MAGIC NUMBERS

Mathematics, "like the crest of a peacock, like the gem on the head of a snake, is at the head of all knowledge," states the Vedanga Jyotisa, a Vedic treatise of 500 B.C. And the mantra of the Bryn Mawr math department is that "Absolutely anything you want to do, you're better off with math."

The department is attracting increasing numbers of majors-29 seniors in 2000-01 and 36 next year-with dynamic teaching and the promise of exciting opportunities in today's job market.

Mathematicians describe their work as beautiful, thrilling, a way of gaining fundamental understanding of universal truths. In the Western world, higher math has traditionally been the preserve of a gifted few and was kept detached from practical and material applications until the atomic age. Socrates sets the tone in Plato's Republic: "It occurs to me, now that the study of numbers has been mentioned, that there is something fine in it, and that it is useful for our purpose in many ways, provided it is used for the sake of knowledge and not for huckstering."

In the Republic of 2001, however, all students need to become quantitatively literate citizens. Technology has made math the essential underpinning not only of science and engineering, but of economics and much of industry; it contributes to the design of computers and special effects in film. (So you want to be in pixels?) The National Science Foundation has launched broad initiatives to reform the teaching of math and science, integrating them into other disciplines and vice versa.

Does having an eye on careers after college compromise the joy of doing math for its own sake? Not at all, say faculty and students. "It seems to be an incentive for students who in the past may have wanted to do math, but just didn't think it would be practical," explains department chairman and Associate Professor Victor Donnay.

The department is also committed to making math appealing and accessible to a wide range of people. Faculty colleagues credit the leadership of Rhonda Hughes, Helen Herrman Professor of Mathematics, who joined the department in 1980. Hughes has been recognized nationally for her efforts to encourage women and minorities to study math, receiving the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America (MAA)in 1998.

"In addition to bringing mathematics to a vast number of Bryn Mawr undergraduates, and in many cases helping them find a new confidence in and love for it, our department has been particularly successful in fostering the progress of our majors interested in doing research and obtaining Ph.D.s in mathematics," says Hughes. "Many distinguished alumnae are active and respected members of the mathematics community." (See sidebar.)

Associate Professor Helen Grundman encourages her students to develop skills needed for graduate school, regardless of their plans. "They will have the background they need to succeed if they make a decision to go to graduate school late in college or even a year or two after graduating," she says. "And if they choose not to go, the skills will still be invaluable.

"I believe that one of the biggest crimes against young women in this country has been to educate them without challenging them and without setting very high expectations," Grundman says. "Thankfully this is much less of a problem in recent times, but it is still common for people to expect less of women, particularly in science and mathematics. I'm thrilled that at Bryn Mawr, my students thrive on the challenge!"

The real numbers
About 1 percent of students nationally major in math, according to the U.S. Department of Education. Last year BrynMawr matched Williams College whose math program has been getting attention in the national press, with 8.5 percent, jumping to 9 percent this year and 11 percent in 2001-2.

Instituted in 1984, Bryn Mawr's quantitative requirement asks even hard-core "humanities types" to pass with a 2.0 one semester of a course such as Conceptual Physics or Introductory Physics, Modal Logic, Introduction to Computer Science; Calculus and Analytic Geometry, or Elements of Probability and Statistics. These are not the only options; there is an ongoing effort to enrich the curriculum with new courses that offer math and quantitative reasoning in various and exciting contexts. In Working with Economic Data, for example, students use spread sheets and other tools to study topics such as unemployment and economic forecasts. Statistics courses are also taught in psychology, economics and sociology; next year, the College will evaluate a request for a new position in statistics that would coordinate various departmental offerings.

Entering students are mailed a math placement test over the summer to determine whether they are prepared to take a 100-level or higher course that satisfies the quantitative requirement. Fundamentals of Mathematics, a review of algebra, analytic geometry and trigonometry, is taken mainly by McBride Scholars more removed in years from their high school math study.(McBride Elizabeth Ferry '97 worked as a nurse before majoring in math at Bryn Mawr and went on to earn a master's in math from Wake Forest University.) Of an entire incoming freshman class, there are usually no more than 20-25 students who need this preparation.

Many of those who are not strong enough to take Calculus I, but who do not need the review of algebra, take Elements of Probability and Statistics. "This is a very practical course for humanities students who aren't necessarily interested in doing a lot of mathematics," Donnay says. "It will help them make them educated citizens who understand, for example, references to statistics they read in the general press."

