We adapt the Lanchester combat model to represent conflict between vampires and humans. It is assumed that vampires attack humans during the hours of darkness whilst humans attack vampires during the hours of light. The right-hand side of the differential equation model therefore depends upon the hour of the day. A key insight is that to answer…

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…

Calculus is perhaps the most widely taught and researched upper-secondary/post-secondary mathematics subject the world over. The research literature is amassing greater clarity around students' understandings of calculus, yet calculus instruction tends to be at odds with this literature, maintaining a focus on procedural aspects of the subject.…

Spring-mass systems are presented as an application of second-order, constant coefficient differential equations in many differential equations textbooks. The phenomenon of resonance can then be analysed after nonhomogeneous differential equations are introduced. With some simplifying assumptions, the movement of the roof of a one-story building…

We introduce a new metric--the classification power--to examine the effectiveness of postsecondary mathematics placement policies. This metric addresses the methodological challenges of contextualizing the effectiveness of a single placement policy and comparing the effectiveness of multiple placement policies across different student populations.…

This study explores opportunities to learn definite integrals in three widely-used textbooks in the US, Definitions, worked examples, and exercise problems were coded using research based cognitive resources in definite integrals to examine if widely-used textbooks provide students with opportunities to explore how two quantities are related to…

Many students struggle to grasp the concept of limits, leading to persistent learning obstacles. This study employed a didactical design research (DDR) approach involving 56 mathematics education students at Universitas Khairun, divided into two groups. Data from tests, interviews, and teaching interventions revealed three key obstacles:…

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…

Recent research on integration has shown the importance of quantities-based meanings for integrals. However, this research body is still in need of detailed empirical accounts of "how" students develop such understandings across an entire unit on integration. This paper contributes by providing one such account, based on a…

Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of…

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…

Background/purpose: The teaching of integral calculus has become increasingly important in higher education due to its role in the development of advanced mathematical thinking. In this context, the definite integral is recognised as a complex mathematical object with multiple interpretations. The present article aims to systematically review the…

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…

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.

Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to…