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The Teaching of Mathematics:
Fierce or Friendly?

How is mathematical practice shaped by the social context in which it takes place? How are mathematicians affected by the beliefs, biases, and values of the subculture in which they work and teach?

In the previous chapter we wrote about the impact of age and gender on the ways in which mathematicians develop and sustain their lives. We emphasized the importance of balance and social connections as sources of support when researchers face societal stereotypes. Some may even find in mathematics an escape from social challenges, from “everyday life, with its painful harshness and wretched dreariness, and from the fetters of one’s shifting desire.”¹ But more often, one’s mathematical life is intimately entwined with the life and conditions of the larger society. The impact of a mathematician’s early socialization and its attending values and world view can be so pervasive that it influences the researcher’s lifelong behavior.

In this chapter we tell about two U.S. mathematicians whose lives and work were deeply embedded in their time and place. For most of America’s history, the lives of Black people and White people were sharply segregated, and yet, paradoxically they were intimately entwined. We have chosen two mathematicians whose character and practice embodied this paradox. Their backgrounds were as different as one could imagine.

Robert Lee Moore and Clarence Francis Stephens

Robert Lee Moore (1882–1974) was Texas-born. His father, Charles, although Connecticut-born, had made himself a southerner and served in the Civil War as a Confederate volunteer, surviving the battles of Shiloh, Vicksburg, and Chattanooga. On his mother’s side Moore was related to Jefferson Davis, the President of the Confederacy. In his later years “by a masterwork of genealogical detective work, he produced a chart of lineage that would embrace two American presidents, the president of the Confederacy and three European royal houses.”²

Clarence Francis Stephens was born in Macon, Georgia, in 1917, the son of Sam Stephens, a chef and railroad worker, and Jeannette Morehead Stephens. He was the fifth of six children. His mother died when he was 2 and his father when he was 8, so he and his brothers and sisters went to live with their grandmother. She died two years later, and then the children were separated and sent to live with different relatives. Stephens lived with his great aunt Sarah in Harrisburg, North Carolina. There were no high schools in Harrisburg, so when he was 13, his oldest sister Irene arranged for him to go to a boarding school in Irmo, South Carolina. She paid the first year’s tuition, but Stephens himself earned his subsequent tuition, working on the school farm and in the kitchen and cleaning classrooms. After a placement test he was put into the 8th grade alongside students who were over 20. He played football and baseball, held the lead role in his senior play, and was elected class president.³

He earned a B.S. in mathematics from a traditionally Black institution (TBI), Johnson C. Smith University in Charlotte, North Carolina, while supporting himself as a deliveryman for a local drugstore at 6 dollars a week. Then he went to the University of Michigan for a master’s degree, expecting to become a high school teacher. But Professor George Rainich encouraged him to continue for a Ph.D. At Michigan, at that time, “there were no teaching assistant positions for African Americans,” so he supported himself by waiting tables.⁴

In September 1940, before finishing his Ph.D. at Ann Arbor, Stephens began teaching at another traditionally Black institution, Prairie View A. & M. in Texas. After Pearl Harbor he joined the U.S. Navy as a teacher specialist. He finished his Ph.D. in the spring of 1943, returned to teaching at Prairie View A. & M., and stayed through August 1947. Then he applied for a position at Morgan State College in Baltimore, in order to be near the math research library at Johns Hopkins. “To his surprise and initial dismay, he was offered the position of Chair of the department, which would put him in charge of a department of mathematicians, all older than he, at the tender academic age of 30. His desire to live in Baltimore overcame his concerns, and he accepted [in September 1947.]”⁵

Like Johnson C. Smith and Prairie View, Morgan State was also a traditionally Black institution. James A. Donaldson, mathematics professor and dean at Howard University, has written: “After termination of the U.S. Civil War in 1865, free Black people and newly freed Black people, fortified by hope and quiet determination, struggled to prepare themselves in every way for full membership in society. Consequently this period, shortly before and after 1865, saw the founding of several educational institutions (Lincoln University in Pennsylvania, Wilberforce, Howard, Shaw, Johnson C. Smith and others which will be called traditionally Black institutions or TBI’s) with the goal of providing higher educational opportunities for newly freed Black people and other people of African descent.”⁶

Moore and Stephens eventually developed two unique, radically different styles of college-level mathematics education. The decentralized nature of the American educational system permits both Moore’s and Stephens’ traditions to survive. Moore’s method is more widely known than Stephens’ (which is generally known as the Potsdam model). Yet, in somewhat different circles than the Moore method, the Stephens method has also received a lot of recognition. The contrast between these two traditions reveals a great deal about the United States, its math pedagogy, its ideologies, and its racism. We will first describe Moore and his method and then return to the career of Clarence Stephens, first at Prairie View and Morgan State and then at the racially integrated college at Potsdam, New York.

Moore and His Method

In 1898 Moore became a student and protégé of George Bruce Halsted at the University of Texas. Halsted “could point with pardonable pride to the fact that the rolls of the College of New Jersey, at Princeton, bore not only the names of his brother and himself, but also those of his father, an uncle, his grandfather, a great-uncle and his great-grandfather.”⁷ Halsted was the first doctoral student of the great algebraist J. J. Sylvester at Johns Hopkins, in the first math doctoral program in the United States. Later, Halsted pioneered teaching and writing about non-Euclidean geometry. He even went to Hungary and found the grave of János Bolyai. The Hungarian authorities were surprised that this obscure grave site interested a professor from Texas.

