SIMIODE EXPO 2024

Abstract: We present a plan to re-envision the Calculus sequence in a modern, engaging, and applications-driven way by making differential equations modeling an immediate and central part of the Calculus II experience, using differential equations models as the course's introduction and regularly referring back to them as a motivating application for the later material. Differential equations are exceptionally applied and provide the opportunity to discuss many real-world uses of calculus and its intersections with other STEM fields, they give the opportunity to show interesting phenomena unlike anything the students will have seen in Calculus I, and they can be studied and understood well before the students possess robust tools for evaluating antiderivatives. We will discuss our efforts to develop and implement this plan at Florida Polytechnic University in Fall 2023, the updates we implemented in Spring 2024, and how you might be able to update your own approach to teaching Calculus.

Abstract: For over four decades, the Consortium for Mathematics and Its Applications (COMAP) has supported teachers to improve mathematics education with an emphasis on student proficiency in mathematical modeling. Come learn about COMAP activity, and how you can get involved. www.comap.org.

Abstract: While numerous educators have employed standards-based grading (SBG) for years, its popularity has surged in the past decade. The adoption of SBG has emerged as a powerful tool to promote equitable learning environments and foster a growth mindset. This talk explores the implementation of SBG in a course on differential equations, shedding light on its potential benefits and challenges.

Firstly, SBG promotes a growth mindset by emphasizing the journey of learning over mere letter grades. While breaking down complex concepts into clear standards and learning objectives, students are encouraged to embrace mistakes as opportunities for growth. This shift in perspective promotes resilience and a willingness to explore challenging topics.

Equity is at the core of SBG. Traditional grading systems can perpetuate disparities, but SBG offers a more equitable alternative. It provides continuous feedback, allowing students to track their progress and take ownership of their learning. This approach mitigates bias and enables all students, regardless of their background, to succeed based on their understanding and effort.

However, the implementation of SBG does come with its challenges. Managing the extensive overhead labor involved in tracking and assessing numerous standards can be time-consuming for instructors. It requires careful planning, frequent updates to assessment materials, and robust communication to ensure clarity.

Abstract:Stability of equilibrium points of ODEs is characterized by so-called Lyapunov functions, that are a generalization of the physical concept of (dissipative) energy. The theory of Lyapunov functions is far-reaching, indeed the qualitative behaviour of a given dynamical system is completely determined by a complete Lyapunov function for the system. In this talk we describe an algorithm to compute a Lyapunov function for an ODE with a stable equilibrium point, that uses numerical integration of the ODE for numerous initial values to approximate a Lyapunov function for the system. This is followed by a subsequent verification of the conditions for a Lyapunov function for an interpolation of these values over a simplicial complex. A simple Matlab implementation and some examples are presented.

Abstract: I will demo a complete set of interactive course materials created using dynamic Jupyter notebooks using Google Colaboratory. Differential equations provide a rich universe to explore models and train students as mathematical experimentalists. Jupyter is a free, open-source platform that provides instructors and students an environment to weave together narrative text, executable code, visualizations, and videos all in one document. Course materials are delivered in Google Colaboratory, a free, cloud-based application where instructors and students interact with course materials. The flexibility to design materials for a variety of course formats and seamlessly integrate everything into one environment help foster an engaging and fun environment for both students and instructors. Jupyter notebooks help equip students with computational tools to gain further insight and intuition into differential equations. All course materials are Open Education Materials (OER) with a Creative Commons license and are freely available to be reused, revised, and redistributed.

Abstract: As more and more learning moves online, it is valuable to share experiences of instructors teaching online. Questions such as, “Is there really a difference teaching differential equations online vs in-person?”; “How was your online course structured?”; and “Which resources (e.g., software and instructional materials, etc.) have you found helpful in teaching differential equations online?” are just some of the questions that will be explored by this panel.

Abstract: In this talk we shall present three modeling scenarios that were adapted from SIMIODE and used in teaching the Differential Equations class in spring 2022 and 2023.  We shall talk about criteria for selecting the projects, their adaptation and the rubric used to grade the projects. We shall also share some of the students' comments from the end-of-semester Course Evaluations on the use of modeling projects in the course.

Abstract: The Richardson Arms Race Model describes the dynamics of arms expenditures of two nations, each of which is driven to increase its spending by the spending level of its rival, but faces internal resistance as its own expenditures grow and experiences external drivers of good or ill will. The classic formulation of the model is a linked system of two first order linear differential equations.

Phase plane analysis of the model early in a differential equations course reveals the possibilities of sinks, sources, and saddle points. Students can find exact solutions using eigenvalues and eigenvectors in the middle of the term. They can explore more realistic models by replacing the constant parameters with their own choices of continuous functions and apply numerical solution methods to simulate the behavior later in the semester. In an advanced DE course, delay differential equations can replace the ODEs as neither nation may be able to change its arms budget instantaneously in response to its opponent.

Abstract: As a part of NYU Tandon’s Sustainable Engineering Initiative we have implemented a set of modeling-oriented differential equations worksheets and projects to engage and empower STEM students to understand and solve problems related to climate change and sustainability. By applying the skills and concepts they have learned in their calculus, physics, engineering and differential equations courses, along with modeling software to issues of climate change, the goal is to

• increase student engagement,
• deepen student understanding of course material,
• prepare students to craft the climate solutions of tomorrow, and
• to create and support a community of practice among faculty involved in these courses.

In differential equations, we are developing worksheets and projects aligned with the above goals. These worksheets and projects are both original and based on materials developed by SIMODE and others. This talk will present an overview of our interdisciplinary initiative’s work, course materials and report on current progress.

Abstract: This talk is not just about introducing ChatGPT as a new tool, but about inspiring a shift in perspective—from viewing AI as a complex, distant technology to seeing it as a friendly assistant that's ready to help us make differential equations and broader coursework more accessible and exciting for our students. ChatGPT is revolutionizing the way we approach teaching so fast that it's hard to keep pace. You'll leave questioning whether it's the students learning from ChatGPT, or ChatGPT learning from the students. Either way, the future of education will never be the same again.

