+ 1 +

Mathematical Beginnings

A Passion for Numbers

How does a child first begin to become a mathematician? Is it a predisposition or some special gift? Does help and encouragement from parents contribute? What makes it possible, finally, to commit one’s life to this risky, forbidding pursuit?

In this chapter, we tell contrasting stories about the childhood, adolescence, and schooling, up through graduate school, of some future mathematicians, both famous and not so famous. We also report the experiences of youngsters in Olympiad competitions, psychologists’ investigations of prodigies, and what the parents of math prodigies are like.

A few famous mathematicians showed their interest and ability before school age. The Hungarian combinatorialist and number theorist Paul Erdős¹ (1913–1996) claimed that he independently invented negative numbers at age 4.

Stan Ulam (1909–1984) (sometimes referred to as “the father of the H-bomb”) wrote in 1976: “When I was four, I remember jumping around on an oriental rug looking down at its intricate patterns. I remember my father’s towering figure standing beside me and I noticed that he smiled. I felt, ‘He smiles because he thinks I am childish, but I know these are curious patterns.’ ”²

Carl Friedrich Gauss (1777–1855), preeminent after Archimedes (c. 287 BC–c. 212 BC) and Newton (1643–1727), could calculate before he could read. In old age, Gauss told about this childhood feat: In a class that was told to add the numbers from 1 to 100, little Gauss produced the answer in a few seconds. (He constructed 50 pairs, grouping 1 with 100, 2 with 99, and so on. Each pair totals 101, and there are 50 pairs, giving the final sum, 50 × 101 = 5050.)

The famous physicist Freeman Dyson wrote in 2004:

One episode I remember vividly, I don’t know how old I was; I only know that I was young enough to be put down for an afternoon nap in my crib. . . . I didn’t feel like sleeping, so I spent the time calculating. I added one plus a half plus a quarter plus an eighth plus a sixteenth and so on, and I discovered that if you go on adding like this forever you end up with two. Then I tried adding one plus a third plus a ninth and so on, and discovered that if you go on adding like this forever you end up with one and a half. Then I tried one plus a quarter and so on, and ended up with one and a third. So I had discovered infinite series. I don’t remember talking about this to anybody at the time. It was just a game I enjoyed playing.³

For some children, during times of personal upheaval, the simplicity and order of geometry and algebra are a comforting refuge. One example is the famous author and neurologist Oliver Sacks. During the bombardment of London in World War II, he was sent away from home and family. He writes:

For me, the refuge at first was in numbers. My father was a whiz at mental arithmetic, and I, too, even at the age of six, was quick with figures—and, more, in love with them. I liked numbers, because they were solid, invariant, they stood unmoved in a chaotic world. There was in numbers and their relations something absolute, certain, not to be questioned, beyond doubt. . . . I particularly loved prime numbers, the fact that they were indivisible, could not be broken down, were inalienably themselves. . . . Primes were the building blocks of all other numbers, and there must be, I felt, some meaning to them. Why did primes come when they did? Was there any pattern, any logic to their distribution? Was there any limit to them, or did they go on forever? I spent innumerable hours factoring, searching for primes, memorizing them. They afforded me many hours of absorbed solitary play, in which I needed no one.⁴

The childhood of the well-known mathematics educator Anneli Lax (1922–1999) was also disrupted by World War II. Mathematics was “the perfect sort of escape: I didn’t have to look up anything; I didn’t have to consult libraries or books. I could just sit there and figure things out.”⁵

The Hungarian-born physicist Eugene Wigner (1902–1995) was diagnosed with tuberculosis at the age of 11 and spent weeks in a sanatorium in Austria. To help get through this difficult period he worked on geometry problems. “Sitting in my deck chair, I struggled to construct a triangle given only the lengths for the three altitudes. This is a very simple problem which I can do now in my dreams. But then it took me several months of concentrated effort to solve it.”⁶ Wigner attended a famous gymnasium in Budapest where John von Neumann (1903–1957) was also a student. They became lifelong friends. “It was likely the best high school in Hungary; it may have been the finest in the world.”⁷ “My heart was with numbers, not words. After a few years in the gymnasium I noticed what mathematicians call the Rule of Fifth Powers: That the fifth power of any one-digit number ends with that same number. Thus, 2 to the fifth power is 32, 3 to the fifth power is 243, and so on. At first I had no idea that this phenomenon was called the Rule of Fifth Powers; nor could I see why it should be true. But I saw that it was true, and I was enchanted.”⁸

Steven Strogatz, an applied mathematician at Cornell, describes the fear and awe he felt when data he was plotting in a physics lab created a curve he had met in algebra class. He was recording how the length of a pendulum string affects the time for the pendulum to complete a swing. As he plotted the data on graph paper, he realized that

these dots were falling on a particular curve that I recognized because I’d seen it in my algebra class—it was a parabola, the same shape that water makes coming out of a fountain. I remember experiencing an enveloping sensation of fear, then of awe. It was as if . . . this pendulum knew algebra! What was the connection between the parabolas in algebra class and the motion of this pendulum? There it was, on my graph paper. It was a moment that struck me, and was my first sense that the phrase “law of nature” really meant something. I suddenly knew what people were talking about when they said that there was order in the universe, and that, more to the point, you couldn’t see it unless you knew math. It was an epiphany I’ve never really recovered from.⁹

We have limited evidence about childhood engagement in mathematics. However, the stories of those who have an early passion for numbers reveal their fascination with the patterns of mathematics. For others who are experiencing trauma, doing problems provides a refuge.

Mathematicians’ Early Teen Years

Most famous mathematicians first showed a strong interest in mathematics around middle school age. For example, John Todd (1911–2007), a well-known international leader in numerical mathematics, said, “My mathematical career started in the following way. I was in a class, a singing class. My singing was so bad the teacher said I was disturbing the class and had to leave! There were some extra classes, ones with national examinations. And so I had to be put in one of them—it was a second-year algebra class! That’s when I started learning mathematics.”¹⁰

Another mathematician, Jenny Harrison, currently a professor at the University of California, Berkeley, was primarily interested in nature and music during her teen years. In an interview with John-Steiner, she recalled her beginnings. Jenny Harrison was born in Atlanta, Georgia, and spent most of her time outside of school in the woods. This attraction to the natural world may have influenced Harrison throughout her life and can be seen in the way she explores mathematical landscapes. She talked about her visual view of the world and her enjoyment in exploring paths in the woods. This strong preference may well have contributed to her later interests in geometry. The influence and encouragement of her older brother contributed to her strength and self-confidence. He got Harrison interested in basic problems in physics when they were teenagers.¹¹

Although she clearly showed an aptitude for math early on (scoring the highest in her state in a take-home competition), her primary passion was for music. She studied the piano and believed for most of her adolescence that she would devote her life to it. Music is still a part of her life, but Harrison is basically a shy person and found that she was uncomfortable performing in public. “I knew I didn’t want to do music and I went over and shut the lid to the piano. I was intrigued by three problems and wanted to try to understand them: the nature of consciousness, time and light. The question was how best to do this. I eventually settled on mathematics as I felt it would give me answers I could trust.”¹²

