6.2. Further Reading

"A mathematician is a machine for turning coffee into theorems."--Paul Erdös

This is an annotated list of books that I have not referred to explicitly in constructing or writing up this unit, but which have influenced my understanding of mathematics, whether by offering offbeat insights, interesting problems, or alternate approaches. Some can be browsed with a minimum of background in the topic while others are "weightier" in that regard.

Annotated Readings

Abbott, E. (1992). Flatland: A Romance of Many Dimensions. New York, NY: Dover Publications.

Introduces the idea of dimensions--with an extension to the 4th dimension--in an ingenious way.

Bell, R. & Cornelius, M. (1988). Board Games round the World: A Resource Book for Mathematical Investigations. New York, NY: Cambridge UP.

This is explicitly directed toward elementary and middle school teachers on various board games (if I recall, some but not all involve probability) and how to use them to teach mathematical thinking. While I saw it several years ago, this gave me the seeds of the idea of using board games to "teach" less-conventional coordinate systems.

Cipra, B. (2000). Misteaks and How to Find them before the Teacher Does. Natick, MA: A.K. Peters, Ltd.

(Yes, the title is spelled "correctly.") A guide to "reasonable answer" checking in calculus. Also sneaks in conceptual discussions of what derivatives and integrals really mean. Its real relevance was that it was the first book I ever encountered that explicitly discussed estimation techniques as an error-catching technique in mathematics, not just calculus.

Davis, P. & Hersh, R. (1999). The Mathematical Experience. Boston, MA: Houghton Mifflin Co.

An eclectic collection of essays and anecdotes on what it is to "do mathematics" and its underlying philosophies. Back in high school this convinced me definitively that math is more than rote memorization and formulas, an impression I have struggled to defend ever since, including in the teaching of this unit.

Deskins, W. (1992). Abstract Algebra. New York, NY: Dover Publications.

This is a great undergraduate-level introduction to abstract algebra. I mention it here because it's a Dover Paperback ($15.95, compared to the $40-$100 textbooks I've usually seen) and because the author works hard at demystifying the notation, language and concepts of abstract algebra with illustrations, examples and counterexamples; the exercises are pretty neat, too. In other words, if you've wondered why commutativity isn't so obvious that it should go without saying, or why we can assume that greatest common divisors exist, this is one way (albeit a heavy-duty one) to see why.

Dickson, H., series editor (1998). I.Q. Puzzles. London, UK: Lagoon Books.

A collection of colorfully-illustrated mathematical brainteasers.

Gleick, J. (1988). Chaos: Making a New Science. New York, NY: Penguin.

Chronicles the discovery of chaos theory, with illustrations and lovely color plates. In particular, this was my first real introduction to fractals and fractal dimension.

Leapfrogs Group (1995). Images of Infinity. Norfolk, England: Tarquin Press.

A multifaceted feast: drawings (including a reference to Escher's "hands drawing hands") and literary quotes (Borges' "Book of Sand" features prominently) intertwine in developing transfinite numbers (a "newer" and not always intuitive category of number), culminating in a visual presentation of Cantor's "diagonal slash" proof that the cardinality of the real numbers is greater than the cardinality of the integers.

Pappas, T. (1989). The Joy of Mathematics. San Carlos, CA: Wide World Publishing/Tetra.

A collection of intriguing and unusual mathematical anecdotes and puzzles, with solutions in the back. The sheer exuberance of Pappas' approach is something I hope to bring to my classroom.

Paulos, J. (1995). A Mathematician Reads the Newspaper. New York, NY: Anchor Books.

While Paulos touches on statistics, this is also more generally a collection of essays on abuses and misconceptions surrounding mathematics when it appears in a "newspaper" context. The real-life relevance is wide-ranging.

Rucker, R. (1977). Geometry, Relativity, and the Fourth Dimension. New York, NY: Dover Publications.

A challenging but offbeat introduction to the "fourth dimension." If you're teaching in conjunction with a physics class, the connections to special relativity are explicitly developed. The first two chapters develop the idea of "dimension," as from the 3rd chapter onward it goes increasingly into physics and technical details. His end-of-chapter problems are great examples of questions that challenge thinking, especially in novel or multidisciplinary ways (though they would definitely need some reworking for high school classrooms). Rucker's discussion in chapter 2 of why Euclid chose his postulates is insightful, and might be interesting if you're teaching proof-based Euclidean geometry; reading it, I understood them for the first time as opposed to just accepting them blindly. Finally, the annotated bibliography is enlightening even if you have no desire to actually go out and read the books mentioned therein.

Smullyan, R. (1992). Satan, Cantor and Infinity: And Other Mind-Boggling Puzzles. (place): Knopf.

Another of Smullyan's collections of linked puzzles, revolving around a "dialogue" between Satan and Cantor, developing transfinite numbers in the process.

Copyright © 2002 Yoon Ha Lee <requiescat@cityofveils.com>
Last updated on 30 May 2002