"Three years ago, the additional work requirement, which gave students the option of taking more advanced language or more math, was dropped," he notes. "So even though there's a bit less pressure for students to take math, they remain captivated. We have been attracting record numbers of majors because, we hope, what we do in our introductory course and in our 200-level sophomore courses tempts them. They want to continue and then decide to major or minor."

""We've always made great efforts at the introductory level to make math appealing, to give women who have had bad math experiences previously a new start and a fresh chance," says Senior Program Coordinator and Instructor Mary Louise Cookson. "By 1990, we were averaging 20 majors a year, but in the last couple of years we've made more of an effort to publicize and educate students about the opportunities available for people with strong math backgrounds. Associate Professor Lisa Traynor organized a lecture series that brings back alumnae and others who were math majors and then went on to other types of careers." The series was funded for two years through a gift from CIGNA and subsequently a memorial gift to support the department made by Jack Todd in honor of his wife, mathematician Olga Taussky-Todd, who spent a postdoctoral year at Bryn Mawr in the 1930s studying with Emmy Noether, went on to make lasting contributions in matrix and number theory, and was the first woman mathematician at Cal Tech.

"Part of our immediate success is due to Leslie Cheng '91, who received her Ph.D. from the University of Pittsburgh and is here for her third year as a leave replacement," Cookson says. "She has been a wonderful teacher and an excellent role model for the students. Some of us had her in class, and she was my grader for Calc 1 and 2 one year; it's neat having her back as a colleague."

Cookson generally supervises introductory courses for the major, Calculus I and II. She and Instructor Peter Kasius teach sections of this and Elements of Probability and Statistics. "By Calc II, all of the students are given information about summer internships, math research opportunities, graduate school programs, job opportunities, and salaries," she says. "We talk about it in class and they're invited to come to my office to talk about it in more detail." She even passes out chocolate bars in a wrapper she designed that reads, "Do the math, feel the power."

"Our emphasis on giving out information about career opportunities was partly motivated by feedback that we got from our majors," Donnay explains. "A graduating major who had been in the corporate recruiting process told us that when one consulting company interviewed her and found out that in addition to the good writing and oral communication skills she had coming from Bryn Mawr, that she was also a math major, they snatched her up."

Everyone's a star
"It's great to have very strong students who, it's clear from freshman year, have the ability to get a Ph.D. in math, but we want all types of students to do as well as they can," Donnay says. "There is a lot of room in the math world for a wide range of interests and abilities. The College gives us the resources we need such as graders and TAs (teaching assistants) to hold help sessions and provide a safety net. We also encourage cooperative learning, in which students work together in structured groups.

"One of the charming characteristics of our department is that we don't believe that you have to be a superstar to make worthwhile contributions to the mathematics community-or to any field-after you graduate," he says, adding, "Of course, you also might say that any student who is talented enough to come to Bryn Mawr is a star!

"We try to find ways for students to be useful while they are still here-the students who are very keen on advanced mathematics can be doing a senior research project, and those who are interested in teaching run problem sessions for lower courses or work as graders," Donnay says. "Seeing that you already know enough to teach can help motivate you to keep going."

Several courses, such as Donnay's multivariable class, have labs in which upper level math students play a major part in presenting the material, helping the students through the computer commands and producing a product.

The Bryn Mawr College Community Service Office's America Counts math tutoring program places Bryn Mawr students as tutors with children in grades K-9. Some of the tutors volunteer, some go as a field requirement for education course, and others are paid through the Community Service Work-Study Tutoring program. "Our students work with students in an after school program at the Ardmore Avenue Community Center in Ardmore, or are assigned to work with a classroom teacher in school during the class day," says Community Service Director Jennifer Nichols. "We work with Bluford Elementary School in the Overbrook Cluster of Philadelphia and with the five elementary schools in Haverford Township. The program at Bluford is called BEST (Bluford Elementary School tutoring) and the program in Haverford is called Rhythms (because the schools use alternative algorithms to teach mathematics). We currently have 24 students working or volunteering at the three sites."

A nurturing machine
"One of the nice things about the math department here is that there are many chances for students to be involved in doing math outside classes," says senior major Anna Hu '01. "I've worked for the math department tutoring other students, grading papers for professors, occasionally TA-ing problem sessions, and as part of the Customs' Week Mathematics Committee, which serves to advise and help determine incoming students' math course placements."

