April 2007 Archives
Physics has put me in a pure pain eigenstate.
One of my professors mentioned this article today in a meeting. All of us who were rejected by MIT at some point in our academic lives will find humor it in.
"I did bad notation."
--Ted Sanders's physics TA
"I have abused notation. I am sorry."
--Tudor Costin, my physics TA.
Between my quantum mechanics class and my directed reading project on Fourier analysis, I have experienced an incredible synergy of functional analysis, real analysis, algebra, and physics. Fourier analysis is a really interesting amalgam of these areas, and quantum mechanics takes on its full beauty in the terminology of Hilbert spaces and Dirac notation. While these subjects have been extremely fulfilling, I have experienced some minor but annoying problems parsing the zoo of verbiage and notation associated with these topics. Moreover, there is an incredible inconsistency in terminology and notation between physics and mathematics. This raises the paramount issue of the role of notation in math and physics.
This may initially seem like a trivial issue. After all, notation merely stands for some idea or operation; how one indicates the ideas seems to be irrelevant. However, I contend that notation actually affects the way in which a person can visualize and understand an idea. There is a very good reason, for example, that physicists have a strong appreciation for the Dirac notation in quantum mechanics.
This raises the question of what characteristics make notation good. I haven't come up with a lot of specifics, but I have some ideas. Perhaps I should start with what makes bad notation and bad terminology, since that is the area in which I have recently had experience. The first thing is inconsistency. Physics and mathematics have an incredibly gross amount of inconsistency in notation and terminology. Inner products in mathematics are represented usually with corner brackets and commas between entries. Occasionally they use paretheses. But physics always uses corner brackets, and uses the horizontal bar between entries only in quantum mechanics. Some of this is due to specialized (and elegant) notation for quantum mechanics, but these excessive standards are silly. The parenthesis notation is also particularly bad because it generates confusion with notation for vectors and coordinates. Complex conjugation is denoted by a bar above the argument in math but a superscript asterisk in physics. Fourier coefficients are sometimes denoted by hats above the name of the function, but other times they are just written as a or b with a subscript for the index. Laplace transforms are represented by several notations. The Laplacian is represented by a delta in some branches of mathematics, but it is represented by a nabla with a superscript 2 in other branches. The superscript convention in economics is disastrous. Rather than using subscripts on variables, economists will occasionally (but not always!) use superscripts, leading to catastrophic confusion with exponents (this is particularly bad because economists like to use the superscripts a and b, which are also frequently used as exponents).
Terminology is just as bad. Mathematicians call the space of square-integrable functions L2. Physicists call it Hilbert space. But mathematicians call complete normed inner-product spaces Hilbert spaces. So L1, L2,..., L∞ are all Hilbert spaces. This may seem trivial, but it makes an enormous difference. The self-adjoint operators in mathematics become Hermitian operators in physics, and adjoints become Hermitian conjugates. It's one big disaster.
So what makes good notation? Clearly consistency is a major necessity, as is uniformity. Notation and terminology should also explicitly reflect the ideas they represent. Saying L2 and self-adjoint is much better than referring to Hilbert and Hermite, regardless of how famous they were. Naming mathematical objects after people is bound to generate confusion. A less obvious example of good notation versus bad occurs in limits. Although it is longer, the "lim" notation is desirable because it is obvious what is means ("lim" is very close to "limit") and the subscript indicates the limiting variable and what it approaches. On the other hand the convention of writing an arrow to denote a limit is hideous. It is far less explicit and potentially generates confusion with the symbol for "implies."
Still, the limit notation is very bulky. Notation should be as brief as possible. I think the integral is a good example of this. It is easy to write and read, versatile in indicating boundaries and variables, and it has an obvious intuitive significance.
Notation is clearly one of these areas, like typesetting or displaying data, which is very much artistic but essential to facilitating understanding. The current state of notation is absolutely horrific, but I believe that common sense can go a long way in improving the way mathematics is written.
After moving on to page 17 of this week's problem sets, and failing to get more than 6 hours of sleep for about the 10th day straight, I have concluded that school feels vaguely reminiscent of being punched in the face repeatedly.
"Ah, they don't make them like they used to."
--Helmut, the machinist in the Research Institutes, commenting on the declining physical strength of physics students, as exemplified by my own inability to loosen bolts.
I've recently been rethinking my analysis of grade inflation. As far as I am concerned personally, the issue is a trivial one. I generally care much less about the actual grade itself than I do about the comments, criticisms, and flaws of my work. The indicator that the teacher chooses to assign to the work is meaningless to me outside of its instrumental value (which is still considerable... I can't pretend to be that idealistic). So the question that really interests me is whether teachers have become less critical of their students' work. Along with grade inflation has there been any kind of criticism deflation? I have absolutely no idea, and it is probably almost impossible to test.
My typical running route has been to Promontory Point since I have been in Chicago. I decided today, that I was silly for not having branched out sooner. So I jogged over through Jackson Park today, and it was rather pleasant. There is an interesting mixture of lagoons with waterfowl and large grassy expanses abutting the lake. I also noticed some remarkable historical indicators. There were a couple of odd statues of people who are apparently famous in some capacity (I am consistently amazed by the amount of historical significance in Hyde Park and the South Side). There was also a sign describing a missile launch site that existed in Jackson between roughly 1950 and 1970. I'm not sure who they were planning on bombing... I know we always suspected that Michigan would defect to the Soviets... Sadly the missile site was built over a meadow that was completely destroyed by the activity. Only today is the area beginning to recover from the decades of soil compaction. It is an interesting experiment in natural succession.
