Would Math professors ace PhD qualifying exams?

[Gjmdp]

I have seen that most PhD Math students struggle a lot with these examinations. It is fair to assume that any PhD student is already very talented and commited to the field of maths, so I am wondering whether professors (including those who write the exam) could actually ace the exam (get a score close to 100%).

 

[Vanadium 50]

I don't know about Math, but legend is that Oregon students complained to their department chair that the qual was too hard, so he took it. Cold. Lost one point out of a hundred.

 

[kuruman]

I don't know about Math, but legend is that Oregon students complained to their department chair that the qual was too hard, so he took it. Cold. Lost one point out of a hundred.

I hadn't heard this legend. So he lost a point even though he wrote the exam? The one I heard is that the passing line at Princeton was 14%. That was decades ago, I don't know about now.

My impression from being both an examiner and an examinee is that qualifiers are not designed to be a "feel good" experience but a means to push the candidates to their limit until they cry "uncle". The idea is not to determine what they can do but what they don't know and/or are not able to figure out. Qualifiers are in a league of their own and should not be treated or thought of the same way as hourly tests or final exams.

 

[symbolipoint]

Does the "Qualifying Examination" difficulty or set of content depend on the particular institution or department?

 

[Mondayman]

It brings to mind the intense selection and training programs that special forces units have. Only those that are determined to put in the work and sacrifices will make the cut.

I am not at that level so I can not comment on what the quals are like, but I like the idea that the people teaching these subjects have an extremely rigorous program to pass. Being taught physics and math in high-school by education majors/masters was a terrible experience.

 

[Vanadium 50]

So he lost a point even though he wrote the exam?

The chair doesn't write the exam. There's a committee for that.

 

[Infrared]

I think it depends on what you mean by "qualifying exam". Many math PhD programs have two sets of exams: one that tests basic undergraduate/first year graduate material and one that tests tests advanced topics related to the examinee's prospective area(s) of research- and which is called the "qualifying exam" varies by university. My guess is that basically all math professors would do very well on the first kind, but maybe not on the second kind if the areas were outside their specialty.

 

[kuruman]

The chair doesn't write the exam. There's a committee for that.

I understand. I have been on such committees. Forgive the feeble attempt at humor.

 

[kuruman]

Does the "Qualifying Examination" difficulty or set of content depend on the particular institution or department?

I think both.

 

[mathwonk]

It is really hard to write a fair and useful exam. A basic topic like 1st year grad algebra is taught over 9 months and covers possibly hundreds of pages of difficult stuff from an advanced book. How do you choose a few questions, answered in 3 hours, that measure mastery of all of that stuff? (Often people telegraph what the questions will be, but necessarily only to ones own students, hence penalizing others taking the test.)

Also there is an audience for a given test. A PhD prelim test is written for grad students who have recently taken a detailed course in a given topic, and is often written by someone who taught that course. A typical faculty member is someone who may have not thought about some of the topics for a long time. So although that faculty member may have done the hard work to be sure of passing the prelim at the time they took it, they may not still know all that stuff 5-10 years later.

I myself, when I was a faculty member, could hardly pass any prelim outside my own area without preparing for it. Thus you have to ask yourself what is being tested? Is it the knowledge of material years later, or is it the ability to knuckle down and do what is asked at the time? I.e. are you measuring hard work or knowledge?

When I myself wrote the algebra prelims, I was not appreciated, because I wrote questions that tested exactly what was required. I.e. we had a syllabus that stated exactly what one should know, and I tested exactly that. Why was this despised? Because the students did not read and study according to the required syllabus, rather they just looked at the old exams and practiced answering the same questions that had been asked in the past. Those questions focused on only a small fraction of the syllabus. Since I had to work hard to cook up questions that actually measured knowledge of the majority of the syllabus, the students who took my tests were surprized at what was asked.

