rev-ball.qxp 12/6/99 10:04 AM Page 51 Book Review The Number Devil Reviewed by Deborah Loewenberg Ball and Hyman Bass The Number Devil terested in matters of Hans Magnus Enzensberger chance, insensitive to Metropolitan Books, 1998 orders of magnitude, ISBN 0-805-05770-6 and impatient with Illustrations by Rotraut Susanne Berner even a modest use of mathematical termi- Translation by Michael Henry Heim nology or symbolic no- tation. Mathematical Engaging A Mathematically Disengaged ability is widely be- Public lieved to be innate, In 1959 C. P. Snow poignantly described the chasm mathematics a pure between the sciences and the humanities in his “hard” science. Other provocative Rede lecture, The Two Cultures. Strad- fields are seen as “soft”. dling this chasm himself, he noted stereotypes One result of this which are held by members of each culture and chasm is that fewer which shape profound misconceptions about the and fewer American students voluntarily enroll work and endeavors of the other. Forty years later in mathematics courses past high school, and the divide which Snow lamented is as wide as ever. the number of American mathematics majors in John Allen Paulos’s popular books illustrate ways our colleges and universities has dropped steadily. in which many otherwise well-educated Ameri- Another equally serious result is that the public cans lack fundamental quantitative sensibilities is unengaged with mathematics, and matters that and are alienated from mathematics [1]. The same require quantitative reasoning or sense are poorly person who would never dream of saying, “I can understood. A natural response to this problem barely read,” remarks lightheartedly, “I was never has been widespread and often ambitious efforts good at math.” The same person who can be fas- to improve mathematics instruction at all levels. cinated with a linguistic detail or a fine point about One wave after another of reforms has swept sociology or politics or music is frequently unin- over our nation’s schools over the past hundred years, leaving a mixed residue of changes and Deborah Loewenberg Ball is a professor in the School of some improvement. These efforts, usually fo- Education at the University of Michigan. Her e-mail ad- cused on curricular change, have been the sub- dress is [email protected]. ject of controversy and debate and have con- Hyman Bass is professor of mathematics education and fronted reformers with the sprawling nonsystem Roger Lyndon Collegiate Professor of Mathematics at the of American schooling and the independent com- University of Michigan. His e-mail address is plexity of creating educational change which per- [email protected]. meates practice as intended. JANUARY 2000 NOTICES OF THE AMS 51 rev-ball.qxp 12/6/99 10:04 AM Page 52 Another path to changing America’s ambiva- The Number Devil: Enzensberger’s lent relationship with mathematics, less often pur- Mathematical Proselyte sued than educational reform, is to produce math- Hans Magnus Enzensberger is a well-known and re- ematical exposition for public consumption. At spected European intellectual and author with their best such efforts can make good mathemat- wide-ranging interests. He gave a speech on math- ics accessible to the public and engender fascina- ematics and culture, “Zugbrücke außer Betrieb, tion with the subject. Consider, for instance, the oder die Mathematik im Jenseits der Kultur—eine writings of John Allen Paulos, Reuben Hersh, Philip Außenansicht” (“Drawbridge out of order, or math- Davis, Martin Gardner, Keith Devlin, Ian Stewart, ematics outside of culture—a view from the out- and Brian Rotman, among others. Some authors— side”), in the program for the general public at the Lewis Carroll, Anno, Marilyn Burns—have written International Congress of Mathematicians in Berlin books for children. In The Phantom Tollbooth, for in 1998. The speech was published under joint example, Norton Juster tucked fascinating math- sponsorship of the AMS and the Deutsche Mathe- ematical excursions and lessons in and around matiker Vereinigung as a pamphlet in German with the adventures of Milo, a bored young child who facing English translation under the title Draw- has “nothing better to do.” [2]. Writing books in- bridge Up: Mathematics—A Cultural Anathema, tended for children can produce an interesting with an introduction by David Mumford. Enzens- twist, for adults often read such books to chil- berger’s knowledge of mathematics is discerning dren. Parents and teachers, grandparents and and appreciative; he is an adventurous amateur babysitters thereby also engage with the mathe- whose treatment of mathematical topics stays matics of these books. Anno’s Mysterious Multi- within reach of ordinary language and yet is sure- plying Jar offers a glorious visual and textual por- footed without fundamental confusions or con- trait of the mathematics of factorials. Traveling ceptual dissonance. In this