I hope that someone can correct me if I am wrong: The first public school in New York City was founded in the eighteen forties. It was done to get young criminals off the streets. Prior to that all schooling was privately funded.
The Northwest Ordinance of 1787 provided for free public education in the territories where states were forming, a basically the Great Lakes area. Sections of land were dedicated to the support of education through being leased to farmers for growing crops. Free public education predates the ratification of the Constitution.