Teaching determinants poses significant challenges to the instructor of a proof-based undergraduate linear algebra course. The standard definition by cofactor expansion is ugly, lacks symmetry, and is hard for students to use in proofs. We introduce a visual definition of the determinant that interprets permutations as arrangements of…

In this paper I discuss my experience in using the inverted classroom structure to teach a proof-based, upper level Advanced Calculus course. The structure of the inverted classroom model allows students to begin learning the new mathematics prior to the class meeting. By front-loading learning of new concepts, students can use valuable class time…

Students struggle with proving, a fundamental activity in upper-level undergraduate mathematics courses. Learning how to prove is a difficult transition for students, as they shift from largely computation-based to argument-based work. In response, mathematics departments have instituted courses, introduction or transition to proof, designed to…

At the University of Paderborn, the course "Introduction into the culture of mathematics" is required for all first-year students who enter the study program for future mathematics teachers at lower secondary level (grade 5-10). In this inquiry-based transition-to-proof course, we use four different kinds of proofs (the generic proof…

This study investigates one Calculus student's meanings for quantifiers in Calculus statements involving multiple quantifiers. The student was asked in a two-hour long clinical interview to evaluate and interpret the Intermediate Value Theorem (IVT) and three other statements whose logical structure was similar to the IVT except for the order of…

Rouché's Theorem is a standard topic in undergraduate complex analysis. It is usually covered near the end of the course with applications relating to pure mathematics only (e.g., using it to produce an alternate proof of the Fundamental Theorem of Algebra). The "winding number" provides a geometric interpretation relating to the…

Specifications grading is a version of mastery grading distinguished by giving students clear specifications that their work must meet, and grading most things pass/fail based on those specifications. Mastery grading systems can get quite elaborate, with hierarchies of objectives and various systems for rewriting and retesting. In this article I…

Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students…

The majority of American college students pursue a degree outside one of the STEM fields (US News & World Report, 2015), yet these same students must still complete a minimum level of coursework in mathematics for graduation. The intent of liberal arts math courses is to help students who are majoring in one of the liberal arts successfully…

This study aims to evaluate the university lecturers' teaching content preferences in different teaching environments at the undergraduate level of mathematics. The content that lecturers use in teaching the concept of integral is evaluated through a comparison of face-to-face and virtual classroom settings. With comparative case study being used…

We hope to initiate a discussion about various methods for introducing Cauchy's Theorem. Although Cauchy's Theorem is the fundamental theorem upon which complex analysis is based, there is no "standard approach." The appropriate choice depends upon the prerequisites for the course and the level of rigor intended. Common methods include…

We describe a successful collaboration between mathematics and psychological science faculty members to create a learning community for our students that linked sections of introductory mathematical reasoning and psychological science courses. The students in our learning community were in their second or third semester. The learning community is…

Instructors often write feedback on students' proofs even if there is no expectation for the students to revise and resubmit the work. It is not known, however, what students do with that feedback or if they understand the professor's intentions. To this end, we asked eight advanced mathematics undergraduates to respond to professor comments on…

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, to…

We describe some classic experiments on the Möbius strip, the projective plane band, and the Klein bottle band. We present our experience with freshmen college students, college teachers, high school students, and Mathematics Education graduate students. These experiments are designed to encourage readers to learn more about the properties of the…