26
$\begingroup$

When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 years ago.

But when teaching a calculus class for a mix of physics, engineering, economics, business, pre-med, and miscellaneous students, I lack anything resembling this clarity. On the one hand, I can teach them how to solve computational problems, but the majority of these students (at least at the two universities where I have taught) have had calculus in high school and already know how to do the computations, making this painfully boring for them and for me. On the other hand, I could prove theorems. Experience shows that the vast majority of students are not interested in proofs, and it's hard to make the argument that knowing how to prove the Mean Value Theorem will help them in their future non-mathematical careers.

Of course, students can potentially benefit from learning proofs because through this, they will learn how to think in a new way. However, the same could be said of learning almost any academic subject, so this does not help us with the question at hand: what material to teach, how to teach it, and what to emphasize.

If I could succeed in convincing my students that proofs were interesting and/or fun, I could make the class a worthwhile experience for them. But I find that this is a losing battle for the vast majority of students, even bright ones. There will usually be one or two students in a class of 50 who become (or were already) interested in the theoretical side of things, but catering the class to the interests of such a small minority seems like a bad choice. Again, with a class for math majors, I don't have to worry as much about the students' interests (though it's always good to keep in mind) because I know what they need to know, and why. I have no such "moral high ground" with the proof of the Mean Value Theorem.

The standard answer is to ride the fence between the two extremes (computations vs. proofs) by focusing on concepts while mostly avoiding rigorous proofs. I have found it very difficult to adopt this attitude in practice, falling most of the time into one extreme or the other. The one useful concept that I instill in all my students is: stop and think about a problem before trying to compute the answer. I accomplish this by showing them cases where the standard methods fail, and giving them such problems on exams. Most students absorb this lesson, and that's fine. But other than that, I feel I have nothing useful to tell these students, and judging by student feedback on evaluations, they feel the same way. (Which my colleagues all tell me not to worry about, because evaluations are typically mediocre at best for freshman calculus courses. But couldn't this be indicative of a general problem?)

To try to make this question more focused: a good answer would give a big-picture idea of what benefits non-math majors (in particular, those who have already learned the computational methods in high school) can take away from calculus, and/or specific examples of concepts that are "just right" for emphasis in a calculus class, and good ways to teach these concepts.

There is a related question here: How to make Calculus II seem motivated, interesting, and useful? But I am more interested in what should actually be taught, and with what emphasis, than in motivating the material per se.

After posting, I noticed the closely related question The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects. My question is more practical and specific: what concepts, or what type of concepts, can and should non-math majors take from calculus?

$\endgroup$
  • $\begingroup$ A similar question, but from the student's point of view: matheducators.stackexchange.com/questions/5586 $\endgroup$ – Jasper Sep 8 '15 at 5:12
  • $\begingroup$ Most people who have had no more mathematics than first- and second-year calculus and a few more advanced courses have never suspected that mathematics is a field in which, every day while learning it, one is working on understanding why things in mathematics are as they are, rather than just learning dogmas. That includes most have have straight "A+"s all the way through high school, and by high school I mean all of the prerequisites to first-year calculus. And that is something they'd know if those high-school courses were taught honestly. The reason they're not taught honestly is..... $\endgroup$ – Michael Hardy Mar 30 at 21:10
  • $\begingroup$ ....that people have been slow to understand the empirical reductio-ad-absurdum of the false proposition that the way to foster understanding and competence in mathematics is to coerce all high-school students to prepare to take calculus later. Professors who've had their noses rubbed in that particular blindingly obvious proof by contradiction every day for fifty years or more haven't noticed it yet. $\endgroup$ – Michael Hardy Mar 30 at 21:13
10
$\begingroup$

I attended an interesting talk by Timothy Guasco, who teaches physical chemistry at a US college. He's reduced some of the high demands for mathematical skills in PChem by displacing some math derivations with computation and modeling. Then he needs the students to understand the concepts of calculus, but they do not need to be adept at calculating. E.g., they need to be able to understand perturbation theory—partial derivatives and $\nabla$—while relying on computation after the model is detailed.

