How (Not) to Communicate Mathematics
Allison Henrich
In grad school, one of the things I had to learn was which researchers’ work was readable and which I should avoid. There are some (wonderful!) researchers in my field who rewrote important, yet incomprehensible papers in a more understandable way–a way that motivated the problem being solved, reviewed background material, used illustrative examples, and illuminated connections between ideas that had been obscured in the original work. Without these benevolent translators, the ideas would have only been accessible to the top experts in the field. It would have taken years longer for the results to be disseminated to the research community broadly, delaying the work’s impact.
Communicating well is fundamental to the growth of mathematics and to advancing our goal of inclusivity. If a research paper is written only with the two most eminent mathematicians in the field in mind, that paper will likely have its designed impact: it will reach two people. If research is communicated in a way that invites a larger audience to engage with the ideas, more people will have access to research in that area, and the field can flourish.
The founders of the MAA were keenly aware of the impact good exposition can have on welcoming more folks into the math community. The first publication of the MAA, the American Mathematical Monthly, is one of the most widely read math journals in the world. Its founder, Benjamin Finkel, wrote, "Most of our existing journals deal almost exclusively with subjects beyond the reach of the average student or teacher of mathematics or at least with subjects with which they are familiar, and little, if any, space is devoted to the solution of problems." Though the Monthly was founded in 1894, Finkel’s words are true today. The MAA journals are among a small collection of venues for publishing mathematics research in a way that is broadly accessible. While articles written for MAA journals are written with a more diverse audience in mind than those typically written for discipline-specific research journals, there’s a lot we can learn about communicating mathematics from the inviting writing in these journals.
I was first encouraged to read and submit work to MAA journals by Joe Gallian when I was a new PhD. As an MAA Project NExT fellow, I signed up for Joe’s “Getting Your Research Off to a Good Start” workshop at least three times. In addition to sharing many useful pieces of advice, Joe talked about the importance of good exposition and the massive impact survey papers can have. His most-cited paper, “A Dynamic Survey of Graph Labeling,” has 4,460 citations as of this writing (according to Google scholar). Originally published in 1997, Joe has written 24 editions of this paper, with the most recent edition published in December of 2021. How might the combinatorial subfield of graph labeling be different if Joe’s dynamic survey didn’t exist?
When someone writes a survey article, they are not just writing for the top experts in the field. The act of writing a survey article itself is an attempt to make mathematics more accessible, inviting more people into working on problems in the area. As I reflect on the common elements of the best research papers I have read, I realize that they all include introductions that are like mini-survey articles. They contain all of the relevant references, but they don’t outsource all of the work of communicating fundamental background material to those references. They give a brief recap of the important definitions and major results that will be needed to understand the results of the paper. They masterfully explain complex ideas using simple examples. The effect is that the mathematics is approachable. The reader is given tools to develop much of the same intuition about the mathematical objects that the author has, and, therefore, more people are invited into the conversation.
Why don’t more people write this way? First of all, it’s much harder. You need a high-level view, not only of the specific problem that your paper aims to solve, but also of the literature that has shaped the historical landscape of this and related problems. Then you need to be able to concisely share this story with your readers. It takes a lot more thought, which takes time. To write well, you need to think about your research as a teacher might. How can I best teach my reader to understand my results? (Not simply: How can I write down my technically correct results?) When you’re teaching, it’s a common pitfall to focus your attention on the two best-prepared students in class while other students languish in confusion. Similarly, when you’re writing, it’s a pitfall to write for those two mathematicians who already understand your work well. To teach well, you have to care about your students and invest your time and energy in their learning. To write well, you have to care about your readers and invest your time and energy in their understanding.
Secondly, many authors are subconsciously afraid that if they make their ideas too accessible, their work will appear trivial. Indeed, if we do our job of communicating well, many of our deep ideas will look simple. On the other hand, if our work appears complex and technical and our papers impenetrable, it will make us seem smarter, right? This is a more pernicious problem that we need to address not necessarily as authors, but as a community of consumers of mathematical ideas. We, as a math community, need to place value on clear communication so that it’s less risky to illuminate our ideas.
So, what if we invested the time in writing well? There are dangers, to be sure. If you need to get tenure in two years, you don’t have the time to take an extra several months fine-tuning the exposition of your research paper and soliciting feedback. You need to submit your paper to a journal yesterday so it can go through the 1-24+ month process of being reviewed (and then, perhaps, be declined and revised/resubmitted elsewhere) so that you can *fingers crossed* report that it has been accepted for publication by the time you submit your tenure file. Given constraints like these, maybe we can take baby steps towards improving our writing. If one of our goals is to invite a broader readership, we could start by incorporating one or two more examples or an additional helpful image into our papers. Or maybe we could include an extra paragraph on the history of the problem we’re solving. Even an enticing abstract can go a long way towards motivating readers to work through our introduction and try to understand the statements of our main results. So, we could start there, too. Baby steps.
Addressing the second issue is harder, but we can make incremental progress on this as well if we act collectively. As readers (and journal referees, reviewers, editors, etc.), we should prioritize clear communication, acknowledging the difficult work that goes into making ideas accessible. When we read papers that are well-written, well-motivated, and accessible, we should share those papers with colleagues, holding these papers up as models to be emulated. When we are refereeing research papers for journals, we should not shy away from requesting that the authors revise their work if it’s too obscure and unmotivated, suggesting edits that would help a broader group of readers understand the work. Too often, we let poor exposition slide when reviewing papers, perpetuating the impression that if the writing is impenetrable, the research must be good. This is dangerous. It provides nourishment for the exclusionary elitism that exists in many mathematical circles.
And that’s just it. Why write well? Because it helps us achieve our goal of including more people in the math community. It gives people different entry points into working on cutting edge research. A well-written paper can motivate. It can inspire. It can spark math joy. Have you ever read a math paper that put a smile on your face? Have you ever written a math paper that put a smile on someone else’s face? Maybe you could.
Allison Henrich is a Professor of Mathematics at Seattle University and the editor-elect of MAA FOCUS.