High School Mathematics

by Brian on September 13, 2003

Over at Calpundit there’s an interesting discussion going on about the stresses that contemporary high school education places on students. In the comments Kevin expresses surprise (at least I think it’s surprise) that there are students who take two years of calculus in high school. I was rather surprised that this is surprising.

Where I went to school (in a fairly good suburban Catholic school in Melbourne) the median student did two maths courses with hefty calculus sections before graduation, and a sizable minority (about 15 to 20%) did four such courses. And I didn’t think this was particularly unusual. It certainly didn’t strike me as an outrageous amount for high school students to complete.

Because there’s next to no philosophy taught in high school in America (or Australia) I’ve never had to pay much attention to how much incoming college students have learned. So I’ve got no idea really how to compare American and Australian students. But my (quite possibly erroneous) impression is that the demands of American high schools are much less onerous than their Australian equivalents.

If you want some more specific info on what Australian high students are expected to know, here’s the final exams from the last three years given to final year high school students in Victoria. At my school 50% or more of graduating students would have taken the course ‘Maths Methods’, and another 15 to 20% the course called ‘Specialist Maths’. (Back in my day they had different names, but the syllabus doesn’t look to have changed dramatically.) Quickly flipping through the VCAA website it seems the numbers across the state for how many took those two courses are more like 40% and 15% respectively, and you can get some detailed reports on how they did here.



Keith M Ellis 09.14.03 at 12:16 am

American secondary education is notoriously poor. However, I have a question for you about Australia. For the most part, the US doesn’t have seperate tracks for college-bound and “vocational” students. It’s my understanding that in a lot of European countries, there’s a fair amount of sorting that goes on as early as middle-school and students move into seperate tracks, even seperate schools, with completely differenct curricula and even duration. Am I correct about this? What is the situation in Australia? I’ve just imagined that this plays — or at least historically played — a role in the relatively less rigor of American secondary education. When every student is expected to get essentially the same high school education, the standard currilculum is not going to include two years of calculus.

For what it’s worth, the notoriously crappy small-town New Mexican high school I graduated from in 1982 offered only a single calc class for the very few students who wanted it; and the majority of students only took a year of algebra I and a year of geometry. A foreign language was not required; and, anyway, the only language offered was Spanish.

That said, my experience years later at several different universities and colleges (some moderate quality large state schools, a moderately high quality small science/tech/engineering school, another a high quality tiny liberal arts school), even the students that ostensibly had a strong high school background in calculus seemed to have forgotten everything they knew. Which makes me wonder how much difference this all makes, anyway.


Brian Weatherson 09.14.03 at 12:24 am

I’m not sure what the state of vocational education is in Australia these days. When I started high school (er, we don’t have middle school, so this is about what level you’re at age 12) there was this kind of streaming. By the time I finished high school it had become much less important, and I wouldn’t be surprised if it is even less important now. What’s happened to some extent is that the vocational training has been incorporated into the regular curriculum. There is, for example, a non-calculus mathematics course in final year. This won’t look too impressive on your college application, but it is much more useful for students not going to college – especially since it teaches skills they are somewhat less likely to forget as soon as they leave the classroom.


clew 09.14.03 at 12:24 am

How many courses in an Australian high school year?

Most people have to take calculus twice; once for familiarity with the terrain, the second time to learn either the foundations or the applications properly (depending on whether you’re expecting to study more math or more science). The short description for the MIT OpenCourseware calc course is an explicit example.


Brian Weatherson 09.14.03 at 12:33 am

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.


Mary 09.14.03 at 12:52 am

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.


Jonathan Ichikawa 09.14.03 at 1:30 am

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.


Brian Weatherson 09.14.03 at 1:49 am

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.


Shai 09.14.03 at 2:00 am

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.


Dan the Man 09.14.03 at 5:38 am

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.


Doug 09.14.03 at 11:43 am

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?


Chris 09.14.03 at 1:00 pm

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.


duncan 09.14.03 at 1:05 pm

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.)


Nicholas Weininger 09.14.03 at 6:18 pm

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.)


Matt McIrvin 09.14.03 at 6:25 pm

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.


J. Ellenberg 09.15.03 at 4:23 am

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!


eszter 09.15.03 at 5:05 am

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)


Brian Weatherson 09.15.03 at 6:28 am

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.


Antoni Jaume 09.15.03 at 9:14 pm

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.


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