(I'm sorry I didn't reply to this earlier.)
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.
I'm not conflating them. I think there's a ton of value in letting him work on tough problems, and I have at home when we've done math. I'm even somewhat on board with the idea that he can progress in problem solving ability at this stage without a tutor, just by cutting his teeth on problems that he finds hard. I just don't think that it's the whole picture of math development, and I want him to be taught something at school. I don't think that the sorts of math teachers at this school, from what I've seen, are going to increase his ability to think mathematically by coaching him on problems; I'd settle for their not artificially limiting his progress through new concepts in efforts to keep him closer to his current classmates.
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.
I've thought about just doing AoPS and teaching him at home, but that would involve giving up on getting something out of the school day. I agree with you.
for us, significant amounts of time afterschooling only arise outside the school term.
Same here.
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.
Good point. I've been keeping him engaged lately more with computer programming while we decide what to do.
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 agree totally with you in general. What I've found with the local school system is that some of the school admins and teachers are so anti-acceleration that they will seize upon any seeming gaps at all in an attempt to deny the need for advancement. At the most recent meeting, for example, DS's third grade teacher (who is not supposed to be taching him math) was present, and said that she had decided to give him two "math minute" speed drills on basic addition and subtraction facts, and that while he had gotten 50/50 on the addition, he had only gotten 40/50 correct on subtraction, so would need some practice that she'd be giving him. (Not. And it turned out that he had simply gotten bored or tired and stopped writing, but had perfect accuracy. I told him to refuse all such drills in the future, at least in that classroom. It's disheartening for him to plod through such stuff, especially when he's bored plodding through long division and other sorts of arithmetic practice too, but at least it's not so insultingly below his level of skills development.)
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.
I think what I might do over the holidays, if this isn't resolved by then, is have him finish fifth grade in Singapore Math, then just start on Age of Problem Solving and ask again to send work with him to school. Partial homeschooling might also be in our near future. A lot depends on what happens in the next 1-2 months with our educational consultant. It's just a rough fit even with the school trying to do their best.
I like your method for tracking your son's progress. Thank you for all your insights; as usual they are very valuable to me.
