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In this note we explain the importance of clarity and give other tips for mathematical writing. Some of it is mildly opinionated, but most is just common sense and experience. [ABSTRACT FROM AUTHOR]

The article offers the author's insights on guides for research papers including in organizing principles, imagery and metaphors, and story of the development. Topics discussed include the research literature management, the meta-data analysis, and the project management. Also mentioned are the modes of reasoning, the human expression, and the mathematics education management.

For high school and college instructors and students, this paper connects number systems, field axioms, and polynomials. It also considers other properties such as cardinality, density, subset, and superset relationships. Additional aspects of this paper include gains and losses through sequences of number systems. The paper ends with a great number of activities for classroom use. [ABSTRACT FROM AUTHOR]

This paper discusses our new approach to assessing student learning. This approach includes the use of a final project rather than a cumulative exam. In particular, we measure learning by assessing students' ability to connect mathematics to the real world via a final project. We suggest that students taking a deep dive into one particular math concept and being able to make connections between that concept and the real world are educational achievements during this pandemic. We also argue that there is value in online learning because students who learn online choose to use library resources and develop their own interests by attending office hours, all of which benefit their learning. [ABSTRACT FROM AUTHOR]

Mathematics can be used as a tool to question and critique society and, in doing so, give us more information about the world around us and how it operates. This however, is not a common perspective that is conveyed to students during their undergraduate mathematics coursework. This paper contributes to the understanding of how undergraduate mathematics students question and critique society via mathematical modeling tasks. In two courses at two universities, 27 mathematics majors and secondary preservice teachers engaged in the modeling process situated in authentic contexts to learn specific concepts and make mathematical connections across domains and disciplines. Both courses culminated in a final project in which students created and investigated solutions to their own modeling tasks. In this paper, we describe how our courses (1) centered justice as pedagogy, (2) were environments for student agency and exploration, and (3) explicitly demonstrated how mathematics and social justice are intertwined. Drawing on frameworks of mathematical modeling for social justice, we present and analyze student-created tasks to showcase how they utilized mathematics as a tool to question and critique the world around them. [ABSTRACT FROM AUTHOR]

In this essay, I describe the unexpected ways I achieved some milestones in the early years of my career [ABSTRACT FROM AUTHOR]

Information plays a key role in life and in complex biological systems, and dynamical systems underlie and can be used to represent many complex systems. Indeed, dynamical systems and information processing capabilities may be the hallmarks of life-like systems. In this paper we combine dynamical systems with a computational framework to generate art. The framework can be used to generate aesthetically appealing forms of life-like systems. Our work suggests that we may need an "aesthetic sense" to recognize life that we have not seen before. We also provide teaching resources for students in schools and undergraduate institutions. [ABSTRACT FROM AUTHOR]

The definition of who is or what makes a "mathematician" is an important issue to be addressed in the mathematics community. Too often, a narrower definition of who is considered a mathematician (and what is considered mathematics) is used to exclude people from the discipline--both explicitly and implicitly. However, using a narrow definition of a mathematician allows us to highlight, examine, and challenge systemic barriers that exist in certain spaces of the community. This paper analyzes and illuminates tensions between narrow and broad definitions and how they can be used to promote both inclusion and exclusion simultaneously. In this article, we present a framework of definitions based on identity, function, and qualification and explore several different meanings of mathematician. By interrogating various definitions, we highlight their risks and opportunities, with an emphasis on implications for broadening and/or narrowing participation of underrepresented groups in the mathematics community. [ABSTRACT FROM AUTHOR]

In this paper, we, a group of graduate students in mathematics education, discuss some of the metacognitive benefits of the non-traditional teaching methods we observed employed by one of our professors. This professor's methods challenge the common belief that well-managed class time is key for positive learning outcomes. Instead, he orients his teaching to share the exploration and sense-making phases of doing mathematics. The goal of his teaching is to share the idea that learning mathematics is a process of "refining our mathematical thinking". We argue that this approach to teaching helps students see that mathematics is a human endeavor, appreciate the cycles of learning mathematics and the importance of struggle throughout the process, and gain some insight on what it means to be an expert in mathematics. We encourage teachers and professors to move away from obsessing about structured, well-organized class time, and to instead move towards having flexible and risk-oriented class time. [ABSTRACT FROM AUTHOR]

