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Linear Ode Systems Having A Fundamental Matrix Of The Form F(Mt), Kevin L. Shirley, Vicky W. Klima
Linear Ode Systems Having A Fundamental Matrix Of The Form F(Mt), Kevin L. Shirley, Vicky W. Klima
CODEE Journal
We interweave scaffolded problem statements with exposition and examples to support the reader as they explore specific linear systems of differential equations with variable coefficients, $\vec{x}'(t)=A(t)\vec{x}(t)$ and initial value $\vec{x}(0)$. We begin with a constant $n\times n$ matrix $M$ and a real or complex-valued function $f$, analytic at the eigenvalues of $M$ with $f(0) = 1$, and construct a linear system of differential equations with solutions $x(t)=f(Mt)\vec{x}(0)$, where $t$ is a parameter in some interval including zero. In general, the solutions to the resulting non-autonomous system are more difficult to analyze than solutions to the constant coefficient case. However, some …
Differential Equations For Modeling Pathways Leading To Diabetes Onset, Viktoria Savatorova, Aleksei Talonov
Differential Equations For Modeling Pathways Leading To Diabetes Onset, Viktoria Savatorova, Aleksei Talonov
CODEE Journal
This paper presents a mathematical model that explains potential pathways leading to diabetes onset. By utilizing a system of nonlinear differential equations to describe the dynamics of the glucose regulatory system, the model can serve as a pedagogical tool for teaching and learning differential equations, dynamical systems, mathematical modeling, and introduction to biomathematics. Within this framework, students can analyze equilibrium solutions, investigate stability, assess parameter sensitivity, and explore the potential for bifurcations. Theoretical analysis is complemented by illustrative numerical examples. Instructors have the flexibility to adapt and incorporate suggested activities according to their teaching preferences and objectives.
The Conviction Of Miss Prediction, Dane C. Joseph
The Conviction Of Miss Prediction, Dane C. Joseph
Journal of Humanistic Mathematics
Miss Prediction is questioned in a court of law over her involvement in the mischaracterization of linear models when they were inappropriate.
Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau
Polygon Quadrature And Dodecagonal Tessellation With Pattern Blocks, Gunhan Caglayan, Ben Kamau
Journal of Humanistic Mathematics
The age-old challenge of polygon quadrature involves converting a polygon into a square of equal area. In this educational resource, we utilize pattern blocks, commonly employed instructional aids in K-12 education across the United States, to visually demonstrate the transformation of different equilateral and regular pattern block polygons into squares. This is achieved through the application of the area conservation principle and geometric congruence/similarity reasoning.
Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett
Seating Groups And 'What A Coincidence!': Mathematics In The Making And How It Gets Presented, Peter J. Rowlett
Journal of Humanistic Mathematics
Mathematics is often presented as a neatly polished finished product, yet its development is messy and often full of mis-steps that could have been avoided with hindsight. An experience with a puzzle illustrates this conflict. The puzzle asks for the probability that a group of four and a group of two are seated adjacently within a hundred seats, and is solved using combinatorics techniques.
Undergraduate Mathematics Students Question And Critique Society Through Mathematical Modeling, Will Tidwell, Amy Bennett
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 …
Sharing Four Biscuits Between Three People: An Illustrative Example Of How Mathematics Is Intertwined With Human Values, Lovisa Sumpter, David Sumpter
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 …
Gödel's Theorem In The Continuing Education Of Mathematics Teachers, Ana J. Lemes
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 …
Finding Your Mathematical Roots: Inclusion And Identity Development In Mathematics, Linda Mcguire
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 …
Special Issue On Public Policy: Front Matter
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
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."
Applying The Sir Model: Can Students Advise The Mayor Of A Small Community?, Carrin Goosen, Mark I. Nelson, Mahime Watanabe
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 …
Raising Student Awareness Of Environmental Issues Via Writing Assignments With Differential Equations, Michelle L. Ghrist
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.
Blue Whale And Krill Populations Modeling, Li Zhang
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 …
Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, Christopher Evrard, Callie Johnson, Michael A. Karls, Nicole Regnier
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.
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
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 …
Fitting A Covid-19 Model Incorporating Senses Of Safety And Caution To Local Data From Spartanburg County, South Carolina, D. Chloe Griffin, Amanda Mangum
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 …
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.
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
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 …
Ode Models Of Wealth Concentration And Taxation, Bruce Boghosian, Christoph Borgers
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.
