Analyzing Aspects of Difficulty

Physics teachers have a lot to gain by analyzing what factors contribute to the difficulty of the course they teach. For students, the priorities of the course are dictated by these factors, whether or not these priorities are intended by the teacher.

Conceptual understanding of physics is an important facet of a student's physics education, but this is not the only thing we strive to teach in our classes. Aspects of student learning could be categorized as conceptual (e.g. applying Newton's 3rd Law to an analysis of forces), mathematical (solving a quadratic relationship to find the time aloft for a projectile), skill-based (plotting data with error bars), and knowledge-based (knowing that g on earth is 9.8 N/kg), and we could certainly identify other categories as well (problem solving, scientific reasoning). Any physics teacher makes choices to emphasize certain of these aspects over others, based on their own personal priorities and priorities dictated by others (through a standardized test or curriculum or through administrative or departmental influence). How can we use these distinctions to inform decisions about what priorities to set in our curriculum?

I believe that a physics course should be difficult. I also believe that we have a responsibility to our students to examine the "difficulty" of the course we teach through the lens of categories like I mentioned above. In other words, if a task or problem is difficult for our students, why do they find it difficult? Do the concepts being studied conflict with misconceptions? Are students being asked to use unfamiliar math techniques or analytical skills? Are they having trouble recalling or accessing necessary knowledge?

To illustrate a distinction between conceptual and mathematical difficulty, for example, consider the diagrams to the right, depicting two varieties of the "modified Atwood machine." As part of a discussion of Newton's 2nd Law in a high school physics course, students might be asked to, "Find the acceleration of the hanging block."

In my experience, either of these problems would present a challenge for a ninth grade student. It's a challenge to recognize that the acceleration of the two-block system is affected by both masses, though the force of gravity on the large and small masses affects this acceleration in different ways. However, the problem on an incline would present a significantly greater challenge to an average ninth grader. Why is this second problem more difficult? Does it require a require a more sophisticated conceptual understanding, or does it simply require more knowledge of math?

Well, I'd say it requires both*, but I'd also argue that the added difficulty of the second problem in this case is overwhelmingly due to the trigonometry involved in the solution. If we choose to include this problem in our course, it should be because we want to increase a student's familiarity with trigonometry, not because we assume that this trigonometry is inseparable from the physics concepts. Of course, a student's understanding of physics concepts is often connected to their facility with the relevant math (though the connection between math and physics is less direct than we sometimes assume), but we physics teachers have some degree of flexibility to include only the pieces of the "whole story" we deem to be age-appropriate. We don't shy away from discussing gravitational field in a typical algebra-based high school course simply because our students can't compute the line integral required to show that this field is conservative.

In making choices about how to teach and what to teach in our courses, there's a lot to be gained by analyzing how the activities we include in our course develop both skills and understanding. In a Physics First course, our task is twofold (at least!): we are teaching physics and introducing students to high school science. In determining how best to accomplish these parallel tasks, we can continuously ask ourselves what conceptual, mathematical, skill-based, and knowledge-based aspects of physics our course is prioritizing.

*To clarify, it seems to me that vector analysis of the forces on an object on an incline is mostly a conceptual (or perhaps skill-based) task. However, I'd argue that the added challenge of applying this analysis to a calculation of acceleration is mostly mathematical in nature.