Mathematics provides us with tools to capture and explain phenomena in everyday biology, even at the nanoscale. The most regularly applied technique to biology is differential equations. In this article, we seek to present how differential equation models of biological phenomena, particularly the flow through ion channels, can be used to motivate…

Students often instrumentally use variables and unknowns without considering the variational thinking behind them. Using parameters to modify the coefficients or unknowns in equations or systems of linear equations (without altering their structure) involves consciously incorporating variational thinking into problem-solving. We will test the…

This paper describes a project assigned in a multivariable calculus course. The project showcases many fundamental concepts studied in a typical course, including the distance formula, equations of lines and planes, intersection of planes, Lagrange multipliers, integrals in both Cartesian and polar coordinates, parametric equations, and arc length.

As part of an effort to examine students' mathematical sensemaking (MSM) in a spins-first quantum mechanics course during the transition from discrete (spin) to continuous (position) systems, students were asked to construct an eigenvalue equation for a one-dimensional position operator. A subset of responses took the general form of an eigenvalue…

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.

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…

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…

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…

A standard element of the undergraduate ordinary differential equations course is the topic of separable equations. For instructors of those courses, we present here a series of novel modeling scenarios that prove to be a compelling motivation for the utility of differential equations. Furthermore, the growing complexity of the models leads to the…

Experimental research on perception and cognition has shown that inherent and manipulated visual features of mathematics problems impact individuals' problem-solving behavior and performance. In a recent study, we manipulated the spacing between symbols in arithmetic expressions to examine its effect on 174 undergraduate students' arithmetic…

The aim of this research is to investigate how well high-achieving students entering tertiary-level education in Ireland understand school algebra. As part of a larger project, a 31-item test was developed to assess first-year undergraduate students' understanding of basic algebraic concepts. The test was administered online to students studying…

In this case study we explored how a mathematician's teaching of the Cauchy-Riemann (CR) equations actualized the virtual aspects of the equations. Using videotaped classroom data, we found that in a three-day period, this mathematician used embodiment to animate and bind formal aspects of the CR equations (including conformality), metaphors,…

Algebraic thinking is the ability to generalize about numbers and calculations, find concepts from patterns and functions and form ideas using symbols. It is important to know the student's algebraic thinking process, by knowing the student's thinking process one can find out the location of student difficulties and the causes of these…

High requirements for professional training of Defence Specialists were and remain the main guarantee of successful functioning of any military structure. Continuous improvement of the educational process in higher military educational institutions is the basis for its transformation to the conditions of the current situation in the world. In the…