Another semester and another new class - general relativity and thermodynamics

When I was an undergraduate at the University of Pennsylvania, I took a class in general relativity (GR). It was taught by the eccentric Professor Jeffrey Cohen, not to be confused with the equally eccentric Professor Michael Cohen.

Michael Cohen had instilled in me a deep appreciation for truly understanding physics. Just as Michael Cohen felt that he would never attain the depth of understanding commanded by his adviser, THE Richard Feynman , I too feel that I will never approach the physical intuition of Michael Cohen. It is fortunate that the singularities that we call great physicists are born with abilities far superior to their contemporaries.

After more than three decades have passed, I recall little from my undergraduate flirtation with GR. However, some of the mathematical formalism of differential forms has taunted me for much of my career. I recall Jeffrey Cohen mentioning a paper on the topic of the properties of a black hole in some complex geometry that took forty pages of derivations in an article that appeared in The Physical Review. Using the trickery of differential forms, he was able to solve the problem in just a few steps.

The trick was to formulate the problem in a coordinate independent way, then to project the results into the coordinate system that reflected the symmetry of the problem. In contrast, the Physical Review paper used the inelegant brute-force approach of picking the coordinate system up front, and then by necessity painstakingly plodding through all the messy mathematics.

Given the complexity of the problems that we work on as a matter of daily routine in our research, I am always looking for simplifying tools. In teaching various classes, my intention is to sneak in a little bit of differential forms to wet the appetites of my acolytes and to teach my old brain some new tricks. Furthermore, the geometric interpretation of the mathematics adds a deeper layer of understanding.

In the upcoming spring semester, I am teaching graduate statistical mechanics for my first time. As usual, preparing for a new course if filled with grand excitement. You can imagine my elation when I realized that a homework assignment in the textbook could be done with ease using differential geometry. Since then, it has been difficult for me to think about anything else.

The problem is a simple one that normally requires a bit of math. The student is to show that the 6N-dimensional volume element in phase space for an N-particle system is invariant under a canonical transformation. To put this into simple English, the problem seeks to show that a transformation of coordinates does not change the nature of the results. Be reformulating the problem so that the volume element is represented as a wedge product of what are called one-forms, the volume element is shown to be the same when the so-called Poisson bracket yields unity -- the requirement of a canonical transformation. Thus, the problem is solved without the need for messy mathematics.

This realization makes me feel like a kid at Christmas. Ironically, tonight is Christmas Eve, the focal point of my family's celebration. My father has made what may be his last trip to Pullman from Philadelphia. He is 94 and still lives on his own, drives a car, and prepares meals for senior citizens at the Ukrainian Cultural Center in Fox Chase, Pennsylvania. Though still vigorous, his body betrays the telltale signs of wear and tear due to old age. Both of my children are home for the holidays, and all the fragrance from the traditional Ukrainian foods simmering on the stove and in the oven permeate the house. As I write this post, my wife is busily making last-minute preparations.

It is fortunate for me that my family values my passion for physics, and allows me to occasionally be a recluse. Just a few minutes ago, my wife called out a query about my whereabouts. I simply answered, "I am excited about something." Though she undoubtedly had some mundane duty for me to perform, she immediately signaled her understanding of my state of mind, and left me alone. I am truly fortunate to be living with someone who shares in my passions.

The intensity and meaningfulness of spirituality that I derive from physics far exceeds all others, including the times in my distant past when I had embraced religion. As my family turns in for the night, I continue to sit at my desk, full of excitement in my new-found understanding, and looking forward to sharing this understanding with my family and my students. It is a truly privileged life that allows me to rekindles the child-like wonder of Christmas on a daily basis.

Preparing for class

I am spending large chunks of time preparing for a new class that I am teaching this fall. I never found classical mechanics particularly difficult; but, writing lectures and solving problems is still a challenge.

I devote each Saturday and Tuesday to course development. Though I took a couple of breaks to eat and surf the web, I remained true to my schedule. After grading last week's homework assignment and a quiz on Saturday, I continued to plow through the textbook and organize the material for my lectures.

My serious teaching career (excluding my work as a teaching assistant in grad school and an instructor at a community college) started 20 years ago at Washington State University. Those years have not jaded my enthusiasm for teaching. I continue to make adjustments based on my past successes failures.

But how are improvements possible when good teaching is difficult to define? While there is mounting research on pedagogy, in my mind, many of these studies are inherently flawed. For example, while courses that are based on peer instruction and conceptual problem solving certainly lead to better understanding, it makes it difficult to cover the same amount material. Admittedly, it is wasteful to teach lots of stuff that everyone forgets; but, coverage is also important - especially if a course is a prerequisite in a sequence courses. Such trade-offs must be carefully weighed in an undergraduate curriculum, especially for the student who will take one or two physics classes in a lifetime.

