In this study, a description is provided for the development of two undergraduate students' geometric reasoning about the derivative of a complex-valued function with the aid of "Geometer's Sketchpad" ("GSP") during an interview sequence designed to help them characterize the derivative geometrically. Specifically, a particular…

Gravity racing can be studied using numerical solutions to the equations of motion derived from Newton's second law. This allows students to explore the physics of gravity racing and to understand how design and course selection influences vehicle speed. Using Euler's method, we have developed a spreadsheet application that can be used to predict…

The radius of curvature formula is usually introduced in a university calculus course. Its proof is not included in most high school calculus courses and even some first-year university calculus courses because many students find the calculus used difficult (see Larson, Hostetler and Edwards, 2007, pp. 870- 872). Fortunately, there is an easier…

Polynomials with rational roots and extrema may be difficult to create. Although techniques for solving cubic polynomials exist, students struggle with solutions that are in a complicated format. Presented in this article is a way instructors may wish to introduce the topics of roots and critical numbers of polynomial functions in calculus. In a…

Research identifies two domains by which mathematics allows learning physics concepts: a technical domain that includes algorithmic operations that lead to solving formulas for an unknown quantity and a structural domain that allows for applying mathematical knowledge for structuring physical phenomena. While the technical domain requires…

Virtual Reality (VR) is an artificial environment that is created with software and presented to the user in such a way that the user suspends belief and accepts it as a real environment. Users interact with VR applications through on-screen toggles or with optional additional devices such as controllers, keyboard and mouse. However, when it comes…

We perform two approaches of machine learning, logistic regressions and decision trees, to predict student dropout at the Karlsruhe Institute of Technology (KIT). The models are computed on the basis of examination data, i.e. data available at all universities without the need of specific collection. Therefore, we propose a methodical approach…

The "domain-coloring algorithm" allows us to visualize complex-valued functions on the plane in a single image--an alternative to before-and-after mapping diagrams. It helps us see when a function is analytic and aids in understanding contour integrals. The culmination of this article is a visual discovery and subsequent proof of the…

The variance formula derived for a two-stage sampling design without replacement employs the joint inclusion probabilities in the first-stage selection of clusters. One of the difficulties encountered in data analysis is the lack of information about such joint inclusion probabilities. One way to solve this issue is by applying Hájek's…

In this paper, we derive the result of the classical gambler's ruin problem using elementary linear algebra. Moreover, the pedagogical advantage of the derivation is briefly discussed.

These classroom notes are focused on undergraduate students' understanding of the polysemous symbol of superscript (-1), which can be interpreted as a reciprocal or an inverse function. Examination of 240 scripts in a mid-term test identified that some first-year students struggle with choosing the contextually correct interpretation and there are…

Drawing parallels to classical test theory, this article clarifies the difference between rater accuracy and reliability and demonstrates how category marginal frequencies affect rater agreement and Cohen's kappa. Category assignment paradigms are developed: comparing raters to a standard (index) versus comparing two raters to one another…

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…

Test score distributions of schools or demographic groups are often summarized by frequencies of students scoring in a small number of ordered proficiency categories. We show that heteroskedastic ordered probit (HETOP) models can be used to estimate means and standard deviations of multiple groups' test score distributions from such data. Because…

This article takes a close look at Lagrange and Newton interpolation by graphically examining the component functions of each of these formulas. Although interpolation methods are often considered simply to be computational procedures, we demonstrate how the components of the polynomial terms in these formulas provide insight into where these…