Astronomy throughout the ages

Having been born and raised in Mexico, I grew up reading in class about the ancient astronomical discoveries of the native tribes in Latin America before the Spanish conquest. What really amazes me is how precise and accurate their calendars have proven to be even now, several centuries after they were first compiled. I've actually visited the astronomical pyramids and seen how complex and extensive their research was. The Mayan calendar was first developed in 3114 BC and runs up to the year 2012. So far every single important celestial event foreseen in the calendar has proven correct. Not only that, even rather recent natural phenomena has been recorded in there. It's baffling.
How did such an ancient culture gather such knowledge through observing the skies? What is that link that so acutely ties man, through the centuries, with the cosmos?

I'm incredibly curious to cover how this kind of theoretical knowledge in class, and how it has been refined through the Copernican era until now.