Thoughts on qualifying exams ·

So, you’re gearing up for your Qualification Exams, but before you begin, I’d like to share a few thoughts about the exams and the preparation. This has less to do with the course material and more to do with the ’exam’ aspect of it. It is a rather long rant about ‘preparation’. I’m sure most of us are aware of the things listed below. You may even find it boring but I recommend that you read it. It never hurts to remind ourselves of the easier things that we take for granted. There’s no one-size-fits-all preparation strategy. We all learn differently and at our own pace. However, there are general ideas that can streamline your preparation. Even if you don’t find anything helpful here, this should at least encourage you to take a step back and think about ‘how to prepare’ before you start the prep. It is good to have a plan so that you use your time more effectively. The qualifying exams are designed to test your understanding of the material covered in the 5000-level courses. Therefore, it won’t be enough to just know the material if you do not know how to apply it. Memorizing has its place but cannot be a substitute for understanding. Thus, I would like to draw attention to the following aspects of the exams. This should, hopefully, guide your preparation better.

Problem-solving
#

Throughout Qual Prep, you’ll solve a lot of problems. The goal isn’t to do a large number of problems and hope that some of them show up in the actual exam. Instead, through practice, develop habits that’ll help you avoid common mistakes. Numerous books and articles have been written on problem-solving. The resources are endless. You most definitely have methods of your own. That being said, here is what a generic approach should look like:

Make sure you understand the problem. On the face of it, this sounds like the most obvious advice one can give you. However, on a timed exam it is not good to have wasted 10 minutes on a problem only to realize that you misunderstood the problem. Are you supposed to provide a proof or an example/counterexample? Try rephrasing the problem in your own words or converting the problem statement from words to symbols/notation. Maybe drawing a picture helps. Can you think of a simpler case or an example where you can easily verify the problem statement?
Come up with a plan. Once you’ve identified what to do, it is time to devise the master plan. Here are a few questions that should help you in the process. Have you seen a similar problem before? Do the assumptions the problem makes relate to some well-known results? Maybe what you’re supposed to conclude is a consequence of a well-known result. What’s a better method to use here? For example, if you want to directly prove the statement or its contrapositive; maybe proof by contradiction works better here, etc. You should have a convincing plan before you devote a serious amount of time to a particular problem.
Execution/Verification. There is nothing much to be said here. You have your outline and it is only a matter of filling in the details. Revise your plan as you go, if needed.

The planning step may sound trivial to you. It might feel like it is “not worth the time”. However, you should realize that it isn’t simply helping you answer the question but also helping you organize your thoughts. It is easy to get lost in the details while solving a problem. Clarity of your proofs matters just as much as correctness and an outline will make your writing more coherent. You might be on the other extreme. You might think ‘planning’ is easier said than done. Looking at the problem, if you don’t even know where to start, then all you need is some practice. Here are a few things to try. You should look at solved problems and think about how you’d solve them during an exam. While studying from your course notes, you should write a summary of individual sections. Try listing out key ideas and how they relate to each other.

When working with friends, before presenting a detailed answer to a problem, you should give a “sketch” or a strategy. Explain how you would approach this particular problem during an exam.

While practicing, you will revisit the problems you’ve already solved. If you get stuck, you should only look at the sketch from before and see if you can fill in the details.

Although problem-solving is the most important aspect of the exams, we shouldn’t get too focused on it. These are timed exams and being prepared doesn’t simply mean having mastered the subject. Apart from problem-solving skills, you also need to learn how to manage your time and expectations.

Managing time and expectations
#

During the exam, say, you’ve spent 15 minutes on a problem but you’re nowhere close to the solution. You keep going because it feels like you have a better chance of solving this problem than starting a new one from scratch. Much later you realize that you’re going nowhere with the problem and should have switched to a different one sooner. Or take the other extreme case where you spend only a few minutes on each problem but don’t have a solution to any of the problems. I wish there was a golden rule about the least amount of time to spend on a problem and when to move on. It varies from person to person and problem to problem. Over the summer (or winter, whenever you’re preparing for the exam) while practicing, you should be mindful about this and find your comfort zone. Experiment with solving problems under time constraints, and find your rythm.

I should end this note with an important aspect of the preparation which is not entirely exam related.

Collaboration
#

Remember, you’re not alone. You’re all having a similar experience. There is no reason why you should do it alone. I encourage you to work with others. Learning with/from others is only going to make the process easier and faster. Together, you’ll thrive!

These are just some of my thoughts. Your ideas might be better than the ones listed here and you should not shy away from sharing them with others. Good luck and have fun!