Oral exams in mathematics at university

Question

When I went to school, I actually liked mathematics a lot because I never had to study for it outside of class. If I understood a concept, I could apply it, and practicing in the lectures was mostly enough.

In university, though, things changed a bit because I can't just go to a lecture and be fine anymore. You have to push yourself through it by doing lots of problems, reading, and learning the proofs from the lectures. Thinking deeply about everything, etc.

An unfortunate challenge I face is that regardless of whether you do all that and "actually" understand things, it's not enough to get the best possible grade (even though this "actual understanding" is the most important, and without it, nothing else can help you improve more).

To excel in an exam, you have to prepare specifically for the exam. For oral exams, I use flashcards and do well, but I dislike everything that is repetitive. It's such an unnatural way of learning, and I hate doing it.

In written exams, we have to solve exercises, while in oral exams, we have to reproduce the lecture. That is a huge difference in difficulty. I enjoyed solving exercises much more than learning the lecture by heart.

Why are oral exams about the lecture and written exams like exercises?

Answer 1

Not a mathematician here, but Computer Science lecturer.

In my experience, oral exams benefit the student. In a written exam you're stuck with whatever questions/tasks the lecturer came up. Usually not all topics can be covered in a single exam - as a lecturer you have to select some topics and focus on those. From a student perspective this may be good or bad depending on whether your focus on topics aligned with the ones of your lecturer.

However, in an oral exam I, as a benevolent examiner, can shift the focus of the questions I pose. When I notice a student struggling in a particular topic, I usually shift attention to other topics covered in the lecture. While the student likely is still penalized for failing one area, the overall grade can still be salvaged when they excel in another one.

Similarly, you get immediate feedback on your current solution. While in a written exam you might spend a lot of time pursuing a wrong direction, in an oral exam your examiner will stop you rather quickly there. This gives you the option to rethink and maybe correct an error you made before.

^{Sidenote: I'm only considering the outcome with respect to grades in an exam here. In my opinion, other outcomes like an actual increase in knowledge and capabilities should already have been achieved throughout the lecture (and possibly exercises etc). An exam is only required in our day and age where everything has to be quantifiable so that someone can "objectively" compare between outcomes.}

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From the lecturers point of view, there is a tradeoff in effort needed. Personally, I preferred oral exams for smaller groups of students (maybe up to 20 students in a course) and written exams for larger ones. Depending on the exact nature of the exam and the number of people available for grading, written exams are easier to scale but have a higher time investment up front.

As a lecturer I also prefer to verify in an exam whether students actually grasped the underlying concepts and ideas over checking whether they memorized a set of definitions and formulae. While in a written exam it is often about the details, this big picture view is far easier to test in an oral exam. Sometimes it just takes a single what-if-question going slightly of the beaten track of the lecture to really see whether students got the main message. One of my basic principles in teaching is that details can be looked up (and hence don't have to be memorized), but understanding ideas takes time/effort and is way more helpful in the long run.

Answer 2

The main advantage of an oral exam is that it is interactive. It can be used both to correct stupid mistakes and slips of tongue on the fly and to probe deeper into the student's knowledge when you need it. Also, both the student and the examiner can see quite clearly where things stand, so that no doubts or arguments about what was meant versus what was written can ensue. I find these advantages so essential that on the last (written) qualifier exams I allowed the students to check their solutions during the exam once per problem with me. I just read them until the first error or unclear place and told them where that error/place was or confirmed that the solution was correct. If there was an error they had a second attempt to solve the problem (but not to check the solution again during the test) It benefited some of them quite a lot and gave me a much clearer picture of who knew/didn't know what.

The only real disadvantage I can see is that, because of the very same interactive nature, the oral exam cannot be made completely uniform, so an evil minded or just reckless instructor can make it difficult for some people and easy for other. Administering and grading an oral exam is a skill and some sort of an art, while grading a written one is mainly an algorithmic activity: 1 point for this, 3 points for that, add/subtract/multiply the points and compute the score. Also, of course, it is more time consuming if the class size is large. Still I prefer to do it whenever possible (for instance, I did it during the COVID quarantine semesters when written exams were totally useless because you could have no control whatsoever over the possible cheating). I do not make them exhaustive: just check a few main points like the knowledge of a few random definitions and proofs of a few theorems, and, if a student wants an A, give a medium difficulty problem to solve in the end.

