[anatoly]

I'm not sure I see how that'd work with the really smart children. They can often get their homework done with the fraction of effort required from other kids. Praising the effort will then look ridiculous and seem like a lie. "Oh, I see that you've finished your math homework in 10 minutes. You must have worked really hard on that!"

[hugs]

So let them finish their homework in 10 minutes, and then find something that will take the child more than 10 minutes to do. And praise them for working on that. If praising the 10-minute homework effort is ridiculous, then don't do it. Raise the bar. There's got to be something in the whole field of mathematics that would take the kid more than 10 minutes. (General disclaimers apply, I'm sure there are limits to how high you can raise the bar before the kid goes homicidal on you.)

[dkarl]

This is absolutely correct. There always has to be more and harder stuff for kids to accomplish. If there's a ceiling, then smart kids will quickly hit the ceiling and then start measuring themselves by how much effort is required to stay there. That's the way it was at my school: all the honors/AP kids learned the same material, and in every class there were at least a couple of kids who got high As on most things. The only way to distinguish yourself was to do it without working hard, because you couldn't distinguish yourself by accomplishing more and harder things.

[sethg]

...or, praise them for working hard in a subject where they aren't talented enough to whip through the homework in ten minutes. If the kid is razor-sharp in math but merely above-average in reading, then their efforts in reading are the ones that particularly need encouragement.

[anatoly]

(despite the elaborate caveat below, thank you for the detailed answer to my question; it seems like a very promising thing to try)

But if you treat the 10-minute homework as something not worthy of praise or mention, the child might feel cheated. Other kids are being praised for much less impressive efforts. I know I'm doing better than most in my class; why is this overlooked? Am I being penalized for my smarts? Perhaps I'm better off making the homework seem like a huge deal, and finishing it in two hours of seemingly herculean effort?

The parent may think it right to raise the bar as far as the child's aptitude allows, but the child won't necessarily feel this policy to be fair. And most children are naturally apt at gaming parental policies.

[MtL]

Then give the children a chance to develop themselves further; more advanced tasks, other variations of the same task, ask them to explain to you how they think when they solve the task, give them something completely different to spend 20 minutes learning/doing, etc.

[theblackbox]

I highly recommend juggling and balancing (a broom stick on the chin is a personal favourite) - it can be taught easily, requires quite different mental gymnastics and helps generics such as focus and determination. Plus circus is all about collaboration and entertainment, things that are rarely taught outside of a competitive framework in school. Good preparation for the development of an "interesting" personality, but I would say that!