A few days ago, as part of his valedictory (about which I'll have more to say Sunday), Kevin Drum posted a brief item on how high school girls are taking as many math classes as boys, and, perhaps more surprisingly, how they're more likely to take advanced math. Kevin's question doesn't focus on this as much as it does the result that about 33% of all students take pre-calculus or calculus, compared to: "I'd guess that in the 50s, roughly 0% of high school students took pre-calculus or calculus classes."
I'm not sure how much of my own history with math I've given in this blog, and I don't feel like searching back through the archives, so a brief recap. Competitive mathlete (sic) in high school at a time when there were few competitions, then I became a substitute for a year after grad school. At the same time, through an odd set of circumstances, I took on the role of head math coach. The team won the state championship that year, and, once I had moved to the role of assistant coach, kept on winning, eventually putting away eight straight victories.
Of more importance to this topic is my own experience with math in the public schools. At 9, I took a summer college course, and my mom found a tutor for me. Ridiculously, I would still have to sit in the standard math classes, though I wouldn't have to pay attention or take tests. Then, a couple of years later, in eighth grade, we stopped the tutor thing (not sure why), and I, based on a lot of fighting the system by my mother, got to walk from the junior high to the high school to take geometry. This was a pretty good school system, one of the best in Michigan, but they had tremendous problems letting a junior high school kid take a class two years above grade level.
In 10th grade, we moved back to the Chicago area, and I ended up in one of the best high schools in the country (at least it's considered such every time such things are measured). I should have been in calculus then, but the school was worried about the two years after, so I had to sit through a year of senior math, a kind of time-wasting catchall that allowed students to get four years of high school math without scaling the peaks of calculus. After taking calculus my junior year, I was pretty much left on my own for my senior year. I ended up auditing a math class at Northwestern, a thoroughly unpleasant experience that displeased everyone involved, and was likely responsible for my departure from the study of mathematics.
Now this was the '70s, and, even at that time, a well-regarded high school was nonplussed at the thought of having to educate anyone in math beyond the calculus level. (To their credit, they now offer a class in multivariable calculus and linear algebra, so they've made progress...not that it does anything for me.)
I can't speak to Kevin's opinion that, in the '50s, no one took pre-calculus or calculus. I actually doubt that, given that pre-calc tends to include things like logarithms and trigonometry, and students were certainly learning about those. But he's probably right to the extent that calculus was pretty rare in high schools of the time.
Still, is it all that surprising that more students take these courses now than in the pre-Sputnik world? We pretend to prepare students for the high-tech world of tomorrow, so we offer classes that appear to do that.
But, as I understand from my lingering connections to the field, these classes are not at all equivalent to the delta-epsilon studies of my youth. The advent of symbol-manipulation graphing calculators have apparently changed the coursework in huge measure. For me, calculus was my first exposure to "real" mathematics, where underpinning concepts were to be mastered (geometry offered some of this, but in a very easy format). Apparently it has become, facilitated by the little brain-boxes, essentially an extension of algebra in which formulas are used to solve applied problems.
This is the danger of standards-based education (and a danger of private enterprise education). There is a tendency to offer what customers demand, whether it be a college requirement of four years of math or a desire to gain a fairly easy masters degree. If every university requires four years of mathematics for only vaguely technical fields (and three for every other field), high schools will be sure to provide it.
But the students are not necessarily systemically better, so the only way to get them to meet the requirement is to lower the rigor of the field. If we're going to use the study Kevin cites to justify a belief that schools today are better than ever, we better just move on, because a lot more work needs to be done to prove any such thing.
I'm not sure how much of my own history with math I've given in this blog, and I don't feel like searching back through the archives, so a brief recap. Competitive mathlete (sic) in high school at a time when there were few competitions, then I became a substitute for a year after grad school. At the same time, through an odd set of circumstances, I took on the role of head math coach. The team won the state championship that year, and, once I had moved to the role of assistant coach, kept on winning, eventually putting away eight straight victories.
Of more importance to this topic is my own experience with math in the public schools. At 9, I took a summer college course, and my mom found a tutor for me. Ridiculously, I would still have to sit in the standard math classes, though I wouldn't have to pay attention or take tests. Then, a couple of years later, in eighth grade, we stopped the tutor thing (not sure why), and I, based on a lot of fighting the system by my mother, got to walk from the junior high to the high school to take geometry. This was a pretty good school system, one of the best in Michigan, but they had tremendous problems letting a junior high school kid take a class two years above grade level.
