Physical Sciences and Mathematics Commons^™

Undergraduate Mathematics Students Question And Critique Society Through Mathematical Modeling, Will Tidwell, Amy Bennett

Journal of Humanistic Mathematics

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 …

Aesthetic Approaches To Symmetric Functions, John M. Campbell

Journal of Humanistic Mathematics

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions.

Mathematicians Going East, Pasha Zusmanovich

Journal of Humanistic Mathematics

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 …

Sharing Four Biscuits Between Three People: An Illustrative Example Of How Mathematics Is Intertwined With Human Values, Lovisa Sumpter, David Sumpter

Journal of Humanistic Mathematics

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 …

The Automathography: A Humanistic Autobiographical Writing Assignment For Mathematics Courses, Colton Sawyer, Ron Buckmire

Journal of Humanistic Mathematics

In this article, we discuss an autobiographical writing assignment that we call an “automathography” which can be used in different types of mathematics and mathematics education courses, and by extension, in other disciplines as well. This assignment can enhance student-instructor interactions, develop student communication skills, and provide outlets for student creativity by leveraging the lived experiences of students. We have deployed the assignment in different kinds of classes (general education [service] courses and major-only seminars) at different kinds of institutions (a private, open-admission, large university on the East Coast and a private, highly selective, small liberal arts college on the …

Finding Your Mathematical Roots: Inclusion And Identity Development In Mathematics, Linda Mcguire

Journal of Humanistic Mathematics

This paper details a semester-long course project that has been successfully adapted for use in mathematics courses ranging from introductory level, general-education 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 …

Gödel's Theorem In The Continuing Education Of Mathematics Teachers, Ana J. Lemes

Journal of Humanistic Mathematics

The notion of dépaysement épistémologique (epistemological disorientation) aims to capture the sense of disorientation when a learner is led to question their prior assumptions and understandings, generating uncertainty in a context in which they thought they had certain knowledge. This article describes an activity used with a group of practicing mathematics teachers in Uruguay that integrates elements of the history of mathematics related to Gödel’s incompleteness theorem, with the aim of provoking in the participants the experience of dépaysement épistémologique. Results show that several of the teachers participating in the activity felt dépaysement épistémologique, and this feeling triggered …

Our Histories, Our Values, Our Mathematics, Mark Huber, Gizem Karaali

Journal of Humanistic Mathematics

No abstract provided.

Front Matter

Journal of Humanistic Mathematics

No abstract provided.

Special Issue On Public Policy: Front Matter

CODEE Journal

The Front Matter contains the Editor-in-Chief's Foreword, a Dedicatory by Associate Editor Douglas Meade, a Preface by the Special Editors Bev West and Samer Habre, and the Table of Contents.

Full Issue - Engaging The World: Differential Equations Can Influence Public Policies

CODEE Journal

This is the full issue (front matter and all papers) of the Third CODEE Special Issue, with the theme, "Engaging the World: Differential Equations can Influence Public Policies."

Differential Equations For A Changing World:How To Engage Students In Learning And Applying Differential Equations, Biyong Luo

CODEE Journal

In this article, I share my decade-long experience teaching an intensive five-week summer Differential Equation course covering complex topics and tips for creating an interactive and supportive learning environment to optimize student engagement. This article provides my detailed approach to planning and teaching an asynchronous course with rigor and flexibility for each student. An interactive teaching approach and variety of learning activities will augment students’ mathematical fluency and appreciation of the importance of differential equations in modeling a wide variety of real-world situations with special attention to ways differential equations can be relevant to creating public policy.

Odes And Mandatory Voting, Christoph Borgers, Natasa Dragovic, Anna Haensch, Arkadz Kirshtein, Lilla Orr

CODEE Journal

This paper presents mathematics relevant to the question whether voting should be mandatory. Assuming a static distribution of voters’ political beliefs, we model how politicians might adjust their positions to raise their share of the vote. Various scenarios can be explored using our app at https: //centrism.streamlit.app/. Abstentions are found to have great impact on the dynamics of candidates, and in particular to introduce the possibility of discontinuous jumps in optimal candidate positions. This is an unusual application of ODEs. We hope that it might help engage some students who may find it harder to connect with the more customary …

Ode Models Of Wealth Concentration And Taxation, Bruce Boghosian, Christoph Borgers

CODEE Journal

We refer to an individual holding a non-negligible fraction of the country’s total wealth as an oligarch. We explain how a model due to Boghosian et al. can be used to explore the effects of taxation on the emergence of oligarchs. The model suggests that oligarchs will emerge when wealth taxation is below a certain threshold, not when it is above the threshold. The underlying mechanism is a transcritical bifurcation. The model also suggests that taxation of income and capital gains alone cannot prevent the emergence of oligarchs. We suggest several opportunities for students to explore modifications of the model.