The department sponsors several clubs including the Math Shakespeare Reading Group and The Distressing Math Collective (DMC), both started by Helen Grundman. DMC members take turns giving presentations on apparent paradoxes not usually addressed in courses. A memorial fund for John Oxtoby, a distinguished mathematician who taught at Bryn Mawr from 1939-79, enables the department to send majors to math conferences. Jane Holsapple '02, who studied at the University of Nebraska at Lincoln last summer under the College's Summer Science Research Program, organized a group of seven students to attend the Nebraska Conference for Undergraduate Women in Mathematics held there this February. Several gave talks, having practiced them first at DMC. The Math Club sponsors events that supplement departmental colloquia, most recently bringing Margaret M. Murray, associate professor of mathematics at Virginia Polytechnic, to discuss her book, Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America. (Read a review of "Women Becoming Mathematicians: Creating a Professional Identity in Post-World War II America," by Margaret Murray.)

An annual December math party, also started by Grundman and supported through the Taussky-Todd Fund, features mathematical favors and games, and has become quite an anticipated event helping to build camaraderie.

Last semester's "Poster Night" showcased final projects for Math 201: Multivariable Calculus. "The assignment was to encourage students to present math ideas in broader and creative contexts, relating them to other topics," says Lecturer Anne Schwartz. "The Great Hall was full of students who were making mathematics come alive in the most imaginative and playful ways," says President Nancy J. Vickers. "The energy in the room was palpable."

A university atmosphere
Katharine E. McBride Professor Emeritus of Mathematics Frederic Cunningham Jr., who came to Bryn Mawr in 1956 and is a resource of departmental memory, continues to teaches one course each semester. The graduate program, which was a top producer nationally of women Ph.D.s though the 1930s, also helps create the university atmosphere of a community of mathematicians at different levels. Graduate students work with undergraduates as teaching assistants and in summer programs, and qualified undergraduates may take graduate seminars.

"Our graduate program is small but very important," says Grundman. "We pride ourselves in giving opportunities to nontraditional students, particularly those with unusual backgrounds who might not be supported at larger institutions. The success we've experienced with some of these students really makes it worthwhile. It also works well to have students just out of college working with older ones as well as a mixture of cultural and economic backgrounds. The undergrads benefit, too, I think, from working with this wide variety of people.

"Most of our M.A.s go into college teaching, and each of our four Ph.D.s in the last decade has gone directly into a good position at a liberal arts college. This was during a time when many graduates from the 'top' P.hD. programs in the country were failing to find academic jobs," notes Grundman.

"It is a well-oiled machine of nurturing," Cookson says firmly. "As individuals and as a department, we've given every part of the program a great deal of thought to best orchestrate giving the students the maximum benefits for their education in mathematics classes. We do nothing randomly or haphazardly."

-JAN T. TREMBLEY '75

For more information, see the mathematics home page. Recommended readings include The Crest of the Peacock: The Non-European Roots of Mathematics by George Gheverghese Joseph (Princeton University Press, 2000). See biographies of Bryn Mawr mathematicians, Charlotte Angus Scott, who created Bryn Mawr's department, Anna Pell Wheeler and Emmy Noether.

Thinking on two levels
Lisa Duffy '01, of Staten Island, NY, loved math in high school but says she decided to major because "I felt so welcomed and encouraged in my first calculus class here. The department as a whole is very interested in what each student thinks and really helps you to get the most out of the math major."

She has taken a number of different math classes and enjoys the quantitative as much as the abstract. For the past three years she has been a calculus teaching assistant for Mary Louise Cookson. "I loved the opportunity to help those people who were struggling with some concepts in calculus while at the same time brushing up on my own skills," she says.

Duffy plans to attend medical school and pursue her lifelong dream of becoming a pediatrician. "I think being a math major will help me to think on two different levels, qualitatively and quantitatively. The problem solving skills I learned will also help," she says. Duffy has completed a minor in Russian, having had the opportunity to begin studying it in high school. She also works in the admissions office as a clerical assistant, and is an active member of the pre-med club, of which she was president last year.

Coming up gold
In the spring of 1963, McBride Scholar Lynn Litterine '96 burned her Algebra II textbook in her parents' driveway after completing what she thought would be the last math class she would ever have to take.

