@yearstogo
Except group theory is intended for students who have completed the high school sequence and don’t have access to post-secondary courses.
From AOPS,“This class is aimed primarily at students who have mastered the standard high school curriculum and do not have access to a strong post-secondary curriculum.”
http://www.artofproblemsolving.com/school/course/grouptheory
^^Perhaps he meant Number Theory, not Group Theory.
Mom2aphysicsgeek, He claims to have taken all the introductory and intermediate courses and calculus in the Art of Problem Solving web site but I can see his grades only for his latest course which is AIME where he has not done particularly well with average 7-8 challenge problems done out of 10 so clearly he will have to repeat the course. I can see that he has enrolled himself in Olympiad Geometry and Group Theory for summer.
Fwiw, you can’t gauge performance of 7 out of 10 as the equivalent of a 70. Their classes are incredibly difficult and students are not expected to make 100s. I am not overly familiar with the competition courses, bc my son was far more into physics and theory than competition math. But, if he has taken the core courses, making top grades in an typical class should be easy (at least that is our son’s experience.) He should also not need to repeat the courses.
I would be curious over how many yrs he completed all of their courses. Completing all of them would take a few yrs, even for top students taking them at the rapid AoPS pace. (The geo, intermediate alg, pre-cal, and cal classes, for example, are all 22-25 week courses. Their counting and probability courses and number theory are shorter and many kids double up with those on top of the core courses.) Some of their challenge problems can take a couple of hrs to solve. They are not courses that can be completed without commitment if the student is actively completing the assignments.
For deciding how to place him next yr, you could contact AoPS and ask if they can provide him a transcript (I believe they are now accredited, so the transcript would have that weight. They weren’t accredited when our ds was taking their classes.) You could also have him take the post-tests for pre-cal and cal. If he passes their criteria, he would most likely be very bored repeating the coursework.
@LateCut - It has been a while since I looked at the AOPS website, so please double check what I’m saying for yourself. But a score of 7-8 challenge problems out of 10 at the AIME level for an 8th grader may be a very very fine performance, maybe even a super performance. These problems aren’t drill exercises so repeating until he gets them all right isn’t the right mindset.
To give you a calibration, getting 11 out of 15 problems right on the actual AIME would put him around the top 400 of all math students nationally in all grades <= 12. It all depends on how hard the problems they assigned him were. If you post a link to the problems I can tell you for sure.
Olympiad Geometry looks like a great course for him. The Group Theory one might be a little advanced for him, but you said he’s getting help from a professor on group theory so they could give you an opinion. But if your son enjoys the challenge I’m sure it’s ok as long as he doesn’t get discouraged. Even if he doesn’t comprehend everything just getting the exposure is great - keeping him challenged and interested is the goal.
The only thing you’ve said that gives me some pause is your son’s pre-Calculus knowledge. To be honest, it seems out of whack for a kid who’s going to learn the basics of Galois theory over the summer. I looked at the test you gave your son at https://math.berkeley.edu/sites/default/files/pages/F03_Final_Exam-B.Johnson.pdf.
Most of these problems seem quite basic; nothing that a diligent community college student wouldn’t get a 100% on except maybe for the bonus problem. If your son is as advanced as he seems you should definitely hold him to a much higher standard in terms of difficulty and challenge. Just because the test comes from Berkeley doesn’t mean anything.
Mom2PhysicsGeek, Al2Simon,
On this point (AIME) I respectfully disagree. My rule of thumb is that if someone scores routine 100 in an exam at home, they have a good shot at coming close to100 in the real life pressure situation of taking the exam live but if they are scoring 70 percent then they will likely scored 40 percent in the real live exam. That is 6 problems in a live AIME exam and when I had him take the 2013 AIME II that’s about what he got or 7. I am a strong believer in the “practice over and over again” school of thought as you may have understood by now so I think he should keep taking the course over and over 3x a year till he is blindly solving all the problems without any hitch. But that is only if he wants to do well in AIME which will be his call.
