Abstract
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.
Highlights
Physics students are often required to perform integrals in contRexts ranging from mechanics to introductory electricity and magnetism to thermodynamics by a line of
She connected the diagram, equation, and experiential representations into a single unified argument, and she did this without any hints about the correct representations to use. Another way to assess Amber’s skill is to notice the aspects of her presentations that were most innovative and least like the representations she had recently learned about. Her depiction of a narrow box-shaped volume element in lesson five was unlike anything that she had seen in these lessons; she may have transferred her knowledge from calculus to the physics context
The student must be able to relate each layer of the integral to several different representations, including graphical, verbal, equation, and concrete physical representations
Summary
Physics students are often required to perform integrals in contRexts ranging from mechanics (finding displacement using vdt) to introductory electricity and magnetism (findingR uRsing the k dq r2 electric field produced ) to thermodynamics by a line of (finding work charge usingPdV). Research by Meredith and Marrongelle has shown that students may rely on cues to decide when an integral is necessary, including recall, dependence (a quantity that varies with another quantity), and parts-of-a-whole cues [1]. Helping students to differentiate between a change dA and an area element dA, and to use these infinitesimals correctly, may require some instruction targeted at the student’s physical intuition about infinitesimals. It is this kind of understanding that we addressed in this research
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