E.g. instead of making them to show a group of a given order was not simple, I might ask them to answer a true or false question that tested whether they did or did not know the statement of some theorem that they had been told to know. I.e. the syllabus said to know the sylow theorems, but the old tests only asked specific problems using some of those theorems. I would ask questions testing their knowledge of the statement of those theorems. I would ask questions that could be answered if one only knew the statement of the basic spectral theorems, exactly what was required on the syllabus, but my students were stuck, only having practiced diagonalizing a given matrix.

I did not purposely sabotage my students, rather I meant to help them by asking them to answer exactly what they had been told to know on the syllabus, but they assumed I would just ask the same questions that old exams had asked. Apparently I was the only person who ever naively took the requirements at face value, and the department never asked me to write the algebra exams again.

I have told the following story before but it may still be amusing, about my own experience with prelims. I was at two grad schools, Brandeis and Utah, and in between I just walked in and sat for the prelims at Univ of Washington. So I had passed them at both Brandeis and UW, when I arrived at Utah, where I asked on day one, how I should sign up for the prelims. This question stunned the staff there who said:

"What! You want to take the prelims? no one has ever asked that! Our students usually want to avoid them as long as possible."

I said: "Well, if I can't pass, maybe I should be seeking a different career, so I want to find out right away." They answered: "wait a minute, have you ever taken them somewhere else?'

I admitted, "well yes, I have passed them twice already, at Brandeis and UW."

the answer: "oh, then you don't need to take them. you pass."

Of course as you have guessed, when i saw the prelims that year, there were plenty of difficult group theory questions I could not answer. (They had a strong specialized group theory section there, who tested their own favorite topics.)

So apparently in those days at Utah, the prelims partly measured fear of them.

 

[mpresic3]

I suspect some faculty would score well, and some would not. Some faculty teach graduate and undergraduate material year after year, as well as conduct research. They would do well. Some faculty advance research as well as teaching, so the faculty divide themselves and apportion themselves to varying degrees between research, testing, and administration. The teaching professors would likely do better than the researchers and administrators, who may not have seen the core material less recently.

I asked some faculty whether they could pass the exam. The most common answer that I thought was most honest was, they suspect they would have trouble, but if they were given the time to study and the core courses (often the instructors for these courses are the same as on the qualifying exam committee, so the homework problems and the qualifying exam problems are similar) that they grant the graduate students for study of the exam, they could pass it.

In general, the graduate student is in his/her mid 20's and has been studying physics 5-10 years. The average faculty member is in his/her mid 40's and been studying physics perhaps 25-30 years. I know from my own experience, that I can do better on these types of exams based on my experience in solving problems, and not necessarily because I am more clever than I was in my 20's.

As far as acing the test, the test is graded somewhat subjectively. The idea that one point out of 100 was taken off is unlikely. Some of the graders would take off for something trivial, and some would not. For the most part, the only result of note is whether you pass or fail. I once found out I lost 2 points out of 10 because I used latitude in my spherical coordinate system, and not co-latitude. I was returning to school after a period of employment in geodesy where latitude is generally used. Consequently I had cosines in my answer in place of sines. Because it did not determine the ultimate outcome, I did not press the matter, but I did think the grader should have been more careful.

More often, you never get to see your test, and only get to see your result. One professor did say my treatment of a problem, that turned out to be correct, was excellent, and no one else got it. (I suspect but do not know that I probably got "extra" points on a problem where I did not do as well, because of the impressive performance on the earlier one)

For the most part the graders, one even told me this once, say "you can tell just by looking at the papers for about 5 minutes, which are going to pass and which are going to fail". I do not necessarily agree with this but this statement has been expressed to me.

 

[kuruman]

In general, the graduate student is in his/her mid 20's and has been studying physics 5-10 years. The average faculty member is in his/her mid 40's and been studying physics perhaps 25-30 years. I know from my own experience, that I can do better on these types of exams based on my experience in solving problems, and not necessarily because I am more clever than I was in my 20's.

