Calculus and formal reasoning in intro physics

A physicist asked me, “One thing I noticed in most recent introductory physics textbooks is the slow disappearance of calculus (integrals and derivatives). Even calculus-based physics now hardly uses any calculus. What is the reason for that?” Here is what I replied:

Concerning calculus, I would say that I’m not sure the situation has actually changed all that much from when I started teaching calculus-based physics in the late 1960s. Looking through a 1960s edition of Halliday and Resnick, I don’t see a big difference from the textbooks of today.

More generally, there is a tendency for older faculty to deplore what they perceive to be a big decline in the mathematical abilities of their students, but my experience is that the students are adequately capable of algebraic manipulation and even calculus manipulation (e.g. they know the evaluation formulas for many cases of derivatives and integrals). What IS however a serious problem, and is perhaps new, is that many students ascribe no meaning to mathematical manipulations. Here is an example that Ruth Chabay and I have seen in our own teaching:

The problem is to find the final kinetic energy. The student uses the Energy Principle to find that K_f = 50 joules. Done, right? No! Next the student uses the mass to determine what the final speed v_f is. Then the student evaluates the expression \frac12 m v_f^2 (and of course finds 50 joules). Now the student feels that the problem is solved, and the answer is 50 joules.

We have reason to believe that what’s going on here is that kinetic energy has no real meaning, rather kinetic energy is the thing you get when you multiply \frac12 times m times the square of v. Until and unless you’ve carried out that particular algebraic manipulation you haven’t evaluated kinetic energy.

Another example: A student missed one of my classes due to illness and actually went to the trouble of coming to my office to ask about what he’d missed, so he was definitely above average. The subject was Chapter 12 on entropy. I showed him an exercise I’d had the class do. Suppose there is some (imaginary) substance for which S = aE^{0.5}. How does the energy depend on the temperature? I asked him to do this problem while I watched. (The solution is that 1/T = dS/dE = 0.5aE^{-0.5}, so E = 0.25a^2T^2.) The student knew the definition 1/T = dS/dE, but he couldn’t even begin the solution. I backed up and backed up until finally I asked him, “If y = ax^{0.5}, what is dy/dx?” He immediately said that dy/dx = 0.5ax^{-0.5}. So I said, okay, now do the problem. He still couldn’t! His problem was that he knew a canned procedure that if you have an x, and there’s an exponent, you put the exponent in front and reduce the exponent by one, and that thing is called “dy/dx” but has no meaning. There is no way to evaluate dS/dE starting from aE^{0.5}, because there is no x, there is no y, and nowhere in calculus is there a thing called dS/dE.

We are convinced that an alarmingly large fraction of engineering and science students ascribe no meaning to mathematical expressions. For these students, algebra and calculus are all syntax and no semantics.

A related issue is the difficulty many students have with formal reasoning, and here there may well be a new problem. It used to be that an engineering or science student would have done a high school geometry course that emphasized formal proofs, but this seems to be no longer the case. Time and again, during class and also in detailed Physics Education Research (PER) interviews with experimental subjects we see students failing to use formal reasoning in the context of long chains of reasoning. An example: Is the force of the vine on Tarzan at the bottom of the swing bigger than, the same as, or smaller than mg? The student determines \vec{p} just before and just after and correctly determines that d\vec{p}/dt points upward. The student concludes correctly that the net force must point upward. The student determines that the vine pulls upward and the Earth pulls downward. The student then says that the force of the vine is equal to mg! Various studies by Ruth Chabay and her PER grad students have led to the conclusion that the students aren’t using formal reasoning, in which each step follows logically from the previous step. Often the students just seize on some irrelevant factor (in this case, probably the compiled knowledge that “forces cancel”).

This problem with formal reasoning may show up most vividly in the Matter & Interactions curriculum, where we want students to carry out analyses by starting from fundamental principles rather than grabbing some secondary or tertiary formula. We can’t help wondering whether the traditional course has come to be formula-based rather than principle-based because faculty recognized a growing inability of students to carry out long chains of reasoning using formal procedures, so the curriculum slowly came to depend more on having students learn lots of formulas and the ability to see which formula to use.

Coming back to calculus, I assert that our textbook has much more calculus in it than the typical calculus-based intro textbook. This may sound odd, since we have had students complain that there’s little or no calculus in our book (we heard this more often from unusually strong students at Carnegie Mellon than at NCSU). The complaint is based on the fact that we introduce and use real calculus in a fundamental way right from the start, but many students do not see that the sum of a large number of small quantities has anything to do with integrals, nor that the ratio of small quantities has anything to do with derivatives. For formula-based students, \Delta \vec{p} = \vec{F}_{\text{net}}\Delta t has nothing to do with calculus, despite our efforts to help them make a link between their calculus course and the physics course.

Bruce Sherwood

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