Mathematics Commons^™

The Urysohn Lemma: Munkres Variation, Irida Altman

Journal of Humanistic Mathematics

There is a tradition of Mathematics contributing to the understanding of Music. This article sets out to contribute in the other direction. It uses musical sensibilities to feel its way through a piece of mathematics called the Urysohn Lemma, suggesting the presence of rhythmic, melodic, and harmonic elements within standard proving practices of contemporary Western mathematics. This article employs a literary-philosophical technique of viewing a text in one discipline (mathematics) through the conceptual lens of another (music). As such, this article reports on no experiments—reading the article is the experiment. Each reader performs it for themselves.

Prime Numbers Are Written On Your Hand: You Simply Have Not Been Counting Correctly, Nona Servais Kamdem Simo, Awa Traore

Journal of Humanistic Mathematics

Prime numbers have resisted systematic description for over two millennia. We propose that part of the difficulty may lie in an insufficiently attentive examination of one’s own hands. In particular, we assign the natural numbers to the joints of the human hand according to a cyclic traversal protocol, and make a handful of observations which can naturally be translated into number-theoretic properties of prime numbers. We show that these properties lead us to natural activation thresholds of a unified three-phase sieve algorithm — the resulting framework provides, for the first time, a zone-based partition of the admissible space with explicit …

Developing A Learner-Centered Teaching Routine, Hyeeun Jang

Journal of Humanistic Mathematics

This paper shares my story of developing, from Fall 2022 to Spring 2025, a learner-centered teaching routine for my undergraduate mathematics courses. I use four steps: an opening question, a short mini lecture about meaning, structured small group work, and a short end-of-class exit check, at times supported by quick visuals in Mathematica and, when useful, Python. Instead of relying on extended traditional lectures, the routine emphasizes collaborative problem solving and invites students to articulate and check their reasoning together. I used this routine mainly in my elementary statistics classes, but also in calculus and linear algebra. The evidence I …

Numbers, Patterns, And Memories: Early Experiences Of Mathematicians And Mathematics Education Researchers, Constantinos Xenofontos

Journal of Humanistic Mathematics

This paper offers a reflective analysis of early mathematical experiences shared by fifty-six mathematicians and mathematics education researchers, including my own, as part of Cambridge Mathematics’ “Seven Questions with . . . ” interview series. Drawing on thematic analysis, I identify five key patterns across participants’ recollections: the context of early mathematical engagement, the influence of significant others, emotional responses, shifts in mathematical thinking, and mathematics as a social or competitive experience. Many accounts emphasise informal, everyday encounters with mathematics—often preceding formal education—as central to developing interest and confidence in the subject. The findings challenge the notion of innate mathematical …

Preparing Undergraduates To Philosophize With Children About Mathematics, Lawrence M. Lesser

Journal of Humanistic Mathematics

This paper shares my reflections and insights from preparing and facilitating a session in a weeklong inaugural study away program for undergraduate students (especially philoophy majors). My session was a vehicle for helping these students experience or visualize how the program’s Philosophy for Children (P4C) approach could be applied to exploring and discussing mathematics with children.

The Dynamics Of Vincent Van Gogh, Monica Morales Hernandez

Journal of Humanistic Mathematics

This article explores the interplay between Vincent van Gogh’s distinctive artistic style and principles of fluid dynamics, particularly through metaphorical comparisons. Van Gogh’s swirling, turbulent brushstrokes in works such as The Starry Night and Wheatfield with Crows evoke the chaotic and dynamic behaviors found in fluid motion, like vortices and turbulence. Although there is no documented scientific influence from fluid dynamics on van Gogh’s art, the parallels between his emotive, flowing techniques and the mathematical principles governing fluid behavior offer a unique lens to appreciate both his artistic innovation and the science of fluid dynamics.

Mathematicians Against The Tide, Nicola Ciccoli

Journal of Humanistic Mathematics

Mathematicians in popular culture are often depicted as people obsessed by their studies but with little connection to society, poor social skills, mental issues and narrow interests, and more generally a dull life. Adding to these explicit stereotypes, subtler and more implicit ones also abound, regarding mathematicians’ gender, ethnicity, age; they are all supposed to be white middle-aged men. In this paper I describe a project aimed at disrupting such stereotypes that culminated in a book containing short comics and biographies of twentieth-century mathematicians that were, with respect to the perceived image of mathematicians, against the tide.

