We ended the last post on this subject in the middle of the fifth century B.C. At that point, the Persian Wars were thirty years in the past and the Golden Age of the Athenian Polis was underway. Without classical schooling, however, the likes of Pericles, Sophocles, and Phidias had to bring the Athenian culture forward on the back of their own limited grade school education. There was no “modern” school in existence to teach them philosophy and science like there would be later.
This is an excellent example of the time lag that always exists between culture and education. Culture is driven by experimentation and the spontaneous output of new outlooks, while education is a more conservative process of defining those advances in the form of a routine.