Um, well, I was simplifying. Algebra 1 I learned between 2nd and 5th grade, mostly incidentally but not especially quickly, in the course of asking my dad about probability, basic number theory (rational and irrational numbers; modular arithmetic and divisibility facts; etc.), the limit of 1⁄2 + 1⁄4 + 1⁄8 + …, and other topics of interest. (Algebra 1 was harder for me than, say, Bayes’ theorem; which I don’t think is the case for most high schoolers. It’s as though some cognitive skills were online and others weren’t. Especially, the formal/schematic ones weren’t.)
Algebra 2 I learned in sixth grade, in a normal course (for 8th graders in the gifted program). It wasn’t too slow, though, or not by much. I came in with less than a full Algebra 1 worth of background, struggled a bit the first semester, did fine the second. Geometry I learned in 7th grade, working from a book (they let me do my own thing in math class) with help from my dad, and doing more exploration and proofs than the book included. I spent maybe 2/3rds of the year on it, then did some trig and basic discrete math.
Which suggests a fairly normal rate of learning, though with deeper exploration and at a younger age. I would non-confidently guess that many of those who go on to study math would have been similar as kids, if given the opportunity. Kids have less ability to hold formal scaffolds in their heads, and, as Douglas Knight notes, it’s hard as adults to see how large the cognitive distance is.
Um, well, I was simplifying. Algebra 1 I learned between 2nd and 5th grade, mostly incidentally but not especially quickly, in the course of asking my dad about probability, basic number theory (rational and irrational numbers; modular arithmetic and divisibility facts; etc.), the limit of 1⁄2 + 1⁄4 + 1⁄8 + …, and other topics of interest. (Algebra 1 was harder for me than, say, Bayes’ theorem; which I don’t think is the case for most high schoolers. It’s as though some cognitive skills were online and others weren’t. Especially, the formal/schematic ones weren’t.)
Algebra 2 I learned in sixth grade, in a normal course (for 8th graders in the gifted program). It wasn’t too slow, though, or not by much. I came in with less than a full Algebra 1 worth of background, struggled a bit the first semester, did fine the second. Geometry I learned in 7th grade, working from a book (they let me do my own thing in math class) with help from my dad, and doing more exploration and proofs than the book included. I spent maybe 2/3rds of the year on it, then did some trig and basic discrete math.
Which suggests a fairly normal rate of learning, though with deeper exploration and at a younger age. I would non-confidently guess that many of those who go on to study math would have been similar as kids, if given the opportunity. Kids have less ability to hold formal scaffolds in their heads, and, as Douglas Knight notes, it’s hard as adults to see how large the cognitive distance is.