What you might not know about Theoretical and Applied Mechanics

Professor Tim Healey wouldn't say there's a direct link between Laika the space dog and the evolution of the Department of Theoretical and Applied Mechanics (TAM), of which he is now chair. But had that Russian dog not gone into space, thus lighting a fire under America's space program, who knows how things would have turned out? "A lot of departments like ours sprung up in the sixties," Healey says. "And a big reason was Sputnik. After that happened, there was a tremendous push for more science and math in engineering education and more fundamental research."

Almost five decades after the Russian space launch, the department is still in the business of fundamental research — research, Healey explains, in which methods are general and the results are often applicable to phenomena in widely disparate fields. Case in point: in 2002, the fourteen faculty members in TAM collaborated on projects with faculty from ten other departments — including physicists, neurobiologists, astronomers, and researchers in textiles and apparel. Not surprising then is how diverse the areas of research being done in TAM are: topics include planetary dynamics, the mechanics of walking and running, mathematical modeling of disease dynamics, DNA elasticity, insect flight, flow and segregation of granular materials, fracture and failure in metals and composites, and complex small-world networks, which includes the study of the collective behavior of simultaneously flashing fireflies.

"Sometimes when people hear about us," Healey says, "they think we sound like the department of misfits." But, he hastens to add, "While our specific fields of application are typically quite diverse,  what we have in common is an approach to research based upon the fundamental and grounded in modern science and mathematics — we all speak the same language."

One might assume, since there's no undergraduate major in TAM, that the faculty would be far removed from the everyday lives of undergraduate engineering students. Healey says, however, that one of the hallmarks of the department is outstanding undergraduate teaching, particularly in the area of first- and second-level courses in engineering math (Math 191,192,293,and 294). This kind of commitment from an engineering department to math instruction is rare; in fact, of all the Theoretical and Applied Mechanics or Engineering Mechanics departments in the country —Healey estimates that fewer than ten stand-alone departments now exist — Cornell's department is the only one to participate in math instruction.

The relationship between TAM and the Department of Mathematics in the College of Arts and Sciences extends back nearly forty years. In the mid-sixties, as Edmund Cranch, professor emeritus, former TAM chair, and former dean of the college, explains it, a treaty was made between the two departments to share in the instruction of the second-level mathematics courses for engineers, partly in response to a desire shared by many engineering faculty members to see math instruction geared to the interests of future engineers. "This was, at the time, a groundbreaking development in the instruction of mathematics for engineering," Cranch says. "It worked off the notion that engineering faculty are in fact qualified to teach the beginning mathematics courses — and vice versa, of course, that the mathematicians have something to contribute to the upper level of courses. There was an element of trust and willingness to learn that was implicit in this."

This unique relationship soon gave rise to a book-length set of notes co-written by Cranch, his TAM colleague Professor David Block, and two professors from the math department — Bob Walker and Peter Hilton. The set of notes, never formally published, was the first program of study of its kind, in that it integrated the subjects of differential equations, linear algebra, vector calculus, and infinite series. "It really was a pioneering step," says Cranch. Not long after it became the standard issue text for second-level math at Cor- nell, engineering professors at other schools caught wind of it and asked to use it in their classrooms.

With the advent of computer science and the evolving needs of engineering students, this set of notes is no longer being used, but TAM and the Department of Mathematics continue to collaborate in the management of these courses. Two or three times a year, the Math Liaison Committee, comprising two members from each department, convenes to discuss potential changes in curricula, syllabi, and textbooks. Says Professor Ken Brown, the current chair of the Department of Mathematics, "We just want to make sure we're all on the same page and meeting the needs of the students."

The next big innovation in the teaching of engineering math at Cornell came at the tail end of the 1980s, when the College of Engineering decided that its students would benefit from small-section courses in Math 191, first-semester calculus. "Bill Street, then dean of engineering, talked a lot about wanting to make first-year courses more friendly," Healey says. It was felt by many professors that some students were alienated by the large class sizes and were too intimidated to come ask for help.

"Everyone thinks of Cornell students as being extremely able, but even the most competitive ones sometimes feel like deer caught in the headlights," says Joe Burns, professor and department chair in TAM when the small classes were established."We were keeping only sixty or seventy percent of our students and part of it was that they weren't getting to see engineering faculty early on." The math department was already having success with the small-section calculus course it provided for students in the College of Arts and Sciences; why not provide the same services to engineers?

The project was given the green light. A named assistant professorship was added in the mathematics department, professors in TAM as well as other engineering departments were recruited to teach, and a new policy was instituted requiring that freshman advisers offer an hour of tutoring to their advisees every week. Six years later, under the guidance of the chairs of TAM and Math at that time, Professor Jim Jenkins and Professor Bob Connelly, small-section courses for Math 192, second-semester calculus, were added as well.

The program has been, not surprisingly, an unequivocal success. Although it's difficult to measure the direct correlation, the attrition rate in the College of Engineering has shrunk by almost half since the small-section courses were instated, and results of a Student Satisfaction Survey taken after these curriculum changes were instituted show that students felt their professors were more approachable for help and cared more about their academic success.

"What's great about these classes," Healey says, "is that you have a real opportunity to make a safety net. Everyone who teaches these courses says that. You can be there for the struggling students, and if you have real crackerjack students, you can be there for them,too. It's that personal touch that really has a big effect on the bottom and the top."

