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The eight papers making up this special issue on promoting women in mathematics consider the underrepresentation of women in mathematics and the experiences and observations of authors who are immersed in this work. In this editorial, we identify common themes we see across many of these papers, framing our discussion with the "social identity framework of mathematical development" presented in the first paper. This framework consists of three beliefs conducive to the pursuit of mathematics, and we discuss how the authors address each of these beliefs in their work. We also present data to illustrate that the underrepresentation of women in mathematics continues to be an issue, and we hope the papers in this issue will be informative for others who want to lead gender-based initiatives. [ABSTRACT FROM AUTHOR]

This article describes the outcomes of a faculty partnership between a math professor and a composition professor to improve the final paper in a research-focused senior capstone math course. Much attention has been paid in recent years to the importance of undergraduate research experiences and how to best create and support such endeavors. However, little space has been dedicated to helping math faculty design and scaffold writing assignments to support their students' development as scholars. We describe how our partnership took place, the changes we made to existing assignments and the creation of new ones, and an assessment of our efforts completed by an interdisciplinary group of faculty and students, which showed marked improvement in the areas of information literacy, critical thinking, and quality of writing. [ABSTRACT FROM AUTHOR]

In this paper, several research topics are presented that provide productive avenues of exploration for students in an undergraduate cryptography course. The topics are intended for a variety of skill levels and interests. In some cases, a student might be inspired to try to solve the proposed problem or write computer code. Other topics might motivate them to learn new mathematics or algorithms, or research a historical event or person. In all cases, a short description of the topic is given along with several references that can be used as a starting point for student research. The author also describes some challenges and recommendations when teaching a course where a project is a main component. [ABSTRACT FROM AUTHOR]

This editorial introduces the special issue, Resources for Undergraduate Cryptology. We begin by describing possible roles for cryptology in the undergraduate mathematics curriculum together with a brief overview of the subject. We conclude with a brief preview of each paper included in this issue. [ABSTRACT FROM AUTHOR]

The Mathematical Sciences Department at Shenandoah University focused on increasing the participation of women in the mathematics program by fostering a welcoming learning community and promoting double majors. Data from the past decade is presented. During this time, the program saw an increase in women undergraduate mathematics majors from 11% to as high as 71%. This paper describes the initiatives taken to make these changes. [ABSTRACT FROM AUTHOR]

The University of Minnesota Talented Youth Mathematics Program (UMTYMP) is a selective, five-year accelerated mathematics program for students in grades 6–12. During the program, students take college mathematics courses on University of Minnesota campuses, starting with algebra and continuing through logic and proofs, linear algebra, and multivariable calculus. The majority of UMTYMP students come from two demographic groups: White and Asian/Asian American. In 2020, we surveyed UMTYMP students to understand the impact of model minority stereotypes (MMS) on Asian/Asian American students, particularly girls, who are labeled as "gifted" and/or "talented." In this paper, we reveal the preliminary results of this study, discuss their implications, and provide recommendations for addressing the MMS in mathematics programs. [ABSTRACT FROM AUTHOR]

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum. [ABSTRACT FROM AUTHOR]

A project involving data related to socioeconomic indicators was administered in an introductory statistics course. As part of this project, student groups investigated a data set from the United States Department of Agriculture Food and Nutrition Service website with information on recipients of Supplemental Nutrition Assistance Program (SNAP) benefits. Pre/post data from the Miville-Guzman Universality-Diversity Scale (M-GUDS) short form were collected to measure changes in diversity awareness. The survey administered also included demographic information and open-ended questions. Based on study results (n = 48), there was moderate evidence of a change in the diversity of contact subscale of the M-GUDS short form. Written feedback indicated that students demonstrated empathy for individuals with potentially different lived experiences. The focus of this paper is to describe the study results and provide information on the importance of incorporating elements of diversity into curriculum. [ABSTRACT FROM AUTHOR]

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 recognize a continuous function from its graph. To help mitigate this problem, these visuals can be used alongside the definitions in analysis courses, and in calculus courses where the same definitions are used. The content is arranged into a series of pictures accompanied by questions, with suggestions for the instructor on how to guide students in their learning. The design allows students to make their own discoveries and connections. The investigations crescendo to a surprising fact: that almost all continuous functions are differentiable nowhere! [ABSTRACT FROM AUTHOR]

