The historic total global human population dataset is available on Wikipedia and provides an opportunity for modelling with simple models such as the exponential and logistic differential equations for population. Using the per-capita population growth rate (PPGR) predicted by these two models and estimated PPGR from the data, we are able to…

The logistic differential equation is ubiquitous in calculus and differential equations textbooks. If the model is developed from first principles in these courses, it is usually done in an abstract mathematical way, rather than in one based in ecology. In this short note, we look at examples of how the model is derived in mathematical texts and…

Research into didactics of calculus has maintained a long-standing interest in students' grasp of the relations between definite integrals and areas. This study comes to contribute to this line of research by unpacking how students use the concept of area to find definite integrals. Specifically, we focus on mathematical situations where the…

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…

Adjusting the Calculus I curriculum by putting modelling and differential equations literally at its centre leads to a better-organised and better-motivated course. The biggest change is including a section on "qualitative" and "numerical" solutions to ordinary differential equations between the customary sections on…

We have introduced a novel approach to competency-based education in mechatronics from the undergraduate to the postgraduate level. What distinguishes this approach is the integration of modeling and control of sampled systems right from the beginning of the undergraduate education. It is achieved by changing the structure of the first-semester…

On 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near…

We present an example of one of the modelling projects we assign to students in our differential equations classes. Students are asked to determine how to run a cost-efficient hot water heating system. We consider a cylindrical tank filled with water and heated by a heating element immersed in it. Together with students we discuss physical laws…

The main objective of this study is to develop a mathematical modeling framework for a deeper understanding of dynamics of math anxiety as a contagious process. Borrowing from theories of the spread of infectious disease, we develop two classes of mathematical models representing the spread of math anxiety in math gateway classes. The first…

Mathematical modelling is an interlinking process between mathematics and real-world problems that can be applied as a means to increase motivation, develop cognitive competencies, and enhance the ability to transfer mathematical knowledge to other areas of science, such as engineering disciplines. This study was designed to investigate the effect…

We studied aspects of undergraduate STEM majors' mathematical reasoning as they engaged in mathematically modeling a predator-prey scenario. The study used theoretical viewpoints on quantitative reasoning to inform scaffolding moves that would assist modelers in overcoming blockages to their mathematization of real-world problems. Our contribution…

In this paper, we discuss mathematical modeling opportunities that can be included in an introductory Differential Equations course. In particular, we focus on the development of and extensions to the single salty tank model. Typically, salty tank models are included in course materials with matter-of-fact explanations. These explanations miss the…

Prior research on students' productive understandings of definite integrals has reasonably focused on students' meanings associated to components and relationships within the standard definition of a limit of Riemann sums. Our analysis was aimed at identifying (i) the broader range of productive quantitative meanings that students invoke and (ii)…

In the precalculus curriculum, logistic growth generally appears in either a discrete or continuous setting. These actually feature distinct versions of logistic growth, and textbooks rarely provide exposure to both. In this paper, we show how each approach can be improved by incorporating an aspect of the other, based on a little known synthesis…

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…