Waiting For The Magic: A Historical And Mathematical Study Of Queues In Disney Theme Parks, Nina E. Becket

Journal of Humanistic Mathematics

This paper, written as part of an honors research project in high school, provides an introduction to the line and queuing systems at Disney theme parks. We begin with a brief history of Disney World and its creators. We then introduce the basics of queuing theory and queuing notation. Finally, we examine modern line systems at Disney (as of 2023) and the future of its different queue systems.

An Exploration Of Mathematical And Musical Connections, Jeffrey Pair, Egypt Pannell, Dani Rodriguez, Jerico Sarmiento

Journal of Humanistic Mathematics

In this paper, we present an introduction to some of the mathematical principles underlying music and mathematically develop a connection between the twelve musical notes and the group of integers modulo twelve. Our discussion touches on the simple mathematical ratios used to calculate consonant musical tones on the length of a string, the frequencies of tones and the method by which the twelfth root of two provides the foundation of equal temperament tuning, some algebraic connections to musical concepts like the euphonious circle of fifths, and a curious connection to twin primes.

Impact Of Covid-19 Pandemic On Mathematics Achievement And Problem-Solving Ability: Insights From Secondary Schools In India, Laishram Nirtish Singh, Anand Jyoti Sanasam, Jocyline Thokchom, Ningombam Cha Cogent, Laisom Sharmeswar Singh

Journal of Humanistic Mathematics

The COVID-19 pandemic disrupted students’ engagement with mathematics. This study aims to contribute to our understanding of the extent of this disrupting by examining differences in mathematics achievement and problem-solving ability between adjacent cohorts who learned Class 8 under different schooling conditions in India: Group A (pandemic-exposed Class 8 cohort) and Group B (post-reopening Class 8 cohort). Using a descriptive design in a natural setting, we selected a stratified sample of 1,200 students from Class 9 and Class 10 across 21 secondary schools in Manipur, India. The sample was stratified by gender, family type, school type, and locality to ensure …

"Math Is Like A Rollercoaster": Exploring Math Similes And Metaphors, Carmen M. Latterell, Janelle L. Wilson

Journal of Humanistic Mathematics

In this article we explore the metaphors and similes that pre-service elementary teachers use to describe mathematics. Employing qualitative content analysis, we separately coded the metaphors and similes into conceptual categories and analyzed themes that emerged. This grounded theory approach led to the creation of seven categories of metaphors and similes and three overarching themes. Math was viewed as a journey, a puzzle, a personal experience, a creative process, an algorithm, infinite, and an embodied process. Pre-service elementary teachers viewed mathematics as a process that takes effort. Overall, mathematics is a positive experience for them. In addition, mathematics involves fitting …

Cultivating Mathematics Teacher Educators’ Hearts, Critical Consciousness, And Professional Praxis Through Book Study, Mary Raygoza, Missy D. Cosby, Sheila Orr, Barbara King, Justin T. Burris, Siddhi Soni

Journal of Humanistic Mathematics

At a time when executive orders and state legislation are attempting to ban practices promoting equity, belonging, and social justice from education, specifically STEM education, in the United States, mathematics teacher educators (MTEs) remain committed to humanizing mathematics teaching and learning practices and seek opportunities to learn how to adhere to their convictions while preparing mathematics teachers for the classroom. This paper offers reflections and considerations for how the authors engaged interested MTEs to shift or strengthen their individual and collective professional praxis through a book study. Specifically, we examine the design (the what) and process (the how) of a …

Piracy, Terrorism, And The Law: Differential Equations In Hostage Situations, Gabriel Hallevy

Journal of Humanistic Mathematics

Pirates have taken the crew of an American ship hostage. They promise to release the hostages only if another pirate who is held in an American prison for commission of piracy crimes against American citizens, is released. Should the U.S. government enter into negotiations with them? Should they send armed forces and risk the hostages? Should they release the prisoner immediately and unconditionally? The article models and analyzes possible policies regarding sensitive situations involving hostages and other related risks using differential equations. The solutions are surprisingly simple, but not necessarily intuitive. Our analysis aims to demonstrate how powerful mathematics is …