Despite convincing arguments by mathematicians, philosophers, sociologists and machine learning practitioners to the contrary, there remains a widespread notion amongst many members of the general public (and some practitioners) that mathematics is neutral, that it is free from human values. One reason why this notion persists is that we lack clear-cut examples that demonstrate how mathematics and values are intertwined. In this paper, we offer one such example. In particular, we show that when sharing four biscuits between three people, several possible mathematical and ethical frameworks can be used. We demonstrate that different solutions--hiding one biscuit, arbitrarily sharing the extra biscuit, randomizing allocation, dividing the extra biscuit into three parts, and successively dividing it into smaller and smaller parts--involve different mathematical methods and evoke different human values. We discuss the construction of quantum biscuit splitting devices and the use of machine learning to divide biscuits. We argue that the multitude of different mathematically-correct solutions to this problem (each with its own ethical justification) might influence the values held by practicing mathematicians. The example we propose here has been used in teaching to help students understand why mathematics cannot be cleanly separated from human values. [ABSTRACT FROM AUTHOR]

This paper details a semester-long course project that has been successfully adapted for use in mathematics courses ranging from introductory level, generaleducation classes to advanced courses in the mathematics major. Through creating aspirational mathematical family trees and writing mathematical autobiographies, this assignment is designed to help battle belonging uncertainty, to challenge students to self-situate in relation to the history of mathematical and scientific knowledge, and to make visible a student's developing identity in mathematics and, more broadly, in STEM. The construction and scaffolding of the project, assignments, examples of student work, foundational readings, assessment and outcomes, and adaptation strategies for various classroom settings are discussed in detail. [ABSTRACT FROM AUTHOR]

The algebra classroom in urban public high schools in the United States is a complex space, ripe with many challenges and opportunities. In this paper I introduce the notion of a heart-centered stance for the teacher and the educator and a method of engaging in creative expression for reflection and introspection toward individual change in the rich context of the high school algebra classroom. My evolving relationships with two high school algebra teachers, observations of their classrooms, as well as my own self-study and professional growth, are incorporated into this paper as I introduce and exemplify two tenets of a heart-centered stance: multidimensionality of experience and receptivity to relatedness. This study suggests the possibility of using creative artistic expression and a self-study approach to support the transformation of educators’ perspectives toward research, creative activities, and outreach that are receptive to the mathematical experiences of teachers and students in our local communities. [ABSTRACT FROM AUTHOR]

The COVID-19 pandemic inspired reflections on the importance and challenges of teaching exponential growth, such as doubling, to varied audiences. [ABSTRACT FROM AUTHOR]

This article makes available some early letters chronicling the relationship between the biologist Joseph H. Woodger and the logician Alfred Tarski. Using twenty-five unpublished letters from Tarski to Woodger preserved in the Woodger Papers at University College, London, I reconstruct their relationship for the period 1935–1950. The scientific aspects of the correspondence concern, among other things, Tarski’s reports on the work he is doing, his interests, and his — sometimes critical but always well-meaning — reactions to Woodger’s attempts at axiomatizing and formalizing biology using the system of Principia Mathematica. Perhaps the most interesting letter from a philosophical point of view is a very informative letter on nominalism dated November 21, 1948. But just as fascinating are the personal elements, the dramatic period leading to the second world war, their reaction to the war events, Tarski’s anguish for his family stranded back in Poland, the financial worries, and his first reports on life in the East Coast and, as of 1942, at the University of California, Berkeley. [ABSTRACT FROM AUTHOR]

The artist's vision helps decide what should be created; the mathematician's insight what can be created. Yet most people view art as merely decoration, or a reflection of existing reality, while they think of mathematics as just a tool for accurate scientific description. Can more people learn to value and enjoy both art and mathematics? And spend a lifetime exploring them and appreciating them for their own sakes? [ABSTRACT FROM AUTHOR]

