Jeremy Côté

A Different Kind of Final Exam

What does a final exam look like in mathematics?

I bet you can tell me exactly what happens. There’s a room full of students, each isolated at their own desk, working against the clock to answer a bunch of problems. Usually, the students don’t have access to any sort of notes or results that were seen in class. In the exam, it’s every student for themselves, with no outside help.

They certainly cannot collaborate with any of the other students. After all, if they collaborated, the mark on the test wouldn’t reflect the knowledge of the student, but some union of everyone who pitched in. This would be unfair, since students who aren’t as strong could benefit by working with students who are much better than they are. As such, the only fair way to do things is to have everyone work on their own without any sort of resources, right?

But there’s another way.

It’s not fool-proof, and it can’t be implemented in all classrooms, but I know that it does work. Furthermore, it creates a much better environment for the students.

As I write this (but a year later when you’re reading), I’ve just finished writing my final exam in my abstract algebra class. For the exam, my classmate and I (we were only two in the class) were allowed to work together. We had to write our own solutions, but collaboration was allowed, and encouraged. This is a huge shift from the usual routine of writing final exams, and I had been looking forward to this opportunity for the past few weeks. I wanted to see how it compared to the regular exam format.

Here are my thoughts.

The first benefit is one that anyone can agree with. Collaborating gets you out of your own head. When faced with a problem, we often try to tackle it in a specific way. If this method doesn’t work, it can be difficult to pull back and reevaluate. We end up getting stuck in our thinking, wasting time and not getting any closer to a solution. With a partner though, the situation changes. If you find yourself stuck in one way of thinking, you can voice that thought out loud. Your partner can tell you if your idea is crazy, or if you might be on the right track. I can’t underscore how helpful this is. Often, we are nervous for an exam, but being able to talk with someone makes such a difference. For this reason alone, more professors should consider trying to change the format of their final exams.

The second benefit is that you don’t have to remember every single fact from the semester. I find it so ironic how teachers will often tell students how great collaboration is, yet then they base most of our grade on a situation where no collaboration is allowed!

Have you ever gotten to a final exam, seen a question in which you knew there was a result or theorem that was seen in class, yet you can’t remember the precise wording? This is frustrating, because you can’t proceed to answer the question. For myself, this has led to much agonizing during a test over what is the “correct” word, leading to a huge waste of time. With a partner, you can just ask them. It’s as simple as that, but it’s amazing how nice it is to be able to confirm these little things. It lessens the load in one’s mind to remember every single theorem and result that was seen in class. The load gets spread out over two people, making it easier on both of them.

There’s also the fact that one can voice their thoughts out loud. A small thought from one person can spark the other to think of something that ends up being the key to the problem. This is exactly what happened during my exam. One of us would say something out loud, and the other would get an idea from it. Bouncing ideas off of each other would not be possible in a traditional exam format, but it’s something that should be encouraged much more.

I hope you can tell from the above paragraphs that I’m excited by the prospects offered by this experience. It’s not a huge change in the final exam format, but it’s enough to make me believe in the viability of exams again (though I do have other ideas too). This transforms the exam format from making students memorize and regurgitate facts to collaborating in order to tackle problems.

I do want to note some caveats. Clearly, this would be difficult for a larger group, since it could create chaos in the classroom during the exam. The larger problem though is one my professor brought up himself when he suggested this to us. He said, “I can’t usually do this with other groups, but since you’re only two and you are both strong students, it should work. I think it’s more in the spirit of mathematics anyway.”

This is an inherent limitation of the idea, and one I highlighted at the beginning. If you have one student who is a lot better than the other, than it can skew the results of the “weaker” student. Whether or not that’s a problem, you can decide. For my experience, I found that I was the weaker of the two students (but not by too much). And yet, I still found multiple instances to contribute.

I’m convinced that this is a great way to do a final exam. As my professor said, it is much more in the spirit of mathematics than the traditional exam ever will be. It allows the professor to bring collaboration back into the classroom, turning an exam into a problem-solving session instead of a nerve-wracking experience.