I have to disagree with this. I’m in my 3rd year of electrical engineering and my 2nd year of physics. I’m terrible at calculus. However, its not because of a lack of understanding of the concepts, its because not enough time is spent teaching the mechanics. Its all theory, theory, theory with very little time devoted to going over how to actual DO the work. IMO, more time should indeed be spent on the “tricks”.
For engineers, absolutely. You can take math classes if you want the theory behind the tricks (and you should be familiar with it).
Obviously, that’s not good for the theory crowd.
For engineers, absolutely. You can take math classes if you want the theory behind the tricks (and you should be familiar with it).
Obviously, that’s not good for the theory crowd.
Exactly. Let the engineers teach the tricks in their courses and let the mathematicians teach mathematics.
In fact, about a hundred years ago this is exactly what happened. Physicists taught their own calculus courses and told their students to to pay attention to mathematicians and their theory. I mean, the mathematicians were going on about such insane things as “differential manifolds” and “Hilbert-space operators”, which couldn’t possibly have anything to do with the sensible real world of physics…
Oh, wait…
Oh Yes. Hard to explain to someone who has not experieanced the Ah Ha aspect! I now understand why the earth is round, why time is money, why there are limits on possibilties, why I must save for my children’s education and my retirement. The universe is finite, but infinte at the same time. Hard to grasp a concept.
Maybe GW should take (again?) a course in Calculus. :wally
In my experience, those engineers who took “engineering physics” (usually taught by GTAs) were far less versed and less prepared for upper division classes requiring physics knowledge than those who took the A&S physics courses. I advised my then-girlfriend to take “real” physics on her second go-around (and also, to retake Calc II, even though she passed it with a D the first time), and despite her limitations as a student she was far ahead of other students in her classes.
Ditto for math; engineers and physicists (and other scientists as well) really need to understand how the tools they are using fundamentally work. OTOH, I think that (in the case of physics) an applied mathematics class–demonstrating applications of these tools–is an extremely valuable supplement to traditional math classes. In my physics curriculum, for instance, we had an “advanced physical mathematics” class (I forget the exact name.) I didn’t take it 'cause I was a semester out of sequence with everyone else, but I did look through the syllabus, which showed it covering in detail things like Fourier and Z transforms, numerical simulation using the FORTRAN libraries, tensor applications, differentation under the integral, et cetera, i.e. things that aren’t normally covered well in math classes, and that usually end up taking a significant amount of time in standard physics classes like A-Bomb and Electricity & Masochism. (Sorry for the sophomorisms, Mathochist. 
)
The value isn’t just in knowing the tools, either, but in learning the process of logical thinking. It’s a pity there isn’t a specific class on solving proofs; for all that proofs are hated (even by math students) they produce a way of approaching problems that is rigorous and critical. These processes (should) apply equally to other areas of science, as well. IMHO, every engineer and scientist (and frankly, any college graduate in the liberal arts or sciences) should be required to take an introductory logic course. The world would be far better off if more people were educated in thinking skills.
Stranger
Strangers post has merit, to a point. Calculus concepts are not hard to grasp. It’s just in how you are taught to apply the concept. Engineers are taught (basically by rote, hopeing that they grasp) to do certain functions. But the beauty is in understanding the big picture.
It’s like most everything in life. The more you understand, the deeper you grasp its true meaning. You almost have to study it to death to find out that it is truely simple. That’s why universities make people study it.
Simply put, teach your child to add and subtract. They will thank you later! 
Well, there was more than a tinge of sarcasm in my answer. To be more precise, classes in calculus are for the theory and engineers and physicists should take them and take them from the mathematicians (in case anyone didn’t get it, the two fields I mentioned are the underpinnings of general relativity and quantum mechanics, respectively). The engineers should teach the tricks in their own engineering courses, assuming their students have learned the general theory from the mathematicians.
Oh, and don’t knock grad students’ teaching abilities a priori. They’re often more motivated and more interested in teaching than any professor is.