This paper describes the implementation of a STEM scholarship program which utilized a holistic approach to providing a multi-dimensional student support system. The program has been successful in encouraging and supporting women in Applied Mathematics and Computer Science by offering a diverse suite of extracurricular opportunities, actively engaging them in organized events, research projects, and participation in STEM communities, and helping them achieve higher GPAs and shorter times to graduation. The supported women also benefitted from close mentoring relationships with the faculty mentors. The program emphasized the development of empowering settings for women's engagement and achievement, which act to sustain and expand interest in mathematics and computing, and thereby help them to see themselves as future professionals in the field. [ABSTRACT FROM AUTHOR]

Sonia Kovalevsky Days (SK Days) are nationally-held outreach events that are commonly organized by local chapters of the Association for Women in Mathematics (AWM) to encourage young women to continue their study of mathematics. The AWM Student Chapter at Worcester Polytechnic Institute hosted virtual renditions of SK Day for middle school girls in Spring 2020 and 2021. This paper discusses the implementation of these events, including the virtual program logistics and five interactive online mathematics activities. Participant feedback suggests a successful transition from an in-person to online format of SK Day, with benefits to both the middle school participants and the undergraduate student volunteers. The activities and online resources described in this article have the potential to be utilized in future hybrid outreach events. [ABSTRACT FROM AUTHOR]

With this paper, we share an activity for the undergraduate mathematics classroom called The Harp Project that leverages the aesthetic nature of both the mathematical and the musical arts. This project was conceived as a STEAM/PBL project with the added feature that it was carried out in pieces by an entire class. Anecdotal evidence suggests that the uncommonly high level of students' interest and engagement in the experience and the communal feel of their participation can be attributed to this collective quality of the project's design. For this reason, we invite others to consider this curricular experience as it appears viable for supporting efforts to cultivate a broader population of students with an affinity for mathematics. [ABSTRACT FROM AUTHOR]

This paper reports on a novel corequisite design and implementation for College Algebra at Florida International University. The corequisite course uses online, just-in-time, prerequisite assignments delivered on an open-educational platform. Students get help from near-peer learning assistants inside a math emporium environment. The course remains three credits after the redesign and now supports all students who would otherwise be taking Intermediate Algebra as a prerequisite course. [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]

The Miller–Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography. [ABSTRACT FROM AUTHOR]

In this paper, I analyze the impact of culture and metaphor on cryptology education. I will compare and contrast the historically grounded metaphors of cryptology-is-warfare and encryption-is-security to a set of counter-metaphors: cryptology-is-privacy and encryption-is-communication. Using this explicit understanding of conceptual metaphor, I present design recommendations and metaphorizing activities for educators building their undergraduate cryptology course. [ABSTRACT FROM AUTHOR]

This curated collection covers a selection of PRIMUS articles published over a roughly 12-year period that focus on modeling and applications. The collection includes sections on individual projects, courses with a significant modeling component, and modeling and applications in extracurricular settings and throughout the curriculum. [ABSTRACT FROM AUTHOR]

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]

We studied aspects of undergraduate STEM majors' mathematical reasoning as they engaged in mathematically modeling a predator-prey scenario. The study used theoretical viewpoints on quantitative reasoning to inform scaffolding moves that would assist modelers in overcoming blockages to their mathematization of real-world problems. Our contribution is a set of four scaffolding moves contingent upon and responsive to participants' in-the-moment quantitative reasoning that guided them toward a meaningful model for a predator-prey scenario. [ABSTRACT FROM AUTHOR]

A multi-section coordinated math for social sciences course at a Hispanic-Serving Institution went through a major revision which included the addition of many resources. One resource was educational songs, including a lyric video on the three possible outcomes when solving a system of linear equations. The learning objective associated with this topic is one where learning (if not also teaching) may fall short, thus motivating a creative way to increase engagement. We discuss results from anonymous surveys of students and instructors to assess this educational song's potential to support student engagement (and hopefully learning), contextualized with qualitative data such as classroom observation. Recommendations for future assessment and class use are provided. [ABSTRACT FROM AUTHOR]

