Mathematics Commons^™

Data-Driven Discovery Of Canonical Models In Undergraduate Differential Equations Education, Devon L. Garls, Raina Majumder, Scott A. Strong

CODEE Journal

Roughly a decade ago, Brunton, Proctor, and Kutz built upon the re-

sults of classical control theory, compressed sensing, and symbolic regression

to create a method for Sparse Identification of Nonlinear Dynamics (SINDy),

allowing one to infer a dynamical system from time series and bring mathe-

matical dynamics into the modern era of large data. This paper is designed for

a reader concurrently enrolled in a first course in ordinary differential equa-

tions and begins with a technical overview of least-squares regression and

finite-difference approximations to derivatives, which is enough to perform

an example hand-calculation on three data points. The …

The Use Of Gadgets And Learning Motivation Among Grade Xi Students In Biology Learning At Sma Negeri 5 Palu, Yulia Citra, Mohammad Jamhari, Masrianih Masrianih, Lestari M.P Alibasyah, Mursito S. Bialangi, Yulia Windarsih

Jurnal Pendidikan Sains

This study aims to describe the profile of gadget use and learning motivation among Grade XI students in biology learning at SMA Negeri 5 Palu and to explore students’ and teachers’ experiences of gadgets as learning support tools and potential sources of digital distraction. A qualitative descriptive design was employed, supported by descriptive questionnaire profiling. A total of 74 students completed a 50-item questionnaire on gadget use and learning motivation, while 10 students and two biology teachers participated in semi-structured interviews. Questionnaire data were analyzed using frequencies and percentages, while interview data were analyzed thematically. The descriptive findings showed that …

Differential Equation Modeling For Sustainable Resource Management: A Steady-State Optimal Harvesting Approach, Iordanka N. Panayotova, Aleksei Talonov

CODEE Journal

Mathematical models based on differential equations provide a powerful framework for connecting real-world data to informed decision-making. In this work, we present a student-accessible project that uses an optimal-control framework to study the sustainable management of biological resources.

Motivated by fisheries management, we examine a predator--prey system in which harvesting decisions must balance ecological and economic considerations. The model is formulated as an optimal control problem that seeks to maximize the total discounted net revenue from harvesting. Rather than solving for the complete time-dependent harvesting trajectory, we restrict the analysis to positive controlled coexistence equilibria and characterize an interior stationary …

Using Approximations Of Practice To Support Calculus Instructors’ Enactment Of Equitable Active Learning Teaching Practices, Kaitlyn Stephens Serbin, Karina Uhing

School of Mathematical & Statistical Sciences Faculty Publications

We implemented a semester-long professional development workshop series to help Calculus I and II instructors implement active learning and equitable teaching practices. The workshops actively engaged instructors in approximations of practice (Grossman et al., 2009) in which they learned to enact various teaching practices like designing tasks; eliciting, noticing, and responding to evidence of student thinking; orchestrating class discussions; fostering inclusive social norms; and equitably managing student participation. After they participated in the workshops, we interviewed the instructors about their decisions to implement the teaching practices from the approximations of practice in their teaching. We analyzed the instructors’ resources, orientations, …

Motivating Second-Order Differential Equation Models Beyond The Spring–Mass System, Katherine E. Rutherford, John C. Borger

CODEE Journal

Students in introductory differential equations courses often learn to associate specific equation types with familiar physical systems. While this approach supports procedural fluency, it can inhibit transfer learning: students recognize familiar forms rather than construct models from first principles. This paper presents a modeling-first instructional approach designed to help students recognize the shared structure of second-order differential equations across diverse contexts. Students derive governing equations from fundamental physical laws for multiple systems, including an Inductor, Resistor, and Capacitor (LRC) electrical circuit, a simple pendulum, and viscous pipe flow. Through guided comparison, students identify common components such as inertia, damping, restoring …

Pedagogical Moves For Provoking Prospective Teachers’ Intellectual Needs While Connecting Abstract Algebra And Secondary Mathematics, Kaitlyn Stephens Serbin, Estrella Johnson, Sthefanía Espinosa

School of Mathematical & Statistical Sciences Faculty Publications

Researchers have exhibited ways in which prospective secondary mathematics teachers can connect content from abstract algebra with secondary mathematics in ways that reshape their understanding of secondary mathematics and thereby support their teaching of that content. However, there has been very little research done on what mathematics teacher educators can do in the classroom to help prospective teachers make connections between abstract algebra and secondary algebra in ways that support such reshaping. This study focuses on a mathematics teacher educator who guided a class of prospective teachers to connect properties of algebraic structures with equation-solving procedures by directing their focus …

