Steven Jones recently published an article titled “Scalar and Vector Line Integrals: A Conceptual Analysis and an Initial Investigation into Student Understanding” in the Journal of Mathematical Behavior. Steven has answered a few questions about this article below:
Who would you say is the target audience for this article?
The target audience would be those interested in thinking carefully about how big ideas across the undergraduate mathematics curriculum connect together, as well as those who want to understand how students make sense of such big ideas (especially calculus instructors).
What is the big problem you hoped to address with this article?
The big idea of line integrals is notoriously difficult for multivariable calculus students to understand and reason about. Further, because integrals stretch across much of the undergraduate curriculum (including Calc I, Calc II, Calc III, Diff Eq, and so on) this big idea is not isolated to one class. Rather, it has connections to how integrals are taught, understood, and reasoning about throughout the undergraduate curriculum. Thus, this paper examined both how multivariable calculus students understand line integrals, to better identify what some of the problems are in learning about them, and how these difficulties are not isolated to multivariable calculus classes.
What are some of the key ideas in the article?
There are many different meanings that one can ascribe to definite integrals, including as the “space” contained underneath some type of graph, or as an instruction to compute an anti-derivative, or as the summation of a quantity evaluated within many small pieces. These meanings are quite different, so if a student uses one meaning to try to make sense of a line integrals, they will come to quite different understandings than a student using a different meaning. It turns out that some of the meanings, while not incorrect, are limited in how deeply they allow a student to make sense of line integrals. In particular, students most commonly tried to use “area” meanings to make sense of line integrals, but while that worked for scalar line integrals of two variables, that meaning appears inadequate for comprehending vector line integrals and even scalar line integrals in higher-dimensional space. The anti-derivative meaning only helped students recall facts about needing to parameterize the variables in the line integrals, but could not explain what the integral meant or why it was structured the way it was. By contrast, the idea of integrals as summing up a quantity across many small pieces could have been very useful in making sense of these integrals, but the students struggled to use that meaning and often could only invoke elements of that meaning that were insufficient for fully making sense of the line integrals. Similar phenomena have been reported in other areas of calculus education and science education, leading to strong support for foregrounding this “adding up pieces” meaning for integrals during calculus instruction, as opposed to area or anti-derivative meanings. This is not to say area or anti-derivative meanings are incorrect or inappropriate, but should not be the primary meaning students adopt for the integral.
What are some of the main ideas you hope your audience will take from the article?
I hope the audience will realize first of all that there are many different meanings that can be associated with definite integrals, and that while experts may be able to manage them fluidly, students often focus on only one or two meanings at the exclusion of the others. The meanings they often focus on are “area” and “anti-derivative” meanings. My hope would be that instructors would realize the conceptual benefits of having students engage in the process of imagining the domain partitioned into small pieces, a quantity being evaluated within each piece, and a summation across those pieces leading to the idea of an integral. If students have the chance to do this, they can develop much stronger meanings for integrals that apply to a wide range of contexts.
The abstract for the article follows:
This paper adds to the growing body of research happening in multivariable calculus by examining scalar and vector line integrals. This paper contributes in two ways. First, this paper provides a conceptual analysis for both types of line integrals in terms of how theoretical ways of thinking about definite integrals summarized from the research literature might be applied to understanding line integrals specifically. Second, this paper provides an initial investigation of students’ understandings of line integral expressions, and connects these understanding to the theoretical ways of thinking drawn from the literature. One key finding from the empirical part is that several students appeared to understand individual pieces of the integral expression based on one way of thinking, such as adding up pieces or anti-derivatives, while trying to understand the overall integral expression through a different way of thinking, such as area under a curve.