An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts

- Students' understanding of non-kinematics derivatives are explored.
- Six commonly occurring categories of understanding are identified and described.
- Some connections between these categories and kinematics derivatives understanding are explored.
- First-year calculus instruction may benefit from incorporating non-kinematics contexts more often.

Much research on calculus students' understanding of applied derivatives has been done in kinematics-based contexts (i.e. position, velocity, acceleration). However, given the wide range of applications in science and engineering that are not based on kinematics, nor even explicitly on time, it is important to know how students understand applied derivatives in non-kinematics contexts. In this study, interviews with six students and surveys with 38 students were used to explore students' “ways of understanding” and “ways of thinking” regarding applied, non-kinematics derivatives. In particular, six categories of ways of understanding emerged from the data as having been shared by a substantial portion of the students in this study: (1) covariation, (2) invoking time, (3) other symbols as constants, (4) other symbols as implicit functions, (5) implicit differentiation, and (6) output values as amounts instead of rates of change.