Musings

Jacques,

I know the feeling. I am teaching basic mechanics to engineering students at the University of Ottawa and some of them never saw some concepts of linear algebra and calculus. In Ontario, the provincial Government chopped one year of the high school curriculum two years ago to save money… What a bad mistake! Students can get into science and engineering without a single high school physics course and without a full calculus course…

Cheers!

Charles

I’m not that surprised that linear algebra is separate course. It is that way here at RIT, though they get a little bit in differential equations. What I find is that even if they have covered the “eigens” in a course you still get blank stares. There are even times when I hear the crickets chirping in a first-year physics course and I talk about calculus. It usually comes back to them as you use it in class, but I can’t count on using certain maths right out the box.

Frankly,I’ve never heard about linear algebra being bundled with calculus. In most science curicula, linear algebra is always taught as a separate course.

I’ve never heard of linear algebra being taught in calculus either. Actually I was one of those people with a blank look on his face during my first quantum mechanics course as I hadn’t taken any linear algebra yet. I was quite frustrating to be so in over my head at first.

For sure linear algebra is in the calculus sequence at Caltech, but I would bet that it’s separate in most American undergrad programs.

There seems to have been a general decline in educational standards over the years, in parallel with grade inflation.

I went to a public high school where the senior year math courses were linear algebra and calculus, along with a third course which was a hodge podge of topics like probability, statistics, combinatorics, and some linear programming. The linear algebra course was almost identical to the freshman university version, except it didn’t cover the proofs as extensively. The calculus course was almost identical to the freshman university version, except it didn’t cover the more formal things like epsilon-delta limit proofs. These courses were heavily computational more than anything else, where the teachers largely tested us on how fast we could do the calculations without making too many mistakes.

I guess at the high school level that’s about the most the teachers could do considering more than half of the students didn’t care about the subject, but had to take it because they wanted to go to university. Many universities seemed to require these two or three math courses for admissions, even for majors which used very little to no math.

When I took freshman university physics, I came to the realization that the high school physics courses I took were largely watered down versions. It was a bit of a shock for me at the time when I found that many of the freshman classical mechnics and electromagnetism problems required a lot of effort on my part. In comparison, the problems we were assigned in high school physics were relatively trivial and didn’t require a lot of heavy effort.

It could have been a fluke that I just happened to have tough math teachers and “easy” physics teachers, at the public high school I went to. Years later I spoke with several of my old high school teachers, where many of them thought that things were getting “watered down” over the years in the senior level courses. My old physics teachers mentioned that when they first started teaching high school physics in the 60’s or 70’s, the level of difficulty was almost at the same level as freshman university physics courses. Over the years things got watered down when more and more students were flunking the junior and senior level physics courses.

I would suspect that the “watering down” problem seems to have origins even before high school.

Things are quite different in Canada. We don’t do linear algebra in High School at all. Calculus is available, but only as one of the “locally developed” courses, meaning that the Province and School Board take no responsibility for the content. Its all left to the teachers to develop, test, and deliver.

At university, there are two semisters of calculus in first year. In second year, we start with a semister introduction to multivariate calculus, followed by differential equations. Linear algebra is left to a different class entirely, often taught in parallel. The “advanced calculus” course in third year uses matrix and vector algebra with calculus for the first time.

Having just slogged my way through the third year calculus course, I can attest to the “memory drain” that happens over the summer. It was painful.

Texing up some brief handouts on the math
methods you need to cover as they come up
–and getting straight to the point–might be the best way to get them up to speed, maybe with some problems or worked examples. I know its a pain doing this but usually if you want something done right or explained right it’s best to do it yourself.

Here in the UK there are complaints that each batch of new incoming science students to universities seem less well prepared with each passing year. Lecturers are usually having to devote a lot of time to explaining fundamental material that should already have been covered. It is not just in the sciences either. Some basic literacy skills are often missing. There is also a big problem here in that fewer and fewer students are actually taking courses in the hard sciences, opting instead for soft option ” vocational degrees” in things like “interior design” or “media studies”. (What the press has dubbed “Mickey Mouse” courses.) As a result some 90 physics and engineering departments have actually closed in recent years. A lot of chemistry departments are closing too. It’s a depressing scenario. Science courses should never be dumbed down though. Something is only worth pursuing if it is difficult.

Fortunately as a math buff when I was younger I had most of those classes by the time I took the upper tier physics classes as an undergrad. Of course sometimes thats even more difficult, as one is used to different notation and handling of things. Often physics has its own peculariaties that takes getting used too, and quite a few hours of puzzling over things that are by name identical, but mathematically formally nothing to do with one another.

The one exception (and something that drives me nuts with undergrad curricula). I had little to no understanding of probability theory, outside maybe knowing what standard deviations were. When I took stat mech, that was one of the hardest and most confusing moments of my entire career. Hell learning parts of string theory and QFT was easier.

Functional analysis or at least probability and combinatorics should be requisites for any physicist. Its totally ridiculous that they are not prereqs for some of the upper tier physics classes.

I think Math has really changed in Canada as they got rid of honor’s classes. When I was in High School in the late 80’s we did a little bit of linear algebgra. Nothing that advanced though. We also did a lot on series. In fact when I went to Dalhousie probably the first half of my calculus class I’d done in grade 12. Down here in the states though I’ve noticed that most people going into the sciences though have graduated with AP calculus and related courses that seemed a bit more advanced than what we did even in honors calculus in Nova Scotia. But my brother tells me they’ve definitely dumbed down all the classes up there the last decade or so.

Regarding pre-requisites, in my physics classes you nearly had completed a math major before you started taking upper division physics classes. I think the first half of mechanics (Lagrangian and perturbation theory) was taking concurrently with linear algebra and ODEs. The advanced mechanics was taught in the physics department and was basically more perturbation theory, PDEs, and doing lots of line integrals. We then had an optional class on tensor analysis which was basically just calculating lots of different change of coordinate systems to get an easy solution. However it definitely came in handy with QM which really isn’t that intense mathematically if you’ve encountered the basic kinds of math before.

I’m sure Cal Tech is a bit more advanced than what we did. But it does seem odd not to require linear algebra for an advanced mechanics class. However to be fair even our first advanced mechanics class had a bit that some people struggled with if they weren’t enrolled in linear algebra alongside it.

I do think that if one can assume more math background that it would enable Junior level physics to be taught a little better. I don’t see the problem with requiring most sophomores to be taking advanced mathematics - especially given the fact that most science students come in a sophomores due to AP classes. If they aren’t quite ready for it then they end up staying an extra year. But so what? I suspect at most universities it is hard for a physics major to graduate in four years.