Refining Calculus Pedagogy

TL;DR

This paper proposes integrating computational tools like Gaussian quadrature and series analysis into calculus teaching to enhance understanding and accessibility, especially in multivariable contexts.

Contribution

It introduces easy-to-develop computational tools and methods, such as Gaussian quadrature, to improve calculus pedagogy and address gaps in student comprehension.

Findings

Gaussian quadrature can be developed using basic linear algebra.

The method extends to higher-dimensional integration.

Series discussed may be more useful than traditional ones.

Abstract

There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and use. Doing so also addresses a different way to view calculus, and attempts to fill the gaps in students' understanding of both differentiation and integration. We will describe the basics of both these topics in a way that might be much more useful and relevant to students, and hence possible ways in refining calculus pedagogy to make calculus more accessible to them. For integration, an elementary development of Gaussian quadrature using basic linear algebra is presented. This numerical method can be extended to integrate functions over various domains in higher dimensions, a subject that is not currently well covered in multivariable calculus or other Mathematics courses. We also briefly discuss series that may be more useful than some of those taught in current calculus courses.