In early 2015, the Mathematics-Literature Creativity Award was offered through competition for technological and vocational college students in Taiwan. This paper focuses on the two winning works, the poem Parabola and the flash-fiction, Love Letters of a Science Student, by analyzing the mathematical notions involved and their uses to enhance literary expression. The introduction surveys briefly the various ways of combining mathematics and literature in a piece of creative writing that enriches and supports communication in both disciplines. [ABSTRACT FROM AUTHOR]

In this paper, we utilize duoethnography -- a research method in which practitioners discursively interrogate the relationships between culture, context, and the mechanisms which shape individual autobiographical experiences -- to explore what may be beyond ethics in the context of data science. Although ethical frameworks have the ability to reflect cultural priorities, a singular view of ethics, as we explore, often fails to speak to the multiple and diverse priorities held both within and across institutional spaces. To that end, this paper explores multiple perspectives, epistemologies, and worldviews that can collectively push researchers towards considerations of a data science education that is equity-minded both in concept and practice. Through a set of dialogues which examine our positionalities, journeys, ethics, local cultures, and accountabilities, this paper explores the contextual realities rooted in the authors' educational settings. These conversations focus on the humanity of our students, the communities from which we come from and serve, as well as the unintentional harms and possibilities associated with the development of data science programs across institutional types. We take a set of five core questions to examine how we made, and continue to make, sense of our diverse cultural perspectives on data science education and equity with/in relation to others' realities. Broadly, this paper seeks to offer reflections on the related but differing functions of ethics and equity in data science education. [ABSTRACT FROM AUTHOR]

Here I narrate the story of the last few days of my graduate program in mathematics. After the completion of the thesis and the delivery of the defense, several twists and turns await in the hours and even minutes before the last deadline. [ABSTRACT FROM AUTHOR]

This is a personal account of an intersection of the schools of research in foundations of geometry founded by Alfred Tarski and Friedrich Bachmann. Their academic lineages and the origins of the schools are also described, as well as the mathematics that resulted from this intersection. [ABSTRACT FROM AUTHOR]

In this paper we address the questions: What is synesthesia? What support(s) can teachers provide for students who have synesthesia? Nicole, a future mathematics teacher who possesses this synesthesia "superpower", describes how it has impacted her learning. We collected data for this case study through an audio-recorded and transcribed interview, as well as from subsequent email correspondence between the three authors. We asked Nicole three kinds of questions: questions she is frequently asked, questions she would like to be asked, and questions teachers (like Shelly and Beth) might ask. Results indicate that synesthesia may have helped Nicole learn English as a second language as well as memorize certain mathematical formulas and procedures. Questions arose that, if answered, may influence the learning of not only other synesthetes in the mathematics classroom but also their classmates. [ABSTRACT FROM AUTHOR]

This paper reports the implementation of open educational resources (OER) in a university trigonometry class, with an emphasis on the pedagogical considerations and academic labor involved. To date these two matters have been underreported in the literature on OER. I provide an account of the work involved both in choosing an open textbook and in creating hundreds of accompanying homework exercises for an online learning platform. I also present the pedagogical lens that informed this implementation, discuss how it informed my adoption of an open textbook, and provide specific examples of how it guided the creation of these exercises. Based on my experiences I make some observations for those who might be considering OER in their own classes, and I present the results of a survey I gave my students on the use of OER in their class. [ABSTRACT FROM AUTHOR]

Existing literature widely reports on the value of collaborations between mathematicians and mathematics educators, and also how complex those collaborations can be. In this paper, we report on four collaborations that sought to address what mathematics is and who gets to do it. Drawing on the literature and from the careful and intentional work of the collaborators, we offer a framework to capture the richness of those collaborations -- one that acknowledges the importance of acknowledging and welcoming the extensive personal and professional experience of each person involved in the collaboration -- and a look at how collaborations built with that intentionality and acknowledgment can be impactful for students and institutions and be personally and professionally rewarding for the collaborators. [ABSTRACT FROM AUTHOR]

