As I have said before, teaching a child is simply providing him with the tools to learn, and Macaulay shows this in her outline of a curriculum based on "living books" (literature and other works written by a single author truly interested in sharing their knowledge, as opposed to textbooks) and a love of learning. The three R's are indeed important to the creation of a free citizen, which is why we still insist on teaching them, but children must be allowed to love reading, to love writing, and to love the intricacies of mathematics. How else are they to educate themselves if they do not possess this love, and how are they to be free citizens if they cannot or will not educate themselves?
One of the impossibilities of public schools is the growing expectation that teachers can take the place of parents--that is, that everything can be taught in school. In contrast, homeschooling gives the teachings of morality, ethics, "self-awareness", sexual education, etc, proper context. Children learn by example; thus, a lesson in honesty is much more effective when a child sees his parent tell the truth than when he watches a video or demonstration in class. Also, as Macaulay explains, children are rarely confused by differing viewpoints in such matters when they are kept at home. They more easily develop will and reason through this contextual, exemplary teaching and through encouragement to figure things out for themselves.
Homeschooling provides such a richness of learning that children become not only academically stronger, but wholly stronger as well. This is what education is supposed to be.