Angie Thuy-Anh Mai
University of Portland
In Hans Magnus Enzensberger's The Number Devil, translated by Michael Henry Heim, a boy and a devil interact with each other through a series of dreams, uncovering mathematical abstractions around them. Enzensberger creatively tells a story of Robert, a young boy with mediocre math skills, and his fantastical journey through the world of mathematics under the tutelage of the number devil, a black-bearded, irascible, elderly red man whose colorful outbursts lead to mathematical treats.
Throughout the book, Robert and the number devil challenge each other mentally, psychologically, and emotionally. At the inception of their relationship Robert is skeptical of the elderly red man and is unsure how he should respond to the devil's behavior, brushing off comments the man makes. As their story unfolds, the number devil exposes Robert to many mathematical adventures. Robert helps the number devil see beyond the problem and to challenge accepted yet seemingly baseless notions. With each dream their relationship grows and deepens, and soon Robert eagerly awaits his next encounter with the number devil.
As the story progresses, Robert's attitude about math changes from indifference to interest. At first Robert is reluctant to cooperate with the number devil. He is in no mood to deal with the number devil or his numbers. Eventually Robert becomes accustomed to dreaming about the little red man. In the fourth chapter, he stops dreaming of the number devil for a while and waits and waits for the number devil to reappear in his dreams. He anxiously anticipates the dreams in which he and the number devil would have the chance to discover truths and tricks of math again. In the last chapter when Robert stops dreaming again, he finds himself sleeping sounder than he has in a while. He however does not like the idea that the number devil may have forgotten him. Because of his maturing relationship with the number devil, Robert grows to enjoy and respect numbers, something he probably did not think would occur in his life.
This paper aims to examine and interpret three different symbols of the story, though many others can be found throughout the book. They are the characters, the vocabulary, and the dreams. In the following sections, I have provided my interpretations of the symbols found in the text. Henceforth, any citations refer to the English translation of the text.
Robert and the number devil represent pivotal people involved in a mathematical learning experience. They develop a scholarly relationship while Mr. Bockel and Mother play secondary roles that serve to enhance the relationship between the main characters.
Robert represents a typical middle school math student who fails to see how math relates to his life. He is not interested in his math classes and would not dream of dreaming about math.
The number devil represents a math mentor who is passionate about mathematical concepts. The number devil represents a general mathematician. He is often frustrated and does not understand how a student could not grasp or delight in the concept he is trying to demonstrate.
Mr. Bockel represents the traditional textbook math teacher who teaches math without connecting ideas to life. He spends much of his time preparing lessons and chasing students to finish their work without inspiring students to learn math.
Mother represents a parent who does not understand her child and who does not intellectually discuss her child's interests with him. She thinks that her son feels ill when he brings up mathematical concepts and encourages him to play with his friends rather than to pursue his interests. Mother even says it is not normal for Robert to think about numbers (p. 125). She is portrayed as someone who is not a part of the mathematical world and therefore does not understand her son's experiences.
The real-life mathematicians presented in the book are given special names and are organized alphabetically by last name. The names are formatted as follows: 1) True names are in bold italic. 2) Names in normal text are those in the German text. 3) Names in (parentheses) are in the English translation. A description of each name follows.
Georg Cantor – Professor Cantor (Professor Singer). The German Kantor literally translates to the English singer.
Leonard Euler – Eule (Owl). The German Eule is a play on the name Euler and literally translates to the English owl.
Fibonacci, aka Leonardo da Pisa – Bonatschi (Bonacci). Both versions are variations of the name Fibonacci.
Carl Friedrich Gauss – Professor Grauss (Professor Horrors). The German Grauss is a play on the name Gauss and literally translates to the English horrors.
Felix Klein – Dr. Klein (Dr. Happy Little). The German Klein literally translates to the English little. The Old Latin felix literally translates to the English happy.
Johan van de Lune – Johnny vom Mond (Man in the Moon). The French lune and the German Mond literally translate to the English moon. The name in both versions is a play on moon.
