Requiring students to know all the proofs on an oral exam

Question

I'm asking this question as a student, wondering what various pros/cons to the given formula for oral exams could be. Let me give some context first. I am a first year mathematics student at a university. The are two approaches available to students for every subject: There is a "basic" group and an "advanced group" and we were able to choose which one we wanted to attend at the beginning of both semesters with the possibility of moving from one group to another if we found the level inadequate for 2 weeks since the semester's start. The difference between groups is really large, to the point where the only people left in the advanced groups are ones who have either studied a significant amount of university-level maths in their free time in high school. So, in short, attending the advanced groups is completely voluntary and the courses can't be really treated as ordinary introductory ones, because while they might theoretically assume any background knowledge, they do require quite some mathematical maturity which is bound to come with the backgorund knowledge.

After this long rant we arrive at the actual question. In these advanced groups, students are required to know all the theorems together with complete proofs of all the theorems/propositions presented during the entire semester in the given course. Learning all that is obviously not an easy task - not only is that a lot of material, it is also significantly harder than expected from an ordinary first year course. What do you think about such a formula? Is it too much? Or is it actually a good thing to expect this much from students who are highly above "average"?

I think I did somewhat benefit from learning all the proofs in the first semester, but on the other hand the task seems like a bit unreasonable, since even professors use notes when presenting proofs during lectures.

Answer 1

This will depend in some part on the course you are taking. If it is an introduction to proofs class, then probably it is reasonable to hold such an expectation. For higher level courses, some of the details that go into a proof can make memorizing a lot of theorems difficult. (Concrete example: I think the proof that permutation groups can be partitioned into "even" and "odd" elements is the sort that, if there were a lot of these to be memorized for an Abstract Algebra final, would constitute asking too much of the student.)

An important difference in how advanced vs. novice students think about proofs is related to how they "chunk" the information. An advanced undergraduate who has finished Calculus in high school might be taking a first course on Real Analysis by semester two of their first year; in such a case, consider the theorem that says every closed and bounded infinite subset of $\mathbb{R}$ contains a limit point.

When I first saw this proof, I was very caught up in the details; I could not chunk it very well, having no real familiarity with this style of argument. Today, I certainly would not memorize how it is proved verbatim. Instead, I would think: Put subset into an interval (boundedness); divide in half and choose left half if it contains infinitely many points, otherwise choose right half (infinite); repeat indefinitely. This zooms in on a limit point [essentially by giving its binary representation if the interval is $[0,1]$, where "choosing left" gives a $0$ and "choosing right" gives a $1$] and that limit point is in the set (closed).

Even in the paragraph above, I have written more details than I keep in my head. My actual thinking is something like: divide in half repeatedly and use the three givens (boundedness, infinite, closed).

To keep many proofs (especially ones that are not being written out formally, but rather explained in an oral examination where they will probably give you hints if you get stuck) in one's working memory is a doable task. But it does, in your own language, "require quite some mathematical maturity."

Summary: This task could be unreasonable if the proofs are overly involved and too many; but, more generally, the ideal for such an examination is that students can grasp the key concepts and reconstruct proofs or proof sketches without needing to store word-for-word arguments in their working memory.

Answer 2

I'd certainly expect a student to be able to remember the more important proofs, and remember enough of more run-of-the-mill results to be able to come up with a proof (sketch) on short notice. But I'd also expect quite a bit more, in terms of being able to use them to solve (somewhat) similar problems that weren't covered in class. Given time constraints on exams, I'd probably leave out most of the "state and prove theorem XYZ" type questions. Some memorization is OK, but the focus should be elsewhere.

But then again, this is first term, and I'm spoiled by more advanced students anyway.

[In the above you can freely replace "proof" with whatever analogue may be applicable to some other subject area, be it Computer Programming or Italian Renaissance Poetry.]

Answer 3

This is a good way, in my opinion the best way, to teach somebody that actually wants to study mathematics, especially if they should have the goal of becoming a mathematician. I use the word teach above, since while the question is about the exam, this style of exam also impacts the way students (should) learn the material.

I can see nothing wrong with it. In various systems it is also not at all unusual, and not limited to some "elite" groups or institutions. Obvioulsy, in order to judge if some course is hard or not one would have to know more. But, just the general layout is not that unusual.

The reason why I consider this as a good way is that it makes it necessary for students to really engage with the theoretical core of the material, and not just learn how to do some excercices as they often are used to from secondary education. This is often a fundamentally different approach, and the first year of university seems like a good point for this shift of style.

Answer 4

I myself am not very interested in seeing whether students (whether exceptionally talented or not) can reproduce long, complicated proofs. Or do long, complicated computations. I am interested in their awareness of what is true, and of the relative difficulty of proof, and of the prerequisites for a good proof. And awareness of the utility of the fact(s). To my mind, "the standard results" are the context in which one does mathematics. Re-establishing the standard context, as opposed to demonstrating awareness, is not very interesting or useful.