Sorry if this is the wrong section of the forum, but I didn't find a proper section to post this. I am having a really bad time in trying to understand how I should study. Usually, I put a paper sheet on the solution of the problem I am solving and I try to solve it myself. If it is (at least for me) too much difficult, I usually try to think about it for the next days while and, in the mean time, try to solve other exercises. When I am stuck for too much time, I usually read just one line of the proof and try to go on alone and I repeat this process until I can solve the problem (or, in the worst case scenario, until I have read all the solution). I do this mostly because I think that this is the best way to learn math: that is, struggling yourself against the problem. However, often I feel like I am spending too much time on problems and that I am just stubborn and I don't want to be "defeated" by the problem. The point is that I always had troubles with math in my youth and I discovered the beauty of it only after highschool. Because of this, I feel like I must train myself in a very hard way to become successful in this field. However, there is a part of me that realizes that I can't solve all the problems I encounter and sometimes I should see how other works and learn the problem-solving strategies in a "less active way". I am afraid to "get old" and go nowhere because of this attitude. This has become toxic, because sometimes I even do this with major theorems of the textbooks I am reading. Do you think that reading solutions after struggling on problems/theorems can become a bad habit and, sometimes it is even more useful read the solution/proof of a problem/theorem after spending a reasonable amount of time on it and split my time in "active learning" (that is, solving the problems myself without external help) and "passive learning" (that is, seeing someone else do the problem and study its solution)? Isn't this latter attitude a risky habit that could make me waste my possibility of becoming a professional mathematician?
You seem to being doing everything correct. To be a mathematician (in my opinion) you need to be thinking about math (almost) all the time. You're doing correct by thinking about a problem for days. This makes you think about the problem in many different ways, which is a good way to start thinking like a mathematician. Looking at a line of the proof is ok IF you thought about the problem for a while. Otherwise you may never see the proof. For me, studying math was difficult but extremely enjoyable. If you don't love math, then it is not the smartest idea to try to become a mathematician.
For the record, I am not a mathematician as I do not have a PhD in math. I do however have an undergraduate degree and a masters degree in math.
This would be the right way to work on an exercise you are given (which is there to give you practice using what you have been taught), but not necessarily for a worked example (which is presumably there to show you how to do something), and definitely not for a proof of a major theorem (which may have been the life work of some past mathematician, and for which you are not expected to invent the solution on your own -- though it's true that sometimes a modern approach can make what was originally a great challenge relatively accessible).
The more preparation you have been given for a problem, the more valid it is to try to do it entirely on your own. But sometimes you do need a hint.
@Steven G, and Dr Peterson, thank you. I really needed a confront (since I am studying alone) and your words helped a lot, I am very grateful to both of you. I do like a lot mathematics, and I like spending a lot of time on problems, is just that sometimes I am insecure and concerns start pass through my head. I hope I will find some kind of stability sooner or later.
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You seem to being doing everything correct. To be a mathematician (in my opinion) you need to be thinking about math (almost) all the time. You're doing correct by thinking about a problem for days. This makes you think about the problem in many different ways, which is a good way to start thinking like a mathematician. Looking at a line of the proof is ok IF you thought about the problem for a while. Otherwise you may never see the proof. For me, studying math was difficult but extremely enjoyable. If you don't love math, then it is not the smartest idea to try to become a mathematician.
For the record, I am not a mathematician as I do not have a PhD in math. I do however have an undergraduate degree and a masters degree in math.
This would be the right way to work on an exercise you are given (which is there to give you practice using what you have been taught), but not necessarily for a worked example (which is presumably there to show you how to do something), and definitely not for a proof of a major theorem (which may have been the life work of some past mathematician, and for which you are not expected to invent the solution on your own -- though it's true that sometimes a modern approach can make what was originally a great challenge relatively accessible).
The more preparation you have been given for a problem, the more valid it is to try to do it entirely on your own. But sometimes you do need a hint.
@Steven G, and Dr Peterson, thank you. I really needed a confront (since I am studying alone) and your words helped a lot, I am very grateful to both of you. I do like a lot mathematics, and I like spending a lot of time on problems, is just that sometimes I am insecure and concerns start pass through my head. I hope I will find some kind of stability sooner or later.
I think you ought to expand your definition of 'mathematician' such that you become a mathematician, Steven G!
[imath]\;[/imath]
In my heart, I feel that a mathematician is someone who has earned a PhD in mathematics or a mathematician is a blind man in a dark room looking for a black cat which isn’t there.I think you ought to expand your definition of 'mathematician' such that you become a mathematician, Steven G!
[imath]\;[/imath]