Pythagoras – Pythagoras (Pythagoras).
Bertrand Russell – Lord Rüssel (Lord Rustle). The German Rüssel means snout, and the English rustle means to move softly, crackly. These names were meant to look close to the name Russell.
The terms are presented in the order at which they appear in the book. Organization is as follows: 1) Words in bold italic are the English words. 2) Words in parentheses are the German words. 3) Words following the hyphen are the true words. An interpretation of each word follows, many of which are plays on the words.
Minus Numbers (Minus Zahlen) – Negative numbers.
Hopping (Hopsen) – Raising a number to a power. A visual description of raising numbers to powers.
Unreal or Imaginative Numbers (Eingebildete Zahlen) – Imaginary numbers.
Unreasonable Numbers (Unvernünftige Zahlen) – Irrational numbers.
Garden Variety Numbers (Hundsgewöhnlichen Zahlen) – Composite numbers. Garden variety refers to the commonly seen. This phrase may also refer to the natural numbers.
Prima Donnas (Prima) – Prime numbers. Prima donnas are special divas of operas, and the comparison is being made between prime numbers and the importance of divas.
Ordinary Numbers (Gewöhnlichen Zahlen) - Natural numbers.
Rutabaga (Rettiche) – Root or square root. Rutabaga is a play on the word root and literally is a root. Rettiche literally is a radish.
Quang (Quang) – An arbitrary unit as arbitrary as the word quang.
Coconuts (Kokosnüsse) – Triangular numbers, e.g. 1, 3, 6, 10 …. The number devil uses coconuts to visually illustrate triangular numbers to Robert. Thereafter, triangular numbers are referred to as coconuts.
Bonacci Numbers (Bonatschi Zahlen) – Fibonacci numbers.
Number Triangle (Zahlen Dreieck) – Pascal's triangle.
Changing Places (Platztauschen) – Permutations. Changing places visually describes permutations in Robert's dream.
Vroom! (Wumm!) – Factorial. Vroom! is the visual description of the speed at which factorial numbers grows.
Handshakes (Händedrücke) – Combinations without repetition. Handshakes are the visual description of combinations without repetition, which is the way Robert learned combinations without repetition. A group of people shaking hands without shaking hands with the same person more than once is used to illustrate combinations without repetition.
Broom Brigade (Besenkommando) – Combinations. Broom brigade is the visual description of combinations because Robert learned about combinations in the context of having any three students, no matter their order, sweep the playground.
Rocks (Steine) – Steps of a proof. Rocks are the visual description of the steps of a proof when describing how to cross a raging stream (or write a proof).
The Number Devil uses the motif of dreams to tell its story. Robert's dreams may be compared to a student's experience with mathematics. His dreams describe his math journey in seven chronological stages: 1) Incompetence: student struggles independently on math, 2) Support: mentor teaches student, 3) Incompetence: student again struggles independently on math, 4) Support: mentor teaches student once again, 5) Connection: mentor shows importance of relationship with student, 6) Competence: student is confident without mentor, 7) Accomplishment.
These stages follow Robert through his experiences as a math student. His first recurring dreams suggest that he is overwhelmed by anything math related. Once he meets the number devil, his dreams shift to more pleasant surroundings in which he feels comfortable with math. Then there is a period of time when Robert does not dream of the number devil and regresses back to having overwhelming dreams. Once the number devil reappears, Robert is again comforted and eased back into the world of math. At one point, Robert falls ill with the flu, and the number devil appears in the dream by his bedside with a glass of water, signifying that a relationship beyond math is established between mentor and pupil. After recovering, Robert stops dreaming altogether and sleeps more soundly than he has slept in a while. In his final encounter with the number devil, the number devil takes Robert to the secret math headquarters, where Robert is accepted into the Order of Pythagoras, Fifth Class, which is the lowest level of number apprentices of the math society. After he is awarded with a medallion, Robert and the number devil part ways. When Robert awakes, he is pleasantly surprised to find the medallion dangling from his neck, evidence that proves his status as a number apprentice.