Well, as the math departments see it, there is: advanced calculus. Sometimes it’s called “analysis”, but in general it’s the second pass through a calculus sequence at the late-sophomore or junior level. This time everything gets proved and it’s intended to cement the notion of proof, which the students have been slowly acclimated to by seeing more and more done here and there through the classes leading up.
Alternately, most computer science departments’ upper-level algorithms classes accomplish the same goal: a revision and extension of earlier material provides an opportunity to learn how to do proofs on material the student already knows the general shape of.
True. I had a professor for Circuits for Non-Believers (a big lecture hall class for all non-EE students) who was a rambling, drunken malcontent who often lectured on information that was far and wide from the class material (read: bizarre conspiracy theories.) On several occasions, he failed to make it to lecture, or was extremely tardy, which was a great benefit to everyone because then one of his TAs would teach, and teach well. One time, when the prof walked in halfway through the TAs lecture and started to take over, half the lecture hall booed him. Tres amusing.
Stop it! You’re making me want to go back to school and get all the stuff I missed the first time!
Stranger
I am very glad I found this thread. There have been insightful remarks all around. I would add some of my own experiences, however gratuitous they probably are.
I absolutely agree. I took such a class in grad school. Mathematics for Political Scientists was a great hulking mongrel of a class: we covered proofs and proof strategies, set and number theory, calculus (with an emphasis on optimization problems), probability, and almost certainly a few areas that I have already forgotten. Its goal was to acquaint non-mathematicians with the theory and toolbox typically used by economists and quantitative social scientists, and was a prerequisite for some of the advanced formal theory and game theory classes.
In many respects it was far below the level of an intermediate undergrad math class. Since my mathematical background was limited to AP Calculus AB in high school five years before, it was a bit of a reach for me.
The value I derived from this class is exactly as you described above, Stranger. Knowing the tools is helpful, but learning how to attack a problem with structure and quantitative rigor was an intellectual awakening. In my current field, I use econometric solutions to solve problems that require very little mathematical heavy lifting. Nevertheless, because of this class, I am motivated to take night courses in calculus, probability theory, and even physics. I found that I enjoyed slogging through mathematical writing, and definitely miss it now that I finished my degree and am a working stiff.
My teacher emphasized teaching mathematics as a language and fashioned her pedagogy accordingly. That’s what lit my internal burner. I was never a mathematician: I was in the advanced track in high school, but I did not really distinguish myself. I always reified mathematics and believed that one’s intellect was either hardwired for science and mathematics of for language and literature. My path was the latter, where I was able to do better work than I have ever done in a quantitative field. So amid much kicking and screaming, I applied some of my language learning techniques to mathematics and was surprised at the outcome. By concentrating on learning math as a language used for crafting arguments and proving results, I was able to make a great deal of progress.
Internalizing the principles of calculus was the catalyst for my own development. It became much more than a tool to find the areas of oddly shaped solids, but a useful way of describing the world. I do not think that any other area of math has had such an influence (save probability, which calculus utterly pervades) on the way I apprehend phenomena. Its pedagogical importance, nay, exaltation makes sense to me in this context.
I think all you people who like and enjoy pure maths are getting caught up in your own interests.
Sure, I don’t want to just plug and chug to solve problems. But if we’re being marked on our ability to solve problems, yet not taught how to solve problems, well that can make being a math student (of any variation) very tedious.
No one is suggesting that you not be taught to solve problems. We’re just saying that you need to understand the theory behind the problem solving.
There’s a difference between being taught how to do the basic calculations as applications of the theory and the sort of tricks I’m talking about engineers and physicists using.
A calculus sequence teaches what the integral of a function is and the basic technique of antidifferentiation. It even goes into a fairly large number of detailed techniques for various situations – disks/washers, cylindrical shells, coordinate transforms – and a moderately deep study of functions like hyperbolic trig functions that almost never show up except in applications. We’re already bending over backwards. What we’re not here to do is give more than mild samples of how these tools get applied to your work. If a calculus professor is asking you to solve engineering problems without teaching you how, complain to your heart’s content. I doubt this is happening anywhere, though.