Physics WITH Calculus

A hat tip to Chad at Uncertain Principles for a blog about calculus "based" physics that jumps off from a very good article by Thoreau. (He also points to another article by Thoreau that I intend to blog about, but first things first.) I won't quote extensively from either Chad's or Thoreau's articles, so I'd recommend reading Chad's article for background and another point of view.

As I said in the comments about Chad's article, I teach a physics "with" calculus class that, like the one Chad teaches, has the first semester of calculus as a co-requisite. This significantly reduces, but does not eliminate, the use of calculus in the first semester course - as I will explain. The second semester is another kettle of fish entirely. I should also say that I use (and love) Wolfson's "essential" textbook, but like some of the pedagogy in books by Knight and Redish. For example, I wish Wolfson did momentum before energy like Knight does, and resistance before capacitance, like Redish does. I don't like how any of them use calculus. By way of background, I should also add that I first taught out of the "blue bible" edition of Halliday and Resnick. which used little calculus at first, but I learned physics (with lots of calculus, since differential equations was a co-requisite) from the Berkeley physics series of textbooks.

So, what are the issues, from where I sit?

1. One issue is whether calculus SHOULD be a prerequisite for calculus-based physics. That one comes up regularly in discussions with my colleagues, although the fact that nearby Wannabe Flagship (where most of my students transfer) has calc as a co-requisite makes it pretty much of a moot point. It's not like my class has less calculus in it than theirs. (I think it might have more.)

The main concern about the math requirement concerns "success" rates. Would students do better if they had passed calculus before taking physics? The answer is a hearty MAYBE, because "passing" calculus is a surrogate for "knowing algebra and trig". This is supported by research, by the way. Students who are taking Calculus 2, which is harder than Calculus 1, will fail (or drop) if their grades indicate weak algebra and trig skills, while those taking Calculus 1 will thrive if they got an A in our pre-calc and trig class. Problem solving skills and the ability to follow a non-trivial sequence of algebra steps is the key to success in physics and calculus, and would not be any less important if we had more calculus in physics.

One way of putting this is to say that students would do much better in physics (and calculus) if my math colleagues failed 70%, rather than merely 50%, of the kids who take pre-calc and trig at our institution. This would not make the powers that be very happy, however.

A related concern has to do with getting them into physics in a timely fashion. Students at a CC tend to be there because their math is weaker than the norm. Although Wannabe Flagship has its share of students who enter needing to start in college algebra or pre-calc, they get a lot more of the ones who are allegedly prepared to take Calculus 1 and Chemistry 1 during their first semester. (Allegedly. Some of them end up at our CC shortly thereafter.) When a student has to take three (or more) classes before calculus, they are almost done with their other AA courses before they even get to "freshman" classes in the pre-engineering curriculum! Some actually run into problems with financial aid due to some restrictive rules put in place a few years ago. They aren't ready for "junior" classes in their major until they complete 3 semesters of calculus and two of physics, so anything we can do to help them fit math and physics into their remaining time with us will help them succeed. And, as I noted above, if they actually learned algebra at some point, they do just fine.


2. A key issue, the one raised by Thoreau and implicit in Chad's article, is the importance of calculus to physics. Is it important enough that students need to know calculus before taking physics so it will be part of the subject from the first day?

Well, it must be important if Newton had to invent it to do some key problems he developed within the Principia, but having had just such a course (see above), I am not convinced that it has to be central to Physics 1. Solving the anharmonic oscillator was fun and interesting, but it didn't add much to the physics content of the first semester mechanics course. It is important enough that it must be in there, but not so important that it becomes a math class.

However, my experience says that calculus DID add a lot to the other parts of the course, and I let that inform the way I teach the second semester of physics. To give just one example, I solve the simple, undriven, undamped harmonic oscillator problem in Physics 1 using basic methods from differential equations, since everyone can do the chain rule with sines and cosines by that point. That is pretty minimal calculus content. However, in Physics 2 I solve the undriven, damped LRC circuit using complex exponentials, and make the explicit connection to the classical mechanic problem I skipped over. I also give them plenty of homework and exam problems where they have to do "simple" derivatives and integrals as part of the solution.

Is that enough to jump straight into Lagrange's mechanics? Probably not. But where I was an undergrad (and at Wannabe Flagship) they got (and get) less than that in their first-year course, so Intermediate Mechanics had to start off with a review of things like the driven-damped problem. In addition, they also need a review of Lagrange multipliers. Even if it got taught in their differential equations class, the odds are approximately zero that the math prof teaching the class emphasized the importance of this method in physics or engineering.

