This is basically a brain dump. I don’t expect it to be perfectly coherent.
Since the summer AAPT meeting in College Park, I’ve been thinking a lot about how mathematics is approached in introductory physics. James Gates pointed out something I’d not realized before. If we consider mathematics a language, then in physics, we don’t speak the same dialect that mathematicians speak. We speak a “pidgin” dialect of math that we more or less invented for our own purposes. While I agree with this (look no further than the Dirac delta function), it bothers me in that I have been looking for ways to increase the carryover from introductory calculus into introductory physics for my students. Is this a mistake? Now I’m second guessing myself and my intentions. Am I setting students up to hit a brick wall, one that isn’t really intended to be broken through?
Prior to the College Park meeting, I had already formulated some ideas about increasing the mathematical content in calculus-based introductory physics. I feel that vector analysis isn’t given a complete enough treatment. We show students how to calculate dot “products” and cross “products” (note the quotes…that’s another future post) but to my knowledge, we never show students how to “unwrap” these constructs and solve them for unknown vectors. We don’t fully exploit the geometric properties of vectors and how that geometry encodes essentially all of classical physics. What about calculus itself? We know that students see infinitesimals differently depending on the context (see Von Korff and Rebello, Am. J. Phys. 82, 695 (2014) and references therein). We frequently set up integrals differently in physics compared to how we do it in calculus. Nature doesn’t always provide us with sanitized functions like those found in calculus textbooks so derivatives look differently in the wild. As Gates pointed out, every mathematical thought in physics must reflect and account for some physical reality and that isn’t the case in a mathematics class.
But I also landed on another idea, and that is that maybe we create problems with the mathematical aspects of introductory physics by assuming that students will inevitably find the mathematics difficult and use it as an excuse for not learning the physics. Do we subconsciously water down the mathematical content in physics (let’s include astronomy in this thought too) because we don’t think students can handle it? I don’t know. Maybe. Maybe not. I think it’s worth thinking about and I confess to having spent a great deal of time with the notion.
So what can we do to improve the situation (assuming we agree that it needs improvement)? This past summer, I concluded that one way of framing the approach to mathematics in introductory physics is to first point out to students what Gates pointed out. Physics surely includes math, but a different dialect of math and we sometimes even invent new math on the fly to accommodate some physical behavior. Furthermore, it seems to me that we should point out to students how the physicist’s dialect of math can still be thought of as an extension of what the standard math course sequence contains. Students have ostensibly learned to solve systems of equations and this can readily be put to use in solving vector equations (e.g. solving a cross product for one of the input vectors). Students have ostensibly learned symbolic algebra and should therefore be able to apply the relevant properties (e.g. commutativity, distributivity, associativity, inverse operations and their absence, etc.) to vector analysis. Students have previously seen scalars, vectors (perhaps within the context of a trig course), and matrices and here we have a perfect introduction to tensors framed as a superset of all these entities with a generalized way of manipulating them. Maybe this can, in turn, ease the introduction of four vectors in relativity.
In summary, what I’m trying to say, I think, is that by pointing out to introductory physics students that “physics math” is not always quite the same as “math math” and that much of the math in introductory physics is an extension of what they have already seen in previous courses, we may both ease the transition into introductory physics and also increase the level of mathematical content in our courses, thus giving earlier exposure to content students will need in subsequent physics courses.
I hope that made sense. What do you think?