Graduate programs, on the other hand, are populated by motivated students who have both an interest in and an aptitude for physics. PhD programs like ours are based on coursework and research. All students take a set of core classes in their first two years. To become a PhD candidate, they must pass the PhD qualifying exam, which is given at the end of the forth semester.

The qualifier exam committee solicits two problems from each faculty member. A list of topic areas is used as the basis of the solicitation to ensure that the test is well-balanced. Thus, the qualifier exam reflects what the faculty as a whole believe is the core competency of a PhD in physics, and is not necessarily limited to the material covered in class. However, past exams are made available to the students as a study aid.

I find this system to be the ideal game-theoretic approach to ensure buy-in by all parties. It reflects poorly on me as a professor if the students were to do poorly on the part of the exam that is associated with my course. This makes me think very carefully about how to most effectively cover the material to get the optimal balance between breadth and depth. The looming qualifier exam motivates the students to learn the material for long-term understanding, not just for a particular test. Everyone works harder as a result.

My approach is to stress the fundamentals and give lots of examples. I gloss over topics that the students can learn on their own. When preparing a lecture, I first read the material, sometimes many times, until I feel I have a good understanding of the concepts. I then identify what I think are the important points, and use them as anchors in my lectures. I post readings and homework assignments on the web well ahead of the lecture dates so that the students are well prepared for class.

In crafting my lectures, I select problems from the end of the chapter on the basis of how well they complement my notes and whether or not they challenge the student to think more deeply. To select appropriate problems requires me to first suffer through the calculations. This time-consuming task give me ideas on how to fine tune my notes to address potential misunderstandings.

The night before my class, I go over my notes to make sure that I stress the important issues. The morning before my class, I copy my notes to scrap paper, going over in my mind the flow of my presentation. I pay particular attention to the anchor points.

Finally, I reproduce my lecture on scrap paper without looking at my notes. My preparation is complete when I am able to navigate from one anchor point to the next based on general principles of physics and logic.

So to finally get back to my diary entry after this long-winded diversion, I spent most of Saturday reading the textbook, writing notes, and working physics problems. This reminded me of how learning physics requires an intense and prolonged effort. I am thankful to my employer that my teaching responsibilities force me to spend time thinking very deeply about the subject that I love so dearly. And then, I get to share my enjoyment with a group acolytes who share my enthusiasm for learning.

Student Evaluations

Just this morning, while clearing off my desk, I leafed through my teaching evaluations before filing them away. As I do with declined proposals, I usually let some time pass after the end of the semester before reading them.

Our department very reasonably allows each instructor to use a unique format that each of us feels most appropriate. My large introductory class evaluations are designed like a product survey, where I ask the students to rank various aspects of my teaching, followed by an area where they are free to make comments. I take the numerical rankings with a grain of salt, but read the comments carefully. They can be inane, but sometimes I get useful comments. For example, one student pointed out that color blind individuals had trouble seeing the red markers that I sometimes used. I wish that someone had pointed this out earlier.

In the most advanced classes, I solicit an essay rather than have graduate students check boxes. Being conditioned to fill out surveys, even some of the most sophisticated students are compelled to rate me numerically.

Our university is moving in the direction of a standardized forms. Some colleges at my university already have done so. When I complain that it makes no sense to have freshman students who are taking a general education requirement, and graduate students who are taking advanced quantum mechanics to answer the same survey questions, I am condescendingly reminded that I can add additional questions or leave room for an essay.

Contrary to firm denials by administrators, teaching quality is often quantified by one number. I was once a member of a tenure and promotion committee in a college that uses standardized forms. Faculty X, who had an average evaluation of 3.2/4, was deemed a better instructor than Faculty Y, who had an average score of 3.1/4. The fact that courses taught by these two faculty members were at different levels and had students of differing demographics was not taken into account. It is not unexpected that a major taking an advanced course will rate an instructor differently than an introductory student who is in a crowded lecture hall with hundreds of other students.

Hiding behind numbers are a convenient way to excuse oneself from the responsibility of making a judgment. When I did annual reviews for our faculty, I read every comment - even in classes with 200 students. One particular faculty member had a low numerical score for his teaching. A small majority of comments said that the course was too hard. Another set of responses stated that the course was challenging but very interesting, and that this particular professor took extra time in class to explain difficult concepts. Also, he was available in his office whenever the students wanted to talk. The comments as a whole painted a much more positive picture than his scores alone. In contrast, a high-scoring faculty member had comments to the effect that "this class was fun." There were few comments that mentioned anything about learning or being challenged. I fear that focusing on a single number generated by students on a standardized form will encourage superficial teaching. We are already sending a message that if the material is hard, there is something wrong with the teacher.