Providing math instruction has also helped TAM stay as small, diverse, and autonomous as it is. The trend over the past few decades, Healey says, has been for small, non-standard engineering departments like TAM to be either broken up or absorbed by bigger departments, like mechanical, civil, and aeronautical engineering.  "You're vulnerable," he says. "The math teaching has a lot to do with our stability."

TAM has many outstanding teachers. Among others, three who have consistently received rave reviews from students for their teaching in both first-and second-year engineering math courses are Professors Chung- Yuen (Herbert) Hui, Richard Rand, and Steven Strogatz. Hui, who joined the faculty in 1981 and Strogatz, who arrived at Cornell in 1994, have doctorates in applied mathematics; Rand, who arrived at Cornell in 1967, has a doctorate in engineering mechanics.

The key to their success is no mystery, says Healey. "They have excellent command of the material, they are enthusiastic about their subject, and they really care whether students learn or not, and that comes through loud and clear." The comments students write in their evaluations — "Don't let this one go!"  "Give this guy a raise!"  "Best math teacher I've had." — speak volumes, Healey says, about how well these professors are reaching their students. Their success is all the more impressive, Healey adds, considering how difficult it often is to get students excited about these basic, required classes. "The easier courses to teach are the ones that students are already interested in. That's not what's going on here. They have to take these classes."

One of the benefits of having professors from TAM teach math courses to engineers, says former chair of the math department, Professor Bob Connelly, is that they bring an engineering viewpoint to the material. Adds the current chair of the department, Professor Ken Brown: "They understand what courses [their students ] will be taking later and what's important for their future careers as engineers."

"The engineering crowd typically tests well above average in mathematics but they're not motivated to study mathematics purely as an intellectual endeavor," Rand explains. "They need to be motivated by examples." So Rand gives examples. When students are stumped on, say, a vibrations problem, Rand might bring up the paper he and Professor Bob Cooke, a colleague from biological and environmental engineering, wrote in which they calculated the optimal modes of oscillation for apples to be shaken (or vibrated), with their stems and without their stems, off a tree branch. Or when students studying matrices are struggling to visualize an abstract ten-dimensional space, Rand might talk about a simple three-dimensional car engine, composed of ten moving parts, each associated with a certain variable, to help them make the mental leap.

More generally, Rand says, the challenge in teaching both large- and small-section math courses is to reach everybody regardless of their skill level. "What I'll do," he says, "is aim for the middle and even spend a little bit of each lecture going over the material again, in order to pull in a few more people. Occasionally, I'll give a lecture aimed at the more accomplished students and will reassure the class that this lecture will be something that's interesting but it won't be on the test. This way I hope to not discourage any of the students."

While Hui doesn't find it possible to lift many "for instances" directly from his own research (currently he's studying how geckos adhere to a variety of surfaces), he does try to bring across interesting ideas to his students that are related in a tangential way to his own research, as well as make connections between the material they are studying in his class and things they are learning in other engineering or physics classes.

Hui finds teaching the smaller sections to be easier and less stressful. "In a small class if you make a boo-boo, it's much easier to correct. You can just apologize and say ‘I made a mistake.’  You can also make eye contact with everyone, and if you see eyes glazing over, you can slow down. But in a large class you just can't afford to make mistakes in a lecture. In a large class, if you lose [the interest of ] ten percent of the students, it's a disaster, because they're going to make noises and pretty soon the noises are going to propagate throughout the hall, and you can't give an effective lecture."

Hui also notes the importance of presenting material in a way that will get undergraduates' attention. For example, when Hui gives a lecture on a probability principle called Markov chains, he chooses to illustrate his point by setting up a problem dealing with the predictability of a hypothetical boyfriend or girlfriend's bad mood as opposed to the predictability of the weather or certain stocks or bonds — examples that he feels are well-worn and alien to most college students.

Strogatz also says it's difficult to mine his research (which includes human sleep-wake dynamics and mathematical investigations of the "six degrees of separation" phenomenon) for real-life scenarios helpful to beginning-level math students; there are simply too many prerequisites to understanding it. But he does occasionally share questions that his former high school math teacher sends him —questions that ask students to solve problems associated with kayaking across a fast-moving river, or designing a swimming pool, or slicing up cylinders of heating oil. In the small-section courses especially, he makes a point to ask his students not only to solve the problem, but to convince their classmates of their reasoning. "To do that takes an extra degree of understanding."

And even though he understands that most of engineering students taking calculus or linear algebra or differential equations see the courses as hurdles on the way to receiving their degree, Strogatz feels its worthwhile to bring up the big questions that orbit the disciplines of mathematics and engineering, those fundamental questions that his colleagues ponder as well: "I'll sometimes ask the kids: ‘What is the nature of this enterprise? Is it art? Are we trying to create something beautiful? Is it science? Are we trying to figure out something about how the world works?’  And the answer,of course, is that it's all those things. At the same time it's a fight. The doing of math as opposed to the appreciating of math is a real struggle. There's this thing that's resisting. And of course the real struggle is with yourself."

That struggle to define the enterprise is reflected at the department level as well. With its diversity of research topics and its lack of a field-specific undergraduate degree, TAM is often referred to — even by its members — as a "nontraditional" department in the College of Engineering. Yet the faculty is internationally recognized for its fundamental research in engineering science and applied math, and they are routinely honored for excellence in teaching. And that amounts to what Healey simply calls "the best department you never heard of."

Mark Rader is a freelance writer and a recent graduate of Cornell's master's program in creative writing.