John Winfrey’s thought-provoking letter in Physics Today’s April 2020 Readers’ Forum (page 10) makes two points regarding the physics curriculum and teaching materials. First, he notes that gaps in understanding originate in the undergraduate curriculum and persist into faculty teaching; second, he suggests that they are part of a problem with physics textbooks and pedagogy not adhering to well-known discoveries in cognitive science.
Gaps are not a problem in just the physics curriculum; they are pervasive throughout undergraduate education. One of us (Kevin) has taught the mathematics sequence from college algebra through differential equations and has seen the problem most clearly exhibited in trigonometry courses. Despite its elementary nature, trigonometry is an important recruiting ground for physical-sciences and engineering students: Class rosters are filled with students who show mathematical and scientific talents but have had poor guidance about how to apply them.
A typical trigonometry course is divided into three parts: trigonometric ratios and functions; analytic trigonometry, with its identities and equations; and applications and advanced topics. The courses tend to overemphasize part two, with classes wallowing for week after week in identities. As a result, advanced topics such as complex numbers, polar coordinates, and vectors aren’t covered at all, and an opportunity to introduce concepts that physical scientists and engineers use extensively is squandered. Similarly, algebra courses will skip important material later in the textbook, such as an introduction to exponential and logarithm functions, because of lack of time. In a differential equations course, operational mathematics might be skipped.
In physics classes, instructors may eliminate topics such as hydrostatics or some of the introduction to fields—especially quantities related to the magnetic field—to make room for advanced topics that may be of more interest to faculty and students but actually do most students little good. Moreover, introductory courses in physics and in engineering will present vectors in somewhat different ways. Mechanical engineering students may not even take Physics I because the material is ostensibly covered in their statics and dynamics courses. So various cohorts of students entering Physics II possess different ideas and tools.
Winfrey posits that gaps in understanding result from instructors’ attempts to build from specific to general ideas, and because of time constraints in most courses, the students never reach the general material. That approach, he writes, ignores the primacy effect: Material presented earlier is mastered better than material presented later. Textbook authors should therefore take the primacy effect into account and go from broad, general concepts to specifics.
Winfrey’s suggestion runs into the somewhat unsettled realm of educational theory. Every teacher recognizes that students learn early course material best, but it isn’t clear why. Many theories, all with some supporting evidence, attempt to explain the effect. For example, some researchers propose that information is easier to retrieve when it is subjected to occasional tests of recall, and early course content is tested more often.1,2 Another theory holds that later course content exceeds the cognitive load that students are able to successfully process and store in their long-term memory.3
An introductory physics course will seek to teach students the foundations of electrostatics, in which time derivatives are zero. Winfrey’s more general formulation of Coulomb’s law brings in dynamical quantities. Although that formulation is in keeping with his general-to-specific paradigm, it runs counter to the idea of reducing unnecessary complexity in order to avoid cognitive overload.
Whatever the true sources of cognitive barriers in instruction turn out to be, all of us who teach mathematics, physics, and engineering can do better by learning what our customers—the students—need most and reordering or reemphasizing instruction to meet those needs. Possibly we can add big-picture generalizations, as Winfrey suggests, while also removing redundant material to avoid adding to the cognitive load. It is our responsibility as instructors to determine, in coordination with other departments, what is germane for each course we teach and to design our instruction accordingly.