Fast-Track Ode: A Student-Centered, Game-Based Summer Differential Equations Course, Chamila D. Malagoda Gamage

CODEE Journal

This article describes a student-centered and game-based redesign of an Elementary Differential Equations course taught during a six-week summer session. In this compressed setting, students had to move through the standard course content quickly while still developing procedural fluency, conceptual understanding, and confidence with applications. To support these goals, the course used real-time feedback tools, collaborative problem solving, applications connected to students' fields, and mathematical games. The paper describes the course context, major activities, implementation details, and student feedback. Student responses suggest that the activities were well received and helped students stay engaged, participate regularly, prepare for exams, and see …

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …

Rockin’ Rover On The Rainbow Road, Michael Kolta, Lawrence Burgee, Ying Yuan

Transformations

This paper presents a progressive series of age-appropriate lesson plans for grades K-12 that all use the same interdisciplinary activity to educate students about Science, Technology, Engineering, Art, and Mathematics (STEAM) simultaneously. Technology from Texas Instruments (TI) was employed including a TI Nspire graphing calculator that can run Python programs, a TI Innovator Hub, and a TI Rover. The TI Rover is a small, robotic car that has sensors and is controlled by the calculator via the Hub hardware interface. A Python program was developed that uses the color sensor in the Rover to detect the color on colored paper …

From Reluctance To Resilience: The Link Between Attitude And Growth Mindset In Calculus Students, Olivia Valente, Kathryn E. Pedings-Behling, Amy N. Langville

Journal of Educational Research and Practice

Many postsecondary students enter general education mathematics courses with negative attitudes, shaping their willingness to engage with the content. This study explores the relationship between students’ attitudes towards mathematics and their growth mindset, revealing effective strategies to enhance learning experiences, particularly for students who are reluctant learners. The research investigates two questions: (1) How do students’ Attitude Toward Mathematics Inventory (ATMI) scores and Growth Mindset Scale (GMS) scores relate? and (2) Do gender or course modality affect this relationship? This study addresses a gap in literature by examining how enjoyment, motivation, self-confidence, and perceived value connect with growth mindset. The …

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 …

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 …

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 …

Coupled-Pendulum Modeling In An Ode Class: An Assignment On Fourier-Initialized Gradient Descent In Machine Learning, Huy Truong, Andrew Bennett

CODEE Journal

As data-driven methods are increasingly used in science and engineering, students benefit from learning to integrate machine learning techniques with traditional mathematical modeling. We present a hands-on extra-credit assignment for an undergraduate ordinary differential equations (ODE) course that enables students to compare classical analytical methods with data-driven approaches on the same physical system. Using a coupled-pendulum system---two pendulums connected by a spring---with real experimental data acquired via video tracking of a real physical setup, students work through three models in a guided Jupyter notebook with all code provided. First, they fit a neural network with Fourier features as a purely …

A Differential Equation–Based Epidemiological Model Of Post-Operative Chronic Pain In Scoliosis Patients With Data-Driven Analysis, Paige Zhu, Padmanabhan Seshaiyer

CODEE Journal

Chronic post-surgical pain (CPSP) is a common and often overlooked complication following surgical correction of idiopathic scoliosis, impacting long-term patient wellbeing despite improvements in surgical outcomes. This project introduces a novel epidemiological framework to model the progression of CPSP using a compartmental structure. By applying a coupled system of nonlinear differential equations, we simulate pain trajectories over time and assess the effectiveness of surgical interventions. The model is implemented for a single-cohort population and extended to a two-cohort design to compare outcomes between Posterior Spinal Fusion (PSIF) and Vertebral Body Tethering (VBT) procedures. Further stratification by patient age enables us …

Toward A Didactical Phenomenology For The Completeness Axiom, Sean Larsen, Tenchita Alzaga Elizondo, Kristen Vroom, Stephen Strand Ii

School of Mathematical & Statistical Sciences Faculty Publications

The study is part of an instructional design project focused on introductory real analysis. The goal of the project is to develop a theoretically grounded and empirically supported instructional approach that builds on students’ experiences in the calculus sequence to engage them in the reinvention of the rigorous foundations of the calculus. An essential aspect of this foundation is the completeness of the real numbers. Drawing on the didactical phenomenology heuristic from the theory of Realistic Mathematics Education (RME), we conducted an iterative instructional design study focused on the completeness axiom. The work proceeded in two phases. First, we conducted …

