What are the legal boundaries of a parent's right to direct their children's education in terms of a private school or homeschooling curriculum?

Question

All states in the US have some form of compulsory education law with allowances for private schools and homeschooling for children roughly between the ages of 5 and 17. There are some existing exemptions from compulsory education for special situations, but let's set aside those exemptions and assume we're talking about a child without special physical, mental, religious or geographical considerations when it comes to education.

In many states the curriculum for the non-public education must cover a number of specific topics, like language arts, math, history etc. Sometimes the law requires the schooling offer an "equivalent" education to the public school education. How comprehensive and/or specific can the government's dictates be with regard to specific topics?

For example, some states have banned teaching certain concepts like "That the United States of America and the state of Iowa are fundamentally or systemically racist or sexist. Currently, the law only applies to "governmental agencies and entities, school districts, and public postsecondary educational institutions", but what if the state refused to recognize a parent's proposed curriculum as equivalent or failed to certify them as a home school teacher because they knew a parent was going to teach the banned concepts?

What prevents a state from requiring certain topics that some parents might object to on non-religious grounds? Can the state insist on a particular version of history for example? If the state requires that the curriculum include teaching that "Jimmy Carter was widely regarded as the handsomest US president." do parents that believe that is not true have any recourse?

I realize that state law varies widely on how a parent can go about homeschooling their child, and I'm not trying to express an opinion on what children should or should not be taught. I'm interested in how far the government can go with compulsory education to ensure that its populace receives an education that is reality-based (we don't have winter because Persephone is in the underworld), and what safeguards are in place to prevent the government from using public education to teach things in its own self-interest like "only bad people need a lawyer when being questioned by the police".

Answer 1

The state has wide discretion on what to require that a school teach to children, particularly in the K-12 grade range. There is little case law where a parent or private school has challenged such requirements. However, for a state to forbid that certain things be taught would probably run into a first amendment problem, because the school has free speech rights. In addition, if the school is a religious one, forbidding teaching its dogmas and views might also impact the free exercise clause.

Note that a state need not permit homeschooling at all, and some states do not.

When a state requires that a school teach something that it disagrees with, it can often follow a line such as "Many people think X {Standard state supported concept} but we believe Y instead. Then it has taught X, but not endorsed it. Some religious schools have handles the teaching of evolution in this sort of way.

If a state mandated teaching matters of opinion as fact, such as "CARTER WAS THE MOST HANDSOME president EVER" there might well be a challenge, and I am not at all sure how it would be resolved.