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Mathematical Vistas: From a Room with Many Windows

Peter Hilton, Derek Holton and Jean Pedersen
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Marvin Schaefer
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Rarely does there appear an undergraduate textbook that begs to be taught from even as it leaves its reader wanting to take the class from its authors. Peter Hilton, Derek Holton and Jean Pedersen have produced a gem in this genre. The authors have written a set of independent mathematical essays that explore topics and interrelationships for which there is rarely room in the undergraduate curriculum.

Mathematical Vistas: From a Room with Many Windows (MV) is a companion volume to the authors' 1997 work Mathematical Reflections: In a Room with Many Mirrors (MR). Because of the close interrelationships between MV and MR, this review discusses aspects of both texts. The authors' intended audience is readers having at least the mathematical maturity of bright secondary mathematics students (for MR), bright college or university freshmen, and "adults wishing to update and upgrade their mathematical competence". There is a clear invitation for teachers of mathematics to read these volumes, particularly chapter 9 of MR in which the authors have written an essay on how students best may learn mathematics and how teachers may apply their "principles of mathematical pedagogy".

This series is much more than it would seem since many descriptive terms can mislead one to expect less of these books than there is. The Library of Congress In-Publication Data categorizes it as "1. Mathematics — Popular works", but one would be very wrong to confuse this series with the rash of simplistic popularizations that one finds on today's marketplace. The authors state that these books are intended to convey "what mathematics is about, and how it is done". And so they are — but the about part includes the relevant details and proofs, while the done part concretely addresses the processes of synthesizing prior theorems and attempting to formulate and prove generalizations.

To be sure, there is a lot in these books that forces reader participation and that is, frankly, fun. Magic tricks based on the Fibonacci series, paper folding and polyhedra construction, and matching boys and girls at dances are examples the authors use to lead into deeper, and sometimes surprising, mathematical topics. The authors often speak directly to the reader. They point out areas where the reader may have become confused or where the reader needs to look for subtleties in the exposition. Problems and thought-provoking questions are interspersed as "Breaks" at appropriate points in the exposition. Clearly, many of these breaks are intended to lead the reader to anticipate the direction in which the text's investigation is about to turn.

Each book contains nine more or less independent chapters that can pretty much be read in arbitrary order. In MV, these concern:

  • mathematical (mostly statistical) paradoxes;
  • a history of elementary approaches to proving Fermat's Last Theorem with a commendable summary of Wiles' proof;
  • deriving properties of the Fibonacci and Lucasian numbers (as an extension to a comparable chapter in MR);
  • relationships between number theory, paper-folding and polyhedra-building (also an extension to a preliminary chapter in MR, but culminating in a proof of the General Quasi-Order Theorem);
  • explorations of graph theory and its use in proving the Four Color Theorem;
  • generalizing from binomial coefficients and the Pascal Triangle to multinomial coefficients, the Pascal Tetrahedron, and more as extensions to a comparable chapter in MR;
  • the Catalan numbers and their generalization;
  • symmetry, group theory and a motivated statement (without proof) of the Pólya Enumeration Theorem; and
  • cliques, matching, pigeonhole principle applications, Erdös probabilistic proofs, and Ramsey theory.

Needless to say, there's a lot here. (In addition to three chapters cited above, MR also covers linear recursive definitions; modular arithmetic; quilting as a motivation for tessellation of the plane; large numbers through orders of infinity; and the mathematics of fractal geometry).

The level of treatment is appropriate to the undergraduate audience. Arithmetic and algebraic manipulations are treated in close detail, often so that a method of manipulation is pointed out that can be used as a toolbox technique for proving or deriving similar theorems. In many cases the exposition is similar to the way a teacher would perform at a blackboard, using cartoon bubbles that point to portions of an expression that have a special characteristic or that will be manipulated in a special way. Conceptually difficult passages are clearly marked as such for the reader. The authors devote a lot of space to instructions illustrating the manipulations of paper folding and polyhedron-building so that the student can produce and work with three-dimensional models. Such models can, I believe, prove to be particularly helpful in explaining the concepts behind the Pólya Enumeration Theorem. Relevant references are cited in each chapter. Sometimes, the extended chapters come close to becoming tedious as properties of a mathematical structure are examined through a suite of related manipulative derivation and proof techniques. But the reader can feel the enthusiasm of the authors for deriving new results and the joy of generalizing upon them. Such pedagogical enthusiasm is quite contagious!

The authors' personal feelings about "how mathematics should be done" are central to the text of both volumes. They are stated, with illustrations, in chapter 9 of MR, and are worthy of educators' attention. I believe it is worth listing a few of them here:

  • Mathematics is only done effectively if the experience is enjoyable.
  • Mathematics usually evolves out of communication between like-minded people; human beings can only communicate interesting ideas informally.
  • Never be pedantic; sometimes, but by no means always, be precise (the "Principle of Licensed Sloppiness").
  • Elementary arithmetic goes from question to answer; but genuine mathematics also, and importantly, goes from answer to question.
  • Algorithms are first resorts for machines, but last resorts for human beings.
  • Use particular but not special cases.
  • Symmetry is a pervasive idea in mathematics; always look for symmetry.
  • Use appropriate notation and make it as simple as possible.
  • Mathematics should be taught so that students have a chance of comprehending how and why mathematics is done by those who do it successfully ("Basic Principle of Mathematical Instruction").

In some developments, the authors lead the reader to a point where it appears that a lot of tedious algebraic expansion, collection of terms, and other manipulation will be required. With sparkling freshness, they instead let the reader know that they don't enjoy "slogging" through a proof, particularly if mechanical slogging doesn't lead to achieving understanding. And so, appealing to a principle of being optimistic, they show the reader a variety of useful algebraic techniques that lead to the construction of conceptual proofs. The applications of chapter 9 make MR and MV a joy to read, and I'm certain that they will prove to be a delight to teach from!

There are only a few unfortunate typos in MV or remaining in the 2nd [Corrected!] printing of MR.

There is some duplicated text between the three pairs of corresponding MR and MV chapters mentioned above. But without question, the continuations in MV provide genuine extensions to the foundations laid in MR. Although it is possible to read MV without first having MR in hand, it is this reviewer's opinion that to do so would not be easy for the targeted audience — and it would also deprive the reader of a tremendous number of mathematical treats and insights. The writing is seamless, thought-provoking and entertaining. You and your students deserve to read both books!

Marvin Schaefer ( is a computer security expert and was chief scientist at the National Computer Security Center at the NSA, and at Arca Systems. He has been a member of the MAA for 39 years and now operates an antiquarian book store called Books With a Past.