Julia Robinson (1919–1985), who became famous for helping to prove that Hilbert’s 10th problem is unsolvable (specifically, that there is no formula or computer program that can always decide whether an arbitrary polynomial equation with whole number coefficients has a whole-number solution) wrote, “One of my earliest memories is of arranging pebbles in the shadow of a giant saguaro, squinting because the sun was so bright. I think that I have always had a basic liking for natural numbers. To me they are the one real thing.”¹³

Sophie Germain (1776–1831), who was to make an important contribution to proving Fermat’s Last Theorem, was born in Paris. As a girl, she had to fight hard for the right to read mathematics. Her interest in mathematics began at age 13, during the French Revolution. Because of the battles going on in the streets of Paris, Sophie was confined to her home, where she spent a lot of time in her father’s library. There she read about the death of Archimedes. It is said that on the day that his city, Syracuse, was being overrun by the Romans, Archimedes was engrossed in a geometric figure in the sand and ignored the questioning of a Roman soldier. As a result, he was speared to death.¹⁴ If someone could be so engrossed in a problem as to ignore a soldier and die for it, thought Sophie, the subject must be interesting.

Sophie began teaching herself mathematics, using the books in her father’s library. Her parents did all they could to discourage her. She began studying at night to escape them, but they went to such measures as taking away her clothes once she was in bed and depriving her of heat and light to make her stay in her bed at night instead of studying. Sophie’s parents’ efforts failed. She would wrap herself in quilts and use candles she had hidden in order to study at night. Finally her parents realized that Sophie’s passion for mathematics was “incurable,” and they let her learn. Thus Sophie spent the years of the Reign of Terror studying differential calculus without the aid of a tutor!¹⁵

Sofia Kovalevskaya (1850–1891) was the first woman to achieve full status as a professional mathematician. Born in Moscow in 1850, her interest in mathematics was stimulated in childhood by the wallpaper in her nursery. Kovalevskaya wrote, “When we moved to the country from Kaluga the whole house was painted and papered. The wallpaper had been ordered from St. Petersburg, but the quantity needed was not estimated quite accurately, so that paper was lacking for one room. Considering that all the other rooms were in order the nursery might well manage without special paper.”¹⁶ Some paper that was lying around in the attic was used for the purpose. By a happy chance the paper for this covering consisted of the lithographed calculus lectures of the analyst M. V. Ostrogradsky that her father had acquired as a young man. “It amused me to examine these sheets, yellowed by time, all speckled over with some kind of hieroglyphics whose meaning escaped me completely but which, I felt, must signify something very wise and interesting. And I would stand by the wall for hours, reading and rereading what was written there. I remember particularly that on the sheet of paper which happened to be in the most prominent place on the wall, there was an explanation of the concepts of infinitesimals and limits.”¹⁷

Figure 1-1. Sofia Kovalevskaya, pioneer Russian analyst. © Institut Mittag-Leffle

Kovalevskaya recalled her uncle speaking to her about the quadrature of the circle: “If the meaning of his words was unintelligible to me, they struck my imagination and inspired me with a kind of veneration for mathematics, as for a superior, mysterious science, opening to its initiates a new and marvelous world inaccessible to the ordinary mortal.”¹⁸

Kovalevskaya, too, had to overcome the resistance of her family. In fact, in order to study mathematics, she had to leave Russia for Germany. To do this legally, she had to marry, and so at age 18 she married “in name only,” another idealistic student, Vladimir Kovalevskii.

The Uruguayan mathematician José Luis Massera (1915–2002), who in his fifties became an international cause célèbre as a political prisoner, described his discovery of mathematics as “a revolution.” Massera wrote:

When I was fifteen the revolution began in my home which lasted for several years. My father had a Hispanic American Encylopedic Dictionary of several volumes and of sufficiently high level. One day, on returning from German class, it occurred to me to look in the dictionary for one of the words that had been used, probably “equation.” I found myself with an enormous quantity of different equations, whose existence I hadn’t even suspected, nor how to approach them. My curiosity being amply satisfied with the algebraic equations, I went to look for one of the others of the dictionary. Thus, day by day and word by word, I began a review, entirely chaotic, no doubt, that was providing me with a harvest of mathematical terms and valuable information about them that I was accumulating and conceptualizing slowly.

During a family trip they passed through Paris where he accompanied his father to a great bookstore. There they found two books: one on classical geometry and the other on trigonometry. Massera devoured them in a short time. He liked the way the material was sequenced and practiced. Subsequently, with the help of a dictionary, he began to read a larger volume on projective geometry. He met his colleague and lifelong friend Rafael Laguardia in high school. He and Laguardia established a study group of young people interested in mathematics and shared their knowledge with them and each other.¹⁹

One of the most astounding childhood accomplishments in mathematics was achieved by Louis Joel Mordell (1888–1972), who was chair of the math department at the University of Manchester from 1922 to 1945, and then successor to G. H. Hardy in the Sadleirian chair at Cambridge University. This pillar of English pure mathematics was a self-taught boy from Philadelphia. He was the third of eight children of a Hebrew scholar. Louis became fascinated by mathematics while in grade school. He learned on his own, buying second-hand mathematics books from a bookstore in Philadelphia for 5 or 10 cents when he was 13 years old. These books contained problems from the Cambridge tripos examinations (written tests for undergraduates), so Louis came to think of Cambridge University in England as the mathematical summit. He later wrote, “I conceived what can only be described as a thoroughly mad and crazy idea of going to Cambridge and trying for a scholarship. . . . I had no idea of the necessary standards. I was self-taught mathematically and had never participated in a competitive examination.”²⁰ He earned the money for his passage to England by tutoring his fellow pupils for 7 hours a day. In December 1906 he went to Cambridge and competed in the university scholarship exam. He placed first! He could only afford a one-word telegram back to his father. It read simply, “Hurrah.”

The Russian, Israel Moiseevich Gelfand (1913–2009), is one of the most illustrious mathematicians of the 20th century. In the small town near Odessa where he grew up, there was only one school. The mathematics teacher there was very kind, but Israel Moiseyevich quickly surpassed him. Gelfand said, “My parents could not order books in mathematics for me; they had no money. But I became lucky. When I was 15 my parents brought me to Odessa for an appendectomy. I said that I wouldn’t go to the hospital unless they bought me a book on mathematics.”²¹ Before he read that book, he had thought that algebra and geometry were two different mathematical subjects. When he saw Maclaurin’s formula for the sine, he suddenly realized that there was no gap between the two. “Mathematics became united. And since then I understood that different areas of mathematics together with mathematical physics formed a single whole.”²²

Figure 1-2. Israel M. Gelfand, one of the foremost mathematicians of his time. © Mariana Cook 1990.

