what period of history is that?
The Archaic period
Maya civilization
The Archaic period, prior to 2000 BC, saw the first developments in agriculture and the earliest villages. 
how far back can we trace the mayan culture As far as circa 2000 BC
wiki:15310873
Maya civilization [SEP] Introduction
The overarching term "Maya" is a modern collective term that refers to the peoples of the region, however, the term was not used by the indigenous populations themselves since there never was a common sense of identity or political unity among the distinct populations. Today, the Maya peoples number well over 6 million people, speaking over twenty-eight surviving Mayan languages and residing in nearly the same area as their ancestors. The Archaic period, prior to 2000 BC, saw the first developments in agriculture and the earliest villages. The Preclassic period () saw the establishment of the first complex societies in the Maya region, and the cultivation of the staple crops of the Maya diet, including maize, beans, squashes, and chili peppers.