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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]

College algebra fulfills general education requirements at many colleges in the United States. The study reported here investigated differences in mathematics achievement between undergraduates in college algebra classes using one of two homework methods:WeBWorK, an open-source system for web-based homework, or traditional paper-and-pencil homework. We assessed learning for 439 students in 19 college algebra classes at a large public university in the United States. Twelve classes usedWeBWorKand seven had traditional paper-and-pencil homework. Analysis of covariance revealed no significant differences in algebra performance or achievement gain by homework group, ethnicity or gender when statistically controlling for previous mathematics achievement. Results support the conjecture thatWeBWorKis at least as effective as traditionally graded paper-and-pencil homework for students learning college algebra in moderately sized lecture-based classes. [ABSTRACT FROM AUTHOR]

As calculus instruction moves further from lecture-based instruction toward student-centered pedagogy, support services must align with the changing goals for the course. This paper describes the work of a department as they collectively design and redesign a co-requisite Calculus I course, in an effort to better promote students' academic and social integration into active, student-centered calculus classrooms. We highlight the inclusion of more cognitively demanding and group-worthy tasks, re-negotiation of support for procedures and fundamental concepts, and promotion of collaborative participation. Our discussion is supplemented with reflections from faculty, graduate student instructors, and students. [ABSTRACT FROM AUTHOR]

This section calls for the submission of papers for occasional special issues of the journal "PRIMUS."

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 case study examines how online mathematics homework impacts student learning, attitude, and efficacy in a college Calculus I class. This research is important because it provides data on student perceptions for how to effectively implement online homework. Participants for this study completed homework using a blend of online and paper formats. We use a mixed methods approach to analyze quantitative survey data and qualitative data collected in focus groups and interviews. Our findings indicate students benefited from using a combination of paper homework and online homework. Recommendations are made for how instructors can utilize the benefits of both approaches to support student learning, attitude, and efficacy. [ABSTRACT FROM AUTHOR]

The idea of cutting out a curve while looking at the direction of cut of the scissors is used to explore concepts related to curves, tangents and derivatives, especially tangent line, concavity and points of inflection, curvature, and a curve as envelope of tangent lines. [ABSTRACT FROM AUTHOR]

The articles discusses the call for submission of papers for the publication Primus' Special Issues, including the requirements and process of selection for the articles that will be published, guest editors like Gavin Larose and Krista Maxson, and the journal's themes.

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]

We discuss an upper division applied statistics course that has been structured to satisfy our college's writing intensive requirements within the mathematics major. In this course, students complete two projects, performing both a statistical survey and a controlled experiment, and then write up their work on each project in a formal paper. After each paper, the students read a group of papers written by their peers and then critique and rank these papers. Next, the students meet in groups, having read the same papers, and together they must form a consensus about their group rankings. Finally, after analyzing and discussing several papers from their peers, the students revise their own papers. [ABSTRACT FROM AUTHOR]

Several major themes related to rethinking assessment emerged from a search of the papers published in PRIMUS. Authors explored ways to change the stakes of assessments, often with outcomes-based grading and testing structures. They explored changing the structure of assessments by considering media beyond written exams and by configuring them as group activities. They explored ways to change the focus of their assessments by asking students to synthesize and engage messy applications/projects and by making metacognition part or all of the assessment activities. And they explored issues around assessment as part of a larger system by considering approaches to grading in general, alignment with other course activities, workload, and perceptions and expectations about assessment in mathematics. In this PRIMUS Curated Collection, I discuss these themes and offer some editorial perspective. [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]

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]

As studies repeatedly show the benefits of students participating in undergraduate research, off-campus research opportunities have become increasingly competitive. In order to provide our students with research experiences, students from Murray State University and Centre College engaged in a year-long cross-institutional undergraduate research experience. In this paper, we summarize our experience and provide some useful perspectives in mentoring students in cross-institutional undergraduate research. We present effective methods for improving distance collaboration and strategies for eliciting the best research from students. [ABSTRACT FROM AUTHOR]

The article offers information on a special issue of the journal "Primus" featuring the topic of teaching inquiry, in relation to the Contributed Paper Session on Teaching Inquiry of the Joint Mathematics Meetings (JMM) in 2015.

