Mathematics Commons^™

The Butterfly Effect In Economics: Exploring Chaos With A Simple Financial Model, Sandy Hs Herho

CODEE Journal

Wild swings in financial markets need not result from external shocks like earthquakes or wars—they can emerge from deterministic chaos. This article introduces kalimusada, an open-source Python library that lets students and instructors explore this phenomenon through a simple three- equation model of financial dynamics. The model couples interest rates, investment, and prices through nonlinear feedback, generating bounded but unpredictable oscillations characteristic of chaos. Tiny differences in starting conditions—smaller than any measurement could detect—grow exponentially until two initially identical economies follow completely different paths. The library provides ready-to-use tools for visualizing this “butterfly effect” in economics, computing divergence metrics, and …

The Math Teaching Atlas: Trails, Anchors, And Compass In Action, Ivan Z. Feng

Journal of Humanistic Mathematics

This paper advocates a shift grounded in mathematical reasoning: teach and write math through explicit routeways rather than narratives, anchor new ideas to familiar concepts on a roadmap, and use motivation and concrete examples as a compass and simulation for discovery.

Creating Art With The Generalized Pythagorean Theorem, Ángel Colín

Journal of Humanistic Mathematics

The Pythagorean Theorem and its generalizations have been widely studied. In this article we focus on one particular generalization that asserts that the area of a polygon with one side the hypotenuse of a right triangle is the sum of the areas of similar polygons whose corresponding sides are the other two sides [4]. Using this theorem together with interactive software or simply pencil and paper, we create some visually enticing artwork. This activity, we believe, can contribute positively to geometry teaching at all academic levels.

Let's Play Uno! Training Mathematical Thinking With An Inclusive Card Game, Thierry Meyrath, Clara-Ioana Mincu, Antonella Perucca

Journal of Humanistic Mathematics

We explore probability exercises based on the game UNO. Excluding the Wild cards, the deck contains precisely 100 cards, which makes it convenient to express probabilities as percentages. We discuss the mathematical relation between cards that can be played after other cards. We also propose an algebraic exercise that relates several quantities, such as the number of cards left in the losing player’s hand. Finally, we argue that UNO is a very inclusive game: it is logistically easy to play, allows for players of different skill levels, and can be adapted to meet special needs. Beyond being playful exercises, our …

How Did Cauchy Prove His Integral Theorem? A Historical Reconstruction For Education Research, José Gerardo Piña-Aguirre, Rosa María Farfán Márquez

Journal of Humanistic Mathematics

In his article “Mémoire sur les Intégrales Définies, prises entre des limites imaginaires” of 1825, Cauchy proves that the integral of a complex function f defined on complex numbers is independent of the path of integration if the function f remains finite and continuous along the chosen path. In contemporary literature that analyzes the mathematics of Cauchy, the specific mathematical process that allowed Cauchy to prove this result is not explicitly presented. The purpose of this manuscript is to communicate a symbolic-algebraic derivation that justifies the narrative argument used by Cauchy to support his proof, thus hopefully filling this gap. …

Using Provided Guided Notes In Coordinated Introductory First-Year Mathematics Courses, Jennifer L. Huber

Mathematics Dissertations

The goal of this study is to investigate how standardized guided notes shape instructional practices and student engagement in coordinated introductory first-year college mathematics courses at a large public university. The researcher explored three multi-section introductory mathematics courses with overlapping learning objectives. Each course required students to purchase a student workbook as part of the instructional materials for the class. The instructors taught primarily from the workbook containing guided notes created by a former coordinator of the course. The researcher used a mixed-methods approach. Instructors and students participated in surveys, class observations and provided class meeting notes. Instructors shared additional …

I Think, Therefore I Can: Self-Efficacy, Self-Regulation, And Sense Of Belonging In Introductory Physics, Danielle Christine Maldonado

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cognitive and affective beliefs in physics classrooms are important indicators of student success beyond traditional measurements of academic performance. This body of work explores major beliefs and processes, including self-efficacy, self-regulation, and sense of belonging, as they relate to academic achievement among introductory undergraduate physics students. First, this dissertation explores the development of physics self-efficacy and physics sense of belonging, two highly correlated variables. This work identifies unique sources of physics self-efficacy and sense of belonging development from current and prior academic experiences, personality factors, and demographic variables. Then, relationships between early- semester self-efficacy and sense of belonging, mid-semester …

Test Retakes In Introductory Math Courses, John Hird, Chris Mcclain

2026 Scholarly Teaching Conference: Poster Session Papers

In this poster, we describe the implementation of an exam retake model using specifications grading for math courses taken by non-STEM majors. This system was implemented by two faculty members over three years in two sequential courses. During that time, we tried several versions of allowed retakes, with varying restrictions on partial credit. Some of the challenges that we faced were scaling the system for use by different faculty members and with different courses, managing faculty workload on writing and grading multiple exams, and managing student expectations.

