The subject of mathematics is important for many other subjects, including physics. The questions we ask in physics often need a mathematical translation to be graphically interpreted and transformed. Students learn physics when interacting with physical questions; questions from physics also give insights and discoveries in mathematical problems.…

This article considers starting with an existing SIMIODE modeling scenario [Winkel, B. (2015). 1-031-CoolIt-ModelingScenario. SIMIODE (Version 2.0). "QUBES Educational Resources." https://doi-org.bibliotheek.ehb.be/10.25334/3WG8-EC31] that develops Newton's law of cooling by considering data on the cooling of a beaker of water in a room, and expanding upon…

The process of diffusion is intimately related to random molecular displacement and is central to many problems in chemistry. Here, we provide an intuitive derivation of the diffusion equation based on the analysis of the random trajectories of particles generated and plotted by simple MATLAB and Python scripts. The codes are very simple and…

This paper describes two examples of teaching situations in which the idea of infinity arises, and supports the conclusion that infinity is not a physical reality but a very powerful and useful mathematical device which facilitates modelling and the solution of problems in physics.

Too often, mathematics and science are taught in isolation from each other and from meaningful problems that matter to students. This book draws on the authors' experiences with teacher colleagues, including time spent in their classrooms co-developing and refining lessons. The core of their approach is to encourage learners to pursue solutions to…

How might fourth graders address the problem of measuring the size of a leaf? And why might their classroom teachers choose such a shape to measure? One group of fourth-grade students in an urban Australian school were interested in knowing the sizes of leaves on a passionfruit vine growing at their school. The children were excited after a visit…

Synthesis and diffusion of a pigment molecule can be simulated using deterministic equations in computer software. These lesson materials describe how tiger stripes emerge from manipulations in this code, and how students can engage in mathematical inquiry by exploring these reaction-diffusion equations.

Is a suction cup at the bottom of a bathtub subject to an upward force from the surrounding water, even if there is no water under it? A student question, posted in a teacher facebook group on a Monday morning, led to a discussion involving 21 comments with 225 replies offered by 16 teachers during the next few days, including several simple…

The integration of biology with mathematics and computer science mandates the training of students capable of comfortably navigating among these fields. We address this formidable pedagogical challenge with the creation of transdisciplinary modules that guide students toward solving realistic problems with methods from different disciplines.…

This paper introduces a course design for a post-Calculus undergraduate course on mathematical modeling for the life sciences. The course includes differential equations, discrete, and cellular automata approaches (among others) to modeling questions in the life sciences, as well as a variety of biological topics. The course focuses on model…

This paper demonstrates how the transcendental number "e" may be arrived at by observing the discharge of a capacitor through a fixed resistor and then modelling the system using a simple step-wise procedure. The experimental phase makes use of the Arduino microcontroller, while simple modelling of the system is carried out by means of…

Biological systems exhibit strong dynamic variations; for example, a system may have a tendency towards either a steady state, periodic behaviour or chaotic oscillations. These findings, explained by dynamic systems theory, have allowed researchers to formally relate a wide variety of systems, such as the theory of rumour propagation, glucose…

Learning to create, use, and evaluate models is a central element of becoming a scientist. In physics, we often begin an analysis of a complicated system with highly simplified or toy models. In introductory physics classes, we tend to use them without comment or motivation. Some students infer that physics is irrelevant to their understanding of…

Excluded-volume interactions are ubiquitous to modeling the average size of polymers in solution. This paper shows how simulations can be used by students to explore the emergence of mathematical scaling relations from excluded-volume interactions. Simulations provide robust visual representations of the system, and can be used to investigate a…

Inspired by the approach first employed by C.S. Holling in his classic "disc experiment," this article provides a sequence of learning activities that increase students' understanding of the mechanisms behind saturating effects in predator-prey scenarios. The proposed lesson is recommended for inclusion in courses that address…