Leading by Counterexample

By Neel Modi, Physics

Teaching Effectiveness Award Essay, 2025

Physics, as a field of study, lives somehow at the intersection of two contradictory paradigms. On the one hand, it is highly abstract and technical–arguably one of the most math-heavy STEM fields you can choose to study in college. On the other hand, unlike pure mathematics, physics is not totally bound by the rules of logic. The central balance is to arrive at powerful, true conclusions, while minimizing the introduction of formalism and technicalities. This can lead to a very natural, often difficult-to-express confusion among students: “what am I allowed to assume, and what am I supposed to deduce?” I propose that a highly effective way to prevent this confusion is by emphasizing counterexamples.

I first became aware of the extent to which this confusion might be occurring when I had students expressing it to me in intro-level courses such as Physics 8B and Physics 7A. For example, one point of concern was about the justification (or rather, lack thereof) for certain kinds of initial conditions in physics problems. The points these students made were definitely valid; but I was surprised, because genuinely answering them would require advanced, proof-based mathematics. In contrast, the prerequisite math for these courses is rooted in intro-level Calculus. Although it has been rare to see students manage to articulate this kind of difficulty, these experiences have allowed me to gain a little bit of clarity on how I could improve my own teaching.

The challenge is that we need students to be able to consistently arrive at the correct physical conclusions without giving them a full course on higher-level mathematics. To an extent, this minimalist approach is already the goal of an introductory Calculus course; and, more broadly, the field of Physics in general. (After all, it was with Physics in mind that the techniques of Calculus were developed.)

But a common pitfall among students, in a workaround attempt to mitigate this difficulty, is the rote memorization of equations or example solutions. Once the confusion kicks in, this resort is often automatic, recognizable on exams as “using the wrong equation” or “plugging in the wrong variables.” It’s not uncommon for the confusion to turn to frustration, as students lose points “even though they plugged into the equation for the electric field.”

It is with these future mistakes in mind that I started trying to teach by counterexample. The idea is to emphasize a modification to the nature of solving problems. Instead of jumping straight to the correct solution, which many students are used to seeing (but struggle to replicate), I will first write down an example of a solution that looks like an application of what we reviewed, but actually contains an error. The students’ challenge is to figure out what error(s) have been made. And then there are important variations to this scheme; for instance, using one of my own real-time mistakes as the counterexample to show that even my thought process has mistakes along the way.

This rephrasing of the problem-solving process as an iterative improvement on initial ideas–with wrong turns along the way–is a lot closer to what students will face on their own exam papers; and the structure of “these are the basic concepts, but this one doesn’t work in that case because of X, Y, and Z,” is more in line with how physics knowledge is usually conceptualized in practice.

The effect of this method seems to have been successful based on recent courses I’ve taught, with substantially less confusion from students when it comes to making a good effort on exam problems. In addition, the feedback from students on evaluations and polls has spiked particularly positively on the subject of clarity of explanations–even in some cases when the explanation is purely through counterexamples! Indeed, to understand any concept in physics is to understand when it fails.