Critical Mass typically concentrates on the problems plaguing the academic humanities and some of the social sciences--that's mostly a matter of making sure I write about what I know, and it's partly a matter of assuming that there's more there there in the hard sciences. I still think that's true. But in response to my call for readers to send in their thoughts about graduate school attrition, I've gotten quite a bit of mail from people in the sciences and what they have to say dovetails quite neatly with many of the ongoing themes of this website.

I've written a lot about how humanities education today is departing from the idea that there is a more or less stable canon of literary works that should form the basis of study in the field, and how it is replacing that idea with an emphasis on theory and on inclusion. Graduate education today is shaped increasingly around reading secondary and even tertiary works--though most students arrive at grad school without a thorough grounding in, say, English and American literature, their coursework does not center on ensuring that they get this grounding. Instead, it is frequently centered on introducing them, via a series of disconnected, overfocussed seminars on sundry rarified topics of special interest to the professor, to the current theories--of race, class, gender, sexuality, nationhood, power, oppression, desire, performance, and, of course, reading. The training in theory--itself scattershot and unanchored in the history of ideas--becomes a substitute for the thorough grounding in literature that the students, and a growing number of younger professors, never had. It ensures that they will not only never get this training, but also that they are equipped with a strong "theoretical" rationale for why their lack of grounding does not matter ("grounding" is never possible, and what is "grounding" anyway? How can we speak of a literary tradition in which to become grounded without becoming ourselves complicit in the oppressive and exclusionary power structures upon which that tradition is based? And so on). It's enough to make you throw up your hands in despair, watching graduate education in the humanities short-circuit itself.

But it's not just this way in the humanities. It's also, apparently, this way in math. Here's what a woman who is completing her Master's in algebraic geometry says about why she won't be going on to get a Ph.D.:

My story: I'm a third-year masters student in algebraic geometry. The math masters program here is a two-year program, and it's not unusual for students to take three years to finish their programs. IÝexpect I will be finished my masters degree by the middle of the summer. Two and a half years ago, I fully expected to completeÝmy Ph.D. here. Now, I am struggling to keep myself motivated enough to complete my masters essay, and it is only in recent weeks that I became confident that I would get even that far. Last semester, I abandoned the research topic on which I'd made almost no progress during the year and a half I'd worked on it, and am now working on a topic that is more interesting, under an advisor who is more helpful.

For a year and a half, I worked - sort of - on a research topic that my advisor chose for me. This is not unusual for math students; in fact, choosing a topic for their students is one of the primary roles of thesis supervisors in math, as students are generally unable to tell the difference between a tractable (and thesis-worthy) problem that they doÝnot yet understand, and a very difficult problem that they do not understand. Indeed, most math students do not understand course descriptions for their classes until they've finished taking the course, whereas I, someone with no background in, say, history, can easily understand the description ofÝan upper-year history course - so it's not surprising that math grad students are so dependent upon their advisors. Anyway, my research problem (to learn some theory that would allow me to extend some theory he'd developed in a paper of his) was one that I found pretty much impossible at first, but I was told that this is common for math grad students. So I stuck with it.

I held nothing back in asking my supervisor for help and direction. Half my questions were dismissed - I was assured that it was okay that I didn't understand why Theorem A or Proposition X was true; it mattered only that I knew how to apply them. I soon learned that this too is typical in math - the rush is to produce "original work" (highly overrated; in all likelihood adding together the numbers 35143514361361564452561 and 2514905146146095714870 is something no one's done before, but who cares?), which doesn't leave much time to learn the material. If I can just apply other people's results, then I will be able to produce new results that can be written up into a thesis - never mind that I would not have the mathematical certainty that I crave when I work on math problems. Sure, my thesis would be one that I won't understand very well,Ýbut at least I'd have a thesis, and isn't that the point? (Besides, I soon realized - my *supervisor* would understand my thesis, which would contain extensions of *his*Ýtheories.)

