Comments on: High School Mathematics

By: Antoni Jaume

Antoni Jaume — Mon, 15 Sep 2003 21:14:36 +0000

In Spain as of now we have a common secondary schooling from 12 to 16, which is not supposed to distinguish what will be the posterior schooling of the students. Then they have 2 years of, lets say, high school, or go to professional formation. DSW

By: Brian Weatherson

Brian Weatherson — Mon, 15 Sep 2003 06:28:06 +0000

Jordan – yep, but you did much better than I. Well done! It’s funny that an Olympiad post should come up just before a post about a high school curriculum that doesn’t stress calculus. Because (at least when I was in it, and I think this is still true) the Olympiad didn’t involve calculus, but did put quite a bit of emphasis on things like coordinate geometry and combinatorics. (Actually, to tell the truth it didn’t put much emphasis on coordinate geometry, but normally the only way I could solve a geometry puzzle was to whack a pair of axes down at start solving equations. So I have a memory of it being all coordinate geometry all the time.) There’s certainly something to be said for that approach. I suspect the Hungarian approach will do more to teach students about how to solve problems, and provide them more effective intellectual exercise, than a calculus based course. (I have no data to back this up, so I could be wildly mistaken.) But I suspect Eszter’s point that people identify calculus with rigour in the curriculum would prevent a change of this kind occuring in Australia (or America) even if it were valuable.

By: eszter

eszter — Mon, 15 Sep 2003 05:05:17 +0000

I’ve always found the focus on “calculus” per se very interesting for the purposes of determining the level of rigor in math training. It is not part of the general math curriculum in Hungarian high schools (although it is available on the advanced math track for those pursuing math, econ or physics later), but that certainly does not imply that math isn’t taken very seriously in Hungarian high schools. In general, my impression is that math is taught very differently from how it is taught in the States. We (as in Hungarian high school students) spent months on things like coordinate geometry and combinatorics. We would first go through proofs and then have problem sets as homeworks each of which would take an hour or two to do. I’m afraid I cannot offer an English translation of our required end of high school math exam, but this Web site is interesting and some of it you can get without understanding the text (well, maybe). (“gimi” stands for regular high school, “szakközép” for vocational)

By: J. Ellenberg

J. Ellenberg — Mon, 15 Sep 2003 04:23:23 +0000

Not related to this thread at all, really, but I just had to say to Brian—wow, we were in the 1989 IMO in Braunschweig together!

By: Matt McIrvin

Matt McIrvin — Sun, 14 Sep 2003 18:25:54 +0000

In my US public-school education I only had a one-year math course that was called “calculus.” However, the previous second semester of “functions/analytical geometry” was actually an introductory differential calculus course, so that was really a year and a half of calculus. In the terminology of Advanced Placement tests (which could get you credit later on in some universities—I don’t know if this is true any more), the semesters were called A, B, and C, and the class I took my final year, and the associated exam, was actually Calculus BC. And since I went to a science/math public “magnet school” for my last year of high school, I knew many students who had skipped ahead of the normal curriculum earlier and were able to take a second year of multivariable calculus or differential equations.

By: Nicholas Weininger

Nicholas Weininger — Sun, 14 Sep 2003 18:18:56 +0000

dan the man: yes, some private high schools do offer linear algebra, usually to give exceptional students who get done with calculus in 10th or 11th grade something to do. The one I went to, Phillips Exeter Academy, offered linear algebra, differential equations and a two-semester discrete mathematics (= combinatorics) sequence. Now, Exeter is in no way typical of anything at all. And note that I didn’t avail myself of these opportunities; I was not one of the exceptional math students and took only one semester of calculus there, and now I’m working on a math Ph.D. thesis. So the amount of difference such offerings actually make to one’s longer-term prospects remains questionable. Also, note that in some areas, exceptional students have colleges nearby at which they can take advanced courses, so the high schools’ offerings don’t really reflect what’s available to them. During my junior year in college, a local high school student was a classmate of mine in a two-semester real analysis sequence. Lebesgue measure theory in high school—now there’s precocity for you… (BTW, I share your opinion that linear algebra is easier than calculus.)

By: duncan

duncan — Sun, 14 Sep 2003 13:05:30 +0000

The British system keeps changing, but in my day (the 1980s) if you chose to stay at school past 16 then you mostly studied just three subjects, and of course maths did not have to be one of these. By that time I had already studied some calculus. I can’t remember how much. (And the old system of one set of schools for the college-bound and another for the plebs had almost completely disappeared by then.)

By: Chris

Chris — Sun, 14 Sep 2003 13:00:29 +0000

Yes, I think that’s correct. Under the GCSE (pre-16) regime there’s no calculus and then only those who do maths to A-level (post-16) would get to study it. Under the old O-level/A-level regime there were 2 possible O-levels, a basic one and a further one and the further one dealt with probability and calculus. So, back then, at least some non-scientists (such as me: English, History and Econ to A-level) did some calculus. But nowadays not.

