# Recommended Reading

Add your own favourite book of extension material for teachers and lecturers of mathematics students aged 16 to 19, with a brief description. Follow the style of the entries below.

##### Felix Klein, *Elementary Mathematics from an Advanced Standpoint* (Dover 2004), 2 Vols

These are the first two volumes of the three-volume German edition, *Elementarmathematik vom höheren Standpunkte aus* (J. Springer, Berlin, 1924-1928).

##### László Lovász, *Trends in Mathematics: How they could Change Education?* (2006)

This paper is especially recommended. IMU President László Lovász' paper delivered at the *The Future of Mathematics Education in Europe* in Lisbon, 2006. You can download it here: PDF, TEX

##### H. Behnke, F. Bachmann, K. Fladt (Eds.), *Fundamentals of Mathematics* (MIT Press, 1974)

Translation from the German of *Grundzüge der Mathematik*, Vandenhoeck & Ruprecht, Göttingen, 1962. 3 Vols.

The book is the translation of a work commissioned by the ICMI in order to support the scientific foundations of instruction in mathematics, which was one of the topics chosen by the Commission at a meeting in Paris in 1954, in preparation for the International Congress of Mathematicians in Edinburgh in 1958. The English book has three volumes with 42 chapters: the first volume concerns the foundations of Mathematics, the Real Number System and Algebra; the second volume is devoted to geometry and the third to Analysis. There are, in general, two authors for each chapter: one a university researcher, the other a teacher of long experience in the German educational system. And the whole book has been coordinated in repeated conferences, involving altogether about 150 authors and coordinators.

##### Martin Gardner, *Mathematical Puzzles and Diversions* (Penguin, 1956)

The first of his series of books derived from his columns in *Scientific American* from 1956 to 1981 [1]. Categorised as “Recreational Mathematics”, these are accessible to those with secondary school mathematics and a willingness to explore mathematical ideas.

##### Albert Cuoco, *Mathematical Connections: A Companion for Teachers* (2005)

This is a resource book for high school teachers. See the review by Steve Maurer.

##### Douglas R. Hofstadter, *Gödel, Escher, Bach: An Eternal Golden Braid* (Basic Books, 1979)

From Wikipedia:

GEB takes the form of an interweaving of various narratives. The main chapters alternate with dialogues between imaginary characters, inspired by Lewis Carroll's "What the Tortoise Said to Achilles" , in which Achilles and the Tortoise discuss a paradox related to *modus ponens*. Hofstadter bases the other dialogues on this one introducing characters such as a Crab, a Genie, and others. These narratives frequently dip into self-reference and metafiction.

Word play also features prominently in the work. Puns are occasionally used to connect ideas, such as "the Magnificrab, Indeed" with Bach's Magnificat in D; SHRDLU, Toy of Man's Designing" with Bach's Jesu, Joy of Man's Desiring; and "Typographical Number Theory", which inevitably reacts explosively when it attempts to make statements about itself. One Dialogue contains a story about a genie (from the Arabic "Djinn") and various "tonics" (of both the liquid and musical varieties), which is titled "Djinn and Tonic".

One dialogue in the book is written in the form of a crab canon, in which every line before the midpoint corresponds to an identical line past the midpoint. The conversation still makes sense due to uses of common phrases that can be used as either greetings or farewells ("Good day") and the positioning of lines which, upon close inspection, double as an answer to a question in the next line.

##### Zalman Usiskin, Dick Stanley, Anthony Peressini, Elena Anne Marchisotto, *Mathematics for High School Teachers: An Advanced Perspective* (Prentice Hall, 2003)

##### Philip Davis & Reuben Hersh, *The Mathematical Experience* (Birkhauser, 1981)

From Wikipedia:

[This book] discusses the practice of modern mathematics from a historical and philosophical perspective. It won the 1983 National Book Award in the Science category.

It is frequently cited by mathematicians as a book that was influential in their decision to continue their studies in graduate school and has been hailed as a classic of mathematical literature. The book drew a critical review from Martin Gardner, who disagreed with some of the authors' philosophical opinions, but was well-received otherwise.

A study edition and also a study guide for use with the book have been released, both co-authored with Elena A. Marchisotto. The authors wrote a follow-up book, *Descartes' Dream: The World According to Mathematics*, and both have separately written other books on related subjects, such as Davis' *Mathematics And Common Sense: A Case of Creative Tension* and Hersh's *What is Mathematics, Really?*

##### Philip Davis & Reuben Hersh, *Descartes' Dream: The World According to Mathematics* (Houghton Mifflen, 1986)

##### Martin Aigner & Günter Ziegler, *Proofs from THE BOOK* (Springer-Verlag, 1998)

##### Lynn Arthur Steen (Ed), *Mathematics Today: Twelve Informal Essays*. (Springer-Verlag, 1978)

L.A. Steen has written many books and articles on [http:// mathematics], from this very early one to a follow-up *Mathematics Tomorrow* (Springer-Verlag, 1981)