Andrew Wiles, who is now famous for proving Fermat’s Last Theorem, first became fascinated by that problem at the age of 10. He loved doing math problems in school and would take them home and make up new ones of his own.²³ When he was 10, Andrew found Fermat’s Last Theorem in Eric Temple Bell’s classic Men of Mathematics. Thirty years later, Wiles remembered, “It looked so simple, and yet all the great mathematicians in history couldn’t solve it. Here was a problem that I, a ten-year-old, could understand and I knew from that moment that I would never let it go. I had to solve it.”²⁴

Psychological Themes

David Feldman, a developmental psychologist at Tufts University, studied child prodigies and didn’t find many prodigies under the age of 10. He wrote, “Mathematics fosters far fewer prodigies than I thought would be the case when this work began.”²⁵ With the help of Julian Stanley at Johns Hopkins University, he did identify Billy Devlin. When Devlin was 6 he scored in the high school range on the Scholastic Aptitude Test. He became one of the participants in Feldman’s study at age 7. But as he grew older, he turned away from mathematics to physics and astronomy.

Recently, Terence Tao received international attention as a 2006 Fields Medal winner. “Terry is like Mozart; mathematics just flows out of him,” said John Garnett, professor and former chair of mathematics at the University of California, Los Angeles, “except without Mozart’s personality problems; everyone likes him. Mathematicians with Terry’s talent appear only once in a generation. . . . He’s an incredible talent, and probably the best mathematician in the world right now.”²⁶ He is a prodigy who started to play with numbers at 2 years of age. His father, Billy Tao, is a pediatrician who, together with his mother, carefully supported their child’s great gift. They were successful in setting up an individualized program for him where he was able to acquire each subject at his own pace, quickly accelerating through several grades of math and science while remaining closer to his age group in other subjects.²⁷ His home environment was supportive, and today he continues this legacy by being a thoughtful and devoted parent. At the same time, he produces an extraordinary range of important mathematics. He is a wonderful counterexample to the popular image of the eccentric mathematician.

In the psychologist Ellen Winner’s comprehensive work on gifted children, she writes of Ky Lee, a toddler who loved letters and numbers, starting at age 18 months. By age 2, his favorite toys were plastic numbers and blocks with numbers on them. He said the numbers over and over as he handled the objects. When he was 3, he was on a camping trip with his parents. When the ranger at the park gate asked for their license number, neither parent remembered it, but Ky Lee answered with ease, “502-VFA.”²⁸ Ky Lee also could do mental addition, a feat shared in childhood by a few famous mathematicians.

Another recent prodigy was Ganesh Sittampalam, who revealed exceptional insight into subtraction at age 5. He was coached by his father, who had a Ph.D. in mathematics. His father said, “I felt it was very important that he didn’t learn anything by rote. He had to understand the conceptual and logical structure behind the whole thing. I made sure he thought for himself and I always stopped short of telling him what the next step was.”²⁹

Some researchers suggest that ease with calculations comes from living with numbers, being fascinated by them, and representing them as a mathematical landscape—being on familiar terms with numbers and knowing them inside out.

In the most comprehensive study done so far of mathematicians raised in the United States, William Gustin interviewed 20 mathematicians who had won a Sloan Foundation Fellowship. The majority of their fathers had advanced degrees, and their mothers were also well educated. The parents had gone to college during the Depression, and they had a deep commitment to learning. The value of intellectual and educational achievement was transmitted to the children. Working hard, doing well, and being precise were values that their parents taught them.³⁰

Steve Olsen studied U. S. participants in the 2001 International Mathematical Olympiad. The participants’ parents reported their children’s early interest in puzzles and Legos. One mother remembered that her son, Tiankai Liu, had been fascinated with the size and shape of manhole covers. Spatial visualization and an interest in patterns are two of the most often noted interests of future mathematicians. Gustin quotes the father of one Sloan award winner, “He would spend hours building a tower of blocks, precariously balanced. There would be a wail of exasperation and anguish when it finally collapsed. And then he would start redoing it.”³¹

The parents of gifted children are often described as highly attentive, actively stimulating and teaching their children. The families are usually stable and harmonious, with great warmth and nurturance. Independence and autonomy are encouraged.³² Young people raised in such families work up to their potential more often than those from less supportive homes, and they are more independent and original.³³ Many of the mathematically gifted children in the United States (or their parents) are immigrants from other cultures. Margaret Murray found that one-third of the women mathematicians who received Ph.D.’s in the United States in the 1940s and 1950s were immigrants, or children of immigrants, from eastern and central Europe. These immigrant families brought with them the respect for learning and culture that was the hallmark of many European societies and strongly emphasized in the Jewish tradition. In many of these families, the dream of a better life in a land of opportunity was an extremely powerful motivating force for both parents and children, encouraging them to achieve in work and at school.³⁴ Some of the women Murray studied were daughters of highly educated professionals. Others had at least one parent who had attended some college.

More recently, a high number of young participants in science and mathematics competitions are from East Asian families, either native-born citizens or immigrants. A study by Stuart Anderson of the 2004 Intel Science Search found that 7 of the top 10 award winners in the year’s contest were immigrants or their children. Of the top 40 finalists, 60 percent were the children of immigrants.³⁵ One immigrant child, Tiankai Lu, explained one reason why he chose mathematics during his early schooling: “I wasn’t super-duper in English, partly because my parents didn’t know English very well . . . so I decided that maybe I could do math.”³⁶

Some parents spend many hours tutoring and encouraging their children in mathematics. Two who succeeded were Leo Wiener, the father of Norbert Wiener, and Tobias Dantzig, the father of George Dantzig (1914–2005), the inventor of the simplex method of linear programming. Leo Wiener was a self-educated scholar who became a professor of Slavic languages at Harvard. In the July 1911 issue of American Magazine, his strong belief in early training as the source of precocious mental development is described.³⁷ The article reports that Leo Wiener followed his strong convictions by making his children the subjects of an education experiment. He said to the reporter, “It is all nonsense to say that Norbert (and Norbert’s sisters Constance and Bertha) are unusually gifted children. They are nothing of the sort. If they know more than other children of their age, it is because they have been trained differently.” In Norbert’s autobiography, Ex-Prodigy, he wrote of his father’s algebra instruction. “He would begin the discussion in an easy, conversational tone. This lasted exactly until I made the first mathematical mistake. Then the gentle and loving father was replaced by the avenger of blood. . . . The very tone of my father’s voice was calculated to bring me to a high pitch of emotion. . . . My lessons often ended in a family scene. Father was raging. I was weeping and my mother did her best to defend me, although hers was a losing battle.”³⁸ Even while Norbert was a student at Tufts College, Leo continued to monitor his son’s homework. It required putting the Atlantic Ocean between them to emancipate the son from the father. Wiener’s student Norman Levinson wrote of his teacher, “[E]ven forty years later when he became depressed and would reminisce about this period, his eyes would fill with tears as he described his feelings of humiliation as he recited his lessons before his exacting father. Fortunately he also saw his father as a very lovable man and he was aware of how much like his father he himself was.”³⁹

