This is related to what people usually think of as 'homeschooling philosophy.' A question like "What is your homeschooling philosophy?" would automatically bring to mind whether you're an unschooler, a Montessorian, a classical educator, or a child-centered (as opposed to hands-off unschooling) homeschooling family. Those methods are most likely how you are trying to accomplish your goal, but may not actually be your true goal, just how you're trying to get there (pedagogy versus goal). The philosophy I'm talking about may be different (though the method could very well lead to certain goals that you may or may not have anticipated).
What is it that you really want for your children? What is so important that you've decided that you need more control over your child's schooling? Those questions should help you answer "What is the primary purpose of education?"
I've been listening to Philosophy of Education, by Leonard Peikoff. He touches on this point in his introduction. It is fascinating to hear the different end-products and how that would drive pedagogy (how to teach). It makes sense that your ultimate goal for how you want your child to be as an adult will drive how you teach them. As an example, if you're primary goal is a child that is a creative being, you will allow them to do pretty much whatever they want whenever they want because you wouldn't want a lot of structure to dissuade them from doing whatever might strike their fancy. If they happen to pick up a lot of information on the way, that's fine but what you're really after is the creative part. If your ultimate goal is knowledge, then you pretty much just want them to read, read, read, listen, read, memorize, read, cram, cram, cram. If that child also learns some other stuff on the way, no problem, but your goal is to 'stuff them full of facts,' as stated by Peikoff. Another ultimate goal often put forth by education (and this is one of John Dewey's primaries so this is generally what the public school is trying to do) is socialization. Everything else that may come from going to school (like information or thinking skills) is secondary to getting kids to work together and, indeed, to rely on the group for everything. The kid may learn how to read but what is most important is making them feel impotent without others doing the same thing. Thinking skills are a recent bailiwick of current (the last 60 years or so) educational thought as well. Thinking skills, according to this idea, are more primary than anything else--including information. If a child struggles with blocks long enough they will develop the thinking skills needed to add numbers together. Their interest does not include facts or creativity, but if those come along, well, it's not the end of the world.
The stand in Philosophy of Education, by Leonard Peikoff is that the primary purpose of education is to teach knowledge and thinking skills together--it does not have to be a dichotomy of one or the other. The other goals listed above (socialization, creativity, and there is also individuality) can be included as nice side effects of a good education, but the PRIMARY goal of education should be to teach knowledge and thinking skills. These ideas are not independent because in learning facts in the proper order one can see how human knowledge progressed and thus understand the thinking that occurred to get us from point A to point B. Once students have seen good thinking demonstrated, they learn good thinking themselves.
Look at how your children are currently learning. What is it that they are doing? What traits and understanding do they take away from their education being structured in such a way? Does that type of environment mate with what you believe is the ultimate goal of your homeschooling journey?
Are they flitting from activity to activity with each whim that happens to suit their fancy at the moment? Are they becoming encyclopedias that can recite facts but don't understand how human kind got that information? Are they spending a lot of time discovering that red and blue make purple when it is something that should only take two minutes to be taught and demonstrated?
This is the homeschooling philosophy that I'm thinking about this week.