Struggling to both understand and remember maths! I hope it is OK to post this question here...
I'm currently in my 2nd year of maths at uni (doing cal 2, discrete, prob/stat) and hadn't studied maths formally for about 15 years beforehand. In my first year I did somewhat OK and scraped a pass in all of my modules (alg 1, calc 1, proofs) with thanks to distance exams. Otherwise, I would have absolutely failed.
I enjoy maths... the whole problem solving/challenging aspect is what drives me. However, I cry almost everyday because I'm so frustrated at how little I actually "get" it and it's actually starting to depress me (I dedicate at least 40 hrs p/week, 7 days a week to it). On the other hand, my boyfriend can look at a question, investigate it a little online and it makes sense to him with little effort.
In terms of learning methods I've found that notes do zilch. Perhaps my notetaking is just bad but I've found it wasted my time. My comp is full of ebooks as recommended by the staff too. So I have many sources of info at my hands but yet my intuition or core understanding is roughly the same as when I started.
Exams are in 5-6 months, I'm already getting myself stressed about it and I have to improve. I know maths is challenging, that's whats great about it! But when you find yourself staring at a question that involves bayesian theorem, wondering why the heck you have to even use division, then there's something missing.
Would it be most effective to simply do as many exercises as possible? How do you conceptualise theorems, etc and apply them to problems?
Any tips would be hugely appreciated :) Thank you in advance.
 A: Also not quite sure whether this is the appropriate place to discuss your problem. However, it is very relatable! During my first year of mathematics I already was coming up with a backup plan (working as a train driver) just in case I would not make it. I kept that backup plan all the way up to receiving all my points in the first year. After that I was still not sure whether I was cut out for it. A few years ago I received my BSC and I will be going for my MSC next years.
Even though you have doubts, if your passion really lies with mathematics, don't give up! You proved that you can do this by scraping a pass in all of your modules in the first year. You could say that is thanks to distance exams. The main reason for you passing your exams is you though. YOU did it, not someone or something else.
There will be some that understand certain concepts of mathematics just by reading about them once and there will be a whole lot more that have to do exercises in order to get a better understanding. Unfortunately that is the truth. What might help is to try to find a study group in which you can comfortably do homework together and talk about these issues. Believe me; you will find that most of your fellow students feel the same way. I know it's rather difficult today due to COVID, but try to find some way online or something. It might help.
A: I am also someone new in formal mathematics, so I hope to be of use.
First, don't compare yourself to others; the best way you can measure your performance is by looking at how much you've grown in math.
My advice is to try to exchange ideas with friends or colleagues; you can also start the homework problems trying to see them from an "informal" point of view and then have a general idea and be able to formally write the solution / proof.
If you have no idea how solving the exercise tries to "discover" what is the meaning of the exercise and thus be able to invest in a specific way on that topic by reading similar proofs or seeing some solved exercises.
Also, take a break from time to time, I understand that you want to level yourself but mental health and well-being is important, so take breaks from time to time and enjoy math.
A: First of all, let me notice that your post sounds as you have the right attitude and motivation for doing maths. Of course, there is no one-fits-all recipe that I can give here, as learning maths is an individual task for everyone. Therefore, I try to focus on things I observed and that were helpful for me, when I started doing maths.
For understanding the topics covered in school-maths it often was sufficient to be talented, which is in high contrast to how it is with university-maths, which is hard-work (that, in the best case, is supported by talent). Hardly anyone understands everything just by hearing/reading about it. Especially in the first semesters of study there are many students pretending that they understand things easily. According to my experiences; be sure that this is not true for 90% of them ;-)
For me, taking notes from lecture is very important and the quality of my notes usually gives immediate feedback of how good I indeed understood the lecture. As many lectures currently are recorded due to the pandemic situation take this as an advantage. Try to watch the lectures first catching the main points and then (if you feel that handwritten notes might be useful for you) try to reproduce the statements and definitions. Writing something down helps you to find out whether you undestood something well or just think that you got the whole idea.
When you learn about a theorem/lemma, try to group the statement in prerequisites and statement. Make yourself clear what is demanded and what is the conclusion of the theorem. Try to find out why you need every single prerequisites, i.e. why does the theorem/the proof not work if you remove this part.
It is hard to learn all the proofs you see during the semester by heart, so try to grasp the idea of the proof, i.e. that what remains if you forget about the technical details. Often, you can find similar ideas in other proofs again and again. So, once you build up a certain "toolbox" of ideas you understand fundamentally, it turns out that other things become easier for you, too. This is also a great strategy to solve assignment problems. Very often assignment problems can be solved if you understood the proof-idea of a theorem in the lecture (of course, this does not work every time, but helps you to get started with math problems).
Finally, math is not just learning definitions and theorems. Most of the time, you are going to spend with solving problems (or, at least, trying to). Again, you have to build up your personal "toolbox" where you store all ideas that worked for previous problems. Adding ideas to your toolbox is nothing but hard work. Try to solve as many problems as possible on your own and if you see solutions from others, try to extract the main idea behind the proof and try to proof it on your own afterwards, without copying the whole solution, as, again, this shows whether you understood the proof or just think that you understood the proof.
Summing up, don't be daunted by the speed others solve (or pretend to solve) problems and grasp ideas. Many people do not admit how much effort it was to get there, as it is more impressive to pretend to be the 'genious'. Don't compare yourself to others, compare with your younger self and think about your progress. For me it sounds like you have a great attitude and a honest fascination about maths, which, in my eyes, is the most important thing.
I am going to finish with a quote from one of my professors, that was helpful for me in my first year of maths: "Maths is about failing 95% of your time and the rest is being incredibly glad and proud that you haven't given up on your way. Mathematicians live from the hope that this 5% of pure happiness is going to outperform the other 95% of suffering." ;-)
I hope that helps at least a bit :)
A: Richard Feynman described his approach (in Surely You're Joking, Mr Feynman, my emphasis):

I can’t understand anything in general unless I’m carrying along in my mind a specific example and watching it go.  Some people think in the beginning that I’m kind of slow and I don’t understand the problem, because I ask a lot of these ‘dumb’ questions: “Is a cathode plus or minus?  Is an an-ion this way, or that way?”
But later, when the guy’s in the middle of a bunch of equations, he’ll say something and I’ll say, “Wait a minute!  There’s an error!  That can’t be right!”
The guy looks at his equations, and sure enough, after a while, he finds the mistake and wonders, “How the hell did this guy, who hardly understood at the beginning, find that mistake in the mess of all these equations?”
He thinks I’m following the steps mathematically, but that’s not what I’m doing.  I have the specific, physical example of what he’s trying to analyse, and I know from instinct and experience the properties of the thing.  So when the equation says it should behave so-and-so, and I know that’s the wrong way around, I jump up and say, “Wait!  There’s a mistake!”

I too find that very abstract stuff just leaks straight out of my brain without making much impact — but if I can visualise an example and follow it through, I have a much better chance of understanding and remembering.  (It's not always easy to find good examples, but it really helps.)
