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This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included. [ABSTRACT FROM AUTHOR]

Connecting and comparing across student strategies has been shown to be productive for students in elementary and secondary classrooms. We have recently been working on a project converting such practices from the K–12 level to the undergraduate classroom. In this paper, we share a particular instantiation of this practice in an abstract algebra setting. Students compare across two common proof approaches to showing that the Abelian property is preserved by isomorphism. We share a complete sample lesson where students make sense of the theorem and the two proof approaches, then leverage the differences between them in order to modify both proofs and mathematical statements. We conclude with the students' reflections on the activities, and share our learnings from adapting best practices from K–12 to this new setting. [ABSTRACT FROM AUTHOR]

This paper presents an extended project that offers, through American football, an application of concepts from enumerative combinatorics and an introduction to proofs course. The questions in this paper and subsequent details concerning equivalence relations and counting techniques can be used to reinforce these new topics to students in such a transition-to-proofs-type course or to offer an application of known theoretical mathematics to students with more advanced understanding in the major. [ABSTRACT FROM PUBLISHER]

Practices such as making connections between topics, prioritizing content, and identifying broad themes are essential to learning mathematics. This paper describes a project designed to integrate these synthesizing activities into an abstract algebra class. Students used a classroom wall to record and organize their collective learning and developing knowledge, with an emphasis on visual connections. The evolution of the wall reflected the development of students’ thinking throughout the course, and helped to personalize the classroom space. Suggestions for implementing similar projects in different courses and spaces are also included. [ABSTRACT FROM AUTHOR]

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 start writing proofs. This is the technique of writing proof frameworks, which is based on the logical structure of the statements of the theorems and associated definitions. Also, in order to unpack the conclusion and know what is to be proved, students need definitions to become “operable.” [ABSTRACT FROM PUBLISHER]

In this editorial introduction to the Special Issue on Mathematics and the Arts in the Undergraduate Classroom, we explore how the explosion of interest in connections between math and the arts over the past several years has moved into the classroom. We describe papers in this special issue that report on a range of classroom experiences combining math and the arts in ways that enhance student learning and expand student appreciation of both mathematical and artistic ways of thinking. [ABSTRACT FROM PUBLISHER]

This paper details an inquiry-based approach for teaching the basic notions of rings and fields to liberal arts mathematics students. The task sequence seeks to encourage students to identify and comprehend core concepts of introductory abstract algebra by thinking like mathematicians; that is, by investigating an open-ended mathematical context, identifying patterns, and venturing conjectures. A sequence of open-ended instructional tasks that aim to capitalize on students’ prior experiences with equation solving is provided along with notes and sample student responses for prospective instructors. [ABSTRACT FROM PUBLISHER]

The value of using manipulatives to teach mathematics is well-documented by education research. Even at the undergraduate level, concrete models can enable students to explore mathematical concepts with hands-on activities. Using food as a manipulative is a time-honored tradition in my courses and adds some extra interest and motivation for students as they learn mathematics. In this paper, I describe some successful applications of “food for thought” with examples from courses such as Precalculus, Calculus, Statistics, Geometry, and Proof-Writing. [ABSTRACT FROM PUBLISHER]

Secondary school mathematics teachers often need to answer the “Why do we do that?” question in such a way that avoids confusion and evokes student interest. Understanding the properties of number systems can provide an avenue to better grasp algebraic structures, which in turn builds students’ conceptual knowledge of secondary mathematics. This paper investigates the number systems from K-12 mathematics, discusses the properties held in each, and diagrammatically establishes connections between the number systems and algebraic structures. [ABSTRACT FROM PUBLISHER]

In this paper we discuss an interesting class of noncommutative rings which can be constructed using finite directed graphs. This construction also creates a vector space. These structures provide undergraduate students connections between ring theory and graph theory and, among other things, allow them to see a ring unity element that looks quite different from the ones they are accustomed to using. [ABSTRACT FROM PUBLISHER]

Mathematics is a creative process, and unfortunately, that process is often hidden from students of the discipline. This is certainly the case in the area of mathematics commonly referred to as abstract algebra. Current pedagogy conceals from the student many of the great ideas generated by significant problems in the history of the discipline. In his famous paper Memoirs on the Conditions for Solvability of Equations by Radicals [1, Appendix 1], Evariste Galois introduced the mathematical term group and identified a specific property of groups, now known as solvability, which enabled him to translate his original problem from the theory of equations into an equivalent problem within the newly established theory of groups. This article will explore Galois' definitions of group, conjugate group, normal subgroup, and solvable group and suggest how they can be used to enhance the learning of students as they encounter these concepts in a first course in abstract algebra. [ABSTRACT FROM AUTHOR]

Even in the highest-level mathematics courses at undergraduate institutions, most students do not read mathematical research from professional journals. Such depth of content is often reserved for “independent studies” or summer REU's. In this article I describe a course that required students to read research papers, all of which centered on a single theme. The course covered advanced group theory topics while conveying the notion that there is a mathematical community that works together to advance mathematical knowledge. Using a single story as an integrative thread is an approach that can be adapted to any topic in mathematics. [ABSTRACT FROM AUTHOR]

This paper describes writing assignments from a modern algebra course which were used to foster the development of proof-writing skills. Each assignment consists of proofs and mathematical work supposedly written by fictional characters, where the proofs are designed to illustrate common misconceptions, difficult points, and typical writing flaws. Students critique the fictional work and write short essays on their analysis. The response from students has been very favorable, especially among future teachers. [ABSTRACT FROM AUTHOR]

Mathematics majors' study of abstract algebra should provide these students with opportunities to connect what they are learning to their prior experiences with algebra in high school. This paper illustrates how such connections can be used to motivate the notion of binary operation and the axioms for a group. [ABSTRACT FROM AUTHOR]