Everything Tigerle said - excellently put!
You are really looking for two things: flexibility and open-mindedness.
As Tigerle notes, most private schools seem to think their program is so awesome, they actually tend to be more rigid than public. So hunt for the exceptions, whether public or private. You're looking not for "the best", but for the one which is the best match for your unique kid, at this particular time. Also, be aware that many schools that focus heavily on high achievement also tend to place top value on compliance and conformity - which is an uncomfortable fit for some (but by no means all) gifted kids.
