I sometimes wonder if there might be too much teaching in a "crowding-out" sense. There is only so much time and teaching takes it away from independent learning. If one is assigned sufficiently many books to read and techniques to learn, one cannot pursue independent thoughts to the possible detriment of new ideas. However, if one is not assigned anything, one might conclude that one should not do anything. What is the optimal mix between independence and "teaching"?
Dr. Johnson's adage that "a man should read as his fancy takes him for what he reads as a chore will do him little good" seems to me to be almost self-evident, though many will require some assistance in getting the most out of their fancies. For instance, there are exceedingly many books worth pursuing and one's fancy may find difficulties in choosing among several at the top. The guidance of seniors, such as teachers assigning books to read, may then prove helpful.
This approach regards fancy as rather finely divisible, since one is allowed guidance to pursue it, in contrast to the route of choosing a subject (say, in graduate school) and being offered guidance there according to often fairly stringent limits: these very courses and these specific books or papers. But again, maybe that is the right approach?
Here is a radical idea that I wish would be implemented to test the relative merits of the finely-divisible fancy and the block-fancy: Let students take whatever courses they like (if any) and judge their progress by paper output after three or four years. Students taking courses may begin writing papers more slowly, but to the extent that courses help them their papers will be of a much higher quality when they begin. Students who want very narrow specialization can avoid several courses and, if they were right in doing so, will benefit from the freedom to pursue their own projects which reduced course work allows.
Maybe there are some programmes of this variety, only I have not been acquainted with them yet. It would be interesting to attempt to measure the calibre of the students they graduate as compared to that of other programmes.