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.