Fostering A Growth Mindset In Mathematics: Faculty And Student Experiences, Yolanda G. Rush

Theses and Dissertations

According to the Center for Community College Student Engagement (2019), many students attending two-year institutions need productive persistence strategies, including the development of a growth mindset. Although some growth mindset interventions have been effective in improving academic achievement among students (Boaler, 2016; Canning et al., 2024) and persistence (Lewis, 2019) among students, especially those with developmental needs (Suh et al., 2019) and those in mathematics, little is known about the experiences of students and teachers (i.e., students’ perceptions of teachers’ intentions and implementation) as teachers work to foster a growth mindset culture (Murphy et al., 2021). In this dissertation, I …

A Conceptual Framework For Cognitive Engagement And Human Agency In Ai-Mediated Learning, Sam Rhodes, Zuhal Yilmaz, Terrie Galanti, Younggon Bae, Victoria Delaney, Zareen Gul Aga, Stephen Hutt

School of Mathematical & Statistical Sciences Faculty Publications

The purpose of this chapter is to propose a conceptual framework that unpacks AI use in instructional practice across three interrelated dimensions: human agency, cognitive demand, and AI orchestration. We ground this work in historical perspectives on technology's mediating role in education. We then propose the conceptual framework and three characterizations of the use of AI in school instruction: cognitive load reducer, cognitive capacity builder, and cognitive companion. We then illustrate a worked example of each, including a curricular example and sample student prompts, and conclude with questions that educators might ask themselves as they consider if and how to …

An Exploratory Approach For Teaching Systems Of Linear Odes With Real Repeated Eigenvalues, Biyong Luo, Li Zhang

CODEE Journal

In this paper, we present a project that aims to pique students' interest and facilitate their engagement in learning. Students will be encouraged to take an active role in their learning of systems of linear ODEs with real repeated Eigenvalues. Through discussion and completion of a set of guided questions given in the project, students will gain a deeper understanding of the topic. Also, students will use the technology tool Bluffton App to draw solution trajectories in a phase plane, and then learn necessary skills for explaining the movements of the trajectories and their long term behaviors. This project may …

Analysis Of Solutions To Research Methods: Navigating The Infinite Content Of Mathematics, Jesus Navarro

Student Scholar Symposium

In the infinite realm of mathematical research, understanding and communicating complex concepts require structured and logical approaches. However, a variety of research methods are employed in mathematical studies; these may differ in approach, historical context, and other important considerations which can contribute to confusion and a lack of effective communication. The genesis of this research began with personal confusion during my research on "Counting the Uncountable: Life in a World Without Numbers", setting the stage for the central challenge: How do mathematicians explain their research? To answer this issue, we analyzed a variety of mathematical research articles to highlight their …

From Textbook To Classroom: Gaps In Mathematical Representation And Translation In Nepali Education, Deepak Basyal

Mathematics and Statistics

Translating real-world problem contexts into mathematical symbols is a key skill in curricula worldwide. This study examines Nepali students’ abilities to translate words into symbols and how Nepali textbooks from earlier grades support the development of these skills. To investigate this phenomenon, a sequential explanatory study was conducted with a test for 10th graders and an analysis of Grades 6—9 textbooks. Results show that students struggle to convert real-world contexts into mathematical symbols, while textbooks provide limited opportunities for practice. In particular, students have difficulty translating symbols back into real-world situations, suggesting a link between textbook opportunities and achievement. These …

Connections On Manifolds: A Modern Approach To Classical Differential Geometry, Jed Allen Wilshire

Theses and Dissertations

The purpose of this thesis is to connect modern differential geometry concepts to the classical differential geometry study of surfaces, specifically spheres. This thesis will cover the topics of Riemannian manifolds, tangent spaces, vectors, vector fields, the connection of Levi-Civita, the Christoffel symbols, and Riemannian curvature. Each concept will be demonstrated on the well known unit sphere manifold.