By: Doug

Doug — Sun, 14 Sep 2003 11:43:14 +0000

Here in Germany you're tracked into one of the three main paths of secondary education at about age 10. It's appalling, and more broadly based schools (Gesamtschulen) are seen as weird, lefty creations, especially in Bavaria. This is a key reason that, according to a recent article in Time Europe, only 8% of German children whose parent do not have a university education will attend university themselves. (The article has since disappeared into pay-per-view.) I don't imagine that the kids in the Hauptschule or the Realschule get much calculus. On another, er, track, I seem to remember that university-bound high school students in the UK specialized in their chosen field of study before leaving school. Thus someone intent on doing modern languages at university was unlikely to have calculus at all, while someone doing physics was likely to have quite a bit of advanced mathematics. Any enlightenment from the Home Counties?

By: Dan the Man

Dan the Man — Sun, 14 Sep 2003 05:38:31 +0000

I heard that some private schools teach Linear Algebra in high school. I’m not too surprised since I’ve always considered Linear Algebra to be easier than Calculus. Calculus all too often reduces to memorizing a bunch of formulas while one can reach a certain amount of understanding in even rudimentary Linear Algebra.

By: Shai

Shai — Sun, 14 Sep 2003 02:00:07 +0000

There is, or was at one point philosophy classes in high school here in Ontario. On the first day of an intro to philosophy class here at u of toronto the prof asked how many people had taken a philosophy course in high school and about 1/3 raised their hand. That said, judging by the questions in class and tutorial I don’t think anyone learns much in high school philosophy class.

By: Brian Weatherson

Brian Weatherson — Sun, 14 Sep 2003 01:49:12 +0000

Much thanks to Mary for the info about NSW. What I said about vocational courses was wrong as a generalisation about all states, and after reading through what she said I suspect it is probably wrong about Victoria as it now is. (There’s been a few changes to the Victorian system since I finished, and it now sounds a lot like the system Mary describes.) When I went through the system a few years back there was an explicit classification of subjects into academic and vocational (though I think the groups had less PC names like “Group 1” and “Group 2” respectively) and the vocational couldn’t count for university entrance ranking. But her information is much more up to date, and it’s helpful to know. On Jonathan’s point, it is possibly worth noting that I was a teenage math geek, which in Australia at least seems to be a common route into philosophy.

By: Jonathan Ichikawa

Jonathan Ichikawa — Sun, 14 Sep 2003 01:30:45 +0000

A rather minor point which I find interesting: I have noticed in two weeks of knowing you that you are very much better at moderately-complicated mental math than the vast majority of Americans I know, Brian. You don’t make a big deal about it, and I assumed you were just being modest, but in light of this post, it’s possible that you aren’t aware that most of the people around you can’t compute tips, or identify inverse exponential relations, as easily as you can.

By: Mary

Mary — Sun, 14 Sep 2003 00:52:11 +0000

Here are some Australian data points from someone who finished high school five years ago. This is information about the New South Wales system, as the Australian high school curricula are set at a state level). Most students take five and six courses, although taking the highest level of mathematics (or English) is the equivalent of two normal courses. In NSW, it is required that English be one of those courses. Mathematics is the second most popular course, around 85% of students take a mathematics course. NSW does not have separate tracks for college-bound and vocational students, although there are separate courses. The distinction is that it is quite possible to take courses from both vocational and academic courses, and there is no firm dividing line between a college-bound and vocational student imposed from the outside. In NSW, you need to complete a certain number (10 units, or 5 “courses” in the sense I used the word above—most courses are 2 units, or 120 hours per year) of “academic” courses in order to achieve a university entrace score. But I could, for example, have taken a lot of vocational courses (metalworking, woodworking, hospitality) in the middle of high school and then taken academic courses in the final two years and gone to university. Many university bound students take a vocational course, and almost all students who have no intention of going to university will need to complete a few academic courses. Finally, we don’t have a “college application” in the American sense. All the academic subjects have a state-wide exam, which, after considerable manipulation of means based on the performance of students across all their subjects, allows all the subjects to be compared. The upshot of all of this is a score out of 100 representing your performance in academic subjects (100 is the best score, 99.95 the next…) compared to the rest of the state. For the vast majority of university courses (as you can imagine, music degrees and creative arts degrees are among the exceptions), entrance to the course is competitive and completely based upon this score. So if you have achieved a very high score taking unimpressive subjects—and it is possible to do so if you did very very well in all those subjects, although recent reforms have attempted to encourage good students into harder courses—then you will be preferred over someone with a lower score, regardless of the subjects they took. As far as I know, at least, Brian is wrong in thinking that certain courses might look “bad” to universities. Or at least, they may look bad, but they don’t take that into account when admitting students.

By: Brian Weatherson

Brian Weatherson — Sun, 14 Sep 2003 00:33:19 +0000

Most students in final year take 5, maybe 6 if you’re really enthusiastic, subjects. In penultimate year it’s more common to take 6. (At least when I was going through the system, in Victoria, this was the standard. I’m not sure how much it’s drifted in the last 10 years, or how different it is in the other states.) I wouldn’t be convinced that high school students even after having 4 of these courses (2 in each of last 2 years) being calculus-heavy that they particularly understand calculus, but they are at least in a position to follow college courses that do things rigorously.