Did it take a year because it really took that long to understand the material, or because the class took a year to present it to you?
Of course, age is also a factor, an adult can concentrate on a subject for longer than a child can. This might be better illuminated by change in the rate of self directed learning.
It really took me roughly that long, although it was more conceptually deep than most algebra courses. I learned most of my math at my own pace, with help from my dad. My non-confident guess is that most mathematically talented people encounter algebra and other subjects long after they’re ready for them, and therefore learn them fairly rapidly but at the cost of having wasted time earlier on. But I may just have been slow.
In any case, even restricting to bright college students other than me, I’ve watched multiple individuals get much faster at learning math over the course of undergrad.
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.) and other topics of interest. (Much of algebra 1 was harder than, say, Bayes’ theorem, which is not the case for high school students. It’s as though some skills were online while others, especially formal/schematic others, 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.
My non-confident guess is that most mathematically talented people encounter algebra and other subjects long after they’re ready for them, and therefore learn them fairly rapidly but at the cost of having wasted time earlier on.
That might explain my experience in tutoring my cousin in math. I find he is able to catch up quickly once I explain the background material a given concept is based on. So, if he had been ready for some time to learn the background material, then learning it when I present it is not a big deal and doesn’t even noticeably detract from the effort and focus he needs to understand the new concept he is supposed to be learning.
Did it take a year because it really took that long to understand the material, or because the class took a year to present it to you?
Of course, age is also a factor, an adult can concentrate on a subject for longer than a child can. This might be better illuminated by change in the rate of self directed learning.
It really took me roughly that long, although it was more conceptually deep than most algebra courses. I learned most of my math at my own pace, with help from my dad. My non-confident guess is that most mathematically talented people encounter algebra and other subjects long after they’re ready for them, and therefore learn them fairly rapidly but at the cost of having wasted time earlier on. But I may just have been slow.
In any case, even restricting to bright college students other than me, I’ve watched multiple individuals get much faster at learning math over the course of undergrad.
Just how old were you when you studied it?
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.) and other topics of interest. (Much of algebra 1 was harder than, say, Bayes’ theorem, which is not the case for high school students. It’s as though some skills were online while others, especially formal/schematic others, 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.
That might explain my experience in tutoring my cousin in math. I find he is able to catch up quickly once I explain the background material a given concept is based on. So, if he had been ready for some time to learn the background material, then learning it when I present it is not a big deal and doesn’t even noticeably detract from the effort and focus he needs to understand the new concept he is supposed to be learning.