Just my 2 cents.

Answer 3

In my previous job, when we interviewed recent graduate candidates for certain roles, we gave them a problem to solve that included mathematics, economics and lot's of common sense. They had 45' preparation time for this and then had to explain their solution to the interviewer. Once I had a candidate who had done absolutely nothing. When I asked what happened he said he didn't understand the problem at all. I then suggested we did it together from the start. As soon as he understood the issue he solved the problem brilliantly and answered all my follow-up questions. We not only hired him, but he also became one of the best hires I ever did.

Had the test been written, we would never have considered him. An oral exam is really the only way to see how well a student understands the subject and to identify excellence from mediocrity.

Answer 4

This is just a couple of anecdotes but they may provide some enlightenment.

My doctoral advisor (Math, in US), when he was teaching undergrad courses always used only oral exams. He was from Europe. The advantage was that it was very quick for both he and students. Five minutes was all it took to explore the knowledge of the students. Moreover he could give hints when a student stumbled a bit and could move up or down the difficulty scale quite quickly to get a good sense of the student's level of insight. He also told each student how they did at the end.

For my own part, I once had an important oral exam with a committee. I was asked a question I couldn't answer. But I was able, at the moment, to explain my reasoning and the nature of my block. The committee, I later learned, was just as impressed as if I'd answered it correctly. It was clear that I wasn't depending on memory alone, but could both develop and explain the reasoning, even if it led nowhere.

Answer 5

The primary benefit of oral exams is that they are interactive. In the best case, this allows to more rapidly identify the level of understanding of a student.

One can ask a question, and if it's clear they understand how to approach it, you can stop them without requiring them to work through the details of the derivation, you can ask them to write out the details of just one key step; or you can follow up with a deeper, harder question. Alternatively, if you ask a question and it's clear they are struggling, you can give them a hint to get them unstuck and see if they can progress or switch to a different topic or ask an easier question. If you start to get a suspicion/hypothesis that the student might have a particular misunderstanding, you can ask a question that quickly tests that hypothesis. In this way, empirically, often you can quickly calibrate the depth of understanding of a student.

A secondary advantage is that oral exams are more resilient to issues in the problem statement. If there is an ambiguity or flaw in the problem statement, this can be identified and immediately corrected. On a written exam, you have to be extremely careful to make sure you have identified all possible ambiguities and avoided any errors in the statements of the questions, as a single issue can be uncorrectable or lead students to waste a lot of time on it.

There are also some disadvantages of oral exams. They don't scale well if you want to test a large number of students. They are more subjective to grade. It is more challenging to make them fair. Some students don't do well in oral exams (e.g., nervousness, time pressure, style of presentation), so their ability might be masked by other issues.

Overall, they are another tool in the arsenal for assessment. It's not that they are better or worse than written exams; rather, they have a different set of tradeoffs, and what is appropriate will depend on your specific setting.

Answer 6

What is the benefit of oral exams at all? Why don't they just examine with written exams?

The utility of an oral exam is the flexibility of the format, and the tightness of the feedback loop. This is the same reason as someone might wish to talk to their colleauge in their office directly (or on a video call) instead of a long email exchange.

When one prepares a written exam, they are making a bet about a number of things. For example, (1) the list of questions they have prepared will help them assess whether the student has understood the material; (2) The student will not misunderstand the questions; (3) The examiner will understand all the answers; and so on. When the assumptions go wrong, a tight feedback loop reduces friction in the evaluation process.

If the exam is an oral exam, there are many chances for the examiner to adapt their examination strategy. Many of the other posts have given examples, but here are some examples anyway:

• Perhaps the student is making a small arithmetic error, which will make the rest of their argument invalid. One could argue that the student still understands the material, but the miscalculation was an accident. In this case, the examiner could potentially just fix the mistake in the spot, if he notices it.
• Perhaps the student produces a complex proof. It is better for the examiner to be able to quickly ask for clarification, instead of spending a lot of time to figure out whether the proof works.
• Perhaps the examiner realizes that it would be fair to phrase the question somewhat differently.

