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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 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 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.

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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 . 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

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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. plains these imaginative terms. A warning there ex- When Robert admits he was guessing, the Number plains that when Robert and the Number Devil Devil huffs that “mathematics is an exact science” talk about mathematics, they use some “unusual where no guessing is allowed. expressions.” The reader is cautioned to use stan- Their explorations continue in this vein, span- dard terms when not in the fanciful world ex- ning a range of mathematical fascinations. On the plored by the Number Devil and Robert. third night the Number Devil takes up factoring, We turn next to a closer look at the adventures introduces “prima donnas” (primes), and cautions, of Robert and the Number Devil and the mathe- more or less dogmatically, that by zero is matical territories of their explorations. strictly forbidden. The sieve of Eratosthenes is il- Devilish Mathematics lustrated with the numbers 1–50, inviting the The book opens with Robert, who sleeps fitfully, reader to complete the task. Then, gladly quelling plagued by strange puzzle-filled dreams. Deter- Robert’s initial satisfaction, the Number Devil asks, mined to banish the tricks his mind plays on him “What if you have a number like 10,000,019, or in the darkness of night, one night he dreams of 141,421,356,237,307? Is it prima donna or isn’t it?” a meadow. Suddenly he notices an elderly man He notes that “even the greatest number devils have the size of a grasshopper perched animatedly on come to grief over it.” And later, “The more sim- a leaf, staring at him with bright eyes. “I am the ple-minded number devils use giant computers,” Number Devil!” asserts the small creature. Dubi- a curious characterization. ous, Robert dismisses him, saying there is no such To Robert’s plea, “What’s the point of it all?” the thing as a Number Devil and, “Besides, I hate every- Number Devil replies, “You do ask stupid questions! thing that has to do with numbers!” When the Luckily you’ve got me to initiate you into some of Number Devil inquires why, Robert replies that he the secrets.” He then asserts (without discussion) must never have gone to school. “Or perhaps you that between any number larger than one and are a teacher yourself?” twice that number there is a prima donna. “And After some tussling back and forth, the Num- when I say always, I mean always!” Then the Num- ber Devil endeavors to show Robert that what he ber Devil enthusiastically goes on to announce thinks of as mathematics from what he has seen that any even number larger than 2 is the sum of in school is a far cry from the mathematics that two prima donnas. Robert tries a few examples and the Number Devil could show him: “The thing sees what the Number Devil is saying. But when that makes numbers so devilish is precisely Robert asks why, the Number Devil confesses that that they are simple. And you don’t need a calcu- “no one knows why [this Goldbach Conjecture is lator to prove it.” He starts out: 1+1, 1+1+1, true].” 1+1+1+1. “Any idiot” can see that these go on On the fifth night, after an encounter with the forever, infinitely. “Have you tried it?” asks Robert, infinity between 0.0 and 1.0, the Number Devil curious. “It would be a waste of time,” says the warns of the existence of “unreasonable” (irra- Number Devil dismissively. Here begins their char- tional) numbers, which “refuse to play by the rules.” acteristic pattern of interaction: the Number Devil By way of motivation he recalls the “hopping” triumphantly shows Robert interesting things, but numbers—2, 4, 8, 16, 32, etc.—and proposes the when Robert challenges or asks questions, the problem of “hopping backwards”, which he calls Number Devil is derisive. When Robert asks more “taking the rutabaga” (square root). This leads to questions about the infinity of numbers, for ex- the square root of 2, which the Number Devil ample, the Number Devil shouts at him impatiently asserts is unreasonable. He does give a rare and and proceeds to show there are infinitely small beautiful geometric proof (sans Pythagoras)√ that numbers too. Robert does not seem to under- the diagonal of the unit square has length 2. On