In this paper we analyze the distribution of musical note frequencies in Hertz to see whether they follow the logarithmic Benford distribution. Our results show that the music of Johann Sebastian Bach and Johann Christian Bach is Benford distributed while the computer-generated music is not. We also find that computer-generated music is statistically less Benford distributed than humancomposed music. [ABSTRACT FROM AUTHOR]

During the summer of 2021, I experimented with a new way of getting children excited about mathematics: math poetry. "Math" can be a trigger word for some children and many adults. I wanted to find a way to make learning math fun -- without the students knowing they're doing math. In this paper I describe some activities I used with students ranging from grades K-12 to the college level and share several poem examples, from students in grades two through eight. [ABSTRACT FROM AUTHOR]

This paper investigates creativity in students' constructions of everyday examples about basis in Linear Algebra. We analyze semi-structured interview data with 18 students from the United States and Germany with diverse academic and social backgrounds. Our analysis of creativity in students' everyday examples is orga- nized into two parts. First, we analyze the range of students' creative products by investigating the mathematical variability in the more commonly mentioned examples. Second, we unpack some of the collective processes in the construc- tion of students' examples. We examine how creativity was distributed through the interactions among the student, the interviewers, and other artifacts and ideas. Thus, in addition to contributing to the process vs. product discussion of creativity, our work also adds to the few existing studies that focus on collec- tive mathematical creativity. The paper closes with connections to anti-deficit perspectives in mathematics education and some recommendations for individual and collective creativity in the classroom. [ABSTRACT FROM AUTHOR]

I was recently asked to contribute a paper to a forthcoming Foundations for Undergraduate Research in Mathematics (FURM) volume [1]. The article contained, in addition to several lengthy case studies, a list of brief recommendations for undergraduate research supervisors. Several colleagues suggested developing these general principles into a separate, self-contained article. This is that article. [ABSTRACT FROM AUTHOR]

In 2014 Luke Wolcott created the User's Guide Project, in which a group of algebraic topologists came together to write user's guides to coincide with their research papers in hopes of making their research more accessible. We examine the role of this innovative project within the greater mathematics community. We discuss the structure and history of the project, its impact on the community, and its value to the participants of the project. We end by encouraging the math community to recognize the value of the project and expand the User's Guide Project to other subfields. [ABSTRACT FROM AUTHOR]

While studying the computable real numbers as a professional mathematician, I came to see the computable reals, and not the real numbers as usually presented in undergraduate real analysis classes, as the natural culmination of my evolving understanding of numbers as a schoolchild. This paper attempts to trace and explain that evolution. The first part recounts the nature of numbers as they were presented to us grade-school children. In particular, the introduction of square roots induced a step change in my understanding of numbers. Another incident gave me insight into the brilliance of Alan Turing in his paper introducing both the computable real numbers and his famous 'Turing machine". The final part of this paper describes the computable real numbers in enough detail to supplement the usual undergraduate real analysis class. An appendix presents programs that implement the examples in the text. [ABSTRACT FROM AUTHOR]

We bring to the forefront of educational thought a specific attitude toward the COVID-19 crisis that harnesses the symbolism of wind and water to navigate the cultural storm interfering upon our mathematical and pedagogical craft. The purpose of our paper is to open up space for opportunities in mathematics education using integral mindfulness as the rudder to readjust our bearings. More specifically, through conceptual analyses and making explicit the currents of change, disorder, and technology, we can apply discernment to these metaphors that intersect our pedagogy to re-align efforts and attitudes toward an integrated (aperspectival) culture of mathematics education. Through shared responsibility during these tumultuous times, we can once again strive toward the pursuit of excellence in mathematics education. [ABSTRACT FROM AUTHOR]

Mathematical visual poetry is a poetic genre whereby metaphorical expressions are created using mathematical structures. Within the structure, the poetics are understood by the cross-mapping of numerous conceptual domains including visual, lexical, and mathematical. Here I focus on one particular mathematical visual poetic structure: what I call a Similar Triangles Poem or Proportional Poem. To illustrate the ideas discussed, I present “Pandemic Meditation,” a mathematical visual poem; in particular, I discuss how this mathematical poem uses the mechanisms of poetic metaphor in the context of the embodied mind. The intent of this paper is not to explain “Pandemic Meditation,” for explanations of poetry serve only to kill it. Instead, the intent here is to give the reader the tools to access similar triangles poems in general, and this expression in particular, and to show how it functions within the definitions of poetic metaphor. This paper can be used as a template to study all similar triangles visual poems, and more generally, as a source to study visual poetry. [ABSTRACT FROM AUTHOR]

