Integrating Math and Students
It is said that some of Harvard's most renowned professors are so concerned with their own academic research that they have little time to devote to undergraduates. However, one Harvard math professor, who many call one of the most brilliant members of the department here, can often be found in Lowell House dining hall or in his office talking with undergraduates on all sorts of topics.
Hollis Professor of Mathematiks and Natural Science Andrew M. Gleason, who 30 years ago solved one of the most difficult math problems proposed this century, devotes much of his time today to thinking about ways to improve the teaching of math at Harvard and around the world.
"He started his career by solving the famous Hilbert's fifth problem as part of a team, which is quite an outstanding achievement," says colleague Raoul Bott, Graustein Professor of Mathematics. "But he's always been very interested in the undergraduates as well. His office is always full of them."
Gleason, who was granted tenure without a Ph.D. after he had solved the notoriously difficult Hilbert problem, focuses his academic work on topology and space mathematics. David Hilbert, a German mathmetician, proposed a series of about 10 problems he considered the most pressing math questions unsolved at the turn of the century. Math scholars are still working on solving some of the problems, which define much of 20th-century math.
"Considering the fact that he's such a superb researcher in math," says Deborah Hughes Hallett, senior preceptor in mathematics, "Andy's done a considerable amount with teaching, which he certainly hasn't had to do. He's done lots as far as undergraduate education and has been extremely important in a number of ways as far as getting things started."
The professor has taught and studied almost every math course in the catalog from Math B, an old algebra course, to Math 212, "Functions of a Real Variable," a course on abstract set theory.
Gleason, who has written two math textbooks, says he most enjoys teaching a freshman honors course, Math 25, "Honors Intermediate Calculus." "It's actually a more sophisticated course than the title suggests," says Gleason. "And it's enjoyable to present the material to students who are excited about the work and able to grasp the concepts easily."
Chairman of the Math Department Arthur M. Jaffe says that Gleason also gets undergraduates interested in math by promoting their participation in the Putnam Math Competition, a national math exam taken by the brightest young math scholars in the U.S. Gleason won the contest three consecutive times as an undergraduate.
QRR or Calculus
At Harvard, Gleason was also the founder of the Core Curriculum's Quantitative Reasoning Requirement (QRR), a test on basic statistical and math abilities that all undergraduates must pass by the end of their freshman year. He currently serves as the QRR's chairman and as a representative to the Core's Board of Advisers.
When the Core and QRR were being formed, several math professors believed that all Harvard freshmen should be required to take basic calculus courses. Gleason instead decided about 10 years ago to focus the QRR on the numerical skills an average student would need to survive college-level courses.
"He did the whole QRR singlehandedly," says Bott. "And while there are some of us who belong to the old school and feel all [students] should have to take calculus, we're very grateful for his new ideas, that he is concerned with those sides of math."
"He was responsible for determining what kinds of math and reasoning would need to be included on the QRR test," says Hughes Hallett. "He has been very important to people in math courses, but he's also been concerned with teaching math to people who are not concentrators as well."
Gleason says that it was not until he began to work with the Core and the QRR, in fact, that he realized the extent to which many students loathe and even fear working with numbers.
"Deborah Hughes Hallet felt that I didn't have a good idea of what students outside the math concentration were like and suggested that I teach some students who were taking math only as a requirement," says Gleason, who will be head tutor of the Math Department this spring.
"You can't expect everyone to be into math, and certainly if everyone was excited about everything they did it would be strange, but a lot of students just really feel like they can't do math," he says. "And that's too bad."
Gleason says that the American public seems to be convinced that doing math--even something as simple as multiplying fractions--is a matter of ability and not of hard work.
"People seem convinced that being able to do math is having a style of thought which is conducive to math, " he says.
To remedy these problematic attitudes towards mathematical education in America, Gleason has become involved with the Mathematical Science Education Board, a subgroup of the National Research Council (NRC), a Washington-based organization designed to promote the study of math and science in American schools and industry.
Gleason attends NRC conferences several times a year and discusses education policy in the U.S. with other professors and government officials.
He says that the board is trying to improve communication among leaders in math education nationwide and to promote changes in the way math is taught in American schools.
"In the early 1960s it seemed as if math was going to improve, but then Vietnam came along as a political preoccupation, and everything collapsed. There needs to be a new operation aimed at every angle," he says.
"Our object is to coordinate communication and leadership to do something good for math education in America," he says.
Gleason says that American school children today perform much worse on standardized math tests than do students from other countries, partly because of the attitude here that the ability to do math is an inherited trait--not something that can be acquired.
To explain such differences in ability, "people will point to Japan and say that Japanese children are in math class longer than American children, but that [assertion is] not really true," Gleason says. "If you look around a foreign country and ask parents what they think, they say that if you work hard, you can do math."
"But here the philosophy seems to be that if children are born with the ability then they can do it, and if not, don't worry," he says.
Gleason says that as the number of Americans who cannot perform higher math increases, the nation is put at risk--not only in terms of mathematical research, but also in terms of factory workers' ability to perform competently.
"Kids are coming out of school not knowing anything," he says. "The number overall is a small percentage of the population, but still it is disturbingly large. People in assembly line jobs at chemical plants and such are now needing extra training to deal with the math they need to just read the meter to see if the chemical is above the danger mark. There are too many of these people."
To follow up on these concerns, Gleason and Hughes Hallet are currently working on a project to study ways to improve the teaching of calculus in American high schools.
A Life of Math
Gleason says he has been interested in math problems and numbers for as long as he can remember. "As a child, I had learned how to do long division and that kind of thing, but it was not something that I particularly enjoyed. I was fascinated, though, with some games my father used to play, which involved numbers and cards."
When he was taking a high school geometry class, Gleason says he suddenly realized he understood more about math than did his teacher. While his contemporaries were having difficulties solving proofs and determining the volumes of spheres, he understood the subject well enough to help his teacher with the explanations in class--an experience that led him to his decision to become a researcher and a teacher.
Gleason majored in math as an undergraduate at Yale, and after a stint in the Navy during World War II, he became a junior fellow at the Society of Fellows program at Harvard.
"In the orignal days when President Lowell began the Society of Fellows, the purpose was to break the stronghold of Ph.D. s, which seemed to be oppressive in American academia at the time," says Gleason, a New York native. "So I don't have a doctorate in mathematics and have never written a thesis, but with the fellowship I could study anything I wanted, and I studied math."
Gleason then returned to the Navy and served two years in the Korean War. He returned to Harvard in 1952 to become an assistant professor of math, a full faculty member in 1957 and the Hollis Professor of Mathematiks and Natural Sciences chair 12 years later.