Are you so sure about this? Why does the suggested degree plan for mechanical engineering list the same standard battery of undergraduate mathematics (a year of calculus followed by courses followed by a year of differential equations / linear algebra and multivariable calculus) that I've seen at every community college and university?

Are you saying that the mathematics department doesn't get a budget to staff its own courses?

And yes, Gian-Carlo Rota had a very dry wit.

Engineering students at my alma mater had the same math course codes, but usually separate sections. English, however, was a separate business literacy course instead of the introductory-level literary analysis courses everybody else was forced to take.

I’m not sure if those courses were taught by or billed to the engineering faculty or math/humanities.

The concern about engineering subsidizing mathematics didn’t hold at my alma mater at least. Other than the honours math courses they were always packed.

Eh, yeah lower level courses are the same, but some upper level ones diverge. Reviewing the mechanical engineering courses:

• ME 118 Mechanical Engineering Modeling and Analysis (4)

As well, in electrical engineering:

• EE 105. Modeling and Simulation of Dynamic Systems (4)
• EE 114. Probability, Random Variables, and Random Processes in Electrical Engineering (4)

Which basically will cover the only mathematics an EE needs to know, in context of electrical engineering. The 146 differential equations series is usually only taken by physicists and mathematiicans.

UVA starts sequestering courses much sooner:

https://rabi.phys.virginia.edu/mySIS/CS2/page.php?Semester=1182&Type=Group&Group=APMA

Beginning in calculus, engineers take different courses.



Not exactly that, it's more that mathematicians teach diffeq courses that aren't geared for engineering, and engineers have their own differential equations courses focused on solving them in that particular context.

I should specify, there is really no "applied math" group at UVA, applied math is part of engineering, so the course name doesn't relate to the department and aren't touched by anyone in the department.

But Rota was taking about lower division calculus and differential equations. Every engineering and physics student takes these.

And, well, there are a hell of a lot more of them than there are math majors.

This, that's why I took that to be a joke.

But yeah, I teach many parts of first order differential equations (no proofs, just methods to solve) in Calculus 2 here, and I think it's kind of pointless. There's no interesting problems to solve using it, and is misleading about what PDE research is like from what I know, where research is just plugging numbers into Matlab.

Edit: that last line is meant to be sarcastic.

Was he? I thought much of the curriculum he mentioned described a year-long upper division differential equations course.

Anyway per Jon's comment and Reid's explanation of how things work at UVA this clearly wildly varys by institution.

The truth is, though, that there are a HUGE number of people who have had to take calculus over the years, and clearly not just engineering. Do you really think that the majority of those courses weren't taught by mathematicians over the years? What did all those extra PhD's do for all those decades?

Don't tell me you've never heard the old canard about your job security as a mathematician (well, with the glut on the PhD market this might no longer hold as it may have in past decades).

Not at all. Everything he talked about in that paper I learned in my sophmore differential equations class. And in the essay, Rota refers to such a course literally as the "sophmore differential equations course". Every physicist and engineer takes this course.

Yes, people in mathematics usually teach calculus. And yes, much of the calculus curriculum is taught very poorly.

I'm not sure what the proper solution to that is. It's not just the fact that mathematicians are poor teachers. Which is often true, don't get me wrong, but I don't think it works well as a sole explanation.

Part of it is probably financial. When the chair of the department goes to the deans for funding, how do you think they're going to present their courses? Which sounds more convincing: "we have a course that inculcates students in the ideas of differential equations," or "we have a course plan that students can learn at this level which teaches X Y and Z"? The latter is going to sound better to a person trying to maximize profits, as it sounds like more material is being taught for less course time, even if it breeds less understanding.

And, from what I understand discussing politics with people at UCR, well I can't get into too much, but the department's not likely to review coursework and update classes anytime soon. Nor would it fly with the higher-ups to fund math courses centered around understanding differential equations.

Studies make it pretty clear that students learn on their own, anyway, classroom teaching has little effect: "good teaching" vs. "bad teaching" isn't the source of student's understanding. Changing the coursework is probably the only way to even change that, but people are pretty restricted.

Unless you're tenured as ****, like M Lapidus, who does whatever the **** he wants in math courses.

If you go back to the essay by Rota, the most essential and non-useless part about the sophmore differential equations is basically an application of linear algebra: the theory of linear first order ordinary differential equations.

It is true that integration is at its heart the solution of a differential equation, but imparting this point would mostly be a matter of culture. I do think it's a good point to make, and it would tie well into applications of integration to physics problems, which of course are differential equations, and, well, are the reason calculus exists. But beyond that? Either you'll need linear algebra for the meat and potatoes of the sophmore differential equations stuff, or, as Rota argued, you'd be venturing into material that should never be taught anyway. So maybe a little goes a long way here.

Yeah, I reviewed an intro differential equations course and you're right.

I see the problem, the only thing I'm not clear about is what a feasible solution would be. All the students know at that level is a usually poor understanding of calc 1 and 2, and to get into any real understanding of differential equations must be preceded by some actual understanding of analysis. If you could get motivated students and the funding, teaching basic analysis into differential equations would be great. But good luck selling that to the profit-maximizing optimization machines up top.

