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.
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.
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.
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.
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