Why aren't public schools unconstitutional?

One of the most satisfying moments in my teaching career came when a student said, late in my First Amendment class, "Now I understand why you said on the first day that public schools ought to be unconstitutional!" (By "ought," I mean "by a consistent application of current First Amendment doctrine to the existence of public schools," not that this would be a good idea. Indeed, I think that one of the rules of modern constitutional law is that public schools are constitutional -- another is that Brown v. Board of Education was correctly decided -- and if we explicitly worked from that premise rather than skipping over the whole embarrassing issue, other areas of First Amendment law might see interesting changes.)

I don't think public schooling is an establishment/free exercise problem as some do (see also here). I don't think education is inherently a religious issue, at least not for all subjects (I've never heard a plausible explanation of why math class could interfere with someone's religious freedom). The basic problem is the government judging any idea or statement of fact at all. Suppose I believe that the reason nitrogen nodules cling to the roots of plants is love; if I write that answer on a biology test, I haven't committed fraud or otherwise harmed someone else, and yet the state can still punish me for my idiotic belief, because it's idiotic. And let's not even get started on what the state can tell me in history or social studies ...

Someday I hope to write the article about starting from the premise that public school is constitutional, and seeing how First Amendment doctrine would have to be altered to conform.