During the month of October, my 17 year old son has been accepted to college, awarded a significant scholarship, and been invited to interview for a full ride. He also improved his rating in fencing to C, which puts him into Division I. His 15 year old sister won a national level horse judging competition, and may be the youngest person ever to win it.
I write this not to brag, well not completely anyway ;) The reality is my kids are not that special. Neither is endowed (as far as we know) with genius level intellect, or a prodigy type talent at music, art, athletics, or something else. Their parents certainly aren't geniuses; and we are not rich and have not provided them the finest tutors money can buy. In fact, the stuff my kids are passionate about is stuff that neither me nor my wife had any experience in before our kids got involved. Even academically, my son's deep, deep interest in history comes from somewhere inside of him. I hated history in school. I see now, that was primarily because of the way it was taught. In many ways, my kids look and act like most other kids their age. In public, you'd be hard pressed to call them out as homeschoolers.
The one thing we have given our kids that has made the most difference is time. Since they don't go to school, they have time to dive deep into the stuff that interests them. It's not studying, and it's not "homework," when you are enjoying the process of learning. When you are enjoying the process, you'll spend more time at it. The high school kid failing math because he won't do his homework has no problem spending hours a day mastering the intricacies of a video game. Instead of making him repeat math, maybe we should asking if we are doing a good job teaching math, or questioning if he needs that math class at all. Trigonometry and calculus just are not that important for most people. If you plan to pursue engineering, it's important, but not not so much for the future history major.
School curriculum is just another expression of currents fads and trends. School administrators think they've identified the core knowledge every kid needs to know. School administrators 50, 100, and 150 years ago all thought the same thing. If you look back, you'll find a few common threads. The ability to read well, express yourself in both written and verbal forms, and math up through about Algebra I and Geometry have been important throughout history. After that, everything is an elective, subject to the whims of the day. Interestingly, if you have that core base down, you can teach yourself just about anything else you want to know. I don't think that is a coincidence.
Lifelong learning is important, but there is no one answer to the question of what should be learned. For my kids, it's history for my son, and equine science for my daughter. How much of those subjects could either of them get in school? I think many, many kids could do cool stuff, if only they had the time. It's imperative that we as a society start finding ways to eliminate the busy work and drudgery of school, and unleash the high achieving "geniuses" residing in our kids.