Abstract: Slopes is a mobile application with an intuitive interface that is designed to visualize solutions to differential equations and support active learning in the classroom. By making slopefields, phase planes and numerical solutions more accessible, students are able to engage in higher level discussions of mathematical models that incorporate differential equations. Slopes is currently available for iPads, iPhones, and Android phones, which are highly portable and feature larger touch screens that allow students to view and manipulate content easily. I will discuss in-class activities that emphasize a visual understanding of mathematical models in order to introduce and reinforce key concepts in differential equations. I will also discuss how students used Slopes as a primary tool for semester-long modeling projects. In a recent study, we found that students used Slopes to visualize solutions, aid in discussion and cooperation, build prototype models, and demonstrate understanding of differential equations concepts.

Abstract: Specifications grading is a relatively new paradigm of grading that addresses several flaws in the traditional weighted-average grading scheme that relies on partial credit. Specifications grading was made popular in Linda Nilson’s 2014 book, “Specifications Grading: Restoring Rigor, Motivating Students, and Saving Faculty Time.” In this talk, I will describe my implementation of specifications grading into college-level mathematics courses. I will also show other examples of using specifications grading from courses that involve projects and assignments that have students working together.

Abstract: Have you thought about offering undergraduate level courses for high school students in your state? Do you already teach college mathematics courses to high school students? The Distance Math Program (DMP) at Georgia Tech offers undergraduate level mathematics courses to over 900 advanced high school students across Georgia, many of whom have exhausted their options for math courses before their senior year of high school.

Four core mathematics courses are taught in the DMP: linear algebra, multivariable calculus, applied combinatorics, and differential equations. Students in this program take undergraduate classes, take in-person proctored exams at their high schools, and engage with college faculty without ever having to travel to campus.

Students who complete DMP courses gain Georgia Tech credit while still in high school and gain exposure to college culture, and many DMP alumni go on to enroll at Georgia Tech.

In this talk I will give a deep dive into how this program runs behind the scenes, current challenges, possible future directions, and some of the unexpected benefits we have seen from offering this program. I will focus especially on the online differential equations course that we teach for high school students.

An opportunity to dig deeper into uses of SLOPES with the creator, software developer, and (most important) teacher.

Consider, how best to model time/wages not utilized when employees have to wait in line to use the only one wheelchair-accessible stall; compare to having 6 instead of 8 restroom stalls where all are wheelchair-accessible and so there would never be a waiting line for just one stall.

More generally model stochastic processes using differential equations in addition to 1-027-StochasticProcesses-ModelingScenario (https://dx.doi.org/10.25334/BA33-M747)and 7-008-MachineReplacement-ModelingScenario (https://dx.doi.org/10.25334/N3RA-R937).

Abstract: A crucial question in fire engineering is to understand the ignition of solid fuels. What is a solid fuel? In the context of a building fire it is any material that can catch fire, i.e. everything in a room. A common way to investigate the burning of solid fuels, such as synthetic polymers, is radiative testing.

The burning of solids is a classic problem in applied mathematics, we can make the model as complicated as we like by building in increasing amounts of chemistry and physics. The simplest plausible model assumes that the material is chemically inert and ignites when the surface temperature reaches a critical value.

Assuming that the test sample is `thermally thin' the model consists of a first-order linear ODE. Solving this we find that ignites occurs if the applied heat-flux is sufficiently large, i.e. there is a critical heat-flux below which a sample does not ignite. For heat-fluxes greater than the critical value we can find the time-to-ignition as a function of the applied heat flux.

For thermally thick materials the equivalent model is a linear PDE. We make life easier by only considering the steady-state temperature distribution in the sample. We can no longer find the time-to-ignition, but we can find the critical heat-flux.

In this presentation various simple extensions of these models are discussed. These models are suitable for students who can solve a first-order linear ODE or the steady-state heat-conduction equation.

Abstract: Density dependent population models are of the form:
p′ = r(p)⁢p     or     p(n+1) = (1+r(p(n))⁢p(n),
where the per capita growth rate, r, is a function of the population size. Under the assumptions: 1) there is an intrinsic per capita growth rate, 2) there is a carrying capacity, 3) r is decreasing, the most common functions for r are linear, rational, or exponential.

The well-known logistic model, in which r is linearly decreasing, gives clear contrasts between the discrete and continuous: The continuous model can always be solved and has a stable positive equilibrium population size, while the discrete model a) cannot be solved for most r’s, b) exhibits period doubling to chaos, and c) solutions go to negative infinity if the slope of r or the initial population size are too large.

The discrete model is more realistic if we also assume, 4) r decreases to negative one. This leads to four of the classic discrete models. Solutions for these models cannot, in general, be found. Solutions for the rational r in the continuous case can theoretically be found, but in practice, cannot realistically be found nor do the solutions give useful information.

Analysis of the continuous models can be done easily using phase-line analysis, resulting in stable and unstable equilibrium population sizes while for the discrete case, we must also use r′ at equilibrium to determine stability. In fact, many discrete models, like the logistic model, exhibit period doubling to chaos.

Abstract: Mathematical modelling provides useful insights into real-world problems and is a valued activity across many disciplines, including biology, computer science, and engineering. Differential equations are a valuable tool used in modelling. Modelling provides a way for students to engage with differential equations within a contextual environment. Teaching mathematics in context has the potential of giving students something to anchor the mathematics to and hence act as cognitive roots (Blomhøj, 2019). In light of this, how can lecturers use mathematical modelling to give students a contextual learning experiences of differential equations?

A New Zealand study was carried out involving three case studies. Each case study compromised of a mathematical modelling course, lecturer participant, and student participants. Modelling examples and activities that involved differential equations were part of all course content. In this presentation I will present information on each course, the main modelling activities each course used, and examples of students’ use of differential equations for these activities. Detailed Schedules discovered into the common practices of the three lecturers in regard to using mathematical modelling to teach differential equations will also be shared.

