Firday’s quick and sarcastic post came about because I thought the Dean Dad and his commenters had some interesting points in regard to high school math requirements, but we were spending the afternoon driving to Whitney Point so I could give a graduation speech. I didn’t have time for a more detailed response.
Now that we’re back in town… well, I still don’t have time, because SteelyKid has picked up a bit of coxsackie virus, meaning that nobody in Chateau Steelypips is happy. But I did want to offer at least a partial response to some of the comments both here and elsewhere.
To start off, Tom raises a good point about correlation and causation: it may be that, as the Dean Dad notes, students who took four years of math did so because they do well in math, and that math aptitude is the reason for both their extra preparation and their success in college math.
There’s probably something to this. Unfortunately, ability to do well in math is only imperfectly correlated with desire to pursue courses of study for which math is a prerequisite. And, more importantly, I think there are different ways to do four years of math classes.
This is also my response to the comment by Jim at the Dean Dad’s (the comment permalinks are broken– it’s at 10:20 am, about halfway down):
Math, like Foreign Languages, but unlike almost every other high school subject, builds. If you just scrape through Algebra I, you start Algebra II behind the eight-ball. Algebra II expects you to know the stuff from Algebra I. In contrast, if you barely scrape through Earth Science, you start Biology even.
There’s some truth to this, but it seems to me that this is a problem that can be solved by offering different flavors of Algebra II (or whatever)– students who just scrape through Algebra I should get a version of Algebra II that spends a significant amount of time reviewing and reinforcing the material from Algebra I, while students who breeze through Algebra I go directly into Algebra II.
The Math department at Union does something like this with calculus. The “default” calculus sequence is two courses (two of our three academic terms, so not quite a full year) covering differential and integral calculus. Students who have good-but-not-great AP scores get a one-term version of the same material, while students whose preparation is a little weaker can take a three-term version of the same thing. This creates a few problems with classes like intro physics because our prerequisite enforcement is nonexistent, but from a straight math perspective, it’s a good system.
Of course, that would never really work at the high school level, because the overhead is too great. Not only would you need more teachers to cover the different versions of the same course, you would create some administrative overhead in trying to keep track of who took which version, and it would play hell with the notion of standardized testing of all students. In an ideal world, though, I think that would be the way to go– set a minimum standard, but provide tracks for students with different levels of math aptitude to meet that standard in different amounts of time.
Rather than the current version of the requirement where a student gets through Math II in four years by taking Math I twice, and Math II twice, have a four-year sequence covering the material of Math I and Math II, which would give them a better chance of retaining some of what they’re supposed to be learning.
I’ve switched to Math I and Math II in the above paragraph, because there’s also a valid point raised by harlan, and also in this TED talk by Arthur Benjamin (via… Jennifer Ouellette on Facebook, I think), namely that there’s no particular reason why the curriculum has to be structured the way it is. As convenient as it is for physicists to have the entire high-school math curriculum aimed at calculus, which is really about physics, there’s no reason why things have to be that way. And you could make a good case that teaching every high school student probability and statistics would be more useful to society as a whole than teaching everybody calculus. If you’re going to start imagining pie-in-the-sky reforms of math education, you might as well do it from the ground up.
The final frequently-raised point is one that almost doesn’t deserve a response, namely “None of these students are ever going to use {math topic}, anyway…” (sometimes in the “I took {math topic} back in the day, and I’ve never used a bit of it” variant). In addition to the standard annoyed responses (“Yeah? Well, then, take English Literature and stick it up your ass, ’cause they’re never gonna need that, either…”), though, there’s a response in the Dean Dad thread (Mikey at 7:58 am) that’s worth highlighting:
The fact of the matter is that most people don’t retain all of the math that they’ve ever learned. For someone in a developmental math course it might be something like 50%. For a grad student, it might be something like 80% (as a former math grad student, I can tell you that I retained significantly less than that). So if we need someone to be very comfortable with arithmetic to function in society, stopping at the end of arithmetic won’t be enough for them to retain it. They’ll need a bit more in order to retain everything.
Since math does build on itself, we can start teaching these students algebra. In using algebra, students will be challenged to use their arithmetic skills in many different ways, forcing them to actually own arithmetic. It will give them the practice of not only performing arithmetic, but also knowing when it’s appropriate to use different operations.
Finally, the problem solving and even logic skills one employs in an algebra class are really a dimension higher than that in an arithmetic class. Being introduced to a different way of thinking (using variables to represent unknown quantities, as well as using symbolic manipulation) is immensely helpful in any sort of problem solving arena, whether or not it involves math. Everyone needs problem solving skills: if jobs didn’t require them, everything would be run 100% by computers.
The problem-solving skills bit is familiar, but the retention argument is new to me. I like it, though, and I think it’s a good point. And so, I mention it here.
And that’s about all I have on math education at the moment. So, I’m off to check on SpottyKid…