I love the phrase "free range curiosity"!
In response to Katharine's question, I'll share my daughter's story. She had no direct instruction, ever, in math or any other typical academic subjects when, at age 10, she decided to try a traditional public school. We spent three weeks going over the math, grammar, and such that she would be expected to know. No, not three years, three weeks.
The result? She had no problem whatsoever successfully doing grade level work. In other words, she mastered 4 - 5 years of curriculum with only three weeks of direct instruction in her ten years of life. I can't help but feel sorry for all her classmates who spent five of those same years sitting in a desk following someone else's agenda, while she played to her heart's content and simply pursued her passions all that precious childhood time.
She attended school for 5th and part of 6th grade. Then she took a break until freshman year of high school and did not receive any further "formal" education beside a short hands-on Algebra class that met once a week for about 6 weeks.
She decided to try high school part-time to see if she would like it and did very well academically. She ended up getting the highest grade in all of her teacher's Algebra classes. He called at the end of the year to let me know he would have nominated her for student of the year if she had been a full-time student.
She did not continue in high school because she felt it was pretty much a waste of time. At age 14, my daughter decided to try community college, a class or two at a time. She has been successful in this setting as well, even to the point where one instructor told my daughter that her final project was one of the best she had ever received. I had to laugh, considering no one ever taught her how to write papers, take notes, do research, or any of the other supposedly necessary instruction she should have received to be successful in college.
Is my daughter some type of genius? A child prodigy? I don't think so. I think she's a normal kid who happens to have been given complete freedom, trust and respect her whole life. As a result, she has naturally learned and developed the skills necessary to succeed without needing years of classroom instruction. She is a perfect example of how faulty the assumptions of the traditional educational system are.
I hope some day more than a handful of people in our society will be able to think outside the box and we can have some real, significant changes in our educational system.