It is taking me a long time to figure that out and to be quite honest, I'm still not sure. Of course I read every homeschool blog ever and even listen to podcasts to try to figure out what is best for my Bubby.
I guess really the main philosophies I have come across are classical, Charlotte Manson, curriculum based (kind of like doing traditional school, but at home), unschooling and eclectic homeschooling. I like aspects of all of these which basically makes me an "eclectic homeschooler." I am very drawn to the classical style and unschooling which happen to the polar opposite of each other... of course, that makes sense for me.
I love the idea of classical education because of what children learn and how they learn it. We have joined Classical Conversations, which is a local co-op centered on classical education. Classical education is based on three stages of learning... the grammar stage, dialectic stage and the rhetoric stage. In "elementary" school children are in the grammar stage. Starting at 4 years old children begin memorizing some crazy facts. The point in this is that they create a "file" in their brain for this information so that when the dialectic stage comes along they will have all the facts already and will be ready to expand on them. Four year olds are just little sponges and they have the most fantastic memories. Really anything they hear repeatedly will stick (good or bad). Why not expose them to the Bible, foreign language, history, science, english grammar, music and art? Oh my how much easier it would have been for me to learn in the upper grades if I had been given the gift on knowledge as a small child. As I have said before I just ache to give Jamie the best opportunities in life. I believe this is one of them. I am learning so much too, but Jamie picks it up so much quicker than I do. I have to ask him to remind me what we have learned. As the children progress to the dialectic and rhetoric stages there is a lot expected of them. I have more learning to do about those stages. But for now we are quite excited about it!
I also love the idea of child led learning, which is more of the Charlotte Manson and unschooling. An example of this would be Jamie's recent obsession with Jimmie the giraffe at the zoo. So then my job would be to provide him with every opportunity to learn about giraffes. I can then find ways to incorporate all the other areas of learning, centered on giraffes. Using this method children also learn through everyday experiences. A big one for us is cooking. Jamie is learning fractions and practicing his reading by following recipes while we cook and bake together. If Jamie is "choosing" what he is studying it creates intrinsic motivation and therefor he learns more.
I don't know what the future will hold for us, but right now these styles of homeschooling work for us!