I've asked Dana, a member with Duke/TIP experience, to come in and share her experiences. Both of us have a background in education (I'm an ex-school psychologist, and Dana is a math professor), but we approach homeschooling from different perspectives. I'm on one end of the spectrum, where I promote letting children take control of their education as much as the children and parents feel comfortable with. It's worked well in my family. My children have been able to really focus on their interests in high school and afterwards (college-wise, career-wise, etc.) and are successful. Dana is more on the other end of the spectrum (parent-led track), and it's been working well for her. (We've had great discussions about our different educational/homeschooling philosophies.)
Anyway, she's great about helping parents with math-related and Duke/TIP/AP questions, so I'll wait for her to chime in.