In this paper, we describe our extension of the action, process, object, schema (APOS) theory to capture the complex interplay of mathematics and physics. We do this in the context of the 2D heat equation. We describe a hypothetical learning trajectory of the 2D heat equation, a preliminary genetic decomposition that stresses the conceptual…

Undergraduate science students face difficulties using mathematics in their physics courses. Choosing an institutional perspective, we consider that these students experience a permanent transition between mathematics in their mathematics courses and mathematics in their physics courses. We refer to the anthropological theory of the didactic and…

Research on the interaction between curricula and textbooks--particularly the textbook development process--is critical for understanding their reciprocal influence. While mathematics textbooks are typically developed in response to curriculum revisions, textbook research can also inform and shape subsequent changes to the curriculum. This study…

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum…

Four variations of the Koch curve are presented. In each case, the similarity dimension, area bounded by the fractal and its initiator, and volume of revolution about the initiator are calculated. A range of classroom exercises are proved to allow students to investigate the fractals further.

Given the increased focus of educational research on what works for whom and under what circumstances over the last decade, educational researchers are increasingly turning toward mixture models to identify heterogeneous subgroups among students. Such data are inherently nested, as students are nested within classrooms and schools. Yet there has…

In confirmatory factor analysis (CFA), model parameters are usually estimated by iteratively minimizing the Maximum Likelihood (ML) fit function. In optimal circumstances, the ML estimator yields the desirable statistical properties of asymptotic unbiasedness, efficiency, normality, and consistency. In practice, however, real-life data tend to be…

Currently, dynamic structural equation modeling (DSEM) and residual DSEM (RDSEM) are commonly used in testing intensive longitudinal data (ILD). Researchers are interested in ILD mediation models, but their analyses are challenging. The present paper mathematically derived, empirically compared, and step-by-step demonstrated three types (i.e.,…

The dynamics of a simple pendulum are often presented to undergraduate engineering students in introductory courses in dynamics. It is usually the first dynamic system considered by students that is modelled by a differential equation. This paper presents the standard material given to students. It is fair to say that students are accepting this…

The relativistic addition of velocities is usually introduced early in the study of Einstein's special theory of relativity. The equations are simple enough, but randomly chosen velocities lead to unwieldy calculations that can dishearten the student. This paper presents tables of velocity components in two dimensions, composed of five or fewer…

Transfer of structural math knowledge to physics is difficult for students. While research suggests various improvement techniques, enhancing parallelism of algebraic structures used in physics to those studied in mathematics courses seems underrepresented. This paper proposes an alternative way of introducing wave functions as a set of parametric…

In this study, we focus on a specific visual representation that is used across several mathematics and science content areas: the 'partitioned square' (PS). Previous research has examined PSs in single content areas in isolation, such as for mathematics polynomials or biology random mating, where the PS was generally in the service of other…

Background/purpose: This study aimed to investigate the differences among the equations used in estimating the reliability coefficient using the half-split method. These equations demonstrate Spearman-Brown's, Rulon's, Guttman's, Mosier's, Flanagan's, and Horst's. Materials/methods: The study instrument was a 43-item scale for evaluating the…

Research literature about visually impaired students' approach to mathematics is still very scarce, especially in the case of algebra, even though mathematical content is becoming increasingly accessible thanks to assistive technologies. This paper presents a case study aimed at describing a blind subject's process of algebraic symbol manipulation…

Electricity and magnetism are fundamentally interconnected, as represented by the symmetry in Maxwell's equations. Much of the research on Gauss's and Ampere's laws has focused on their application in calculating electric or magnetic fields. However, there remains a significant gap in the literature in exploring these laws in a broader…