There is another important distinction here summarized by "breadth vs. depth". When a grad student prepares for the qualifier, the pressure is on to maximize breadth. Once that grad becomes a faculty member, the pressure is on to maximize depth in order to be competitive in research. It's like a rectangle of fixed area one side of which is breadth and the other depth. The area itself is a monotonically increasing function of the native intelligence of the individual and only weakly related to years of experience. Most Nobel winners did their prize-winning research at the early stages of their career.

 

[mpresic3]

I had a math professor who told us the story when she was on a oral examination committee to give the student a candidacy exam. She asked the student the question (I am not a math major and I do not remember the specific question but I am told it was a fair question).

She failed the student and he had to take the candidacy exam again. Some months later (I do not remember the specifics), she had the same student in front of her. She asked exactly the same question. He told her he never thought she would ask the same question again, and he was not prepared at all to answer the question. She failed him for a second time.

She ended the story there, and we never found out if he was dismissed, or whether he eventually passed, or transferred etc. The moral of the story is : if you are going to fail, at least do not fail on the same question.

 

[kuruman]

To me the student's reaction in this story shows that he didn't make the transition from the undergraduate to the graduate student frame of mind. Undergraduates try to guess what "they are responsible for" and study to it. The transition to the graduate student frame of mind comes with the realization that one has to work towards the goal to know (or be able to figure out) everything. Of course "everything" is a tall order and one has to back off and settle for "as much as possible".

A professional would not try to game the system but would welcome the opportunity to fill a perceived gap in his/her understanding. Where is one going to make the attitudinal transition from undergraduate student to professional if not in grad school? I wouldn't be surprised if the student in the story were dismissed for not being professional material rather than not being able to answer the same question the second time it was asked.

 

[Vanadium 50]

A professional would not try to game the system but would welcome the opportunity to fill a perceived gap in his/her understanding.

One of the questions on my own exam was (oversimplifying) "A. Write down the Lagrangian for this system. B. Write down the equations of motion." While they intended to use (A) to get (B), it was in fact easier to treat (B) as a freshman physics problem than a more advanced problem. So that's what I did, and I got full credit for it. As I should have.

Two days later, at the oral portion, the very first question was "I see you found an interesting shortcut to problem #6. So, using that Lagrangian, please write down the equations of motion."

 

[kuruman]

One of the questions on my own exam was (oversimplifying) "A. Write down the Lagrangian for this system. B. Write down the equations of motion." While they intended to use (A) to get (B), it was in fact easier to treat (B) as a freshman physics problem than a more advanced problem. So that's what I did, and I got full credit for it. As I should have.

Two days later, at the oral portion, the very first question was "I see you found an interesting shortcut to problem #6. So, using that Lagrangian, please write down the equations of motion."

It's an art to write an exam question covering all the loopholes without making it read like (or have the length of) a legal document.

 

[Vanadium 50]

That is true.

However, I don't feel even a little bad about the shortcut - which is maybe more descriptive than "loophole".

1. The faculty had another shot it it, which they took. I also got this right, but of course I knew the right answer already and had a suspicion I would be asked this.
2. The test was to see if I knew any physics. I would argue that recognizing that the problem was not that hard shows more knowledge of physics than not recognizing it.

 

[Frabjous]

It is not always appreciated that teaching classes is part of the reason that professors are so knowledgeable. Year after year of dotting I’s, crossing T’s and responding to student questions gives them a very solid foundation.

 

[FactChecker]

I have no doubt at all that the professors would pass the tests with flying colors if they were given a short time to review subjects that they did not specialize in. But perfection and 100% is not required of anyone. The candidate's approach, knowledge, and clarity of thought, are evaluated. I doubt that a score on a 0%-100% scale would even be possible to assign.

 

[fluidistic]

One of the questions on my own exam was (oversimplifying) "A. Write down the Lagrangian for this system. B. Write down the equations of motion." While they intended to use (A) to get (B), it was in fact easier to treat (B) as a freshman physics problem than a more advanced problem. So that's what I did, and I got full credit for it. As I should have.

Two days later, at the oral portion, the very first question was "I see you found an interesting shortcut to problem #6. So, using that Lagrangian, please write down the equations of motion."