In 10th grade, we moved back to the Chicago area, and I ended up in one of the best high schools in the country (at least it's considered such every time such things are measured). I should have been in calculus then, but the school was worried about the two years after, so I had to sit through a year of senior math, a kind of time-wasting catchall that allowed students to get four years of high school math without scaling the peaks of calculus. After taking calculus my junior year, I was pretty much left on my own for my senior year. I ended up auditing a math class at Northwestern, a thoroughly unpleasant experience that displeased everyone involved, and was likely responsible for my departure from the study of mathematics.
Now this was the '70s, and, even at that time, a well-regarded high school was nonplussed at the thought of having to educate anyone in math beyond the calculus level. (To their credit, they now offer a class in multivariable calculus and linear algebra, so they've made progress...not that it does anything for me.)
I can't speak to Kevin's opinion that, in the '50s, no one took pre-calculus or calculus. I actually doubt that, given that pre-calc tends to include things like logarithms and trigonometry, and students were certainly learning about those. But he's probably right to the extent that calculus was pretty rare in high schools of the time.
Still, is it all that surprising that more students take these courses now than in the pre-Sputnik world? We pretend to prepare students for the high-tech world of tomorrow, so we offer classes that appear to do that.
But, as I understand from my lingering connections to the field, these classes are not at all equivalent to the delta-epsilon studies of my youth. The advent of symbol-manipulation graphing calculators have apparently changed the coursework in huge measure. For me, calculus was my first exposure to "real" mathematics, where underpinning concepts were to be mastered (geometry offered some of this, but in a very easy format). Apparently it has become, facilitated by the little brain-boxes, essentially an extension of algebra in which formulas are used to solve applied problems.
This is the danger of standards-based education (and a danger of private enterprise education). There is a tendency to offer what customers demand, whether it be a college requirement of four years of math or a desire to gain a fairly easy masters degree. If every university requires four years of mathematics for only vaguely technical fields (and three for every other field), high schools will be sure to provide it.
But the students are not necessarily systemically better, so the only way to get them to meet the requirement is to lower the rigor of the field. If we're going to use the study Kevin cites to justify a belief that schools today are better than ever, we better just move on, because a lot more work needs to be done to prove any such thing.
4 comments:
I'm curious, are you implying maybe we should roll back some of the sexier advanced math classes and concentrate more on upgrading the basics? For an "average" high school kid, what would you recommend? Algebra I in 9th Grade, Geometry in 10th Grade, Algebra II in 11th Grade and Something or Other their senior year?
Your scheme is the classic one, but I'm not sure that there's any reason (other than a lack of qualified teachers) that we can't move those up a year (algebra is probably able to be mastered by most 8th-graders). The question is what do we do with the students their junior and senior years. We could make calculus a two-year or three-semester sequence, try to get students to really understand it. We could offer courses like statistics or linear algebra, but that is again subject to the limitations of teacher ability. Or we can just recognize that, for a lot of students, the sequence through Algebra II is sufficient. I can't say I've thought through the alternatives fully, but those are some initial thoughts.
Thanks for your thoughts. I'm seeing kids do, to my untutored eye, in their 7th and 8th grade "math" classes, an awful lot of stuff that looks like "algebra" to me.
I noticed our high school has a dizzying array of math subjects, which I suspect may be being taught over the course of a few semesters rather than an entire year, or being taught according to student abilities, or a combination of the two. However, "trigonometry" is noticeably absent. I know the subject matter is being rolled into other classes, but do you have any idea why "trigonometry" seems to have fallen out of favor?
I think trigonometry was always problematic, living somewhat at the cusp of algebra and geometry. It never really took hold in geometry, which kept its proof methodology somewhat pure, so it ended up in Algebra 2, which resented the inclusion of a geometric topic in its realm.
Furthermore, I suspect that the modern calculator has rendered much of trig apparently useless (not a philosophy to which I subscribe; if you let technology push out everything in the curriculum, we wouldn't need schools at all). Knowing that tan 60 = sqrt(3) pales next to the wonderful exactness of 1.7320508, even if the former has the benefit of being accurate.
And what is the value of trig anyway, unless you plan to become a surveyor or a sailor? Better to fill up students' time with discrete math, in which we teach kids how to "do" statistics, but not how to read a newspaper article with statistics in it.
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