When was the public school system established?

The idea for public education came from the Catholic missionaries to the colonies. They would teach the children for free.

The Land Ordinance of 1785 established a mechanism for funding public education in the United States. Until at least the 1840s, however, most schools continued to be privately owned and operated.
Coeducation and the emergence of modern high schools; the expansion of compulsory education. The growth of extracurricular activities (1890s-1950s). The principle of equalization becomes a standard to achieve.
The United States Supreme Court's ruling in Brown v. Board of Education was a hallmark in education. It forced previously segregated schools to integrate, and led to the rise of desegregation busing across the country.