There is currently a lack of diversity in the discipline of statistics. In order to provide opportunities within the discipline of statistics to diverse groups, the present study presents a detailed description and evaluation of a one-credit workshop and mentoring program for undergraduate women interested in exploring opportunities within statistics. The key features of this program include career education, mentoring, and student–faculty interaction. A description of the program, along with key artifacts, is provided as a model for researchers and educators interested in creating similar experiences. The main components of this program were weekly class meetings; weekly reading assignments and written reflection; weekly near-peer mentoring meetings; and a final career portfolio. Initial evaluation of the program is conducted based on student and mentor feedback and results indicate that the program was generally well received. Areas for future work, including additional recruitment efforts and extending the program to additional populations, are explored. [ABSTRACT FROM AUTHOR]

Drawing on a social identity framework of mathematical development, the authors present a model, Improving Girls' Math Identity (IGMI), designed to address two key "leaks" in the female STEM pipeline: undergraduate and middle school. IGMI involves a supportive professional development network for undergraduate women preparing to transition into mathematics-related careers, and a mentorship program connecting middle school girls with these undergraduates to develop metacognition, problem-solving, and spatial skills. Preliminary evidence demonstrates that the model successfully strengthens undergraduates' problem-solving abilities, improves resilience and persistence in pursuing STEM fields, and increases investment in supporting future cohorts of women in mathematics and science. [ABSTRACT FROM AUTHOR]

Since its founding in 2013, the Kutztown University Association for Women in Mathematics student chapter has built a supportive community of women students and faculty on campus and beyond. This article aims to share our successful experiences with the broader community by presenting our pre- and mid-pandemic AWM activities including Mathematics Day for Girls, Women's Empowerment Day, Beyond Pink-Collar Jobs, and our Kutztown University AWM Virtual Lecture Series. Feedback from past and current participants of our AWM student chapter is also included. [ABSTRACT FROM AUTHOR]

Math Girls $ \mathbb {R} $ R ock! is a year-long, two-tiered mathematics mentoring program that prepares female undergraduate mentors to facilitate high school girls' engagement in challenging mathematics concepts through a dynamic after-school program. In this article, we describe the distinct educational component of this program in which female faculty members mentor female undergraduate mathematics and mathematics education students in researching and developing the content to be presented at the high schools. In addition, we discuss some of the feedback collected from program participants about various aspects of this program relating to the involvement of the undergraduate students. In conclusion, we share advice for those interested in starting and running similar outreach programs. [ABSTRACT FROM AUTHOR]

There has been and continues to be a substantial amount of work done to encourage, support, and retain women in mathematics. However, very little has been written about the faculty leading these gender-based initiatives, most of whom are women. This dynamic—women advocating on behalf of other women—is connected to identity-based activist movements. In the following, I describe how I have started to see gender-based work in STEM as a form of institutional activism. I then describe how this change of conceptualization led me to rethink the landscape of gender-based work in STEM. I reflect on literature from institutional activism and describe how that literature has changed my own engagement in gender-based service. My goal in this article is to provide math practitioners with some ideas for reflecting on their own actions in contesting historical forms of sexism. As we begin to understand the rules, norms, and cultures that constrain us, we will be better equipped to strategically contest them and create the sorts of institutional change that we want to see. I conclude the article with a series of questions to challenge all of us to think strategically about the future of our work supporting women in STEM. [ABSTRACT FROM AUTHOR]

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]

The number of undergraduates placing into developmental or "remedial" coursework continues to increase. This rise is concerning because developmental or "remedial" coursework creates barriers for many students. This mixed method study examines a holistic approach to placement. After one-on-one interviews with mathematics faculty, students (N = 8) within two points of passing our mathematics placement test (MPT) were given the opportunity to enroll in college-level coursework (Business Calculus) instead of the developmental coursework (Algebra). There were no significant differences in final exam and course grades between these students and those who initially placed into the course. Additionally, the qualitative data revealed many common themes and future ideas for intervention. [ABSTRACT FROM AUTHOR]

We investigated the effects of a major revision of the differential and integral calculus curriculum, the primary goal of which was to improve STEM retention. The revamped curriculum has greater emphasis on the power of computing to help visualize patterns and gain insights to better prepare students for STEM majors, and less emphasis on traditional and theoretical calculus content, such as the limit definition of the derivative. We investigated a comparison between 338 students who had taken the traditional sequence and 328 who had taken the revamped sequence during the Fall 2020 and Spring 2021 semesters. STEM retention, i.e., following through with the initial intent to major in a STEM field, was enhanced by 7% in the revamped group. STEM majors in the traditional group were found to over-perform in their other STEM classes by a 0.18 GPA margin, while non-STEM majors in the revamped group over-performed in their other STEM classes by a 0.17 GPA margin. Both differences are statistically significant. Focus groups were also conducted to gather and synthesize student perceptions. Results may encourage similar innovations to core math curricula in other universities to foster enhanced STEM retention. [ABSTRACT FROM AUTHOR]

