Developing student understanding: Contextualizing calculus concepts

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

The ability to make connections is a second indicator of understanding as explanation. Throughout the study, students consistently identified the explicit connections between the concepts of calculus and physics and the importance of those connections. One student's description typifies the responses of the students who were interviewed about their insights into the two subjects:

I see many connections between physics and calculus. The one that sticks out in my mind the most is having to do with taking the derivative of equations that we learned in physics. For example, if you take the derivative of the equation for motion, you get the equation for velocity. If you take the derivative of that, you get the equation for acceleration.

A significant connection that students understood related to the relationship between the derivative and antiderivative. In order to facilitate application to physics, derivatives and antiderivatives were presented together as opposite operations. Students first recognized this relationship in physics when they numerically analyzed data generated by a constantly moving vehicle. As they worked, one of the groups began to make the mathematical connection. One student described this insight: One of the groups determined that if you [use] the position equation that we were looking at, it gave you the same - like by plugging in for the time value, you get the same equation... as you did by figuring out the area under the shaded region [relative to the velocity graph].

The physics instructor was also aware of the students' insight into this process. He observed, "They were given a velocity-time graph, and they were asked, 'How does one get displacement from this?' and they looked at it and, somehow, something clicked and they realized that was an antiderivative soon to be called the integral."

The link between different representations of the same problem scenario also emerged as an important connection, which served as a third indicator of understanding by explanation. Students consistently displayed their ability to use the Rule of Four and represent ideas and information numerically, algebraically, graphically, and verbally. This ability was apparent in the manner in which students approached a physics test problem. Students were given a set of data points (numeric), which they had to graph (graphic). Then they were asked to explain (verbal) how they would find velocity and acceleration. The physics teacher observed, "Some of them actually, from the data points, deduced that the equation [algebraic] describing the motion was x = t^sup 2 ."

In her comments on the course evaluation, one student also noted an advantage of approaching problems and concepts from a variety of vantage points: "Since we learned how to do things in different ways, like graphically or algebraically, really helped, too, since we all learn differently." During a videotaped discussion, another student recognized the importance of translating a concept into a visual representation: "You can make the image of velocity or changing velocity, and things like that. It makes it a lot easier to actually grasp it and not just do the processes involved." When responding to specific prompts related to understanding derivatives, one student noted the importance of "the definition of derivative both algebraically and graphically." He also emphasized the need to make connections between the "graphical and algebraic forms of a function and its derivative."