Trattato Dell’Alcibra Amuchabile (Anonimo): A Guided Translation, Gary Towsley, Olympia Nicodemi

Geneseo Authors

The Trattato dell’Alcibra Amuchabile is a pre-modern algebra text from c. 1365. It is written in a Tuscan dialect of Italian and is situated in the abbacus school tradition, schools that taught the mathematics needed for a mercantile society. Like all the algebra written in Italy at the time, it was inherited from al-Khwarizmi and, like his, written with no symbols—no x’s, y’s, plus signs, etc. It was what is sometimes called “rhetorical algebra.”  There is very little source  material available in English from this important era in the history of algebra. This book helps fill that gap. …

Centering Culture, Language, And Identity In A History Of Mathematics Course For Preservice Teachers, Luis Miguel Fernández, Tenchita Alzaga Elizondo

School of Mathematical & Statistical Sciences Faculty Publications

This article presents the redesign of an undergraduate History of Mathematics course for preservice teachers at a Hispanic-Serving Institution. Grounded in Community- and Family-Centered Pedagogy (CFCP) and Marshall’s framework for culturally sustaining mathematics teaching, the course intentionally moved beyond Eurocentric narratives to center students’ linguistic, familial, and cultural resources as integral to mathematical meaning-making. Two core assignments anchored the redesign: (1) evolving Padlet reflections that connected global mathematical histories to students’ family and community practices, and (2) bilingual Cuentos Matemáticos that wove mathematical concepts with personal and cultural narratives in English and Spanish. Selected student reflections illustrate how these experiences …

From Cork To Coasting: A Multi-Stage Ode Model Of Water-Rocket Flight, Viktoria Savatorova, Patryk Kustra, Ethan Dyer, Connor Carlson, Aleksei Talonov

CODEE Journal

Water rockets provide an affordable and engaging context for exploring applications of differential equations. Motivated by outreach activities conducted with undergraduate students, we develop a four-stage mathematical model of vertical water-rocket flight that is suitable for use in an ODE or mathematical modeling course. The model includes the cork-release phase, water-thrust propulsion, air-thrust propulsion with compressible and potentially choked flow, and the final ballistic stage with quadratic drag. While retaining key physical features, the model can be formulated as a system of ordinary differential equations that can be integrated numerically using tools familiar to students. We compare model predictions with …

You Mean What? Commognitive Conflict In Resolving Contextual Logarithm Problems, Endrayana Putut Laksminto Emanuel, Fatkul Anam, Radhitya Duta Pradana, Anik Kirana, Sikky El Walida

Jurnal Pendidikan Sains

Students’ approaches to solving contextual mathematics problems involving logarithms vary significantly due to differences in their prior mathematical understanding. These differences may trigger commognitive conflict during the interpretation and reasoning processes when students attempt to construct mathematical meaning from contextual situations. This qualitative study aimed to explore how commognitive conflict emerges as a mechanism of mathematical interpretation in students’ discourse while solving logarithmic contextual problems. Twenty students participated in the study and were grouped based on their performance. One student was selected as the main research subject for an in-depth analysis. The findings reveal that commognitive conflict appeared in two …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Beyond The Best Fit: Fitting Sir-Style Models To Data While Prioritizing Biological Meaning, Meredith L. Greer

CODEE Journal

Many students find comfort in mathematics because math classes have been a place where they can find the “right answer” to exercises. Modeling courses typically emphasize more nuance, teaching students to try modeling approaches, compare back with real-world mechanisms and data, and then refine their models, in an ongoing cycle. One topic, however, can cause students to quickly revert into “right answer” mode: fitting a model to data. Phrases such as best fit and minimize the distance suggest there is one optimal solution that can be determined by an algorithm, and these algorithms typically have no relationship to the biology …

Course Portfolio: Elements Of Physics Phys 151, Evan A. Rich

UNL Faculty Course Portfolios

This course portfolio documents the instructional design, teaching methods, and ongoing assessment efforts for PHYS 151: Elements of Physics, an algebra-based introductory physics course at the University of Nebraska-Lincoln. The course serves a broad undergraduate population, including architecture, construction management, and life science majors. The portfolio describes the teaching framework that integrates pre-lecture video preparation, active in-class engagement through iClicker questions, and collaborative weekly recitation sections, all unified around a structured six-step problem-solving approach. A central concern of the course is building students’ self-efficacy in physics, particularly among those with math anxiety or limited preparation. Two assessments are reported: a …

1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, ..., Annemarie Torresen

Masters Theses

1

I invite you to see it and feel it and sit in it and breathe it in and

hold it in your lap. Art isn’t too scary, and neither is math.

