Someone said this:
"Memorizing the entire Swedish dictionary will never make you mastering Swedish. Same to Maths, the only way is to actively think & use basics you absorbed so far to practice in advanced environment and explore fast. Learning by doing will make you speaking maths fluently, flexibly and intelligently. Babies are the best learner, they neither play with single words too much before trying to use them to speak, nor waiting too long to absorb new words to expend the zone."
And our professor said
"one should be able to use and prove main theorem of Calculus/Analysis/Linear Algebra/Algebra/Probability/Topology even without thinking like breathing every second. Especially one want to do research on mathematics or even applied mathematical aspect of some natural science"
I think it's a good idea, even though Euclid said "there is no Royal Road to geometry", but there're still good and bad paths to go I think.
But when I try to implement this when reading textbook, it seems even though trying to do the exercise after each chapter especially the ones that just train to put values into the formula or simply using concepts, no matter how many exercises to do with these, it doesn't help a lot to understand things. well, the opposite trial, by using textbook not too seriously, just like a manual to PC game, a little faster "scanning" to get the main idea and def/theorem, then try to self-motivately construct the def/theorem by myself and try to prove theorems on my own even having the big "picture" in mind to construct instead of trying to recall the details, only refer to the book when proceed next or when get stuck too long for getting main idea. When this is doing , it seems the brain works more actively, and the def/theorem is absorbed, like get the hammer controlled on hand instead of only read the manual of hammer.
The question is firstly to hear some comment on it especially the two quotations on the top.
And for the "practice makes perfect", will there be some selection principle of "good practice" to more actively learn certain mathematical textbook ? Since simple value-in-formula-use or basic concepts problems really doesn't help too much. Or say, should one put time doing more proof problems skipping some too simple questions even though the formula is newly learned. Because try to prove on own, it seems more using something actively, but value-in-formula-use, no matter how many problems to do, it seems the tool still not belonging to me.