Aside from the stages themselves, Robert's dreams are also noteworthy. From among his numerous dreams, I will focus on one of the themes that weave through Robert's dreams. Early in Robert's mathematical experience, he dreams of a raging stream (p. 89). He feels the need to cross the stream but finds no bridge, only rocks. When he tries to cross the stream, all the rocks surrounding him disappear. He can neither move backwards nor forwards. However, because this is his first experience with the stream dream, he does not fully understand its significance and dismisses it as a nightmare he has during the number devil's absence.
This stream later appears in another dream (p. 218). The number devil inquires as to whether Robert has ever tried to cross a raging stream. Robert cynically affirms that he has. The number devil then informs Robert that he cannot swim across the stream because the current is too strong and asks Robert how he plans to cross. Robert notices the rocks in the middle of the stream, and he replies that he is going to jump from rock to rock until he reaches the opposite shore. It is here that the number devil explains the meaning behind the stream. Robert is trying to make his newfound knowledge make sense.
Robert's first stream dream occurs during a period of time when he stops dreaming of the number devil. It is in this dream where he attempts to figure out math on his own but discovers that he is still not ready for proofs and reasons. He is not able to organize all of the information in his dream, hence the vanishing rocks. This is very frustrating for him, but this dream shows Robert's developing maturity in mathematical thinking.
In the second stream dream, the number devil explains that the rocks are the steps of a proof and that Robert should leap onto rocks that are closer together, i.e. find steps that closely follow or relate to each other. The steps of the proof are important for the validity of a theory. That is why the number devil mentions to Robert that one cannot just swim across the stream. With the number devil's help, Robert is able to cross the raging stream and to understand the mechanics of a proof.
As mathematicians become competent with proofs, the raging stream analogy can be extended to show that sophisticated number devils can leap expertly from rocks that are far away from each other, having gained the experience from much painstaking practice. As Robert matures in mathematical thinking, he too can attempt to leap to farther rocks without falling into the stream.
I was very intrigued by the author's portrayal of a mathematician as a devil. If one would consider a devil someone who had tricks up one's sleeves, then a devil would be fitting for mathematics. Those who are not familiar with the nature of numbers may consider math a bunch of puzzles and tricks, even black magic. However, a devil may also be one that neither plays by the rules nor cares about the well being of others. To be fair to the term, the author portrays the number devil as an ill-tempered magician who cares for Robert and yet leaves Robert in the end to pursue other work with no more than a "good bye." He often appears in places of sanctuary to lure Robert to math, in such areas as in the palm tree amidst a desert, in a warm cozy chair that looks out at the freezing snow. In spite of these portrayals, the number devil does show a softer side by Robert's bed with a glass of water when Robert falls ill with the flu. Though the number devil tends to push Robert mentally, he also allows Robert time to rest. For these reasons, the use of the word devil was chosen to describe the mathematician's sly nature rather than any wickedness.
Although the author did a fine job of writing a tale about a child's experience with math, it would have been more effective had the main character been female. As the number devil mentions, mathematics has long been considered man's work (p. 245). By perpetuating this view with male characters, the author fails to offer the possibility for gender equity in the field of mathematics and the possibility that female children can be as successful in changing their perceptions of math as Robert is. Mother's character in the book is also stereotypical. She is portrayed as a female figure who acts ignorantly when Robert talks about math. Rather than discussing his newfound knowledge intelligently, she shrugs off his ranting and believes her son has fallen ill. As a female mathematician and educator, I find that it is hard to keep females interested in the field of mathematics because male peers, teachers, or society can discourage them from succeeding. Without more support from the school system and intellectual world, female mathematicians may be discouraged from continuing their hard work. This book perpetuates this portrayal of male dominance in mathematics and gives little encouragement for female mathematicians.