As a professor in Texas, Halsted engaged in legendary fights with the university’s board of regents. His star pupil was young Robert Lee Moore. He mentored Moore through a bachelor’s degree and then sent him on to the University of Chicago. At Chicago Moore became friends with fellow student Oswald Veblen, who would become a powerful professor, first at Princeton University and then at the Institute for Advanced Study. As students, Veblen and Moore became interested in the axiomatic foundations of geometry, a popular research subject after David Hilbert published his famous Foundations of Geometry. After his studies at Chicago, Moore taught at Northwestern, at the University of Pennsylvania, and finally back at the University of Texas in Austin. “With his snowy white hair immaculately combed, his piercing blue eyes always seeking exciting new proofs to complex problems, and his well-muscled boxer’s physique clad in dark three-piece suits and old-fashioned, handmade laced-up black boots, he was a commanding presence on the campus of The University of Texas for 49 years.”⁸ (The “well-muscled boxer’s physique” is no mere metaphor. In his young days Moore did enjoy and excel at the sport of amateur boxing.)

At Pennsylvania Moore supervised the thesis of John R. Kline (1891–1955), who remained there and ultimately was department chairman from 1933 to 1954. Kline also was secretary of the American Mathematical Society (AMS) from 1936 to 1950—we will say more about him in a little while.

Moore early turned from geometry to what he called “point-set theory.” This was part of point-set topology, which was a very active field in the early 20th century. A central question was the question of “metrization.” A topological space is one where there are “neighborhoods.” A topological space is called a “metric space” if the neighborhoods are defined in terms of a “metric” or “distance.” For instance, an ordinary surface such as a sphere becomes a metric space if distance between points is measured along the “geodesic,” the shortest curve on the surface connecting them. But some topological spaces do not permit a metric or distance to be defined between points. What extra conditions on a topological space will permit a metric to be defined?⁹

Moore’s fame rests less on his own research than on his teaching method, and on the amazing number and accomplishments of his students. Through his 50 Ph.D. students he has 1678 doctoral descendents! Three of them followed him as president of the American Mathematical Society; three others became vice-president; and another served as secretary of the AMS. Moreover, five served as president of the Mathematical Association of America (MAA), and three, like Moore, became members of the National Academy of Sciences.

The essence of the Moore method is easy to describe. He recruited students beginning their first year so that he could control their mathematical education from the very start. They were not permitted to have any previous knowledge of topology. Also, they were forbidden to read any books or articles about it. Also, they were even forbidden to talk about it outside of class.

Figure 8-1. Robert Lee Moore. Courtesy of Dolph Briscoe Center for American History, The University of Texas at Austin. Identifier: di_05554. Title: R. L. Moore and Mike Profit. Date: 1970/01. Source: Halmos (Paul) Photograph Collection

At the first meeting of his graduate class in point-set theory, Moore would present his axioms for point sets and give the statement of theorem 1, without proof or hints. (Perhaps he might also state theorems 2 and 3.) That’s it! Nothing else would happen, in that class or the next, until one of the students claimed he or she could prove theorem 1. The student then presented the proof at the blackboard. The class and the professor reacted to the proof. Eventually, theorem 1 was proved, and the class was ready for some student to propose a proof of theorem 2.

This method is reminiscent of a well-known old method of teaching swimming called “sink or swim.” Moore persisted in this teaching method at Austin until 1969, when, at age 87, he was finally, with great difficulty, forced to retire.

What happened in class when nobody was ready to offer a proof? Moore did not take the opportunity to lecture on some mathematical topic. Not at all. Instead, he simply chatted about any casual matter that happened to come up. In his chit-chat he sometimes expressed low opinions about women, about Black people, and about Jews. When his student Gayle Ball told him that her husband, Joe, was planning to go to a northern university after graduation, Moore warned Ms. Ball that she might “have to, perhaps, get on a bus and sit down next to a person of color. He thought this would be very offensive. . . . He said, ‘What would you do, Mrs. Ball, if this happened to you?’ She answered, “I would say: ‘How do you do?’ ” Moore’s reaction is not recorded.¹⁰

Another of his students, Mary Ellen Estill, later known to the world as Mary Ellen Rudin, would direct 16 doctoral theses and become a vice-president of the American Mathematical Society. In her interview for the book More Mathematical People, she talked frankly about her mentor Robert Lee Moore.

MP. So you had a course from Moore every semester?