After talking to some other students and professors in my field, I learned that virtually everywhere in math, the rush to be original andÝto publish original results before anyone else publishes them (as well as the rush to finish a thesis) trumps the motivation to understand - or teach - mathematics deeply. Consequently, the body of newly-published mathematical papers is a mess: every paper contains references to dozens theorems in other papers, and as often or not, following the trail of citations fails to yield an actual proof of the theorem or theoremsÝin question. Proofs in math papers are often mere notational sleight-of-hand, failing to prove anything at all. One professor at my school estimates that around 20% of math papers published today are "fatally flawed" - that is, their key results rest upon theorems that are incompletely proven, inadequately proven, or just plain false. Refereeing math papers is a thankless task; consequently, it is seldom done.ÝStories abound of newly-minted Ph.D.'s who land tenure track positions for their groundbreaking theses - which, years later, are shown to be garbage. One subfield of algebraic geometry, according to my former supervisor, is without any textbook; proofs of the most frequently cited theorems in this field appear absolutely nowhere in print. Apparently they're true, but the retired emeritus professor who came up with them never bothered to write up the proofs. Part of what attracted me - and many other grad students - to math, was the assurance that at least in this field, we wouldn't have to defer to authority in assessing what was true. We had rules, and they would guide us. I have heard the phrase "trust me, it's true" so often that I feel like apologizing to my field.

Mathematics as a subject is cumulative, more cumulative than any other academic subject, and I am convinced that this makes it distinctly unsuited to the publish-or-perish ethic that dominates academia. A typical undergraduate math curriculum includes courses such as group theory, real analysis, linear algebra, and abstract algebra. Fifty years ago, a typical math grad student would write a thesis in advanced group theory, or advanced real analysis, or advanced linear algebra, or advanced abstract algebra, building on the background they acquired as an undergraduate. Today, those fields have been more or less exploited, and so graduate students much reach higher. So a typical undergraduate curriculum still includes group theory, real analysis, and so on; but when an undergraduate student enters grad school, they don't have *time* to learn advanced group theory, advanced real analysis, orÝwhat have you. They must take the results in those fields as *given* - no time to actually *study* them!Ý- and build on them. Though I do not claim to know more math than my professors or even my peers in math graduate school, I find that I have a broader knowledge of the subject than most of them. I am consistenly saddened by how few of my peers are acquainted with the enduring, elementaryÝresults of ancient mathematicians - or even hundred-year-dead mathematiciansÝ- simply because *those* results are no longer hip, and hence are not as profitable to someone looking to produce a thesis.

So as soon as I get my M.Sc., I will be leaving. I have seen what it takes to be successful in my field - work that is originalÝbut uninteresting is valued above lasting results. I fell in love with the math of Martin Gardner, Ian Stewart, and William Dunham, and Underwood DudleyÝ- expository math writers with a respect for their predecessors, mathematicians who unify the whole of the field, dazzling math enthusiasts of all levels. They would not last ten minutes in today's grad schools, with their emphasis on boring, ill-understoodÝnew math at the expense of the enduring-but-no-longer-profitable old math. The mathematics being published today will not last longer than it takes another stressed-out professor or grad student to quote it; it will be forgotten before the next generation of students enters university. I would rather learn deep mathematics than crank out shallow mathematics built upon a shaky foundation that I am not supposed to bother myself w ith building.

So that's my story - I've written more than I originally wanted to, and almost certainly more than you wanted to read. But it's why I won't be around here next year. I aspire to write expository math one day - but I'll have to look outside the university system for the training and the resources to do so. (I've asked a number of professors for advice, and their recommendations have been depressing: "Finish your Ph.D., in a field that doesn't interest you. Get a postdoctoral position, then a professorship. When you're 40, you might have tenure, and if you're not yet burned out and jaded - you'll have time to do the math you *really* want to do.")

UPDATE: I got some excellent mail from mathematicians in response to this post. It's long, so I've put them up as comments.

