At that summer program, I worked with several kids who completed Algebra I (which used to be a ninth grade course, but is now frequently an eighth grade course) in three weeks during the summer between fifth and sixth grades, and their parents were worried about how they were doing in school, because they had trouble doing fifth grade math the way their teachers taught it.

One of the big things there is understanding how variables work in algebraic equations. Some kids just get it. Other kids need to be taught it. And a lot of the stuff used in elementary school and pre-algebra to lead up to the concept of variables just looks really confusing to kids who already understand it -- it's trying to explain things without explicitly using variables, but for the kids who understand variables, these methods seem totally counterintuitive, because their intuition is something that the book assumes they don't know.

We had 8th grade algebra in my adequate public school 20+ years ago -- it was how you got to calculus in high school.

they had trouble doing fifth grade math the way their teachers taught it.

In college one of the requirements was this layman's math class for dummies, and I Could. Not. Do. It.

All the steps, givens, proofs, gone. I could do the math, given a piece of paper and a pencil, but I immediately became the dumbest one in the group because I couldn't do it in my head. To assume that one hadn't screwed up a step in their head drove me bonkers. But, but, but, you could be *wrong*!

And of course it was only those of us in the "smart class" who took algebra in 8th grade, but how else do kids get to calculus?

I take crappy inside pictures: [link]

At my school, in the late eighties when my sister was in Jr. high, Algebra I was a ninth grade class for the "regular" kids and an eighth grade class for the "accelerated" kids -- to take algebra in eighth grade, you had to either have a recommendation from your seventh-grade math teacher or have your parent sign a special waiver. About 40 of the 100 kids in her grade took it in eighth grade. By the time I was there, four years later, eighth grade algebra was standard, with just the "slow" kids taking it in ninth grade, and I had to take a special test to get to take it in seventh grade. Five of the 150 kids in my grade took it in seventh grade, and about 130 of the rest took it in eighth grade.

I can't believe I didn't take pictures of the DR and its interesting window config.

And of course it was only those of us in the "smart class" who took algebra in 8th grade, but how else do kids get to calculus?

Calculus in high school is a relatively recent thing -- I don't have the statistics right here, but it wasn't until the mid eighties or so that taking calculus in high school became something for the regular-smart kids, rather than just for the couple of really advanced kids.

Ooh, that looks nice, Sara!!

(I am having cut-and-paste issues, but:) Ah, I was taking calculus in high school where it was somewhat common (I was in AP, but I think there was regular calc as well), but it was early 90s, so that makes sense with the time frame you're describing.

And of course it was only those of us in the "smart class" who took algebra in 8th grade, but how else do kids get to calculus?

Yeah, in my school you had to double up on math one year to get to AP Calc. Maybe about 10% of our class did that. Regrettably I took an intensive PE class instead...OTOH, between weight-lifting and calculus, only one of those comes up on a biweekly basis.