This editorial provides an introduction to the special issue, Infusing active learning into precalculus and calculus courses: Insights and lessons learned from mathematics departments in the process of change. We begin with providing the background for the special issue, including an overview of the current status of active learning in precalculus through calculus, what many of the papers in this issue mean by the term "active learning" (inquiry based mathematics education), and a brief overview of the literature on institutional change. We then offer an overview of each of the three issues, explaining how various papers relate to each other within each issue. We conclude the editorial with some reflections on the institutional changes efforts in light of the COVID-19 pandemic. [ABSTRACT FROM AUTHOR]

In this article, I will discuss a collective research project that I designed for my History of Mathematics course. My students, who are by and large pre-service teachers, explored online, digital versions of 18th-century British almanacs that contained question-and-answer sections for mathematics. In a multi-stage research process, they explored the primary sources, investigated selected mathematical problems from the almanacs, conducted prosopographies of the contributors to the almanacs, and finally surveyed secondary sources. They then presented their results in pairs to their classmates. Through these presentations, as well as the prosopographies and notes they shared on our online classroom management system, the working pairs of students wrote research papers that placed their findings within the context of the research of the whole class. This collective research project allowed my students to begin focused and purposeful research within the first week of classes. Moreover, this project encouraged them to become highly engaged with their primary sources and enabled them to work as a team. This collective research project framework could be applied to a variety of groups of primary sources from the history of mathematics (such as textbooks and journals), now widely available online, to a variety of mathematics courses. [ABSTRACT FROM PUBLISHER]

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]

The two parts in this special issue address interdisciplinary conversations involving mathematics courses in the first two years of undergraduate work. The special issue was inspired by work funded by the National Science Foundation under a grant titled "SUMMIT-P." The special issue includes papers written about projects both from SUMMIT-P and from outside of SUMMIT-P. Part I focuses on frameworks and models of interdisciplinary collaboration. [ABSTRACT FROM AUTHOR]

In this paper, the authors discuss their experiences with ungrading at a small public university in the U.S. and a large public university in Germany. The courses described are Calculus 1 and a content course for pre-service secondary teachers of mathematics. The professors teach differently: one with lecture, the other with inquiry-based learning, and choose to use elements of ungrading in different ways. In one case, the professor lets students self-evaluate their homework, while in the other the professor uses a mix of specifications grading and ungrading for the final grade. We outline and compare our assessment approaches and discuss student performance and feedback. We also present some patterns relating to gender: in both cases, we observed that women students chose lower self-evaluations than men students. We end this paper with a general reflection on our teaching experience and our plans to use ungrading in the future. [ABSTRACT FROM AUTHOR]

Texas A&M University-Commerce has implemented supplemental instruction (SI) for Calculus since the fall of 2013. A direct display of the ABC rates from fall 2005 to spring 2021 only presents fluctuation from semester to semester, and one can hardly deduce that the ABC rates are increasing. In this paper, the "moving average" technique is used to reveal the effectiveness of SI in improving the percentage of students earning a grade of A, B, or C (ABC rates) in Calculus. Our longitudinal data under the "moving average" also shows that the SI implementation increased the ABC rates of major ethnic groups. [ABSTRACT FROM AUTHOR]

We provide an introduction to the special issue on Teaching Inquiry, through its motivation and themes, focusing here on Part II: Implementing Inquiry. [ABSTRACT FROM PUBLISHER]

Mathematics is not just the queen of the sciences; it permeates through art, commerce, and in fact, every aspect of modern life. However, while we as mathematicians see beauty in the mathematical way of thinking and problem-solving, our students tend to think of mathematics as a set of rules and abstract computations. Our pleasure of doing mathematics usually does not transfer over to our students. In this paper, we describe two liberal arts general education courses which seek to develop a broad appreciation of mathematics for students who may not require any technical mathematics course for their academic major or chosen career path. We present mathematics as it relates to everyday life, with the goal of improving student appreciation of mathematics. [ABSTRACT FROM AUTHOR]

This paper introduces a course design for a post-Calculus undergraduate course on mathematical modeling for the life sciences. The course includes differential equations, discrete, and cellular automata approaches (among others) to modeling questions in the life sciences, as well as a variety of biological topics. The course focuses on model set-up and revision for these different scenarios and approaches as well as contemporary science approaches. Changes in student ranking of biology and math as well as student feedback are included. This paper may be of interest to any faculty member designing or revising a similar course or inserting modules on these topics into any related course. [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 presents ideas for constructing mathematics homework assignments based on lessons from the cognitive science of learning and memory. In particular, we focus on two popular techniques from the field: spaced practice and mixed practice. The paper describes the techniques and supporting research, and then discusses various straightforward ways to incorporate them into mathematics instruction via homework assignments. The ideas presented can be applied at virtually any level of instruction and may be of interest to educators looking to implement research-based practices from cognitive science. [ABSTRACT FROM AUTHOR]

In this paper, I share an activity that reinforces students' understanding of translations, reflections, and rotations via body movement. As a result of this activity, students gain a different perspective compared to previous explorations on paper, communicate mathematics, and visualize transformations as rigid motions on the plane. Furthermore, the combination of sensation and perception helped some students assess whether their work was correct. [ABSTRACT FROM AUTHOR]

