Archimedes of Syracuse and Sir Isaac Newton: On the Quadrature of a Parabola

Good mathematics stands the test of time. As culture changes, we often ask different questions, bringing new perspectives, but modern mathematics stands on ancient discoveries. Isaac Newton’s discovery of calculus (along with Leibniz) may seem old but is predated by Archimedes’ findings. Current mathematics students should be familiar with parabolas and simple curves; in our introductory calculus courses, we teach them to compute the areas under such curves. Our modern approach derives its roots from Newton’s work; however, we have filled in many of the gaps in the pursuit of mathematical rigor. What many students may not know is that Archimedes solved the area problem for parabolas long before the use of algebraic expressions became mainstream. Archimedes used the geometry of the ancient Greeks, which gave him a vastly different perspective. In this paper we provide both Archimedes’ and Newton’s proofs involving the quadrature of the parabola, trying to remain true to their original texts as much as feasible.

Modernizing Archimedes’ Construction of π

In his famous work, “Measurement of a Circle,” Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining π eschew his method and instead use arguments that are easier to justify, but they involve ideas that are not elementary. This paper makes Archimedes’ measurement procedure rigorous from a modern perspective. In so doing, it brings a rigorous and geometric treatment of the differential properties of the trigonometric functions into the purview of an introductory calculus course.

Rolling Railgun: A Lab Activity for Introductory Electromagnetism

We describe a laboratory exercise for an introductory calculus-based electricity and magnetism course in which students construct and study the performance of a “rolling railgun” formed by two small coin magnets connected by a ferromagnetic axle which carries current from one rail to the other. This exercise can be scaled up from a simple, mostly qualitative activity to a more comprehensive comparison between theory and experiment that will challenge students’ calculus skills. The required components are small and inexpensive enough to mail to students who are taking the course remotely. We report on our initial success in incorporating this lab into our curriculum at Towson University.

Physics at the Molecular and Cellular Level (P@MCL): A New Curriculum for Introductory Physics

ABSTRACT In this article, I describe a new curriculum for introductory physics for the life sciences, a 2-semester sequence usually required of all biology majors. Because biology-related applications on the macroscale are complex and require mathematics beyond introductory calculus, the focus is entirely on applications from molecular and cellular biology. Topics that are more relevant for engineering have been removed, and topics relevant to biology have been added. The curriculum is designed around 2 main themes: diffusion and electric dipoles. Diffusion illustrates the concepts of conservation of momentum and energy and provides the framework for introducing entropy from the perspective of statistical mechanics. Electric dipoles illustrate the basic concepts of electromagnetic theory and provide the framework for understanding light waves and light interactions with biomolecules. These themes are supported by small computational activities to help students understand the physics without advanced mathematics. This curriculum has been piloted over the past 4 years at Michigan State University and should be applicable to many colleges and universities.

Agile and Active

It is well documented that the use of active learning strategies increases student learning (Freeman et al., 2014; Prince, 2004; Springer, Stanne, & Donovan, 1999). A key difficulty in innovating college mathematics is identifying and sustaining what works for both students and the faculty. This study discusses efforts to innovate and sustain curricular change in introductory calculus at a private, elite institution. To examine if incorporating active learning strategies made a difference in student performance, student grades in the redesigned course and performance in subsequent courses were analyzed. Using Austin’s 2011 framework to understand the context in which the course redesign took place, individual faculty and contextual barriers and “levers” to sustain change are discussed. Findings are applicable to other STEM disciplines and to colleges and universities in general. Next steps in this research include identifying how to scale change, including, perhaps, networks of faculty to implement and spread the reform on campus.