Abstract: This talk explores the pedagogical strategy of employing traffic flow models to enrich the teaching and understanding of differential equations. It begins by establishing the significance of modelling real-world phenomena, specifically traffic flow, to demonstrate the practical applications of differential equations. The presentation will outline various traffic flow models, such as the kinematic wave model, and discuss the mathematical concepts they incorporate, including relationships between flow, density, and velocity. It aims to show how integrating these models into the curriculum can make abstract mathematical concepts more tangible and relevant, thereby enhancing students' conceptual understanding and problem-solving skills.

The talk will delve into the specific pedagogical approaches used, including interactive simulations and case studies, to integrate traffic flow modelling into differential equations coursework. It will present the outcomes of this approach. The conclusion will summarize the effectiveness of using real-world applications in teaching complex mathematical theories and suggest future directions for curriculum development.

Abstract: A standard element of the undergraduate ordinary differential equations course is the topic of separable equations. We present here a series of novel modeling scenarios that prove to be a compelling motivation for the utility of differential equations. Furthermore, the growing complexity of the models leads to the development of various types of separable equations. In this way all the topics you normally want to cover in a unit on separable equations can be addressed in context through a single coherent modeling scenario.

Abstract: I am trying to construct a new version of our introductory ODE course at Tufts University. Some of the principles I am trying to adhere to:

1. Almost never write down any ODE that doesn't mean anything in any application that I can explain in class. Applications should be primary.
2. All applications should be simple but honest. The modeling should lead to, or suggest at least, non-obvious conclusions.
3. Applications should come from a broad range of fields, including both natural sciences and engineering and, even in a course for engineering students, social sciences.
4. The students should be enabled to graph solutions themselves, without computing formulas first.

I'll give examples, with special emphasis on differential equations models in the social sciences that I have recently written articles about. I'll also discuss difficulties that I am struggling with, and hope for helpful comments from the listeners.

Abstract: In the fall term of 2023, we participated in the annual differential equations modeling challenge SCUDEM VIII hosted by SIMIODE in which we received the highest distinction—the Outstanding Award.

We undertook the challenge of addressing Problem B—Punishing Babies, which involved a study focused on infants (6–19) months old and their tendencies to ‘punish’ or exhibit aggressive behaviors towards other infants who were thought to be acting out negatively towards other infants.

Our mission was to then develop a model that could effectively model different populations with varying propensities to act out against those perceived as engaging in aggressive or similar behaviors towards others. Additionally, we explored beyond this initial problem and looked at how separate real-world examples might follow this trend. In our talk, we would be delighted to introduce why we chose our problem, present our modeling/iteration approach and results, discuss the specific general and technical challenges we encountered, and lastly reflect on how the SCUDEM experience has contributed to our development for future real-world applications of the mathematical concepts we learn in the classroom.

Abstract: Our group participated in the SCUDEM VIII Challenge hosted by SIMIODE in the Fall of 2023. The prompt we chose, prompt 3, led us to generate a model of a dog named Fritz and various food objects being thrown at him to determine if Fritz was bad at catching or if the owner was purposefully throwing food at Fritz unfairly. We thought about the physics behind the problem, developing a model using python. To give the owner further advice, we utilized a learning curve to draw a conclusion. We accounted for Fritz's position being fixed in place, his reaction time, and his ability to learn to catch. We concluded that Fritz simply struggles with catching and that the owner does not purposefully throw the treats incorrectly. With time, we believe that Fritz can improve his skills. We intend to build on the judge's feedback, explain what we would have done differently if we did the problem again, and how we can update our current model to improve our understanding of the problem by examining the errors we made.

Abstract: Our group participated in the SCUDEM VIII hosted by SIMIODE in the Fall of 2023. We selected the prompt, “Punishing Infants,” which based its premise on the research done with the punishing behaviors of infants. Our model and subsequent solution revolved around the Lotka-Volterra model, which we modified to account for shifting societal trends and an additional punisher group. We concluded that societies with a greater punisher population and greater severity of punishment contained fewer aggressors over time. Refer to the link below to view our entire presentation.

Abstract: In this presentation, we discuss our insights from SCUDEM 2023 Problem C: Dog Cannot Catch, where we employed a systematic methodology for mathematical modeling. We modeled the mouth as a damped-driven harmonic oscillator and compared the predicted behavior to data taken by an AI tracker. Then, we used the linear prediction assumption to calculate when and where the dog will open its mouth to catch the food. Finally, we used the dog’s behavior and the actual trajectory to derive an inequality that must be satisfied for a catch to be successful. To conclude, we acknowledge some shortcomings of our model and discuss possible extensions and implications.

Abstract: Problem C: Dog Cannot Catch, from SCUDEM VIII 2023 was the problem our team investigated. Here, we attempt to answer the question of why Fritz, a dog from the popular YouTube video "Fritz Learns to Catch," cannot catch food that is thrown at him. This presentation will discuss the background work and techniques that ultimately led to our final presentation. Background discussion will include the various approaches our team considered such as models taking root in statistics and multivariable calculus. Then, we will discuss our chosen model, which was divided into two distinct components: the kinematics of the flight of the food object and the decision-making tendencies of Fritz. The model for the flight of the food object was bounded by differential equations considering air resistance. To model the dog's decision-making, our team utilized the NEAT (NeuroEvolution of Augmenting Topologies) method to train an AI model. Finally, the presentation will discuss our results and future improvements our team would take the provide a more accurate model of Fritz.

Abstract: In the Fall of 2023, we participated in SCUDEM VIII, out of an interest to apply our knowledge in differential equations to a real-life situation. The real-life situation we were given was a dog named Fritz that was struggling to catch treats thrown by its owner. We were asked to determine whether the dog was bad at catching or if the owner was bullying the dog.