I was rather surprised to see the entry (title above) on a letter you received from the 3rd year masters student. There's several points I think need to be made, however, regarding that reader's comments. Basically, most of what she says is a distorted view of the mathematics community in general.

Let me address some specifics of what she writes.

"For a year and a half, I worked - sort of - on a research topic that my advisor chose for me. This is not unusual for math students; in fact, choosing a topic for their students is one of the primary roles of thesis supervisors in math, as students are generally unable to tell the difference between a tractable (and thesis-worthy) problem that they doÝnot yet understand, and a very difficult problem that they do not understand."

I would say the typical situation is that an advisor will encourage the advisee to pick his/her own topics, which if the student is motivated is enough and up to par is not a problem. Sometimes a student can't just come up with the whole project and in that case a compromise is what happens, like in pretty much any other subject. The advisor *advises*, sometimes this leads to suggesting some specifics, but only as a last resort will a professor tell a student, "You will do this and this." That is only done when the professor considers the student basically hopeless in pursuing his or her own ideas.

Typically, many masters theses are of that kind. Unlike what you may imagine, it is not necessary to get a masters before a Ph.D. in mathematics. Actually, there are many mathematicians who never got a masters, and in fact, the best schools in mathematics do not offer a masters, except in certain cases as the equivalent of an "All But Dissertation". Indeed, masters students have a lower status at graduate schools in mathematics, at least in the U.S.; they are often seen as those not capable of getting a Ph.D., rather than those who have chosen a masters for some other reason.

Part of the reason for this perception is based on reality: many masters students spend much of their time taking undergraduate classes to make up for their weak undergraduate backgrounds. Actually, it's not uncommon for this to happen for Ph.D. students also, particularly for the universities with a weak mathematics program.

"I held nothing back in asking my supervisor for help and direction. Half my questions were dismissed - I was assured that it was okay that I didn't understand why Theorem A or Proposition X was true; it mattered only that I knew how to apply them. "

I can't know for certain what's occuring in this person's situation, but I have an explanation that explains many other situations that occur. First, it should be noted that often one needs to understand what "Theorem A" is before understanding why it's true. Part of this understanding is garnered from being able to apply "Theorem A"; this helps with familiarity with all the terms, definitions, and concepts involved. Secondly, understanding mathematics is not a simple case of just "getting it" and then one knows it completely. There's often higher levels of understanding a concept. One never completely understands something. At the most basic level, "understanding a theorem" is often thought to be an ability to follow the proof, i.e. a line of logical reasoning that explains why the theorem results from other known theorems.

I can see two reasons why an advisor would discourage a student from trying to understand the proof of "Theorem A". The student may not be at the stage where she can understand the proof. Advising such a student to first be able to apply it is sound advice. The sad thing is that the student may be even incapable of this. In this case, I don't see how, short of kicking the student out of the program, one can advise a less than capable student to understand all the theorems; probably the best thing in that situation, is just to have the student be able to operate at a very basic level: be able to use the theorems to do some kind of masters-worthy work. Another reason is that some proofs are just basically very technical, but eventually as one garners experience in the methods typically used in the subject, they become utterly transparent; the technicalities disappear, mainly because one is no longer intimidated by the fancy jargon or notation. In this case, when one is on a time-table, to spent one's efforts on this one proof of a theorem, is inefficient use of time.

"If I can just apply other people's results, then I will be able to produce new results that can be written up into a thesis - never mind that I would not have the mathematical certainty that I crave when I work on math problems. Sure, my thesis would be one that I won't understand very well,Ýbut at least I'd have a thesis, and isn't that the point? (Besides, I soon realized - my *supervisor* would understand my thesis, which would contain extensions of *his*Ýtheories.)"