"At that very good public high school, this was not a gender issue," she wrote in a column for The Philadelphia Inquirer Magazine (10/11/92): "Brains were a valued commodity, in boys and girls, and intellectual ambition fueled the upper reaches of our social scene. We talked about Vietnam, or All The King's Men, even quantum jumps, but there was nothing to say about numbers, no establishment to overturn in geometry, no showy self-expression in solution sets. Real life got played out in the humanities. Math, like its department chairman, was stone-faced, a monolith we approached under duress and left as quickly as possible."

When Litterine decided to finish her college degree at Bryn Mawr 28 years later, she took Fundamentals with Mary Louise Cookson and "came up gold." She decided to take calculus, drawn by its beauty and knowing that Cookson would be teaching her again. "I completely trusted her judgment of what I could do," she says. "I think the two most important qualities Professor Cookson has as a teacher-and she is among the most gifted teachers I've ever had-are the ability to teach any math concept in half a dozen different ways and the instinct to defuse any anxiety we might feel by humanizing math," Litterine says. "The first quality means that she explains a topic several times over, each time differently, so however you learn, you've got a shot at it. It might be a spatial description, a shape you can see in your mind. It might be an abstracted process, a technical, law-driven explanation. It might even be a joke.

"The calc lecture hall is large for Bryn Mawr, maybe 60 seats, and I can remember her standing down in front-she's not a large person-in a bright red Minnie Mouse sweater charting a function that just got lower and lower and never reached an end. She was singing the chorus of a Bruce Springsteen song-'I'm goin' down, down, down, down.' You don't forget that function at exam time.

"She said at the start of the course that all of us could pass if we came to class every day and practiced until we were 'blue in the face.' She was right, but I only had to practice until I was red in the face; I never actually stopped breathing."

'Abstract, beautiful'
Jane Holsapple '02 of Lincoln, NE, knew she wanted to major in math before coming to Bryn Mawr, and a telephone interview with Mary Louise Cookson encouraged her to consider the combined A.B./M.A. program.

Algebra is her favorite area of math. "It's abstract and can be quite difficult at times, but it can also be beautiful," she says. Holsapple took Discrete Math (Math/Computer Science 231) her freshman year and confesses that she "just about gave up! The class was very difficult for me at first and was completely different from anything I'd seen before. However, I adjusted and learned to love higher math, and eventually algebra, more than the geometry and calculus I'd learned in high school." (Discrete mathematics has important applications in technology and communication. Covering a multitude of topics that include counting techniques, graphs and trees, and election theory, it may be described simply as the mathematics necessary for decision making in noncontinuous situations.)

Holsapple takes ballet and dance composition classes and works for Computing Services and the math department. She definitely plans to take some time off after graduation, perhaps to teach. "After that, I'll either go back to school for a Ph.D. or try to find a career in which I would use the math that I've learned," she says.

Calming knots
Anna Eiformosa Hu '01 of Murrysville, PA, has loved math since the first grade. She has also enjoyed studying foreign languages (French in particular) as well as music and had considered majoring in one of those areas while in high school. "But when it came down to deciding what I wanted to focus on as a major in college, I realized that I wouldn't feel the same if I were not doing math every day," she says.

"What's interesting about math at the college level (and beyond) is that it is very different than the math you're exposed to in elementary and secondary school. Lots of people think there is not really any new research to be done or problems to solve in mathematics, which is far from the truth! I didn't realize this before choosing to major in math, and I think the same is true for many others. It may sound funny, but I find sitting down and working on math very calming.

"One of the most enjoyable math classes I've taken at Bryn Mawr has been Knot Theory with Associate Professor Lisa Traynor. I found that, for me, it required a different type of thinking than in many other areas of math, and it is very visual. This year, I have been doing independent work studying knot theory with Paul Melvin, Rachel C. Hale Professor in the Science and Mathematics, so I can learn more about it."

Hu is completing a minor in education. She is not entirely sure what she wants to do after graduation, but is attracted to consulting work. Eventually, she would like to pursue a higher degree in math, and/or possibly teach at the secondary school level. She also works as a student and Help Desk operator at the Computing Center, plays clarinet in the Swarthmore Woodwind Ensemble, participates on the executive board of the College Democrats, and has in past years enjoyed running on the Track and Cross Country teams.