It could well be that this tradeoff between mathematical derivations and computation will be more widespread in the future. If so, it suggests that understanding the concepts of calculus is more important than the ability to solve calculus problems or prove theorems.

$\endgroup$
10
$\begingroup$

I'll address one piece of your question: how to think about proofs in a class for non-majors. I've tried to approach this by assuming that, whether or not a proof is good for the students, they haven't learned how to read and understand proofs, and can't be expected to do so on their own.

So any time I include a proof, I need to not only teach them the proof, but teach them how to think about this particular proof. Assuming the students are comfortable with calculations, that means only worrying about the proofs that are one conceptual step above the calculations, so that I can explain its value by pointing back to the calculations. Usually these end up being proofs where either the proof is mostly a worthwhile calculation itself ("this is a good example of the method we just learned...") or where the proof illustrates the use of an assumption. If I can't explain the purpose of the proof concretely ("Well, this proof helps us understand why...") then it's too abstract.

$\endgroup$
  • 2
    $\begingroup$ It would be interesting to see an example (or sketch) of a proposition/proof and the sort of accompanying explanation you'd give for its purpose... $\endgroup$ – Benjamin Dickman Sep 3 '15 at 22:28
  • 1
    $\begingroup$ @BenjaminDickman: From my current class (linear algebra): next week we'll do the proof that an upper triangular matrix has determinant equal to the product of the diagonal elements. The justification is that it illustrates looking at the determinant as a sum of components and analyzing the components individually. $\endgroup$ – Henry Towsner Sep 4 '15 at 0:22
  • 1
    $\begingroup$ One more example that illustrates the other big class: the week after we'll do the proof that the solutions to a homogeneous linear equation forms a subspace. The explanation is that the proof illustrates why the assumption of homogeneity is necessary, because it's easy to see all the steps fail with non-homogeneous equations. $\endgroup$ – Henry Towsner Sep 5 '15 at 14:45
7
$\begingroup$

What does an engineer need to get out of their calculus classes?

  • The Reynolds Transport Theorem. All of their Physics, Chemistry, and Engineering classes have lots of hard conservation law problems. Teach them what all of those problems have in common, and their lives will be much easier.

  • How to set up a story problem -- drawing a picture, labelling axes, directions, and dimensions, identifying knowns and unknowns.

  • Comfort recognizing field conditions and boundary conditions.

  • Comfort working with variables. It is much better to be able to derive an answer using variables, than to only be able to punch numbers into a calculator.

  • Comfort using units and dimensional analysis. (For example, distances in meters, speeds in meters per second, accelerations in meters per second per second, jerks in …)

  • How to check their work.

  • Significant figures and error analysis. How accurate are the inputs? Does the algorithm mitigate or enhance errors? How accurate are the outputs?

  • How to clearly explain an answer to someone who never took a calculus class.

$\endgroup$
6
$\begingroup$

Engineers and physical scientists need to know a lot of maths, even if just to sensibly drive the software they will use in professional practice. However engineers (whom I teach) are driven by the need to design and make things rather than to understand things. If I have to pray to the full moon to get something to work, I'll do so and worry about the theory later. This is, I think, the wrong way round for mathematicians. So, to motivate engineers you must quickly relate the mathematics to a real-world application with which they are familiar. This tends to mean going from the particular to the general, which is again (I think) the wrong way round for mathematicians.

Things that will make engineering students' eyes glaze over:

  • 'here are some axioms/definitions - let's see what follows and then we'll examine an application'. This is the wrong way round for engineers and '...and it's applications' textbooks often fall into this trap.
  • 'a tensor is a 'mapping from a vector space to itself' or even 'something that transforms according to...'. I introduce tensors as 'a thing like stress', which they know from Mechanics 1, at least in the uniaxial case.
  • eigenvalues are not 'solutions of the equation $A\lambda=v\lambda$'. They are (the squares of) natural vibration frequencies.

We could write a book about this. Maybe later!

