College-level calculus burned me so much, I didn't had the algebra background necessary to grok it, much of it is memorizing heuristics and "tricks", and college professors (specially in my public uni) didn't had much patience either. I even did well in other math-related classes, but calculus was a chimera. Unfortunately, I fell so much behind the curriculum that I had to drop out of college to start working full-time.
Lately I've been picking up other math-related online classes, and I finally realized that I had never had real exposure to mathematics, and that high-school curriculum is something else. So, there's that too... we are mostly scammed into thinking we've learned something during high-school, and then life happens and you actually figure out you don't know shit. Hopefully I'm a curious person, so I'm always learning something by myself, mostly to broaden my knowledge in other topics, but to not let the gears grind too.
It is very rare for an incoming calculus student to know algebra. In fact, I often tell my students that the first goal of college-level calculus is to make them actually learn algebra. It's an extremely sad state of affairs for me (a mathematician).
I can vouch for your article. I believe you captured the problem with school-level math.
As an anecdote for you, since you're a mathematician: during school I had zero patience for math. I would often study just enough to get a passing grade. On the other hand, I enjoyed physics classes a lot, it involved more problem solving and holistic vision than memorization (since you can just memorize a couple formulas and derive everything else you need - e.g. the second and third equations of motion from the first). Maybe I would have enjoyed math classes more if it involved more intuition and less rote? I'm sure you'll come across kids like I've been during your voluntary lectures.
My high school curriculum was deliberately set up so that Calculus is where we really learn algebra. 9th and 10th grade were designed to teach mathematical thinking. In 11th grade you have the choice of taking 11th grade math or Calc 1 (with permission of your 10th grade math teacher), both of which would instill algebra (technically, you are aloud to take any math elective, but I think everyone takes at least one of these two). The concept is that, because algebra is being used as a tool to solve a more complicated problem, there is no choice but to actually learn it.
Having said that, we did go into it being passingly familiar with algebra, which may be more than you observe in your students. Also, I don't think it is possible to go through the 9th and 10th grade courses without developing some feel for meaningful symbol manipulation.
EDIT: I just read your article, I think our math program is like you describe it should be.
Thanks for posting the article. I wish I had known more about this when I was in high school. I have recently found myself becoming much more interested in mathematics when thinking about very simple concepts and asking why it is a certain way vs. memorizing the rule. I think my 10+ year aversion to the subject is rooted in the fact that it felt like nothing more than rote memorization in high school, and thus a dreary subject to learn anything about.
I just want to say, as someone who is now re-educating themselves in Math, I really love your Math ∩ Programming site. Thank you for this lovely resource.
-- says that in 2009, just 16% of U.S. high school students took calculus. And a common complaint by college professors is that high school calculus courses teach rote procedures without explaining calculus in any depth -- the students know how to solve calculus problems, but they don't actually understand the subject.
That's my perspective on it. I first took calculus as a junior in high school, and the majority of it boiled down to memorizing what the derivative of `x` type of function is and what the integral of `y` type of function is.
When the AP test rolled around I was really struck. The problems were about the flow rate of water descending into a tank and what happens when you put a drain here with this rate of flow, etc. Completely different from our earlier problems which could be solved by rote memorization. And from my perspective - one of a student who knew a bunch of calculus "rules" but lacked understanding of how calculus really worked - this was incredibly difficult.
I got a 3, which is passing, but opted to take beginner-level engineer's calculus when I got to college anyway. Even if you "learn" calculus in high school, it does not teach the level of mathematical maturity required to understand the higher level manipulations. This kind of thought is the foundation for any good engineer, and I have no regrets about retaking the course.
I definitely remember the first time I could no longer slack off, take 0's on homeworks, and ace tests to get a C+/B and it was in high-school :(
I jumped to AP Calculus without taking the pre-requisite AP class everyone else had taken, and I thought I was so cool and smart. By the weeks end I no longer felt so cool and smart - I was completely lost.
Calculus sucked more for me because here in Brazil the education system doesn't include it in the high-school curriculum, so unless you're on some top private school, you only first hear about "calculus" in college. In the USA and other places there's exposure to it prior to college, which I believe is sensible.
As a side note, I can't stress enough how bad the education system is in Brazil (in the sense of lack of structure, motivation, etc.), even at college level. Almost everything I know is self-taught (including English). I sometimes hear complaints about public school in the USA, and there's a meme that americans are "dumb", but it seems miles ahead to what I had access to.
In India we had pretty advanced calculus in our senior and pre senior year. This included extensive use of calculus in physics courses. Electrodynamics was totally calculus based and we used to have lots of 'word problems' involving calculus. This was pretty standard across most schools, so much so that when I went to college to do my engineering we had zero basic calculus classes and went straight to advanced calculus.
I have no clue if learning advanced calculus this early helped us or not. Or exposing the whole class to these courses was such a good idea. However now they are easing these courses a bit and moving some of the tougher parts out.
Yeah, luckily the area I grew up in happens to have one of the top rated public school districts in the nation, so most things that were offered in private school education-wise were offered in public school as well.
From my experience, there's probably much less calculus-taking going on in American high schools than you think. Of that, even less is anything resembling calculus.
College-level calculus burned me so much, I didn't had the algebra background necessary to grok it, much of it is memorizing heuristics and "tricks", and college professors (specially in my public uni) didn't had much patience either. I even did well in other math-related classes, but calculus was a chimera. Unfortunately, I fell so much behind the curriculum that I had to drop out of college to start working full-time.
Lately I've been picking up other math-related online classes, and I finally realized that I had never had real exposure to mathematics, and that high-school curriculum is something else. So, there's that too... we are mostly scammed into thinking we've learned something during high-school, and then life happens and you actually figure out you don't know shit. Hopefully I'm a curious person, so I'm always learning something by myself, mostly to broaden my knowledge in other topics, but to not let the gears grind too.