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The Heart of Mathematics: An Invitation to Effective Thinking

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MAA Review

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:

• Relevant Math: Books that show how math is relevant to everyday life, especially by emphasizing applications to the social sciences rather than the more traditional emphasis on the physical sciences. COMAP's For all Practical Purposes (Freeman, 7th edition 2006) was the pioneer here and still seems the dominant text.
• Core Curriculum Math: Math facts everyone should know, whether they like it or not. Popular texts here are Staszkow & Bradshaw, The Mathematical Palette (Brooks Cole, 3rd edition 2005); Berlinghoff & Grant & Skrien, A Mathematics Sampler (Ardsley House, 5th edition 2001)
• Fun and Amazing Math: These tend to be mathematically rigorous, although leaving out many details, and are admired by mathematicians. They have fallen out of favor for this kind of course because of a perception (probably accurate) that they are too difficult for the audience. Some prominent books here include: Courant & Robbins & Stewart, What is Mathematics? (Oxford, 2nd edition 1996); Herstein & Kaplansky, Matters Mathematical (Harper & Row 1974, AMS Chelsea reprint 1978); Stein, Mathematics: The Man-made Universe (original publication McGraw Hill 1963, Dover reprint 1998 of 1994 edition); Davis & Hersh, The Mathematical Experience (Birkhäuser 1981, Houghton Mifflin paperback reprint 1982); and Rademacher & Toeplitz, The Enjoyment of Mathematics (Princeton 1957, Dover reprint 1990).

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:

• Each copy comes with a CD-ROM and a set of 3D glasses bound in. The CD-ROM has an extensive set of Java applets that act as manipulatives. They include the Monty Hall problem, the counterfeit coin problem, RSA encryption, and many more. The applets are very well done and add value to the text. One drawback of the computer approach is that there's no transparency in the random simulations — how do we know the computer program is not rigging the results to support the orthodox mathematical opinion? We would have more confidence if we were flipping the coins ourselves.
• The instructor's manual (co-authored with Deborah Bergstrand) provides some additional mathematical background, guidance on topics that have been troublesome, ideas for classroom exercises, and estimates of time needed for each section. It also has solutions for all problems. The supplement is well-done, although experienced and mathematically-knowledgeable teachers would probably not get a lot of extra value out of it.
• There's an optional set of manipulatives that has a few items. I thought this was probably not very useful, since each item would only be used in one or two spots in the text, and most can be constructed from common household materials (constructing them has some value for the students too). One novelty is that the regular polyhedra are built as wire-frame models using pipe cleaners and drinking straws rather than the more traditional method of tracing a squashed version on a piece of paper and folding and gluing it together.
• The companion web site http://heartofmath.com  is primarily a marketing platform for the book but does contain some supplemental material for teachers and students, as well as the complete set of Java applets (mentioned above) so they can be run in a web browser if they cannot be installed on a local computer.
• Various instructor videos and a test bank. I did not examine these.

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.

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