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I have somewhat often found myself in the following situation, especially when self-studying a mathematical subject:

I'm reading a book on a certain topic, and at certain parts I don't immediately understand something, for example a proof. But instead of re-reading it I just skip over it and continue. This then leads to a feeling of uncertainty about everything I subsequently read, because I feel like I don't fully understand what's going on, which then immediately becomes a reality when I try to solve an exercise.

I guess my question can be broken down into 2 parts:

  1. What should I do when I'm reading something and don't understand it? (The answer to this is likely to find another source which might perhaps be more insightful, so I guess the follow-up question would be: what is a good way of finding alternative sources?)

  2. What is the best way for me to patch-up my konwledge on subjects which I have half-studied as described above in the past, which I already have some understanding of but am not confident in?

P.S.: I was slightly unsure what tags I should attach to this question, so if you have any suggestions please feel free to edit.

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    $\begingroup$ Most books have an extensive list of related literature you can dig into if you want another perspective. As for feeling more confident in the understanding of the given subject, I've found it very helpful to solve exercises and problems myself. If there aren't exercises given, try to apply the given statement to something you're familiar with and see if there is a correlation. $\endgroup$ Commented Mar 28, 2024 at 11:21

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Most literature has a bibliography or footnote citing for a lot of the concepts they present. Therefore to answer the question of where to find alternative sources, look to the bibliography or footnotes if available. Otherwise, a lot of mathematical concepts are explained and explained again by a diverse set of folks on the Internet, simply typing the concept followed with "explanation" or "tutorial" into Google should suffice

In terms of patching, the best thing you can do is keep studying as you are in order to locate these gaps. As of now you may not know what you don't know, and going back may unmotivate you to reread large chunks of content you already know. Therefore continue as you are and when you come across something you don't know, then refer to the advice of the first paragraph

Keep in mind that learning isn't linear. Simply traveling from A to B in in a book/resource isn't guaranteed to have you learn it all in one go. Most of the time, it takes several rereadings before you can grasp a concept, and that's perfectly normal and expected - especially in research and very high math education. Personally, it took me well over ten-fifteen rereadings for some research papers for my MA thesis before I fully grasped what they were saying and proving, and that involved skipping sections to read other sections in which those later sections made the previous work easier to understand

Edit:

I've asked some of my colleagues, and they have the following advice in terms of helping to retain or understand concepts from literature

  • Imagine trying to explain concepts to others, as if you're in front of a classroom teaching the topic
  • Ask/answer questions about what's going on in the book
  • Some misc. topics in book sections are not necessary
  • Look for worked-out examples or solutions to problems, as most problems presented have been solved over-and-over again in many different presentations
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