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which is a (well-rationalized) convention, not a theorem. The Number Devil says, “I could prove it to you, but you’d go mad in the process.” He then goes on metaphorically to describe proving as sim- 1 ilar to crossing a stream in which one cannot swim: one must step from one stone to another suffi- ciently nearby. Previous results established by 1 mathematicians furnish sturdy dry rocks, and the d2 = 2 • 1 shore is built of axiomatic types of propositions (such as that any (whole) number has a unique suc- cessor, or two points determine a unique straight line). The Number Devil explains that even though patterns may persist for very large computations, Figure 1. Geometric proof that the diagonal√ of this does not constitute a proof, and exceptions the unit square has length 2. (which are fatal) may always show up when the computations are further extended. The example the diagonal the Number Devil erects another from the first night is mentioned. Robert reminds square, which he illustrates is composed of four him that he became angry when Robert raised triangles, two of which can form the unit square questions about the extension of the pattern. “You (see Figure 1). Thus the larger square has twice the had a good nose,” admits the Number Devil. “Even area of the unit square, whence the claim. though the formula worked well enough for a Next the Number Devil notes that the unrea- while, it collapsed in the end, without proof. In sonable numbers are not rare, in fact there are other words, even a Number Devil can fall on his “more” of them than there are reasonable numbers. face.” Robert protests, since he already knows well that there are infinitely many reasonable numbers, so The Devil Is in the Details: How Well Does how can there be “more than infinity?” No answer The Number Devil Work? is offered. The Number Devil is an attractive and imaginative Robert’s dreams continue, with encounters with fantasy. Robert and his devilish guide visit some triangular numbers, Gauss’s argument for the sum of the loveliest sites of mathematical discovery of the first 100 (or N) numbers, Pascal’s triangle, and achievement. Their banter is entertaining, the Fibonacci numbers, permutations, the golden mean, ideas attractive. Robert is bright in a common and more. Never are these ideas named as such. sense way, and gradually over the course of the Sometimes they are given fanciful names (e.g., book he becomes more and more engaged with and Bonacci numbers) and sometimes left unnamed. expectant of his encounter with his unusual men- Rarely are explanations given. One exception is tor. The Number Devil is sprightly, a bit mischie- on the ninth night, which deals with some infinite vous, temperamental, and cunningly seductive phenomena. The Number Devil seeks to convince with Robert. The book is aesthetically appealing, Robert that all sequences (whole numbers, odd with attractive illustrations and clear and colorful numbers, triangular numbers, primes, etc.) have the fonts and line drawings. But how well would En- same number of members. Next he gives a geo- zensberger’s book work to attract the mathemat- metric proof that 1/2+1/4+1/8+1/16 + =1, ically hesitant or uninterested reader? What is the followed by a proof that the harmonic di- book saying about good mathematics and about verges, by forming groups of terms of length 2n who can do it? What is it offering the reader math- for n =2, 3, 4,.... This is one of the rare instances ematically, and what would it take for readers to where some substantial mathematical arguments profit from it? are executed with a fair degree of completeness. We worry on several counts. First, real mathe- The eleventh night is particularly interesting, matics is repeatedly said to be different from what since Robert himself begins to ask the Number goes on in school. Good enough. This could be an Devil why all the things shown to him are true: encouraging message for those who have been “You’ve shown me things, but never proved them.” turned off by what they encounter in math classes. The Number Devil answers, “So you want to know Robert’s teacher, Mr. Bockel, who never actually ap- the rules of the game, what a mathematician wants pears, is a scapegoat symbolizing oppressive prac- to know.” He continues, “Showing is easy and fun, tices (e.g., long, boring drills) and back- guessing and testing guesses is better, but proof ward attitudes (e.g., forbidden use of calculators). is all.” He explains how important, and how hard, School, it appears, is not where one learns real proving is for mathematicians, often taking cen- mathematics. It is really not Mr. Bockel’s (or other turies and plagued with mistakes along the way. teachers’) fault, for he cannot go “rock leaping” When Robert asks for an example, the Number whenever he wants, as the Number Devil and Devil chooses a curious one: “10, 80, 1000 , all =1,” Robert do, because he has to correct homework.

