Power of the Pen ≥ Power of the Point

August 14, 2019
Kate Thompson, AoPS Grader

While I’ve always loved reading, I have hated writing for a long time. I never had any desire to be a creative writer, and when it came to non-fiction, well…it just seemed pretty self-explanatory. State some facts. Cite some books. Done. My mother was a former high school English teacher and Catholic school principal. She decided when homeschooling me that—whether I liked it or not—I would write regularly and learn the rules of grammar. She taught me how to diagram sentences (#oldschool). Every morning I had to read a news article and write a summary. These assignments weren’t remarkably taxing (except the day she had me diagram the first sentence of the Constitution—THAT was a bear). They were just boring.

In undergrad, I had to write at most four papers (each at most eight pages long) a semester, all for courses that were “general graduation requirements.” Again, it wasn’t that bad since these papers were all basically glorified book reports. No one in undergrad really cared about grammar, sentence structure, or anything like that. Other than creative writing, there were no composition courses offered. You’d earn an “A” just for proving you’d read the book, and for following the page limit and margin requirements. 

There is, of course, ONE course math majors almost everywhere need to take that is writing-based. While every college has a different code name for it, it’s essentially “Intro to Proof.” Though I took it with one of my favorite professors, it was my least favorite class. It seemed to take copy editing to extreme levels. Who cares if equal signs line up perfectly across multiple equations? Do you really need to see a square to tell that a proof is done? We also had to submit everything in Microsoft Equation Editor and on a floppy disk. Come to find out, even back in these dark ages when Pluto was still a planet, neither of these was en vogue. When I started graduate school two years later, everyone except for yours truly was fluent in LaTeX and using thumb drives. 

It didn’t take long to learn LaTeX; however, when I was also busy trying to learn the material to pass my qualifying exams, LaTeX was just another thing added to what seemed like a way-too-long list of topics to master. So I decided if I ever got to teach “Intro to Proof” myself, my students were going to be TeX-ing their portfolios. As for the quality of the writing: math papers are highly technical and are not meant to be entertaining. In fact, in comparison to homework assignments, fewer details were demanded–a lot is left to the reader. So again, it seemed that by just citing literature appropriately and making it clear that your math was new and correct, you had a publishable paper. 

What did seem to be a big deal: the oral communication of mathematics. There are often multiple seminars every day in graduate school. The culture at my department required attending at least two or three seminars every week. Then there would be impromptu “working seminars,” where the goal was actually to get a paper out. But every single seminar was an hour (or longer) talk. Networking at conferences was also a big deal. Being able to give the mathematical research equivalent of an elevator pitch at a lunch table was key. It’s how you convinced other mathematicians that your work is interesting, or that you’re worth working with on a project.

The Student Becomes the Master. 

Fast forward to 2014. I finished my Ph.D., and started a postdoc at Davidson College. I felt very lucky to have this job—not just because it was my dream location and type of school. The last few years of “the market” (aka the job market in academia) had been remarkably rough. Many of my friends, whom I honestly thought were better mathematicians than I, could not get a job. Even today, it’s a buyers’ market—there are many more Ph.D.’s who want to be faculty members than there are faculty positions. And unlike getting into college—in applying for jobs, it’s not good enough to be at the top of your class, or even in the top quartile of all applicants. Since there’s only one slot, you have to be the best candidate. With competition that fierce, anything and everything can be used as a reason to eliminate an applicant.

But that had not yet resonated with me. I was too busy living my dream. In my first year at Davidson, I was assigned two sections of Intro to Proof (code name: “Sets and Proofs”). Payback is sweet. I was going to take all of the good and the bad that I thought I had learned about communicating mathematics and apply it. My students were going to learn LaTeX—that was my big emphasis. And in any more advanced class, I would “incentivize” students (i.e., give minimal extra credit) to those who continued to TeX. No Microsoft. Ever. Again. Also, emphasizing the importance of the “elevator pitch” and utilizing the small class sizes, I had every student give a 20 minute math talk to the class. The students graded each other with a pre-designed rubric on board work, ability to address audience questions, and other general presentation criteria. I thought I was rocking this. Even retrospectively, what I did was not a “waste” of time.

But in 2016, as I left Davidson for my first tenure-track, I also started to work for Art of Problem Solving.

Reading, Riting, and Rithmetic

One thing I have always thought was unique about Art of Problem Solving (and something which I increasingly believe is absolutely crucial) is its emphasis on writing. Around the time I started working for AoPS, I realized I wasn’t really concentrating on the aspects of writing that were most helpful. Writing systems (like LaTeX) and oral communication were not the keys to success. Rather, it was the quality of writing that was truly important. 

