This study investigated how students reason about the partial derivative and the directional derivative of a multivariable function at a given point, using different graphical representations for the function in the problem statement. Questions were formulated to be as isomorphic as possible in both mathematics and physics contexts and were given…

When learning physics, students need more than just an understanding of mathematical and physical concepts. Integrating the two fields is crucial, as research indicates that students often struggle even when they have a strong grasp of both. In this paper, we use the heat equation as an example from higher education. Given the importance of the…

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, [delta]"F"(x), are compared to the regions of area…

A spreadsheet plotting assignment of real-life data from a racecar is employed to help students master the relationships among speed, acceleration, and distance. The racecar application will capture the imagination and interest of the student. The data is obtained by reading speedometer values from a YouTube video for 40 s as a racecar accelerates…

This paper is a continuation of an earlier discussion in this journal about adhering to principles of mathematics while presenting function graphs in physics. As in the previous paper, the importance of the vertical line test was examined, this paper delves more in-depth, and it pinpoints a need for presenting graphs with a continuous rate of…

In this paper, we introduce and discuss a construct called "graphical forms," an extension of Sherin's symbolic forms. In its original conceptualization, symbolic forms characterize the ideas students associate with patterns in a mathematical expression. To expand symbolic forms beyond only characterizing mathematical equations, we use…

This paper is an addition to the series of papers on the exponential function begun by Albert Bartlett. In particular, we ask how the graph of the exponential function y = e[superscript -t/t] would appear if y were plotted versus ln t rather than the normal practice of plotting ln y versus t. In answering this question, we find a new way to…

Theorising STEM Education in the 21st Century is a book that captures the essence of Science, Technology, Engineering and Mathematics and the intricacies of STEM education in the contemporary society. It explores STEM as an interdisciplinary field as well as the individual disciplines that make up STEM. This ensures the field of STEM as a whole is…

Prior research suggests that introductory physics students have difficulty with graphing and interpreting graphs. Here, we discuss an investigation of student difficulties in translating between mathematical and graphical representations for a problem in electrostatics and the effect of increasing levels of scaffolding on students'…

Instead of solving ordinary differential equations (ODEs), the damped simple harmonic motion (SHM) is surveyed qualitatively from basic mechanics and quantitatively by the instrumentality of a graph of velocity against displacement. In this way, the condition b ? [square root]4mk for the occurrence of the non-oscillating critical damping and…

In the context of a generic harmonic oscillator, we investigated students' accuracy in determining the period, frequency, and angular frequency from mathematical and graphical representations. In a series of studies including interviews, free response tests, and multiple-choice tests developed in an iterative process, we assessed students in both…

We have investigated whether and how a categorization of responses to questions on linear distance-time graphs, based on a study of Irish students enrolled in an algebra-based course, could be adopted and adapted to responses from students enrolled in calculus-based physics courses at universities in Flanders, Belgium (KU Leuven) and the Basque…

We report on a study investigating the influence of context, direction of translation, and function type on undergraduate students' ability to translate between graphical and symbolic representations of mathematical relations. Students from an algebra-based and a calculus-based physics course were asked to solve multiple-choice items in which they…

Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven…

Those who come to Cambridge soon learn that the fastest route between Harvard and MIT is by the subway. For many students, this short ride is a quick and easy way to link physics and calculus. A simple, homemade accelerometer provides all the instrumentation necessary to produce accurate graphs of acceleration, velocity, and displacement position…