Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to…

In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as…

We present a free student-facing tool for creating 3D plots and smartphone-based virtual reality (VR) visualizations for STEM courses. Visualizations are created through an in-browser interface using simple plotting commands. Then QR codes are generated, which can be interpreted with a free smartphone app, requiring only an inexpensive Google…

This paper offers instructional interventions designed to support undergraduate math students' understanding of two forms of representations of Calculus concepts, mathematical language and graphs. We first discuss issues in students' understanding of mathematical language and graphs related to Calculus concepts. Then, we describe tasks, which are…

Although modeling is an integral process in applied mathematics, students rarely encounter modeling opportunities in their calculus courses. We introduce a laboratory experience as a starting point for calculus students to investigate multivariable functions. A layered system of coffee and milk serves as a physical model for temperature gradients…

In a series of group activities supplemented with independent explorations and assignments, calculus students investigate functions similar to their own derivatives. Graphical, numerical, and algebraic perspectives are suggested, leading students to develop deep intuition into elementary transcendental functions even as they lay the foundation for…

Why would anyone think of teaching and learning mathematics directly from primary historical sources? We aim to answer this question while sharing our own experiences, and those of our students across several decades. We will first describe the evolution of our motivation for teaching with primary sources, and our current view of the advantages…

As part of a larger research study, this paper describes calculus students' reasoning about the Intermediate Value Theorem (IVT) in verbal, written, and graphical form. During interviews, students were asked to verbally describe the IVT in their own words. They then provided written descriptions, watched video of their verbal descriptions,…

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…

The properties of a function such as limit, continuity, derivative, growth, or concavity can be determined more easily from its graph than by doing any algebraic operation. For this reason, it is important for students of mathematics to interpret some of the properties of a function from its graph. In this study, we investigated the competence of…

The purpose of this article is to present two very active applied modeling projects that were successfully implemented in a first semester calculus course at Hollins University. The first project uses a logistic equation to model the spread of a new disease such as swine flu. The second project is a human take on the popular article "Do Dogs Know…

This study investigated whether students who are studying calculus were able to determine the graph of the derivative of the functions related to the problems they come across in their daily lives or in applied science. A test was given to science and engineering students at a Turkish university. At the end of this study, it was determined that…

Building block toys can be used to illustrate various concepts connected with functions including graphs and rates of change of linear and exponential functions, piecewise functions, and composition of functions. Five brief activities suitable for a pre-calculus course are described.

We investigate the possibility of approximating the value of a definite integral by approximating the integrand rather than using numerical methods to approximate the value of the definite integral. Particular cases considered include examples where the integral is improper, such as an elliptic integral. (Contains 4 tables and 2 figures.)

We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…