Improving Mathematical Maturity
Hello,
I am having some doubts/problems regarding my abilities in math. I am a math major, currently at a junior standing. I have taken calc I & II, linear algebra, probability, history of mathematics and an introduction to proofs class thus far, and I am taking calc III, Differential Equations, and Statistics this semester. I have gotten an A+ in all but the intro to proofs class in which I got a B+. I am interested in improving my "mathematical maturity" as it were. When reading my textbooks, or taking notes in class, when a proof or something similar comes along, I find myself becoming frequently overwhelmed and discouraged as I can sometimes follow along after a bit of rereading but I never really have a feel for where the proof originated, and sometimes I simply cannot follow along at all. I was wondering if anyone had any suggestions that I could do to become more "mathematically mature", and more comfortable doing proofs and the like? 
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Firstly, I want you to know that something like this is very normal. And introtoproofs class will just teach you the vocabulary and grammar of the prooflanguage, but it will not teach you how to speak the language. I remember that I understood epsilondelta proofs quite well, but I couldn't proof it myself if I wanted to!!
If I continue my analogy with a language: in order to speak a language well, you will need to practice a lot!! Here is how I always read a textbook. Others will do it differently, no doubt. But I found my method effective for me. 1) Read the proof in question. 2) Understand every single step in the proof. 3) Try to divide the proof in logical, easy steps. 4) Close the book and try to come up with the proof yourself. Only peek if you're really stuck. Repeat until it "feels good". 5) Ponder about the proof: What is the main idea in the proof? Could this step be shorther? Did I see this technique before? Could I generalize the technique and use it myself later? 6) Ponder some more: Is the converse of the theorem true? (don't worry if you don't find this, it can be a very difficult question sometimes!!) Where are all the hypotheses used in the proof? Can I find a counterexample if I leave out some hypothesis? What theorems did I use in order to prove the theorem (writing a logical diagram of all the theorems in the book is really really helpful)? What happens if I apply the theorem on simple examples? Where can I use the theorem in the exercises? 7) Make easy exercises whose proof is closely related to the relevant definitions, or whose proof is similar to the proof you just analyzed. 8) Make some more challenging exercises. Yes: this is a process that takes up LOTS of time. I realize this. But there is hardly another way to learn mathematics. Finally: there will always be proofs which will be mysteries to you. Some proofs are so beautiful and elegant that it's impossible to know what the author was smoking when he came up with it (most of the time, these are theorems which have a "name"). Don't let this discourage you. 
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If you tell me what class you're taking now, then I will gladly give an examle of the previous method.

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There are different ways of viewing probability and statistics. Statistics is built on top of probabilistic properties and arguments. Aside from the axioms of probability (Kolmogorov), we also have ones that follow from these which include things like the conditional probability formula. Its a little hard to reduce everything to a set of few things because there is a lot of context involved. Maybe if you could give us a few examples like micromass stated, then we would be able to give you a more specific and tailored response. 
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Since the OP brought up the issue of being overwhelmed with the material, can anyone also give some insight in how long to study per day (or what pattern) heavily proofbased mathematics?
I also frequently feel that I'm overwhelmed with the material when I study for long hours, but on the other hand I'm also worried that studying less would even hurt me more. Thanks. 
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The truth is that you probably won't understand everything enough in the time you have in any great detail. It's not to say you won't have any understanding, but a lot of the context won't be there. If you are doing coursework, it's best to just take the guidance of your professor, lecturer, and TA. They have done this kind of thing no doubt for a while so they have a good idea of what to get you to focus on. If you have any doubt as to what should be focused on and what should be prioritised, then ask your professor, lecturer or TA. This probably won't be much an issue because you will be assigned things like assignments, lab projects, exercises, and so on. Some exercises are even marked with an asterix which means that they are harder ones that will challenge students more, but will not be important for assessment. Also I advise you that if you have extra time and you want to learn new things, expand your available resources. Ask a question here on PF, read a textbook from a different author, read math blogs, ask your peers questions, get extra advice from your professor, lecturer, or TA. Nowadays with the internet, we have so much at our fingertips and with the right kind of filtering of information, we have so much access that the scholars of the past could have only dreamed of, so make use of it. 
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All in all, you need to study until you're comfortable with the material. There is no fixed time limit. 
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When it comes to studying, it isn't so much about the length you studied but how effective your studying is. There were times when I cold sit and study 6 hours straight and feel as if my understanding of the concepts only grew by maybe a milometer.
For me personally, I've found that the best way to study a mathematics class, is to review the proofs given in class. First, give a serious attempt to proving it yourself. If you can't do it review the proof given in class and compare why you method wasn't working. After you've done that for the proof given in class, try to find different theorems not proven in class but in the book and attempt the same thing. After that, try extra problems in the book and if you can't solve those ask the professor. I've found that doing all that helps give you insight as to how your initial outlook on a problem may be wrong, what 'trick' you may have overlooked, and also it should force you to review previous theorems and results thus help enforce the,. 
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This stresses me so much that I hate studying for the subject and I'd rather do unnecessary work for other subjects first, which then makes me even worse at maths :S 
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I remember when I first saw the proof of the implicit function theorem, that I found it really really really difficult and I also couldn't grasp the idea behind the proof. I haven't read the proof since then, but I hope that it would be a bit easier right now, but I still expect it to be not so fun to read. How are the smaller proofs?? Those often contain only one idea, so those might be easier. Quote:
You should make exercises in increasing order. First make exercises which are straightforward and only gradually do the tough things. But tough things are tough things: don't expect to find them fast. I find that exercises are a bit overrated at times. I think that exercises are a very good way to reinforce understand, to increase understanding and to test your understanding. But I don't really see much benifits of really tough exercises. You could use that time to research something for yourself for example. Quote:

