Qualifying Exam Review Sessions:

An Exemplary Element Snapshot

Department of Mathematics, University of Southern California

review session for Real Analysis
review session for Real Analysis

Description of the review sessions

In the fall 2004 semester, the USC Department of Mathematics initiated a comprehensive program of qualifying examination review sessions. The primary goal of these sessions was to aid students in preparing for the departments bi-annual battery of screening and qualifying examinations given in mid-September and mid-January of each year. For the Ph.D. in Applied Mathematics, the students are required to take and pass four one hour examinations: Numerical Analysis, Real Analysis, Probability, and Statistics. These exams are traditionally taken after the student has completed one year of graduate study. For the Ph.D. in Core Mathematics, the students take a two hour screening exam in either Algebra or Analysis after completing one year of graduate study. When they have completed their second year, they take two additional two hour exams. One of these exams is chosen from the following areas: Geometry and Topology; Probability and Statistics; and Ordinary and Partial Differential Equations. The other two-hour exam the student must take will be either Algebra or Analysis (whichever one they had not taken previously).

The review sessions in detail

The qualifying exam review sessions are organized as follows. Senior graduate assistants--ones who have already successfully completed their exams and are in the latter stages of thesis research--are assigned to be graders for the principal courses corresponding to each of the exams. Grading is typically considered only a half time appointment (approximately 7 hours per week). For the other half of their teaching assignments, they lead a qualifying examination review session for the exam which corresponds to the course they are grading. The sessions also serve as problem sessions for the course, since it is essentially the students in the course who will be taking the corresponding qualifying exam that semester. The review sessions meet once or twice a week in a variety of settings. The sessions might be held in the department's Math Center, in a university snack bar or even in a local coffee shop. The senior graduate assistant leads the session by providing problems (often from old exams and other texts, or from exams at other universities) to be worked on and discussed at the session. The sessions are meant to function as a participatory workshop rather than as a formal class. The students are also encouraged to bring problems for the group to work on. We consider it important to remind the graduate assistant leading the session that it is not essential that he or she be able to work all of the problems to be discussed at the session. In fact, we believe that students learn more from watching seasoned graduate students think on their feet about a problem and solve that same problem in front of them than they would learn from simply being presented with a solution in its final polished form.

Educational purpose of the review sessions

The primary purpose of the sessions is to help our students prepare to take and pass the department's screening and qualifying exams. By design, the exams cover a body of material rather than just what is covered in a particular course; consequently, our students in the past have found it difficult to prepare for the exams. This is especially true of our domestic students who have come directly from an undergraduate program. The sessions are intended to stress problem-solving skills in addition to fostering an understanding and memorization of basic facts and theorems.

Impact of the review sessions

The department witnessed a significant increase in the pass rate in all but one of the exams offered during the fall 2004 semester. For the first time in anyone's memory all of the students taking the screening/qualifying exam in graduate real analysis (measure theory, Lebesgue Integration, elements of functional analysis, etc.) passed at the Ph.D. level.

Reflection from a faculty member

by Gary Rosen, Professor and Graduate Vice-Chair

Students not passing the screening and qualifying examinations has become a real problem. For a variety of reasons, the students in our program are arriving less well prepared, on average, than they were even a decade ago. This may in part be a result of our more active recruitment of domestic students; it may also reflect the fact that more of the students in our Ph.D. program in Applied Mathematics come to us from undergraduate programs in disciplines other than mathematics (such as engineering, physics, and biology). Regardless of its origins, however, these students' unpreparedness hinders them from passing the exams and thus has become a significant drag on our program. Because the exams are offered only once a semester (twice a year), failure on any exam sets the student back a full semester. Since students can not really begin their research until they have successfully completed the exams, their graduate careers end up in limbo. This state of affairs has had negative consequences for both the students and our program as a whole. Anything that could be done to help mitigate this problem would be welcome. The qualifying exam review sessions, at least for this first semester that we have run them, appear to be quite promising.

Student reflections on the review sessions

by Andrea Jedwab and John Mayberry, Graduate Students

Preparing for comprehensive written exams can be a daunting task, but it can be less intimidating when there are others with whom you go through the process. The Qualifying Exam Review Sessions, which were run for the first time in Fall 2004, were designed to give students an opportunity to meet regularly in order to discuss problems relating to the exams. In addition, a senior graduate student was present at each meeting to provide the students with further guidance and direction.

In past semesters, the results on the exams were not as positive as the students might have hoped. These results stemmed, at least in part, from the students' inexperience in taking comprehensive departmental exams and their lack of understanding about how to prepare for them. In order to help students along, the review sessions, led by a senior graduate student mentor, were meant to create an environment in which the students would be able to discuss problems and learn techniques for preparing for the exams. The long term goal is to expedite the Ph.D process by allowing students to complete this important first requirement on time.

Sessions were held for the Probability, Real Analysis, Algebra, and Numerical Analysis exams, and each had its own distinctive, but informal manner. Our observations are based on actual comments from students who participated in the first two sessions.

Real Analysis: The exam sessions for the Real Analysis exam were held weekly and usually consisted of 2-3 hours of problem solving and discussion. The senior graduate student gave us direction by suggesting problems from past exams and other sources; then collectively as a group, we would work out the key points of the solution. Next, we were encouraged to write out detailed solutions on our own. I found these sessions very helpful to my preparation as they gave me an opportunity to check my own understanding of the material with others and also exposed me to alternate methods of thinking about key concepts. I had previously attempted and failed this exam the previous semester, one reason being I did not know on what material I should concentrate. Having a senior graduate student running the session--someone that had already gone through the process--helped focus my efforts and exposed me to a greater variety of problems. Another benefit of the sessions was that it provided all of us students taking the exam with an opportunity to bond. After all, any formidable task is always made easier when we have company with whom to share our struggles.

Algebra: Going to the algebra review sessions was very helpful for me while preparing for the exam. Most of the time the group consisted of the same three participants. Since we did not otherwise study together on a daily basis, this was the time to come together and share our concerns. I would say it functioned as a type of office hours: each student would bring in questions on problems he or she could not solve, and the rest of the people in the group would try to help. It was a good way of figuring out whether your approach to certain problems was on track, and where your preparations stood with respect to those of the other students. That is always an important concern when you take an exam like this for the first time--as was the case for all of us--because it is hard to know when you are really prepared. Seeing what your classmates are doing gives you an idea on how you are doing and how to face the exam period. Having a mentor available who had already been through the exam was very important, not only because he could help with the problems and in understanding material better, but also because it is always useful to listen to somebody else's experience and be able to share your worries and doubts. Besides, you knew there was that hour in the week when you could count on someone helping you, instead of having to bother somebody else asking them for help.

I think this is a very good resource for students taking the qualifying exams, and the department should continue to offer it.

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