How do I get the best mathematician out of myself? Hello all, I'm an undergrad on the first year of his masters maths program. Right now, I'm really interested in doing a PhD and being a researcher. Not sure what area, but as a first year I gather that's tomorrows problem. My issue is with how I spend my time learning and developing as a mathematician. If I'm going to do this I want to be the best that I can, but I sometimes feel that I'm not going about it the right way. My current philosophy ( in practice right now) is this : Assuming all of the compulsory "bookkeeping" has been done ( by which I mean weekly assignments, lecture notes all written up, lecture proofs gone through) focus your time on asking yourself hard proofs. I provide two examples: 1. In linear algebra, we were taught that " the rank of any two row echelon forms of the same matrix are equal". The proof was omitted so I decided to work on trying to prove it. 2. In calculus, we were taught about integrability of a function, but the definition of this property was not in terms of a limit. I decided to try to unify the definition we were given with one which concerned a limit  specifically that the limit of the Riemann sum of the function existed. For number 1, I spent a few hours toiling before producing what I thought was a proof. I showed it to my lecturer, and low and behold, he spotted a technical issue I wouldn't have. He said to wait until I'd learned more and return to it. For number 2, similar outcome but self  realized. I noticed that I hadn't been told the definition of the limit of a sum, rather the limit of a function. I felt out of depth immediately, even more so because I wasn't even sure my hypothesis was correct to begin with. In both cases, I'm sort of setting myself mini projects to best recreate the conditions of being a senior researcher. My concern is with how I'm developing a knack for biting off more than I can chew. Is it more important to drill the questions set on the degree and get these foundations down like a machine? Should I leave the deep questions until later on? (seeing as I'm usually told the same thing each time) I mean, what are the odds that I'd ask myself a deeper proof which can be conveniently solved all with first year knowledge, that wasn't covered in lectures? In summary, I have two main questions. Should I even be doing this during term time ( when exams on compulsory content are around the corner) and in the summer when I can do my own thing, should I carry on throwing myself off these cliffs without a harness, or just buy some more advanced books and try the proofs in those? Do I continue down the guided path, or is there merit to going large? It would seem going maverick has it's drawbacks. Best regards to all, and a merry Christmas. James. 
I assume you are given sets of homework problems that are subsets of the problems listed in your text. After completing these finish the rest of the problems listed, especially the more difficult problems towards the end or so marked. This way you will explore each subject in depth w/o running into areas you haven't learned about yet. Another option is to get a textbook from your library that is essentially the same course but at a higher/deeper level. You're an undergrad now, so grab yourself a graduate text on linear algebra or whatever and go through that. The difference shouldn't be that significant (except maybe for probability) but here and there you will be exposed to deeper material. I assume you've got access to a library with graduate textbooks, if not it's remarkable what can be found online these days. 
Although you're right to care a lot about getting better in math, you must successfully balance all of your studies and keep in mind that there is life outside of academics. 
I like to think of math as one big accumulating body of knowledge, and to conquer it (or at least, feel like you can) you need to tackle it from all angles. This is at least while you are still searching for some specific area that you are really interested in and really want to pursue further. As romsek has said, i would finish all the prescribed homework and assigned tasks for the unit before venturing out into the unguided, and unknown world. That being said, i believe it's very important to do your so called 'mini projects'. Perhaps you don't need to go all the way with them though. Dive into an attempted proof by all means, but if you begin to become stuck, don't let it consume you. As you begin to learn more you will probably end up going back to the things you were once stuck on, and solve them with ease. 
I think there's a lot to be said for mastering the basics. While it's really fun to jump off the deep end, and try to "rediscover" things, in the beginning, this is really tough. Once you've gained enough math chops to "prove" your way through entire elementarylevel books, that is, read the statements of the theorems, but work out the logic by yourself, I think you'll find your math projects more fruitful. You'll know the definitions, know the right questions to ask, know the difference between a blind alley, and a potential buried treasure. That being said, trying to rediscover and recreate theory on your own is one of the great pleasures of mathematics. If it's not interfering with your course work, have at it. You may want to start smaller, though. I find linear algebra books easy to prove your way through. I'm working through Curtis's Abstract Linear Algebra right now. Haven't hit a proof or problem I couldn't do yet. It's really fun, and a good skill builder. You can even make a game out of spotting and fixing the logical inconsistencies. (pg 11 on the span, anyone?) In particular, you'll have a much easier time reasoning about integrals, limits, etc after taking a proper real analysis course. I try to struggle "just enough" as it were. Try to give yourself as little info as possible as you try to solve a project, but at the same time, if you are stuck to the point of discouragement, let yourself read the next line of a good proof. You could spend years rediscovering integral theory if you are at a basic level, and you're doing it without reference. I just finished undergrad, and I can reconstruct the Reimann theory in one variable in maybe an hour and a half, reproving everything from facts about the real numbers. It feels really cool, and really just took an analysis course to get there. I also highly recommend Spivak's Calculus to let you in on some secrets of proving the theory of calculus. His writing is intelligent, clear, and inspiring. Definitely something special among elementary math books. Russ 
How do I get the best mathematician out of myself? This is a guess: an exorcist? 
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