George Dantzig’s story is happier. His father, Tobias, was a well-known mathematician who had studied with Henri Poincaré (1854–1912) in Paris and who wrote a book, Number, the Language of Science, that is still one of the best popularizations of advanced mathematics. He provided strong support to George, who wrote, “He gave me thousands of geometry problems while I was still in high school . . . the mental exercise required to solve them was the great gift from my father. Solving thousands of problems during my high school days—at the time when my brain was growing—did more than anything else to develop my analytical power.”⁴⁰

Personal Characteristics

Gustin found that both the participants in his study and their parents had the ability to devote long periods of time to a single activity.⁴¹ In Steve Olson’s book, Count Down, the stories of successful Olympiad competitors revealed an extraordinary capacity for mental focus; they were willing to stay with a problem for hours, or even days. If there is one quality that young mathematicians share, it is this power of concentration.

Feldman writes of prodigies’ intense dedication to their field, their extreme self-confidence, and their mixture of adult- and childlike qualities.⁴² Recalling their childhoods, many creative individuals remember persistence, enthusiasm, energy, and determination. Some of them say they had a great thirst for knowledge, an exceptional sense of direction, or even an obsession with the problems they pursued. “They often have an unusual ability to resist the distractions of everyday life, to ignore discouragement or ridicule, or to persist in working toward their goals in the face of repeated failures.”⁴³

The most consistently mentioned quality of gifted young mathematicians is curiosity. One parent recalls, “He asked questions at a very young age—constantly asked questions. He just couldn’t wait to learn.”⁴⁴ There are similar reports from almost all the parents, and what makes parents of mathematicians unique is their response to their children’s questioning. They respond seriously, often encouraging even more questions.⁴⁵ Billy Devlin, the math prodigy studied by David Feldman, had a voracious appetite for information and made astonishing progress in his work with a math tutor.⁴⁶ He had a passion for collecting things and arranging them in order. He knew a lot about natural science, science fiction, geography, arithmetic, and mathematics.⁴⁷, ⁴⁸

Between 1955 and 1956, Krutetskii compared mathematically gifted Russian school children with their peers. He found, as many others have reported, that students who excel in mathematics spend a lot of time thinking about the subject. He emphasizes the students’ flexibility in thinking. While they know how to use a previously practiced solution, they can readjust it when the practiced solution fails. Like the Olympiad contestants and other successful young mathematicians, they look for direct and elegant solutions. They tire less in math class than in more verbal classes. Ellen Winner distinguishes between “globally gifted” children and those specifically gifted in mathematics. One globally gifted child mastered reading when he was 3 years old and showed as much interest in numbers as in words. He would ask his parents to give him addition and subtraction problems to do in his head. He could mentally add two-digit numbers if there was no carrying involved. Because he usually solved math problems in his head, he had difficulty in school when his teacher insisted he write out math problems with conventional symbols.⁴⁹

Winner describes one gifted child who likes being different and doesn’t mind being alone a lot of the time. This is a typical story. Gifted youth spend more time alone than average youth. Like most people, they feel happier when with others, but they do not mind solitude as much as most.⁵⁰ Half of the mathematicians in Gustin’s study seem to have handled isolation and long periods of solitary study quite well. But other mathematicians found being “different” difficult. Their social needs were met only after they had the chance to take college math courses or to meet other young people interested in mathematics. Students like these are helped by summer programs for gifted children, where they find peers who share their passions and perseverance.

Many gifted children do have normal social lives, but still they cherish their independence. Their self-confidence and desire to be in control of their own activities can make schooling a challenge. They like to choose books themselves and to focus on topics not taught in class. They enjoy taking college classes while they are still in high school, and they prefer to have a flexible curriculum.

Radford quotes authors who find that young mathematicians have a passion for counting, and they often include numbers in stories and rhymes. They like to use logical connectives (if, then, so, because, either, and or). They delight in making patterns showing balance and symmetry. They like jigsaw puzzles and construction toys, and they arrange their toys neatly and precisely. They use sophisticated criteria in sorting and classifying.⁵¹

The young mathematicians Olson wrote about found it important to picture mathematical problems visually. Some famous physicists have possessed this ability. In an oft-quoted account, Einstein said that he first thought about special relativity by imagining he was riding on a wave and watching the wave behind him.

Some mathematicians in Gustin’s study were interested in the way things work. They liked to take toys apart and look at the gears, valves, gauges, and dials.⁵² Half of the mathematicians in the study were interested in scientific and mechanical projects, and in building models, before the age of 12. “I think I spent a lot of time by myself as a youngster. The first dollar I ever saved I spent on a model airplane. I sanded it, glued it and put it all together and painted it. I just fell in love with the whole process.”⁵³

Teachers

Many mathematicians’ interest in math was stimulated by a teacher. The exceptional instruction at the Lutheran Gymnasium in Budapest, Hungary, produced aerodynamicist Theodor von Kármán (1881–1963), mathematician John von Neumann (1903–1957), and quantum physicist Eugene Wigner (1902–1995). All became world-famous and major contributors to U.S. science. Wigner fondly remembered his mathematics teacher, László Rátz, who “loved teaching; he knew his subject and how to kindle interest in it. He imparted the deepest understanding. Many . . . teachers had great skill but no one could invoke the beauty of the subject like Rátz.”⁵⁴

On the other hand, some talented girls were not supported in the study of mathematics by their teachers. When Alice Schaefer was skipped from third to fourth grade, her teacher said that although she and one of her classmates were expected to do all right in fourth grade, the teacher doubted that the two girls would be able to do long division. Alice was indignant. She subsequently recounted that she was determined to learn long division in the fourth grade. The experience gave rise to her first feelings about mathematics.⁵⁵ Then her high school math teacher opposed her continuing in math even though she excelled in his class. When Schaefer asked him to recommend her to the University of Richmond, he answered, “If you want to major in mathematics I won’t write for you because girls can’t do mathematics.”⁵⁶ But Schaefer prevailed. Other female students were more fortunate. For instance, the Canadian mathematician Margaret Marchand was encouraged by her teacher, Mr. Robson. He recognized her aptitude and was the first person to suggest that she go on to the university.

