After a brief “Welcome!”, Burger and Starbird’s The Heart of Mathematics opens with mock Internet pages that allow readers to “surf” highlighted topics in each of the book’s main chapters: 1) Fun and Games: an introduction to rigorous thought, 2) Number Contemplation, 3) Infinity, 4) Geometric Gems, 5) Contortions of Space, 6) Chaos & Fractals, and 7) Risky Business. Mathematics for liberal arts students who are successfully able to surf this text are certainly in for a wild, compelling ride.
The content of this text is uniformly rich, inviting, and engaging. It is also quite broad, including topics that are somewhat common in texts on mathematics for liberal arts (e.g., patterns in nature, prime numbers, the infinite, symmetry, and probability) as well as many that are not (e.g., knot theory, the fourth dimension, Julia sets, Penrose tessellations, the art gallery theorem, the RSA algorithm). One cannot help but be struck by the wealth of interesting and significant mathematics that the authors have woven through their major themes. While retaining accessibility, the authors treat this material in a fairly rigorous way. There are carefully executed proofs of Cantor’s theorem of the non-denumerability of the continuum, of the existence of exactly five Platonic solids, of the infinitude of the primes, and many other significant results. For anyone concerned about the level of substantive mathematics in mathematics for liberal arts courses, The Heart of Mathematics certainly provides legitimacy.
The text is infused with wonderful examples and questions. As I looked through the text I found a great many of my favorite things to share with general audiences: fractals via video feedback on TV screens, the dollar bill/paper clip trick, Hilbert’s hotel (called “Hotel California” for some reason), n2 + n + 17 generating 15 primes in a row, etc. The text has many images (e.g. the hypercube, Platonic solids with their embedded duals) that can be viewed with 3D glasses. It is infused with many wonderful quotations and interesting images, including both graphic art and art from well-known artists. It includes some mathematical history, but generally at a level that is superficial. The text is written in a colloquial manner that makes it appear user-friendly.
For these reasons I think that this text is a welcome addition to the genre mathematics for liberal arts. I would encourage teachers of this course to consider this text. However, I would also like to warn potential adopters of two reservations I have about this text: focus and pedagogy.
Mathematics for liberal arts has benefited recently from the availability of high quality texts focussed on real applications of mathematics — mathematics for liberal sciences, if you will. Students are drawn to these applications. When they see how linear programming is used to schedule thousands of flights at international airports they are excited. “I finally see a place where mathematics is really used.” Is there an analogue for mathematics for liberal arts? The promise of the text under review here is a focus on effective thinking. The authors state “We do not have modest goals for this book. We want you to look at your life, your habits of thought, and your perception of the world in a new way.” This is the same clichéd adage we’ve always tried to use to sell mathematical education. I’m not sure I believe it. I know that students don’t pay it much heed — especially the students who generally populate mathematics for liberal arts classes.
We denigrate mathematics by justifying it only on utilitarian grounds — effective thinking included. This is mathematics for liberal arts, why not justify the study of mathematics on its supreme beauty, its remarkable creativity, its profound intellectual insight, and its magical wonders? The text includes many pieces of great artwork and it pays some modest attention to “great ideas” which may “pique your curiosity and tantalize your intellect.” If these parallels to other great arts were more fully developed and used to set the context for the entire text I think it would have made the text stronger. More importantly, based on my experience teaching mathematics for liberal arts, I think it would have provided a context that students would find much more compelling.
My second concern involves pedagogy. Publicity for the text advertises it as “a new approach” in which “every lesson emphasizes the discovery approach, opens minds, and keeps students involved.” This is plainly false advertising. The authors implore students to “read the book… answer our questions… think… be active not passive… have fun.” But this plea is hardly “new” and does not constitute the discovery approach. The Heart of Mathematics is akin to a giant lecture. It’s a fine, well thought out lecture with interesting questions to consider afterwards. While the instructor’s manual, manipulative kit, and available CD-ROM might offer some pedagogical flexibility, the overall style of the text is typical — a book written squarely in the style of the lecture method of teaching.
This brings me back to the beginning, when I mentioned the book’s invitation to surf. I immediately went to http://www.heartofmath.com/ContortionsofSpace, ready to interact with some mathematics, to use Java applets to direct rotating hypercubes, create animated tessellations, make my own Julia sets, etc., as the Internet now enables us. But alas, the sites are only mock sites. It’s ironic, for it seems the authors sense that their audience is really willing to surf mathematics — to get involved, to do mathematics. It is unfortunate that the authors have given their audience, largely, just another mathematics text.
The Heart of Mathematics is a fine, worthy, high quality text which compares quite well to similarly focussed mathematics for liberal arts texts such as A Mathematics Sampler by Berlinghoff and Grant. However, given the advance publicity, I was a bit disappointed. I was expecting a new approach that paralleled some of the significant reforms that have challenged other textbook areas as well as a resource that would give us some firepower to challenge the applied mathematics that is overrunning mathematics for liberal arts courses. I didn’t find these things in this text. Despite its many merits, this text does not represent the next generation of mathematics for liberal arts texts.
Julian F. Fleron (firstname.lastname@example.org) will begin the coming academic year as Associate Professor of Mathematics at Westfield State College. He has broad mathematical interests which he tries to share with his students and both students and citizens in the local community — hoping to change rampant negative perceptions of mathematics. He has developed fairly extensive, discovery learning based materials on the infinite which are used in Westfield State’s mathematics for liberal arts course.