Make Eigenvalues Resonate : SIMIODE Expo 2022

This is a companion blog post for my talk Make Eigenvalues Resonate that I gave this talk on Friday 2/11/2022 from 1:15pm – 2:15pm EST as part of the SIMIODE Expo 2022 Virtual Event.

SIMIODE is “a community of practice focused on a modeling first method of teaching differential equations.” The acronym SIMIODE stands for “A Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations.”

In this blog post, I share all resources I generated for this talk as well as other resources that might be helpful for participants who want to return to these ideas after the talk ends. Enjoy.


We present a learning activity that enables students to apply eigenvalue theory to model a useful physical phenomenon. Specifically, we demonstrate how to build a spring-coupled pair of pendula and how students can analyze this system using eigenvalues. This activity can be designed to enhance student motivation and prepares students to apply differential equations and linear algebra while building authentic experiences with mathematical modeling inside the classroom.

YouTube Recording of this Presentation

Below is a YouTube recording of this talk:

YouTube video recording of Make Eigenvalues Resonate with Our Students talk at SIMIODE Expo 2022

Presentation Document

Below is are two documents useful in this presentation. The first is the Google Document that all participants co-generated together during this presentation. The second is the powerpoint presentation that I used to guide my remarks about this eigenvalue modeling activity.

Resources for Future Learning

How can I access the Make the eigenvalue problem resonate paper?

This talk is a companion to a paper I wrote with Dr. Michael McCusker. This paper, entitled Make the eigenvalue problem resonate with our students, was published in Taylor & Francis’s journal PRIMUS : Problems, Resources, and Issues in Mathematics Undergraduate Studies. Below are links you can use to learn more about this paper:

What other resources do you have available?

Lots. As part of this project, we host a support website:

This support website includes links to online videos that introduce relevant curriculum, a laboratory project prompt that can be assigned to students, sample experiment videos that can be analyzed using image processing software, example spreadsheets that provide analysis for the modeling process, and many other resources to support you in applying eigenvalue theory to an authentic, real-world problem.

How can I build my own McCusker apparatus?

Great question. Below are a few resources that might be helpful in constructing your own McCusker apparatus. We believe this is best done in a team setting. This is a perfect project for a group of motivated students who have an interest in applied learning.

How can I use tracker software?

This project relies on an open-source physics software called Tracker. Below are a number of resources you can use to download Tracker and learn how to use this software.

Do you have any YouTube videos I can use with students?

Yes! This project provides good interdisciplinary modeling activity that can easily fit into introductory courses in differential equations, linear algebra, or applied mathematical modeling. Instructors with an interest in active learning might use these resources in project-based classrooms, inquiry-oriented instruction, laboratory settings, independent study opportunities, or flipped learning environments. In any of these contexts, you might find it helpful to have students watch videos in their individual spaces outside of class and come into class ready to actively engage with this modeling problem and the McCusker apparatus. Below are two playlists you can share with students. Playlist 1 leads viewers through the mathematical derivations used in this modeling process. Playlist 2 explains the context of this activity by highlighting the larger field of eigenvalue theory

Goals for Presentations

By the end of this presentation, participants will:

  1. Co-create a list of challenges and questions that arise working to implement modeling activities
  2. Visit the support website for this project
  3. Be able to identify at least three resources for thinking through classroom design issues for implementing this project
  4. Create a personalized list of next steps for those interested in leveraging this work

Intended Audience

This presentation is designed for teachers and learners of differential equations, linear algebra, or mathematical modeling. The activity described in this work fits nicely into introductory ODE and linear algebra classes that traditionally show up in the first two years of many college STEM degrees. However, motivated high school teachers might also enjoy this activity as an open-ended project to get students engaged in college-level explorations.

Description of Project

In this presentation, we demonstrate how to build an apparatus and develop an experimental framework that enables students to apply eigenvalue theory within the context of a useful mathematical modeling problem. To achieve this goal, we show participants how to design and assembled a spring-coupled pair of pendula. We also provide a modeling scheme that supports students in analyzing the dynamics of this system using eigenvalue theory. The entire apparatus and data acquisition system is easy-to-build, inexpensive, simple-to-assemble, safe, and of appropriate size for in-class demonstrations and student laboratory explorations.

This modeling activity puts students in the driver’s seat as applied mathematicians and empowers each student to evaluate the eigenvalue theory within the context of an observable phenomenon. Moreover, this process highlights the type of applied mathematical thinking that students need to transfer content knowledge from the classroom into their world.

This presentation will engage participants using each of the following: an Icebreaker activity, a detailed discussion of ways this activity might be integrated into classroom learning, a next-steps reflection activity where participants brainstorm about possible next steps, and a question and answer period at the end of the presentation to engage participants in this discussion.

In addition, participants will be given access to a support website that includes online videos to introduce relevant curriculum, a laboratory project prompt that can be assigned to students, sample experiment videos that can be used in the classroom, example spreadsheets that provide analysis for the modeling process, and many other resources for use in the classroom.


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