Many topics in this book engenders thoughts of math education: the pretzel problem (p. 11-12), calculator usage in schools (p. 12), dangers of making assumptions, guessing in mathematics (p. 25), knowing the "inside story" (p. 63), patterns in mathematics, and readiness to understand mathematical theory (p. 216). The pretzel problem is a complicated math problem asking how much time it would take to produce a certain number of pretzels using two pretzel making machines (p. 11-12). This problem illustrates one reason Robert is turned off to math. Though it is good to push students to challenge themselves, it is also important to relate problems to students' lives to help students understand the relevance of the problem. This problem turns Robert off to math because he has difficulty seeing how a pretzel machine has anything to do with his life. Why would students want to learn something that does not have anything to do with their lives?
When the number devil asks Robert to use his calculator, Robert responds that he is not allowed to use calculators and does not have one on him (p. 12). The number devil affirms that Robert should be able to do arithmetic on his own but claims that mathematics is different from arithmetic. Mathematics is the heart of the lessons in schools, but some teachers put more emphasis on the arithmetic instead. Some math teachers prohibit the use of calculators in class because they do not want students to depend on calculators. However, rather than banning calculators, teachers and students should come to an agreement about appropriate calculator usage and integrity. In processes where arithmetic is not stressed, students should use calculators to help them understand the algorithm or concept they are learning. Students should also understand that the calculator must not be used to multiply numbers they can do themselves. Even for slightly harder arithmetic problems, it is healthy to work them out on one's own. A calculator is a tool to aid learning - not laziness.
In the real world, people know that making assumptions and guesses may lead one to misconceptions and mistaken conclusions. This is true in mathematics. The number devil cringes when Robert tells him he guesses in mathematics (p. 25). The number devil retorts that one cannot guess in mathematics because math is an exact science. On the other hand guessing can lead to other discoveries. When should guessing be allowed or encouraged in mathematics? In math education, teachers tend to show students many easily executable patterns but neglect to show students how or why these patterns work, which is probably a cause for guessing. Noting Robert's reaction to discovering the "inside story" (p. 63), it seems that students want to know the mechanics of math problems. When are students ready to learn the story behind the problem? These questions are left for the readers to consider.
In the traditional school system, students must master mathematical concepts in order to be promoted to the next level. Mathematics seems to be linear, but I see it differently. It seems to grow concentrically. The earlier levels of math classes concentrate on skill building because the skill to solve problems is easier to acquire than the ability to understand the nature of a problem. The following levels of math classes merge abstract concepts with skill building. After students have mastered the skills and concepts of mathematics at the high school or college levels, they can then study the mechanics of mathematics and examine how math works via proofs and models. However to reach this stage, students' analytical reasoning must mature, and students must be able to solve problems. Because facts and skills usually have little to do with students' lives, students tend to search for more relevance and connections from their classes. They often receive the true but unsatisfactory response that they need these skills for their class next year. Perhaps teachers should suggest that the best is yet to come.
Hans Magnus Enzensberger has composed an extraordinary book about a child and his journey through mathematics with the mentorship of a mythical character. Though some of the mathematical content the child explores may be challenging for young readers to fully grasp, the book is written clearly and simply. It brings up interesting thought-provoking issues in mathematics, which can appeal to adult readers as well as young readers. These issues not only include math topics but the portrayal of math in the global society.
This paper examined three symbols of Enzensberger's story among many other examples of symbolism throughout the book. Enzensberger uses a plethora of imagery to illustrate the brilliance of mathematics. The chapters introduce several math topics. The characters all have roles in the study of mathematics. The vocabulary visually represents abstract concepts. The dreams are metaphors of the main character's math experiences. The author shares his knowledge and creativity through different venues, creating a book that attracts children and adults alike. Like an onion, The Number Devil is full of juicy mathematical layers that inspire people to examine and consider math in their world. This book is written for all to enjoy and I highly recommend it to anyone.
Enzensberger, Hans Magnus. The Number Devil. New York: Henry Holt and Company: 1997.