## “I don’t know how to do a line integral either”

**If you hang around engineering professors — of any sort — for any length of time, at some point one of them will start complaining about how bad students’ math skills have become.** The other professors will nod in agreement, and each professor will list some tidbit of math they need for the classes they teach, and bemoan how the students look at them as if they had never even heard of such things. Sometimes this leads to grousing at lowered standards in high schools. (At Georgia Tech, occasionally blame is placed on one of our previous Presidents — to be clear, of GaTech, not the US.) **Everyone will agree that the students math skills’ have been declining over time, and that the pinnacle of mathematical educational excellence coincided with the time in which the oldest in this group of professors was an undergraduate.** (I’ve confess that I’ve engaged in these conversations myself; at one faculty meeting I complained about how some of my ECE2025 students didn’t seem to know how to differentiate exponentials.) I conjecture that faculty in the “pure” sciences (physics, chemistry, etc.) indulge in similar pedagogical pity parties. Perhaps mathematics professors even kvetch about their own math majors? (Maybe claims of this decline are well supported by solid evidence, maybe not; I suspect the discussion says more about the professors than it does the students.)

**In electrical engineering, you’ll quite often hear such complaints from instructors teaching electromagnetics.** This is not without reason — out of all the undergraduate EE topics, electromagnetics probably relies on the most sophisticated mathematics an undergraduate engineering student is likely to encounter, namely vector calculus. Electromagnetics also is probably the most inherently challenging subject all EE majors are typically required to study, independent of the complexity of the related math. (It’s the class that our students fear the most — the phrase is “Emag, Remag, Threemag, Management” is a well-worn student slogan.) At Georgia Tech, vector calculus is covered in “Calculus III.” When I was at Washington University, it was part of an omnibus “Engineering Mathematics” class taught by the Systems, Science, and Mathematics department. (Incidentally, SSM majors always complained that their main problem was having to explain to potential employers what an SSM major was.)

**A while back, one of my colleagues, who had recently taught electromagnetics, complained that his students “couldn’t even do a line integral.”**

**I noted, “Well, I don’t know how to do a line integral either. I mean, I’m sure I learned them, and could do at one point, but I couldn’t do one for you now.”**

“Horror,” you holler! Here’s Aaron, a professor at Georgia Tech — an professor of electrical and computer engineering — and he can’t even do a line integral?

Well, yeah. At least as of 15 minutes ago, when I got the idea to write this post. **Line integrals just don’t show up often in the kinds of research and engineering my graduate students and I do. There’s certainly no shortage of funky math that shows up in my work, and for my work I generally have those funky facts “at my fingertips,” but that’s just because I use them a lot, not because I made any particular effort to memorize them.**

One of my other colleagues said something along the lines of “well, Aaron, you may not have needed that lately, but these students have taken Calc III the previous semester.” Yes, but **most of what the students crammed into their heads the day before the final exam probably leaked out of their heads the day after.** Here’s a thought experiment: imagine the students who took a final exam having to take another final exam a month after the first one, with no warning and no chance of study. How do you think they’d do? (A more practical experiment for my colleagues: give a pop quiz half way through lecture and ask the most basic questions you can imagine on what you just talked about for the last 25 minutes. Try not to weep when you figure out how little attention anyone is actually paying to you.)

So, let me pull up wikipedia and type in “line integral.” (Aaron looks over webpage for a few minutes.) Oh, OK. Scalar line integral, vector line integral… derivative of the function goes here… OK, got it.

I wonder. Did seeing that web page jog my memory? Come to think of it, I do remember looking at these things when studying Grenander’s pattern theory, sometime around 1993 to 1995. Or was the memory of line integrals really lost, and I just relearned it from scratch? Or something in between?

Can I tell you something else?

**I don’t know Maxwell’s equations either.**

“Double horror!” you exclaim. How can Aaron claim to be a EE professor when he doesn’t even know Maxwell’s equations? Didn’t I take a class in Emag when I was an undergrad? Well, sure I did, and it ended with Maxwell’s equations. I even got an A in it. Emag is taught as a math class, and I was good at the math (well, good for an engineer), and could pattern match sufficiently to translate the homework and exam problems into the math. But I never developed any intuition for the subject at all.

I have a shirt with Maxwell’s equations on it (with the words “And God said…” in front of them, and “…and there was light” after them), and you’ll see me wearing it on campus sometimes. But I couldn’t tell you what most of the symbols on that shirt mean. Well, I know what dot and cross products are (mostly because I’ve been teaching computer graphics lately). But I couldn’t tell you what a “div” or a “curl” was off the top of my head.

Here’s the thing. I don’t use Maxwell’s equations, or most basic Emag, in my work. This may seem odd since radar is one of my specialties. But I study the algorithmic side of radar, and much of the Emag-ness has been abstracted out of the kinds of mathematical conventions that are typically employed in radar signal processing.

But here’s Wikipedia. And I picked up this fantastic little book called A Student’s Guide to Maxwell’s Equations at our campus bookstore a while back. It’s waiting for me if and when I need it.

In twenty years as a working engineer I drew on my calculus skills once. I needed to find the center of gravity of our liquid oxygen tank when partially full of propellant. It was a slightly rounded cylinder so I set forth to find an equation that would give me the CG as a function of length and truncation angle. I alternated this with working on the “harder” problem of doing the same analysis of the more irregularly-shaped fuel tank. Turned out our CAD program had a function to calcuate the CG of an arbitrary solid. So no need to approximate the rounded-off LOX tank as a pure cylinder, I could just throw the CAD program at it too.

Never did finish that function. Or use calculus again. Algebra and trig got used lots.

I’m studying in a significantly different field, but I believe part of the benefit of studying certain mathematical and physical science subjects is not that you can instantly bring to mind how to do specific problems, but that the pathways have been made in your mind once to think in a certain way. In my introductory physics course for instance I struggled greatly throughout the semester to keep up, and it culminated the night before my final in me finally truly seeing on an intuitive level that the equals sign in equations does not mean that the figures on the right are the answer (as was generally taught throughout all my public school education), but that the two things are equivalent and can be swapped out as needed. I aced the final, and then forgot most of the specifics of the subject over the next month. However when Tara or other friends are studying physics and have trouble with it, I’ll look at it for 10 minutes or so and be able to walk them through the procedure. I could not have done that, had I not had the initial growth experience.

We bemoan the purging of learning after classes, but perhaps that is because we are focusing on the wrong things as measurements of education. And doing so to the extent that many classes never really manage to teach anything, because they are solely introducing information without producing new ways of thinking about anything.. in that case the experience truly is wasted.

In general, I think our testing system needs to go away. What needs to happen is either oral discussions with the professor to ascertain what you really can think and communicate about a subject, or project based examinations where you are given a problem and told to work out a solution, or at least explain how a solution could be thought about. But of course both of these require a bit more on the part of professors, and everywhere we are cutting back on professors because they are expensive.

*laments the lack of focus on education in our country*