Despite the recent bout of good weather, everything is still dead and snow is predicted for later in the week. I have become convinced that Chicago has one of the more inhospitable climates in the US. It seems similar to Siberia, I believe, although tempered somewhat during winter. I guess spring is later here than on the west coast.
Still, Jackson Park is very pleasant. My next running expedition is to Washington Park, following which I want to explore the ghetto directly south of my dorm. I figure that a runner with nothing more than an ID card is a pretty bad target for a mugger. Besides, if you're a mugger, I might just outrun you.
I was thinking about the previous entry, and I noticed an amusing correlation. For the schools I listed below, here are their ranks on the Princeton Review's list of schools with happiest students:
Chicago: Not listed
Stanford: 5
Harvard: Not listed
Princeton: 2
Pomona: Not listed (although I believe Jonathan claimed that it was listed rather high a few years ago)
Brown: 1
UC Berkeley: Not listed
It's probably just a statistical anomaly, but it is amusing. People seem to like being told that they're smart.
Various social commentators and academics tend to make draconian claims about grade inflation and its effects. A google search on the subject exemplifies this impression, with descriptions like "fraud" and "nightmare" appended to titles of articles. The issue even forced Princeton University to enact a policy mandating that no more than 35% of students in a given class may receive A grades. The entire uproar and controversy conveniently distills to two main considerations: the extent of grade inflation and its effects. The strong presence of grade inflation is supported by virtually every empirical indicator that I can find. Nevertheless, while most commentators decry grade inflation as bankrupting the academy itself, I believe that it is mildly problematic, but mostly inconsequential.
The most extensive survey of grade inflation on the web comes from the site gradeinflation.com. Reports have indicated that the effect is most pronounced among the most elite universities. Data seems to confirm this observation. My own institution, the University of Chicago, has long been known for being one of the harshest graders in academia, yet it experienced massive grade inflation in the latter half of the 20th century.
Average GPA:
UChicago
1965: 2.50
1996: 3.26
Comparatively speaking, however, Chicago is a still a somewhat harsh grader compared to similar schools
Average GPA
Stanford
1968: 3.04
1992: 3.44
Harvard
1985: 3.17
2001: 3.39
Princeton
1971: 2.99
2001: 3.40
Pomona:
1970: 3.06
2001: 3.43
Brown
1989: 3.34
1999: 3.47
UC Berkeley:
1986: 2.95
1996: 3.10
These statistics are interesting, but they are not proof of grade inflation, strictly speaking. If students are indeed performing better today than they used to, then higher grades are justifiable. This would not be inflation, but merely the effect of increasing intelligence. I don't have statistics for this possibility, but I find the sheer magnitude and consistency of the rise in grades over time to render it doubtful. On the other hand, it may have some effect. The Flynn effect, for example, is a well-documented phenomenon in which IQ scores increase over time. Indeed, the IQ index is periodically recentered to 100 to account for this rise. So people are getting smarter, but it seems unlikely that they are getting that much smarter. I consequently believe that it is very reasonable that grade inflation exists to a significant degree in college.
The real issue, however, is whether all of this is a problem. I contend that it is, at worst, an annoyance. Most people decry grade inflation: exemplary work can no longer be distinguished from the shoddy, students are given a false impression of success, the GPA becomes meaningless, students cease learning... etc. But I think that these arguments rest upon the false premise that the absolute meaning of a grade is fixed. In other words, people assume that if I attended college in 1965 I would feel the same about receiving a B- as I would today. This is plainly false. A B- is a vastly worse grade today than it was in 1965. Academic agents constantly adapt their interpretation of grades based on prevailing grade level, in the same way that economic agents constantly adapt their prices based on prevailing prices.
I see three possible counterarguments to my line of argumentation, one of which is moot, and the other two of which legitimate. The moot point is that, as grades rise, grades become less detailed and sensitive. In other words, a hypothetical grade distribution might only include grades between C+ and A. In this range, there are only 6 possible choices. If we used the full range between F and A, we would have 12 levels. So there is less detail in the grade. This is not terribly important since, for all practical purposes, the GPA is the most used indicator of a students grades and it has two decimal places of accuracy. Furthermore, relative grades still give some measure of a student's performance.
The second counterargument is more serious. Unlike monetary inflation, in which prices can rise forever, grade inflation has a natural limit: 4.00. It is clearly not useful to have large numbers of students earning 4.00 GPAs. At this point, one really does lose the ability to differentiate performance. Even so, grade inflation only becomes a problem in this case, if we have large numbers of students earning perfect or near-perfect grades. I don't have comprehensive statistics on the matter, but I think this is not terribly problematic in college. At Chicago, for example, roughly 1 student graduates with a perfect record every other year or so. That's about 0.05% of the population. I don't see that as unreasonable. This effect was, however, unreasonable in our high school. We had 17 people with 4.00 GPAs, amounting to roughly 5% of the population (given the difficulty of classes however, most of these students probably genuinely deserved their grades).
The final, most damaging, but most tenuous counterargument to my view deals with the trend in grades. I mitigated the previous counterargument by pointing out that the current number of perfect students is not worrisome. Grade inflation is a dynamic process, so one should really be concerned with what will happen in the future. All indicators seem to suggest that grades are increasing, if they are moving at all. Lacking a firm theoretical basis for why grade inflation has occurred in the first place, this equation of past performance with future changes is not perfectly sound.
For aesthetic reasons, I find "uninflated" grading to be much preferable to the status quo. Nevertheless, I don't think grade inflation at most institutions has been particularly problematic. It is certainly not good, but I don't think that it is particularly troubling.
Speaking of theoretical reasons for grade inflation... I have my own theory about the commodification of higher education, but that will have to wait until another, not-so-late night.