Fitting A Covid-19 Model Incorporating Senses Of Safety And Caution To Local Data From Spartanburg County, South Carolina, D. Chloe Griffin, Amanda Mangum

CODEE Journal

Common mechanistic models include Susceptible-Infected-Removed (SIR) and Susceptible-Exposed-Infected-Removed (SEIR) models. These models in their basic forms have generally failed to capture the nature of the COVID-19 pandemic's multiple waves and do not take into account public policies such as social distancing, mask mandates, and the ``Stay-at-Home'' orders implemented in early 2020. While the Susceptible-Vaccinated-Infected-Recovered-Deceased (SVIRD) model only adds two more compartments to the SIR model, the inclusion of time-dependent parameters allows for the model to better capture the first two waves of the COVID-19 pandemic when surveillance testing was common practice for a large portion of the population. We find …

Raising Student Awareness Of Environmental Issues Via Writing Assignments With Differential Equations, Michelle L. Ghrist

CODEE Journal

In this paper, I discuss two environmentally-focused writing assignments that I developed and implemented in recent integral calculus and differential equations courses. These models of carbon storage and PCB’s in a river provide interesting applications of one-compartment mixing problems. The assignments were intended to focus student attention on sustainability concerns while also developing other essential skills. I discuss these assignments and their effect on my students’ technical writing and environmental awareness. Detailed introductory instructions and mostly complete solutions to these assignments appear in the appendices, to include sample student work.

Modeling Aircraft Takeoffs, Catherine Cavagnaro

CODEE Journal

Real-world applications can demonstrate how mathematical models describe and provide insight into familiar physical systems. In this paper, we apply techniques from a first-semester differential equations course that shed light on a problem from aviation. In particular, we construct several differential equations that model the distance that an aircraft requires to become airborne. A popular thumb rule that pilots have used for decades appears to emanate from one of these models. We will see that this rule does not follow from a representative model and suggest a better method of ensuring safety during takeoff. Aircraft safety is definitely a matter …

Blue Whale And Krill Populations Modeling, Li Zhang

CODEE Journal

We present an intriguing topic in an undergraduate mathematical modeling course where predator-prey models are taught to our students. We describe modeling activities and the use of technology that can be implemented in teaching this topic. Through modeling activities, students are expected to use the numerical and graphical methods to observe the qualitative long-term behavior of predator and prey populations. Although there are other choices of predators and prey, we find that using blue whales and krill as predator and prey, respectively, would be most beneficial in strengthening our students' awareness of protecting endangered species and its impact on climate …

Nonlinear Dynamics Of Mountain Pine Beetle Populations: Discussion Of Forestry Policy, A Survey Of Existing Mathematical Models, And Code Base Demonstration, Scott A. Strong, Maya Maes-Johnson

CODEE Journal

This article presents existing mathematical models associated with mountain pine beetle populations in lodgepole pine forests, whose reproductive cycle requires the destruction of colonized host trees, decreasing timber availability/quality, and providing fuel sources for wildfires. With the existence of a positive-feedback loop with environmental warming, the need for intervention and management is clear. However, the legislative responses to the focusing events from our 2000-2010 North American epidemics are characterized as under-leveraged. While the reasons for this are multifaceted, increasing the capacity of STEM-informed individuals to take part in quantitative modeling of the underlying ecosystem generates awareness and provides pathways connecting …

Applying The Sir Model: Can Students Advise The Mayor Of A Small Community?, Carrin Goosen, Mark I. Nelson, Mahime Watanabe

CODEE Journal

This is an account of a modelling scenario that uses the sir epidemic model. It was used in a third year applied mathematics subject. All students were enrolled in a mathematics degree of some type. Students are presented with the results of a test carried out on 100 individuals in a community containing 3000 people. From this they determined the number of infectious and recovered individuals in the population. Given the per capita recovery rate and making a suitable assumption about the number of infectious individuals at the start of the epidemic, they then estimate the infectious contact rate and …

To Open Or Not To Open: Developing A Covid-19 Model Specific To Small Residential Campuses, Christina Joy Edholm, Maryann Hohn, Nicole Lee Falicov, Emily Lee, Lily Natasha Wartman, Ami Radunskaya

CODEE Journal

In May 2020, administrators of residential colleges struggled with the decision of whether or not to open their campuses in the Fall semester of 2020. To help guide this decision, we formulated an ODE model capturing the dynamics of the spread of COVID-19 on a residential campus. In order to provide as much information as possible for administrators, the model accounts for the different behaviors, susceptibility, and risks in the various sub-populations that make up the campus community. In particular, we start with a traditional SEIR model and add compartments representing relevant variables, such as quarantine compartments and a hospitalized …

Solar Panels, Euler’S Method And Community-Based Projects: Connecting Differential Equations With Climate Change, Victor J. Donnay

