Developing student understanding: Contextualizing calculus concepts

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

Another student's statement about the problems he selected to include in his portfolio summarizes the importance of the components of understanding and the significance of making those components explicit in learning experiences.

I feel that the ... problems presented here demonstrate my understanding of calculus and how it has developed over the course of the semester. The group collectively shows my understanding of limits, derivatives and their applications, my connections between physics and calculus, and evaluations of graphs. I have been able to understand and refresh my mind on everything and make connections between that and calculus.

Discussion

This study looked at the practice of one teacher and its impact on the understanding of her calculus students. Providing students with concrete examples from their physics class gave them a contextually rich environment in which to explore the abstractions of calculus. Students' understanding of calculus can be described by examining their ability to explain concepts, apply them in a specific context, and reflect on their learning (Wiggins & McTighe, 1998).

Students were given many opportunities to explain calculus concepts to the teacher during the resource period and to each other in their small and large groups. Four indicators of the ability to explain concepts and processes emerged from this study. Students were able to explain the process they were using to solve calculus problems. They were able to tell an audience what they had done to determine the height from which an object was thrown, for example, as well as give the rationale for why they approached the problem in that way. A second indicator of students' ability to explain their understanding was their recognition of the connections between concepts in physics and calculus. Concepts, such as function, derivative, and the antiderivative, took on new meaning as students grappled with how they were used to determine position, velocity, and acceleration. Students developed the flexibility to explain different representations of the same problem, a third indicator of understanding as explanation. Students were able to explain problems numerically, algebraically, graphically, and verbally. The fourth indicator of understanding as explanation that emerged from the study was the ability to make inferences. Students were able to predict what would come next in the solving of a problem and were able to use this problemsolving strategy at the appropriate time.

In addition to explanation as a measure of understanding, the study also examined contextual application, in other words, students' ability to apply what they knew about calculus in a variety of physicsrelated contexts. Students demonstrated this ability by solving many problems in their calculus as well as in their physics class. Sometimes they chose a physics approach, such as graphing the mathematical model and geometrically calculating the area under a curve. Other times they translated problems into mathematical models, which were solved by calculus procedures. Physics made calculus concrete for students, while calculus provided efficient methods in calculations and deepened students' understanding of physics concepts.