Grationality, With A Spoon, L. Jeneva Clark

Journal of Humanistic Mathematics

I introduced grationality at a 2025 sectional meeting of the Mathematical Association of America to install a handle on a concept akin to rationality of numbers, but in a geometric context. More specifically, we define a nice n-gon to be a regular n-gon with side lengths that are natural numbers; then a number n is grational if and only if there exists a nice n-gon such that its area equals the sum of areas of n congruent nice n-gons. In this paper I offer several examples of grational and non-grational numbers, followed by theorems about how the grationality of a …

The Set Card Game And Partitions Into Maximal Caps: A Partial Difference Sets Approach, John Polhill

Journal of Humanistic Mathematics

More specifically, we revisit the problem of determining the maximum number of cards that can be dealt in the SET® card game without any matches, which is well known to be equivalent to the problem of determining the size of a maximal cap in the affine space AG(4,3). We present a new approach to the problem by using partial difference sets, a method that is equivalent to using strongly regular graphs with a regular automorphism group. The paper includes several possible avenues for future investigations on how partial difference sets might be used in related problems.

A Method For Folding Origami Rectilinear Polygon Extrusions, Everett L. Jin

Journal of Humanistic Mathematics

We present a method for generating crease patterns for folding any rectilinear polygon extrusion with a specified uniform minimum edge-to-edge distance, using a rectangular sheet of paper. We demonstrate that the resulting models are efficient and watertight. In addition, they achieve optimal efficiency, meaning they attain the smallest possible scale factor from the original sheet of paper to the folded rectilinear polygon extrusion. Finally, we provide a code implementation of the method. Our study not only facilitates crease pattern design for origami artists but also offers an innovative, art-driven pedagogical tool for inspiring interest in mathematics.

Manipulating Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Sums Of Consecutive Integers And Sequence Of Divisors, Mushrat Fatema, Abdulkadir Hassen

STEM Student Research Symposium Posters

This work is concerned with two problems in number theory. The first part is to find the maximum number of sequence of integers such that two consecutive numbers in the list are multiple or factor of the other, and integers that can be expressed as a sum of consecutive integers.

Error-Tolerant Metric Dimension, Leander Ten Hoff

Master's Theses

Metric dimension is a graph parameter that measures the smallest distance-based unique coordinate system on a graph. Fault-tolerant metric dimension is a variant that requires redundancy. Inspired by this idea, we propose and investigate a new variant we call error-tolerant metric dimension. We first prove some fundamental results to gain familiarity with this more complex variant. Then, we use these tools to study relative behaviors of error-tolerant metric dimension. We prove explicit computations for simple graph families, including bipartite complete graphs and paths. Finally, we extend extremal results from previous papers on metric dimension, including a full correction of an …

Probabilistic Metric Dimension, Kyle Worley

Master's Theses

The metric dimension of a graph G is the smallest number of unique vertices (often called “marked vertices” or “landmarks”) such that each vertex has a unique distance to the marked vertices. If a set of vertices S ⊆ V (G) when marked gives all vertices unique distances, it is called a resolving set. Hence if G is a graph with a resolving set S, then for all v, u ∈ V (G) there exists s ∈ S such that d(v, s) ̸= d(u, s). The metric dimension dim(G) = min(|S|). Two widely studied variants exist, one where edges are …

The Relationship Between Foliations Of The Plane, Kaplan Diagrams, And Non-Hausdorff 1-Manifolds, Monique Justine Howe

Master's Theses

In this paper we seek to explain the relationship between foliations of the plane, Kaplan diagrams, and simply connected non-Hausdorff 1-manifolds with countable basis that are orientable with an ordering on branch points. We will walk the reader through the definitions of all of these objects, and provide examples with a focus on the motivating example of the Reeb foliation. This paper will describe and define the known bijection from the set of foliations of the plane, F, to the set of Kaplan diagrams, K, its inverse, and the one-to-one map from F to the set of simply connected non-Hausdorff …

Monoids With K-Strictly Local Geodesic Languages, William M. Hong

Master's Theses

A paper by Gilman, Hermiller, Holt and Rees (2011) prove that a group G finitely generated by X is virtually free if and only if the language of geodesics words in G over X is k-strictly local if and only if the word problem is context-free. If M is a monoid finitely generated by X, we define Γu(M,X) to be the underlying, undirected graph of Γ(M,X). We prove that if the language of geodesics in Γu(M,X) based at the identity is k-strictly local, then the monoid and semi-group word problems are context-free.