This does introduce some subjectivity into the evaluation process, but one might argue that evaluation is inherently subjective.

Answer 7

One more advantage of the oral exam: at the beginning of my Analysis I+II oral exam, my math professor asked me what topic I would like to start with. That kind of thing helps immensely by having the student start with something they are confident about. This is hard to do in a written exam (unless it is so comprehensive that it covers all the material, so everyone can start with whatever topic they prefer).

Conversely, the professor can tailor the rest of the exam to the student's demonstrated ability, giving easier questions to less advanced students and skipping the easy stuff for the better students, to dig quickly into the hard parts that distinguish the very good students from the excellent ones. This kind of discrimination is hard to achieve in written tests (though well-designed computerized versions should be able to do this, at least in multiple choice formats).

Answer 8

Oral exams are common in Europe. I find them much better than written ones when i) you are not prejudiced and ii) want to actually see how good a student is.

They will not be completely fair, though - people are judged on (arguably) topics at different levels of complexity, but the advantage is that a good prof will interactively gauge the knowledge of a student.

The prof may also look for portions of the curriculum that a student knows/understands/likes better and go from there.

I find them more representative of the knowledge of students.

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Story time. The weirdest oral exam I had (as a student) was in my last year of MSc and was on a rather esoteric topic in advanced physics. Our group of, what, 10 students knew the prof very well (we were the core of the student's academic physics association and he was the official supervisor). The exam was extremely relaxed and we were all very good students.

He told us that the exam really pisses him off but that rules are rules (sight) and that he has to do it. So he will go through all the topics he taught us and who wants to answer just raises their hand.

He started with ... vectors. How to add them and whatnot. Again, we were after 5 years of intense math and physics studies, five of the 10 students would go for a PhD and ... vectors. I immediately raised my hand and explained how to add vectors, I added for extra points, tongue in cheek, how to subtract them as well.

And then it got more and more complex, exponentially. At the 6th or 7th question we had our eyes wide open because the prof was reading his notes and inventing really hard problems. Obviously the answers were getting more and more vaporous.

We finished, he gave marks and there was an uprising of the ones who were at the end. The prof sighted again and said he would lower the marks of the first ones because, indeed, it was not fair. So the other half started to protest vehemently. Since his wife was waiting for him he gave up and told us to put the marks we wanted because he was in a hurry.

The conclusion is that oral exams help you to pass some courses (and seriously - this was an esoteric course we took for the pleasure and fun at the end of our studies so the marks were completely worthless)

Answer 9

Oral exam, on any level of education, is interactive exam. Written exam or test is one-way do -> submit -> evaluate.

It means that the teacher/professor can directly observe the student and see what errors the student does (lack of knowledge, temporary failure, misunderstanding,...) and correct them on-the-fly. They can assess the approach to the problem. Some students learn poems - they memorize the definitions, derivations etc. character-perfect. Others learn the ideas. In the oral exam you can "poke" the student "off the track" and observe how they react. Do they follow the idea or get stuck untill the "poem" is restored?

The teacher/professor can ask additional questions to assess the depth of knowledge much easier. If the student performs well all the basic questions can be omitted, same if the student performs badly, all the "interesting" questions may be omitted.

Additionally, if you are to talk about some subject you have to form your thoughts in sentences. You are forced to approach the problem from multiple ways. You need to "translate" the question to your thoughts, process and then "extrapolate" to the teacher's/professor's language. On the paper you can just list points or use much shorter sentences or phrases.

Another difference is in "dimensionality" of the test. The oral test shows the time dimension - how fast you perform different tasks. In the test there is no such information, in the written task there is only the sequence of steps and even that is not reliable source of information.

The written exam is good for assessing performance-oriented tasks. Oral exam is good for assessing the understanding of the subject.

Jokingly, I was able to pass basic calculus exam because it was oral exam. One of the questions was simple: Define absolutely convergent sequence and relatively convergent sequence. I missed. Then theprofessors were asking me to label different sequences to catch me I memorised the definition wrongly. After couple of minutes one of them gave up saying "You defined the qualification wrongly and you stick to those definition strictly." In other words, I missed the literal content but proved the understanding of the subject.