I've also talked to some of the engineering and physics faculty at Wannabe Flagship, although the one kid every year or two who becomes a physics major means my concerns are mostly with engineering. Their point of view is that they want their students to know both physics and calculus, separately, and well. They will help them put them together. IMHO, if you can figure out the free body diagrams and write out the correct equations, the rest is "just math" that might be solved with Maple or MathCAD for a real problem. They seem to agree, and have at least one "intro" course that has that as its main objective.

But there is another extreme, and I think that is what Chad and Thoreau are talking about. Back in grad school, I had an instance where a student took the wrong final exam for Physics 2. I was teaching trig-based physics, and he took the calc-based exam that was in a different room in the same common final exam block. He was a C student in my class, and he passed the final exam in the calc-based course! We gave him credit for it, and made a mental note that the engineering students were not exactly being challenged more than the pre-meds were. I don't think that is the case for the exam I give.


2A. Why can't they fix the calculus curriculum? I'd be a lot happier if they covered all of basic differential and integral calculus in the first semester, skipping all of the functional analysis, and put limits and continuity into the third semester when developing partial derivatives. That would also make more room for applications problems in calculus.



3. Finally, how does one use these books, which uniformly minimize calculus, if you want calculus to be in the class? Well, I push it in there, kicking and screaming, first in lecture and later in homework and exams.

In Physics 1, I put problems that require taking the derivative on the later exams and/or the final. Thoreau is right about the dearth of good problems in the books we use, but I write my own when they don't suffice. But my emphasis is on problem solving, not math. I'm more than content to ask a problem where they have to SET UP the differential equation or integral. That is the physics part of the problem. Similarly, I always have problems that only ask them to draw the correct free-body diagram(s) and write the set(s) of equations to be solved, without solving them. I can ask much harder, messier problems if I don't expect them to solve them - and don't have to try to grade an attempt to solve the indeterminant set of equations or non-integrable function they came up with in error!

My emphasis is on problem solving. That is a big enough challenge, along with other basic skills. I liked the comment about torque from an engineer in the discussion of Thoreau's article. You see, I know that engineers call it the moment, M, and tell my students that. That is also a subtle way of telling them they need to be able to evaluate the torque next year as well as this year.

But I will also add calculus-based examples as needed. I insist that they read the book, so I don't have to repeat something that is adequately covered there. I can use examples to fill in the gaps, showing how the basic method can be extended to more interesting situations.


4. I'll close with one quotation from Thoreau's article, an observation that works as a segue to the next article, about teaching concepts. Thoreau writes:

As near as I can tell, they [people interested in physics pedagogy] believe that calculus (a subject to which no less of a physicist than Newton made huge contributions) doesn’t have much place in the freshman physics classroom.

[Side comment: Hey, I'm from the school that says Newton invented it, even though I know Leibnitz did it independently and the idea itself had been around before either of them.]

That certainly would explain why certain textbooks push a thousand pages and still manage to have very little calculus in them. They clearly think developing conceptual understanding comparable to that of an expert on physics pedagogy is as important as problem solving, and that both are much more important than using advanced mathematics within the course. I think this is wrong mainly for the reason that led me to tag this article in the "prerequisites" category: If we don't use calculus in our physics class, we are telling our students that they should forget calculus as quickly as they learned it, that they don't need to retain it for next year. I use it, and tell them as often as possible (and sometimes get my "alumni" to tell them as well) that the "next level problem" would use calculus here, or there, or there, and give them an example when feasible.

And (as I will elaborate later) I think emphasizing that idea, that the basic skills from physics and calculus will be used next year and need to be learned and retained, is much more important than some of the concepts that are displacing calculus from the textbooks. After all, some of the most important concepts and skills have NOTHING to do with calculus. Consider torque. Sure, you have to do an integral to figure out the torque due to a force distributed over an object (like wind load on a building), but is that integral more important than knowing how to handle a cross product? I don't think so. You can learn how to add the integral if you can evaluate the torque due to a simple force, but you can't if you have no memory of how to compute a torque correctly.

PS -
It won't be long before I teach Physics 1 for the 20th time, tweaking it as I go. My plan for this summer includes hanging out at the nearby engineering campus and getting some feedback from recent students and their faculty on what does, and does not, need more emphasis in my class.