Exams

I fucking hate exams. I'm also not really a big fan of homework. These might be rather odd statements to hear from someone who really does quite like school, and who plans to go into academia. But, they are true. At this point in my life, I am well and truly of the opinion that exams are a thoroughly horrible pedagogical tool that actively hinder the educational process, and do not give a very good indication of one's knowledge or understanding of the material. The best exams are the ones that do the most to minimize the problems that exist within exams; even then, I don't exactly think that they're good, per se.

I'm going to begin by going back in time a little bit to my teenage years. Though I'm currently in America, I did not grow up here; instead, I lived in a country that had a strong over-reliance on exams in their educational system. The exams themselves came from British colonialism. For two years you would do nothing but study for those exams; nothing else mattered. Your entire sense of self-worth as a person becomes tied up in your academic performance, and your academic performance is solely reliant on how you do over a series of intense examinations - in my case, when I was 16, I had 28 hours of exams across 6 subjects, over the span of a couple weeks. And they're graded on a curve; not the American 'graded on a curve' that's designed to bump people's grades up, but the 'you need to be a standard deviation above average to get a B' sort of graded on a curve.

It was a frankly quite toxic educational environment, no matter how supportive my teachers and tutors were of me. When I recieved my grades - over half of which were As, then I had two Bs and a C - I wanted to kill myself. I'd done significantly better than average on almost all subjects, and managed to pass everything. For some reason, the colours that I reported in the chemistry practical were all wrong, and that grade instantly killed my passion for the subject which I'd previously enjoyed. One of the Bs, as well, resulted in a rather lukewarm response from my principles of accounting tutor, who had developed quite high expectations of me.

Despite that, the exams in my home country were difficult enough that I didn't need to study for the American SAT to get a good grade on that. And thus, I was able to get into some universities in America, which started my US undergraduate experience.

Exams were definitely less intense throughout undergrad than they'd been when I was in the equivalent of highschool. However, this was counterbalanced by the existence of homework. Before undergrad I never needed to do weekly homework submissions; I'd do practice problems, but they were on my schedule. Now, the pendulum has swung from everything hinging on one exam to everything hinging upon my quick ability to figure out time management skills and get homework and projects done in time, and balance 7 different expectations for 4 courses simultaneously.

Despite my ADHD, I still see this as an objectively better system. It's not a good system by any means. I feel like time moves faster for me; I lose track of hours and days quite easily, so I was consistently falling behind in schoolwork. What I would typically do is alternate between my four courses; I'd spend a week doing four week's worth of homework in computer architecture, then I'd spend the next week doing four week's worth of homework in abstract algebra. (By "week", of course, I strictly mean the weekends - I was too busy attending lectures to have the mental bandwidth for homework during the week itself). There are exceptions; I found intro physics to be a rather boring subject, so I did none of the homework (and still got a B+), and I did all of my probability homework over the course of a single 20-hour binge homework session (and still got a B+).

The more egregious example of my struggles with homework, however, came in the form of the courses that I ended the semester with most of the homework undone. They certainly existed. Typically what would happen is the professor would just award me an incomplete without my asking them to. I would proceed to spend a month or three panicking about how I should work on my incomplete, without getting any work done - I simultaneously did not rest, and also did nothing. Then, with a few weeks left until the next semester, I would just go ahead and do all of the work.

There are examples in the other direction. Once I ran out of courses to take (which was not difficult to do, at a small, 4-year LAC), I started doing independent studies with professors that I liked. Self-directed reading, focussing on subjects that I liked, that I was interested in - scarcely a day would go by where I didn't do an exhorbitant amount of work for these. I once did half a semester's worth of a graduate-level applied functional analysis textbook in the span of two weeks (while ignoring all the homework for my other classes, of course!). Getting to control what I studied, when I studied, and how much I studied tended to result in me studying significantly more, and learning significantly more, and retaining significantly more.

So, with all this in mind, grad school sounds like either a perfect fit for me, or hell on earth. Certainly, my undergrad professors thought that I was smart enough to go to grad school - near the end especially, it felt like they were bending the rules of the syllabi that they assigned in order to give me straight As (ignoring deadlines as much as I did, etc.), and they told me that they wrote some of the strongest letters of recommendation they'd written for me. So, I decided to give it a go.

First year grad courses feel like hazing, but I won't talk about that. Instead, I will talk about what is truly hell - qualifying exams. It feels like I'm a teenager again, to a large degree - everything is dependent upon a series of exams. The exam stress started to get to me. I had a panic attack in a midterm (because despite quals, we still have midterms!). Ultimately, the stress of grad school - not of research, but of preparing for some fucking tests - wound up resulting in a suicide attempt and a hospitalization, that significantly impaired my ability to study for the exams that I was stressed about to begin with. Ultimately, I did not pass them - though the reason I didn't pass was because I kept fucking up basic facts that I struggled to memorize, not because of any difficulty I had with the material. (For example, there is a standard example of an L^2 function which is not L^1, and where in the real world I would have just looked it up in a book, on the test, I was punished for just not remembering it). I still have another year in which to make my attempts, but I just want to be done with tests.

