Mathematics, as a discipline, seems to have garnered far more than its fair share of stereotypes (however untrue): It's difficult (especially for women and assorted minorities); it's dry and boring; it's the province of socially deficient nerds; students only take it because it's a requirement. (Well, O.K., that last one might be less of a stereotype and more of a ... preponderance of anecdotal evidence.) But why does mathematics seem to inspire such strong -- frequently negative -- opinions and emotions, and what realities can be teased out from among all the myths and knee-jerk reactions?
A new book, Loving and Hating Mathematics: Challenging the Myths of Mathematical Life (Princeton University Press) takes a look at some of the most common (mis)conceptions about mathematics and mathematicians, addressing their origins and assessing their truth value in a somewhat unexpected fashion. Rather than amassing data on PISA and SAT scores, analyzing the race and gender breakdowns of degrees awarded or tenure and promotion rates, or perhaps administering Enneagram tests to math majors, authors Reuben Hersh and Vera John-Steiner focus on the lives and experiences of mathematicians, past and present. At times, the book reads almost like an encyclopedia of prominent scholars, grouped by category: collaborating mathematicians, women mathematicians, African American mathematicians, crazy mathematicians....
While the scientific validity of the authors' conclusions might be questionable, the insights and anecdotes are strangely fascinating. Austrian topologist Leopold Vietoris "completed his thesis while a prisoner of war" after being captured by the Italians during World War I; Hungarian mathematician Pál Turán, forced to serve in a Fascist labor camp during World War II, was recognized by a guard who had read his papers, and who arranged for Turán to be assigned to the least arduous work available -- giving the mathematician the time and mental energy to solve difficult problems in his head while he worked.
Hersh, professor emeritus of mathematics, and John-Steiner, professor emerita of linguistics and education, both at the University of New Mexico, also tackle the issue of mathematics education, offering their thoughts on how to get more American students loving -- and, with luck, a lot fewer of them hating -- mathematics.
Inside Higher Ed interviewed both authors by e-mail to discuss the themes and ideas of the book. (Skip to Question 3 if you just like anecdotes.)
Q: What inspired the two of you – who come from very different disciplines – to write this book and to collaborate on it?
John-Steiner: We were both experiencing a transition from full-time teaching to a greater diversity of projects and decided to pursue a collaborative endeavor. In choosing a joint project, we were aware of the strong interest in mathematics and mathematicians in the public media. While there are films and plays on the subject, there are few social scientists who have examined the complexities of mathematical lives. We thought that bringing our complementary perspectives to the task would provide a fresh approach.
Hersh: Vera said, "Let's do something together."
Q: The book's style is rather unusual – you delve into a number of broad questions about mathematics and mathematicians largely through anecdotal discussions of the lives of various prominent mathematicians. What is the idea behind addressing the book's themes this way?
John-Steiner: Our primary emphasis is anecdotal because it is through the life stories of mathematicians that we were able to discover neglected themes such as their partnerships, their communities, the joys and challenges of doing mathematics. There are very few studies of mathematicians, and, when relevant, we included them. In recent years there has been a shift in the social sciences, from purely quantitative, short-term studies to what has been called narrative psychology. We were also influenced by the great success of Mathematical People and More Mathematical People, which clearly address interest in the people behind the formulas. In addition, as we were concerned about issues of gender, ethnicity, and age, we were eager to show changes in mathematical culture -- which was characterized, over a hundred years ago, as “a young man’s game.”
Hersh: The first goal in writing a book is to make it INTERESTING. Telling stories is the best way to get someone's attention or interest.
Q: The book contains a vast number of anecdotes about famous mathematicians. Could each of you share one or two of your favorites?
John-Steiner: I particularly like the stories about Julia Robinson, who worked with little recognition in Berkeley, California until the day she was admitted as a member to the National Academy of Science. (She made a major advance in solving Hilbert's Tenth Problem, which was a major achievement.) When a member of the academy called to notify her of this honor, the departmental secretary said, “Oh, you mean Professor Robinson’s wife?” It wasn’t until her national recognition that the University of California offered this great woman mathematician a permanent position on its faculty.
Hersh: Israel Moiseyevich Gelfand was one of the most creative and influential mathematicians of the 20th century. His family in the Ukraine were too poor to buy him even a single book. But when he was 15 he had appendicitis, and he refused to go to the hospital unless they bought him a math book. That was how he discovered that algebra and geometry are really the same subject!
Louis Mordell in Philadelphia was also a child of poverty. He earned money by tutoring his fellow students, and was able to buy cheap second-hand math books to study. He conceived what he later called the "thoroughly mad and crazy" idea of competing for a scholarship at [the University of] Cambridge. He came in first on the exam! His telegram home had one word: "Hurrah!"
Bella Abramovna Subbotovskaya in Moscow conceived and created a Jewish People's University, to help Jewish students who were excluded from the math department at Moscow University. It operated successfully from 1978 to 1983. Then she was called in for questioning by the KGB, and spoke to them defiantly. Soon afterwards, she was killed by a hit-and-run truck.
Q: The book goes into some detail about the array of obstacles that women mathematicians faced in earlier eras. And even today, you write, "[m]any women still feel isolated and embattled in their roles as mathematicians." What is your view of how gender roles continue to affect mathematicians in their own lives and mathematics as a discipline? What problems must still be addressed?
