Tapping Physics Education Research for a graduate-level curriculum: A novel approach for a Ph.D. qualifying exam preparation course
Authors:
Warren Christensen, Iowa
State University
Larry Engelhardt, Iowa
State University
Introduction
Every summer a painful ritual is undertaken by many would-be
physicists in classrooms across the country.
A comprehensive written examination, although it has been modified or
even removed at certain institutions, is still a key measure used by many
schools to determine who is qualified to continue on a quest for a Ph.D. in
physics. Over the last two years, the
authors of this article have created a Ph.D. qualifying exam preparation course
that utilizes several research-proven methods.[1] These methods include, but are not limited
to, peer-led instruction, training in problem-solving skills, and the use of multiple
representations. The teaching of
upper-level undergraduate and introductory graduate physics content in this
manner was not only engaging and interesting, but also gave students an
opportunity to improve their understanding and, as is borne out in our data, an
improved chance to pass the exam.
Exam Background
At Iowa State University (ISU), the physics Ph.D. qualifying
exam is administered in two four-hour-long exams in a large lecture hall, on a
Tuesday and Thursday morning during the same week, approximately 2 weeks before
the start of fall classes. The first
exam, known as the “Classical Exam,” covers questions on Newtonian mechanics,
Lagrangian mechanics, electricity and magnetism, relativity, optics, as well as
qualitative questions about experiments or scientific ideas. The “Modern
Exam” (the Thursday exam) includes problems on quantum mechanics, condensed
matter physics, high energy physics, nuclear physics, and astrophysics, as well
as other modern topics.
Problems range in difficulty from introductory level
concepts to advanced graduate material, with most of the exam being at the
advanced undergraduate and first-year graduate school level. Until recently, students were required to
pass both exams in the same year in order to continue on with their
studies. Now, however, students can pass
the two parts in subsequent years and still continue toward their Ph.D. Graduate students who are new to ISU, entering
without a Masters degree, are expected to pass the exam within their first two
years, and students entering with a Masters degree are expected to pass after
one year. Those students who fail to
pass the exams in their allotted number of attempts become ineligible to
continue working towards a Ph.D. in physics at ISU. In most cases, these students choose to
finish a Masters degree from the department and then transfer to another
department on campus or to another university in a similar field of study.
Authors’ Background
We both entered graduate school having received degrees in
physics, but discovered that we were unprepared for the breadth and depth of
the exam’s topics. Upon the
recommendations of a graduate student who had previously failed his two
attempts to pass the qualifying exam, we confined ourselves in a classroom and
collaboratively worked problems from old qualifying exams for roughly 20-40
hours a week for multiple months.
Initially, we had many questions regarding the material, and we
discovered that the answers to our questions could not be efficiently found in
textbooks. We were provided solutions to
these old exam questions, but they were often limited to algebraic solutions
with very little written explanation.
Thus, determining how or why we were supposed to apply a certain method,
and interpreting details about underlying assumptions or approximations, was
nearly impossible. We found that
discussing ideas between ourselves was an effective method for studying but
lacked efficiency. What we really needed
was an “expert” to direct our conversations, leading us not only to correct
answers but to correct understanding as well.
Not surprisingly, neither of us was successful in passing
the exam on our first attempt, although the material that we encountered in
future courses became much more accessible and comprehensible. The following summer, we adopted a different
approach that we believed would improve the efficiency of our studying. We focused our attention on the key concepts
behind each particular problem and strived to look at a larger number of
problems. We met weekly with a larger
group of people to discuss specific worked problems, but also did a great deal
of independent studying. With these
revisions incorporated into our study tactics, we both passed the exam on our
second attempt, much to our own delight.
Course initiative
After an external review of the department and several
meetings between the department chair and the graduate student body, it was
obvious that the qualifying exam and, in particular, the lack of assistance in
passing it, was a significant cause of distress among graduate students. The department chair thus determined that
instruction directly focusing on the qualifying exam was desired by the
students, and he approached us with the idea of creating such a course. Having painstakingly developed our own
successful study techniques, and being familiar with proven pedagogical
techniques used in physics education research (PER), we enthusiastically
agreed.