Rudin. Every single semester during my entire career at the University of Texas. I’m a mathematician because Moore caught me and demanded that I become a mathematician. He schooled me and pushed me at just the right rate. He always looked for people who had not been influenced by other mathematical experiences, and he caught me before I had been subjected to influence of any kind. I was pure, unadulterated. He almost never got anybody like that. I’m a child of Moore. I was always conscious of being maneuvered by him. I hated being maneuvered . . . he maneuvered you in order to build your ego. He built your confidence that you could do anything. I have that total confidence to this day . . . having failed 5,000 times doesn’t seem to make me any less confident. . . . We were a fantastic class. [The class of ’45 included, besides Mary Ellen Rudin, R H Bing, R. D. Anderson, Gail Young, and Ed Moise!] Each of us could eat the others up. Moore did this to you. He somehow built up your ego and your competitiveness. Actually in our group there was another, a sixth, whom we killed off right away. He was a very smart guy—I think he went into computer science eventually—but he wasn’t strong enough to compete with the rest of us. Moore always began with him and then let one of us show him how to solve the problem correctly. And boy, did this work badly for him! It builds your ego to be able to do a problem when someone else can’t but it destroys that person’s ego. I never liked that feature of Moore’s classes. Yet I participated in it. Very often in the undergraduate class, I mean, I was the killer. He used me that way, and I was conscious of being used that way. . . . He viewed his two earlier women students as failures, and he didn’t hesitate to tell me about them in great detail, so I would realize he didn’t want to have another failure as a woman. . . . I only knew the mathematics that Moore fed me. At the time I wrote my thesis I had never in my life seen a single mathematical paper! I was pure and unadulterated. The mathematical language that he used was his own. I didn’t know how mathematical words were used at all.

Figure 8-2. Mary Ellen (Estill) Rudin. Source: More Mathematical People: Contemporary Conversations. Eds. Donald J. Albers, Gerald L. Alexanderson, and Constance Reid.

MP. How did you feel about your mathematical education later?

Rudin. I really resented it, I admit. I felt cheated because, although I had a Ph.D., I had never really been to graduate school. I hadn’t learned any of the things that people ordinarily learn when they go to graduate school . . . I didn’t even know what an analytic function was.

MP. What feelings toward Moore, as a person, did you develop over time?

Rudin. Oh, I had a very warm, enthusiastic feeling for him, although I also had very negative feelings. I was conscious of both levels. I was aware that he was bigoted—he was—but I also was aware that he played the role of a bigot sometimes in order to get our reactions, maybe even to keep us from being bigots. I’m never sure to what extent that was true. Moise, for instance was a Jew. Moore always claimed that Jews were inferior. I was a woman. He always pointed out that his women students were inferior. Moise and I both loved him dearly, and we knew that he supported us fantastically and did not think that we were inferior—in fact, he thought that we were super special. On the other hand, he wanted us to be very confident of ourselves in what he undoubtedly viewed as a somewhat disadvantaged position. Now then, did he play the role of a bigot to elicit a response? I have no idea.¹¹

Moise managed to survive the Moore method, including Moore’s Jew baiting. He became a Harvard professor, vice-president of the American Mathematical Society, and president of the Mathematical Association of America.

Although the Moore method is usually thought of as his special way of teaching graduate topology, he also taught calculus and trigonometry. Mary Ellen Rudin testified that she saw no distinction between his way of teaching, whether in trigonometry and calculus or in his most advanced graduate course. She said that in his class you didn’t end up knowing much calculus because you had no training in the standard problems. So the course was not useful to a physics or engineering student. And as for the poorer students, they “dreaded and hated class for the simple reason that it was a painful experience for them. Because they had difficulty presenting things and because the things that they conjectured were often wrong, they were used as an example of how you could be wrong. It wasn’t necessarily a pleasant experience for them. In fact, I think that even for the good students, it wasn’t necessarily a pleasant experience altogether. You didn’t enjoy seeing people fail.”¹²

Another one of his students, John Green, received two Ph.D. degrees, the first with Moore and the second from Texas A. & M. in statistics. He subsequently became a principal statistician at a DuPont laboratory. Green recalls that early in the course Moore announced that “he wanted us to think about his class all day, every day, to go to bed thinking about it, to wake up in the night thinking about it, to get up the next morning thinking about it, to think about it walking to class, to think about it while we were eating. If we weren’t prepared to do that he didn’t want us in his class. It was also quickly evident that he meant exactly what he said.”¹³ Green reports that Moore would never tell the class how to prove a theorem or construct a counterexample. If the students didn’t do it, it wouldn’t be done. Green found that the self-reliance learned in Moore’s classes was invaluable in the chemical industry. “One benefit of working with Dr. Moore,” he writes, “was learning how to work with the intimidation factor. First, there is the man himself. He was a very imposing figure, he dominated his environment. We were all in awe of him on many levels. And having worked under him, no one else can intimidate me.”

Moore’s bitter fights with his colleagues at Texas take up many pages of Parker’s biography. There were “fisticuffs” against Associate Professor Edwin Beckenbach in 1944. His most prominent mathematical colleague was Harry Schultz Vandiver, a determined and persistent attacker of Fermat’s Last Theorem. (He proved it for all primes less than 2000.) He won the Cole Prize of the American Mathematical Society for his research in number theory and, like Moore, was elected to the National Academy of Sciences. Indeed, he continued producing massive amounts of research into his eighties, unlike Moore, who had long given up research in favor of producing students to become researchers. The origin of the enmity between Moore and Vandiver is unclear, but “They fell apart sometime in the late 1930s and soon they had reached the point of no contact. Their joint efforts to avoid ever having to converse with each other, let alone being in the same room if at all possible, became the talk of the university.”¹⁴

There was even an “incident” when Moore threatened Vandiver’s son Frank with a handgun. Moore owned a revolver and had been involved with gun club activities.

