Calculus II students know that many alternating series are convergent by the Alternating Series Test. However, they know few alternating series (except geometric series and some trivial ones) for which they can find the sum. In this article, we present a method that enables the students to find sums for infinitely many alternating series in the…

One technique for identifying and addressing common student errors is the method of classroom voting, in which the instructor presents a multiple-choice question to the class, and after a few minutes for consideration and small group discussion, each student votes on the correct answer, often using a hand-held electronic clicker. If a large number…

Mathematics teachers constantly encourage their students to think independently. The study of integration in calculus provides an excellent opportunity to encourage inventive investigation. In contrast to differentiation, which is predominately mechanical, integration is a more creative process. One such possibility is offered by the study of the…

This article points out a simple connection between related rates and differential equations. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.

The Common Core State Standards (CCSS) provide teachers with the expectations and requirements that are meant to prepare K-12 students for college and the workforce (CCSSI 2010b). The Common Core State Standards for Mathematical Practice (SMPs) emphasize the development of skills and conceptual understanding for students to become proficient in…

Considering the nature of the complex prose that K-12 students today must learn from, in light of the Common Core State Standards, students need to read informational texts on a meaningful level-and with enthusiasm. Teachers, Brozo says, need to achieve three goals: motivate students to read informational texts, expand students' background…

John Lamb, a professor of mathematics education and a teacher of high school precalculus, describes how he developed a way to use the elements of the game Angry Birds® as a platform to engage his students with the concepts of parabolas and vectors. The game could be categorized as a type of microworld game in which students interact with the…

In this paper we discuss flipping pedagogy and how it can transform the teaching and learning of calculus by applying pedagogical practices that are steeped in our understanding of how students learn most effectively. In particular, we describe the results of an exploratory study we conducted to examine the benefits and challenges of flipping a…

This work introduces a distance between natural numbers not based on their position on the real line but on their arithmetic properties. We prove some metric properties of this distance and consider a possible extension.

Mathematical elegance is illustrated by strikingly parallel versions of the product and quotient rules of basic calculus, with some applications. Corresponding rules for second derivatives are given: the product rule is familiar, but the quotient rule is less so.

We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…

In mathematics, as in baseball, the conventional wisdom is to avoid errors at all costs. That advice might be on target in baseball, but in mathematics, avoiding errors is not always a good idea. Sometimes an analysis of errors provides much deeper insights into mathematical ideas. Certain types of errors, rather than something to be eschewed, can…

The purpose of this classroom note is to provide an example of how a simple origami box can be used to explore important concepts of geometry and calculus. This article describes how an origami box can be folded, then it goes on to describe how its volume and surface area can be calculated. Finally, it describes how the box could be folded to…

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

Flipping the classroom refers to moving lectures outside of the classroom to incorporate other activities into a class during its standard meeting time. This pedagogical modality has recently gained traction as a way to center the learning on students in mathematics classrooms. In an effort to better understand the efficacy of this approach, we…