Search

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 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]

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 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 they supported active and broad student participation. By guiding student exploration of a few key example groups and structuring the exploration to identify recurrent structure, students formulate key lemmas that they can combine to prove Lagrange's theorem. [ABSTRACT FROM AUTHOR]

The algebraic group $ \operatorname {Spin}_{3\times 3} $ Spin 3 × 3 arises from spinning collections of the numbers 1–9 on a $ 3\times 3 $ 3 × 3 game board. The authors have been using this group, as well as a corresponding online application, to introduce undergraduate students to core concepts in group theory. We discuss the benefits of using this deceptively simple, toy-like puzzle in terms of student learning and engagement. Practical exercises as well as use cases outside the abstract algebra classroom are provided at the end. [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]

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]

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]

Though posters have been used as a pedagogical tool in a variety of fields, the literature suggests that the use of posters as an educational assessment tool by the mathematics community has been limited. This is unfortunate given the numerous potential benefits of a poster, including the development of oral communication skills and the ability to examine a mathematical topic at a deeper level. We describe the use of a poster project in an introductory abstract algebra course with specific implementation guidance including a schedule of assignments and assessment criteria. We also include reflections on the success of this project over several semesters as well as advice for adapting the project to a remote course. Samples of student posters are available upon request to the authors. [ABSTRACT FROM AUTHOR]

We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor's use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor's simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students' referents, amplified students' contributions to develop conceptual fluency, and linked students' body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete. [ABSTRACT FROM AUTHOR]

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]

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 (to the students) mathematics. In this way, students come to creatively establish new concepts that they may take ownership of. I provide insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing the lesson. [ABSTRACT FROM AUTHOR]

We present a detailed description of a Cryptography and Computer Security course that has been offered at Knox College for the last 15 years. While the course is roughly divided into two sections, Cryptology and Computer Security, our emphasis here is on the Cryptology section. The course puts the cryptologic material into its historical context and also includes reference to ciphers used and discussed in literature. Through lectures and assignments on cryptanalysis, the course motivates students to learn a variety of topics in advanced discrete mathematics, including discrete probability, elementary statistics, number theory, and abstract algebra. The students have found this approach motivating and we believe our experiences are relevant to instructors teaching this material in a variety of different courses and institutions. [ABSTRACT FROM AUTHOR]

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]

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]

The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around group activities that build intuition for abstract concepts while avoiding excessive technical machinery. Teaching this course presents some unique challenges, as much of the content typically appears in a second course in abstract algebra, yet the students exhibit a wide range of mathematical preparation. [ABSTRACT FROM AUTHOR]

As mathematics educators we want our students to develop a natural curiosity that will lead them on the path toward solving problems in a changing world, in fields that perhaps do not even exist today. Here we present student projects, adaptable for several mid- and upper-level mathematics courses, that require students to formulate their own questions and to begin to develop the basic research skills needed to answer these questions. These projects, where each student is given an individualized object to study, allow students to take ownership over their own learning while introducing them to the joy and challenge of discovery and research. Each student is directed to use the concepts and techniques presented in class as a set of tools to guide the investigation of their object. We discuss our experiences–both positive and negative–with these inquiry-based projects. [ABSTRACT FROM PUBLISHER]

We describe the design and implementation of lessons in undergraduate abstract algebra that integrate applications to teaching high school mathematics. Each lesson consists of a pre-activity, class activity, homework set, assessment questions and detailed instructor notes. The included applications intend to bridge advanced mathematics concepts to secondary school mathematics content for prospective secondary mathematics teachers, as well as provide instructors with a useful resource. Each lesson includes four guiding features: addressing meaningful content; making connections to mathematics teaching; emphasizing active learning; and providing comprehensive notes for instructors. Interviews with instructors and their students document how the instructors adapted the lessons to their classrooms and how the lessons influenced undergraduates' mathematical understanding. [ABSTRACT FROM AUTHOR]

Almost all finite groups encountered by undergraduates can be represented as multiplicative groups of concise block-diagonal binary matrices. Such representations provide simple examples for beginning a group theory course. More importantly, these representations provide concrete models for "abstract" concepts. We describe Maple lab assignments which explore group actions, subgroups, normality, cosets, quotient groups, homomorphisms, isomorphisms, and kernels. [ABSTRACT FROM AUTHOR]

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number three, whose proofs are explanatory in the sense of Steiner. Explanatory proofs transform what might initially seem mysterious or even magical into lucid mathematics. We propose possible ways to use these statements and others similar to them in a course in abstract algebra emphasizing proofs, or in an introductory course on proofs based on group theory. [ABSTRACT FROM AUTHOR]

The purpose of this article is to share a new approach for introducing students to the definition and standard examples of Abelian groups. The definition of an Abelian group is revised to include six axioms. A bullseye provides a way to visualize elementary examples and non-examples of Abelian groups. An activity based on the game of Jenga is used to help students determine for themselves the sets and operations that combine to form Abelian groups. A second collection of more advanced examples and non-examples of Abelian groups is also represented on a bullseye. This article is intended for those who teach undergraduate abstract algebra. [ABSTRACT FROM PUBLISHER]

The Parity Theorem states that any permutation can be written as a product of transpositions, but no permutation can be written as a product of both an even number and an odd number of transpositions. Most proofs of the Parity Theorem take several pages of mathematical formalism to complete. This article presents an alternative but equivalent statement of the Parity Theorem, replacing “transposition” with “adjacent transposition,” as well as a proof based on an inversion count technique. To build an intuitive understanding of the concept of inversion and to help students discover the Parity Theorem on their own, this article presents a series of five classroom activities based on the group-building initiative “Telephone Pole Shuffle.” [ABSTRACT FROM AUTHOR]

Algebraic structures are a necessary aspect of algebraic thinking for K-12 students and teachers. An approach for introducing the algebraic structure of groups and fields through the arithmetic properties required for solving simple equations is summarized; the collective (not individual) importance of these axioms as a foundation for algebraic thinking is discussed by way of an example of an abstract group. Results of this approach in a mathematics for teachers course are presented, with K-12 teachers (n=12) having increased their content knowledge both about arithmetic properties and abstract algebra, as well as having their teaching beliefs and intended practices influenced. Implications for classroom teaching about a vertical slice of the K-12 curriculum related to algebraic structure are examined. [ABSTRACT FROM PUBLISHER]

This article reports on techniques of teaching abstract algebra which were developed to achieve multiple student objectives: reasoning and communication skills, deep content knowledge, student engagement, independence, and pride. The approach developed included a complementary combination of inquiry-based learning, individual (not group) homework problems, oral exams, and student portfolios. Students in this class peformed better on the final exam and showed a greater understanding of concepts and proving techniques than students in previous semesters. [ABSTRACT FROM PUBLISHER]

We describe a simple photographic assignment appropriate for an abstract algebra (or other) course. Students take digital pictures around campus of various examples of symmetry. They then classify these pictures according to which of the 17 plane symmetry groups they belong. [ABSTRACT FROM AUTHOR]

Carry Groups are a wonderful collection of groups to introduce in an undergraduate Abstract Algebra course. These groups are straightforward to define but have interesting structures for students to discover. We describe these groups and give examples of in-class group projects that were developed and used by Miller. [ABSTRACT FROM AUTHOR]