In this paper I describe an activity based on a 1903 Sherlock Holmes murder mystery, in which a substitution cipher is used to encrypt secret messages. The story provides a fun and interesting way to talk about frequency analysis, and can be used as a segue into mathematical constructs such as modular arithmetic and computation. The activity is accessible to ages twelve and above, and has been successfully used in mathematics outreach and popularization efforts as well as in general education and mathematics courses. [ABSTRACT FROM AUTHOR]

In this paper we propose a thematic focus on aesthetics in the context of an interdisciplinary collaboration between mathematics and literature (Language Arts) as a way to further students’ reflections on and deeper understanding of what characterizes the two subjects. Furthermore, we argue that approaching aesthetics through the perspective of literacy can potentially strengthen students’ understanding of ways of thinking particular to specific (academic) disciplines; ways of thinking that are otherwise often hidden when teaching focuses on more pragmatic aspects. G. H. Hardy’s A Mathematician’s Apology from 1940 serves as the recurring illustrative example in our discussions of the pedagogical potentials of an interdisciplinary rendezvous between mathematics and literature. [ABSTRACT FROM AUTHOR]

In this article, dictated by Maohua Le and arranged by Yongzhong Hu, Professor Le briefly recounts his legendary experience of self-study mathematics, which reflects the life experiences of his generation of Chinese people. [ABSTRACT FROM AUTHOR]

Not everyone is a great artist, but we don't often hear, "I dislike art." Most people are able to appreciate visual arts, music and sports, without necessarily excelling in it themselves. On the other hand, the phrase "I dislike math" is widely prevalent. This is especially ironic in our current society, where mathematics affects our day-to-day activities in essential ways such as e-commerce and e-mail. This paper describes the opportunity to popularize mathematics by focusing on its fun and creative aspects, and illustrates this opportunity through a brief discussion of interdisciplinary topics that expose the beauty, elegance and value of mathematics within and beyond the typical K-16 curriculum. We share practical outreach methods inspired by real mathematics, and our experience with a liberal-arts 'math for poets' course which aims to develop math appreciation without losing its rigor or depth. While many of us are not comfortable doing live performances or making fancy videos, we all can use these methods to encourage a more positive mindset about mathematics in our communities. [ABSTRACT FROM AUTHOR]

This paper argues that interactive theorem provers (ITPs) could play an important role in fostering students' appreciation and understanding of proof and of mathematics in general. It shows that the ITP Lean has three features that mitigate existing difficulties in teaching and learning mathematical proof. One is that it requires students to identify a proof strategy at the start. The second is that it gives students instant feedback while still allowing them to explore with maximum autonomy. The third is that elementary formal logic finds a natural place in the activity of creating proofs. The challenge in using Lean is that students have to learn its command language, in addition to mathematics course content and elementary logic. [ABSTRACT FROM AUTHOR]

This paper presents a study on the extent to which creative literature been used as a vessel to carry forward the development of mathematical thought. The role of mathematics as a driving force for literature is highlighted, and while many examples exist that clearly show an attempt to disperse mathematical ideas, with Lewis Carroll, OuLiPo and ancient poetry considered, the argument that the sole purpose of the writings was for the sake of mathematical development is not clear-cut. [ABSTRACT FROM AUTHOR]

This is a review of What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf Coles (Cambridge University Press, 2017). In this collection of sixteen chapters, philosophers, educationalists, historians of mathematics, a cognitive scientist, and a mathematician consider, problematise, historicise, contextualise, and destabilise the terms 'mathematical' and 'concept'. The contributors come from many disciplines, but the editors are all in mathematics education, which gives the whole volume a disciplinary centre of gravity. The editors set out to explore and reclaim the canonical question 'what is a mathematical concept?' from the philosophy of mathematics. This review comments on each paper in the collection. [ABSTRACT FROM AUTHOR]

