A well-written exam will test students’ understanding of the concepts covered during class, with a series of questions that might take various forms, including: multiple-choice, true false, calculations, proofs (in the case of mathematics exams), and essays.

To really be prepared for an exam, regardless of the exam format, students need to be able to boil down the material from class and the textbook into key concepts and skills that should be practiced over and over, in different contexts.

As one progresses through the educational system, the problems tend to become less about rote memorization (“plug-and-chug”) and more about synthesizing concepts and applying ideas to new contexts, though the rate at which this happens depends on where one receives their education.

In any case, I want to talk now about the various merits of writing your own problems if you’re a student preparing for an exam. While I’ll talk about the advantages of this, some of the reasons I do it personally, and why I implement this into my teaching, remember that the fundamental idea is to continually rearrange and manipulate the concepts that are important for understanding the material.

Why I started writing problems, and why I still do it

Even as an undergraduate, I started writing problems for fun; that’s really what it came down to before I had fully realized the benefits. Even in such a “left-brained” discipline, I always try to tap into it with a pinch of creativity.

While I did this for classes, I also wrote problems for high school math competitions on a volunteer basis; this was related to my service as a co-founder of the Carnegie Mellon Informatics and Mathematics Competition, affectionately known by its abbreviation, CMIMC.

Eventually, I realized that what I was really trying to do is read the minds of my professors. I’m putting myself in the teacher’s shoes when I write problems by asking: “if I were the teacher, what concepts would I want to assess, and how would I do so meaningfully?” Writing problems forces me to figure out what the recurring themes were.

My efforts in the art of problem-writing reached a peak last summer as I was studying for my two written qualifying exams at the University of Tennessee. The first exam covered real and complex analysis, and the second exam covered partial differential equations.

As I had taken the corresponding courses during the year prior, I had realized that the subject matter in the two areas carried a substantial amount of overlap, and I used that to my advantage. Since I was studying for both exams at the same time, I tried to treat my endeavor as studying for one larger exam as much as possible.

Thus, alongside working through hundreds of problems from books and past exams, I tried my hand at writing a large collection of my own (and of course solving them all as I went along).

If you’re interested, my efforts can be viewed on this page of my website, under “Practice Problems and Practice Exams.” As I created the problems, here are some of the questions I asked myself:

1. What formulas and theorems are repeatedly used within each topic? How can they be used for immediate consequences, and how can they be incorporated into more complicated arguments?
2. Given an existing problem, what else can be asked that’s related? Will the same concepts and ideas work as did before? If not, why not?
3. Can I combine concepts from the two subject matters I was being separately tested on?

Needless to say I passed both exams on my first try. But even if you aren’t at the graduate level, this thought process can still be helpful as you synthesize and embrace the concepts being taught in your math classes.

In contrast to Ph.D qualifying exams, I actually use problem-writing as the basis for a review day activity for the entry-level quantitative reasoning course I teach at the University of Tennessee.

Students enrolled in this class traditionally work in groups of three to four on day-to-day assignments. In the class period before an exam, I break the unit of material into lessons and ask each group to write and solve a problem related to the content from a specific lesson, ideally so that all lessons are covered once.

The natural instinct of the students is to go back to their notes from that lesson and start searching for ideas that can be crafted into a problem.

Many of them go further and look at specific problems we talked about in class, or old homework problems related to their assigned lesson. Sometimes they have a little fun and make goofy word problems out of it, and that gives all of us a good laugh.

The activity has generally been well-received by my students, and as a bonus, they can take their newly written problems with them to use as a study resource.

To sum this up, I remind them that they are trying to read my mind. If they were the teacher, what would they put on the exam? It’s not uncommon for students to produce problems that are similar to those on the actual exam.

Let me conclude by discussing some practical aspects of including problem-writing in your exam preparation. It’s important to actually work out the problems you create.

Copy-pasting a textbook problem and changing two numbers doesn’t do any good unless you can repeat the procedure of the textbook problem, and one can develop a false sense of confidence if they cut corners here.

On the other hand, at the higher levels, where writing mathematical proofs enters the picture, it’s important to write out your reasoning carefully so you don’t accidentally presuppose something to be true that is actually false.

This trial-and-error can really challenge and enhance your understanding of the material. If your logic reaches an apparent contradiction, then either a mistake was made or you haven’t reached a contradiction at all, and at that point you should double-back and clear things up.

It’s also perfectly acceptable to ask another student or your professor about your confusion. I’ve prefaced such questions to my professors with the phrase, “I was thinking about how these concepts interrelate, and I came across a possible issue,” or something along those lines.

In my experience professors are happy to answer these types of questions, and are often pleasantly surprised by them since I’m not simply saying “help, I’m stuck on a homework problem” (though those types of questions are acceptable as well).

Finally, if you can convince your classmates to embark on a similar studying endeavor, you can trade questions and make for even more rewarding group study. In the event where you and your classmates disagree on whether a solution is correct, you debate it and work it out (people inherently like to be right and will vie to defend their ideas).

If an impasse is reached, you can go to your professor or a TA for clarification.

Disclaimer: I designed the review activity for my quantitative reasoning course when I was teaching it over Zoom. Now that my teaching is being reverted to in-person, I may need to adapt this activity’s structure and logistics if I wish to continue using it.