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 biannual battery of screening and qualifying examinations
given in midSeptember and midJanuary 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 twohour 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 assistantsones who have already successfully completed their
exams and are in the latter stages of thesis researchare 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 problemsolving 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 ViceChair 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 23 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 sessionsomeone that had already gone
through the processhelped 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 timeas was the case for all of usbecause 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.