Untitled Unmastered: Initial Conditions And Emergent Properties Of The Mathematics Classroom, Reginald L. Mcgee

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Expanding Ode Examples: Introducing Gene Regulation Dynamics Through Hill Functions, Maila Hallare, Jane M. Santamore

CODEE Journal

Gene regulation is a fundamental biological process that controls gene expression. It can explain phenomena such as cell differentiation, circadian rhythms, disease progression, and metabolic control, among many others. Despite their importance in mathematical biology, gene regulation models are rarely featured in traditional ODE textbooks, which focus mainly on examples from engineering, physics, chemistry, and population biology. This article introduces gene regulation dynamics as a valuable addition to ODE curricula, presenting the models from a mathematical perspective and building on properties of the Hill function. These models deepen the understanding of biology-inspired ODE applications, provide accessible research opportunities for students, …

Follow Me Into Math: Near-Peer Influencers Take The Stage, Aaron T. Wilson, Sergey Grigorian, Mayra Ortiz Galarza, Xiaohui Wang

School of Mathematical & Statistical Sciences Faculty Publications

Young people like to be entertained, and to entertain others. What if students were to invest enough of their time and effort into learning topics from our mathematics courses to become experts in the eyes of their peers? In this study we positioned college students and high school students as mathematical performers on the stage in front of their nearpeers, for whom they became math influencers, developing and presenting MathShows. The paper shows readers how we positioned students on the stage, using a 7-step MathShow Design Process informed by a Neer-peer Mathematical Mentoring framework. Of import for educators, researchers and …

Growth Mindset In The Mathematics Classroom: Studying The Effects Of A Growth Mindset Intervention On High School Students In A Geometry Classroom, Katelyn Strauss

Mathematics Graduate Theses

Having a growth mindset is crucial to being successful in a mathematics classroom. However, many students believe themselves to be either “mathematics people” or “not mathematics people”. When students believe they are not a mathematics person, they are far less likely to engage in material where they face struggles, and thus they miss out on valuable learning. A review of the literature has show that students have been able to shift their mindset from fixed to growth through a series of interventions. I used a unit in Geometry to see how a growth mindset intervention would affect the mindset of …

(Si15-140) Designing Bayesian Double Sampling Plans Based On Zero Inflated Poisson Distribution, Priyadharshini R., Shalini K., Hemalatha R., Sangeetha S.

Applications and Applied Mathematics: An International Journal (AAM)

The implementation of attribute-based sampling inspection serves as a quality control technique used across numerous industries to evaluate items or workflow processes. When the data exhibits a substantial number of zero counts, the zero-inflated Poisson (ZIP) distribution serves as an effective model for accommodating this zero-inflation. Double sampling plan (DSP) is a quality check method where the decision to approve or decline a batch comes after examining two samples, providing more conclusive information compared to a single sample plan (SSP). In practice, effective decision-making regarding submitted lots considers both within-lot and between-lot variations, which can be addressed through the use …

The 3d Double Spherical Pendulum: Modeling, Analysis, And Simulations, Alexander Gofen

CODEE Journal

This paper presents an inquiry-based research project that develops and analyzes the system of ordinary differential equations governing the motion of the 3D double pendulum, blending computational experiments with applied and theoretical mathematics. We formulate the Lagrangian for the three-dimensional double spherical pendulum and use Maple to derive the four coupled ordinary differential equations (ODEs) in angular variables; for completeness, we also present an equivalent Cartesian formulation. We then compare and visualize the models using the Taylor Center high-order Taylor-series solver, which delivers high-accuracy trajectories and real-time animations in 2D and anaglyph 3D (red/blue). We place the system in context …

Project-Based Learning With Odes: Modeling Straw Rocket Motion With Air Resistance, Viktoria Savatorova, Ethan Dyer, Aleksei Talonov

CODEE Journal

This paper presents a hands-on project that guides students through building and validating a mathematical model of projectile motion. The project starts with the idealized case of motion under gravity without air resistance and then introduces air drag : first as a linear force, and then as a nonlinear quadratic force, with the Reynolds number providing the justification for the quadratic model. Students perform experiments with vertical and angled launches, capturing and analyzing motion data using video analysis software. Vertical launch data allows parameter estimation via least squares fitting of the nonlinear drag model, yielding values for initial velocity and …

Let’S Teach More Accurate And Inclusive History! The Case Of Islamic Contributions To Mathematics And Science, Nuh Aydin

Journal of Humanistic Mathematics

Schools commonly teach a history of mathematics and science that is Eurocentric. This selective version of the history of science, called Classical Narrative by some, is rooted in colonial times and mindset, and has reinforced certain negative opinions about other cultures. It ignores or downplays contributions to mathematics and sciences from non-western civilizations, and falsely attributes many scientific discoveries to European scholars. Medieval Islamic Civilization, which had strong connections to Renaissance Europe, serves as a clear example of how this narrative is distorted. Based on research on primary sources since the middle of the 20th century, we now know that …