My teaching style has changed over the years, especially in the advanced graduate classes. I used to prepare my notes very carefully and tried to be very precise in my delivery. Given the complexity of the subject matter, I was effectively copying from my notes to the blackboard to avoid errors: I was not thinking, but rather reciting. Though I had very good teaching evaluations, I felt that the students were simply creating a transcript of my lecture. They were not thinking in real time.

As I got older, and felt less obligated to please the students, I changed my style. Rather than read from my notes, I prepare very carefully for each lecture, making sure that I understand the ideas that I am trying to convey as well as the critical parts of derivations. Since the mathematics is quite complex, it is often difficult to remember the logic when on my feet. For this, I use the students to help me.

Because I am thinking as I am teaching, the pace is just right for the students to think along with me. When I get stuck or make a mistake, the students are always more than happy to point out my error or to help me out. This approach is difficult on me, because when I screw up, sometimes it may take a while for the students or me to notice. The longer it takes to find the error, the more painful is the process of getting back on course.

I believe that the students are learning far more than in the days when I meticulously followed my notes. However, I invariably get a sprinkling of comments that I tended to make mistakes or that I was sloppy (or that I was unprepared!). In the past, I would have cringed at this criticism, but I am now confident enough in my methods to discount them.

Recently, a colleague passed along an interesting essay about class evaluations that appeared in his opinion piece in the New York Times titled Deep in the Heart of Texas, by Stanly Fish. The author writes:

"But sometimes (although not always) effective teaching involves the deliberate inducing of confusion, the withholding of clarity, the refusal to provide answers; sometimes a class or an entire semester is spent being taken down various garden paths leading to dead ends that require inquiry to begin all over again, with the same discombobulating result..."

I was intrigued by this analysis, not only because I was coincidentally musing about the same topic, but it got me thinking more about the corporatization of academia. How does one balance the need to assess teaching with the reality that the seeds of our labor do not take root until many years after graduation?

In a perfect world, the academic would be an expert teacher who understands how teaching techniques today lead to the desired outcomes years or decades later. Since the benefits of a good education appreciate over time, no single teacher can live to fully observe the fruits of their labor. Such information is necessarily accumulated over the generations.

Societies with highly-educated individuals tend to have the highest quality of life characterized by low crime rates, material comforts, good health, vigorous economies and functional institutions. As such, how does one ensure great teaching and efficient learning when it is so difficult to judge a teacher based on direct observation of the results?

Students who enjoy learning will find ways to succeed even when the mode of instruction is sub-optimal. If prosperity is to flourish in the general population, we must think of ways to educate the large number of students who see education solely as a route to increasing earning potential. In my perhaps biased sample, I find this large cohort's approach to education as would a patient undergoing an unpleasant medical procedure, thinking "I want to get through this process with the minimum pain."

Such a student views the diploma as a membership card entitling the holder to a well-paying job. From the perspective of the employer, the diploma is evidence of an education, whose quality is commensurate with the reputation of the granting institution. As educators, it is our responsibility to stand behind our product.

To understand good teaching, it is important to view it in light of the mission of the institution. The goal of a university is to produce individuals who have demonstrated the ability to think when confronted with novelty, being able to analyze and assess complex issues, and devise thoughtful long-term solutions without rigidly adhering to preconceived notions. These traits are necessarily learned within some context, which is provided by the student's major program of study.

I would thus associate good teaching with an atmosphere of continual challenge. Learning should be uncomfortable. Challenging problems do not have cook-book solutions. One must try new approaches that often lead to dead ends. Students often complain when they are not given recipes for getting from point A to point B. Some yearn for facts and feel that any amount of confusion, however small, is an indicator of bad teaching.

I will not attempt to answer the difficult question regarding ways of insuring good teaching. It is far easier to identify a perverse reward system, such as the mean numerical score of student evaluations as the basis of faculty compensation. If we want our students to think outside of the box, we are sending the wrong message by taking seriously the results of simple-minded surveys whose design is the antithesis of the skills that we endeavor to teach. It is reasonable for Starbucks to take seriously my displeasure with getting a cold cup of coffee; but, if ensuring the long-term efficacy of our educationally system is the goal, longer-term metrics based on thoughtful study need to be developed and implemented.