Course structure
Our 12-week course covers a different subject each week,
alternating between topics in classical and modern physics. In a given week, two class meetings
occur. A one-hour introduction to the
material takes place early in the week, and later in the week the students
spend two hours presenting the solutions to assigned problems. In the first meeting, we introduce the weekly
topic in a brief PowerPoint® presentation, lasting no more than 20
minutes. We purposely minimize lecture
instruction for the following reasons: 1) Developing lecture instruction at an
appropriate level for everyone was impossible due to the diverse background
(and content knowledge) of our graduate student population. 2) Although it has
not been rigorously tested at the graduate level, the PER community has
provided overwhelming evidence that standard lecture instruction is not an
effective method of learning physics for the majority of students.[2]
The remainder of the first meeting has students working in
small groups, solving problems from old qualifying exams. These specific problems are chosen because
they satisfy two criteria: They are relatively straightforward, and they
clearly showcase the key aspects of the weekly topic. At the conclusion of this meeting, the
students are assigned five problems which are to be presented during the second
class of the week. The second class
period, which lasts two hours, involves students taking turns working problems
out at the board, spending 20-30 minutes per problem. Each student leads a discussion of the
solution, responds to questions, and is asked to elaborate on the concepts of a
particular problem in various ways that are discussed in the following section.
Another key feature of our course was the administration of
full-scale practice exams to students throughout the summer. One of the underlying challenges of passing
the exam is the context in which it is taken: A four-hour time limit, an 8 AM start time, and a formal test-taking
environment. This makes for a very
different experience when compared with a student’s typical problem-solving
environment (i.e., casually working problems often with readily available
resources). We therefore schedule four
different sets of exams that are administered to students in a classroom, at eight o’clock in the morning, on Tuesday and
Thursday mornings throughout the summer.
As one of our students stated, “They were quite helpful in forcing me to
sit through a full exam early in the morning in cramped conditions. The practice exams were also useful in that
by the time the real qualifying exam came by, it was old hat and I was quite
relaxed, which helps.”
Course Goals and
Methods
Our primary goal for the course is quite simple: To enable
students to pass the qualifying exam. In
order to succeed in this goal, there are a number of strategies that we
employ. Some of these strategies are
aimed at learning physics, by developing both our students’ conceptual
understanding and their problem solving abilities. Other methods focus on preparation and
test-taking tactics for the specific
type of exam for which they are studying.
In this section, we describe some of the specific methods that we use,
as well as our motivation for choosing them.
Efficient and
effective use of study time
Since the exam consists of solving written problems, it
seems obvious that the most appropriate means of studying is also to solve
problems. However, we found that many
students relied primarily on reading physics books to prepare for the exam. We therefore placed an enormous emphasis on
working problems, both during the class hours and throughout the rest of the
week. Solving problems, however, is quite
challenging if one does not already have a firm grasp of the different topics
and methods that should (and should not) be employed to solve the myriad of
problems. Simply being told “work
problems” can lead to hours of painfully inefficient studying as we discovered
during our first summer of preparation.
The alternative, reading books, has the advantage that one
can easily make progress, but it is a highly ineffective means of studying for
this type of exam. The central strategies of our course are
therefore to provide the students with summaries of the most relevant physics
content (in the form of our 20-minute presentations) and to provide immediate
feedback on their progress (in the remaining 160 minutes of weekly class time).
If we could focus students’ time on
working problems in an open group environment that allowed for immediate feedback,
we were confident that we would give them the best chance to succeed.