You know what, in my undergraduate degree in electrodynamics we were asked what was the charge density of something (can't remember whether it was a charged line or a charged chunk in 3D space) that was moving with respect to an observer. I described that there was a contraction in the direction of motion (with the gamma factor) and so I quickly obtained the correct charge density they asked for. However I got 0 credit (failed exam) because they apparently expected us to use a matrix representation of a Lorentz tensor (denoted by lambda, as I recall now), to obtain the result. But it was nowhere stated that one should employ this more tedious and more involved (in algebraic terms) way to obtain the desired result.

So, of course I complained, having no credit whatsoever for a correct (and not by accident) answer. I had used what I had self-learned in Purcell's textbook... for no recognition whatsoever. The reply I got was "Hey, we insisted so much during the course about these new mathematical methods that we expected you to use them on the exam, even though we didn't write it explicitly."
I was disheartened, but inside me I knew what I had done was good.

Now I remember when I was around 10 years old in elementary school when we had to perform calculations with the prefixes "kilo", "hecto", "deca", etc. But there were kilograms, and kilometers and other units. I drew a single table with only "kilo" (without specifying if it was meter or gram), and I performed all the calculations in that table. I received no credit despite reaching the correct answers, and I was told "you were the only one drawing a single table. What did you want to achieve? Save ink?". But now that I'm over 30 years of age, I realize that it was actually nice that I realized it didn't matter to put the suffix to perform the calculations. I don't care about academic points, as long as I can still reach the goals I set to myself.

 

[PAllen]

That is true.

However, I don't feel even a little bad about the shortcut - which is maybe more descriptive than "loophole".

1. The faculty had another shot it it, which they took. I also got this right, but of course I knew the right answer already and had a suspicion I would be asked this.
2. The test was to see if I knew any physics. I would argue that recognizing that the problem was not that hard shows more knowledge of physics than not recognizing it.

Of course this brings up the old chestnut of : “How would you determine the height of tall building with the aid of a barometer?”

1) I’d walk into the building facilities office and ask “I’ll give you this nice new barometer if you tell me the height of this building”

When requested that they were looking for knowledge of STEM,

2) On a sunny day I would measure the ratio of shadow length to barometer height, then pace off the building’s shadow, and compute its height.

When requested that specifically physics knowledge was sought,

3) I would throw the barometer off the top of the building and time its fall. Then compute its height.

 

[kuruman]

Of course this brings up the old chestnut of : “How would you determine the height of tall building with the aid of a barometer?”

1) I’d walk into the building facilities office and ask “I’ll give you this nice new barometer if you tell me the height of this building”

When requested that they were looking for knowledge of STEM,

2) On a sunny day I would measure the ratio of shadow length to barometer height, then pace off the building’s shadow, and compute its height.

When requested that specifically physics knowledge was sought,

3) I would throw the barometer off the top of the building and time its fall. Then compute its height.

It would be fun to upgrade the old chestnut and ask “How would you determine the height of a tall building with the aid of a cellphone?”

 

[PAllen]

It would be fun to upgrade the old chestnut and ask “How would you determine the height of a tall building with the aid of a cellphone?”

That greatly increases the number of methods ...

 

[kuruman]

That greatly increases the number of methods ...

Considering all the apps out there? Yes.

 

[symbolipoint]

Topic is drifting. I like fluidistic's discussion post #20. People in power sometimes make bad assumptions and the supervised suffer. Some of the credit for learning and adapting is unclear.

 

[Dr.D]

Does the "Qualifying Examination" difficulty or set of content depend on the particular institution or department?

It must vary from place to place; each department at each school makes up their own exam. It also varies from year to year, and I'm pretty sure the exams today would be quite different from what I took many years ago.

 

[jbergman]

I think it depends on what you mean by "qualifying exam". Many math PhD programs have two sets of exams: one that tests basic undergraduate/first year graduate material and one that tests tests advanced topics related to the examinee's prospective area(s) of research- and which is called the "qualifying exam" varies by university. My guess is that basically all math professors would do very well on the first kind, but maybe not on the second kind if the areas were outside their specialty.

