How to know if you understand and interpret something right?` This has happened to me a lot of times and I'm sure a lot of you might have experienced this as well.
A lot of times, whenever I learn some new concepts, I think that I completely understand it and have no doubts regarding it. My understanding and interpretation of that particular concept happens to be consistent with everything given in the textbook and also with the understanding of my peers and teachers. But, many a times, I encounter a question (or a paradox) that makes me question my entire understanding of the topic. I feel like I don't have clear fundamentals and am unable to decide between which part of my understanding is right and which one's faulty.
It comes to a point where I can't figure out what exactly it is that I don't understand. There are generally two solutions that come to my mind in such cases : First, forget it, be delusional that you understand the topic and move on happily and second, try harder to figure out where you're stuck (doesn't usually work, by the way) or start all over again.
So, I always choose the latter but when I start again, I'm still not confident if I'm doing it the right way. This has been driving me crazy since long and right now, when I thought I got rid of it, it stuck again. This time, it chose Calculus, my favorite part of Mathematics.
I can't let this happen anymore and would appreciate any help.
Thanks :-)
 A: I can only speak from personal experience so I hope it's helpful to your cause. In exact fields it helps a lot to do prescribed exercices.
Write out your answers step by step, as small a step as possible, and convince yourself that every step is a logical consequence of the previous.
If you cannot convince yourself at a specific step, then it's either wrong, or you're not making a necessary connection: reread text that comes before it or the chapter corresponding to the exercise.
In case you are reading a proof or anything like it, do the same thing. If a step is not obvious, try inserting small steps on a separate piece of paper.
Of course, you cannot endlessly do this. However, after you have proved for yourself that some basic theorems hold, once they come up in more advanced methods or theorems you will be able to go 'ah yes I know that is correct' despite possibly not being able to reproduce it on the spot.
That is, for me, the gist of it. Although I do want to add that sometimes a certain reasoning only 'clicked' years later. It happens.
Good luck! (And have fun!)
EDIT:
For some reason I forgot my other main gauge for understanding: explain the topic at hand. There are some levels to this: explaining to a peer with similar knowledge, explaining to a friend with lacking relevant knowledge, explaining to your mum/dad/uncle/etc. (assuming they don't have knowledge on this, but would still listen to you), and explaining to your significantly younger sibling/cousing.
It is not necessary to actually go up to any of those and explain whatever you are struggling with. It often suffices to think about how you would simplify the complex concepts and if its possible to draw similarities with real life examples. Personally I usually only explain to fellow students that I'm studying with at the moment.
Your ability to explain in a clear and concise manner will indicate your grasp of the topic. It's not something I invented myself of course, but it's something that I realised on the go.
A: Achieving mastery is possible just with lots of practice. Not every question is easy to answer, maybe you forgot a result you need maybe it's a theoretical fact full of definitions which confuse you.
In my opinion, to become good ad something you should spend a lot of time (uniformly distributed) on a long period.
So don't be too hard with yourself.
Restarting the subject from zero, I think it's a bit a loss of time ; a better approach is to understand deeply the work of someone else on that question so that you will probabily find the things you missed to arrive to the result.
