Nick Henscheid
Mathematician

Tips for Success on Exam Day

Everyone has their own exam taking strategy. Some like to do problems one-by-one in order, some like to jump around, some like to do the hard problems first, some do the easy ones. While you should trust your own instincts, I think the qual is a different enough exam that I’d like to provide a few tips that I’ve gathered. That’s what this post is about.

Get Your Mindset Right

This really is the most important thing you can do. This exam is not intended to get rid of you (that would cost the program a lot of money!), embarrass you, or make you feel inadequate. It’s a way to get you to work hard, bring your strengths and weakness to the surface, and make sure you’re ready to move on to doing independent research. So, don’t be afraid to mess up a few questions! Showing reasonable progress on a problem, explaining where you got stuck and where you see possible weaknesses in your argument is infinitely better than leaving a question blank because you’re not 100% sure your solution is correct. Trust yourself, seriously - you’re in a mathematics PhD program, you know a lot more than you think you do!

Survey the Exam and do Triage

The first thing I always do during an exam is glance over all the problems. For the qual, I suggest going a half step further: for each question, write a one or two word description of the problem such as “contour integral”, “analysis proof” or “SVD”. This way you can make a snap judgement about which problems to tackle first and which to leave off until the end. Don’t spend too much time reading each question though, because you want a relatively clear head.

Do Some Warm-up Problems then Plan Your Attack

Probably your biggest enemy on the exam is yourself, most notably your confidence level. I strongly suggest choosing one or two problems to do first that you’re fairly confident you will be able to nail right away - if you try to start with a harder question, you might end up getting stuck and frustrated and that will nuke your mindset for the rest of the exam. For instance, I felt most confident with analysis proofs and linear algebra questions, so I made sure to do a couple of those first to boost my confidence to take on the harder questions. I always put contour integrals and numerical ODE questions off until the end because I knew they were tedious but (usually) straightforward, so I could do them with a tired brain. You should think about your own strengths and weaknesses and formulate a strategy that you think will work, keeping in mind that your goal should be to complete as many problems as possible - 6 “pretty good” solutions and 6 “OK” ones is better than 3 great solutions and 9 terrible ones!

Sketch Your Solution First Before Committing

If you’ve ever graded an exam - which you all probably have at this point - you know why I’m saying this. Don’t use the exam itself for scratch paper…figure out the key components of the problem separately, then write up a reasonably tidy final version. This makes it clear what your steps are and where you got stuck (if you do), so consequently will lead to a more positive assessment.

Don’t Get Hung up on Small Details (or, the “95-5 rule”)

I’ve mentioned this many times over the summer. A lot of times, you can get 95ish percent of a question in less than half the time if you allow yourself to be a little bit (but not too) sloppy. For instance, with distributions, you can usually “get the right answer” by working formally/symbollically instead of with test functions. Later on you can (and absolutely should) go back and figure out what needs to be made more rigorous, but carefully - you can eat up a huge amount of time trying to perfect every answer. Sometimes, if you feel pressed for time, it’s OK to just write “I know I should do X here, but I want to work on another problem”. This isn’t to say that mathematics should be done sloppily, just that solutions to hard math problems come in multiple stages, and usually the final stage (making it perfectly rigorous) takes the longest amount of time.

Leave Time at the End to Tidy Up

Make sure you leave time at the end (up to 30 minutes) to go back over the solutions you think are correct and double check that you have addressed every part of the question (sometimes I forget to do a part (b) or something). Then, on the problems you haven’t been able to figure out, you need time to cut your losses, explain your approach and where you got stuck.