In this talk, we would like to introduce this challenge, present our models, our results, the feedback we received from the judges, and share how and why SCUDEM was such a beneficial experience. Our work on this problem received an outstanding award and can be seen on the SIMIODE YouTube channel.

Abstract: Throughout the fall of 2023, we participated in the SIMIODE Challenge Using Differential Equations Modeling VIII (SCUDEM VIII), in which we received the highest distinction (Outstanding Award). The prompt we explored, “Punishing Babies,” is rooted in a recent study that models the intrinsic inclination of infants to punish aggressive behavior observed in others. We created a mathematical model to depict fluctuations in subgroup populations of aggressive and nonaggressive infants, which is also applicable to modeling their impacts on society at large. We used Lotka-Volterra population modeling equations as a base to create our model with subgroups of varying aggressiveness levels. In this presentation, we explain the development of our project, the implications of our model and its extended applications, and the tremendous growth we underwent in our mathematical modeling journeys as a result of this competition.

Abstract: In this work, the modeling, simulation and long-term dynamics of a real-world problem that addresses the intriguing phenomenon of infants displaying a tendency to punish anti-social behaviors and questions its implications in broader societal interactions.

This was one of the three problems of the SCUDEM VIII 2023 titled, “Problem B: Punishing Infants”. The solution methodology employed an innovative approach of adapting the SIR epidemiological model for infectious disease dynamics to introduce two models to study Problem B: one simple and another complex. The simpler model explores how likely the population is to pursue diplomatic action, like importing sanctions or sending humanitarian aid, or retributive (or unretractable) action, actively getting involved in the conflict. The complex model, however, splits diplomatic action into direct action, such as sanctions or public condemnation, and indirect action, such as diplomacy and supply donations.

A rigorous stability analysis of the model provides insights into long-term behavior. With hyperparameter tuning, using data from the Global Sanctions Database, refines the model’s accuracy. To validate the model’s applicability, real-world case studies are used, the Israel-Hamas and Russia-Ukraine conflicts, in which the United States has pursued different courses of action. The findings offer an understanding of the diverse propensities and corrective actions, with implications for societal change and attitudes towards punishment.

Abstract: This presentation utilizes a mathematical model to graph a dog’s catchability difficulty when receiving a food object. Our study focuses on some key factors, namely initial velocity, final velocity, and time in flight to calculate the difficulty the dog experiences based on the collective impact on a dog’s success in catching these thrown objects. Our presentation assumes that all dogs have an intrinsic ability to catch thrown objects due to their physiology and evolved instincts. To quantify this skill, we have developed a graphical model that captures the behavior of an object when tossed based on the velocity and time in flight.

We examine scenarios where a dog may struggle to catch the food item based on a high final velocity or a short flight time, resulting in less time to react. More specifically, the model uses the initial velocity, representing the speed of the item at which the food projectile is thrown; the final velocity, representing the speed of the item just before it reaches the dog; and the time in flight. The presentation calculates an equation that relates initial velocity to final velocity, as well as an equation that relates vertical displacement to initial velocity, given conditions of initial angle thrown and horizontal displacement. In conclusion, our presentation offers valuable insights into the intricate relationship between initial velocity, final velocity, vertical displacement, flight time, and the difficulty for a dog to catch a projectile.

Abstract: The equation of motion of a system representing the same input-output motion of a physical model can be solved using classical ordinary differential equation solutions. We participated in the SCUDEM VIII 2023 challenge and modeled the motion behavior of Fritz trying to catch the object being thrown at him. We derived the equations using Newton's Laws of Motion in two dimensions and simulated the equations including air drag and dimensions of a golden retriever in MATLAB Simulink and MATLAB live editor. Additionally, we visualized the motion by integrating both the CAD model of the dog and equations in MATLAB Simscape.

Abstract: Today, the creation, simulation, and fitting of mathematical model equations, particularly those with Ordinary Differential Equations are done with different programming languages, each with their own syntax. This can be a stumbling block to young students who are just trying to learn mathematical modeling concepts but lack the programming knowledge, or have bugs in the code that prevent them from generating the correct simulated output. Furthermore, models that have discrete inputs at specific times must be explicitly programmed to start and stop at each of these time steps as ODE integration routines such as Runge-Kutta have an adaptive time step. Fitting the model parameters to experimental data adds an additional level of complexity with an iterative, non-linear least squares fitting routine programmed in a separate script outside the model definition.

Abstract: Recent events and technological advancements have led to many innovations on how students are instructed and assessed in mathematics courses. Teaching a traditional differential equation course during the COVID-19 pandemic forced me to think of innovative and interesting ways of assessing the students while making sure I covered all the required topics. Inspired by SIMIODE materials, I helped the students design public education projects. The goal of this project was to use a SIMIODE article to introduce the general public to an application of differential equations. An important punch line of the project description is: ‘Your target audience is a high school student. Your job is to teach this student without intimidating him.’ The projects were disseminated in long form via YouTube, and in short form via instagram and tiktok.

In this talk I will discuss my takeaways including (1) feedback from students; (2) areas that need improvement; (3) scaling beyond differential equations.

Take this opportunity to discuss and experience features and possibilities for using WikiModel in your classroom with your students.

We need more collaboration between the STEM disciplines and the arts and the humanities. We offer an opportunity to discuss about collaborations or ideas for collaborations between SIMIODE EXPO participants and their colleagues in the arts and the humanities. Although collaborations involving DEs are certainly of particular interest, we should look more broadly at collaborations at the undergraduate level.

Last year at SIMIODE EXPO 2023 we had a Keynote speaker Lorelei Koss, Mathematics, Dickinson College, Carlisle PA USA give a talk Connecting Differential Equations with the Arts, Music, and Literature (https://www.youtube.com/watch?v=tamyBIjFOlc) and there was interest, but nothing happened formally.