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Did you find studying the proofs in books alone good enough though? We're using Spivak & Munkres now (mainly Spivak), and the exercises were either too difficult for me, or either useless because they were easy in the sense that they were more or less computational questions. (or expanding the expression and using the definition type of proof) I used to like Maths, when the questions were at least "doable". By "doable", I don't mean solving linear systems or ODEs, but for challenging proof questions were you could at least write something down, juggle around with it, break it into separate cases and figure out the answer if you'd think hard enough for a day or two. I enjoyed it and it was a triumph to me when I actually solved the question. But for now, I barely understand the question. For instance there was one assignment were we were had to prove Lagrangian Multiplier's theorem over a manifold from scratch, and I was completely clueless how I should have started. It then stressed me out because I felt I was going to lose a lot of marks and it started to make me feel unmotivated or even escaping from the subject. 
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And maybe "calculus on manifolds" isn't really your thing. It's certainly not my thing: I'm not very good in the subject. Struggle through the course and hope that your next courses might be more fun. Everybody has to go through some difficult/notfun courses some times. 
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One thing I like to do when I'm feeling motivated enough is to try to prove the theorem BEFORE reading the proof. Also, doing exercises before reading the proof or concurrently with reading it can help. The point here is that often, something seems trivial to the person who is saying it, but not for the person on the receiving end. This usually doesn't mean that the person on the giving end is smarter. It just means that you understand your own thoughts better than someone else's.
Einstein said something to the effect that if you read too much, you develop lazy habits of thought. I have some lazy habits of thought, myself. Quote:
To get some intuition about the implicit function theorem, for example, try to think intuitively about how it might apply to a surface in R^3. Not only does this give you some idea of why it is true, it also illustrates what the theorem is good for. Secondly, you ought to analyze the linearized version of the implicit function theorem. That's just linear algebra. If the linearized version is true, then it's plausible that the differentiable, nonlinear version is true. Note that the principle behind both of these approaches is to try to tackle the simplest case first. Being able to prove the theorem yourself takes a little more than having some intuition like this, but having a little intuition for something is usually more valuable than knowing the whole proof, anyway (the only pitfall being that if the intuition is too far from the rigor, then you might not know how to put it to use directly). With change of variables, the idea is just that the Jacobian can be interpreted as the factor by which volume changes under a mapping. But the details are very technical. If you understand the main ideajust an intuitive proof, it's good enough for most purposes. My adviser, who is a famous mathematician, said that most people start with the most complicated and general case, and it's wrong. So, part of the answer is that you are being cheated out of real explanations of these things. Another thing is that you might not quite know how to search for the real explanations. 
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Part of my problem is that I am not well versed in the language of mathematics. This can be extremely problematic and frustrating for me because it is very difficult to understand many topics/concepts in math without being comfortable with the language of math itself. Does any one have any ideas on how to become more acclimated with abstract thinking in math, because up to this point in my math education, I really have only been exposed to computational ideas.

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