Rewriting The Calculus 3 Workshop At San José State University, Moorea Lippert

Master's Theses

Calculus 3 at San José State University is a very important course for any student pursuing a STEM degree. However, the course is very challenging, so the university offers a supplemental workshop course for Calculus 3 students. Unfortunately, this workshop has not aligned with modern research in math education. This thesis documents efforts to recreate the workshop to better align with this modern research.

Fermat's Last Theorem Over Quadratic Number Fields, Richie Tay

Master's Theses

Fermat’s Last Theorem states that the equation xn + yn = zn has no nontrivial integer solutions for n ≥ 3. While the rational case is completely solved, the situation over quadratic number fields is not as straightforward. For some exponents nontrivial solutions exist, while for others they do not. This thesis studies Fermat-type equations over quadratic number fields, combining classical methods with more modern techniques from the theory of elliptic curves. The thesis first reviews the rational cases n = 2, n = 3, n = 4, then develops the necessary background on quadratic number fields, including rings of …

Rational Non-Euclidean Triangles And Elliptic Curves, Tyler Morales

Master's Theses

We say a triangle △ is rational if its side lengths are rational. For a geometry of constant curvature κ, we say △ is a rational triangle if the generalized tangents of its side lengths are rational. We are able to parameterize rational triangles having the same inradius and semiperimeter on an plane curve, write a bijection between the rational points on this curve and triples of side lengths, and show that this curve is an elliptic curve. Then, we write a general transformation for our plane curve to short Weierstrass form to compute ranks and add points easily using …

Math 141 Calculus I (Queens College), Anisha Clarke

Open Educational Resources

This syllabus provides the outline for a calculus one course using OpenStax Calculus Volume I and no graphing calculator.

Fundamental Solutions To The Fractional Heat Operator, Jacob Flores

Math Theses

In this thesis, we are interested in showing the existence of a fundamental solution to the fractional heat operator. The fractional heat operator is a nonlocal linear operator used to model the time evolution of anomalous diffusion processes whose applicability arises in a wide variety of fields in the physical sciences, engineering, economics, and finance. Fundamental solutions to a partial differential operator are a class of generalized solutions formulated with rich mathematical analysis grounded in classes of well-behaved smooth functions referred to as test functions and their continuous linear functionals referred to as distributions. A primary tool that we use …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution - Part B: Professional Development Framework And Institutional Implementation, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

This paper presents the professional-development and institutional-implementation component of a three-paper program on introducing data-driven dynamical systems into undergraduate ordinary differential equations instruction at a primarily undergraduate institution. The companion paper \cite{SelvitellaAnderson2026} develops the mathematical and computational content, including regression, regularization, Dynamic Mode Decomposition, Sparse Identification of Nonlinear Dynamics, and a Van der Pol oscillator activity. The present paper asks a complementary implementation question: what departmental structures, graduate teaching assistant roles, professional-development activities, and course-support pathways are needed for such materials to become teachable within local undergraduate settings?

We describe a three-phase professional-development model for graduate teaching assistants in the …

Cultural Traits May Replace Human Mobility Data In Forecasting Covid-19 Mortality: A Deep Learning Approach, Saif Abbas, Tamer Oraby, Michael G. Tyshenko, Samit Bhattacharyya

School of Mathematical & Statistical Sciences Faculty Publications

The COVID-19 pandemic highlighted the need for accurate epidemic forecasting to support public health decision-making. Most existing approaches depend heavily on human mobility data, while largely neglecting population behavior shaped by socio-cultural norms. In this study, we analyze daily COVID-19 mortality and Google mobility data from 72 countries during the first 130 d of the pandemic, a period characterized by high uncertainty and behavioral heterogeneity. In particular, we examine whether Hofstede’s country-level cultural dimensions can serve as latent behavioral forecasters of mortality in lieu of dynamic mobility indicators. Using 100 d for training and 30 d for forecasting, we employ …