Some women did not choose mathematics until college. Growing up in the 1950s, Judith Roitman never imagined herself as a mathematician. She started writing poetry at age 8 and was also a minor music prodigy, but as a girl she was discouraged from a career in mathematics. “Roitman remembers feeling that no matter how well she did in mathematics she could never achieve real understanding, because real understanding was, by definition, something only boys could have.”⁵⁷

Planning to become a high school English teacher, Roitman went to Sarah Lawrence, an all-women’s college at the time. There she changed her beliefs about what she could and couldn’t do. First, she studied poetry and language. Then she moved to philosophy. Finally, she turned to mathematics because the fact that mathematicians were constantly inventing new ways of thinking appealed to her.⁵⁸ But in the late 1960s and early 1970s gender bias followed her through graduate school. The climate was a challenge for women in mathematics. Despite such an environment, Roitman succeeded, with the support of a peer group of female mathematicians, graduate students, and postdoctoral instructors, and with Mary Ellen Rudin as her mentor. (Rudin is a leading topologist. We quote extensively from her in chapter 9.) Rudin provided Roitman with an existence proof, that is, an example of a woman with a salaried professional life as a research mathematician.⁵⁹ Roitman became a leading researcher in set theory and a professor at the University of Kansas.

In her study of how women became mathematicians, Margaret Murray (2000) found that nearly all the women she interviewed had at least one college teacher who encouraged and influenced them, even at a time when women in mathematics challenged the prevalent social norms. Pregnant women were discouraged from attending class. It took a committed teacher to encourage a pregnant student to persist in her studies. Students in women’s colleges benefited from the mentorship of female professors. At Bryn Mawr College, the lasting influence of Emmy Noether (1882–1935), one of the most distinguished mathematicians of the early 20th century, lasted even after her death. (Noether was one of the creators of modem abstract algebra.) One Bryn Mawr graduate recalled, “They were still very much under the spell of having Emmy Noether there. There were still stories going on about her.”⁶⁰

These accounts and studies can assist us in regard to how mathematical talent originates and develops. It appears unexpectedly. It is not created at will by parents or teachers, but their support is crucial for developing it. Intense interest in mathematics does not usually manifest itself until the age of 10 or 12. Often a future famous mathematician does not find his or her vocation until their late teens, or even later. It seems that some developing inner inclination or aptitude becomes visible only when a certain intellectual maturity has been achieved and a supporting environment has been provided.

Contests and Competitions

Melanie Wood was the tutor for the U.S. Mathematical Olympiad team in 2001. She first became interested in mathematics in kindergarten, and remembers that she got in trouble for it. Once she was playing with flash cards that had numbers on them, and she sorted them into evens and odds. “I was realizing things like when you add two odd numbers, no matter which two they were, you always got an even number. And when you added an even number to an odd number—things like that. I got in trouble because I wasn’t supposed to be playing with those flash cards. I had already passed that level and I was supposed to be playing with some other flash cards.”⁶¹

Wood’s mother played math games with her. When Melanie was very young, her mother did not think that she was particularly outstanding in mathematics. But by 7th grade Melanie was in an accelerated math class, although math was only one of her many interests. That year she was invited to take part in a national competition called Mathcounts. Participants compete at first in individual schools and then advance to regional contests. The four top individual scorers in each state go on to the national competition. In the 7th grade Melanie Wood had little advanced preparation for Mathcounts, but she finished first in Indiana. “That really flipped my world around. In terms of making math something important in my life, and changing my view of who I was and what I was good at.”⁶²

Figure 1-3. Melanie Wood, Olympiad contestant and coach, mathematical researcher. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach.

When Melanie entered high school, she was asked to train for the International Mathematical Olympiad. At the training camp she was the only girl. This was an emotional challenge to her, which she overcame with the help of a Bulgarian woman who was teaching mathematics at Harvard. “Having a role model like that was a big deal in my life. Previously I never knew a mathematician that I could look at and think, in 10 years I want to be like that person. And so it was hard to imagine becoming a mathematician.”⁶³ In subsequent years she attained many honors and is now a Szegö assistant professor at Stanford University.

Problem competitions arouse great excitement among young mathematicians. Gábor Szegö (1895–1985), who ultimately became chair of the mathematics department at Stanford University, remembered his excitement as a student in the years from 1910 to 1912 at the monthly arrival of the Hungarian high school math newspaper. “I would wait eagerly for the arrival of the monthly issue and my first concern was to look at the problem section, almost breathlessly, and to start grappling with the problems without delay. The names of the others who were in the same business were quickly known to me, and frequently I read with considerable envy how they had succeeded with some problems which I could not handle with complete success, or how they had found a better solution (that is, simpler, more elegant or wittier) than the one I had sent in.”⁶⁴

In the United States, many promising students also thrive on math competitions and the Intel Science Talent Search (formerly the Westinghouse Science Talent Search). This is in contrast with their high school experiences, where many reported that their teachers were not ready to meet the challenge of a highly motivated student. Some of the young mathematicians receive individualized tutoring or attend university classes at an early age. Sixteen of the 20 mathematicians in Gustin’s study did independent work in mathematics while they were in high school. They read books that their parents or older siblings had used in college, and some read scientific magazines. But they found few opportunities to talk mathematics with gifted peers in high school. This is why summer math camps were established in the 1950s. These were more challenging and interesting than regular class. There they explored fascinating new topics and developed their own techniques for solving problems. They discovered the excitement of doing something well and being recognized for it.⁶⁵

Mentors in Graduate School

The most critical socializing influence on future mathematicians is graduate school. A mathematics department’s reputation is determined mainly by the quality of its research. A student goes to a top-ranked department hoping to meet outstanding professors and fellow students deeply involved in mathematics. A graduate school class in mathematics lets the student see a mathematician at work.

Figure 1-4. Norbert Wiener. Courtesy of Smithsonian Institution Libraries, Washington, DC

In graduate school it is possible for students to establish personal relationships with teacher mentors. A powerful example of such a relationship was that of Norman Levinson (1912–1975) and his mentor Norbert Wiener (1894–1964). Levinson writes, “I became acquainted with Wiener in September 1933, while still a student of electrical engineering, when I enrolled in his graduate course. It was at that time really a seminar course. At that level he was a most stimulating teacher. He would actually carry on his research at the blackboard.”⁶⁶ As soon as Levinson showed some slight comprehension of what Wiener was talking about, Wiener handed him the manuscript of a treatise he was writing. Levinson found a gap in one of Wiener’s arguments and supplied the missing reasoning. “Wiener thereupon sat down at his typewriter, typed my lemma, affixed my name and sent it off to a journal. A prominent professor does not often act as secretary for a young student. He convinced me to change my course from electrical engineering to mathematics. He then went to visit my parents, unschooled immigrant working people living in a run-down ghetto community, to assure them about my future in mathematics. He came to see them a number of times during the next five years to reassure them until they finally found a permanent position for me. (In these depression years positions were very scarce.)”