Changing Times

The possibility of Moore’s having to admit a Black student to his course did not arise until the late 1950s, because up to then Black Texans seeking a university education were admitted only to traditionally Black state colleges.

As a result of the Supreme Court’s Brown vs. Board of Education of Topeka decision in 1954, and another specifically against the University of Texas in 1956, the regents of the university declared in June 1956 that thereafter they would strive toward total desegregation. However, Robert Lee Moore never ceased defying this policy of the regents and decision of the U.S. Supreme Court.

In spite of Moore’s intransigence, African-American students were admitted to the University of Texas. Walker Hunt, A. N. Stewart, and L. L. Clarkson became the first African-American students to earn Ph.D.s in mathematics from the University of Texas. Hunt reports: “I also wanted to take Robert Lee Moore’s famous Foundations of Point Set Topology. However, that was not to be. The reason: I was Black! His words were, ‘you are welcome to take my course but you start with a C and can only go down from there.”¹⁵

African-American Pioneers in Mathematics

Several of the Black mathematicians who faced institutional and personal racism in their early careers interacted with Moore or his students. The first female African-American student at the University of Texas to earn a Ph.D. in mathematics was Vivienne M. Mayes. She also was denied access to a course she wished to take with Robert Lee Moore. Subsequently, Dr. Mayes became a professor at Baylor University, in Houston, Texas. In 1971 the Baylor student congress elected her outstanding faculty member of the year. (See chapter 7.)

Raymond Johnson, now a professor at the University of Maryland, earned a B.S. at the University of Texas and a Ph.D. at Rice, in Houston, Texas (the first African-American to do so). He writes, “The image of R. L. Moore in my eyes is of a mathematician who went to a topology lecture given by a student of R H Bing. The speaker was to be one we refer to as a mathematical grandson of Moore. When Moore discovered that the student was black, he walked out of the lecture.”¹⁶

Nevertheless, Moore had at least three African-American mathematical grandsons! Interestingly, one of these, Beauregard Stubblefield, had been a student of Clarence Stephens at a TBI, Prairie View, where he earned a bachelor’s degree in mathematics in 1940. He continued at Prairie View and received a master’s in 1944.¹⁷

Figure 8-3. Vivienne Malone-Mayes. Courtesy of Mathematicians of the African Diaspora website.

Stephens writes:

I discovered Stubblefield in my college algebra class during my first semester as a college teacher. At the beginning of the course, I realized that he knew the mathematics I was to teach in the course. I told him that I was giving him a grade of “A” for the course and I would give him a special mathematics assignment, so that he could benefit from his studies. However, I requested that he attend my college algebra class. At the time, students were assigned problems to work at the blackboard. I told him that I wanted him to help me check the solutions of students at the blackboard. Many students in the class had finished the same high school that Stubblefield had finished. I wanted to demonstrate that a student who had finished the same high school that many of them had could achieve well in mathematics.¹⁸

This was at a time before the University of Texas accepted Black students. So Stephens recommended the University of Michigan for Stubblefield’s doctoral study. Three of Moore’s famous White students had become professors at Michigan. Stubblefield, once enrolled at the University of Michigan, took Moore method courses from Ray Wilder and Gail Young and had many mathematical talks with Ed Moise. His thesis was supervised by Gail Young. Obviously, while Moore’s students may have, at least for a time, followed Moore’s style of mathematical research and teaching, they were by no means infected with his racism! Stubblefield never met Moore himself, but he heard a story about him: “One time a Black got into his class and sat for a while. R. L. Moore said “I am not going to teach any of this course until a certain person leaves my class.”¹⁹ “Stubblefield said he felt entirely comfortable with the Moore method and carried much of it into his career in mathematics, principally at Appalachian State University, and, after his retirement, as a research mathematician concentrating on number theory.”²⁰

Obviously, when Stubblefield and other direct or secondary descendents of Moore used the Moore method in their own teaching, it was not the unadulterated virulent original version. It was modified and adapted to circumstances, and its sheer ferocity, one might even say brutality, was toned down and humanized. Certainly Stubblefield did not hide from his students all mathematical knowledge except what they had created themselves under his control. When speaking of the Moore method used by anyone but Moore, one should think of it as an adapted Moore method.

Professor Stubblefield participates in annual meetings in Austin of an organization called the Education Advancement Foundation (EAF). This group established a Legacy of R. L. Moore Project, whose purpose is to “help advance studies of the mathematician Robert Lee Moore (1882–1972), thereby promoting the study of more effective methods of learning and teaching at all educational levels and in all subjects.”²¹ Dr. Nell Kroeger, who was Moore’s last Ph.D. student, informed R.H. that “The EAF is deeply concerned with getting women and minorities more involved in advanced mathematics.”²²

In addition to Parker’s full-length biography of Moore, there is plentiful documentation of his teaching by his students. A movie of Moore in action is available from the Mathematical Association of America. On the web you will also find an entry about him at a site called Mathematicians of the African Diaspora.

