Gestures are profoundly integrated into our communication. This study focuses on the impact that gestures have in a mathematical setting, specifically in an undergraduate calculus workshop. There was strong correlation between diagramming and the two types of gestures identified in this study (i.e., dynamic and static gestures). Dynamic and static…

Space-filling curves are related to fractals, in that they have self-similar patterns. Such space-filling curves were originally developed as conceptual mathematical "monsters", counter-examples to Weierstrassian and Reimannian treatments of calculus and continuity. These were curves that were everywhere-connected but…

We discuss common difficulties in upper-division electricity and magnetism (E&M) in the areas of Gauss's law, vector calculus, and electric potential using both quantitative and qualitative evidence. We also show that many of these topical difficulties may be tied to student difficulties with mathematics. At the junior level, some students…

Gender differences in student learning in the introductory, calculus-based electricity and magnetism course were assessed by administering the Conceptual Survey of Electricity and Magnetism pre- and postcourse. As expected, male students outgained females in traditionally taught sections as well as sections that incorporated interactive engagement…

We designed a sequence of seven lessons to facilitate learning of integration in a physics context. We implemented this sequence with a single college sophomore, "Amber," who was concurrently enrolled in a first-semester calculus-based introductory physics course which covered topics in mechanics. We outline the philosophy underpinning these…

This article is the story of a very non-standard, absolutely student-centered multivariable calculus course. The course advocates the so-called problem method in which the problems used are a bridge between what the learners know and what they are about to know. The main feature of the course is a unique conceptual story that runs through the…

This article introduces an optimization task with a ready-made motivating question that may be paraphrased as follows: "Are you smarter than a Welsh corgi?" The authors present the task along with descriptions of the ways in which two groups of students approached it. These group vignettes reveal as much about the nature of calculus students'…

This study examined Advanced Placement Calculus students' mathematical understanding of rate of change, after studying four years of college preparatory (integrated or single-subject) mathematics. Students completed the Precalculus Concept Assessment (PCA) and two open-ended tasks with questions about rates of change. After adjusting for prior…

The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

We derive the energy levels associated with the even-parity wavefunctions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the…

We propose an alternative to the standard introduction to the derivative. Instead of using limits of difference quotients, students develop Taylor expansions of polynomials. This alternative allows students to develop many of the central ideas about the derivative at an intuitive level, using only skills and concepts from precalculus, and…

We describe a set of manipulatives developed to help students learn three-dimensional Cartesian, cylindrical, and spherical coordinate systems. (Contains 6 figures.)

Augustus De Morgan (1806-1871) was a significant Victorian Mathematician who made contributions to mathematics history, mathematical recreations, mathematical logic, calculus, and probability and statistics. He was an inspiring mathematics professor who influenced many of his students to join the profession. One of De Morgan's significant books…

High-quality physics instruction is contextualized, motivates students to learn, and represents the discipline as a way of investigating the world rather than as a collection of facts and equations. Inquiry-oriented pedagogy, such as problem-based instruction, holds great promise for both teaching physics content and representing the process of…

Over the last decade there has been a rapid diffusion of cellular technology. Though cell phone use began as a business tool, it has now become part of popular culture. Because of the near ubiquitous presence of cell phone technology among teens in the United States, schools are beginning to explore the use of cell phones as a learning tool. This…