Further there are supporting resources, e.g., the journal Journal of Mathematics and the Arts.

Abstract: ChatGPT is the most recent and currently most in the news flavor of artificial intelligence. In this talk we will distinguish GPT (generative pre-trained transformer) and LLM (large language models) AI from algorithmic AI and from other forms of AI based on extremely large data sets. This talk focuses equally on two things: How we as modelers, mathematicians and mathematics educators at the ODE level can understand how GPT works and how we can use GPT to do and to teach mathematics and modeling.

The most visible aspect of ChatGPT is an illusion — talking with ChatCPT feels like talking with a flesh-and-blood human being. The extent to which this is an illusion depends as much on the nature of flesh-and-blood intelligence as on the nature of GPT intelligence. We are learning more about both. The work of Daniel Kahneman and his book Thinking, Fast and Slow is particularly helpful.

Like all scientific advances, GPT builds on other work. One important example is "word vectors." The Ars Technica article "A jargon-free explanation of how AI large language models work" is particularly helpful. Further, the recent article Forbes article "What is the best way to control today's AI?" is a wonderful article. As modelers, we know that how we represent the features of the real world in our models is crucial and representing words as vectors is particularly brilliant. We are all here today because of our passion for modeling and as educators at the ODE level. But, ODEs are only one modeling paradigm and this talk will argue that for most students the traditional ODE class should be replaced by linear algebra and more broadly inclusive modeling.

The purpose of modeling is building understanding of the world in which we live and how we can change our world for the better. GPT builds in part on extremely large scale data analysis and like its predecessors and in stark contrast to algorithmic AI provides little or no understanding. Cathy O'Neil's popular book Weapons of Math Destruction is particularly helpful.

In short, this talk has three goals: Increasing our understanding of GPT, helping us use GPT effectively, and changing our mathematics courses at the undergraduate ODE level — both for math majors and for other majors. Words are just one form of modeling and writing-across-the-curriculum needs a new partner modeling-across-the-curriculum.

This is entirely optional but if you happen to have a pair of red-cyan 3D glasses please have them handy. These glasses are often packaged with 3D books.

By the time students take differential equations, many have been beaten down by three calculus courses that taught them how to do mindless manipulations — integrate sec³(x) or ¹⁄_(x²(x²+4)) for no apparent reason, but no need to understand how the total amount of water discharged from a river into an ocean could be seen as an integral; differentiate arcsin(sqrt(xe^x)), but no need to understand Euler's method for a simple ODE, or Newton's method for a nonlinear equation.

We must not only teach them something about differential equations, but also to change their understanding of what "mathematics" means. Should we compromise, and do some sets of separation of variables "drill exercises" for instance (to make the students less uncomfortable — those problems may not have intrinsic value)? Or should we ask students to live with the discomfort of NEVER getting the sort of problems that they expect based on their previous experience?

A discussion of adoption and uses of the SIMIODE text, Differential Equations: A Toolbox for Modeling the World with author Kurt Bryan.

Abstract: Ben Jeffers Executive Director of Prison Math Project will lead a presentation about the meaningfulness to individuals in prison to support mathematics inquiry and the offerings and opportunities in PMP for audience members to engage in mentoring and sharing mathematics with those imprisoned.

Abstract: First, the only goal of this All-in-One Taylor integrator [1] for challenging ODEs was scientific research and numeric experimentation [2, 3].

Later, I began collecting illustrious problems in applied ODEs—now a vast list [4] of simulations (without textbooks indeed).

Yet the very method of the real-time graphics suggested to utilize it as a generator of simulations or a virtual lab (Exploratorium [5]). Now it consists of only a few topics in my own expertise: didactic texts with directives what to watch in the respective simulations. When readers encounter a name of a simulation, there is a mechanism for running it by selecting its name.

In fact, any textbook for applied ODEs may be enriched with such simulations. Consequently, instead of still images, the textbook will provide a live real-time animation. And this is the main message of my talk today, and of the Workshop [6].

1. The software. http://TaylorCenter.org/Gofen/TaylorMethod.htm

2. R. Montgomery. Dropping bodies. The Mathematical Intelligencer. 2023. https://link.springer.com/content/pdf/10.1007/s00283-022-10252-4.pdf

3. M. Frenkel et al. The Continuous Measure of Symmetry as a Dynamic Variable. Symmetry. 2023, 15, 2153. https://doi.org/10.3390/sym15122153

4. Simulations (though without textbooks), http://TaylorCenter.org/Gofen/Teaching/Samples.htm

5. Exploratorium, i.e. textbooks with the respective simulations, http://TaylorCenter.org/Exploratorium

6. The Workshop, http://TaylorCenter.org/Workshops.htm

Abstract: You have just written a great textbook that you are certain become very popular.  What now?  You can always work with a commercial publisher, or you can choose to publish the textbook yourself.  If you self-publish, you have decisons to make.   Do you plan to have both an online version and a print version?  If you have an online version, you will need a website.   If you offer a print version, you will need to worry about ISBNs and inventories.  If you offer both a web version and a print version, what is the best way to keep both versions in sync?  Will your book be accessible to users who might need a screen reader, a braille version, etc.  Do you make your book open-source or do you intend to earn royalties on your work?  What are the options for open source?   If you plan to charge, how do you put your online textbook behind a paywall?  We will discuss the choices and options.

Abstract: Tim Pennings, mathematics professor and PMP mentor and Jesse Waite, who was released last summer and is resuming and building his life now, both mathematically and for real, published an article on teaching in prison in MAA FOCUS, and is working with Tim Pennings who mentored him in PMP and Brian Winkel on a book about using SIMIODE materials.

Abstract: Incorporating modeling activities into a differential equations course offers numerous benefits, including enhancing student understanding, increasing engagement, and improving both subject knowledge and critical thinking skills. In this talk, I will discuss how I implemented modeling activities and projects in my class and provide insights into my rationale for choosing these activities and my course preparation. Furthermore, I will share reflections on success stories, lessons learned, and potential improvements/variations. Student performance and their enjoyment in investigating the intersection between theoretical knowledge and real-world applications in the realm of differential equations will be explored through a reflection on student feedback.