Moore had two more Black mathematical “grandsons” whom he refused to acknowledge. They were Dudley Weldon Woodard (1881–1965) and William Schieffelin Claytor (1908–1967). Both were point-set topologists, both students of John Kline, who had been Moore’s first Ph.D. student. What burden did these men bear, being rejected and denied by their own mathematical grandfather? They were honored in 1999 at an exhibition, “Pioneering African-American Mathematicians,” at the University of Pennsylvania.

The older of these grandsons, Woodard, earned a B.A. at Wilberforce College and a B.S. and an M.S. in mathematics at the University of Chicago. He taught at Tuskegee Institute and at Wilberforce and joined the mathematics faculty at Howard University in 1920. He was already a professor at Howard University, and also a dean, when he decided in 1927 to go beyond his master’s degree and pursue a Ph.D. in mathematics. For this purpose he went to the University of Pennsylvania, in Philadelphia, which had admitted Black students since 1879, and became a student of Kline. Kline not only was a student of Moore but also continued to be a close friend and colleague of his after Moore went to Texas. After Woodard completed his doctorate in 1928, he continued teaching at Howard, and established a master’s degree program in mathematics there. In 1929 he was happy to find an exceptionally promising student at Howard, William Schieffelin Claytor. He advised Claytor to go to Pennsylvania for doctoral work under John Kline, Woodard’s old professor.

Claytor was a brilliant graduate student. He won the most prestigious award offered by Penn at that time, a Harrison Fellowship in Mathematics. His dissertation achieved a significant advance in the theory of “Peano continua”—a branch of point-set topology in which Kline was an expert.²³ Yet the only university job Claytor could get in 1936 was at a TBI, West Virginia State College. In 1937, on Kline’s recommendation, Claytor obtained a Rosenwald Fellowship and spent a year working with Wilder and a group of topologists at Michigan. Forty-three years later, in 1980, the National Association of Mathematicians inaugurated the William W. S. Claytor Lecture Series. At that time Raymond Wilder wrote a letter about Claytor:

Toward the end of his stay at Michigan the question of where he could get a “job” came up. We topologists concluded he should join the University of Michigan faculty. Today, I’m confident there would be no hesitancy about this on the part of the Michigan administration, but that was about thirty years ago . . . it was to no avail; the administration was simply afraid (I am sure this was the case more than racial prejudice.) I finally wrote to Oswald Veblen, head of the School of Mathematics at the Institute for Advanced Study, who quickly replied that he’d find a place for Bill at the Institute. However, when I told Bill this, he shook his head and replied, “There’s never been a Black at Princeton, and I’m not going to be a guinea pig.” I’ve always felt this was the turning point in Bill’s life, and a great mistake on his part. I knew how he felt and argued with him, but he was adamant. I am sure that if he had accepted, he would have found lots of friends, at the Institute, and that his future would have been quite different.

Samuel Eilenberg wrote, “I very deeply felt the tragedy of the situation.”²⁴ Gail Young wrote, “The two papers on which his reputation rests are brilliant. That he could not get a job in any research-oriented department was tragic.”²⁵

The web site, Mathematicians of the African Diaspora, reports that “Claytor did make presentations at meetings of the American Mathematical Society, but he was never allowed to stay in the hotel where the meetings were held. Instead, a home of local ‘cullud’ persons was found for his stay.”

Claytor gave up mathematical research. According to the UPENN web site, “. . . his unfulfilled promise was a great disappointment for John R. Kline and his generation of colleagues at Penn.” It’s an irony of American mathematics that Princeton’s loss became Howard’s gain. Claytor joined the Howard University faculty in 1947, a year after Woodard’s retirement, and remained there until taking early retirement in 1965. So Robert Lee Moore, by way of his mathematical son, John Kline of Pennsylvania, generated two of Howard’s leading mathematics professors.

Wilder’s efforts on behalf of Claytor were commendable. He probably didn’t appreciate the difficulties Claytor would have confronted in 1937, just finding a place to live in Princeton. Although Princeton is in “the North,” Princeton College then was a preferred place for wealthy southerners to send their sons. In Princeton in 1937, Jews and Italians were also unwelcome in the “better” sections of town. There were Black people in Princeton—domestic servants in the homes of upper-class whites. Of course, they lived in “their own” part of town. Where would a Black professor have fit in to the Princeton of 1937?

The war against Nazism ultimately led to the United States’ official renunciation of racism. It was during that war that the famous probabilist Joseph Doob came from Urbana, Illinois, to visit the Institute for Advanced Study in Princeton. With him was his brilliant doctoral student David Blackwell. Princeton University as a matter of course made its libraries and seminars open to members of the Institute for Advanced Study, but in Blackwell’s case they said “No.” Blackwell later wrote, “I think it was the custom that members of the Institute would be appointed honorary members of the faculty at Princeton. When I was being considered for membership in the Institute, Princeton University objected to appointing a Black man as an honorary member of the faculty. As I understand the story, the Director of the Institute for Advanced Study just insisted and threatened, I don’t know what, so Princeton withdrew its objections. Apparently there was quite a fuss over this, but I didn’t hear a word about it.”²⁶ Blackwell later was for many years chairman at Howard, and then joined the statistics department at Berkeley. He became chairman there and was elected to the National Academy of Sciences.

