Teaching integration with layers and representations: A case study

We designed a sequence of seven lessons to facilitate learning of integration in a physics context. We implemented this sequence with a single college sophomore, “Amber,” who was concurrently enrolled in a first-semester calculus-based introductory physics course which covered topics in mechanics. We outline the philosophy underpinning these lessons, which characterizes integration in terms of layers and representations. We describe how Amber learned to give oral presentations in which she told a story about how integration comes from products, sums, and limits in a variety of physics contexts. We conclude that by the end of our lessons, Amber was able to conceptualize and explain integrals using multiple representations. In one case, she was able to solve a novel problem about integration in an unfamiliar context (center of mass.) Based on our previous research about integration, we suggest that these achievements would have been unattainable with the use of a single one or two hour lesson.

Figure 1

A network of process-object routes to understanding integration.

Figure 2

Sketch of the blackboard after Amber’s lesson 4 presentation. (Our videotaped screen shot would be difficult to read.) Amber uses both “$T$” and “$t$” to represent time.

Figure 3

A diagram of the trough. The water in the trough has a parabolic shape, $y=a{r}^2$, and in the third dimension (into or out of the page), the trough has a constant depth $D=0.6\mathrm{cm}$. The trough is sitting on a turntable, which is rotating.

Figure 4

Sketch of the blackboard after Amber’s lesson 5 presentation.

Figure 5

Hinged door and channel for lesson 7.

Figure 6

“Sequence diagram” of a car used in lesson 2.

Figure 7

Diagram of an airbag used in lesson 4.

Figure 8

Equation used in lesson 4.

Figure 9

Discrete center-of-mass diagram used in lesson 6.

Figure 10

Diagram of a rod used in lesson 6.