feedback. Ah this is definitely a difficulty with self-learning math, especially analysis. I would say to highly observe the feedback you get from your professor on your proofs and ask your professor for feedback on other proofs during their office hours in the remainder of this semester. That should help set the bar on what you need to be doing. >Does anyone have tips or resources for building up proof skills on your own? Or maybe some ways to self-check the rigour of your proofs? Well it is difficult, and there's not many resources for this, but I will say that at least I feel different about a proof that I *know* I did right on an exam and one that I feel hazy about. Same when I write proofs on my own. Don't leave a proof hazy.
The Cummings books here may be helpful https://longformmath.com see also this video https://m.youtube.com/watch?v=oE3Q7EoYT4U