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On Monday, 2021 May 17, I passed my oral qualifying exam at Rutgers University. This means that I have advanced to candidacy. In other words, I have cleared the necessary hurdles to be a “real” PhD student, and am “ready” to be a researcher.

Of course, this implicitly means that I was incapable of doing research before passing my exams. The paper I published with my advisor didn’t count. The paper I co-authored with my post-qual friend was phoney. The collaboration my advisor and I had with a computer algebra wizard was fraudulent. Even the expository article I wrote can only be described as pre-candidacy material1.

Thankfully, now that I am a “PhD candidate” as opposed to a mere “graduate student,” the tides will turn. Indeed, with some luck, both the quality and quantity of my work will improve because I am not wasting my time on formalities.

The purpose of a PhD program is to produce competent researchers. The Big Lie about qualifying exams is that they support this purpose. In truth, qualifying exams are mostly redundant. They measure things that the department already knows. They exist only to place students into high-risk, low-reward situations for long periods of time, which detracts from their opportunities to do research.

I propose a modification to the qualifying exam structure, but let’s first see how they work at Rutgers now.

Written Qualifying Exam. This is a sequence of three 2.5 hour-long exams on the three courses you take in your first semester. In each, you are asked five random questions, and have to answer three correctly (on average) to pass. You have three attempts, which will take you a full year to exhaust. The written qualifying exam supposedly measures your technical ability in “basic” mathematical subjects.

“But wait,” I hear you ask. “The exams are on the courses you took in your first semester? Didn’t you get grades in those courses? Don’t those already measure your ability?”

I am sorry to tell you that the current program mostly disregards your coursework. “Your grades don’t matter” is a common refrain among graduate students because it is true. Slave over every homework assignment? Go to every office hour? Get perfect marks on every midterm? Congratulations! You have achieved nothing. One exam, completed under threat of expulsion, is the department’s ultimate measure of technical mastery.

Of course, your grade is a measure of your ability. You spent an entire semester completing homework assignments, taking exams, and interacting with your professors. The department would already know who puts in work and who is “technically competent” if it cared to look.

I see two possibilities:

1. The department is out-of-touch with student assessment or does not trust professors to evaluate students.

2. The department wants to haze you.

Which do you think it is?

Oral Qualifying Exam. This is an 80-120 minute long meeting with a committee of four faculty members. You will be questioned on a syllabus that you wrote in consultation with your committee. In theory you have two attempts, but it is exceedingly rare to fail. Supposedly this exam ensures that you have not “overspecialized” before beginning research.

Unlike the written qualifying exam, there is no “standard” oral exam. Your experience is highly dependent on your area and advisor. This gives it some merit: Your syllabus is likely relevant to your proposed research, and the exam is graded subjectively by your committee members. However, this subjectivity makes the exam, again, redundant.

Suppose that your committee is confident that you will pass. Perhaps you’ve been meeting with them regularly to discuss your syllabus. What, in this case, is the purpose of the exam? Don’t they already know your knowledge? Suppose instead that your committee is confident that you will not pass. Then why on Earth are you taking the exam? In both cases, the exam merely makes you demonstrate what the committee already knows in a high-pressure situation.

The only situation where the exam has true merit is when the committee does not know if you are ready. I posit that this situation is nearly nonexistant. If it were a common occurrence, then there would be a significant number of failed oral qualifying exams. There are not.

The benefit of the oral exam process is having four faculty members sign off on your skills. This has nothing to do with the exam, and everything to do with how well the committee knows you.

The Problem. Qualifying exams take time that you could be doing research. If you demonstrate mastery of your first-semester courses, then you should spend your second semester and the subsequent Summer meeting with potential advisors. Doing anything else is a waste of time, as far as producing researchers goes.

The Solution.

Here is a proposal:

• Offer a written qualifying exam to students as soon as they arrive.
• Pass? Skip relevant first semester required courses.
• Fail? Take relevant first semester required courses.
• First semester required courses.
• A? Continue in the program.
• Less than an A? You must pass a written qualifying exam by the end of your first Summer.
• Oral exams are replaced with a requirement that four faculty members agree that you know your syllabus sufficiently well. It is up to them to decide if a formal exam is necessary.

Nothing I’ve said seems specific to Rutgers. I assume that this silly exam process happens at most mathematics departments. Perhaps the situation is similar in other STEM areas, and perhaps even in the humanities. All the more reason to evaluate the evaluators.

1. I don’t think that my outlined contributions are particularly good. I didn’t prove $P = NP$ or the Riemann hypothesis. But I did do research in some small form.