$\endgroup$
  • 3
    $\begingroup$ I'm not sure where you went to university, but if you explain eigenvalues as "the squares of natural vibration frequencies" you will have completely lost most people in the class. $\endgroup$ – user5108 Sep 8 '15 at 16:47
  • 1
    $\begingroup$ Having shown the natural frequency of a 1-DoF mass-spring, I simply say that "Vibration is an example of a class of problem known as an eigenvalue problem...<snip>...natural frequencies are sometimes expressed...<snip>... as eigenvalues , where...": This introduces the word into students technical vocabulary in a simple and familiar context. I return to eigenvalues in several other practical engineering contexts (buckling, plasticity, fracture mechanics, etc). I don't find that students get lost. $\endgroup$ – rdt2 Sep 13 '15 at 14:58
5
$\begingroup$

This depends hugely on what calculus class you are teaching. I'll speak to a non-major-focused calculus class (as opposed to non-majors in some other calc class).

  • Idea that we can get a huge amount of information about functions' behavior from calculus ideas.
  • The concept of marginality - not necessarily using that word, though it's great if you have econ/business majors. This is far more fundamental than we really understand, in my opinion.
  • Totals and rates are inverse operations, in some sense.
  • Assuming you have people with the right interests, that even simple models using rates (derivatives) can be useful (either symbolic or numerical integration, which doesn't have to be as intense as Runge-Kutta but could be essentially Euler's method in a spreadsheet). Models from fisheries to ice sheets to adoption of technology are all appropriate.
  • The notion that how to talk about infinity might matter. (Cue their intro philosophy class and Aristotle or Hobbes, a theology class and Aquinas, or a history class and Galileo.) Everyone seems to have something to say about Zeno, even if not everyone thinks it's solved by calculus.

(Which my colleagues all tell me not to worry about, because evaluations are typically mediocre at best for freshman calculus courses. But couldn't this be indicative of a general problem?)

Yes. Calculus is a course looking for a reason to exist, in many contexts - it's more of a gatekeeper than anything else. If your students don't know why they should care about functions once they are done memorizing them for an exam, it's trouble. That's why the modeling idea is key, though unfortunately not very easy to implement. See the Manga guide to calculus for some very interesting ones that unfortunately probably won't make sense to your students until after they've actually had calculus.

$\endgroup$
0
$\begingroup$

If you replace computations with proofs (since the students don't like it, haven't had it), than you are substituting beets for spinach. ;-) (Worse for bad, if you didn't get it.)

If the students did not validate the course than they have problems with the material. For those who can easily learn the computations, fine, it will be an easy A. But I would be much more wary of kids who NEED more practice since they struggled the first time through. Your counsel to them should be very much on the need to practice, to stay up to speed, and to NOT get overconfident because it is easy at first.

In class and after class drill is what the students need. If they are going to be engineers or physicists or chemists, they need to just be able to think of calculus as no big deal within a derivation or some homework problem in fluids, electrostatics, or P chem.

This isn't just me being a hardass, this is the counsel from Feynman or lots of physics grad students. Lots of classroom and after class drill is the best medicine.

FYI, you admit your lack of experience with what general technical students need, but I can honestly tell you that I have broad background here (I probably don't know what the math student needs!) general engineering at collged, chemistry through Ph.D., mechanical engineering (in the field), officer Navy nuclear power school, applied psychology (heavy stats), etc. And that included classes (but not concentrations) in EE and control systems. You need the basic calculus building blocks to be comfortable when there are derivations or homework problems in technical classes (and many students will at least need to get through ODE. If you aren't solid on integrals, good luck with ODE.)

Teach the kids basic classical Granville. A few proofs along the way are fine. As motivation, examples, etc. But not as the reason. Please don't jump down the "real analysis during calculus" path. There is no reason kids can't go back and do that later. And lots of very high level theoretical physicists say it doesn't help them.

Finally, I encourage you to expand your horizons and get some general technical education. It will make you a better teacher of engineers or the like. One example of a very powerful mathematician who still had an understanding of what technical students needed, and even a sympathy of wanting them to learn it, was Kreyszig. He wrote a very classical mathy (but pedagogical) Advanced Engineering Math text. But he also wrote books on functional analysis and differential geometry. So it's not like he didn't know the fancy stuff.

$\endgroup$
  • 1
    $\begingroup$ Hey now, some of us LOVE both beets AND spinach! $\endgroup$ – Brendan W. Sullivan Jul 24 '17 at 16:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.