When elementary ordinary differential equations (ODEs) of first and second order are included in the calculus curriculum, second-order linear constant coefficient ODEs are typically solved by a method more appropriate to differential equations courses. This method involves the characteristic equation and its roots, complex-valued solutions, and…

This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.

This paper describes one instructor's experiences during a year of flipping four calculus classes. The first exploration attempts to understand student expectations of a math class and their preference towards a flipped classroom. The second examines success of students from a flipped classroom, and the last investigates relationships with student…

This article argues for a shift in how researchers discuss and examine students' uses of representations during their calculus problem solving. An extension of Zazkis, Dubinsky, and Dautermann's (1996) Visualization/Analysis-framework to include physical modes of reasoning is proposed. An example that details how transitions between visual,…

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…

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 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 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.)

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…

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…