This last sentence is just a crock. If this student really thinks her advisor couln't very easily extend his theories in the very specific way he is advising, she is in her own dream world. It's very common for a mathematician to give this kind of easy project -- to extend his/her theory in some minor way that is within the abilities of a student. By "minor way" I mean that it's not difficult to do, nor is it really important as a piece of research. These kinds of extensions are often very obvious to the advisor, which makes his/her job much easier, since s/he knows how the so-called original research is going to proceed. This is basically charity on the part of the advisor. Not some kind of ploy to get slave labor like your reader suggests. Of course, it may be misguided charity; it may just be better to tell the student that she is better off doing something else besides mathematics. But there are many forces at work that create this kind of situation.

What's funny is that her line of thought above, that the point is just to get a thesis, is probably why her advisor is proceeding in this manner. Undoubtedly he would feel bad in just telling her, that after all the time she's spent in school, she just doesn't have what it takes. After all, since she's only a masters student, she's not going to go into research and embarass him or the school. But certainly a masters would help her out in getting a job.

"After talking to some other students and professors in my field, I learned that virtually everywhere in math, the rush to be original andÝto publish original results before anyone else publishes them (as well as the rush to finish a thesis) trumps the motivation to understand - or teach - mathematics deeply."

I don't know what I can say to this, but that every researcher I've met is a counterexample to this statement. Undoubtedly her frustration at her lack of progress is the cause, so I feel I should be somewhat sympathetic. On the other hand, when someone who has obviously not been in the research environment spouts off like this....

"Consequently, the body of newly-published mathematical papers is a mess: every paper contains references to dozens theorems in other papers, and as often or not, following the trail of citations fails to yield an actual proof of the theorem or theoremsÝin question."

This is just false. Following a trail of citations is a very good way to learn the basic proofs and theorems in your field. I've certainly not had this experience. When I see a citation to a published work that claims the cited work contains a proof, with only rare exception has it not been true. Will there be mistakes? Sure. Will some citations be wrong? Sure. But the picture painted above is just false.

"Proofs in math papers are often mere notational sleight-of-hand, failing to prove anything at all."

Again, just false. Statements like this make me think this person has not actually read many math papers at all. As apparent from her letter, she takes these kind of statements on faith, after hearing them from others. If she simply took the time, to ask anybody that told her this, "Give me one example of this", she would undoubtedly get a flustered look, and stammerings of "Well...I remember some time ago, I found a mistake..." False proofs are nowhere near as common as she claims.

"One professor at my school estimates that around 20% of math papers published today are "fatally flawed" - that is, their key results rest upon theorems that are incompletely proven, inadequately proven, or just plain false. Refereeing math papers is a thankless task; consequently, it is seldom done.Ý"

These kinds of estimates may be true, but it's hard to say either way. Certainly I've run into papers that contain crucial typos, or not enough proof, or sometimes (but much rarer) contains false results. But it's also true that the more important the result, the more likely it is to be in the "typo" or "not quite enough proof but still true" categories. "Fatally flawed" seems a strange way to describe the kinds of mistakes in these categories. After all, these mistakes can be easily fixed.

The truth of the matter is that publication is not how mathematical verification takes place. A mathematical result is accepted as true when the experts in the appropriate field(s) feel they understand the result and the result has been simplified enough so that many people (even some outside that area of expertise) can understand it. Publication is often just a final step in the verification process. It's rather anti-climactic.

Seen in this light, publication is often not required. The mathematical community is a living entity where news and knowledge travels quickly. Famous mathematicians, in particular, sometimes feel they don't want to waste time with the arduous publishing process when the results have already been disseminated to their colleagues and have been verified.

"Stories abound of newly-minted Ph.D.'s who land tenure track positions for their groundbreaking theses - which, years later, are shown to be garbage. One subfield of algebraic geometry, according to my former supervisor, is without any textbook; proofs of the most frequently cited theorems in this field appear absolutely nowhere in print. Apparently they're true, but the retired emeritus professor who came up with them never bothered to write up the proofs. Part of what attracted me - and many other grad students - to math, was the assurance that at least in this field, we wouldn't have to defer to authority in assessing what was true. We had rules, and they would guide us. I have heard the phrase "trust me, it's true" so often that I feel like apologizing to my field."