From Shudu [Sudoku 數獨] To Dushu [Learning Of Mathematics 讀數]?, Man Keung Siu

Journal of Humanistic Mathematics

The puzzle game Sudoku has been popular for over two decades, and even at one time became a craze for the public. This paper will briefly describe what the game is, a little bit of its history, and how to play it, but will leave the details to the many published books about the game. Some interesting and difficult mathematical problems relevant to the game will be highlighted, again leaving the detailed discussion to specific books published on this issue. The main point the paper tries to focus on is a discussion on the cognitive and the psychological aspects of …

Math Anxiety And Shame Exemplified On The Reality Show Survivor, James Drimalla, Dru Horne

Journal of Humanistic Mathematics

In an episode of season 45 of the reality show Survivor, three contestants competed in a challenge involving mathematics. In their reflections and responses to the challenge, the contestants discussed feelings of anxiety, fear, and shame. In this article, we describe (a) the challenge, (b) the three contestants’ attempts to complete the challenge and their responses to the challenge, (c) responses to the challenge in the media, and (d) relevant details of our personal experiences and related backgrounds. We then discuss themes of math anxiety, shame, and virtues, and encourage leaders in mathematical spaces to place a higher value on …

Introducing The Linear Poem, Cristian Ramirez Rodriguez

Journal of Humanistic Mathematics

In this article I introduce the linear poem, a poetic form I developed in fall 2023 as a pedagogical tool for high-school students. The linear poem is a poem which represents an integer valued linear function, treating the line numbers as inputs and the linguistic units in a line as outputs. Two examples of linear poems are given as well as the steps taken to produce them. By interpreting poetic forms as integer-valued functions as is done in this article, interdisciplinary pedagogy can be achieved, allowing educators and students to express mathematical ideas such as slope-intercept form in a creative …

Exploring Commutativity In A Hindi Language Classroom, Kumar Gandharv Mishra

Journal of Humanistic Mathematics

Commutativity is an important concept in mathematics. However, not much emphasis is given to it in the elementary or secondary school curriculum, as there is a lack of interesting noncommutative examples at this level. Since most examples of binary operations provided deal with numbers, commutativity appears trivial. In this article we offer an interesting example of noncommutativity that is familiar to Indian school children from a different context. Students in India study a chapter based on the joining of vowels in Hindi grammar. This process is analogous to producing commutative as well as noncommutative operations. When we bring in mathematical …

A Proposal To Explore Geometry With Geogebra: Graphical Exploration And Formal Demonstration Of The Collinearity Of The Barycenters Of A Polygon, Saulo Mosquera Lopez, Marlio Paredes, Walter Castro

School of Mathematical & Statistical Sciences Faculty Publications

This paper illustrates an example of a mathematical activity that teachers and students can replicate to create an experience that resembles professional mathematical activity. We extend the property “Consider a triangle ABC, any straight line and let A’, B’, C’ be the reflections of the points A, B, C on the straight line then the barycenters of the triangles ABC, A’BC, AB’C and ABC’ are collinear and the line of collinearity is perpendicular to the straight line” for any quadrilateral. It is proved that there are four additional triangles, for a total of eight, whose barycenters are collinear and that …

The Other Side Of The Equation: De-Simplification, A Prerequisite For Calculus, Stephen L. Brown

ACMS Conference Proceedings 2005

No abstract provided.

“What Makes It Eigen-Esque-Ish?”: A Form-Function Analysis Of The Development Of Eigentheory Concepts In A Quantum Mechanics Course, Megan Wawro, Kaitlyn Stephens Serbin

School of Mathematical & Statistical Sciences Faculty Publications

Eigentheory concepts are central in mathematics and physics; they serve multiple functions, such as symbolizing physical phenomena and facilitating mathematical computations. Words and meanings associated with eigentheory develop and vary over time, as do their associated symbols. In this study, we investigate how “eigen” develops over time in one quantum mechanics course by analyzing form-function relations (Saxe, 1999) for eigentheory concepts over 22 class sessions. We share results concerning our microgenetic and ontogenetic analyses of the creation of form-function relations and their shifts over time by characterizing the continuity and discontinuity of the various functions and forms associated with concepts …

Culture And Context: How Frames Of Teaching And Of Learning Mathematics Form And Change For Graduate Student Instructors, Johan Benedict A. Cristobal

Department of Mathematics: Dissertations, Theses, and Student Research

Why do instructors teach the way that they do? This question is key in understanding how and why asset-based or deficit-based practices are enacted in the classroom. In consequence, this question also gives insight into how the environment where students learn is shaped. Mathematics education research, in particular, has been concerned with how instructors respond to students’ contributions in the classroom and how these ways of responding can (dis)empower students.

One way to answer this question is to understand how instructors frame their own teaching and their students’ learning. Frames are mental constructs which help individuals filter details, interpret information, …