Hersh: In TV, movies, etc., neither a manly man nor a womanly woman is depicted spending time on algebra, geometry or calculus. The negative connotations of math create a difficult obstacle for a girl or young woman who is inclined to study mathematics. In academic life, scheduling and assessment of faculty should accommodate to the needs of people who have to take time to care for young children or elderly parents.
John-Steiner: The number of women who are preparing for a mathematical career has substantially increased in the last decades. But, stereotypes change slowly. Girls and women continue to struggle with the dichotomies that our society favors between working with abstract ideas and being a caring and nurturing person. Mathematicians are deeply connected to their legacy and hold the major contributors to their field with enormous respect. These icons are overwhelmingly male, which makes it harder for both men and women to see women as making transformative contributions to their discipline.
Q: What lessons can we draw from the "Moore method" and the "Potsdam model" of education?
Hersh: Robert Lee Moore in Austin required his students to rediscover, on their own, completely without any help, the proofs of the basic theorems about sets of points. Of those who survived his courses, some went on to become successful research mathematicians. Clarence Stephens, in Potsdam, N.Y., insisted that ANY COLLEGE STUDENT who wanted to learn math could succeed, in the correct supportive environment, given all the time they needed, and with teachers who were confident in their ability to succeed. These two opposite conceptions of math education force us to recognize the two opposing ideologies in our culture, our society, and our schools. On the one hand, reward the elite and discard the laggards and slowpokes. On the other hand, the empty slogan, "No Child Left Behind." But there need be no conflict; it is possible to recognize and develop the mathematically talented, while still meeting the needs of the rest of our children.
John-Steiner: These are two very different approaches to teaching mathematics. The Moore method focuses on a deductive approach and was practiced in the field of topology. Students taught by Moore were and are good problem solvers, but often lack the breadth of mathematical knowledge currently required in the field. Moore’s personal attitudes were prejudicial to African Americans and women. One lesson we can draw from his impact is how a very dominant figure can achieve success in teaching at the expense of respect for ethnic and cultural diversity. The Potsdam model of Professor Stephens is the very opposite. He emphasized the role of caring and support and the deep belief in every student’s potential in his teaching which was enormously successful with minorities, women, and all other students. In the sustained debate about the future of mathematics education, the dialogue focuses on curriculum and test scores. We forget the enormous impact of other aspects of the interaction between teachers and learners, particularly the roles of emotions, values, and trust.
Q: What are some of the key changes you would like to see in mathematics education at the primary and secondary levels, and why are they needed?
Hersh: The most important is to pay math teachers enough so that the public schools can compete for mathematically talented people in the job market. Then, with enough excellent teachers available, to provide excellent teachers in all our public schools, including the barrio and the ghetto. And then to do away with the "teaching to the test" straitjacket that has been imposed on our teachers by high authorities, and let talented teachers freely devote themselves to teaching their students, our children.
John-Steiner: I would like to give students more of a sense of the usefulness of mathematics in their daily lives. The learners who are strongly visual in their approaches may benefit from broad experiences with measurement and geometry. Student s who are more conceptually oriented may benefit from opportunities to formulate problems, debate possible approaches, and apply math to other domains they may be interested in like biology, architecture, etc... There are many excellent reform proposals in the literature, but they have not been given a chance to be fully tried and refined over a sustained period of time. These programs need to be locally adapted and modified to the particular cultural preferences of a community and the children’s shared experiences. Instead of the sense of panic that surrounds mathematical achievement in the USA, we need to make the field and its practitioners more accessible to the public, less forbidding, and that is part of the objective of our book.
Q: And at the postsecondary level?
Hersh: First of all, understand that teaching a math class is not just an unwelcome interruption in the life of a mathematician, but actually an encounter with other human beings! A valuable opportunity to get to know them and to help them!
Get rid of the calculus requirement for admission to business school, medical school, and so on, so that calculus classes could contain mainly students voluntarily studying calculus. For students not desiring to study calculus, provide a humanistic form of math education, designed to introduce them to the nature of mathematical thinking and its importance in science and in our society. Use games and puzzles, historical background, and concrete applications, to give those who are not mathematically inclined a glimpse into what mathematics is all about, and what makes it interesting and attractive. Avoid topics whose principal raison d'etre is as preparation for calculus. Pay attention to probability and statistics, and to computers and computing.
John-Steiner: The current use of calculus as the filter for admission to graduate programs is counter-productive. At the college level, students should have a choice in what they study, and if they avoid mathematics during this period of their lives, they should have opportunities later in their career to return to the subject in ways that are suited to their particular domain. I have become increasingly convinced that we are endowed with diverse forms of intelligence as developed by Howard Gardner’s research. Building upon intellectual strengths such as verbal abilities, or musical talents, is important during the early adult years when students are confronting a lot of self doubt and fear of failure. Mathematics should not add to these problems but should be made exciting to those who like it, feel comfortable with it, and will definitely need it as engineers, scientists, teachers, and mathematical researchers. In addition, classes should be available for those who need a lot of support in mastering mathematical concepts, but who cannot keep up with the competitive pressures of fellow students who have great strength in the area. We need to experiment with new models of instruction which do not misuse mathematics as a measure of intelligence and academic success, but see it as an important area of mastery for many, but not for all.