I think Wyrdolphin actually makes a pretty good point here. First of all, math isn't science. Science is empirical and math isn't. There's a lot of science you can't do without doing experiments. This may not be a huge issue at the gradeschool level, but what about at the highschool level? Where do homeschoolers even learn this kind of science? I wouldn't be able to teach higschool chemistry nearly as well as my highschool chemistry teacher, and I doubt most parents would be able to either. I'm barely even remember my chemistry (and that goes for my college chemistry, too).
As for the performance of public schools, there are a lot of really bad areas that really drag down the scores of other schools. Illinois always outperforms the national average, and the schools in my area always well outperformed the state average. The schools where you live might not be quite as bad as averages would indicate.
What happens when it comes time to study chemistry that goes beyond cooking, though, like highschool chemistry? What are you going to do when you need to create specific chemical reactions and learn good laboratory procedures?
How much time should be devoted to visiting botanical gardens, though? A field trip like that in graofftopicr school was usually considered a day of relaxation and fun, and even at that, we always had a small assignment to complete.
Well, we didn't specifically have biology before highschool, but again, what will you do when it comes time to learn about triglycerides, fatty acids, and DNA recombination? Biology was taught freshman year at my highschool, so I don't remember a huge amount of it, but from what I remember, it's beyond the ability of most parents to teach. I don't remember the labs extremely well, but I doubt most households have all of the equipment either.
Well, I'll admit we never learned anything about geology after graofftopicr school, but again, how much time should be devoted to this? Is this really going to prepare the student for college, or even highschool? My cousin has a PhD in geology, and it actually sort of started unintentionally when he got his bachelors in chemistry and realized with a few extra classes, he could add a degree in geology. This indicates to me, at least, that in the long term, studying chemistry, and properties of media, like birefringence, are better if your interested in geology than just studying rocks.
And if a kid is interested in astronomy, he can get a tellescope and study it on his own, as at least one of my friends did (who now is also a physics major, but infinitely cooler than me).
Once again, is this really the best use of his time? In my highschool, something like that would more likely have been done in an art or shop class than a science class. I can't see it being all that usefull for college. I'm sure the kid had fun making the weapons, but that's not really the point, is it? It might be better as a hobby than part of class time.
Even in the humanities, isn't there something to be gained from, say the discussion of literature with peers, or the insight of a teacher who really knows the work well, or even a class structure that's based on preparing kids either for college classes or exams to get out of taking the college classes?