Case in point? By the time I started my first tenure track, I had applied to around 350 jobs between my two job cycles (and now with a different tenure-track, I can say conservatively that I’ve applied to close to 700 jobs). With every job for which I’ve been a finalist, the quality of the writing in my letters of recommendation have been mentioned; excerpts from my teaching statement have been quoted back to me. I started to see that I was never going to have the chance to give my elevator pitch if I couldn’t convince someone first to hold the door open.

From 2015 thru 2017 I co-ran a National Science Foundation sponsored summer program which allowed me to choose 8 students annually from a national applicant pool to conduct mathematical research. I hoped there would be a lot of interest, and I got exactly what I wished for; but with an applicant pool of between two and three hundred every summer, suddenly I was rivaling Harvard in the “shockingly low acceptance rate” category. Beyond obvious sieves like “is the application complete” and “does the student have relevant coursework,” what did my collaborator and I use to determine who made the next cut? The quality of the students’ writing. How effectively did they communicate their enthusiasm and ability to work with others in their personal statement? When they wrote about math, did they express sweet yet completely naive dreams of proving the Riemann Hypothesis during our eight-week program, or had they actually read the materials on our site (which, sadly, did not strictly-speaking involve L-functions) and expressed interest with those topics? It came down to their writing and—trust me—you’d be shocked at how poor the writing was.

A mathematical vase made by the author with an actual L function value
An actual L-function value

AoPS recognizes writing is crucial for true mathematical development. At least as I’ve always seen it, the separation of style and math points makes it clear that it’s not enough in certain AoPS problems to have the correct numerical answer. While the submission doesn’t have to be TeX-ed, it must feature complete, grammatically correct, English sentences that begin with capital letters and end with appropriate punctuation. It must read like any other writing assignment in paragraph form (as opposed to writing equations and steps one line at a time like you’re writing a to-do list). It must have appropriate citations of any theorems or formulas, and certainly of any extra help used. The why and how and not just the what of the problem must be addressed.

The rest of the L-function value

It seemed so simple that content and format and style should matter more than pretty fonts and special symbols. And yet it took me years of being on both sides of classrooms and interviews to see that.

WLOG and ASFOC that TLAs are NBD

Fast forward to fall of 2018. I was teaching Intro to Proof to over 300 students at Carnegie Mellon University (code name: “Concepts of Mathematics”). Clearly, the logistics of having each student give a 20-minute math talk was just never going to happen. So then it came down to determining what I wanted the writing emphasis of the course to be. 

This is what 300 (cumulative) finals look like

I did not care if students TeX-ed their homework (though those who did received minimal bonus points for doing so). But I did want them to write everything out. Even though it meant selecting results in class carefully due to length and time, I wrote on the chalkboard in complete sentences and encouraged students to correct any grammar mistakes they noticed. The students and TAs were both instructed never to use acronyms—from the more common WLOG (“without loss of generality”) to the Carnegie Mellon specific AFSOC (“assume for (the) sake of contradiction”, a phrase so awkward you wouldn’t even write it out anyway). 

Kate Thompson teaching at Carnegie Mellon
Kate teaching the Carnegie Mellon course referenced here

Some students rolled their eyes at first at the attention to detail, but they learned fast and were fine. The students especially loved pointing out any errors on the board. Some colleagues (not even specifically at Carnegie Mellon) thought I’d taken things to an extreme. A few argued that I needed to recognize that for the large number of (insert population, including but not restricted to “freshmen” and “foreign students”), acronyms are helpful. They thought that putting this much of an emphasis on writing in a math class would take away from the actual math.

For better or worse, I stuck to my guns. And I wouldn’t have thought it was so important, or even something I could “get away with,” had it not been for AoPS. One frequent thought in college lesson planning since starting my contracting is “well, ten-year olds can do this, so clearly (insert college class) can too.”  While sometimes that totally backfires (not every college student is into competition math, for instance), it certainly seems spot-on when it comes to the writing standards. Had it not been for AoPS, I wouldn’t have come to the realization that in terms of what my students have to accomplish after my class, it’s easier to learn LaTeX than to unlearn what can only be characterized as lazy writing habits. Had it not been for AoPS, I would not have been so bold in breaking the honestly nation-wide traditions created by previous iterations of this course in being so nit-picky about writing things out.

Even if in later math classes students could get away with short cuts and $\Rightarrow$s or even $\ergo$s, that was specifically not the point; for 14 weeks of their lives and in a course that was supposed to teach the fundamentals we were all going to do things the formal, correct way. Learning jargon and methods that are not universal will not serve you once you leave your current institution’s bubble (#EquationEditor #floppydisk); not ever being forced to practice the habits of good writing will make it very hard if you ever need it (#jobapps #internships). 

Remembering as a 10-15 year old how I felt diagramming sentences I understand that some (if not many) AoPS students may think the style comments and scores are unnecessary and beside the point—which is to solve the problem. Unlike me, I hope it doesn’t take the students another decade-plus to realize that it’s this attention to detail and attention to proper and formal communication that will separate them from the rest of the pack.

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