Figure 8-4 (above left). David Blackwell, Berkeley statistician and probabilist. Courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach

Figure 8-5 (above right). William Schieffelin Claytor. Courtesy of Mathematicians of the African Diaspora website

Figure 8-6 (left). Clarence Stephens and Harriette Stephens. Courtesy of Mathematicians of the African Diaspora website.

Clarence Stephens and the Potsdam Model

Clarence Stephens was mentioned in Beauregard Stubblefield’s life story. Stephens was at Prairie View, Texas, before going on to the chair at Morgan State in Baltimore. He published two papers on nonlinear difference equations and thereafter devoted himself to teaching. His teaching philosophy was the most complete contrast to Moore’s imaginable.

Stephens writes:

When I tutored my fellow schoolmates in mathematics during my high school days, I learned that many students can learn mathematics if the learning environment is favorable. Early in my career as a college teacher, I came to the conclusion that any college student who wanted to learn college mathematics could do so if the learning environment was favorable. So when I was given the opportunity of leadership as the Chair of a mathematics department, I was determined to prove my conjecture. My main difficulty in proving my conjecture was that students, faculty and administration did not believe my conjecture was true. However, in my effort to prove my conjecture, I received support from each group. The students I was trying to help gave me the best support. The results we obtained at Morgan State College and at SUNY at Potsdam proved, at least to me, that my conjecture was true.²⁷

At Morgan State he found that nearly all students were required to take a 6-semester-hour general mathematics course reviewing elementary and high school mathematics. This policy was based on the results of placement exams. As a consequence, very few students, even mathematics majors, could get to true college-level mathematics courses until their sophomore year. In contrast, Stephens believed that one of the best ways to prepare students for graduate study in mathematics was to place them in calculus courses as early as possible. Stephens expected that a student previously labeled “underprepared” could achieve well in such courses if the atmosphere in the department and in such courses was nurturing, and if there were role models to show that success was possible. Stephens instituted an undergraduate honors mathematics program that exposed undergraduates to first-year graduate mathematics. It drew a large percentage of Morgan State’s best students to major in mathematics. Morgan State had had no student go on to obtain a Ph.D. in mathematics during the 90 years of its history before Stephens’s arrival, but at least nine students who passed through its mathematics program during Stephens’s years on its faculty eventually obtained that degree. One of them, Vassily Cateforis, was born in Greece. Cateforis got a Ph.D. in algebra at the University of Wisconsin and ultimately became Stephens’ successor as chairman at Potsdam.

After Stephens retired from Morgan State, he taught from 1962 to 1969 at the State University of New York at Geneseo. Then he became chairman of the department of mathematics at the State University of New York at Potsdam, a small town in far northern New York State.

In March 1987, an article appeared in the American Mathematical Monthly with the title “A Modern Fairy Tale.” It described an amazingly successful undergraduate mathematics program at a little known college, Potsdam College of the State University of New York. The author, John Poland, was in the department of mathematics and statistics of Carleton University in Ottawa, Canada. He wrote:

Tucked away in a rural corner of North America lies a phenomenally successful undergraduate mathematics program. Picture a typical, publicly funded Arts and Science undergraduate institution of about 5,000 students, with separate departments of Mathematics and Computer Science. While the total number of undergraduates has remained relatively fixed over the past 15 years, the number of mathematics majors has doubled and doubled again and again to over 400 now in third and fourth year. They don’t offer a special curriculum. It is just a standard, traditional pure mathematics department.

More than half the freshman class elect calculus, because of the reputation of the mathematics department carried back to local high schools. And of the less than 1000 Bachelor degrees awarded, almost 20% are in mathematics. In case you are unaware, 1% of Bachelor degrees granted in North America are in mathematics. These students graduate with a confidence in their ability that convinces prospective employers to hire them, at I.B.M., General Dynamics, Bell Laboratories and so on.

Do they just lower their standards? Mathematics teachers in the university across the street [Clarkson Institute of Technology] say “no.” They see no significant difference between their performance and that of their own students.

The students say the faculty members really care about them, care that each one can develop to the maximum possible level. It is simply the transforming power of love, love through encouragement, caring and the fostering of a supportive environment. By the time they enter the senior year, many can read and learn from mathematics texts and articles on their own. . . . They graduate more women in mathematics than men. They redress a lack of confidence many women feel about mathematics. In the past ten years, almost every year the top graduating student at this institution, across all programs, has been a woman in mathematics.²⁸

It is rare in any mathematical publication to find words like “love” and “caring” used in this way. A welcome contrast, to the more widespread discussion of “math avoidance” and “math phobia”! Poland further writes:

What must a mathematics department do to attain this success? The faculty must love to teach, with all this means about communication, caring for students and for their development. They would teach at a pace which allows students time to struggle with the problems and resolve them, rather than primarily to cover material. . . . They would recognize that students need time to build the skill, understanding and self-confidence to handle most advanced mathematics. The faculty would encourage and reward the success of the students, bringing all or most of them to a high level of achievement (and high grades), rather than using the grade to filter the brightest and quickest students into further mathematics studies. The recipe for success at Potsdam is very simple: instill self- confidence and a sense of achievement through an open, caring environment.²⁹

This atmosphere and attitude at Potsdam are largely the creation of Clarence Stephens.