Abstract: In its discussion of mathematical modeling, the Mathematical Association of America delineates between strong and weak modeling. MAA defines strong models as those which have poorly defined problems and many possible approaches. We discuss the merits of introducing a process to: guide a student’s modeling, help students organize their thoughts, and know what to do next. We also explore different ideas about what this framework should look like and how to integrate this modeling process into an ODE course. We close with some thoughts on the importance and value of teaching modeling to students.

Abstract: For instructors and others who choose the content and format of an introductory Ordinary Differential Equations class, there are many interesting and consequential choices to be made. For example: (1) To what extent should the course focus on modelling? (2) To what extent solving those types of ODE's which are simplest and easiest to solve via pencil and paper be taught? Of course, the answers depend largely on the students taking the class and their educational goals. We will discuss pros and cons of various choices.

Abstract: Most projectile motion and free fall models are based on the assumption that gravity is the only force acting on the object. In this talk, I discuss a SIMIODE modeling scenario centering around the construction of a free fall model that accounts for the force of air drag. In the module, students develop, solve, and analyze a second order nonhomogeneous differential equation model for free fall which incorporates air resistance. Students solve the model using two different methods – reduction of order and separation of variables, and method of undetermined coefficients. Using the solution, students derive an expression for the terminal velocity of the object as well as a prediction of the maximum height of the object (assuming it is fired directly upwards). The model is then parameterized for a Nerf dart by an experiment performed by students. The terminal velocity and muzzle velocity of the dart are measured using video frames of the dart’s motion. Finally, the model is validated by an experiment wherein students fire their darts upward and measure the assent time for comparisons with their predictions.

Abstract: Neurons communicate through electrical and chemical signals. Intracellularly, the propagation of electrical signals, facilitated by charged particles, enables rapid conduction from one end of the neuron to the other. Communication between neurons occurs at specialized structures known as synapses. Within the synapse, two neurons are involved, with one transmitting information to the other. In the process of neuronal communication, chemical signals known as neurotransmitters are released by one neuron traverse the synapse and bind to specific molecules on the receiving neuron known as receptors. These receptors receive and process the message before forwarding it to the next neuron. These signals journey to another neuron, initiating a new electrical wave within that cell. Biological neuron models, often referred to as spiking neuron models, entail mathematical representations of neurons. These models specifically elucidate the temporal dynamics of the voltage potential across the cell membrane. Mathematical models are crafted to articulate fundamental behavioural processes with a level of precision. Taking advantage of the mathematical modelling, we present a simple neuronal model which helps to better extract all the complicated dynamics of this single neuron model. Neurons serve as the fundamental units of the nervous system. Through the mathematical modelling of neuronal behaviour, we gain valuable insights into the intricate dynamics of their interactions.

Abstract: Imagine a paradigm shift in higher education: where instead of tossing mathematically apt and advanced students into Calculus II or III as their first college class, suppose that we put them into Differential Equations instead. Furthermore, what if we made that Differential Equations class a modeling class as well, where students would have to simultaneously learn how to model as they learned various techniques including variation of parameters, integrating factor, Euler’s Method and more? What if I told you that we are doing just that? The framework we’ve developed for teaching our advanced population of students involves a parallel learning schematic of mathematical modeling and differential equations. As we scaffold the techniques of solving first-order, second-order, and systems of ODEs; we also introduce modeling at its basic levels of transforming the variables, through the solve step and ultimately interpreting meaningful solutions. We do this as a three-part process: by engaging in in-class lectures and associated practice problems; with smaller, engaging modeling application days where students focus on parameters in ODEs; and finally with a large scale project and presentation at the end of the course on a differential equation modeling problem of their choice. During this talk, we will discuss how we facilitate this innovative educational journey, where students don’t just learn mathematics; they become mathematical explorers.

Abstract: Climate change is the most pressing problem of our time, and in order to address the crisis we need `all hands on deck’, including mathematics students. Standard second year courses in ODEs generally contain little to no real climate change content, but, as it is a first course in modelling, we decided that meaningful climate-ODEs content could be created. We created a series of group work projects that related climate change models the course content in a standard second-year Introduction to ODEs course. In this talk, we present the learning goals for the new content, and outline the group work projects we developed. In addition, we discuss challenges encountered in introducing the content, and student responses to the group work projects. We close with some thoughts for future implementation of climate change content in second year ODEs.

This is an opportunity for further discussions on the constructive understanding and uses of AI in coursework and beyond into society.

We continue conversations about how to effectively use this modeling first and throughout textbook, highlighting its many features.

Abstract: Differential equations successfully model numerous theoretical and real world problems, with examples often drawn from physics and, especially in the covid era, mathematical biology. We discuss a beautiful problem in a very different area that can be done easily in a first calculus class: modeling the famous Battle of Trafalgar, where Nelson's strategy led to one of, if not the, most decisive and lopsided naval victories ever. This example illustrates the power of creating good mathematical models that capture enough of the key features, and having techniques to solve the resulting differential equations.

Abstract: The dynamic of cancer, including its response to therapy, is a topic that students majoring in fields in the Biological Sciences are generally interested in. Students with a mathematical or computational academic background enjoy mathematical concepts being introduced using a practical approach. These interests made teaching an interdisciplinary mathematical modeling of cancer course possible. We introduce a technique for building a system of ordinary differential equations modeled up from data, diagrams, and assumptions. Our model is about tumor evolution under therapy, and we illustrate how this system of ordinary differential equations is developed starting with data representing malignant tumor growth. Initial methodologies employed to estimate parameter values consist of dynamic parameter estimation, and model fitting techniques including different versions of sum of squares minimization using Solver. These strategies are then compared with machine learning’s outcomes. Initial tumor growth models are then modified according to diagrams and assumptions to reflect the inclusion of therapy for tumor reduction. Along the way a variety of methods, R software codes, and free resources are mentioned, and some are employed to illustrate outcomes.