The impact of some particularly gifted teachers goes beyond the subject matter—it communicates the passion of those with experience to the novice. In recollecting his early years, Herbert Robbins (1915–2001)—a famous statistician and coauthor with Richard Courant of What is Mathematics? said of his Harvard professor, Marston Morse:

There was something going on in his mind of a totally different nature from anything I had seen before. That’s what appealed to me. . . . He was a father figure to me—my own father died when I was 13. Marston and I were about as different as two people could be, we disagreed on practically everything. And yet, there was something that attracted me to Marston that transcended everything I knew. I suppose it was his creative, driving impulse—this feeling that your house could be on fire, but if there was something you had to complete, then you had to keep at it no matter what.⁶⁷

(Morse theory studies the properties of gradient vector fields. It connects with many parts of pure and applied mathematics.)

Stan Ulam’s (1909–1984) career was stimulated by his rapport with the set theorist Kazimir Kuratowski in the Polish city of Lvov:

From the very first lecture I was enchanted by the clarity, logic, and polish of his exposition and the material he presented. From the beginning I participated more actively than most of the older students in discussion with Kuratowksi. . . . I think he quickly noticed that I was one of the better students; after class he would give me individual attention. . . . Soon I could answer some of the more difficult questions in the set theory course, and I began to pose other problems. Right from the start I appreciated Kuratowski’s patience and generosity in spending so much time with a novice. Several times a week I would accompany him to his apartment at lunchtime, a walk of about twenty minutes, during which I asked innumerable questions. Years later, Kuratowski told me that the questions were sometimes significant, often original, and interesting to him. . . . At that time I was perhaps more eager than at any other time in my life to do mathematics to the exclusion of almost any other activity.⁶⁸

Figure 1-5. Richard Courant and Herbert Robbins. Mathematical People: Profiles and Interviews. Eds. Donald J. Albers and G. L. Alexanderson. Boston: Birkhauser, 1985. Pg. 285. With kind permission from Springer Science and Business Media.

Sometimes a chance encounter is a catalyst for a new understanding of one’s field. Paul Halmos (1916–2006) described having lunch with the famous probabilist Joe Doob at a drugstore in Urbana, Illinois. “My eyes were opened. I was inspired. He showed me a kind of mathematics, a way to talk mathematics, a way to think about mathematics that wasn’t visible to me before. With great trepidation, I approached my Ph.D. supervisor and asked to switch to Joe Doob, and I was off and running.”⁶⁹

We have already mentioned the father and son Tobias and George Dantzig. After reading papers by the famous statistician Jerzy Neyman (1894–1981), George went to study in Neyman’s department at Berkeley.

During my first year at Berkeley I arrived late one day to one of Neyman’s classes. On the blackboard were two problems which I assumed had been assigned for homework. I copied them down. A few days later I apologized to Neyman for taking so long to do the homework—the problems seemed to be a little harder to do than usual. I asked him if he still wanted the work. He told me to throw it on his desk. I did so reluctantly because his desk was covered with such a heap of papers that I feared my homework would be lost there forever. . . . About six weeks later, one Sunday morning about eight o’clock, Anne and I were awakened by someone banging on our front door. It was Neyman. He rushed in with papers in hand, all excited: “I’ve just written an introduction to one of your papers. Read it so I can send it out right away for publication.” For a minute I had no idea what he was talking about. To make a long story short, the problems on the blackboard which I had solved thinking they were homework were in fact two famous unsolved problems in statistics. That was the first inkling I had that there was anything special about them.⁷⁰

Dantzig became one of the major statisticians in the United States and received the National Medal of Science.

In Budapest, in the years before and after World War I, Lipot Fejér (1880–1959) was the mentor to a whole generation of Hungarian mathematicians. He had been a great problem solver in high school. László Rátz often opened his problem session at the Lutheran Gymnasium in Budapest with the announcement, “Lipot Weisz has again sent in a beautiful solution.” Weisz, who later changed his name to Fejér, won second prize in the Eötvös competition. (That competition is the grandfather of Mathcounts and the U.S. Mathematical Olympiad.)

Fejér became a professor in Budapest in 1911. George Polya wrote, “Almost everybody of my age group was attracted to mathematics by Fejér. Fejér would sit in a Budapest cafe with his students and solve interesting problems in mathematics and tell them stories about mathematicians he had known. A whole culture developed around this man. His lectures were considered the experience of a lifetime, but his influence outside the classroom was even more significant.”⁷¹

One of Fejér’s students, Agnes Berger, recalled that he gave very short, very beautiful lectures that lasted less than an hour. “You sat there for a long time before he came. When he came in, he would be in a sort of frenzy. He was very ugly-looking when you first examined him, but he had a very lively face with a lot of expression and grimaces. The lecture was thought out in very great detail, with a dramatic denouement. It was a show.”⁷² Fejér was a major contributor to Fourier analysis (the expansion of general functions in series of sines and cosines).

Another quality of good mentors is that they understand the tensions graduate students confront—the need to balance discipline and commitment as a learner with the complex responsibilities of adulthood. The Berkeley professor S. S. Chern (1911–2004) was able to respect and work with the challenges his students faced. Following are two tributes from the book, S. S. Chern: A Great Geometer of the Twentieth Century.⁷³

Chern had agreed to take on Louis Auslander as a student at the University of Chicago before Auslander took his Ph.D. qualifying exam in the spring of 1952. But the night before the geometry part of the exam, Auslander’s wife arrived home with a newborn son, and hemorrhaged. The next morning he performed poorly in geometry. But when Auslander saw Chern and asked him if he would still be his thesis adviser, Chern

conveyed the understanding that examinations were not important—it was now the time to do mathematics. Then began a process of education, an apprenticeship, by indirection. Chern would say things like “Would you look at Finsler geometry?” or, “It would be very nice if we meet in my office one day a week and talk things over.” No matter what I presented, Chern would listen politely and almost silently. On occasion he would say, “I do not understand.” I soon learned that “I do not understand” was a euphemism for “That’s wrong!!” Somehow Chern conveyed the philosophy that making mistakes was normal and that passing from mistake to mistake to truth was the doing of mathematics. And somehow he also conveyed the understanding that once one began doing mathematics it would naturally flow on and on. Doing mathematics would become like a stream pushing one on and on. If one was a mathematician, one lived mathematics . . . and so it has turned out.⁷⁴

Figure 1-6. Shiing-Shen Chern. Mathematical People: Profiles and Interviews. Eds. Donald J. Albers and G. L. Alexanderson. Boston: Birkhauser, 1985. Pg. 285. With kind permission from Springer Science and Business Media.

