Readers of the College Mathematics Journal don't need me to explain this book to them, and will probably run and buy a copy for themselves without my say-so.
For the rest of you, here's the scoop. For several years now, Ed Barbeau has been editing a regular column in the CMJ called "Fallacies, Flaws, and Flimflam." The column collects interesting examples of mistakes, fallacies, and other mathematical howlers. Some come from students, some from publications, others from contributors who have invented their own swindles and fallacies. This book (which acquired a "Mathematical" in the title to accomodate readers unfamiliar with the column) is a kind of "Best of FFF." It collects over 150 articles from the column, organized by topic into sections ranging from "Numbers" to "Advanced Undergraduate Mathematics," with some "Parting Shots" thrown in for good measure.
There are some wild errors and some quite subtle ones, some things we recognize from many past student productions, others that are quite original, errors that reveal a creative mind at work. Collected together, they make a book of the sort one reads compulsively from cover to cover. I suspect it can be very useful in our teaching, since some of these creative mistakes would make great discussion starters. At times it's also a little daunting, because the author doesn't always explain what's wrong, leaving the reader with an interesting problem to think about. Other problems are discussed more carefully. For example, many articles give examples of arguments that "look right" and give the right answer, but are in fact incorrect; in these articles, the author often goes on to analyze under what circumstances this method will yield a correct answer. Mathematical Fallacies, Flaws, and Flimflam is definitely worth your time.
Fernando Gouvêa (email@example.com) is Associate Professor of Mathematics at Colby College in Waterville, Maine. His special interests include number theory, history of mathematics, science fiction, and Christian theology.