(As background context, I’m a mathematician, and I’ve taught gifted children math over a span of ~10 years.)Something that I’ve noticed lately is a widespread implicit acceptance of the norm for gifted children to learn math at grade level, or just 1 year above grade level. My experience has been that even moderately gifted children (IQ ~130) can learn algebra in at age ~11, and that highly gifted children (IQ ~145) can learn algebra at age ~8. Moreover, I think that there are strong arguments in favor of this.Developmental capacity

• It’s not uncommon for moderately gifted children to be 2+ years ahead in reading and for highly gifted children to be 5+ years ahead in reading, so one might expect them to have mathematical potential that’s 2+ or 5+ years ahead of grade level (respectively).
• IQ was for a time believed to be “mental age divided by chronological age” multiplied by 100. This notion has (rightly) fallen out of favor, but it’s sufficiently close to the truth for people to have believed it. Under this assumption, a 10 year old with IQ 130 has mental age 13 and a 10 year old with IQ 145 has mental age 14.5, and these 10 year olds are correspondingly cognitively ready for curricula aimed at people of their mental age.
• I know of people of IQ ~160 who have learned calculus at age 7: this suggests that in some respects IQ understates “mental age.”

Desirability

Some people have suggested that it’s better for gifted children to learn a broad range of things rather than accelerating, because if they accelerate then they’ll be out of sync with their peers. I think that this is true in some contexts. But I don’t think that the benefits of being better in sync with one’s peers outweigh the benefits of accelerating through the K-12 math curriculum specifically.

• Grade school math is key to understanding the natural sciences, statistics and economics. Remaining at grade level in math substantially delays a gifted child’s ability to understand these things.
• Learning math well builds general reasoning ability, which has benefits across domains.
• Many gifted children find math especially enjoyable once they become deeply involved in it.
• Being far ahead in math can build confidence on account of being an unambiguous signal of intellectual ability.

In The Calculus Trap Richard Rusczyk at Art of Problem Solving argued that rushing through the standard curriculum is not the best answer for mathematically talented young people, suggesting that students should instead focus on learning how to solve complex problems. I agree with him that learning how to solve complex problems is more important than acceleration through the standard curriculum. But the two things aren’t mutually exclusive: gifted children can both learn how to solve complex problems and accelerate through the standard curriculum.

Learning precalculus and calculus was a transformative experience: it allowed me to understand physics, it gave me a thrill, and it made me better understand myself on account of tapping into my latent mathematical ability. It was when my intellectual development really accelerated. I was 16 at the time. I wish somebody had encouraged me to start earlier. A sizable minority of the most intellectually impressive people who I know I know had similar experiences.

There are large potential returns to gifted children learning more math earlier on.