So in other words, teaching lower division math is a cash cow for mathematics, evidenced by the apparent motivation to market it as useful to as wide an audience as possible, even if it betrays the quality of the exposition.

I for one thought I got shafted by a crappy calculus book at my community college. It wasn't even Stewart, but Stewart lite. An elite institution would likely have a separate course for math majors which is taught out of Spivak, which is an analysis book in disguise.

Well, if you take Rota at his word, the solution is to never write a book on lower division differential equations, because then you'll be forced to teach the same obsolete material for the next three decades of your career as a professor.

In all seriousness, the solution is pretty simple: teach the course as a dynamical systems course, and make heavy use of computers and plots.

See John Hubbard's paper, "What it Means to Understand a Differential Equation" on what funnels and fences are (he also expanded this material into a book with Beverly West as his co-author). As I said, professor Pugh at UC Berkeley teaches the differential equations course as a dynamical systems course and skips all the non-sense.

Also, real analysis has nothing to do with the sophmore differential equations course. As Rota said, the existence proofs of ODE's are useless.

If you wanted to introduce analysis, you're better off teaching a bare minimum of asymptomatic series and complex analysis, and then focus heavily on Fourier series, Green's functions, and a little bit on the Dirac Delta function and the Laplace transform. This is exactly what the upper division mathematical methods of physics course does, though.

I'm not really sure what you mean by that, that teaching lower division mathematics is a cash cow for the departments. Mathematics funding is not predicated on the existence of one single lower divison mathematics course. If it was a cash cow, you'd have like, a year long differential equations course at that level. It looks to me like they cut down on how much can be taught so much all in-depth material gets cut.



That's just how it is, everywhere. For one, many kids entering college just left high school. For many of them, getting out of bed on time to attend class is a challenge. Expecting them to be able to keep up with the more intense, more abstract, more challenging rigor of a book like Spivak after being babied in high school is probably too much.

Next, good calculus education is time consuming. The reason professors and departments do online homework is to save money. If you do online grading, it saves the professors/postdocs/grad students' time without having to hire graders. Assigning students good problems will also take time to really grade them, and that's just not possible under the economics.

Basically, that good education, the theoretical one from a good book which would prepare you for everything to come, doesn't seem to me to be feasible under the constraints of a typical mathematician in a university. It seems the problems are more complicated than that.

But even in that course, the mathematical physicist who taught it to us simply threw up his arms when it certainly to the material on branch cuts and analyticity, telling us that we'd need a full course in complex analysis to really do things correctly.

I guess it's about how in depth you go into those topics. Fourier series alone is a topic worth a semester class, so by fitting it into another one you're cutting off key details which makes them harder to understand.

That's why I'd be in support of an "advanced calculus" course to bridge the gap between the analysis behind a typical differential equations series rather than the not very useful methods taught in a normal differential equations course. That way, when Fourier series do come up, the understanding is more thorough.

Also, it's not like mathematics courses are taught poorly, then you step into engineering and things are taught super well. Courses cut corners and skip details all around. Many departments try to stuff as much material into as few course slots as possible, and get out some weird ******* courses as a result.

It's probably just more noticeable in math courses, as mathematics is universally used, so weaknesses in understanding there affect a larger group of people.

What are you talking about here? The lower division mathematics curriculum is two years long, and as far as I know, taught for the most part by mathematicians (although at my community college a couple physicists and electrical engineers taught them as well).

I think you are reading far too deeply into a witty remark by someone bitter about having to teach a course the wrong way to mostly non-math majors simply because he made the mistake of writing a book on the topic.

Such courses used to exist when physics and mathematics were still more closely aligned. In particular, so-called advanced calculus courses of the kind that were popular in the early 20th century included material that mathematicians now teach in second semester analysis and then in differential geometry and complex analysis. Physicists and engineers now have their own upper division courses, as I alluded to.

If you trust Rota, the sophmore differential equations course is pedantic and awful for no good reason at all, other than tradition.

Let me put it this way: you apparently haven't taken it, and I don't think this has hurt you at all. What does this say about the relevance of the material?

Uh, I'm not quite sure all the pieces are here. You complained about using Stewart lite over Spivak. Spivak is not a book engineers would find useful. It's really rigorous and introduces topics in a more analytic framework, which makes it super useful for people who want to study rigorous, proof-based mathematics, but that's unimportant for engineers.

In books like Stewart, in fact, much of the reason it's poorly done is it goes out of it's way so hard to make connections with engineering. For instance, we spend a week here teaching volumes/surfaces of revolution. Those aren't interesting from a mathematical point of view, we teach them because physicists use it, and over the years they demanded mathematicians teach it, so it got stuck in the curriculum, and now it's taught to everyone, including computer scientists who will never use it in any context, ever. If they could cut material like that, and instead focus on understanding stuff like limits, then much of the later material would be easier to understand.



Keep pumping away at that thesis. My problem with undergrad math education isn't that it's not tied enough to physics, it's the opposite, it's that too many demands have been made from various departments to teach random, disconnected crap, and that dilutes the key mathematical understanding that's actually relevant.