Philip A. Griffiths, former director of the Institute for Advanced Study at Princeton stated:

Chern is genuinely interested in the work and ideas of students just finding their way. He is encouraging yet is willing to say some idea may not be interesting. He demonstrates a combination of wisdom, mathematical discrimination and tact. He always treats one with respect, as a colleague and equal. In addition to the mathematical relationship, he shows a real interest in the person in a broader sense, asking about his family, career plans, and travel, discussing world politics, history and events with as much wisdom as he shows in mathematical discussions. Long before the concept of “mentoring” came into vogue Chern was a model mentor. For those just embarking on a career in mathematics, as I was those thirty-odd years ago, the experience described above can be decisive. A beginning student needs to learn more than facts and techniques: he or she needs to absorb a world view of mathematics, a set of criteria with which to judge whether or not a problem is interesting, a method of passing on mathematical knowledge and enthusiasm and taste to others. To most fully develop as a mathematician one needs a mentor who can provide what Chern has provided for so many: formal teaching, teaching by example, encouragement, realism, and contacts.⁷⁵

While male mathematicians freely give credit to a mentor in informal conversations, we have found that women publish such acknowledgments more often. At the Courant Institute at New York University (NYU), Lipman Bers (1914–1993), a leading researcher on elliptic partial differential equations and Riemann surfaces, had a strong commitment to women. During the 1950s he helped Tilla Weinstein (later Tilla Klotz Milnor) continue her studies when she became pregnant, while others, including the dean, resisted her efforts. Bers continued to advise and support his students after they left the caring environment of NYU.

One Courant Institute student, the group theorist Rebekka Struik, so much enjoyed her close contact with Wilhelm Magnus, another well-known professor there, that she was disappointed when Magnus told her that her work was complete and that it was time for her to look for a position.⁷⁶ Her father was Dirk Struik, a math professor at Massachusetts Institute of Technology (MIT), an authority on differential geometry, and a Marxist. He was one of the “Massachusetts Ten” indicted in the 1950s on the unusual charge of conspiracy to overthrow the Commonwealth of Massachusetts. The case against the Massachusetts Ten was thrown out on appeal, and Dirk Struik was restored to his teaching position at MIT.

Richard Courant (1888–1972) himself was extraordinarily supportive of women and helped them work out the conflict between their home responsibilities and their studies. There was another female student at the Courant Institute whose father was a well-known mathematician. Cathleen Morawetz is the daughter of J. L. Synge, a leading applied mathematician from Ireland who was long a professor at Toronto. (Her great-uncle was the famous Irish playwright of the same name.) Courant and Synge met at a math meeting and found that the daughters of both of them had recently married. The two distinguished fathers bemoaned the likelihood that their daughters wouldn’t go on in their chosen fields, mathematics and biology, respectively. “Ja, ja, well, you can’t do anything about my daughter,” Courant sighed, “but maybe I can do something about yours. You should send her to see me some time.”⁷⁷ Morawetz did become a student at the Courant Institute, earned a Ph.D. there with a thesis on transonic flow written under K. O. Friedrichs, became a professor there, ultimately became head of the institute, was elected president of the American Mathematical Society, and subsequently became a recipient of the National Medal of Science.

When Hersh started graduate work at Courant, he was enrolled in Morawetz’s course, Introduction to Applied Mathematics, and was hired to grade papers in that course. A student grading papers for a class he attends would be considered irregular in most university departments, but it was no problem at Courant, which was run as a family enterprise. (It was joked there that “nepotism is compulsory.”) Both of Courant’s daughters married mathematicians—Jerry Berkowitz was a long-time professor at the Courant Institute, and Jürgen Moser, famous for work in dynamical systems, spent many years at the Swiss Federal Institute of Technology (ETH) in Zurich. There is an old (somewhat sexist) saying, “Mathematical talent is inherited from the father-in-law to the son-in-law.” In fact, Courant, with only two daughters, managed to have three mathematician sons-in-law! (Some years after Jerry Berkowitz passed away, Lori Berkowitz, nee Courant, married Peter Lax—another important member of the Courant Institute.) The day is surely coming when we will give examples of mathematicians as mothers-in-law and daughters-in-law.

Many mathematicians have said that their greatest satisfaction as professors has been working with thesis students. Nurturing and developing a mathematical mind from small beginnings to full-grown power can be wonderfully fulfilling. When a student you have nurtured comes to you with a fruitful new idea about a problem you have long struggled with, the pleasure of such a collaboration can be even greater than the pleasure of solving a problem on your own. One famous example of such a relation was between Karl Weierstrass (1815–1897) and Sofia Kovalevskaya (1850–1891) about which we write further in chapter 5.

On the other hand, not all relationships between thesis student and thesis supervisor are rosy and cheery. The relation can be close and intimate or it can be distant and formal. One famous mathematician described supervising a dissertation as “research by the professor, done under difficult conditions.” The supervisor of one mathematician we know did absolutely nothing to help him further once he had finished his dissertation. Another told us that he dreaded meeting with his supervisor, who would lose his temper and throw chalk during their meetings. Agnes Berger, whom we quoted above in describing Fejér’s lectures in Hungary, said: “I was greatly amazed when I saw that in America a professor would sit down with a graduate student. Nothing like that ever happened in Budapest. You would say to the professor, ‘I’m interested in this or that.’ And then eventually you would come back and show him what you did. There was none of the hand-holding that goes on here. I know people here who see their students every week! Have you ever heard of such a thing?”⁷⁸

Students receive guidance and inspiration not only face to face but also at a distance. George David Birkhoff (1884–1944), the dominant American mathematician of the early 20th century, received his inspiration from Henri Poincaré of Paris.⁷⁹ Learning from “distant teachers,”⁸⁰ whose impact is through their published work rather than from face-to-face interaction, is a critical aspect of creative development. When creative individuals discover their own teachers from the past, there can be a recognition of an intense, personal kinship, as the work of another invokes a special resonance in them. Once such a bond is established, the learner explores those valued works with an absorption that is the hallmark of creative individuals. In this way, they stretch, deepen, and refresh their craft and nourish their intelligence, not only during their early years of apprenticeship but repeatedly through the many cycles of their work lives.⁸¹

For a student first embarking on a mathematical life, the work of one’s predecessors is an intimidating challenge. The great achievements in the field seem overpowering. Often, when facing self-doubt or discouragement, these men and women lean on the support of their mentors.

Preparing for the Mathematical Life

For students of mathematics it is an intense experience to commit oneself to this discipline, where there is no guarantee of success. Research in mathematics requires deep focusing on a problem for long periods of time. One’s previous accomplishments by no means guarantee success at a new problem. Some participants in Gustin’s study still felt insecure about their ability to do original mathematics, even after receiving a Ph.D. “I really had grave doubts about my ability to do creative mathematics. There is no way of knowing until you do it.”⁸² Sometimes a graduate student chooses a thesis topic only to find out that although initially it was promising, ultimately it becomes a dead end.

What drives such a concentrated effort when the end result is not predictable? “You have to immerse yourself in something. Think about it constantly. You work and work and get these ideas floating around and you have to reach a certain threshold and then some of the problems solve themselves. Some of them come to you two years later, but you have to concentrate, to focus.”⁸³ And even after you succeed, recognition comes only from a small group of specialists.