This paper describes an interdisciplinary activity that crosses over between Mathematics and Rhetoric. The professors who created this activity both sought active-learning opportunities for their students, a sense of realism—even urgency—in what can otherwise be perceived as abstract material, and a meaningful liberal arts experience. Evidence of the power of this experience is seen in the media coverage, both from our college and from the Portland Press Herald newspaper. Both courses described in this paper are at the elective level, taken by majors or minors in their respective disciplines. Students have moderate to extensive backgrounds in their subject areas. However, adapted versions of our activity could involve students at more introductory levels. [ABSTRACT FROM AUTHOR]

Academics and educators across the world are grappling with how OpenAI's new software, ChatGPT, will impact teaching and learning. This essay explores ChatGPT's response to a typical calculus problem as a way of illustrating its functionality and limitations. [ABSTRACT FROM AUTHOR]

I survey emigration of mathematicians from Europe, before and during WWII, to Russia. The emigration started at the end of 1920s, the time of "Great Break", and accelerated in 1930s, after the introduction in Germany of the "non-Aryan laws". Not everyone who wanted to emigrate managed to do so, and most of those who did spent a relatively short time in Russia, being murdered or deported, or fleeing the Russian regime. After 1937, the year of "Great Purge", only a handful of emigrant mathematicians remained, and even fewer managed to leave a trace in the scientific milieu of their new country of residence. The last batch of emigrants came with the beginning of WWII, when people were fleeing eastwards from the advancing German army. [ABSTRACT FROM AUTHOR]

Almost exactly fifty years ago, Friedrich Kambartel, in his classic essay "Ethics and Mathematics," did pioneering work in an intellectual environment that almost self-evidently assumed a strict separation of the two fields. In our first section we summarize and discuss that classical paper. The following two sections are devoted to complement and contrast Kambartel's picture. In particular, the second section is devoted to ethical aspects of the indirect and direct mathematization of modern societies. The final section gives a short categorization of various philosophical positions with respect to the rationality of ethics and the mutual relation between ethics and mathematics. [ABSTRACT FROM AUTHOR]

A steward of the discipline was originally defined as an individual to whom "we can entrust the vigor, quality, and integrity of the field", and more specifically, as "someone who will creatively generate new knowledge, critically conserve valuable and useful ideas, and responsibly transform those understandings through writing, teaching, and application" [8]. Originally articulated for doctoral education, in 2019 the construct of stewardship was expanded so that it can also be applied to non-academic practitioners in any field, and can be initiated earlier than doctoral education [18]. In this paper we apply this construct to the context of mathematics and argue that even for those early in their training in mathematics, stewardly practice of mathematics can be introduced and practiced. Postsecondary and tertiary education in mathematics -- for future mathematicians as well as those who will use math at work -- can include curriculum-spanning training, and documented achievement in stewardship. Even before a formal ethical practice standard for mathematics is developed and deployed to help inculcate math students with a "tacit responsibility for the quality and integrity of their own work", higher education can begin to shape student attitudes towards stewardly professional identities. Learning objectives to accomplish this are described, to assist math instructors in facilitating the recognition and acceptance of responsibility for the quality and integrity of their own work and that of colleagues in the practice of mathematics. [ABSTRACT FROM AUTHOR]

In this paper we highlight the experience of a mathematics teacher educator (MTE) and their preservice teachers (PTs) in a middle school mathematics methods course during the 2020 shift to online instruction due to the COVID-19 pandemic. We believe it is valuable to report how the MTE reflected on their instructional decision-making in response to this massive transition to remote instruction. We also report that PTs needed support and guidance to employ new teaching practices they had learned in the methods course instead of reverting to familiar teaching methods. [ABSTRACT FROM AUTHOR]

This paper describes a class activity based on real data about COVID-19 death rates in California. The activity helps students learn about exponential functions while providing an opportunity to integrate social justice concerns into the mathematics classroom. [ABSTRACT FROM AUTHOR]