Read This!

The MAA Online book review column

Briefly Noted

October 2004

Klein's major goal seems to have been to expose future teachers to a rigorous development of elementary mathematics in the hope that this would improve both their understanding of what they were to teach and the quality of their teaching. One early reviewer highlighted the main point: "The elements of mathematics, presented from an advanced perspective by a creative mathematician rather than a pedant, prove to be vastly more intelligible and fascinating than the orthodox presentation would permit one to suspect." (John M. Reiner, writing in Philosophy of Science in 1941 — isn't JSTOR wonderful?).

The two volumes are actually very different. The first, on arithmetic, algebra, and analysis combines an interest in the logical development of the subject with extensive discussion of pedagogy. Each section asks first "what is the state of our knowledge?" and then "what are the implications for teaching?". The geometry volume focuses rather on giving an overall "take" on geometry as a whole (Klein even uses the word "encyclopedic"), which of course reflects Klein's famous idea that the subject should be organized in terms of the groups of isometries attached to various geometries. In the German edition, this volume included, at the end, a discussion of pedagogy, but this is omitted from the translation.

These books were enormously influential in the United States and in the early days of the MAA (one of the translators of this edition was E. R. Hedrick, the first president of the Association). Klein's account of elementary mathematics is still worth reading and pondering. [Fernando Q. Gouvêa]

An impressive collation of mathematical questions and puzzles, all organized beautifully to illustrate a variety of problem solving techniques! In reading Crossing the River with Dogs I very much enjoyed working through, thinking about, and reviewing many of the chapter example problems along with the multitude of practice problems placed at the end of each chapter. These constitute the strength of the work. The authors cover all the classic problems — connect nine dots arranged in a grid with just four contiguous line segments, river transportation problems, the "sum 15" game, Monty Hall, and the like — and it is nice to see such gems collected in one volume. The text also contains some interesting non-standard problems: Is it always possible to draw a perfect square within a given triangle with all four vertices on the edges of the triangle? Is it possible to stack ten cards numbered one through ten so that if every second card is removed, working down the stack with "wrap around," the ten cards are plucked in order? (This is a clever variation of the Flavius Josephus problem.)

The authors arrange the text into 17 chapters, each reviewing a particular problem solving strategy — the value of diagrams, lists, and elimination techniques, for instance; the power of inductive reasoning, analyzing sub-problems, working backwards, using manipulatives, for example, and even the effectiveness of some simple dimensional analysis, techniques of algebra, and elementary graph theory. Each chapter contains detailed "real" examples of student discussions illustrating the thinking processes behind different problem solving approaches. The book is 490 pages long.

I do have to say that I am worried about the "pitch" of the text. At times the book discusses at great length very elementary mathematical concepts that are appropriate for a much younger and less sophisticated audience. (The examples in the "Make a Systematic List" and "Eliminate Possibilities" chapters, for instance, are too simplistic and the authors often don't move beyond this level of writing in later chapters.) The transcribed student conversations are often tedious to read and the mathematics discussed is not deep.

It is not immediate how one would use this text in a college-level course — even in a non-math majors' course. It may be that as a practiced math puzzle solver I am currently suffering from the "binary nature" of math — when you already know the problem and know how to solve it, then all appears "easy" and elementary; but if a problem or type of problem is new to you, then all appears mighty challenging, if not ridiculously hard. Maybe the authors have in fact done the right thing to keep the material very simplistic so that the focus remains on the strategies of the mind rather then the sophistication of the material. For a college course, perhaps the thing to do would be to only use the problems listed at the end of each chapter, hand them out to a class at the beginning of the week, say, and allow discussion of problem solving strategy to occur informally. Students could read the chapter discussions at their own volition. It would be nice for the instructor to also work with students on generalizing results and to explore with them methods of deductive reasoning and issues of proof. (For example, in the method of finite differences chapter, the authors came close to proving, but didn't quite establish, that if second/third differences are constant, then a relationship is indeed quadratic/cubic.) This, I suspect, would provide a satisfactory experience for all. [James Tanton]

I often tell students how important differential geometry is for understanding modern physics. Of course, I'm not an expert in either field, just a fan. As a fan, my knowledge of the literature is limited. That said, Theodore Frankel's The Geometry of Physics looks pretty good to me.

The table of contents of Frankel's book is impressive: it starts with manifolds and differential forms, progresses to discuss integration and Stokes' theorem, introduces the Lie derivative and holonomy. At that point we're around page 190 or so. But it's a huge book, with a lot more to come, covering most of differential geometry, from the geometry of surfaces in three-dimensional space all the way to Lie groups, vector bundles, topological quantization, covering spaces, and homotopy groups. While doing all that mathematics, physical applications are kept in mind and introduced as necessary. So, for example, the chapter on vector bundles ends with a discussion of the Electromagnetic Connection. Minkowski space and pseudo-Riemannian manifolds (whose metrics are not necessarily positive definite) are kept in view, since they are crucial in Relativity Theory. In fact, Frankel has posted a list of questions in physics and engineering that are considered in the book, probably to convince physics students that learning the geometry is worth the effort.

I suspect that mathematics students would also find this book congenial. Frankel adds to every section title a question that captures the main issue at stake. For example, section 4.1, on "The Lie Derivative of a Vector Field" has the question "Walk one mile east, then north, then west, then south. Have you really returned?" Exactly. And how could anyone resist a section title such as "Geodesics, Spiders, and the Universe". (The question for that one is "Is our space flat?")

Of course, treating all this material, even in a book with almost 700 pages, requires quite a bit of compression. This isn't bed-time reading. On the other hand, it's also not a formal mathematics book. People who like to read that kind of book will probably find Frankel wordy and insufficiently rigorous. I imagine that Frankel's style will appeal to students who are willing to accept a less than completely formal account and to professionals, who can supply the formalism in most cases and really need to hear the core ideas, the "what's really going on" information that's usually only shared orally. If you're looking for a well-written and well-motivated introduction to differential geometry, this one looks hard to beat. [Fernando Q. Gouvêa]

Publication Data

Elementary Mathematics from an Advanced Standpoint: Geometry, by Felix Klein. Dover Publications, 2004. Softcover, 224 pp., $12.95. ISBN 0-486-43481-8.

Crossing the River with Dogs: Problem Solving for College Students, by Ken Johnson, Ted Herr, and Judy Kysh. Key College Publishing, 2003. Softcover, 450 pp., $49.95. ISBN 1-931914-14-1.
See also Problem Solving Strategies: Crossing the River With Dogs and Other Mathematical Adventures, by Ted Herr and Ken Johnson. Key Curriculum Press, 1994. ISBN 1-55953-370-6.

The Geometry of Physics: An Introduction, Second Edition, by Theodore Frankel. Cambridge University Press, 2004. Paperback, 694 pp., $45.00. ISBN 0-521-53927-7.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College, editor of FOCUS and FOCUS Online, and co-author of Math through the Ages.

James Tanton left the college scene to take on the challenges of teaching mathematics in the secondary setting (and what a challenge it is!). He is currently the founding director of the St. Mark's Institute of Mathematics at St. Mark's School in Southborough, Massachusetts.

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• June, 2004: The Grothendieck-Serre correspondence, mathematical biology, olympiad problem books, and units.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.