Active Learning Resources for
Mathematical Methods in Engineering and Physics

by Gary N. Felder and Kenny M. Felder

"The only way a skill is developed—skiing, cooking, writing, critical thinking, or solving thermodynamics problems—is practice: trying something, seeing how well or poorly it works, reflecting on how to do it differently, then trying it again and seeing if it works better."
-Richard Felder

The practice of "active learning" uses student-centered activities in class to encourage them to adopt an engaged and reflective attitude. See for a general introduction to active learning, and a list of relevant publications.

Research has validated the success of this approach for student understanding and retention, but as an instructor you may have found that supporting students in active learning requires a lot of work. Traditional textbooks and other available resources can be at cross-purposes with active learning and require not just your own creativity but many long hours to rework these materials into an active form. The materials that have been developed to support active learning are mostly at the introductory level.

Thanks to a grant from NSF we have been able to develop a full set of active-learning based "motivating" and "discovery" exercises and computer-based problems for math methods courses for physicists and engineers.


These exercises and computer problems are all incorporated in our math methods textbook. Click here for information about the book.


An "exercise" is not the same as a problem. The biggest difference is that an exercise is designed to be done before students learn a topic, in order to help prepare them for it; problems are generally assigned after a topic has been discussed in class, to give the students practice and/or deepen their understanding.

We distinguish two different kinds of exercises.

All of the individual exercises are listed below, but you can also download the entire set as a zip file.

Frequently Asked Questions About Exercises

At home or in class? Alone or in groups?
Mix it up. See what works for you. We sometimes assign them as homework due on the day we are going to cover the material, and sometimes as an in-class exercise to begin the lecture. You can have students do them individually or in groups, or a mix of the two. One professor we spoke to starts them in class, and then has her students finish them at home—an approach we never even thought of. You will probably keep your students' interest better if you vary your approach.

Do I need to assign all the exercises?
No. If you are uncomfortable with the process, you may want to try only one or two. We hope you will find them easy to use and valuable, and over time you will use them more, but you will probably never use them all.

How long do they take?
Some are five minutes or less; some are twenty minutes or even more. Very few of them should take the students more than half an hour.

That was all pretty noncommittal. Do you have any solid advice at all?
Actually, we do. First, we hope you will use at least some of the exercises, because we believe they contribute a valuable part of the learning process. Second, exercises should almost always be used before you introduce a particular topic—not as a follow-up. You can start your lecture by taking questions and finding out where the students got stuck.

The "Linear Algebra" Motivating Exercise (The Three-Spring Problem)

This exercise is very different from the rest. It starts with an Explanation (with nothing for the student to do except read it), and then a set of Problems meant to help the students explore the Explanation. A professor would assign a carefully chosen subset of those Problems, not all of them. In all those senses this is like a typical section of our book, rather than being like an exercise.

But the entire section does not teach the first thing about matrices! It sets up a problem and steps through the solution, leaving holes that will be filled in with matrices. In that sense, it is very much like most of our other motivating exercises.

We find this to be a powerful way to introduce linear algebra and tie together many of the most important topics in that field, but it only makes sense to use this if you keep referring back to it throughout the unit. When you teach them how to use matrices to change bases, have them use that to convert between initial positions and amplitudes of normal modes. When you teach eigenvectors and eigenvalues, have them derive the normal modes of the coupled springs as eigenvectors of the matrix of coefficients. The exercise sets this up by pointing out which linear algebra topics will be used for various parts of the solution.

At the end of the unit you can come back and tie it all together with a section where we revisit the three spring problem, using linear algebra in every step of the solution.


Computer Problems

Computers can be used in a number of ways in math methods courses: to illustrate complicated math problems, to apply techniques to problems too complicated to solve by hand, or to skip tedious algebra steps and focus on the math you're trying to teach. In addition, computer skills may be one of the things you want to teach in your class.

The problems below are all platform independent. The instructions simply tell the students what to do, but the details of how to do it will of course be different if they are using Mathematica, Matlab, Maple, or some other platform.

The computer problems for all of the topics are listed below, but you can also download the entire set as a zip file.


User-Created Supplements

David Slavsky of Loyola University Chicago wrote to us that "I think you have a rare opportunity to use your text to generate a community of learners all contributing to this field." He wrote the following supplement to our book and gave us permission to post it to this site. We appreciate his help, and encourage other users to do the same.

Click here for the home page for the book,
If you have questions, we would love to hear from you. Click here to email the authors at