The present paper explores the relationship between attitudes towards mathematics (ATM) and achievement in college calculus in active learning (AL) and lecture-based (LB) classrooms. Previous work on this relationship has mainly been limited to LB instruction, neglecting the impact of innovative approaches such as AL. Less attention has been paid…

Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In…

The aim of the article is to discuss how university mathematics courses can become more relevant for pre-service teachers (PSTs). We therefore introduce a concept we call mathematical orientation and develop an analytical framework based on Felix Klein's pervasive approach to elementary mathematics from a higher standpoint and the work of the…

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…

Industrial-scale models require considerable setup time; hence, once built, they are used in myriad ways to consider closely related cases. In practice, the code for these models frequently evolves without appropriate notational choices, largely as a result of the lengthy development time of, and the number of individuals contributing to, their…

This study sought to investigate university mathematics teachers' perceptions on their general pedagogical knowledge (GPK) while teaching an applied mathematics course tailored for pre-engineering students at a public university in Ethiopia. Using a case study approach, data were collected through a Likert-scale questionnaire and semi-structured…

We introduce learning modules in cryptography that can be crafted to motivate many abstract mathematical ideas, and we illustrate with a sample module. These modules can be used in a variety of ways, such as the core for a cryptography course or as motivating topics in other courses such as abstract and linear algebra or number theory.

This study investigates when and how university students in first-semester introductory calculus interpret multiple representations of the same function. Specifically, it focuses on three tasks. The first task has students give their definitions of 'function sameness', the results of which suggests that many students understand a function as being…

This article presents an inquiry-oriented lesson for teaching Lagrange's theorem in abstract algebra. This lesson was developed and refined as part of a larger grant project focused on how to "Orchestrate Discussions Around Proof" (ODAP, the name of the project). The lesson components were developed and refined with attention to how well…

The introductory statistics course has gotten better over the years, but there are many content areas in STAT 101 that should be reconsidered.

This study contributes to Action, Process, Object, Schema (APOS) theory research by showing two approaches used by advanced mathematics students to construct relations between higher-order derivatives to solve complex problems. We show evidence of students' ability to perform Actions on their graphing derivative Schema, that is, of its…

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…

The multiplication principle (MP) is foundational for combinatorial problem-solving. From a units-coordination perspective, applying the MP with justification entails establishing unit relationships between the number of options at each independent stage of a counting process and the total number of combinatorial outcomes. Existing research…

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new…