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The Heart of Mathematics: An Invitation to Effective Thinking
Edward B. Burger and Michael Starbird
Publisher: Key College (2005)
Details: 760 pages, Hardcover with CDROM
Topics: Liberal Arts Mathematics, Surveys of Mathematics
MAA Review[Reviewed by Allen Stenger, on 03/08/2008]
This book (HoM for short) teaches how mathematicians work on problems, unlike most liberal arts math books that teach some of the accomplishments of mathematics. Each chapter and major section starts out with one or several interesting and challenging problems, usually not obviously math-related, then shows how to attack them. The book is aimed at college liberal arts students who will probably never take another math course.
The authors are well-known proponents of discovery learning (inquiry based learning) in mathematics. Is this a discovery book? Yes, mostly. Chapter 1 is a series of Conundrums, followed by Nudges and then by Punch Lines. In a more mundane discovery book these would be called Problems, Hints, and Solutions. The remaining chapters start with one or several problems, but the authors give you the solution without forcing you to struggle with the problem yourself. But at the end we have the Mindscapes: additional challenging problems based on the ideas used in the section.
This second edition is similar to the first, with the text carried over with little change. The interior layout has been redone and is much more attractive, and most of the illustrations have been redrawn or replaced. The second edition has an excellent new last chapter on Deciding Wisely (quantifying risks, fair division, election paradoxes). This is presented as the capstone of the book, although most of the material could be presented much earlier. I think this is the most interesting chapter of the book and you should arrange your course so that you cover at least part of it.
The writing is lively, exuberant, and even silly at times. The deep philosophical pronouncements and platitudes are there too, but isolated in the introduction and the marginal notes so you can easily skip over them. It's a fun book!
Who is "the competition" for HoM? I browsed through several popular liberal arts and math appreciation texts but did not find any that competed head-on with HoM. We may actually have a unique and original book here, instead of a re-hash of existing texts! Other texts fall into a few categories:
HoM does not resemble any of these books very much. It's closest to the last category. It might be thought of as a discovery version of those books, that starts with the problems rather than the results.
A number of supplements are available for HoM:
This is a wonderful book! It does have a few weaknesses, or at least things to be wary of. It uses very little mathematical jargon and notation, and this may give some onlookers the false impression that it's not a "real" math book. If your students were given a traditional test on math concepts after the course they probably would do poorly because they wouldn't know the standard methods and vocabulary (this is a problem with all discovery courses). HoM is is certainly not a broad survey of mathematics. It concentrates on geometric problems, or at least problems that have helpful pictures, and is limited to discrete math except for a few intermediate-value arguments. The book is vague about its prerequisites. At various spots it assumes a good command of logarithms, of exponent math, and of complex numbers. The writing and group activity exercises are fairly generic and seem to be tacked on, not integrated with the text. But except for these quibbles it is a wonderful book.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.