# Is there any difference between teaching calculus for math and engineering students?

In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems based on the students' needs in their future career they need different types of calculus. For example in a calculus class for the math students, the teacher should emphasize on the abstract aspects, proofs, generalizations and the interactions with the other fields of mathematics and in a calculus class for engineering students, the applications and calculation are more important features of the course.

Question. Is there any difference in teaching calculus for the math and engineering students? Should math and engineering students attend in separated calculus classes?

• One might also argue that engineering graduates should be able to perform rigorous mathematical proofs in calculus even though that is not a major vocational skill for them. If the engineering department does argue that, then it makes perfect sense to share the course with the mathematicians, the argument for a different emphasis disappears. So maybe the departments should independently describe their preferred calculus syllabus. The similarities/differences between the two will help tell you whether the students should be in the same class, and how much compromise is needed if so. – Steve Jessop Mar 27 '14 at 11:58

In my understanding, multiple representations of a concept benefit conceptual understanding. Hopefully the differences among departments can be negotiated to benefit students. Maybe a question to discuss is how we differentiate instruction for diverse learners in a single classroom. After all, even in a pure mathematics classroom all learners are at different places with their prior knowledge and different potential for learning. So philosophically and in my opinion having students from engineering and mathematics (and physics, if you like) in the same classroom can benefit both groups. It does require competent teacher to be able to handle such class.

I do recognize significance of different contexts and different opinions, so, in reality some negotiations between departments have to happen.

A paper that might be of interest:

Dunn, J. W., & Barbanel, J. (2000). One model for an integrated math/physics course focusing on electricity and magnetism and related calculus topics. American Journal of Physics, 68(8), 749-757.

Some references (although mostly focused on k-12) in support of this kind of thinking:

1. Ellis, E., Gable, R. A., Gregg, M., & Rock, M. L. (2008). REACH: A framework for differentiating classroom instruction. Preventing School Failure, 52(2), 31-47.
2. Nunley, K. (2004). Layered curriculum (2nd ed.). Amherst, NH: Brains.org
• Mara, Thank you very much for your interesting answer and useful references. – user230 Mar 29 '14 at 8:21
• +1 for your interesting answer. For a physical version of this discussion see matheducators.stackexchange.com/questions/10297/… – Ali Taghavi Dec 25 '15 at 14:26

Tentatively, yes.

In the universities I've taught, there are substantial differences between those courses:

For engineering students, the courses in calculus (and linear algebra) provide some supplemental, rather abstract information which helps to understand the techniques taught in other classes, such as mechanics and electrodynamics. This is why I think that a good calculus course for engineering students has a curriculum which is coordinated with the engineering classes, uses a lot of examples from those subjects and is light on hard proofs (uses motivations and rough explanations instead).

Calculus for mathematicians is a main subject on its own; it should be thorough and you can afford not to be in tune with some other classes which cover related subjects. Proofs should be precise, but examples can still help to understand the theorems.

It's worth noting that students of physics deserve their own calculus curriculum as well due the use of more advanced tools in modern theoretical physics compared to the engineering classes.

There is a danger that lies in tailoring a class too much towards a major and it is that it won't broaden the students' worldviews. What follows a calculus course is years of specialization. Why start early when the students will see all this stuff anyway in their degree? Take an engineer for example. They'll be doing engineering and speaking with their engineering peers and professors for several years. A more abstract calculus course can be a great opportunity to build strong foundations for those engineering students and this can only help them later on.

This doesn't mean there shouldn't be an tailoring towards particular fields. More so, I believe that calculus courses should be taught the same way regardless of specializations. Teach everyone rigorously (as possible for such a level) and introduce examples from various fields. Most of the biology, business or engineering students I've tutored suffer from poor understanding of the fundamental calculus concepts and not of how they apply to their field. Once we cover those basics they usually have no problem with the applications.

Perhaps the ideal homework would contain enough abstract examples to teach the concepts, followed by problems from various fields with an emphasis on the students' specialization.

Sure. For example, engineers don't mess with fields apart from the real numbers. They only mess with quite particular Hilbert spaces. They don't mess with Galois fields apart from GF(2) in communication and coding theory. They don't work quaternions or generally non-commutative groups.

They'll not even learn the tools for computational analysis (like generating functions and everything around them). If they do use generating functions, their variable is called $z^{-1}$. If they use Fourier transforms, their domain variable is called $2\pi f$. When they work with integral transforms, convergence is assumed. They learn the formulas for main tension values, main inertial axes, and transformations of quadratic forms separately.

Number theory is not covered, tensors are mostly left to the physicists to play with.

For three-phase electric systems, they have transforms used for diagonalizing circular impendance matrices without recognizing those transforms as a 3-point DFT.

In general, a lot of stuff is learnt multiple times under separate names, with separate scaling and naming constants and sign conventions.

Proofs are not really all that important. It's more important for the engineers to be able to apply mathematical tools rather than to create them: a carpenter only needs rather limited knowledge of plant biology.

• How does this affect calculus classes? That was the question. – user173 Mar 27 '14 at 17:36
• For example, engineers don't mess with fields apart from the real numbers. They use complex numbers very frequently. – Ben Crowell Jun 1 '17 at 14:06

I think a standard sort of tricks and techniques course is fine for the math people. A few proofs can be included but more for motivation and understanding than...well proof. Or even for learning how to prove. After all, the math majors will have theoretical calc later (real analysis).

Also numbers wise, there are way more engineers and scientists than math majors. Their needs should be taken care of more. And then students move around anyways amongst majors. So a typical Granville style calculus class is fine.

• Please don't create multiple accounts. – Dag Oskar Madsen Jun 11 '17 at 22:12

The difference between math students and "the rest of us" is quite real and ought to be acknowledged: the mathematician's goal is to prove a statement from already proven statements while the rest of us try to understand the statement sufficiently well to see what it says about the real world. (You might call that a model-theoretic viewpoint -- not to be confused with, and as opposed to, the modelling approach.)

The math-teaching community, though, is loath to acknowledge said distinction, let alone to do anything about it. For possibly why, see, for instance, Keisler's comments in Section 13. Institutional Difficulties with Teaching Infinitesimals of Vinsonhaler's Teaching Calculus with Infinitesimals. And for another possible approach, Laurent Polynomial Approximations, a special kind of Asymptotic Expansions, see my To Calculate in Calculus.

• Is the proof not important to the understanding? I would argue that while a mathematician will write a proof, it is still important for others to read (and understand, and observe the existence of) the proofs. – Daniel R. Collins May 31 '17 at 18:50
• You must be a mathematician! The answer is "definitely not": I don't need any proof to know that you strongly disapprove of my views. There is a huge difference between a proof and a convincing argument. "The rest of us" operates on the basis of arguments. See The Uses of Argument by Stephen E. Toulmin. – schremmer May 31 '17 at 21:09
• What? I thought a proof was a convincing argument. What is convincing naturally depends on who you're trying to convince. So, the standards for proof with certain engineering students is quite low. As a point of mathematical ethics, we try to help them with this. – James S. Cook Jun 1 '17 at 1:50
• A proof is a convincing argument ... for the cognoscenti. But a convincing argument definitely need not be a proof. Re. "standards for proof": would you have First Graders start with Peano's axioms? May I also remind you that physicists had been using distributions a long time before Laurent Schwartz made them legal---and couldn't care less that he did. Are you saying that what was enough for the best minds of the XVIIIth and the first half of the XIXth century would "with certain engineering students [be] quite low"? And, finally, is it really that ethical to impose one's standards on others? – schremmer Jun 1 '17 at 2:53