They are an elite among the elite, and they can achieve wonders with only a writing instrument and a surface to write on.

They are the students of Mathematics 55a, "Honors Advanced.

Calculus and Linear Algebra," a course intended for students, primarily first-years, who have had, according to the Courses of Instruction, substantial experience with abstract mathematics.

"The theory behind Math 55 is that we wanted to design a course that helps [students] mature as mathematicians rather than as course takers," Professor of Mathematics Clifford H. Taubes says. "People can do wonderfully at passing math but not being good mathematicians."

With a drop rate of slightly over 50 percent, Math 55 seems to be more of a challenge than most entering students expect. According to Assistant Professor of Mathematics Pavel Etingof, who teaches the course, 23 of the 43 people originally in the class dropped out, bringing the number of students down to 20 Harvard students and one MIT student.

But many of those who remain say the course--which scores 4.7 out of a possible 5.0 in the CUE guide--is difficult but ultimately rewarding.

Etingof says the course is difficult but is intended to be that way.

"Unless the students have lots of experience in math, it's not surprising that it's not easy," he says. "It's a select group of people. We don't start from scratch."

**An Overwhelming Challenge**

Joshua P. Nichols-Barrer '00, one of the two course assistants for the class, says that Math 55 is more difficult than most undergraduate mathematics courses.

"Math 55 covers the content of Math 21 with maximal generality and rigor," he says. "It is comparable to a graduate course. The types of assumptions it makes about students are more akin to graduate level than 100-level courses, which you can walk in on with no background on the material."

Daniel A. Stronger '01 took the class last year and ended up scarred by the experience.

"[Math 55] pretty much destroyed my year last year," he says. "I was doing more work for Math 55 than for all my other classes combined, and I wasn't even completing all the work. It was really painful and just too hard."

Eleanor E. Williams '02 started out in Math 55 this semester but dropped out after the first problem set was returned.

"[I took it] for the challenge, because several of my friends are taking it, and probably because my sister had taken it,"

Williams says she found the course extremelydifficult, challenging and work-intensive.

"Problem sets were time-consuming, the lectureswere moving quickly...the breadth of what iscovered in the class is astounding," she says. "Ithink Pavel got reprimanded last year for thesubjects he was teaching. Apparently, they weresomething that no 18-year old should see, forwhatever reasons."

Reflecting upon her decision to drop thecourse, Williams says she is "completely confidentthat my decision was right for me."

"My schedule now is definitely challenging,"she says. "But I feel that I'm getting a strongbackground in the subjects I'm taking. I wouldn'thave had that in Math 55."

Like Williams, Dina Roumiantseva '02 startedout in Math 55 but began having serious doubtsabout the class as it progressed.

Unlike Williams, however, Roumiantseva says shedecided to stay in the class, largely because of areassuring e-mail message the professor sent outto the class after the first exam.

"I was strongly considering quitting after thefirst three to four weeks because the atmosphereof the class was rather antagonistic andcompetitive, but during the week of the add/dropdeadline, the professor sent out the e-mail,"Roumiantseva says.

According to Roumiantseva, Etingof wrote in thee-mail," The required problems will be more orless straightforward and designed to deepen theunderstanding, the definitions and theorems fromclass and sometimes to introduce new notions. Iwill keep the number of required problems asreasonable as I can. "

Roumiantseva says the problem sets aredifficult, too long, and often irrelevant to thematerial covered in class. Problems that introducenew concepts do not explain the concepts well, shesays.

Roumiantseva says she regrets her decision tostay in Math 55a and does not plan on taking ontaking Math 55b next semester.

"I feel like I really haven't learned very muchsince I spend most of the time working on problemsI never really understand and I never develop asolid understanding of the basic and importantconcepts we cover in class," she explains.

Roumiantseva is also taking a 100-level mathcourse this semester. She says she had intended toconcentrate in mathematics but that after takingMath 55, she is strongly reconsidering herdecision.

**Great Expectation**

Some students say they decide to take andcontinue with the course, however, because theyfeel that it is expected of them.

"Math 55 is a legend. I heard about it in highschool," Stronger, who took the class last year,says. "I would have been ashamed not to take it."

Jared S. Weinstein '01, a current student inthe class, concurs.

"You're expected to take it," he says. "I don'tknow if it's such a great expectation."

Etingof defends the necessity of Math 55, whichhe says helps structure the knowledge that manystudents come to Harvard with. According toEtingof, about 20 to 25 students enter with somuch mathematical background that they do not needto take the courses offered at the intermediate orlower levels.

"Before this course was created, we found thatstudents had a lot of knowledge, but that it wasnot systemized," he says. "So, they would jumpinto grad courses immediately, but without a firmbackground."

"We created course in calculus and linearalgebra, basic [concepts] in math, that wouldaddress these concepts in such a nontrivial anddeep way the people would still be interested intaking it," he says.

Nichols-Barrer, however, questions the need fora class like Math 55. He says a controversycurrently surrounds the choice students can makebetween Math 55 and 100-level courses, such as the122/123 track (first-year algebra), which meets atthe same time as Math 55.

"People don't need another calculus coursecoming straight out of high school," he says.

Williams agrees.

"If a person has had enough background so thathe or she can reasonably take Math 55, then he orshe can take upper-level math classes, or evengraduate classes," he says." There is no reasonfor there to be a freshman math class that coverswhat 55 does."

Math 55 is now an open enrollment course,although Etingof sets three prerequisites for hisstudents: "a great love of math, a great deal ofexperience with math and a great deal of time."

Etingof attributes much of the difficulty ofthe course to the homework.

"I give more homework than any otherundergraduate course," he says." I do give a lotof homework, but it's interesting, not technicaland routine."

"The lecture moves fast I don't follow anyparticular book; I rely on people being prepared,"Etingof adds.

Nichols-Barrer acknowledges that how wellstudents do in the class depends largely on theirbackground in mathematics.

"It totally depends on background," he says."It's ironic that the people who get the most outof it have had the most experience.

Students in Math 55 acquire that experience ina variety of ways.

Jaron M. Abbott '02, who is currently in theclass, says he read a lot of mathematics books inthe high school and came to Harvard withessentially the equivalent of an undergraduatemath education at a typical college.

Abbott says he finds the class enjoyable andappreciates the challenge.

"I can get excited about math and stay up allnight to do it," he says. "Math 55 forces you todo that. It's about as challenging as you'll get.

He adds," It's a little beyond my comfortlevel, but that's good. It's the way you learn."

Some of the class's current students attendedsummer programs, where they met other people whoshare a passion for math.

David E. Speyer '02, a student in Math 55,attended the Math Olympiad Program during hissummers in high school. Speyer says he finds thework manageable as long as he works hard andspends time on the homework.

"The whole reason [the problem sets] are hardis that you have to work stuff out for yourself,"he says. "Some new theories are not taught inclass, and you have to work through them to makethem understandable. It's a lot, it takes time,and it's hard, but it's interesting, and that'swhat makes it worth doing," Speyer says.

After computing the homework score for hiscurrent students, he say that the lowest score wasabout 72 percent, with most scores in the 80s and90s.

"Although the problem sets are difficult, weseem to have a very cooperative crowd this year,who work maybe 15 to 20 hours [on each set],"Etingof says. "People are doing much better thisyear. On average, we have stronger students."