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The implication is that math teachers do not have A final issue centers on mathematical warrants the mathematical appetite or experience to en- and what is implied about the basis for learning gage students in “real mathematics”. Mr. Bockel is and knowing mathematics. The mathematical vis- a caricature, who, beyond his intellectual narrow- itor will likely come away from the book aware of ness, is fat and hungry and secretly wolfs pretzels certain facts and truths, but still learning mathe- during class. Implicitly, schoolteachers do not matics by acceptance rather than by conviction. The know much mathematics, and school mathemat- mathematics is well and colorfully staged. The ics is trivial. Of a typical word problem, the Num- scope of the book ranges widely, some fascinating ber Devil says, “That kind of problem has nothing doors are opened, and some surprising connections whatever to do with what I’m interested in. Most are made, such as between the multiple incarna- genuine mathematicians are bad at sums. They tions of the Fibonacci sequence and the golden have no time to waste on them. That’s what cal- mean. Ironically, however, mathematics is pre- culators are for.” sented a bit like magic—intriguing and slightly A second concern is that the book may reinforce mystifying, without explanation or development. an image of mathematics as dependent on inside Hardly any credible approximations to a proof of expertise and restricted to privileged access. Al- anything are offered. The ideas are more like in- though this may be intended to be funny to those vitations to something to be pursued beyond the who have endured deadening mathematics classes book itself, like a preview of a good film, with or to caricature what passes for mathematics in highlights of dramatic scenes intended to lure the school, an insidious message is that teachers are reader into pursuing the full version. The notion mathematically incompetent and, worse, that real of proof is actually one of the topics dealt with in mathematics is found in an insider’s world for a the book, on the eleventh night, but even that is left in metaphorical form rather than offering a sub- select and privileged few—the world into which the stantial example. The book presents only a couple Number Devil entices Robert. The topics explored of illustrations of significant reasoning. We found are interesting, and many are substantial. Appre- the exposition of the mathematics somewhat frus- ciating Enzensberger’s mathematical adventure trating—enticing but never consummated. One depends, however, almost entirely on the reader’s could get the feeling that mathematics was akin to own mathematical experience and knowledge. How a bunch of magic acts, a message surely not de- much a mathematical newcomer might be able to sirable to entice broader interest in the subject. glean is far from clear. Virtually all the mathe- Given these concerns, we hesitate to recom- matical ideas are rendered, with due theatrics, by mend The Number Devil as a book that could be the Number Devil. Robert rarely raises questions, profitably read by most upper elementary and speculates, or uncovers mathematics on his own. middle school children without adult guidance— Thus, mathematics appears as something delivered that is, read for acquiring new mathematical in- by inside experts to a privileged audience. terests, curiosities, and understandings. So we ask A third concern has to do with the images im- instead: Could children and their adult compan- plied by the book’s characters—who they are and ions—teachers, parents, babysitters, grandpar- what that may say about who does mathematics. ents—learn some mathematics from The Number Robert and the Number Devil, both good at math- Devil? What would it take for them to come away ematics, are male. Of course, so is Mr. Bockel, an from their encounters with Robert and his friend unsavory character. But the fact that neither of the with a new appreciation for the subject and their main characters is female is significant. Could a appetites whetted for more? Our answer is, un- bright young girl not have met up with the Num- avoidably, a great deal. One thing it would take is ber Devil instead of, or along with, Robert? Or, knowledgeable guidance and interpretation. Read- more boldly, could the Number Devil have been a ing this book aloud to a class or to one’s child, one brilliant and imaginative woman instead of an would have to be able to explain ideas on which bright, impatient man? The choices of gender in the book only briefly alights. What, for example, the book fall unfortunately in line with cultural went wrong when the Number Devil tried to mul- stereotypes which are unlikely to foster change or tiply appeal invitingly to girls. Neither does the char- 11, 111, 111, 111 11, 111, 111, 111, acterization help break stereotypes of mathe- maticians. The Number Devil is disdainful and im- and why? Why does the drawing of the two squares patient and thinks others are stupid and slow. He show that the diagonal of the first square is the sees mathematics as simple and bears little re- square root of 2, why is this called “unreason- sponsibility for explanation, often disparaging able”, and what is important about the idea of an Robert’s questions or confusions. Is this the kind ? Simply reading the book as is, of being that children should see as typical of a children will (and should) have questions. A sec- good mathematician? Funny as he may be, does he ond thing it would take is good sense about how help attract people to mathematics? to respond to such questions. Which questions

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are worth pursuing? Which might simply be an- swered on the spot? A third thing is that the adult reader would need to understand the imaginative names given to mathematical terms and ideas, see the humor in them, and be able to introduce chil- dren to the conventional names. Most of the con- ventional names have their own interesting histo- ries and bases, something that is left to be discovered in the wake of Enzensberger’s choice to use a fanciful invented lexicon, suitable for dreams but not for mathematical real life. An adult reader would need to have good math- ematical sense and taste to help children use and profit from The Number Devil. The chances are slim that most readers who use this book with children will have such sensibility. This reduces the chances of their learning along with their junior compan- ions and reduces what they can help children glean from this fanciful story. But fortunately mathe- matical sensibility and experience are not the only resources that could support children learning from this book. Good sense about what will intrigue children would help, as would sensitivity to what is, to the young reader, especially complicated or especially accessible. A mathematically curious and adventurous adult might be able to guide chil- dren’s experience of The Number Devil in fruitful ways. Similarly, a mathematically sophisticated adult who is fascinated with young children’s think- ing might also be able to use this book to open fas- cinating doors. Without some modicum of both, however—interest in children and sensibilities about significant mathematics and its pursuit—The Number Devil may fall short of its ambition and its potential.

References [1] JOHN ALLEN PAULOS, Innumeracy, New York, 1988; A Mathematician Reads the Newspaper, New York, 1995. [2] NORTON JUSTER, The Phantom Tollbooth, New York, 1961.

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