Interesting. I would have said the opposite, that professors would ace a qualifying exam but that they might not ace the more general exam depending on their specialization.

For instance, I know of some Professors who know almost zero about Abstract Algebra.

 

[Infrared]

Interesting. I would have said the opposite, that professors would ace a qualifying exam but that they might not ace the more general exam depending on their specialization.

For instance, I know of some Professors who know almost zero about Abstract Algebra.

I said that professors might not do well on a qualifying exam if the areas were outside their speciality. If your hypothetical professor knows almost no abstract algebra, then they certainly have no hope of passing a qualifying exam in an advanced area requiring a lot of algebra (e.g. algebraic geometry or homological algebra).

I also find it hard to believe that many math professors wouldn't know at least some abstract algebra.

 

[jim mcnamara]

I think you need to find a mathematician like Hilbert, Picard, John von Neumann, or maybe Poincaré,
who would have been able to do well on exams at any level in very close to all specialty areas. Note: maybe not all areas.

Folklore sets Hilbert as the last mathematician who was well versed in all areas. I do not know for sure.

This whole exercise depends on the number of math specialties exceeding the available time to learn them all well.

Let's set the maximum to 20 ... fingers + toes, since we are making a very subjective analysis. So that probably means pre-1940 or earlier.

So, no is your answer. Why?

In 2021 there is no mathematician who could do well on all 60 different areas of graduate level exams.
I got 60 from the MSC: https://en.wikipedia.org/wiki/Mathematics_Subject_Classification
There are sub-areas as well. So more areas...

 

[Hornbein]

One of my math professors told of administering a Putnam exam for undergraduates. The professors could do maybe half of it. One of the students got a perfect score.

 

[hutchphd]

A series of odd events led to my teaching the undergraduate quantum course (Gasiorowicz) as I finished up my thesis research. I set a very good (if I do say so myself) 90 minute final exam which progressively increased in difficulty: I expected no one to complete it and said so. One student produced an exam paper which was not only entirely correct but was clearer, cleaner, and more lucid than my carefully posted solution set. To this day I cannot describe my ambivalence !

 

[atyy]

Folklore sets Hilbert as the last mathematician who was well versed in all areas. I do not know for sure.

According to the Wikipedia on Terry Tao, Timothy Gowers wrote "Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics as diverse as partial differential equations, analytic number theory, the geometry of 3-manifolds, nonstandard analysis, group theory, model theory, quantum mechanics, probability, ergodic theory, combinatorics, harmonic analysis, image processing, functional analysis, and many others. Some of these are areas to which he has made fundamental contributions. Others are areas that he appears to understand at the deep intuitive level of an expert despite officially not working in those areas. How he does all this, as well as writing papers and books at a prodigious rate, is a complete mystery. It has been said that David Hilbert was the last person to know all of mathematics, but it is not easy to find gaps in Tao's knowledge, and if you do then you may well find that the gaps have been filled a year later."

 

[mathwonk]

Re:post #30. That is impressive, as apparently only 5 perfect scores on the putnam have occurred in its first 82 years (through 2019).

In these discussions of Hilbert, I am reminded of the famous saying that he apparently did not know the definition of a Hilbert space, which does occur on some exams. So it is easy to ask a question in a way that will be challenging. Galois probably didn't know what a Galois goup was either. And Euclid probably couldn't define a "Euclidean domain", etc...

Re: post #29: I don't think it would be hard to find people who know a lot of those areas. I myself know "something" about 20 or 30 of the 60 pure math areas, and I am just an average low level retired mathematician. I have colleagues from my own school who know far more, and people like David Mumford or Curt McMullen, or Robin Hartshorne, or Yuri Manin, or John Morgan, or Jean Pierre Serre, know essentially infinitely more. Given the opportunity, and 50 or 60 years devoted to it, one can learn a lot. Some of us spent our lives teaching calculus over and over, but some people at elite places spent decades pushing the boundaries.