And then there's my teaching obligations. At least homework and exams are objective, I always thought. They're stressful and anxiety-inducing, but at least with a math test, it's clear and obvious how many points you should get for any problem. This cannot be further from the truth. I was instructed to give out weekly quizzes to the sections I TA'ed for, and to grade them. If a mistake was common enough I would assign the same number of points for it each time, but the students were oftentimes creative in their mistakes. Ultimately, on a five-point question, it came down to - how much do I personally think the student understood the process of what to do? I would look through the working and try to give out the best grades that I could justify, but ultimately it was all just so deeply arbitrary. Homework is the same. Exams are the same. When a student was completely right or completely wrong it's easy, but everything in between - the "partial credit" - feels entirely arbitrary.

So, with that out of the way, I'm clearly biased. I've obviously had bad experiences with exams, and those may be colouring my perceptions of why I think that they are bad. Maybe if I was "good at tests", I would not have the opinions that I do. But, I will still sum up why I think that exams are just bad pedagogy.

1. "Exam Conditions" aren't real. When I'm doing math, aside from when I'm taking a test, I always have references on hand. I can always look up theorems, and proofs of similar results. I always have access to multiple textbooks and sets of notes. I have access to the arxiv, and know how to search for papers that I'm interested in. Sure, I still need to prove results for myself, but I don't need to memorize the Sylow's theorem or all of the conditions of the Schwarz Lemma.
2. Exams are nothing like real research. In particular, exams are a sequence of relatively straightforward problems (often of a specific, algorithmic type - there's a stock Rouche's Theorem question that anyone taking complex analysis has probably seen 50 times) to be solved within a few hours. A friend of mine who passed all of the exams told me that, despite being done with the exams, he has no idea how he'll even begin to do reading and research and get to the point where he can get his degree. Conversely, with my experience doing independent studies, I know how quickly I can learn math in a self-directed manner, and know that I can probably bring myself to the ability of doing research in a year or so - and the exams have done nothing to prepare me for that. The goal of grad school is to make researchers, and yet qualifying exams seem to be almost completely orthogonal to that.
3. Exams cause people stress. I think that a significant portion of the stereotypical American "I hate math!" mentality is a result of people hating exams, and conflating math tests with math itself. I like solving puzzles. If there were no stakes, I even like solving puzzles in a timed setting. I solve Sudokus for fun, and when I time myself I can often do them within 5 minutes (so long as they don't involve anything harder than what Andrew Stuart refers to as a "tough strategy"). That said, if someone told me that a month from now, my life goals will become dependent upon my ability to solve 10 sudoku puzzles in a single hour-long test - I would be stressed out of my mind, and would likely grow to hate sudoku. And of course, making students hate a subject will discourage them from learning anything about it in their own time, which is antithetical to good pedagogy.
4. Conversely, tests can be solved well without possessing a deep understanding of the material. On the complex analysis qual that I took, to get 100%, you need only memorize Cauchy's Residue Theorem, Rouche's Theorem and Schwarz's Lemma, alongside very specific methods of solving very specific problems using those three results. Despite complex analysis being the analysis subject I'm the weakest in in terms of my conceptual understanding, it was the only paper that I got 100% on. This may just be a weakness of the paper I was given and not of exams themselves; but I would argue that it's really difficult to write problems of an appropriate challenge level that are solvable in about 3 hours and that are able to deeply test a student's understanding of the material.

Ultimately, I don't really think that I have a good solution to these problems. Abolishing tests and making everything homework might be a partial solution, but even then I'm not too fond of it - homework has a number of the same problems as tests, just to a lesser extent.

Or maybe we could motivate students to want to learn by showing them the intrinsic value of the subject matter. University students are paying tens of thousands of dollars to go to their university; I think that we can trust them to learn without using an existential threat to force them to. Graduate students are graduate students because they're passionate about the material that they study. They don't need the threat of bad grades to motivate them to study. All exams seem to do is discourage people from wanting to learn, who otherwise would have loved to learn.

Last year, when I was doing my first year of grad school, I had to regularly remind myself why I was there. Exams are horrible, yes, but they're not what real math is like. When I crack open a textbook and start reading it for fun, and doing problems because they look like a cool challenge, and attending seminars that I'm interested in because I'm interested in them, I remember why I love math. This past month I've started doing a research project with a professor that I like, and I'm happier than I've been in years, and I'm working my way through the math that I'm doing at an insanely quick pace. And to a large degree, it's because that research project doesn't come with some arbitrary exam hanging over my head, testing my ability to solve tiny problems in a few hours.