@ Kaibab3
Sounds to me like you are exploring wisely. We met with relevant faculty at every boarding school we visited (gasp, 14 of them) and laid out what D had done by 8th grade, and would want to do in HS, and asked them if she should just go to college. Only a couple of the schools stepped up to the plate in a meaningful way. We did not consider “independent study” with one teacher as promising, but some might. It seemed too risky to us.
Both Andover and Exeter seemed fully capable of further educating a 9th grade student coming in at + AP BC math level. I suggest you look at the course catalogues carefully. For example, at Exeter when students come in with AP calc, they go through “transition” problem solving assessment the first term ('X" series) and then are advised on a level, basically a starting point.
We were cautioned at both institutions that level is chosen by one’s willingness to spend time as well as innate ability. The 40 kids on the competitive math team, for example, put in a lot of time (40+ hrs/week) but those kids love it, and they are a great group. They often arrive already gold medalists in major math competitions.
Common classes that are FULLY ENROLLED at Exeter after BC calc include multivariable calculus, linear algebra, discrete math topics, statistics, special topics (non-euclidean geometry this term I think), and over two years of graduate student level theoretical proof based math. Several kids go down to MIT on the train on Saturdays for work with math professors there. One just won the Davidson award after a summer of math research at RSI/MIT.
Science has several years of post-AP coursework at both institutions, notably in molecular genetics and physics.
I think it is easier with humanities and languages to come in at a high level and find interesting ways to branch out or delve deeply at many good schools.
One note of caution: acceptance into these schools may improve somewhat with high test scores and challenging coursework, but as the discussion boards show year after year, acceptance is by no means assured even for the very bright. And it is a rude awakening to find oneself in a math class with the winners of the IMO!
As I said in my prior post, though, each family and very bright child have different needs and resources to work with, and while challenging because they do not fit the usual mold, parental involvement with these kids really makes a difference, whether they are in dual enrollment, boarding school, early college or homeschool.