CODEE Journal

How does mathematics connect with the search for solutions to the climate emergency? One simple connection, which can be explored in an introductory differential equations course, can be found by analyzing the energy generated by solar panels or wind turbines. The power generated by these devices is typically recorded at standard time intervals producing a data set which gives a discrete approximation to the power function $P(t)$. Using numerical techniques such as Euler’s method, one can determine the energy generated. Here we describe how we introduce the topic of solar power, apply Euler’s method to determine the energy generated, and …

Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, Christopher Evrard, Callie Johnson, Michael A. Karls, Nicole Regnier

CODEE Journal

A Sand Tank Groundwater Model is a tabletop physical model constructed of plexiglass and filled with sand that is typically used to illustrate how groundwater water flows through an aquifer, how water wells work, and the effects of contaminants introduced into an aquifer. Mathematically groundwater flow through an aquifer can be modeled with the heat equation. We will show how a Sand Tank Groundwater Model can be used to simulate groundwater flow through an aquifer with a no flow boundary condition.

The Dual Boundary Complex Of The Moduli Space Of Cyclic Compactifications, Toby Anderson

HMC Senior Theses

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on, Δ^(r,n), the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of Δ^(r,n). We compute its Euler characteristic and …

Why Is Grant Lake A Reservoir? A Brief Geological And Human History, From The Pleistocene To The Present, Robert B. Marks

Eastern Sierra History Journal

Drawing on a range of archival resources and illustrative material, Prof. Marks probes why and how Grant Lake in the Eastern Sierra of California became a reservoir. The process is long and involved, and has much to do with a remote and ancient lake ultimately being developed to serve the water needs of the distant city of Los Angeles.

Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, Krishna Rajesh

HMC Senior Theses

Ron Graham's sequence is a surprising bijection from the natural numbers to the non-prime integers, which is constructed by looking at sequences whose product is square. In this thesis we will resolve a 22-year-old conjecture about this bijection, by construction of explicit sequences in a modified number theoretic context. Additionally, we will discuss the history of this problem, and give computational techniques for computing this bijection, levering ideas from linear algebra over the finite field of two elements.

Examining Course Achievement In An Undergraduate Psychology Statistics Course Through The Lens Of Machine Learning Techniques, Sunny Nguyet Le

CGU Theses & Dissertations

The Introductory to Psychology Statistics course stands as a notable challenge for psychology majors, often acting as a gatekeeper course. This study embarks on two primary objectives using machine learning techniques: (1) to identify the determinants of overall course achievement, specifically course grade, and (2) to investigate the influence of statistics anxiety and statistics self-efficacy, when both are present, on overall course grade. Employing a machine-learning approach, both objectives were effectively addressed. The study involved the development of a self-reported questionnaire consisting of perceptions of statistics anxiety and statistics self-efficacy, along with other demographic and academic background variables. Conducted at …

Steklov Eigenvalue Problems On Nearly Spherical And Annular Domains, Nathan Philip Schroeder

CGU Theses & Dissertations

We consider Steklov eigenvalues on nearly spherical and nearly annular domains in d dimensions where d is any given positive integer. By using the Green-Beltrami identity for spherical harmonic functions, the derivatives of Steklov eigenvalues with respect to the domain perturbation parameter can be determined by the eigenvalues of a matrix involving the integral of the product of three spherical harmonic functions. By using the addition theorem for spherical harmonic functions, we determine conditions when the trace of this matrix becomes zero. These conditions can then be used to determine when spherical and annular regions are critical points while we …

A Smart Energy-Efficient Hybrid Gait Monitoring System, Elsa Joy Harris

CGU Theses & Dissertations

Triboelectric nanogenerators are devices that harvest mechanical energy from the environment and turn it into electricity. By coupling the effect of contact electrification and electrostatic induction between two materials that come into contact and then separate they can convert the irregular, low frequency, waste biomechanical energy of human motion into useful electrical energy to run small body-worn electronics. This has shown promising results in multiple applications such as self-powered motion and haptic sensing, self-charging micro-storage devices, neuromorphic computing, and designing batteryless circuits to power small wearables. This work will investigate a smart energy-efficient hybrid gait monitoring system that is powered …

Effectiveness Of Municipal Collaboration For Watershed Management In The Southern Colorado River Basin, Sasha Geschwind

CGU Theses & Dissertations

There are mixed arguments about the best scale of government for providing various public goods and services. Specifically, the management of a watershed, which can be categorized as a common pool resource, is generally implemented at the local government scale via National Pollutant Discharge Elimination System (NPDES) Municipal Separate Storm Sewer Systems (MS4) permits as issued by the United States Environmental Protection Agency (US EPA). A large body of literature argues for rethinking this scale because this does not match natural watershed boundaries.In this paper, I explore the question: Does collaboration, and type of collaboration, among jurisdictions within a watershed …