In this paper, we study how well SAT scores, high school GPA, and undergraduate precalculus grades relate to undergraduate calculus grades at a 4-year private nonprofit university that does not have a placement exam. In addition, we compare the success in calculus of two groups of students, both of whom require two semesters to pass calculus. The first group takes calculus twice (failing the first attempt) and the second group takes precalculus followed by calculus. [ABSTRACT FROM AUTHOR]

This paper presents a series of basic computational problems that are mathematically and/or graphically appealing, and provides an idea of places one might go in trying to understand what is happening, integrating mathematics, computation, and graphics. The real point of this paper is to make a case, through those examples, for computation as an early and integral part in the undergraduate mathematics curriculum. By computation we mean the ability to rapidly and easily program exploratory calculations, and as well display the results with a versatile suite of graphics capabilities. The problems used here, and the philosophy behind them, derive from a MATLAB-based course in computation, programming, and mathematical applications that the author has taught for over 10 years. [ABSTRACT FROM AUTHOR]

The two parts in this special issue address interdisciplinary conversations involving mathematics courses in the first 2 years of undergraduate work. The special issue was inspired by work funded by the National Science Foundation under a grant titled "SUMMIT-P." The special issue includes papers written about projects both from SUMMIT-P and from outside of SUMMIT-P. Part II focuses on what the results of interdisciplinary collaboration look like in the classroom. [ABSTRACT FROM AUTHOR]

This issue is the second of a special PRIMUS two-part issue collecting articles on undergraduate research from experienced faculty mentors. We offer it as a valuable resource for faculty leading undergraduate research programs. This issue presents a collection of papers offering advice on a variety of specific topics important for leaders of undergraduate research programs. Issues of finding and designing appropriate and accessible research projects, assessing undergraduate research, and publicizing it in the media are addressed. [ABSTRACT FROM AUTHOR]

This issue is the first of a specialPRIMUStwo-part issue collecting articles from experienced faculty mentors. We offer it as a valuable resource for faculty leading undergraduate research programs. It is inspired by the “Regional Faculty Workshop on REU Issues” in 2013 and the Joint Mathematics Meeting 2015 Mathematical Association of America’s “Themed Session on Perspectives and Experiences on Mentoring Undergraduate Students in Research.” This issue consists of seven papers focusing on the larger issues involved in creating, running, and sustaining an undergraduate research program in mathematics. [ABSTRACT FROM AUTHOR]

This paper is written to provide ideas and guide faculty who want to design a mathematical modeling course for undergraduate mathematics majors and minors. We discuss course goals, assignments, and projects that can be used to help students gain experience relevant for careers and mathematical modeling opportunities. The authors designed this course to build students' mathematical thought processes and toolbox, ability to analyze and evaluate mathematical models, mathematical modeling skills, and teamwork skills. The course described is intended as an upper division undergraduate course with a prerequisite of differential equations. [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]

Lesson study, a popular method of professional development in which instructors collaborate to critique and improve a lesson, can be used for curriculum development. This paper describes an implementation of lesson study that went through two iterations across different, related, contexts: an undergraduate mathematics content course for secondary preservice teachers and a professional development session with practicing middle and secondary teachers. We describe the ways in which the lesson study process helped us develop curriculum for advanced undergraduate courses for future teachers and provide our insights into the benefits and challenges of this approach. [ABSTRACT FROM AUTHOR]

Experienced provers employ a host of skills when assessing the validity of a justification, often without names for those skills. This paper offers an introduction to a lens called Toulmin analysis that can help make sense of this process. Then this paper describes both an in-class module to help students learn to apply Toulmin analysis and multiple ways this module can be integrated into courses for mathematics majors and non-majors. Toulmin analysis has helped us bring human dimensions of proving to the surface in our classrooms, and it helps us build stronger and more inclusive entryways for students into mathematical communities. Learning about Toulmin analysis helps our students connect mathematical reasoning to other thinking, supporting transfer in both directions, and it has changed our thinking about the teaching and learning of mathematics even in contexts where we do not devote class time to this model overtly. [ABSTRACT FROM AUTHOR]

This paper describes and assesses the implementation of online homework in a first-semester calculus course. Comparing sections of the course before implementation to those after, we find statistically significant improvements in retention rates, measures of student engagement, and participation on homework. We do not, however, find statistically significant changes in the proportions of students passing the common final exam or earning a C grade or better for the course. We conclude with a discussion of qualitative aspects and implications of our implementation of online homework. [ABSTRACT FROM PUBLISHER]