With
this in mind, we set out to create an environment in our class that would
support students discussing, critiquing, and assisting one another. We had groups of students work problems under
the guidance of experienced exam-takers (i.e. us), with rapid feedback
regarding both their solutions and their solution methods. While working problems, student questions
arise and are often redirected back to the other class members, asking for
volunteers to explain certain techniques or ideas. This not only helps answer the inquisitive
student’s question, but it also allows another student the opportunity to
explain his or her ideas, thereby benefiting both students. In addition, other members of the class
become involved in the process, commenting and asking further questions. Our role as peer-instructors (we are fellow graduate students) further
facilitates these discussions in that students do not hesitate to engage us in
healthy debate. Unlike the previous
alternatives that we described, solving challenging problems in this way is
very efficient, since in a class of fifteen graduate students, someone almost
always knows the answer or method that should be used to solve the problem.
Pedagogical methods
We strive to build on student understanding primarily via the
student-student and student-instructor interactions that occur while students
are solving problems. While these
interactions are present during the first meeting each week, they truly flourish
throughout the second meeting when students are working problems at the
board. Instructor-led discussions cover
all aspects relevant to the problem, with particular emphasis placed on
promoting problem-solving skills. The
ability to identify key ideas and plan an efficient solution strategy is
imperative for success on the exam. There is a vast research base that supports
the notion that use of structured problem-solving strategies is an effective
means of developing student conceptual understanding.[3] Additionally, we explore alternative
contexts, alternative solution methods, and how slight modifications to the
question would affect the solution. The goal is to strengthen the understanding
of the student working at the board by challenging them to think on their feet,
while also eliciting ideas from the class to paint a complete picture of how each
problem fits in with other concepts.
Another pedagogical technique that has been shown to improve
student conceptual understanding is the use of graphical and diagrammatic
representations,[4] both of which
are often required as a part of qualifying exam problems. While initially we felt it was important to
practice such skills to be prepared for these types of questions, we
subsequently realized that substantial knowledge can be gleaned from a proper
sketch, and that improved depth of understanding can result from analyzing
it. Once a sketch has been produced,
questions concerning limiting cases and points of interest (such as equilibria)
are readily tractable. By using
graphical representations, peer-led instruction, and a variety of other
methods, we continually refocus students' attention on their method of approach
to solving problems.
The scope of the exam
A key feature of our course is the highly focused nature
with which we present the material.
During our own exam preparation, we spent a great deal of time
determining what types of questions are commonly asked in order to improve the efficiency
of our studying. To specialize our
course, (and to save our students from unnecessarily investing similar time) we
meticulously cataloged and analyzed the most common topics and problem-solving
methods that have been used in previous years of the exam; we hence determined
which topics should be covered, and in which order. We also provided our students with a detailed
inventory of all 26 years worth of old exam problems. Sorted primarily by topic, this resource
allows students who are looking to practice, for instance, boundary value
problems, to instantly locate 19 previously asked qualifying exam questions.
Language
A few weeks into the first summer of teaching the course, we
become aware that, at times, students were misinterpreting portions of the
questions. This was sometimes as simple
as clarifying the distinctions among scientific words (e.g., constant, uniform,
invariant). Occasionally confusion also
arose when students were trying to interpret the instructions in the question,
such as the difference between “Write down …”, “Determine…”, and
“Derive…”. Students, particularly those
who received undergraduate educations outside the United
States, also had difficulties narrowing the
scope of particular problems. When
discussing problems, we therefore make a pointed effort to address precisely
what each question is asking and what is required for the solution. While this may seem trivial to some,
considering the timed nature of the exam it is important to focus students on
doing the work that will yield the most points.
As one student remarked after taking our course, “As a foreign student,
language is always a barrier… I need to be familiar with the way they ask
questions.”
Additional resources
We also highly recommend the series of books titled “Major
American Universities Ph. D. Qualifying Questions and Solutions” (1998)[5] as
another resource for problems at the appropriate level. A set of these books was purchased by the
department and is on reserve for the students.
All other resources are made available to the students online,
and recently the school produced CDs that contained all of our course material,
including PowerPoint® files, the question inventory, and every qualifying exam
with its solutions in electronic format going back to 1979.
Data
This past August, 37 Ph.D. hopefuls took at least some
portion of the qualifying exam. Of these
students, 17 were new arrivals at ISU and, as such, had limited opportunities
to attend our summer preparatory course.
