The principal agreed that if DS shows mastery of an entire year's curriculum, he must be given new concepts to learn, and that it's not sufficient to pad his time with problem solving practice at a lower level.
I'm a bit late here, but I meant to comment on this. I think there are three things that might be in play here:
1) new concepts: good
2) problem solving practice at a lower level, i.e., not introducing new concepts and not using problems that are genuinely hard for him: totally pointless
3) problem solving practice that doesn't introduce or require new concepts, as such (i.e. he isn't going to have to say "what does that word mean" or be unable to make progress because he doesn't know how to do X) but which is hard for him.
(3) doesn't seem to get a look in in what you write, and I hope you/the school aren't conflating it with (2). There are plenty of maths problems whose statements he could understand now but which are unsolved problems, to give an extreme example. Our experience is that although the ideal would be to introduce new concepts by means of hard problems that motivate them, this is really hard pedagogically: it really needs one-to-one support. We've sometimes managed to have both (1) and (3) in school by having parallel strands of work going on, but I think if you have to choose between getting (1) in school and getting (3), (3) is actually far more beneficial. Either can be provided in afterschooling, and if you have time for an hour or more of maths afterschooling every day, it might not matter which it is, but for us, significant amounts of time afterschooling only arise outside the school term. A few weeks spent without learning new maths concepts, but working on hard problems, is fine (you just get a bit of an itch to learn something new, but you're still getting better at maths in the meantime); a few weeks learning new concepts but never needing to turn your brain on when it comes to applying them is much more stultifying, IMO.
I asked how he'd be exposed to new concepts, then, and the idea was voiced that he might be exposed to new concepts in the course of enrichment projects. This naturally prompted a question from me about how he would receive instruction on the new concepts, instead of being left to his own devices and essentially teaching himself. There wasn't really any answer.
Yeah. The danger is that you get new concepts that way, but only ones that are so easy they didn't need any explanation!
It seems to me that if he encounters new concepts and needs a quick lesson, we parents can give it to him at home. However, we would need enough information to be able to do that as necessary; the new concepts wouldn't necessarily be introduced in a good progression or in such a way as to build a strong conceptual foundation, as they would be in a curriculum; and as a result this would essentially be winding up with what we've requested before, partial afterschooling, except with a poorer instructional model.
Yes, I do think that's a danger. You can end up with weird gaps (although IME weird gaps are easily filled when you do discover them, so it's an oddity more than a real problem, provided systematically introduced new concepts are coming in somewhere). I forget what it was, but there was some problem type DS had met a few times for which there's a standard solution method he didn't know. Being resourceful, he was managing to get the right answers by some kind of trial and error method, and it was quite a while before anyone noticed that he had a gap.
What we currently have, fwiw, is (3), with a good teacher who has time to explain concepts that come up, but without as far as I can see any systematic plan about what concepts to introduce. I'm thinking to have DS work through a couple of ALEKS courses at home during the holidays this year (as he did last year) to make sure he has a tangible forward step in what he knows; acknowledging all the failings of ALEKS they do work quite well in this role, with the non-routine problem solving and the writing out of solutions being handled elsewhere.
The other thing I do, and have had as a habit for a few years now, is to have text files of the detailed syllabuses for things he might be following, if he were following a syllabus (iyswim: for us this means the maths syllabuses for A level and Pre-U, but ymmv). I annotate it by adding notes next to each phrase in the syllabus e.g. [DONE April 2012] or [SHAKY May 2012] or [DONE except needs more practice in...] as I notice that he can/can't do things listed in the syllabus. I find this quite useful as it lets me have an overview of syllabus areas in which he's got most of what would be expected at that stage (bar a few gaps which I might then fill in) or where he's got hardly anything (in which case I might look out for an afterschooling resource in the area), or where he hasn't learned anything new for a while, etc.
