There is no constitutional right for home schooling (or private schooling) as a replacement for public schooling, at least in the strict constructionist sense. Every parent has the right and duty to educate their children, but that right does not imply the right to also keep their children out of public schools if their state (or under the authority of the state, whatever school district they are in) requires their children to participate in public education.

Most states allow home or private schooling in place of public schooling, but states do this because for some families it is a reasonable and desirable thing to allow. But it is within the power of the states to insure that all the child citizens meet some standard level of education, and if in the opinion of the states that includes certification of all teachers and curriculum (public, private, or home school), then in my opinion, parents do not have the right to exclusively home school without proving they meet the minimum standards set by the state. If a state choose to certify only public instruction, that is within the state’s domain of authority under the Constitution but in my opinion not in the best interest of their citizens. But the Constitution deals with constraining governmental process, and does not insure a particular result of the process results. And since our Constitution says nothing about education, presumably the regulation of education is reserved to the states and the people to figure how and if to do it on their own.

I repeat: all parents have the constitutional right to home school their children and/or educate them in private institutions. But there is no constitutional right to avoid state laws that require a given amount of state certified education. The choice between state certified education and state uncertified home schooling is not an either or choice. Parents have the right to home school their children (even if they are not certified as educators) but do not have the right to exempt their children from state laws requiring education from state certified teachers.