Developing student understanding: Contextualizing calculus concepts

School Science and Mathematics, Feb 2000 by Schwalbach, Eileen M, Dosemagen, Debra M

A fourth indicator of understanding evidenced by explanation is the ability to make inferences. In the context of mathematics, this ability is demonstrated when a student is able to predict what comes next in a process or application. Students frequently would make comments or ask questions that showed that they were thinking in this fashion. For example, the teacher was leading a discussion in class about the process of using rectangles to estimate the area under a curve. One student asked if an infinite number of rectangles would give the best answer, inferring the use of limits as they could apply to the scenario. The question itself showed significant insight into the process and how it would ultimately evolve. By posing the question, the student helped move the whole class to a new level of understanding of the process.

Contextual Application

In addition to the understanding of semantic knowledge evidenced by explanation, another element in the framework from which understanding has been evaluated is the successful application of procedural knowledge in a variety of contexts. For the purposes of this study, physics problems specifically related to motion provided the primary arena for this application. For example, students began to apply the concept of derivative by calculating velocity of a moving body, given an equation for its position. Similarly, they were able to determine a particle's acceleration from information about its velocity. To complete problems of this nature, not only were students able to connect the calculus concept of derivative with the physics concept of velocity, as in the first example, but they also selected and performed the appropriate mathematical procedure necessary to arrive at an accurate result.

Because the concept of antiderivative was introduced early in the curricular sequence, students were also able to apply calculus procedures to problems beginning with acceleration. A typical problem of this type involved the motion of a moving vehicle: The driver, for a variety of reasons, is forced to quickly employ the brakes and thus force the vehicle to negatively accelerate. Most students demonstrated their understanding of the calculus and physics involved by translating the problem scenario into a mathematical model, which could be solved through the use of calculus procedures. However, several students chose to solve problems of this type with an approach they first learned in physics: graphing the mathematical model and geometrically calculating the area under the curve it created within the parameters of the problem. Although these students initially selected a physics approach, they were able to verify their results using calculus calculations.

An informal conversation with one student illustrated the value of opportunities to apply knowledge in problem solving situations, such as those described above, as opposed to simply repeating a set of mathematical manipulations to arrive at an answer. The student asked for help with a problem he had attempted involving acceleration and the use of the derivative. He had arrived at a solution, but he identified that the answer "did not make sense" in the context of the application problem. Had the problem been presented as a purely algorithmic one, he may not have identified his answer as incorrect.