Abstract: This work introduces an innovative example of teaching ordinary differential equations (ODEs) by harnessing the captivating world of fictional dragons. Drawing inspiration from popular culture, specifically Game of Thrones and A Song of Ice and Fire, we present a structured methodology for employing ODEs to model the growth, ecology, and energy dynamics of mythical dragons. We will demonstrate how to utilize these mythical creatures as illustrative examples within ODE instruction, facilitating students’ comprehension of ODEs. Our aim is to provide educators with a novel teaching tool that sparks student engagement and deepens their understanding of ODEs while enhancing critical thinking and problem-solving skills. We hope this demonstration inspires educators to explore new modeling topics in ODE classes, fostering students’ interest and encouraging their active participation.

Abstract: The effective power of an image as a teaching tool is widely known. Furthermore, when a graph is also used to describe a physical phenomenon along with its mathematical underpinning the explanation always seems better and clearer. Searching for mathematical formulas which graphs match or closely resemble a particular physical behavior is a desirable pedagogical tool to convey the essence of technical work and make it more understandable. In this respect, mathematical formulas and their graphs serve not only as a channel to teach complex phenomena but as a shortcut to avoid talking separately about diverse cases of similar nature. In this paper the authors consider the solution of a free-underdamped differential equation and its computer-generated graph to study the directivity of a loudspeaker.

Abstract: I will discuss a simplified model describing a dynamics of microplastics with algal biofouling in the ocean. The model teaches students how to construct population dynamics equations and equations of motion. The model can also be used for the study of parameters bifurcation.

Abstract: Mathematicians, with some exceptions, have not had opportunities to describe what it means to practice and use mathematics ethically. This talk describes the integration of ethics into the Biocalculus classroom and ethical issues surrounding the teaching of mathematics at Jarvis Christian University, a small liberal arts HBCU, Hawkins, East Texas.

This paper addresses a Biocalculus course for students in biology, chemistry, and mathematics majors as a redesigned course in the mathematical sciences. The course addresses ethics in mathematics by integrating ethical conversations into the course. The paper discusses how ethics in Biocalculus can be addressed, including reflection or assessment on success, describes challenges and barriers to implementing the integration of ethical conversations in Biocalculus, and shares strategies to address those barriers.

Abstract: The relationship between conspecific density and the probability of emigrating from a patch can play an essential role in determining the population-dynamic consequences of an Allee effect.  In this paper, we model a population that inside a patch is diffusing and growing according to a weak Allee effect per-capita growth rate, but the emigration probability is dependent on conspecific density.  The habitat patch is one-dimensional and is surrounded by a tunnelable hostile matrix.  We consider five different forms of density dependent emigration (DDE) that have been noted in previous empirical studies.  Our models predict that at the patch-level, DDE forms that have a positive slope will counteract Allee effects, whereas, DDE forms with a negative slope will enhance them.  Also, DDE can have profound effects on the dynamics of a population, including producing very complicated population dynamics with multiple steady states whose density profile can be either symmetric or asymmetric about the center of the patch.  Our results are obtained mathematically through the method of sub-super solutions, time map analysis, and numerical computations using Wolfram Mathematica.

Abstract: The Laplace transform plays an important role in the teaching of differential equations because it is one of the few methods capable of dealing with discontinuous forcing functions. A century ago, John Renshaw Carson of AT&T, in placing Heaviside's operational calculus on a sound footing, introduced an integral that looks the same as the modern Laplace transform except that it includes p-multiplication (where Carson, like Heaviside, used p for the differential operator). P-multiplication was used: (i) for consistency with Heaviside's operational forms (no longer relevant today), (ii) because it gives 1 for the operational form of the step function (instead of 1 for the impulse function as in the Laplace transform); and because (iii) p-multiplication can lead to algebraic simplification in some applications.

As p-multiplied Laplace transforms have recently seen greater use (now called the Laplace-Carson transform), this talk explores the advantages and disadvantages of the p-multiplied form of the Laplace transform. Two areas of applications are in the transform representation of discontinuous functions (e.g., those used in wavelet analysis) and in the solution of certain integral equations.  Conversion of differential equations into integral equations is also discussed.

Abstract: Our department uses computer algebra systems, either Maple or Mathematica, in the undergraduate introductory and upper-division physics courses. I will concentrate on how I use Maple.

Key properties of Maple for teaching physics and mathematics:

• Compared to Python or MATLAB, it has a lower learning barrier because:
  • much of its “coding” is via clickable icons,
  • Maple’s input and output are readable to the non-user.
• Maple derives symbolic results, which are more informative than numerical results. * Maple performs the busy work of both derivations and calculations.
• Maple allows for a top-down problem-solving approach, the problem-solving process of physicists.
• Maple can solve real-world problems well beyond traditional spherical-cow-type problems.

With these advantages over by-hand methods, we have included sets of linked first-order ODEs in the first-semester physics courses and, in an applied math course, cover ordinary differential equations for upper-division physics and engineering courses in four weeks.

Students are immersed in Maple:

• Skillsets are learned through ten-minute videos.
• Presentation of content is in Maple.
• All student submissions are Maple
• worksheets/documents.

Surveys show that nearly all the students feel Maple increases their interest in physics. They say they concentrate more on understanding physics principles and less on mathematical busy-work. This produces a more inclusive learning environment.

Abstract: 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.

Abstract: In this talk, I will present a very elegant method for solving constant coefficient second order ODE such y” - 5y’ + 6y = 0 by factoring out operators instead of using auxiliary equations. The advantage of this technique that it is much more direct and does not require any guesswork whatsoever. Moreover, it generalizes easily to inhomogeneous ODE and can even be used to solve some PDEs such as the wave equation.