Summer strengthening
Activity in Park Science Center does not slow down during the summer. Undergraduates in the Summer Science Research Program work with faculty on independent projects, attending weekly seminars and a final symposium. Rhonda Hughes began summer undergraduate research for the math department in 1989, pioneering involvement of freshmen and sophomores in the excitement of independent work. This grew into another summer math program for undergraduates co-directed by Hughes and Professor Sylvia Bozeman of Spelman College that was held at Bryn Mawr and Spelman from 1992-5.

Recognizing that the need to encourage and support rising women graduate students is as critical as steering them towards graduate school in the first place, Hughes and Bozeman developed a new program to strengthen women and minority students entering graduate programs in the mathematical sciences. The summer of intensive coursework, workshops and guest lecturers, begun in 2000 at Bryn Mawr, is followed up by a mentoring program and support network. Called EDGE, "Enhancing Diversity in Graduate Education," the program is funded by the National Science Foundation, the National Security Agency, and the Andrew W. Mellon Foundation. (It will be held at Spelman this summer.)

Last summer, Bryn Mawr also sponsored a program for Philadelphia School District teachers with a grant from the Howard Hughes Medical Institute. Divided into three institutes, participants and Bryn Mawr faculty discussed integrating math and science into all levels of the curriculum, discovery-based learning, ways of drawing on the diversity of the classroom, and the use of information technology.

No boring shapes for mathematicians
In a lecture for the Visual Culture Center series, Victor Donnay surveyed the beauty of mathematical forms, showing examples of the use of the Golden Mean and proportion in architecture, shapes in sculpture, patterns in crafts such as quilting, and fractals in nature.

Turning to one of his own research areas, dynamical systems, he demonstrated "Chaotic Billiards," interactive Java applets written by students during a summer program at Bryn Mawr that illustrate the difference between regular motion and chaotic motion, on billiard tables of different shapes. (See http://serendip.brynmawr.edu/chaos/index.html)

"Because we are mathematicians, we can do billiards in any shape we want-a circle, an ellipse, a polygon; the table doesn't have to be a boring rectangle," he explained. "With dynamical systems you can have regular motion where there's a pattern and it's predictable, or you can have chaotic motion where there's no pattern and not predictable."

In collaboration with Keith Burns of Northwestern University, Donnay has discovered three-dimensional shapes that generate chaotic motion. One of these, called a "geodesic globe," is constructed by attaching rounded caps to the ends of the Schwarz P-surface, a minimal surface discovered by the mathematician H.A. Schwarz in the 1880s. (Minimal surfaces are the most efficient way to span space and hence often arise in nature. If you dip a wire frame into a soap solution, the shape you get will be a minimal surface.) Although mathematicians have known for more than 50 years that straight line motion on surfaces can be chaotic, they have never before found real physical surfaces that exhibit this behavior. If you imagine standing on a huge sphere and walking in what you think is a straight line, you will eventually return exactly to your starting point and could continue to trace, again and again, a "great circle," the path of shortest distance that airplanes follow. But if you were living on Donnay and Burns' bumpy sphere and started walking, you would never come back to your starting point. Instead, you would wander all over the surface, coming arbitrarily close to every point. Such a motion is called chaotic.

"These ideas I learn about and develop for the simple things can then be used by applied mathematicians and scientists in much more complicated real world dynamical systems: asteroids, the weather, population dynamics, global warming, economic forecasting,and modeling the spread of HIV," Donnay said.

"There is a new impetus in math pedagogy to emphasize quantitative reasoning in as many courses as possible," he said. "Even at the elementary school level, everywhere and any time, everything is mathematics is interactive, hands-on experience, whether itmeasuring recipes, cutting strawberries in fractions, painting pictures that have math shapes. In kindergarten at Cooperstown Elementary School, for example, my son learned about symmetry by painting an image on paper, folding and mushing it.

Donnay told faculty, "I'm hoping this forum will give you a chance to think a bit about integrating math into any courses you may teach and perhaps designing some new courses for the College Seminar program." Donnay, who wrote for the Bulletin in 1991 about the art of M.C. Escher and its expression of mathematical and scientific ideas, has taught a course at Bryn Mawr on Mathematical Modeling and the Environment and hopes to develop this into an interdisciplinary offering to be team taught with faculty from other departments.

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