2

bouncing between bounds of a binary

bippity boppity! let’s break brains and bread

balance or belly flop, what’s mine is yours:

bumptious and bumbling and barely able

i offer it broken and let you like it that way

Machine Learning For Modeling In An Elementary Differential Equations Class, Nathan Albin, Andrew G. Bennett, Abhinav Chand

CODEE Journal

Mixing machine learning with modeling is an area of increasing importance. This paper presents a lesson where students model a spring-mass system both using traditional analysis with linear damping and using machine learning to learn the damping from real data. The machine learning is implemented in a Jupyter notebook hosted on Google Colab, allowing students to train the neural network without requiring the students to carry out coding. Students get experience with how machine learning can fail, how it can work, and the time and data requirements for machine learning to succeed, and are asked to apply this knowledge to …

Parameter Estimation In Ode Models Using Least-Squares Regression, Ulrich A. Hoensch

CODEE Journal

We present a method of estimating model parameters for non-linear ODEs using least-squares regression. The coefficient of determination can be used as a measure of model fit. The method is demonstrated using US population data to fit a logistic growth model. Also, a competing species model is used to describe the interaction of two different species of yeast.

Emergent Storylines That Influence Positions And Mathematical Status In Collaborative Small-Group Proof Activity, Brittney M. Ellis, Tenchita Alzaga Elizondo

School of Mathematical & Statistical Sciences Faculty Publications

In this paper, we used positioning theory to examine storylines that emerged in students’ discourse as they collaborated on a proof construction task. We purposefully selected a case of group work from an inquiry-oriented introduction to proof course as prior analyses showed it was highly collaborative (Alzaga Elizondo, 2022), yet power dynamics seemed unbalanced. We hypothesized that positioning theory could provide a useful lens to interrogate such power dynamics. Through this analysis, we identified several implicit storylines that influenced the interaction related to the nature of proofs, the nature of mathematics, writing proofs, the role of an external authority, …

Development Of Problem-Based Teaching Modules On Linear Systems Of Equations And Inequalities To Improve Mathematical Problem-Solving Skills, Elsya Suharnita, Kartini Kartini, Sehatta Saragih

Jurnal Pendidikan Sains

This study addresses the low mathematical problem-solving competency among Indonesian students, as evidenced by the PISA 2018 results that showed 71% failed to meet the minimum mathematical competency level. This study aims to develop a problem-based learning (PBL) module for systems of equations and inequalities content to improve Phase E students' mathematical problem-solving abilities. Using the ADDIE (Analysis, Design, Development, Implementation, Evaluation) model, this study involved expert validation, a small group trial with 12 students, and field implementation using a posttest-only control group design with 24 students in the experimental group and 25 students in the control group. The results …

Cognitive Conflict In Iconic Visual Representations: Comparing Conventional And Ai Approaches, Endrayana Putut Laksminto Emanuel, Herfa Maulina Dewi Soewardini, Radhitya Duta Pradana, Sikky El Walida

Jurnal Pendidikan Sains

Background of study: The rapid integration of Artificial Intelligence (AI) in higher education has transformed mathematical problem-solving practices, yet it introduces potential discrepancies in students’ visual understanding, particularly in geometric contexts, leading to commognitive conflicts in visual mediators. Aims and scope of paper: This study aims to analyze and compare the sources of commognitive conflict in iconic visual representations when students solve flat shape problems using conventional methods versus AI-assisted approaches. Methods: A qualitative design was employed involving 20 first-year mathematics education students divided into two groups (AI-assisted and conventional). Data were collected through students’ written work and semi-structured interviews …

Measuring What Matters: Specifications Grading And Latin* Students’ Mathematics Identity, Luis Miguel Fernández, Mayra Ortiz Galarza, Cristina Villalobos, Martha Asare

School of Mathematical & Statistical Sciences Faculty Publications

This study examined Specifications Grading, an alternative grading system emphasizing clearly defined learning outcomes and revision, and mathematics identity among 846 Latin* Calculus I students at a Hispanic-Serving Institution. Mathematics identity, comprising competence/performance, recognition, and interest, was measured at the beginning and end of the semester. Repeated-measures analyses indicated stable competence/performance and recognition alongside declines in interest. Specifications Grading was associated with increased mathematics identity, and multilingual students experienced smaller declines than their peers overall.

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Este estudio examinó la calificación por especificaciones, un sistema de evaluación alternativo que enfatiza resultados de aprendizaje claramente definidos y oportunidades estructuradas de revisión, …