book he has created a with Milo (The Phantom Tollbooth) to Digitopolis, character called the Number Devil who diaboli- meeting the Dodecahedron, and tracing his en- cally hounds a young boy, Robert, taunting and tan- counters with infinity and precision, readers are talizing him with mathematical mysteries and won- drawn into fascination with and understanding of ders. fundamental mathematical ideas. The Number Devil is a book for older children. One common-sense fallacy is that interest is in- The book jacket announces it as “a gift to readers nate: some people are interested in mathematics; of all ages: a cross between Alice in Wonderland and others are not. However, interests, as John Dewey Flatland, in which we discover exciting secrets argued, are made, not found. A central task of ed- about math, without ever having to do a single math ucation is to engender interest, to help build con- problem.” In fact, the mathematical ideas are sub- nections between a learner’s present and the in- stantial and could easily engage even adult read- vitational potential of the subject. One thing then ers. The book’s illustrations by Rotraut Susanne that popular texts involving mathematical ideas can Berner are nicely and simply done and supportive do is to open mathematical worlds to the public, of the narrative. Even the mathematical notation interesting people in ideas of number and space; and expressions are rendered as illustrations, in pattern and relationships; and the language, tools, soft thick lines and pleasing colors. and ideas that mathematicians use to explore these The story unwinds as a narrative of dreams in worlds. A central responsibility of the mathemat- fantastic settings in which Robert has encounters ical writer is to create these texts in ways that can across twelve nights with the Number Devil. Robert connect with their readers, foreign travelers to and the Number Devil are the only active charac- these mathematical worlds. If successful, these ters of the book, though Robert’s mathematics books should send readers away eager for more teacher, Mr. Bockel, is often mentioned, usually with and surprised by what they have encountered. derision. The dream environments, one for each of We take a closer look here at one such book re- the twelve nights, include such settings as a for- cently published: The Number Devil by Hans Mag- est of trees shaped like 1’s, a cave with wall paint- nus Enzensberger. We begin by providing a snap- ings, and an overturned boat on an empty seashore. shot of the book and then trace its interior, These environments are incidental to the story, ex- following the adventures of Robert and his new cept as colorful conversational props. They take friend, the Number Devil. We stand back then and the place for Robert of some nightmarish experi- comment on the book’s efforts to open mathe- ences—being swallowed up by a slimy fish, falling matics to children and their adult companions. down an endless slide, for instance—which tor- Does it engender interest in mathematics? Does it mented his nights before he met the Number Devil. present mathematics as something in which any- Avoiding these distinctly unpleasant perils serves one can engage? Are some significant mathemat- to keep Robert engaged with the sometimes ical ideas made accessible? What would it take for taunting and aggravating Number Devil. The de- a child or a teacher or parent to read and profit scriptions of the dream environments are brief from this book? and not particularly gripping. They are somewhat 52 NOTICES OF THE AMS VOLUME 47, NUMBER 1 rev-ball.qxp 12/6/99 10:04 AM Page 53 arid as descriptive prose, sketches of surrealistic stand what the Number Devil shows him, which landscapes. It is hard to imagine their being highly makes the Number Devil impatient again, captivating to children. accusing him of being afraid of numbers. He In keeping with the fanciful style of the narra- proposes that he can make all numbers out of tive, many mathematical terms and characters are ones and excitedly presents Robert with the colorfully renamed: Fibonacci numbers are now palindromic pattern, 1 1=1,11 11 = 121, “Bonacci numbers”, primes are “prima donnas”, 111 111 = 12321, 1111 1111 = 1234321, and and irrational numbers are “unreasonable”. “Hop- so on. Robert, following along, becomes skeptical ping” is taking powers of a number, and hopping and, suspecting that the Number Devil is bluffing, backwards leads to taking “rutabagas”, i.e., square asks about roots. Felix Klein gets the pseudonym “Dr. Happy Little”, but we have not figured out why Gauss is 11, 111, 111, 111 11, 111, 111, 111. called “Professor Horrors”. Although there is no This flusters the Number Devil, who tries it. “You table of contents, there is an index, called the were right. It doesn’t work. How did you know?” “Seek-and-Ye-Shall-Find List”, which locates and ex- He is annoyed with Robert for embarrassing him.