Abstract: Active learning is the teaching methodology where lessons encourage students to actively participate, rather than just passively listen. In this highly interactive session, some cognitive science and education theory is presented, followed by a catalogue of examples of using strategies to make math (or quantitative science) classes more engaging. Strategies covered include motivating math through modelling, scaffolding, gamification, and other explicit tools to use in a classroom. By the end of this workshop, the audience may create their own mini-math lesson.

Abstract: Formulating a versatile population growth model affords an opportunity to survey some of the important concepts that are presented during a first course in ordinary differential equations. Initially, the classical Malthusian law allows for a discussion of first order linear differential equations including the notion of an integrating factor. While the full model is nonlinear with respect to the dependent variable, it can be solved explicitly, yielding an implicit representation of the solution using the separation of variables method.

The implicit representation of the solution suggests that an analysis of the long-term behavior of all solutions might be challenging. Fortunately, the fact that there are three equilibrium solutions offers a gateway to the geometrical theory of differential equations. In this context, the phase line, the stability of the equilibria, and the long-term behavior of the entire family of solutions may be explored.

Students are often spellbound by the geometric analysis and the reality that so much information can be obtained just from the differential equation. For many students, this experience offers a compelling answer to their query concerning the utility of calculus itself.

Abstract: As a fundamental course for STEM careers, Calculus I is an ideal place to introduce students to the ethical practice of mathematics. However, it can feel impossible to add extra material to an already-packed syllabus. This summer, as part of the Framing Mathematics as a Foundation for Ethical STEM NSF grant, we developed a series of problems which integrate ethical reasoning and Calculus I topics. We then piloted the materials in our respective Calculus I classes in the fall. In this talk, we will share a sample of the problems as well as our challenges and successes in implementing them.

Abstract: Insightmaker is a free online tool for system dynamics modeling. In this talk we will explore three ways in which using Insightmaker enhances understanding in an introductory ODE course by providing a systematic "ground up" way of constructing ODE models, improving ODE literacy (knowing what an equation says when you have one), and using built-in functionality to better understand concepts that students often struggle with. Illustrations of use will include several SIMIODE textbook scenarios. Resource illustration.

Abstract: In an institution without math majors, Differential Equations is offered as one elective for a few other majors. In order for the students to have more control over their learning and enjoy a student-centered while faculty-guided learning environment, I flipped the classroom with a constructive design, which turned out to be a big success. In this presentation, I will talk about how I flipped the classroom and how it positively affected students’ learning. I will also discuss my thoughts on how to improve it and some cons of flipped classroom based on my experience.

Abstract: The CODEE Journal is a peer-reviewed, open-access publication, distributed by the CODEE (Community of Ordinary Differential Equations Educators) and published by the Claremont Colleges Library, for original materials that promote the teaching and learning of ordinary differential equations. At the 2023 AMS Joint Mathematics Meetings, we announced a Call-for-Papers for a CODEE Journal special issue on the theme "Engaging the World: Differential Equations can Influence Public Policies." Exactly one year after the call, we officially launched and released this Special Issue at the 2024 AMS Joint Math Meetings. The thirteen articles within this issue offer a diverse array of perspectives and contributions, illustrating that mathematics can contribute to making informed public policies via community collaboration and dialogue, guided classroom explorations, and/or research investigations.

Abstract: Ethical reasoning is an essential component of applying mathematical modeling in solving real-life problems, both in research and business settings. Our mathematical answers exist in a context of a larger system and have implications on the lives of others, the planet, and future generations. However, mathematics instruction often treats the mathematical work as if it exists in a vacuum devoid of context and omits careful consideration of stakeholders, validity of data, assumptions made, and limitations of the analysis. In this talk, we present a general framework that can be used to modify any mathematical modeling problem or project in a way to help students focus on the missing ethical reasoning perspective in the problem/project. Our framework is generalized in the sense that it can be applied to any course at any level, including the K-12 instruction. The framework provides flexibility to instructors in terms of the types and level of questions that can be asked, as well as the quantity. In addition to describing the general template, we will demonstrate the use of the framework on a specific example to clarify the application of the framework.

Abstract: Dive into the world of ballistic precision as we develop, analyze, and compare three different models for the flight of a sponge dart moving under the influence of gravity and air resistance. The first two models are based on the common simplifying assumptions of no air resistance and linear air resistance. In the third model we explore how to find the correct physical form of air resistance then create and explore the model. Data is provided both for parameter estimation and for model verification, and the verification data is used to compare results of the models. While many calculations for the first two models can be carried out by hand, technology is essential for the analysis of the third model with physical air resistance.

Abstract: Traditionally, the initial value problem and the boundary value problem are the focus of a course in ordinary differential equations (ODE). However, it is important to present a broader view of the ODEs including algebraic equations or finding optimal solutions. This presentation provides an example of such an approach. It demonstrates the construction of a trajectory of structural change in the national economy providing both locally and globally optimal economic growth, mitigation of greenhouse gas emissions, and decrease in energy consumption. The three variable weight coefficients reflect the relative importance of the corresponding indicators. Three projected gradients guide local optimization, each improving the economic structure in one direction. The local optimal direction of structural change forms the smallest possible acute angle with each projected gradient.

Global optimization aims to reach the final optimal state of the economic system. It determines the appropriate values for the weight coefficients and the speed of restructuring. Mathematically, the problem leads to a solution of a differential-algebraic system of equations with non-negativity constraints. Empirical data for a range of economies are available from the World Input-Output Database (www.wiod.org). Alexander Vaninsky. 2023. Roadmapping green economic restructuring: A Ricardian gradient approach. Energy Economics. 125: Sept 2023. Article 106888, provides resources for student research projects. Artificial intelligence may help find suitable clusters.

Abstract: Somewhat nervously, last semester I included my first ever project assignment in my differential equations course. I will share my thoughts on what went right, what went not so right, and how I can adjust for the better this semester.