Mathematical habits of mind for introductory physics

Steve Kanim

New Mexico State University

Physics education researchers have long been interested in students' conceptual understanding of physics topics. One general finding is that a robust conceptual understanding of the material is not a typical outcome of traditional instruction: For most students this understanding develops only with a change in course emphasis, preferably aided by research-based curricular materials. Recently, physics education researchers have broadened the scope of their investigations beyond initial focuses on conceptual understanding and problem solving to include topics such as student identity and epistemology, and we again see that desirable outcomes require curricular changes based on a research foundation. Both for researchers and for physics instructors, then, it is more important than ever to clarify our intent: What is it that we hope to achieve in our physics courses, and how do we prioritize our course goals?

Important research and curriculum development has emerged from Rutgers University based on a recognition that most students don't really understand how science works, that their understanding is not substantially improved through traditional physics instruction, and that this is an important broader goal both for STEM majors and for students who are taking a terminal physics course. As with conceptual understanding, the development of scientific habits of mind requires an explicit course focus, with curricular materials designed to allow students to observe carefully, develop hypotheses and models, and invent experiments to test their ideas.

• 1) In this talk I would like to describe additional goals also related to habits of mind - those practiced by physicists as they apply mathematics to observable phenomena. Suzanne Brahmia (Rutgers University), Andrew Boudreaux (Western Washington University) and I have been conducting research on student uses of mathematics in introductory physics. These investigations have reinforced a recognition that the ways that physicists use mathematics are different from the ways that students are taught mathematics.
• 2) Most physicists (myself included) believe that one outcome of their professional preparation is that they have learned a mode of thinking that is both productive and generalizable beyond their work life. Unfortunately, describing the specifics of these habits of mind seems akin to describing how one rides a bicycle, and they are rarely articulated as part of stated course goals or curricular materials. I would like to focus here on what it might mean both for our curricular goals and for the design of introductory physics instruction if physicists were to take seriously the idea that a crucial part of what we teach is what it means to “think like a physicist.” I believe that there are useful mathematical habits of mind that can be developed within the context of this course that are important for all students as a component of critical thinking, and for our majors as a first step to becoming physicists.
• 3) I will attempt to outline an initial guess for a set of skills that we develop as part of our professional preparation that allow us to think productively about quantity in the context of physical phenomena. In addition, I will give examples of curricular materials that might serve to introduce these skills. Finally, I would like to argue that for most students in introductory physics, an explicit focus on these skills is necessary if they are to understand would like to argue that for most students in introductory physics, an explicit focus on these skills is necessary if they are to understand physics problem solving beyond rote application of formulas.

• 1) Etkina, E. (2015). Millikan award lecture: ``Students of physics: Listeners, observers, or collaborative participants in physics scientific practices?'' American Journal of Physics, 83 (8), 669-679.
• 2) Brahmia, Suzanne, Mathematization in Introductory Physics, Ph.D. dissertation, Rutgers University, 2014.
• 3) Bing, Thomas J., and Edward F. Redish. "Epistemic complexity and the journeyman-expert transition." Physical Review Special Topics-Physics Education Research 8.1 (2012).