Potsdam dates back to the St. Lawrence Academy, which was founded in 1816. Until 1962, its purpose was to educate students for a career in elementary teaching. It had about seven math teachers who taught mostly elementary math courses; the most advanced was introductory calculus. In 1969 the college was visited by Clarence Stephens. His talk on math and math teaching deeply impressed the math teachers at Potsdam College. They were so overwhelmed by his vision of creating a humanistic environment for learning undergraduate mathematics that they approached the administration to give him the best possible offer because “he is worth more than we can pay him.’ ”³⁰

Stephens told Dilip Datta, a visitor to Potsdam who wrote a book about the Potsdam model:

My primary goal as Chair was to establish the most favorable conditions I could for students to learn and teachers to teach. . . . A team of mathematics faculty members with me as a member was formed to teach students in the early (freshman and sophomore years for undergraduates—first year for graduate students) study of mathematics, “How to Read Mathematics Literature with Understanding and to Become Independent Learners.” A person selected for the team was a person who, in my opinion, had a warm relation with beginning students, strong loyalty to the department and the college. . . . I had confidence that any caring mathematics faculty member could effectively teach the students developed by the team. Also, the students who were developed by the team would help us teach other students as tutors. . . . The indicated method for developing the mathematics potential of students was as effective at SUNY Potsdam as it had been at Morgan State College.³¹

The motto of the math department at Potsdam is “Students come first.” Dilip Datta writes that Stephens “insisted that every faculty office must be furnished with comfortable cushioned chairs for the students. . . . He would constantly make the teachers aware that students have other courses to study and that they need to relax over the weekends. . . . The first thing he would ask a teacher would be something like, ‘Are your students enjoying math?’ ”³²

Some of the things Stephens used to say are:

“Believe in your students, everyone can do mathematics.”

“Know your students well—their names, what they know, their hopes and fears.”

“High standards do not mean having unrealistic expectations so students feel that they have failed.”

“Go fast slowly.”³³

Datta further writes, “The most important component of the humane environment of teaching and learning mathematics is the team of teachers. Without their dedicated efforts, nothing would have been possible.”³⁴

Stephens has received honorary doctorates of science from Lincoln University in Pennsylvania and Johnson C. Smith University, his alma mater, and doctorates of humane letters from Chicago State University and the State University of New York (SUNY). He has been honored by the governors of both Maryland and New York. He is included in the National Museum of American History of the Smithsonian Institution. The Seaway section of the Mathematical Association of America renamed its annual Distinguished Teaching Award the Clarence Stephens Award. That means that his name is announced every year at the section’s award ceremony.

In 2003 the Mathematical Association of America gave him its “Gung-Hu award” (officially called the Gung and Hu Award) for outstanding service to mathematics. The citation quotes Stephens’ description of his teaching methods, and comments, “Though SUNY Potsdam is a relatively small regional state college with a total enrollment of just over 4,000 students during Stephens’ time there, in 1985 the college graduated 184 mathematics majors, the third largest number of any institution in the U.S. that year (exceeded only by two University of California campuses). This represented about a quarter of the degrees given by SUNY Potsdam that year, and over 40% of the institution’s honor students were mathematics majors.”

More important than a teacher’s personal awards is the influence of his teaching methods. Robert Lee Moore certainly had a considerable influence in the United States. From 1950 onward, five of Moore’s doctoral students, and a sixth who studied with him, served as presidents of the Mathematical Association of America. The fact that for decades the MAA was led by a student of R. L. Moore must have significantly affected college education. Moore even affected elementary and secondary education, through his student Ed Moise and through Ed Begle, a student of Ray Wilder who was thus a second-generation Moore student. Moise and Begle were important leaders of the School Mathematics Study Group (SMSG), which for a while managed to push sets and axioms into elementary school math teaching. At the university level, the Moore method, or rather the modified Moore method, survives as a teaching method. It is unusual and nonstandard but respected as a proven way to educate future research mathematicians.

I (R.H.) visited Potsdam in 2002 and found the Stephens spirit and attitude prevailing under Cateforis’ chairmanship. Stephens had retired in 1987, the year John Poland’s article, “A Modern Fairy Tale,” appeared in the American Mathematical Monthly. Stephens’ efforts did not end with retirement. In response to a question, he wrote:

I received invitations to visit colleges and universities in Canada and throughout the United States in order to discuss the Potsdam mathematics program. I visited colleges and universities in the East, Midwest, West, and South. I visited almost all of the California State Universities and received invitations from a few that I did not have the time to visit. I made several return visits. After 4 years of making these visits, I stopped accepting invitations. In California and Georgia, I was offered faculty positions in order to help establish similar programs. From my experiences at Morgan and Potsdam I knew that it was difficult to establish favorable academic environments, so that any college student who desired to do so could learn and enjoy learning mathematics. To establish a successful program depended on creative thinking, time, and place. Hence, I did not accept any faculty position offered to me.³⁵