Other than the comments I just made above, all I can say is that this person has unrealistic expectations, and some misconceptions of how mathematical understanding happens.

Regarding the fomer, it's impossible to check everything by yourself. Even if you did, it's unrealistic to assume you're not flawed and will be able to find every mistake. This is why there is a community of mathematicians. Is this kow-towing to authority? No. It's just reality. Having said that, let me add that I'm amazed by the self-correcting ability of the mathematical community. Nowhere else will you find the level of consensus on what is correct, the low failure rate (in catching big mistakes), and an atmosphere that is so amenable to the idea that you listen to what a person is saying and not who the person is.

Regarding the latter, there are many similarities between apparently different structures in mathematics. That's why it's much easier for someone who has become an expert in one field to learn another field of mathematics, as compared to someone just learning their first area of specialization. Many times, it's not only quite plausible that a result is true, but it just makes sense given one's knowledge of mathematics in general. This is actually the kind of understanding one ultimately is aiming for: knowing how all the pieces fit together -- seeing the big picture. Knowing all the proofs of all the theorems is not the ultimate goal, and neither is it a high level of understanding.

Of course, proofs are important, but primarily their purpose is to convince others of the veracity of your results. It's easy for many people who learn some mathematics to believe that "proof" is really the ultimate goal of a mathematician, but that's a mistaken view of what mathematicians really want.

In a classroom environment particularly, it's near impossible to cover all results in an absolutely rigorous way. It just takes too much time. If one sees the goal of such instruction to be checking the veracity of all results, then obviously the classroom model is a failure. The actual goal, however, is to impart mathematical understanding.

I've ran out of steam so I'll just close with some remarks about a few, selected remarks from the latter part of the letter.

"I am consistenly saddened by how few of my peers are acquainted with the enduring, elementaryÝresults of ancient mathematicians - or even hundred-year-dead mathematiciansÝ- simply because *those* results are no longer hip, and hence are not as profitable to someone looking to produce a thesis."

That's not a fair characterization of why few mathematicians study ancient mathematics. By the way, it's unfair to say few are familiar with the mathematics of "hundred-year-dead mathematicians", since much of undergraduate mathematics consists of precisely that. The plethora of results named after Gauss, Euler, Cauchy, etc., are a reminder of that.

Getting back to why ancient mathematics isn't much studied (except by historians of mathematics): the truth of the matter is that much of ancient mathematics is concerned with specific ad-hoc methods to solve certain problems. These problems are much better tackled, in general, by more modern methods. Is it a surprise that most mathematicians are not interested in how Egyptians multiplied and divided numbers? Especiallly when it seems that a different method is required for every slightly different problem? Mathematics has moved beyond that. The kinds of problems considered by mathematicians today not only are more general (containing many of the classical problems as mere special cases), but of a wider scope and thus more interesting to many people. In addition, science has progressed to the point where it relies heavily on modern mathematics. Why wouldn't someone want to study the mathematics that is relevant to the running of today's world?

This is not to say that considerations of what is "hip" do not play a role, but to dismiss the interests of the majority of mathematicians as simply being a result of fashion ignores the important advances mathematics has made, including those that have impacted the sciences. I'm amazed that someone who is familiar with the history of mathematics would make those kinds of statements.

"I have seen what it takes to be successful in my field - work that is originalÝbut uninteresting is valued above lasting results."

This is completely backwards. What is interesting is valued above all else, including originality. People frequently come up with original ideas that are just seen as not very interesting. Those ideas (and papers) end up being relegated to backwater journals (at best). In terms of her situation, what is probably happening is that it's extremely unlikely she will come up with something regarded as interesting, and so her advisor is concentrating more on the originality aspect. That is, after all, the minimal requirement for a thesis. Ideally one wants something more substantial, but not everyone is capable of that. It's interesting that she has somehow distorted the picture in such a way, that she has this view of originality being valued over substance.

