After I took my first analysis course and learned how to be truly "pedantic," I have always been having a dilemma of balancing myself between being "detailed" and "intuitive". I would like to ask how other Math SE people manage this problem. (If this is a duplicate, I would appreciate the link to the similar question, but I also want to ask some following questions if anyone answers this, so it is not completely duplicate.)
Of course, being detailed and intuitive should not be a duality, and I think anyone needs both aspects when it comes to learning mathematics. So, more specific question is: How do you fill up your details?
This question may be understood easily if one considers a very fast paced course or a book that skips many details.
For me, I have tried
- To write (TeX) up entire topic in my own notations and details (e.g. http://gycheong.wordpress.com/)
- Make details that I could not directly see as easy exercises and just prove them (on papers neatly and collect them).
- Read a chapter very briefly and accept whatever the author says without too much thinking and reread it very carefully.
So far, Method 1 (which I have been using for more than a year) hasn't been working very well. It took too much time and, to me, it started to seem like understanding and writing were not exactly the same. I think Method 1 is good when someone wants to review a course that he/she understood (but not always a good way for the first learning). I have switched to Method 2 for about a month and just started to use Method 3 for some subjects that interest me but are sides. Method 2 has been working very well and it also gives a solid background for dealing with more difficult problems that do not trivially follow from theories (again, this is just my thought as the word "trivial" is different for everyone).
I think I am safe to say I am an "average" undergraduate senior in math and I am hoping to continue mathematics as graduate level this September. I really hope to see various opinions from a variety of people in different stages of mathematics.