He told me that he did the courses over the past 3 years typically taking 2-3 courses at a time. I also had him take the post-course precalc Art of Problem Solving exam and he initially scored very low but then scored a 9 after I printed out a list of all relevant trig identities and gave it to him. So he can use the identitites all right and as I said before derive all of them which I checked but he doesn’t remember any of the identities. This will not work in an exam so he has to memorize them and again the key there in my opinion is to just practice, practice, practice.
http://data.artofproblemsolving.com/course-docs/diagnostics/precalc-posttest.pdf
He got 1/2 point in 3 and 5 but full points in all the other problems. In problem #3 he could not prove the generalization to all n. In problem #5 he couldn’t get all the roots of z^4+1=0.
I do not think he will understand much of Galois Theory but if it is fun for him to take the course then I am all for it. It’s summer and the kid should have some fun. 
as I stated, I know nothing about competition math. In their core course classes, 7 correct would not be the equivalent of a 70.
I asked my son if there was a way to access the course info online after the fact and he said he can still access all of his class forums and see see all the weekly problems and the solutions posted. (And these are from several yrs ago.) To access the history, you go to account and then click on my forums.
You should be able to see every weekly problem set assignment and solution for every class taken. That will give you an idea of what he should have mastered.
(I just tried it for my daughter’s acct bc the system changed after my ds finished taking classes. I could see what he was describing. Easy to access. Back when my son was taking taking the classes they had to type up solutions in LaTex and email them in. So he has all of his challenge problems and the grading. I don’t think you can access that in their new system. I couldn’t see my daughter’s. But you can still see the weekly problems and solutions. That should give you a clearer idea of what he has done.)
There are a whole body of distinguished math people who think math contests are extremely counter productive especially for girls. Cathy Oneil the Mathbabe has a blog called math contests kind of suck Terrence Tao and others posted on this blog. I personally think that they are terrible for girls and it is a mixed bag for boys http://mathbabe.org/2011/07/17/math-contests-kind-of-suck/
I think there are plenty of math folks who would disagree. 
I didn’t find the experience counter productive for our son; he loved his tiny math circle and math counts as club type activities. His experiences with his math coach were hugely beneficial experiences challenging his understanding. He just doesn’t like competitions in general and dropped out them once he became more involved in physics. We didn’t encourage him one way or the other. It was purely his decision.
We have close friends whose dd was active in competitions, including Math Prize for Girls. It was a HUGE blessing in her life bc it was about the only time she got be around peers with similiar interests. (As well as at camps like Math Camp.) I would hate for people to discount the opportunity without their child making an informed decision.
Ellison from MIT wrote an interesting piece on gender gaps in math education. It was titled
The Gender Gap in Secondary School
Mathematics at High Achievement
Levels: Evidence from the American
Mathematics Competitions
■ Glenn Ellison is Gregory K. Palm Professor of Economics and Ashley Swanson is a
Ph.D. student in Economics, both at the Massachusetts Institute of Technology, Cambridge,
Massachusetts. Ellison is also Research Associate, National Bureau of Economic Research,
Cambridge, Massachusetts
He concluded
“several elements in our results are consistent with the view that girls suffer in becoming
high achievers in mathematics because they are more compliant with authority
figures and/or are more sensitive to social environment. In most high schools, even
in the highest-level “honors” courses, it is probably unusual to teach material at the
level needed to bring students to the 99th percentile. If talented girls are less likely
to complain and get schools to make special accommodations, and if social factors
make them less likely to join math teams or take advanced online courses, then they
will be more underrepresented when we examine achievement levels that are further
beyond those developed in the standard classroom setting”
We have a situation where girls are left behind in Math Education. I think Math contests are partially to blame.
There was also an interesting article in todays LA Times about women bailing out of the Tech Industry in droves. Teaching competitiveness rather than collaboration has some significant downsides especially in Math education
I don’t think you should be considering the AIME test in your equation. Competition math is not at all similar to what’s in a typical math class. My son did better than anyone else in his high school at the ACM and AIME, but I think his highest score was a 3 on the AIME when he was taking Linear Algebra and doing very well in it i.e. A+ work.