You knew I was going to do this, but now it's time for me to comment on your remarks.

"Theories of theories of theories, without substantive grounding in fact or the history of ideas behind the theories."

This really doesn't apply at all to mathematics. It doesn't actually make any sense at all. I've yet to see a theory of a theory in mathematics, at least the way you mean it. [There are what are called metatheories, but that is a specific concept in logic that I doubt you were thinking of]

"Students who don't comprehend and professors who tell them it's okay to parrot uncomprehendingly."

This is interesting, and something I think I can comment meaningfully on. Hopefully by now you have an idea of why this is mistaken (cf my comments about understanding). But I'll say something more.

Mathematics is very different from many other disciplines. It can be very difficult to understand, but once you do, there is no argumentation about whether you do or not. It's very apparent to others who also understand. The level of agreement among mathematicians of what is right is astonishing. But it's not just a matter of parroting things that sound right (or that you heard others say). It's really impossible to parrot mathematical statements and make mathematicians believe you know what you are saying. My goal isn't to get into the philosophical reasons of why this is so, but I just want to lay out some groundwork at the moment.

Professors certainly don't want students acting like parrots. However, different students will get the material in different ways in different stages. I don't think I've ever understood something completely from the start. I gradually begin to understand pieces and over time, learn to assemble the pieces into a cohesive goal. In the past, some professors have indeed told me to take something for granted, but time has proven again and again, that they were right to tell me that. I just wasn't ready for the whole picture at that time.

When one learns how to add and multiply numbers, one doesn't also go into defining what numbers *are*. Strictly speaking, if one wants to be absolutely rigorous, one should define what a number is before deducing theorems about them, right? But this would be a disaster in teaching. Which is why most people learn how to manipulate numbers first, and then some eventually become math majors and learn what a number is. The logical progression is oftentimes not the most helpful progression in learning mathematics. Your reader has conveniently forgotten this, and sees it as a betrayal by her professors against the whole discipline.

I've enjoyed reading your website over a year now -- I am a beginning graduate student in applied mathematics entertained notions of studying English at the beginning of his college career. I thought of
writing you a note after I read the email you posted from a math
graduate student; I strongly disagree with the ideas expressed in
this email.

Thing is, math is very interconnected. One area is connected to the
next which is turn connected to the next and so on. Pick any paper
published in a journal today; it will have something like 10-20 references; each of those papers will have 10-20 references; and so on.

Understanding all of these papers is impossible. Theres simply too much material.

So what do you do? Well, you make sure you read and understand the papers that are closely connected to what youre doing; as for the rest, you satisfy yourself with simply knowing what the theorems state and not knowing the proofs.

We end up being very specialized -- knowing our own fields extremely well, and knowing little of what our collegeaus work in. But thats why we have departments where faculty have diverse interests -- if we need to know something from another area, we go to the resident expert in the area and ask.

Tradition is not getting sneered at, like you suggest. Rather, we do
not spend all our time studying the work of the past masters because there is way too much of it and, anyway, we dont need to be intimately familiar with it to conduct cutting edge research.

Your respondent clearly wishes to spend her time simply reading past results and not doing original research. She correctly senses that if she does that, she will not be able to find a job by the time she graduates. And rightly so; would someone who published nothing get hired in any academic field?

By the way: graduate students always think their knowledge is more broader than that of their professors. After all, the professors have had years to get specialized in their own field while the graduate students are just now studying diverse fields to pass their qualifying examinations.

Your respondent also claims that original research is "not interesting." This is the intellectual underpinning of her critique of the "publish or perish" culture in the sciences? I find original research in my field to be very interesting. And a lot of it is pretty damn useful too.

