Overview. One of this book's pedagogical goals is to support classes that are discussion-based and more effectively tuned to students' actual needs than a typical lecture-based class can be. If you have successfully encouraged students to fill out the boxes, they should not need a lecture on the basic ideas (as they have read the text) but will likely have questions that have been raised by their efforts to fill out the boxes. Answering these questions provides them with help where they need it most and gives you a sense of where they are.
How I Spend Class Time. I usually begin each class session by prompting students to ask questions about the boxes or at least tell me which boxes they had trouble completing. As each question or difficulty is raised, I try to determine what proportion of the class is interested in the answer. After compiling a set of problems, I then address them, starting with those of most general interest. If a question or issue involves only a single student, I generally answer in class if it will not take much time, but I usually offer to answer more complicated such questions after class or during office hours.
One of my colleagues does something that may be even better: after settling on which boxes need attention, he asks his students (selected in rotation) to put box solutions on the board (he includes himself in the rotation).
Some difficulties are purely mathematical; students simply get stuck applying the formalism. Answering such questions, even if the answers involve little more than working through the math on the board, is very valuable for clearing up misinterpretations of the abstract notation and for displaying manipulation tricks that students may not have imagined. However, I have often found that questions about the boxes often expose deeper problems that students are having with the concepts; answering these questions (and spotting the deeper problems behind them) can be especially valuable, not only because it helps address students' misconceptions but also iit often leads to real and lively discussion.
I find that in about a third of the class sessions, I have time left over to work an example homework problem.
Preparing for Class. Since I don't spend time preparing lectures, I generally prepare for class by (1) working through the box problems myself (even though I have worked them all before), and (2) selecting and preparing several homework problems to work during class if there is time. Working through the box problems before class helps remind me of the issues discussed in the boxes, review the concepts presented in the text, and anticipate student difficulties. Particularly on the day homework is due, I also spend some time reviewing the assigned problems, so that I am prepared to answer students' questions about the homework.
Teaching a question-oriented class does require being flexible, quick on your feet, and discerning about the deeper issues that lie behind questions raised. Being thoroughly comfortable with the material will improve your abilities in all of these areas. Some of this simply comes with experience, but please feel free to correspond with me via email if there are parts of the book you do not uncerstand.
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Thomas A. Moore has been a professor in the Physics and Astronomy Department at Pomona College since 1987. He does theoretical research on detecting gravitational waves using LISA (now eLISA). Send him a message.