But the excitement of obtaining a new result can be deeply satisfying. “I know for the best two or three things that I have ever done, there is a feeling of awe that is just incomparable. I feel privileged to have added a couple of things to the field. I love the subject, there is nothing I enjoy more than finding the solution to a problem after a lot of work. And even though it is occasionally very painful to fail, after working on some problem for a year or two, there is still in the back of your mind the thrill of the chase.”⁸⁴

The choice of becoming a mathematician presents the young person with joy and fear. In this chapter we have sketched the conditions that support an individual’s ability to take enormous risks in one’s intellectual life. Parental response to early curiosity, inspired teaching, and lively interaction with peers and mentors all contribute to the willingness to live with long periods of searching for solutions. There is also the young mathematician’s extraordinary ability to concentrate, ignore diversions, and follow a sense of direction even in the face of failure. Young women face particularly serious challenges when entering a profession that can isolate them from their peers and family and in which gender bias has been prevalent through most of mathematical history. What makes such risk taking and concentration possible may be the powerful appeal of adding deep, lasting knowledge, knowledge that is fulfilling both aesthetically and intellectually.

Bibliography

Albers, Donald J., & Alexanderson, G. L. (1985). Mathematical people: Profiles and interviews. Boston: Birkhauser.

Albers, Don (2007). John Todd—Numerical mathematics pioneer. College Mathematics Journal 38(1), 5.

Bell, E. T. (1937). Men of mathematics. New York: Simon and Schuster.

Bloom, Benjamin S. (1985). Developing talent in young people. New York: Ballantine Books.

Brockman, John (Ed.) (2004). Curious minds: How a child becomes a scientist. New York: Pantheon Books.

Chang, Kenneth (2007). Journeys to the distant fields of prime. New York Times, March 13, 2007. Retrieved April 10, 2008 from http://www.nytimes.com/2007/03/13/science/13prof.html?_r=1&sq=The%20Mozart%20of%20MAth&st=nyt&oref=slogin&scp=1&pagewanted=print

Csikszentmihalyi, Mihaly, Rathunde, Kevin, & Whalen, Samuel (1996). Talented teenagers: The roots of success and failure. New York: Cambridge University Press.

Dyson, Freeman J. (2004). Member of the club. In John Brockman (Ed.). Curious minds: How a child becomes a scientist. New York: Pantheon Books.

Feldman, David H. (1986). Nature’s gambit. New York: Basic Books.

Gallian, Joseph A. (2004). A conversation with Melanie Wood. Math Horizons 12 (September), 13, 14, 31.

Gustin, William C. (1985) The development of exceptional research mathematicians. In Benjamin S. Bloom (Ed.). Developing talent in young people. New York: Ballantine Books, pp. 270–331.

Heims, Steve J. (1982). John Von Neumann and Norbert Wiener: From Mathematics to the technologies of life and death. Cambridge, Mass.: MIT Press.

Hersh, Reuben, & John-Steiner, Vera (1993). A visit to Hungarian mathematics. Mathematical Intelligencer 15(2) 13–26.

Howe, Michael J. A. (1990). The origins of exceptional abilities. Cambridge, Mass.: Basil Blackwell.

James, I. (2002). Remarkable mathematicians. Cambridge: Cambridge University Press.

John-Steiner, Vera (1997). Notebooks of the mind: Explorations of thinking. New York: Oxford University Press.

Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.

Levinson, N. (1966). Wiener’s life. Bulletin of the American Mathematical Society 72(1, II), 3, 24, 25.

MacTutor web site. Birkhoff.

Marx, George (1999). The Hungarian gymnasium. Europhysics News (Nov./Dec.) p. 130.

Massera, J. L. (1998). Recuerdos de mi vida academica y política (Memories of my academic and political life). Lecture delivered at the National Anthropology Museum of Mexico City, March 6, 1998, and published in Jose Luis Massera: The scientist and the man. Translated by Frank Wimberly. Montevideo, Uruguay: Faculty of Engineering.

Mordell, L. J. (1971). Reminiscences of an octogenarian mathematician, American Math, Monthly 78 952–961,

Morrow, Charlene, & Perl, Teri (Eds.) (1998). Notable women in mathematics: A biographical dictionary. Westport, Conn.: Greenwood Press.

Murray, Margaret (2000). Women becoming mathematicians. Cambridge, Mass.: Massachusetts Institute of Technology.

O’Connor, J.J., & Robertson, E.F. (2003). George Dantzig http://www-history.mcs/st-andrews.ac.uk/Mathematicians/Dantzig-George.htm (p.2) (article from MacTutor, School of Mathematics and Statistics, University of St. Andrews, Scotland, JOC/EFR, April 2003.

Olson, Steve (2004). Count down. Boston: Houghton Mifflin.

Paulson, Amanda (2004). Children of immigrants shine in math, science. Santa Fe New Mexican 813 1/04, p. A5.

Perl, Teri (1978). Biographies of women mathematicians and related activities. Menlo Park, Calif.: Addison-Wesley.

Radford, John (1990). Child prodigies and exceptional achievers. New York: Harvester Wheatsheaf.

Rathunde, Kevin, & Csikszentmihalyi, Mihaly (1993). Undivided interest and the growth of talent: A longitudinal study of adolescents. Journal of Youth and Adolescence 22(4), 385–405.

Reid, C. (1976). Courant in Göttingen and New York. New York, Springer-Verlag.

Reid, Constance (1996). Julia, a life in mathematics. Washington, D.C.: Mathematical Association of America.

Sacks, Oliver (2001). Uncle Tungsten: Memories of a chemical boyhood. New York: Alfred Knopf.

Singh, Simon (1998). Fermat’s Last Theorem. London: Fourth Estate.

Szanton, Andrew (1992). The recollections of Eugene P. Wigner as told to Andrew Szanton. New York: Plenum Press.

Tikhomirov, V. M. (1993). A. N. Kolmogorov. In Smilka Zdravkovska & Peter L. Duren (Eds.). Golden years of Moscow mathematics. Providence, R.I.: American Mathematical Society.

Tikhomirov, V. M. (2000). Moscow mathematics 1950–1975. In Jean-Paul Pisier (Ed.). Development of mathematics 1950–2000. Boston: Birkhäuser, pp. 1109–1110.

Ulam, S. (1976). Adventures of a mathematician. New York: Scribner.

Wiener, Norbert (1953). Ex-prodigy: My childhood and youth. New York: Simon and Schuster.

Wigner, Eugene (1992). The recollections of Eugene P. Wigner as told to Andrew Szanton. New York: Plenum Press.

Winner, Ellen (1996). Gifted children: Myths and realities. New York: Basic Books.

Wolpert, Stuart (2006). Terence Tao, “Mozart of math,” is UCLA’s first mathematician awarded the Fields Medal, often called the “Nobel Prize in Mathematics.” UCLA News, August 22, 2006. Retrieved April 10, 2008 from http://newsroom.ucla.edu/portal/ucla/Terence-Tao-Mozart-of-Math-7252

Yau, S. T. (Ed.). (1998). S. S. Chern: A great geometer of the twentieth century. Singapore: International Press.