This e-mail from the algebraic geometry student is rather depressing (if she is reading this, she should keep reading to the end where I have some serious advice). I really feel for her, since it sounds like she had a lousy advisor, and may be in a messed up department. However, I feel compelled to point out that at many schools, including all of the top schools (which don't have terminal masters programs), things just aren't the way she describes. Her criticisms really don't ring true to me, and are certainly exaggerated due to her bitterness.

She shouldn't feel discouraged regarding her goal of writing expository math books. There's no reason why one needs a Ph.D. in mathematics for that - Gardner doesn't have one, and I can easily imagine writing Dunham-style books without one.

One important issue for the mathematics community is whether there should be a high-level graduate degree not oriented towards a research track. The way things currently work, a Ph.D. from one of the top schools is excellent preparation for a career in cutting-edge research, and a Ph.D. from a less distinguished program is basically an imitation with unprepared students who often have neither the talent nor the inclination for such a career, and even if they have both are often not well served by their education.

The problem is that there is strong pressure to offer such degrees (it is prestigious for the department, and pleasant for the faculty who get to teach advanced courses), and any attempts to offer a different sort of degree are often derided as "dumbing down" the program, even when that's objectively untrue.

There's been no substantial progress in this area, but a few schools now offer a "Doctor of Arts" degree in mathematics. It's a doctoral degree specifically aimed at developing a broad background in future teachers at liberal arts colleges or community colleges. Most mathematicians won't consider it quite as good as a Ph.D., and it won't qualify someone for a position in which doing research is a substantial part of the job. However, it sounds ideal for the disenchanted algebraic geometer: she would have a much more pleasant time in graduate school, and would get a degree aimed at what she wants to do.

I urge her to look into this possibility. I know UIC has such a program (see http://www2.math.uic.edu/~omce/da.html), and I think there are a few others although I'd recommend choosing a reputable school. Maybe this isn't right for her, or she is too upset to be interested, but I really think she should check this out before giving up and potentially feeling bitter or disgruntled for years to come.

I'm an ABD in applied math (praise the lord I did pick up
that Masters along the way to show for my time) who dropped out after 6
years of grad school to start a new career as an actuary.

I don't want to get into all the whys and wherefores of me leaving academia;
of my six years in grad school, only the last one was really wasted from my
point of view. It probably have been a favor to me had I been kicked out at
the 4-yr mark, but that's just hindsight talking.

I do have a problem with the math grad program as it is run in most American
universities. First of all -- the teaching component: I have heard a great
deal of talk about improving undergrad education at the two institutions I
went to, but saw half-assed attempts by the department as a whole. I had
=never= seen a grad student kicked out for incompetent teaching, though I
remember one who just stopped showing up for recitations at 8am. He knew
that little would happen if he didn't fulfill his teaching duties as opposed
to getting his dissertation done. At my undergrad institution, at least you
had to go to a 3-day-seminar and go before a panel of professors who decided
if you could TA. If you failed that process, all you could do would be
grade. So much for teaching.

Secondly, one was supposed to do original, solo work for the dissertation.
Why? What is the value of novelty for novelty's sake? Why spend 3+ years
on research that few are even going to care about? This goes for the tenure
race as well: there are some professors who have published a multitude of
papers, each barely incrementally extending previous research...and this
furthers math HOW? There is such a multiplication of journals with marginal
results that no one really has time to review things that probably have some
intersection with one's own work. Writing a textbook is far more useful to
math as a whole, but it takes longer to write than a 4-page-paper and
requires a great deal of synthesis and thinking.

I am enjoying the corporate life a lot more. The work I've done so far (and
I have been doing original research here) is actually paid attention to
because real live money is on the line; and I'm never working alone. We
check each other's numbers and bounce ideas off each other. It doesn't
matter which of us implements it, because it's more important that the
project get done than who is the one who does it. All our names are on the
final product. It's amazing how work that impacts people lives focuses the
mind...

Oh, and the pay is much, much better.