Modeling Aircraft Takeoffs, Catherine Cavagnaro
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 …
Odes And Mandatory Voting, Christoph Borgers, Natasa Dragovic, Anna Haensch, Arkadz Kirshtein, Lilla Orr
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 …
Solar Panels, Euler’S Method And Community-Based Projects: Connecting Differential Equations With Climate Change, Victor J. Donnay
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 …
The Collegiate Tennis Effect: Measuring The Impact Of College Attendance On A Tennis Player’S Professional Career, Ian Freer
CMC Senior Theses
Our study examines the influence of college attendance on the professional tennis careers of the 502 American male players who earned a ranking point between 2014 and 2023. Initial analysis suggests that attending college may lower a player’s career-high ranking by 159 spots. However, after controlling for career length, we observe a significant improvement of 214 spots among players with similar junior rankings. Our analysis of career trajectories reveals that college players often start their professional careers at higher rankings and achieve their career-high rankings earlier than non-college players. Additionally, the career-high rankings of the most promising juniors tend to …
La Emergencia De La Inteligencia Artificial En El Aprendizaje Del Español: ¿Una Suplementación O Sustitución?, Sam Goldfein
La Emergencia De La Inteligencia Artificial En El Aprendizaje Del Español: ¿Una Suplementación O Sustitución?, Sam Goldfein
CMC Senior Theses
Abstract:
The recent emergence of widespread generative artificial intelligence – alongside the sudden demand for an improved online learning experience – questions the validity of current language learning models: could we improve old language learning models with the new, powerful technological capabilities that we’ve suddenly been granted, or is it time to scrap the old practices entirely and embark on a new technological journey void of professors and traditional classroom environments? This thesis stresses the importance of learning languages and specifically emphasizes the advantages of knowing Spanish as a second language in the United States. To determine the best methods …
Entremezclando Sueños: Reimagining Education Through Community-Centered Program Evaluations, Cassandra Hernandez
Entremezclando Sueños: Reimagining Education Through Community-Centered Program Evaluations, Cassandra Hernandez
CMC Senior Theses
This thesis fuses Chicane Latine Studies (CLS) and Public Policy methodologies to explore how student-centered organizations can incorporate Decolonial Feminism into their program evaluation pedagogy. In today’s society, the political economy, influenced by structural biases, gravely impacts marginalized communities by overlooking holistic wellbeing as a valid priority. Embodying Decolonial Feminism can help community organizations fight back against systemic inequities and redefine for themselves their existence. A starting point is adjusting how organizations conduct program evaluations. Using Chicana Latina Feminista (CLF) Pláticas, concepts from these pláticas culminated into a collaborative decolonial feminist program evaluation guide called "Entremezclando Sueños." This guide serves …
Analyzing A Smartphone Battle Using Bass Competition Model, Maila Hallare, Alireza Hosseinkhan, Hasala Senpathy K. Gallolu Kankanamalage
Analyzing A Smartphone Battle Using Bass Competition Model, Maila Hallare, Alireza Hosseinkhan, Hasala Senpathy K. Gallolu Kankanamalage
CODEE Journal
Many examples of 2x2 nonlinear systems in a first-course in ODE or a mathematical modeling class come from physics or biology. We present an example that comes from the business or management sciences, namely, the Bass diffusion model. We believe that students will appreciate this model because it does not require a lot of background material and it is used to analyze sales data and serve as a guide in pricing decisions for a single product. In this project, we create a 2x2 ODE system that is inspired by the Bass diffusion model; we call the resulting system the Bass …
How To Intercept A High-Speed Rocket With A Pair Of Compasses And A Straightedge?, Yagub N. Aliyev
How To Intercept A High-Speed Rocket With A Pair Of Compasses And A Straightedge?, Yagub N. Aliyev
CODEE Journal
In this paper a nonlinear differential equation arising from an elementary geometry problem is discussed. This geometry problem was inspired by one of the proofs of the first remarkable limit discussed in a typical first semester undergraduate Calculus course. It is known that the involved differential equation can be reduced to Abel’s differential equation of the first kind. In this paper the problem was solved using an approximate geometric method which constructs a piecewise linear solution approximation for the curve. The compass tool of GeoGebra was extensively used for these constructions. At the end of the paper, some generalizations are …
Population Growth Models: Relationship Between Sustainable Fishing And Making A Profit, James Sandefur
Population Growth Models: Relationship Between Sustainable Fishing And Making A Profit, James Sandefur
CODEE Journal
In this paper, we develop differential equations that model the sustainable harvesting of species having different characteristics. Specifically, we assume the species satisfies one of two different types of density dependence. From these equations, we consider maximizing sustainable harvests. We then introduce a cost function for fishing and study how maximizing profit affects the harvesting strategy. We finally introduce the concept of open access which helps explain the collapse of many fish stocks.
The equations studied involve relatively simple rational and exponential functions. We analyze the differential equations using phase-line analysis as well as graphing approximate solutions using Euler's method, …
Fibonacci Differential Equation And Associated Spiral Curves, Mehmet Pakdemirli
Fibonacci Differential Equation And Associated Spiral Curves, Mehmet Pakdemirli
CODEE Journal
The Fibonacci differential equation is defined with analogy from the Fibonacci difference equation. The linear second order differential equation is solved for suitable initial conditions. The solutions constitute spirals in the polar coordinates. The properties of the spirals with respect to the Fibonacci numbers and the differences between the new spirals and classical spirals are discussed.