As I sit here working on calc III homework, I come to the realization once again that I really have no idea what I’m doing.
On one hand, I know exactly what I’m doing. I’m taking double integrals, finding Lagrange multipliers, and finding partial directional derivatives. I can tell you what these terms mean, which make any non-engineering or math major cringe, easily enough. They’re volumes under surfaces, a handy way to find max/mins, and lines tangent to surfaces in a direction. They’re actually not even hard to do, it’s just all the stuff you learned (or learned and forgot often in my case) in calc I and calc II applied to R^3 space.
On the other hand, I have no idea what I’m doing. How would I ever use this in the real world? Just trying to figure out an application is difficult. Don’t start attacking me saying there’s applications for this kind of math all over the place, because I’m trying to believe you, I really am. I just haven’t been taught any of them. Say I have a function f(x, y), and then I want to see how the function changes with respect to x only. Okay, sure, just take a partial derivative. Why would I want to find how the function changes at an angle of 42 degrees from a point though? For that matter, where do these functions come from? How do I model real world applications into the form f(x, y) so I can easily do my calc on them? If you look at the Wikipedia article for a hyperbolic paraboloid, there’s 4 sections of it’s properties and how to do stuff with it, and a footnote at the bottom saying “applications: directional antennas and receivers, such as satellite dishes, telescope mirrors, and directional microphones are paraboloids.” Nifty, why don’t they ever give me cool application problems dealing with directional antennas?
If I get a job in engineering or math, I don’t think my boss is going to come up to me and say, “hey, go find the first and second order derivatives of .” First off, I can plug that into Mathematica and get the answer in a few dozen clock cycles, faster than I can even write the first derivative with respect to x. For that matter, faster than I can write the letter ‘x’.
The really sad part is, I’m in “applied” linear algebra and calculus “for engineers”. That means somewhere at ASU there’s a bunch of math majors even more clueless about the applications of what they’re doing, even if they can prove by contradiction that their numbers are at least logically derived.
If I ever became a math professor, this is what I’d do. First off, instead of spending the first class introducing myself and the syllabus, I’d spend it introducing the subject. I’m 4 weeks into discrete math right now and I still have no idea what discrete math really is. It seems like a hodgepodge of topics and mainly a class to teach proofs in, which seems a bit pointless since the teacher said a good 50% of the material overlaps with the class I’m taking next semester on proofs. Don’t go into details about how to do anything, just talk about the general idea of the stuff. If teaching calc I, explain what derivatives and integrals are from a geometric viewpoint. Explain a few basic uses of them as well. Don’t just jump into the limit definition.
Then, when you get into the material, first play around with some problems and applications before you dive into a proof of some formula that takes half the class. This would at least show the students where you’re going during your proof. More often than not I watch my teachers manipulate expressions for 5 minutes, coming up with an obscure formula, then explaining what the formula actually does. If I’m really lucky, they’ll even explain what it means after that.
Finally, once you know the formula, where it came from, and what it does, do real world problems. If almost all the problems are related to physics, maybe physics and math should be combined into one class. It seems like physics classes just apply formulas to problems, and math classes just make up formulas without showing any problems. It’s hard to deny that a large part, maybe even the majority of uses for calculous involve physics. Of course the classes would move slower, but what’s wrong with spending more semesters if you’re combining two classes?