It's important that a gifted kid can move freely through higher grade material. But when that material is aimed at typical (or struggling) students, the kid will need something else. Two things are (1) tough challenging problems, (2) material not normally covered in the standard school sequence.
So the question is, how should (1) and (2) be done. There are some things you
don't want. You don't want, extra busyworksheets, or so-called "gifted pull-outs" or "enhanced/supplemental coursework" if they are not being done
really well, especially when they are just some extra fluff on the side for the highly able student who is nevertheless locked into the standard slow-paced gen-ed track. When needed acceleration is denied, you can't make up for it (much) with extra stuff on the side.
One should be very skeptical of a school saying they don't need to accelerate because we've got this "super duper stuff on the side" instead.
That said, as pointed out in "The Calculus Trap"
www.artofproblemsolving.com/Resources/articles.php?page=calculustrapthere is a lot of mathematics outside of the standard K-12 sequence. Then it
does make sense to learn about these other topics. And I think it even makes sense to then do the standard K-12 sequence at a less accelerated pace while learning about these other topics, than you would if you were just doing the standard sequence.
However, these extra topics mainly start being accessible after one reaches about the pre-algebra or algebra I level (in US courses). For elementary school level standard courses and up to pre-algebra, I really think the sufficiently able student should just get through these courses as quickly as they are truly able, even though there may be other worthwile stuff to do during this time (hard problems and recreational math). This can put them several years ahead of their age cohort, but that's exactly what should happen.
