Should multivariable calculus be a two- rather than one- semester course?

Question

Single variable calculus is typically (and reasonably) taught over a whole year, with the first semester being devoted to "differential" calculus, and the second semester being devoted to "integral" calculus.

In my own experience, "multivariable" calculus is taught in one semester. That is a course with vector calculus, partial derivatives, gradient and the chain rule. Then onto line integrals, multiple integrals, and Green's Gauss' and Stokes' theorems. That's a lot for one semester.

Would it make sense to break "multivariable up into a two semester course, differential and integral, as with single variable calculus?

That would be a "differential" course with vector calculus, partial derivatives, gradient and the chain rule, Taylor series, constrained optimization, matrices, mappings, and determinants, plus the Jacobian in the first semester. And an "integral" course with improper integration, power series, change of variables, line integrals, multiple integrals, Fourier integrals and Green's, Gauss' and Stokes' theorems in the second semester.

Answer 1

I believe it is true that teaching the material in more time would be better. However, there are a lot of useful things for students to do with their time.

In particular, there will be a lot of pressure from your Engineering faculty against your idea. They might propose the opposite of your plan! -- more calculus material squeezed into fewer credit hours. For example, at the University of Illinois, the standard calculus sequence is:

• 5 credits: Math 220 (Differential Calculus and many Integrals and Applications)
• 3 credits: Math 231 (Integration Techniques, Sequences & Series)
• 4 credits: Math 241 (Multivariable as you described)

However, this 15-credit sequence was becoming too burdensome to fit into Engineering degrees, so a special accelerated intro calculus course was created to make the engineering calculus sequence only 14 credits.

They are not evil!! and their intentions are extremely good -- the engineering people have a lot of things that they want their students to take, and it seems reasonable to them to compress first-semester calculus a bit in order to make room for another credit hour of other valuable things. Many of their students have AP credit anyway, and many of their students are fully able to succeed in the accelerated intro course.

Personally I agree with you, that something like 18 credits for all this material might be better than 15. However, the math instructor's entirely legitimate desire to teach more credits of math will come into direct conflict with everyone else's entirely legitimate desire to teach more credits of everything else. This will be the primary thing that will stop you from expanding multivariable calculus into more semesters.

One possibility might be to teach a two-semester sequence that combines multivariable calculus and the generally-boring linear algebra class into a nice soup. This kind of approach might avoid much of the counterpush by other interests.

Answer 2

At my University for engineering Calc I is two or three weeks of differential calculus while Calc II is multivariable and then we have a differential equations course. Not all engineers are required to take differential equations but most (possibly all) engineering majors have to take linear algebra. It used to be that Calc I was the weed out course now Calc II is. Either way adding more semesters to this sequence would lead to problems with prerequisites for major related classes that require these maths. So I guess my answer is that in the context of an engineering school slowing down this sequence will lead to a domino effect slowing down major matriculation. Anyone taking multi variable calculus should be taking it with an eye toward it being specifically useful for some other course or further math study and as a result it is a probably a class taken only by those with an aptitude for math.

Answer 3

At Columbia University, "calculus" goes from Calculus I to Calculus IV, along the lines I've described. At Carnegie Mellon University, there is a course between Calculus II and Calculus III called "Integration and Approximation," that covers advanced integration techniques (e.g partial fractions), improper integration, Taylor and Power Series, Simpson's Rule, introductory differential equations, and Euler's method. Call it "Calculus 2.5," but it effectively lengthens the calculus sequence to four courses.

At some of the state schools, e.g. Georgia Tech and the University of Pittsburgh, Calculus I, II, and III are (reasonably) four- rather than three- credit courses. In this case, it makes sense to break them up into four three credit courses instead.

Carnegie Mellon does have a 17 course (51 credit) math major, so it comes at the expense of humanities and social science courses. Their engineering program allows for lot of "technical" electives, so the greater emphasis on math comes at the expense of "traditional" engineering for these students.

Answer 4

(1) If you add more time, what do you take away? At least start to address this issue.

(2) I agree that there is a huge amount of material covered in multivariable calc, but I fear that your solution of two semesters (adding lots of extra content!) is not even addressing the issue in terms of developing the topics themselves with more practice and familiarity. In other words, you are adding more time, but also more content. How can I tell if this is better/worse? (2 unknowns with one equation.) Why not do something simple like adding more time for the same topics? At least then I know the directionality.

(3) The two quarter idea is interesting in that it allows performing the idea in (2) [deeper practice with unfamiliar and less intuitive topics like line integrals and div/grad/curl and all that, but not adding a bunch of extra content in.] The question then becomes what do you do with the rest of the calc sequence? ODE really could use 2 quarters for the same rationale. If you have advanced students (above average, exposed to strong pre-calc), you could do some quarter sequence like (1) differential calc (2) integral calc (3) calc 3 part 1 (4) calc 3 part 2 (5) ODE part 1 (6) ODE part 2. The extra time for calc 3 and ODE is nice. Disadvantages would be students that really don't know regular calc that well (after all they did not place out of it!) Also, you stretch multivariable calc over a summer. (that said, there are disparate topics.)

(4) I actually think the fundamental topics of single variable calc (and to a lesser extent diffy Qs) are more important to STEM students. Perhaps covering multivariable calc in some less satisfactory manner is sufficient. Sort of how LaPlace transform is covered in a short manner...so that students have at least seen it...but will need to learn deeper within the context of whatever EE or control systems course needs it later. In other words, not really developing competence (as we do with integration methods), but familiarity.

(4.5) The main need for multivariable calc is in E&M. However, the standard physics 1 E&M basically uses line integrals and not that much else (yes, you may vaguely touch a little of it when hitting Maxwell's equations, but line integrals are the main thing for problem solving). It's not ideal, but probably sufficient for the majority of STEM majors (who all agree that first semester physics is like high school warmed over and then second semester is a pain in the butt...but you get through it and move on to other things.) Physics majors really do have a need to do more applied vector calculus, but this is covered very hard (get lots of review AND practice) in the context of junior year E&M, ITSELF. For instance, look at the first chapter of Wangsness (it's one ~80 page vector calc lesson with almost no physics).

Answer 5

I think that would be redundant. I took Calculus 3 in four weeks and I had no difficulty, neither did any other of my classmates.

Does it really take three weeks to teach students about dot and cross products? My colleague does. Calculus 3 is usually a 200-level course, and by then students should not need any coddling by artifically extending the material. I believe that even in a quarter system, Calculus 3 should still be a one term course, not two. Most students have probably taken linear algebra before or concurrently with Calculus 3.

I know someone that teaches Calculus “3.5”, and they fill the remaining few weeks of class with topics of Taylor polynomials and complex differentiation and integration.

Answer 6

I prefer 1, 4 credit course, but 2, 3 credit courses is fine, especially at the Community College level. What I don't like is when Community Colleges, like in Connecticut, say they are teaching the 4 credit version, but in fact only teach 3 credits worth. They somehow forget about Line Integrals and Surface Integrals, even though it is clearly included in the course descriptions. Why? Put another way, "Why would any College not teach the Fundamental Theorems of Multivariable Calculus in any course named Multivariable Calculus? I've been asking this question to Connecticut Department of Higher Education "officials" and Community College Presidents for a year now. No answer. Shame on them all. Professor Steven J Toce