I did take it, I just don't remember it. And I have agreed with that point already, quite a bit: yes, I think the lower division differential equations course is pointless. I'm not sure what that's supposed to imply, though.

Engineering program accreditation is strict. They're required to take more courses per term than other students, the topics covered by the courses are regulated and even the credits earned by taking those courses is regulated. Engineering faculties teach their own "mathy" courses because the students literally do not have enough time to learn the same material that's taught to science undergraduates.

I have no firsthand knowledge of whether those courses teach mathematics better than the non-engineering courses, but that's why things are done that way.

fwiw I think volumes of revolution is a good intuitive example of how integration can be used in practice. I think it's a good idea to spend 20, maybe 40 minutes on them.

I strongly believe that rigorous, proof-based mathematics should begin as early as possible, at least during high school but ideally even sooner. Unfortunately factory schools have proven to be an exceptionally poor mode for this subject. (For example, the way American high schools teach geometric proofs is a horror show.)

I agree, I think if I actually explain my ideal course plan, it would help.

The key to surfaces of revolution is the idea of arc length. We know the formula, it's given by the integral of √(1+(f'(x))^2) over x. Students can memorize that easily. But do they get why the formula makes sense?

**** no. There's an easy way to explain it, too, with infinitesimals. If we instead work with two new symbols: let ∆x represent some arbitrarily small value, which symbolizes only a change in distance along the x axis. Then we can find the change in y by plugging in a point x_0 and x_0+∆x. This gives us a small straight line between two points on a curve, given by two sets of coordinates: (x_0, f(x_0)) for the first, and (x_0+∆x, f(x_0+∆x)) for the second. If we let ∆ get really small, this line gets infinitely close to the curve. Thus, so will it's length. But what would the length be? That's given by a^2+b^c=c^2. How much you change on x with respect to x is given by ∆x/∆x=1, and for y with ∆f(x)/x, so for pythagoras' identity we substitute those in.

In other words, but cutting up the x-axis into arbitrarily small increments, we can infinitely closely approximate the distance along the curve. There's a strong geometric intuition to it, and I think teaching it this way would make it so clear why it works.

It would be possible to teach limits this way to start, then supplement it later with epsilon-delta proofs to make it rigorous.

Combining this strategy with volumes/surfaces of revolution makes each formula seem trivial, as well. How students seem to get by is trying each style and memorizing the formula. Because this arigorous intuition for WHY it works isn't there. A bit more of that, and a bit less of forcing students to memorize solutions for things they don't understand, would go a long way in calculus.

Also, I think "intuitive" is relative. You're brighter, so it's probably easier for you to see why they're so intuitive. Not all students are that way, and if the goal is to educate all students as well as possible instead of as diversely as possible, helping build strong intuition then covering it with rigorous understanding is IMO a preferred route.



Agreed, it would also help if students came from a background of mathematical reasoning.

And yes, many mathematicians would shudder that I just asserted ∆y/∆x≈dy/dx. That's fine, and ideally starting from ε-δ would be ideal. But, the way Americans are taught, they're not ready to understand that definition. But ****, anything is better than the random crap we throw at students with formulas to memorize.

I think professors are worried about teaching intuition from non-standard analysis specifically because it's more intuitive. Like it's a crutch, students (particularly weaker students) will never let it go.

Usually what happens in calculus at a better school is: the professor will teach ε-δ. Everybody's eyes glaze over, except for a couple kids, usually in the front row, who scratch furiously but don't get it. Later, the front row kids will go home and think about it, and it will make sense for some fleeting moments, but will still escape them. That's fine, though, they've learned something even if they don't know it.

The rest of the students won't think. They'll speak to sophomores, who tell them they don't need to know it. So they forget it, and just pay attention to the "important part": at good schools they might know the limit definition of a derivative, which they can use + some poor reasoning to make some actual calculations. At bad schools, they don't even learn that, fail the limit definition problem on the exam, and just memorize the "important part": the rules of differentiation as though they're handed down by God himself. They then skate their way through the rest of the course by memorization.

I suppose that's true. I'd be open to just teaching ε-δ I suppose, if there was any way to force undergrads to think hard on a topic for longer than five minutes. Anything really is better than handing down the formula from the Gods.

I have no idea either, but I didn't know that about engineering accreditation.

if it fits, i sits

Actually, the way I phrased it, in terms of x+∆x could entirely skip the word "infinitesimal" and be phrased in terms of limits of Riemann sums, making it rigorous (since we know all continuous functions are Riemann integrable).

But I guess that gets back to another point: integration, at the undergraduate level, is entirely formulated in terms of Riemann integrals, in analysis you prove the FTC as a consequence. If students understood that integration isn't a magic formula, but the limit of taking smaller slivers, I suppose that's what I mean by saying the arc length formula can be made intuitive.

But in order to even phrase things that way, you must have: series and limits, both of which students have a hard time dealing with.

I swear to god Reid, every single time on this message board in which I desperately attempt to get you back on track from your initial misreading of a post, you just post more and more endless disagreement on stuff that is completely secondary to the original point.