Investigating graphical representations of slope and derivative without a physics context

By analysis of student use of mathematics in responses to conceptual physics questions, as well as analogous math questions stripped of physical meaning, we have previously found evidence that students often enter upper-level physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. As an extension from this work on students’ mathematical competency at the upper level in physics, we report on a preliminary investigation of mathematical understanding of fundamental concepts of slope and derivative among students in a third-semester multivariable calculus course. Among the first published findings of physics education research are investigations on students’ understanding of kinematics, with particular attention to graphical representations of position-, velocity-, and acceleration-versus-time graphs. Underlying these physical quantities are relationships that depend on derivatives and slopes. We report on our findings as we attempt to isolate students’ understanding of these mathematical concepts.

Figure 1

Slope Ranking Task.

Figure 2

Student response from the Slope Ranking Task indicating a ranking consistent with the average slope between points rather than the instantaneous slope at each point.

Figure 3

Original Derivative Sign and Ranking Task. [Later versions featured a curve with a more clearly positive slope for $h(x)$ at $x=a$; see Fig. 5.]

Figure 4

Example of student response to derivative sign task that started to use 2nd derivative reasoning—note the “positive curvature” in the response—but switched to the correct reasoning.

Figure 5

Modified Derivative Sign and Ranking Task, to have more explicit wording and a greater slope for $h(x)$ at $x=a$.