School boards should be able to determine what is taught as science?
Actually, I suppose they can teach just about any pseudo-science they want. A deranged school board could banish evolution and teach Lysenkoism (Stalin's favorite biology theory). The only constraining rule (other than a state's own constitution) is the First Amendment. They can't use a science class to establish a religious doctrine.
Yes, as a matter of fact. Just as they should be able to decide that only rock music can be taught in their schools. When the citizens of the jurisdiction don't like what is being decided, they elect a new school board. A fine example of a democratic republic in action.