First, I wanted to say a little more about that "It's about the family" thing. It concerns me that this concept is easy to misunderstand. What I mean is that the family's faith, attitudes, culture, habits, personalities and whatnot have a stronger influence on the child's educational outcome than their general educational choices. For example, a family with lots of engineers (ours falls into this category) will probably tend to have strong math students whether those students go to public school, private school or are homeschooled. Also, though our society tends to think that everyone should be strong in Math, the reality is that each person (and each family) has their own unique gifts to be developed and appreciated.
The second thought swirling around in my head today is another educational concept that I started thinking about in high school and still have a similar belief today. I think there are some that will disagree with me, but perhaps if I explain it clearly, some of the disagreement will turn out to be just a misunderstanding.
The way I thought of this in high school is that children sometimes need to be "pushed" (I'm sure a nicer word would be required or even encouraged, but "pushed" is what made sense to me at the time). What I meant by this might best, perhaps, be explained through an illustration I had in my head at the time. I remember having trouble understanding certain math concepts until I had completed a number of problems and then suddenly the concept became clear. While I was wading through the problems, I would sometimes be frustrated, but knowing I had to get through a certain number (even if it was simply to turn in the homework assignment the next day) or sometimes having a parent or teacher encourage me to do a few more was precisely what I needed to get through just a few more until I understood it.
This is part of a tricky educational concept of keeping a balance between freedom and non-freedom (whether that be limits, respecting authority, fulfilling certain requirements, etc.). I've spoken a bit about the freedom aspect of things and I think it's something often underappreciated by society today - particularly in the realm of education. And yet, I think it's possible to go to an opposite extreme too. I always loved Montessori's concept of "Freedom within limits" because it helped me make sense out of some of my questions about whether or not children should sometimes be "pushed".
I don't have a final answer on this, but I've seen some things that work with myself and with my children. The parts with affecting my children are a lot easier to explain.
It seems that my children are more eager to learn when they have some reasonable requirements and some structure to get them interested in things and perhaps keep their brain moving. This plays out in different ways. I've noticed that the freedom of unscheduled days after Christmas actually leads to some wonderful days of learning, but I strongly suspect that this still relates to some requirements on other days and the joy of having time to freely choose what to do. It also seems that if we have a few strong hours of formal school, the rest of our day tends to be used in good ways too.
On the other hand, some of our requirements are very basic, more like "limits" within which we require them to work for a time. For example, this year the kids have six "essential" subjects that I want them to work on a little each day (this applies primarily to the four older children - grades 4-12, though Ria, in grade 12, makes quite a lot of these sorts of decisions for herself at this point).
The different subjects have different kinds of structure, for example:
Reading is required for an hour a day. They get to choose the book, but I have to approve it (the requirement being mostly that I want it to be something at their reading level, somewhat worthwhile for school). So this has a double structure of time and parent approval.
Math varies according to the student and the particular material they're studying. It's generally a certain quantity of math (like a lesson or two pages), but can sometimes be based on the amount of time.
Music - They're supposed to work on a musical instrument for about 20 minutes each day. They get to pick the song and the instrument (piano and recorder are the basics). Though most of them are reluctant to either take lessons or work on it on their own (with another huge exception with Ria who is a completely self motivated piano student), having a little nudge is really motivating for them. After two weeks they're starting to enjoy playing again and are getting back into working on learning how to read music.
A general thing I've learned about myself is that if my expectations are too high, things fall apart quickly because I get overwhelmed with the size of the list. I've found that a balance of requirements and freedom works best for our family and keeps us the most productive. The balance point is affected by things like personality and learning style.