By Gail Corbett
With his new textbook—Differential Equations and Linear Algebra—to be published in May, Gilbert Strang of MIT recently visited SIAM headquarters to plan for distribution of the book. (All of Strang’s books with Wellesley-Cambridge Press are available from SIAM.) An informal interview about the new book with SIAM News soon evolved into a broader conversation about teaching and learning basic mathematics today.
This is first of all a full text on differential equations. That course has a well-established structure, but many of the textbooks are a generation old. The linear algebra part is often limited to matrix operations. That misses the transformation that has moved systems to a central place in our work. One spring or one loop or one equation—that is a start, but it’s not enough. You need coupling and interaction and a matrix.
A big connection is through eigenvalues and eigenvectors. Networks are everywhere, and students have to understand matrices—this is the natural format and the right language for understanding systems and data.
Many departments (including engineering) want students to know more linear algebra. But the curriculum is often quite full. A textbook can make the connections between matrix operations and linear differential equations. To start, their solutions have the same form: complete solution = particular solution plus all null solutions. Without help, most students won’t see it.
Students should answer that question themselves! I will ask the class.
[On returning to MIT, Strang created a new gmail account and posed that question to his linear algebra students.]
The replies were fascinating—I should have done this before. Almost all the students report that they read the text for a first picture of a new topic (like orthogonal matrices). The problem sets bring out “details” that they missed in the reading. I wouldn’t always call them details! But their other courses make demands on their time, and engineering courses really crowd them. They try their best to be efficient.
Students do go to ocw.mit.edu (or YouTube), and they often skip actual class lectures. New tapes are being made this semester in linear algebra (18.06). For differential equations (18.03), Arthur Mattuck’s lectures are a tremendous resource on OCW. He has taught more students than anyone in the history of MIT! The upgrades at 18.06SC and 18.03SC added problem solving and more—closer to a MOOC, but not constrained by a fixed schedule.
Videotapes can be extremely useful. The 18.06 tapes have had four million viewers; it was the right idea to make it freely available, to give it away. Videotapes are not difficult to make! I will add selected topics in differential equations, like delta functions and impulse responses—this is the ODE analog of an inverse matrix. These are such fundamental ideas that it amazes me how often they are missed.
A full-scale MOOC would be much more demanding, with careful assessment every minute. I love teaching but I hate grading.
Absolutely! For the new book, solutions will be posted on math.mit.edu/dela along with sample sections and codes for Euler and Runge–Kutta. The time for printing solutions in the back of the book is over! The beauty of a website is that it grows and improves. Everyone can contribute. Ideas and requests are already coming to [email protected].
Books are still exciting. They develop in new ways. So does our research, and our teaching, and so does SIAM. Probably the SIAM book department will turn this into my first e-book.
Gail Corbett is the managing editor of SIAM News.