Typically, these students have little to no chance of passing the exam
anyway, so we have removed them from our data set. Furthermore, due to the recent change in the
passing requirements, five students taking the exam only had to pass one
portion of the exam (all five did pass).
By also removing those five students from our data, we are left with 15
students, eight of whom regularly attended our course. To attempt to assess the effectiveness of our
course, we have analyzed the performance of those 15 students.
Figure 1. Student
Qualifying Exam Performance
As shown in figure 1, five of the eight students who attended
class regularly passed at least one of the exams, while only two out of seven non-attendees
passed. Although a higher percentage of
our attendees passed, it was not obvious whether this was as a result of having
attended our course, or if the students who attended our course were already
better prepared. In an attempt to shed
additional light on this issue, we obtained the average scores that these two
groups achieved on the GRE Quantitative and GRE Physics Exams which they took
prior to entering graduate school. These
data, shown in figure 2, suggests that our attendees were unlikely to have had
any type of pre-instruction advantage. Given
this very small sample of students, it is impossible to claim any statistical significance
with these findings. However, we believe
that these data suggest that our course is successfully fulfilling its goal,
that is, to enable students to pass the qualifying exam.
Figure 2. Average GRE
Quantitative and Physics Percentiles
Conclusions
We have developed a summer-long course whose goal is to
prepare graduate students for the comprehensive written qualifying examination
that is administered at Iowa State
University. This course is taught using pedagogical
methods from Physics Education Research that have been proven to be effective
at the introductory level, with a particular emphasis on active learning and
peer-led instruction. We also teach
efficient studying techniques and stress their importance in order to
drastically improve our students’ chances of passing the exam in a matter of
mere weeks. Data are presented which
suggest that this course is indeed effective.
We believe that this course could effectively serve as a model, both for
qualifying exam preparation at other universities and for GRE exam preparation
for advanced undergraduates.
Acknowledgments
We would like to acknowledge with gratitude the contributions
of the late Ngoc-Loan Nguyen, in particular his insight and interest in
assisting fellow graduate students in their exam studying. Thanks to Eli Rosenberg,
our department chair, for his drive and financial support in improving the opportunities
(and the quality of life in general) for graduate students at ISU. Also, thanks to David Meltzer for his
invaluable discussions and suggestions, and to Lori Hockett, the graduate
secretary, for compiling the data from student records.
Perhaps
an analogy might better explain our idea:
You and I are going to have a swimming contest in three-month’s
time. I am going to spend that time
reading all of the best books and articles about proper swimming
techniques. Meanwhile, you will go to a pool
and swim everyday for three months. Who
do you think will win the race?
Note
that the data presented in this section were only given to the authors in
summary form in order to protect the confidentiality of the results for those
who took the exams.
[1]
“Resource Letter: PER-1: Physics Education Research,” L.C. McDermott and E.F.
Redish, Am. J. Phys. 67, 755-767 (1999). Surveys a wide
scope of research on teaching methods for and student learning of physics at
nearly all levels.
[2]
“Guest
Commentary, How we teach and how students learn-A mismatch?,” L.C. McDermott, Am. J. Phys. 61, 295-298 (1993).
[3]
“Resource Letter: RPS-1: Research in problem solving,” L. Hsu, E. Brewe, T.M.
Foster, and K.A. Harper, Am. J. Phys.
72 (9), 1147-1156 (2004). Comprehensive
review of relevant research on problem solving.
[4]
“Relation between students’ problem-solving performance and representational
format,” D.E. Meltzer, Am. J. Phys. 73, 463-478 (2005). As part of a
discussion of student difficulties with multiple representations, this paper
contains a thorough listing of references on research in the use of graphical
and other forms of representation in
physics instruction.
[5] “Major
American Universities Ph. D. Qualifying Questions and Solutions,” Y.-K. Lim,
ed., (World Scientific Publishing Co. Pte. Ltd., 1998). There are a total of
seven volumes in this series.
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