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I am a math-major bachelor student. And I want to get some advice about the approach I'm trying now for learning maths, not for efficiency, but for depth and fully-mastered.

Firstly, I want to know how does it look like when the real mathematicians were learning a subject. Because recent a few months when I'm reading the Calculus and Linear Algebra textbooks, I frequently feel sort of unsafe or uncomfortable by just read, try to understand what the author saying and finish the exercises. Then, I tried to do it in another way, close the book, and try to establish all the staffs on the blank papers. During this procedure, there're really many obstacles and difficulties, when I'm confused with a problem for more than ,say, half-hour then refer to book to find ideas. But after all these, when you see you yourself establish not entire but still a small theory-building on papers, then we call it comfortable and safe feelings coming out.

Secondly, the approach used above, I feel it improves some creative feeling or say to be easier to connect different concepts/theorems, better than the classical approach I used before as most classmates do, which is only reading books and doing exercises after each chapter. But apparently, I don't mean doing exercises is not important, it's very important I think. Just like playing music instruments, eg. for me when playing flute, you can't enjoy or even play an advanced music if you don't have extremely strong skills for fundamental fingering 'rule' , same as maths. But the point here is , I don't like or say adjusted with the classical learning approach. Same example with playing flute, in an classical way when a student learns to play flute, just as in ordinary music training school, they learn basic music knowledge, and train basic fingering for a very long time, and from like 'do do do re re re mi mi mi' to other harder permutations, to simple rhythm, to easy music, to... yeah, it's systematic and maybe efficient, but I really don't like this way, when I tried to learn flute in a very first time, I used this classical way but just for a few days, it bored me a lot and cannot even make it sound. But after a few months, I tried my own way, skip all the stuffs, just find a fingering table and a real my favorite music notation, and just try to play directly, at first place I tried every sound(symbol) one by one slowly. But it speed-up dramatically, I remember at that time only after 2-3 hours, I can just play that music, even though it's still slow when try a new music. And by this approach, more than a dozen of songs could be played by flute. So, I think this approach is somehow similar to when a baby try to learn language or something, the babies, they don't have systematic taught, but just be around or say inside the natural, chaotic, complex world directly. Therefore, I'm thinking whether it's also could be perfect way in maths. Because even though I tried this approach for 3 maths courses and I see the similar good results(especially when talking in details with classmates during discussion-session, it's fluent and comfortable to explain materials learned by this way.) , but I still want to know how some other people think whether this approach would be great to improve creative or imaginative ability in the long run, especially ones who are doing research in maths.

Thirdly, connected with the approach above, when dealing with the detail material like definition/theorems, how the real mathematician do that. I mean, when I encounter with a definition or theorem, I always try to 'see' a kind of 'image' in my mind, sometimes corresponding geometric viewing, sometimes algebraic expression, sometimes like 'invisible image', in order to really understand what it really means. For first example, when learning with mean value theorem, first is more geometric, if you get two pairs $(a, f(a)), (b, f(b))$ one could easily calculate the slope. And 'in between', if changing-rate has positive increment, then in order to reach the end point finally, there must be somewhere has negative increment to balance, since it's continuous from positive(negative) to negative(positive), it must cross the slope somewhere, and in this way, it's obvious to see it's a generalization of rolle's theorem which is only special case when $f(a)=f(b)$. But when proof, it's not geometric anymore, I could only imagine the algebraic expression in my mind and try it step by step. And especially, I don't know why, 'imagining' proof or theorem or definition in mind is easier to get 'big-picture' feeling than only writing it down on papers. For second example, when it comes to say Schwarz-inequality, especially proof, I could only 'see' algebraic reasoning, in one way, to get some expression contains both $AB$ and $\|A\|, \|B\|$, then we think a right triangle $AOB$, there could be projection for $A$ to $B$, combined with orthogonal property for dot product $(A-tB)B=0$ to get $t=\frac{AB}{BB}$, then we could construct $A=A-tB+tB$ through Pythagoras theorem we could prove the Schwarz-inequality.

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Too long; didn't read. –  The Chaz 2.0 Apr 7 '12 at 4:50

1 Answer 1

up vote 4 down vote accepted

Indeed, this is a very long question. Maybe you get more to the point if you expect answers. Anyhow. Let me point out some things.

In mathematics you should probably have some interaction with a more experienced person, i.e., teaching assistant, professor, etc. just to be sure you are on the right track. For some people it is difficult to find the right balance between formalism and intuition or just to understand what a proof in mathematical practice really is. So, talk to people (ideally people more experienced than you) and have your exercises corrected etc.

Something that worried me slightly is that you said that if you can't solve a problem after half an hour or so, you turn to books and other sources to get some ideas. I find it pretty normal to think about problems for a couple of days, or even months and years (weeks, months, and years are obviously inappropriate for exercises, but not for research problems).
Work on a problem, and if you don't get anything, let it sit for a bit and try again later.

Finally, you usually develop the best understanding of something if you try to develop the theory yourself. I.e., you read some theorem and think about its proof for some time. You will get some understanding of what is really going on. Then you look at the proof, and usually you only have to see some important ideas and you will be able to fill in the details yourself. This way you only have to remember the crucial ideas, but not every detail.

Also, if you learn something new that builds on things that you learned before, go back to the old stuff, in your memory, and try to reconstruct why these things worked the way they did. In this way you will learn about the connections of various results and you get a complete picture.

I have to say though, that in my own research I sometimes look up what I need and try to understand only the bit that I really need without going through the theory that surrounds it. I simply don't have the time and energy to learn everything that somehow touches my own research. If I notice that some concepts turn out to be important for me, I spend more time on learning this. I often I just go into the details of a specific proof that seems relevant and use other thinks as black boxes before I really need them.
This saves some time and energy, which is also important at some point.

I hope this helps.

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