Our expert says:
Maybe, like some really intelligent children, his problem at school isn't so much lack of ability to concentrate, as boredom, if he is ahead of the class, and feels he's having to mark time ?
Fro your careful observations when seeing him and other children in his class doing homework, maybe they just have different techniques in dealing with homework - he's more focussed, maybe wanting to spend the exptra time thus created on something more enjoyable.
I tend to think that the vaue of homework at primary school is modest, and mainly related to acquiring the habit of homework, of working at something that may be unstimulating and unchallenging, even tedious, but which the system requires you to do - more an issue of acquiring self-discipline than of learning remarkable new stuff. Not worth all the boredom it can induce, but maybe of some small use. And maybe it's different for less able kids, as they need to learn and practice basic skills he gained more quickly.
A special school for gifted children may sound tempting, but I'm not actually aware of much evidence from proper evaluations ( or indeed convincing anecdote ) to convince me they're worth the extra costs.
And however gifted the child, and however much they may appreciate the various forms of extra stimulation, sometimes they may also need the chance to just dumb down a bit and watch cartoons or kick a can about.
Its interesting that home schooling in a way suggests that trained teachers have nothing more to offer than a keen mom, which kinda undercuts the whole idea of their training. Some teachers, definitely add no value to the process, but presumably some do.
I suspect that children are so resilient ( fortunately !) and especially bright kids, that trying to micromanage the entire process may not be as useful as one might want to think
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