you cannot logically be an anarchist and support "public schools." they are state institutions funded by extortion. as basically mentioned, the term "public" usually means "state." in economics "public goods or services" are goods or services which the provider cannot regulate who receives it, such as a radio broadcast. this is tied to the "we the people" propaganda the 'state' uses to make people think that "public property" means "the public is considered the owner (after all, the state demands you pay for it under threat of violence whether you want/use or not o.O)" - but 'the public' is just a floating abstraction. it always boils down to some group, and in this case, it's the state. if 'public schools' werent funded by extortion, if it wasnt considered compulsory that you attend, if the state didnt regulate all the curriculum, and if some people got together and made a school and started teaching (even now just out of the house or wherever) and offered it as a service, fine.. as long as they dont coerce people to pay for it, to attend, etc, it's consistent with anarchist principles (the non-aggression principle, individual sovereignty, "self-ownership," liberty). though "private schools" are still under the boot of the state, they essentially consistent with anarchist principles as they usually gather their funds through voluntary (as can be given the situation of current property relations, what some call "capitalism" and others call "socialism," the states demands, etc) payment of tuition.