Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can…

In a previous article published in the "American Mathematical Monthly," Tucker ("Amer Math Monthly." 1997; 104(3): 231-240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying…

Translation by a vector in the coordinate plane is first introduced in precalculus and connects to the basic theory of vector spaces in linear algebra. In this article, the authors explore the topic of collision detection in which the idea of a translation vector plays a significant role. Because collision detection has various applications in…

The authors describe working with partner disciplines during the process of updating mathematics courses for the general education quantitative literacy requirement. The conversations resulted in significant changes to the business calculus course and a rich listing of applications to be incorporated into the college algebra course from technical…

The equivalent resistance calculation for two circuits in cubic arrangements is solved. Emphasis is placed on the plastic (topological) properties of these circuits. In contrast, the opposite topological behaviour of an analogous arrangement is observed in the calculus of a magnetic field. It is also noted that the solved examples may be used as a…

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under…

Storytelling has proved to be an effective way of passing on information from one generation to the next. Whether the information relates to history, culture, health, or morality, the story provides a framework so that complex ideas can be better understood and retained. In this article, we consider the role of narrative in the design and…

Students often find the study of electrostatics to be abstract, heavily mathematical, and non-intuitive. This paper describes an inquiry-based approach to teaching electrostatics in an introductory calculus-based physics class that emphasizes the primacy of student conceptual understanding.

While some tertiary mathematics educators approximated the familiar invigilated, closed-book assessment regime in the online environment forced upon us all by a pandemic, others either by choice or necessity needed to devise a new way to assess their students' learning. This Classroom Note both synthesizes advice from the literature about what…

This article explores the parallels between improvisational theater, commonly known as improv, and active teaching. Specifically, it focuses in the impact of improv techniques on instructor and teaching assistant professional development. The implementation of an active teaching seminar is analyzed, where improv techniques were used in developing…

In this note, we propose a generalization of the famous Bernoulli differential equation by introducing a class of nonlinear first-order ordinary differential equations (ODEs). We provide a family of solutions for this introduced class of ODEs and also we present some examples in order to illustrate the applications of our result.

The purpose of these notes is to generalize and extend a challenging geometry problem from a mathematics competition. The notes also contain solution sketches pertaining to the problems discussed.

In high school, students extend understanding of linear and exponential functions and explore trigonometric functions. This includes using the unit circle to connect trigonometric functions to their geometric foundation, modeling periodic phenomena, and applying (and proving) trigonometric identities. These ideas are fundamental for trigonometric…

The complexity in understanding the [epsilon-delta] definition has motivated research into the teaching and learning of the topic. In this paper I share my design of an instructional analogy called the Pancake Story and four different questions to explore the logical relationship between [epsilon] and [delta] that structures the definition. I…

Answering a call by the "Common Vision" for instructional material to "establish stronger connections with other disciplines" and "work with professional-level technology tools," the present article shows how various aspects of fitting circles to data fit in existing courses from basic calculus to advanced calculus.