We describe the components and implementation of an activity for multivariable calculus featuring applications to the field of chemistry. This activity focuses on the isobaric thermal expansion coefficient found using partial differentiation of the volume of an ideal gas with respect to temperature as pressure is held constant. Broader goals of this activity include enhancing calculus conceptual understanding, encouraging further interdisciplinary study and exploration, and cultivating mathematicians and scientists with broader and deeper backgrounds. [ABSTRACT FROM AUTHOR]

Over the past three years, a community of faculty and graduate teaching assistants has developed within the University of Kentucky Mathematics department centered around the use and development of student-centered assessment methods. In this article, we discuss the environmental factors that contributed to the growth of this community, our motivations for engaging in this work, and our surprisingly consistent successes. We conclude by sharing suggestions for experienced instructors interested in building communities around the theme of student-centered assessment methods. [ABSTRACT FROM AUTHOR]

Exercise is essential for mastering mathematics, but it faces two major hurdles. First, students are often not motivated to do their homework. Second, checking traditional homework is a manual and labor-intensive process that becomes harder to support as the number of students increases. We argue that computer games could alleviate both problems. In contrast to homework, students are willing to spend many hours playing video games. And because video games keep track of the gamers' performance, they offer a scalable solution to the problem of checking the homework. We describe an experiment to augment traditional homework with a computer game that we have developed for exercising the topics of symmetry groups in an undergraduate college course. We describe the mathematical background of the subject matter, how we have embedded the mathematics into the game, the empirical results of playing the game, and the insights we have gained from this experiment. [ABSTRACT FROM AUTHOR]

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]

A standard element of the undergraduate ordinary differential equations course is the topic of separable equations. For instructors of those courses, we present here a series of novel modeling scenarios that prove to be a compelling motivation for the utility of differential equations. Furthermore, the growing complexity of the models leads to the development of various types of separable equations. Finally, survey data of the student experience of modeling indicates a general appreciation for modeling tasks such as this and a positive indication of their educational value. [ABSTRACT FROM AUTHOR]

In introductory math courses such as precalculus, students often have difficulty with assignments that require explanations or synthesizing knowledge. When students receive low scores on such assignments, instructors often wish to help students understand the material they struggled with. A common strategy is to allow the students to revise and resubmit the assignment. In a study of a precalculus course, I investigate the effects on learning and confidence of two strategies: allowing homework revisions and offering extra points for meeting with the instructor to discuss the assignment. The data indicate that discussions appeared to have the greatest effect on quiz scores and confidence, although both strategies were beneficial when compared to using neither strategy. In addition, this article looks at student perceptions of the strategies. [ABSTRACT FROM AUTHOR]

This article describes the author's experience designing and implementing an inquiry-based learning (IBL) pedagogical approach to an upper-division undergraduate cryptology course. The author shares the course goals and how the IBL style supports their achievement. The article concludes with sample activities – in-class exercises that touch on one-time pad, the Birthday Problem, the Meet-in-the-Middle attack, and Diffie–Hellman key exchange protocol, and a programing assignment on the Vigenère cipher. [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]

We describe Shamir's secret sharing scheme and explain how it can be used for secure and redundant cryptocurrency storage. We include samples of individual and group assignments that can be used in an upper-division cryptology class for students who are familiar with modular arithmetic. It takes about one class to cover Shamir's secret sharing, but additional time can be spent on the described coding project about splitting mnemonic Bitcoin seeds into shares. We also provide references for topics of further study that can you use for student research projects. [ABSTRACT FROM AUTHOR]

We present a concrete situation where there is a difference between the theoretical security of a cipher and its limitations when implemented in practice. Specifically, when entering a PIN, smudges on the keypad substantially reduce its security. We show how the number of possible keys in the presence of the "smudge attack" can be calculated. Specifically, it turns out that in all interesting situations, a key with fewer smudges increases security. Possible extensions for use in an undergraduate discrete mathematics or cryptography class are presented. [ABSTRACT FROM AUTHOR]