In my opinion, those are fundamental ideas for human beings in the year 2010; obviously we have to lower our expectations to match reality, but we shouldn't pretend they are unreasonable. Really, how difficult is the idea of a cosine? That's not exactly an unapproachable level of abstraction.
Remember, these are people who have had 12 years of full-time mandatory schooling, and have signed up for a few more years on top of that. If anyone with an average IQ cared even a tiny bit about such things, they could learn all of it in half that time; and if they don't care at all, parents and teachers didn't do their job.
Of course, I agree with the main thrust of your post. Somehow, you have to actually encourage (and allow) the students to understand what they're dealing with, instead of giving up, moving numbers and letters on paper, and passing to the next class. If that doesn't happen, there's no surprise that they've forgotten it six weeks after the course.
I agree that just drilling on the mechanics of long division doesn't help. But, drilling the concepts can help. The point of the article is that instead of letting kids discover everything on their own and then feeling happy, teachers should direct their learning. In doing so, I would say they should drill the same concept repeatedly and ask questions that test both the ability to do the procedure and understand the concept.
Anecdotally, this is how I taught my sister multiplication and how my friend tutored someone in calculus.
Remember, these are people who have had 12 years of full-time mandatory schooling, and have signed up for a few more years on top of that. If anyone with an average IQ cared even a tiny bit about such things, they could learn all of it in half that time; and if they don't care at all, parents and teachers didn't do their job.
Of course, I agree with the main thrust of your post. Somehow, you have to actually encourage (and allow) the students to understand what they're dealing with, instead of giving up, moving numbers and letters on paper, and passing to the next class. If that doesn't happen, there's no surprise that they've forgotten it six weeks after the course.