And what about the Stephens method, the Potsdam model? When I visited Potsdam, I asked the faculty whether other math departments emulated their method. They answered, “People come here and watch. Then they say, ‘This is great, but we could never do it on our campus.’ ”

However, when I asked Clarence Stephens whether his model had been adopted elsewhere, he named several prestigious schools. Cal Tech—California Institute of Technology! And Princeton! And Dartmouth! And two other selective private schools of somewhat lesser fame: Spelman College in Atlanta (a traditionally Black women’s institution) and Harvey Mudd College in Claremont, California (a high-prestige engineering-oriented school, one of the well-known Claremont Colleges). But it has not been adopted by one state university or community college, where the great majority of American college students are found.

This is disappointing, but understandable. At Potsdam, Stephens had to persist for years in order to win his whole faculty over to his point of view: “Under favorable conditions, any college student interested in learning mathematics can be successful!”

And what are favorable conditions?

“Students come first.”

“Give a student all the time he or she needs, to absorb the material.”

“Have complete confidence that every student can be successful.”

At Potsdam, the faculty is not under constant strong pressure to publish as much as possible. And mathematics is there for any student who is interested in learning it—not only future scientists and engineers. Many of the math majors at Potsdam become mathematics teachers, but others go to work in business or other nonacademic careers. Yet many of them find that their mathematics education is useful in their life.

But the prevalent view in U.S. college education is different. “You major in the subject that will ultimately land you a good job.” If you take calculus but you aren’t a math or science major, it’s in order to get into business school (or medical school or architecture school).

It would be quite a project to convert an ordinary state university in the United States to the Stephens philosophy. Faculty are to be rewarded for putting students first? And any student who is interested in mathematics will succeed? And each student of mathematics feels loved?

However, here is one consideration that might move faculty and administrators to pay attention to the Potsdam model. Many U.S. universities are struggling to increase the success rate of minority students in mathematics. (In chapter 10 we report on Uri Treisman’s successes at Berkeley and Austin.) At any school where the Potsdam model is implemented, one could confidently expect great improvements in the success rates of all students—both minority and majority.

Conclusions

The stories of Clarence Stephens and Robert Lee Moore embody two different, opposed strains in American education: the egalitarian versus the elitist; the cooperative versus the competitive; the heritage of the Declaration of Independence versus the heritage of the Confederate States of America. Their stories reveal that while mathematical life may at times appear to be an ivory tower where we escape from social conflicts, it may at times be a maelstrom where social currents clash. Moore’s barring African-American students from his class was part of a long legacy of racism in the United States, especially in the “slave states” that sought to secede from the Union.

Stephens, on the other hand, grew up in African-American communities that took pride in traditional Black institutions. The beliefs and values of his subculture included mutual support. For him, teaching was more than the transmission of specialized knowledge; it meant full acceptance of each individual in his or her efforts to create a better life.

The southern segregationist subculture to which Moore owed allegiance is now considered defeated and disreputable. This was demonstrated by the election of an African-American President of the United States. But full integration of previously excluded groups is still to be achieved. It requires more than mere legal equality, it demands transformative teaching methods.

Bibliography

Albers, D. J. (1990). More mathematical people. Boston: Harcourt Brace Jovanovich.

Albers, D. J., & Alexanderson, G. L. (Eds.) (1985). Mathematical people: Profiles and interviews. Boston: Birkhäuser.

Datta, D. K. (1993). Math education at its best: The Potsdam model. Framingham, Mass.: Center for Teaching/Learning of Mathematics.

Donaldson, J. A. (1989). Black Americans in mathematics. In Peter Duren (Ed.): A century of mathematics in America, part III. Providence, R.I.: American Mathematical Society.

Einstein, A. Quoted in Holton, G. (1973). Thematic origins of scientific thought: Kepler to Einstein. Cambridge, Mass.: Harvard University Press.

Lewis, A. C. (1976). George Bruce Halsted and the development of American mathematics. In J. D. Tarwater (Ed.). Men and institutions in American mathematics. Lubbock, Tex.: Texas Technical University Press.

Mathematicians of the African Diaspora web site. Retrieved April 5, 2007, from http://www.math.buffalo.edu/mad/

Megginson, R. E. (2003). Yueh-Gin Gung and Dr. Charles Y. Hu award to Clarence F. Stephens for distinguished service to mathematics. American Mathematical Monthly 110 (3), 177–180.

Parker, J. (2004). R. L. Moore: Mathematician and teacher. Washington, D.C.: Mathematical Association of America.

Poland, J. (1987). A modern fairy tale? American Mathematical Monthly 94 (3), 291–295.

Stephens, R. (2006). Personal communication.

Stephens, R., web site. Retrieved March 15, 2007, from http://www.mathsci.appstate.edu/~sjg/womenandmonoritiesinmath/
student/stephens/sephen

Wilder, R. L. (1982). The mathematical work of R. L. Moore: Its background, nature and influence. Archive for the History of Exact Sciences 26, 73–97.