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I'm a student studying Mathematics at a university level. I've completed Single Variable Calculus, Differential Equations, Multivariable Caculus, Real/Complex Analysis, and Linear Algebra and I've gotten decent grades in all of them. I studied Probability and Statistics on my own with videos and textbooks. I'm taking Abstract Algebra and Intro to Algebraic Topology this year (third year) and I was wondering what the best method to studying these courses and Mathematics in general is.

I attend all the lectures and watch additional lectures from a different university (I usually pick one lecture series with the best quality) from beginning to end for review and different insights. I read my textbooks but do not do much problems in it. I usually just try to spend all the time solving p-sets. Is this a good way to study Mathematics? Should I do more problems from the textbook? I usually do only p-sets but I think it might be better to spend some time doing applications from textbooks as well. I also do not bother with proving theorems myself. I look at the proofs in the textbook, read the annotations, and if I think I understand it, then I don't bother doing it myself :( Are doing these theorems independently important in learning Mathematics? Also, are combinations of lectures, video lectures, textbooks, textbook problems, and p-sets enough to succeed throughout undergraduate courses and graduate courses? I really hope I get into a graduate school for studying Mathematics! Hopefully, I can get some feedback on the way I'm studying Mathematics. Constructive critism/suggestions are very welcome.

P.S. It would be helpful if you guys can provide me with good online lectures for courses that I would have to take later (more advanced courses; preferably from YouTube!) and it would also be helpful if you could provide excellent textbooks for these courses!

P.S.2. It would also be helpful if you guys could tell me if studying with 1 textbook is optimal, or studying with 2 or more would be better. Personally, I think that 2 would be best for reviewing and getting different approaches but it might take some time :( I want to know your thoughts on it!

P.S.3. The reason I tagged this into self-learning is that I want to apply this method of learning to courses that I wouldn't take in the University itself but for interest and knowledge only. I am aware that I cannot take all the math courses at the University but I don't wanna miss out on many things so I'm trying to teach myself a few courses that I wouldn't be able to take and be more relaxed when learning (for fun!)

Thank you!

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closed as primarily opinion-based by Martin Brandenburg, 1999, Jonas Meyer, graydad, John Ma Jun 30 at 2:43

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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I know this is totally unrelated, but just so you know, if you want to add a P.S. after a P.S., the second P.S. would be P.P.S. -- this is because P.S. stands for "post script", so P.P.S. is "post post script" -- script that is after the post script. (And a third P.S. would be P.P.P.S.!) :) –  user46944 Jul 21 '14 at 1:48
    
Also just from a personal standpoint Ive learned, especially early on learning the material spend a lot of time thinking of the definitions and theorems that books provide try to come up with why you would want to define things this way, see the motivation in the text. Struggle with the material and after doing that look up some resources that give a explanations of examples, intuitions, and motivations of concepts in mathematics because lots of times when things seem confusing or I am lacking motivation it is because I don't understand the why or the reason for definition and theorems –  Kamster Jul 21 '14 at 1:51
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As a general rule, you'll be much more likely to get an answer if you start with the question instead of your life story. People are loath to read long biographical detail before they know the real nature of your question. –  Thomas Andrews Jul 21 '14 at 2:35

4 Answers 4

I'll hazard an answer. For both plans and videos, notes etc... much can be found from MIT's OpenCourseWare. I would say your general method of studying is wise except you probably already realize it's a bit heavy on lectures. I also wouldn't say it's necessarily wise to just work all the problems. Whatever you do, it needs to stretch your mind. Try asking or answering some questions here which test the boundaries of what you study. Sorry to be vague, but only you know what you don't know. Whatever that is, that is the thing you want to pick at.

Just as a toy example: if you don't really know the analysis which underlies complex analysis, you might ask something like "we used uniform convergence to prove certain theorems of complex analysis, is this special to complex, or is there a more general intuition I should seek out here ?

Finally, for a global plan for an ideal undergraduate sequence, you might find some ideas at MESE useful.

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I think that the only answer I could give is that as long as you like what you're doing, there's no wrong way to study. If you try to change your method and by doing that you don't have any pleasure, stop ! Of course there are boring and tough periods, but if you enjoy it more than it sucks... You'll do it !

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When I want to study a new subject in math, I find myself a good textbook on the subject, buy a new notebook and sit down at the desk. I work my way front to back through the book, taking very descriptive notes in the notebook. This allows me to have the exact balance of algebra and intuition in my notes. I also don't actually write anything down until I understand and can demonstrate why it's true (unless doing so would be pedantic and/or take up too much space. Also, I always explore the topic that I'm currently on outside of my book, and record any useful or interesting findings. At the end of the (admittedly long) process, I am well-versed in the subject. I actually don't listen to lectures, but I can understand why it's helpful to a lot of people. You know yourself better than anyone else on this site, so you know what helps you learn the most efficiently and deeply. I've heard that Khan Academy is high-quality, so there's that if you're looking for something outside of a university setting.

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If you would like a book giving challenging problems in linear algebra, modern algebra, and real analysis and complex analysis, which really put your mathematical reasoning skills to a good test and training, as well as forcing you to understand how/when certain classical theorems are used, I highly recommend "Berkeley problems in mathematics". It is a compiled ~35 year collection of problems, basically 36 problems per year, from old "prelim exams", designed to test new graduates in their undergraduate training and ability to make sure it is "good enough" to indicate the student likely to succeed at the graduate level at UC Berkeley. In fact, if you don't pass the UC Berkeley math 12 problem test within your first year of graduate program (you get 3 tries), then almost always you are kicked out of the graduate math program there. I know from experience, by going through those problems I became more well versed in the classical subject matter, especially by reading through a solution if I was totally stumped and then reviewing a theorem used in the solution if I didn't remember it totally.

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