We automate and randomize the building of linear systems with a parameter, appropriate for assigning to students. When the parameter takes on a specific value, the system has no solutions. When the parameter takes on a different value, the system has an infinite number of solutions.

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…

In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for…

Jordan canonical form (JCF) is one of the most important, and useful, concepts in linear algebra. Mathematics, physics, biology, science and engineering undergraduates often find the first application of real JCF in the discipline of differential equations (continuous models) to solving systems of differential equations. In this work, we apply…

The COVID-19 pandemic, like past historical events such as the Vietnam War or 9/11, will shape a generation. Mathematics educators can seize this unprecedented opportunity to teach the principles of mathematical modeling in epidemiology. Compartmental epidemiological models, such as the SIR (susceptible-infected-recovered), are widely used by…

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with…

In this article we consider two triangles: one inscribed in another. We prove that the area of the central triangle is at least the harmonic mean of the areas of corner triangles. We give two proofs of this theorem. One is based on Rigby inequality and the other is based on the known algebraic inequality, to which we bring a new, geometric, proof.…

The systematic design and analysis of tasks which can be implemented in first-year university courses and point to advanced inner- and extra-mathematically rich issues and their rationales is an open problem in the didactics of mathematics in higher education. Potentials of such tasks can be seen with regard to learning processes in the first year…

The Klein bottle and hairy ball theorem are important concepts in advanced mathematics and they are both examples of the Poincaré-Hopf theorem. Complex theories such as these usually remain unrealized in the minds of mathematicians. In this collaborative work between a mathematician and a designer, we introduce the concept of the hairy Klein…

Textbooks on Dynamical Systems Theory usually present only three types of codimension-one bifurcations occurring in autonomous first-order nonlinear differential equations. These three types are called saddle-node, transcritical, and pitchfork. However, the number of bifurcations is virtually infinite. In this didactic work, the bifurcation…

We present a model for roleplaying (RP) for university students that is aimed at fostering the interactions of each learner with the teacher and coursemates, self- and peer-assessment, as well as an awareness and meaningful learning of mathematics. Specific features of our model are its focus on metacognition and individualized learning, based on…

In undergraduate mathematics education, students' collaborations have gained a reputation as a 'good' learning practice. A more complex image emerges once collaborations are construed as an arena where cognitive, social, and affective matters intertwine in ways that can fuel and impede learning. In this study, I take a close look at a…

Flipped classrooms serve as a new pedagogical approach for teaching and learning, which involves switching from the traditional teacher-centred academic agenda. Unlike the traditional classroom model, the flipped classroom enables students to understand the lesson better while learning at their own pace and enhancing their communication skills…

This paper reports on themes that arose in an investigation of university lecturers' views on the teaching of linear algebra. This focus on themes was the initial part of a study concentrating on four areas: What is important to teach in a first course in linear algebra? Are there teaching methods which are particularly suited for such a course?…

This paper reports on data obtained during semi-structured interviews with students who had recently experienced a guided inquiry approach to learning ordinary differential equations (ODEs) in a service-taught module. Previous research identified the strengths and weaknesses of similar cohorts of students. The results from that research informed…