Interview with Michael Kinyon
Kamila Kinyon

Michael Kinyon is an associate professor in the Department of Mathematics. He earned his B.S., M.S., and Ph.D. degrees from the University of Utah in 1986, 1988, and 1991, respectively. From 1992 to 2006, he was on the faculty at Indiana University South Bend.  In 2006, he was pleased at the opportunity to come to DU and be part of a lively research and teaching environment. His research interests include nonassociative algebraic structures, and his undergraduate teaching interests include the history of mathematics.

Kamila Kinyon: How do you see the role of writing in the teaching of mathematics to undergraduates?
Michael Kinyon: In the field of mathematics, writing is more important than it is traditionally given credit for. The writing process is, for me at least, a big part of doing mathematics. I would be a lousy mathematician if I didnt write papers in mathematics. This leads somewhat naturally to my conviction that ones understanding of mathematics is reflected in ones ability to write it. This doesnt imply that brilliant stylistics is a prerequisite to being a good mathematician; I know many excellent mathematicians who are rather bad at organizing their papers, and I dont claim to be all that good at it myself. What characterizes good mathematical writing is that the chain of ideas is coherently presented so that each proof is clearly a proof, each example illustrates what it should, each definition is necessary without being superfluous, and so on.

So what does all that imply about teaching mathematics to undergraduates? Simply put, learning to write mathematics well is part of learning to be a good mathematician. However, there are clear priorities. As Aristotle indicated, an understanding of logic must precede an understanding of rhetoric. (As an aside, I worry sometimes that in other courses, students might be learning this the wrong way around.) Almost all teachers of mathematics have had experiences with students who cannot tell the difference between the implication if P, then Q and its converse if Q, then P. Are these really issues of writing, or just of logic itself? I would say both, because if a student doesnt have a working understanding of some basic principles of logic, then all the rhetorical devices in the world are still not going to lead to a correct argument. And in mathematics, its not a question of some arguments being more convincing than others. A purported proof either is a proof or it isnt. A purported counterexample either shoots down a conjecture or it doesnt.

In lower level mathematics courses, the emphasis is mostly on problem solving. A typical response to a test problem will be many calculations, not necessarily well organized as a piece of writing, but hopefully leading to a correct result, and if not, then sufficiently decipherable so that the source of error is clear. There are teachers of mathematics who have experimented with introducing more writing into their lower level courses. The argument, which I cant really fault, is that by learning to write about the mathematics they are learning, students will get a better understanding of it. It is a tantalizing point of view, but I am still not comfortable with it. So many students struggle with just the basics of problem solving, that I worry about adding the burden of having to write as well. I am not convinced that writing about the mathematics they are studying transfers over to improved problem solving ability. I might be less worried about this if it werent that most lower level mathematics courses are service courses, that is, their primary audience is students who are not intending to major in mathematics. I am not sure that emphasizing writing in those courses really helps those majoring in other areas. Perhaps faculty in those other disciplines could convince me otherwise.

In upper division courses, where students are usually mathematics majors, the emphasis is more on proofs, and there writing must be stressed. As I already indicated, the highest priority is a basic understanding of informal logic. Hopefully as students learn to put together coherent proofs, they are also learning how to write those arguments in a clear way, so that they can be understood on first reading.

So I guess the question becomes how should a mathematics major learn to write mathematics well. I dont have a good answer. Mathematicians of my generation, and I suspect all later ones, learned to write in a mathematical style by reading a lot of examples. We didnt focus on writerly issues in the classroom; we learned how to write a proof well by reading many polished, textbook proofs. Saying that we absorbed good mathematical writing through our pores would sound ridiculous if it werent that there is a grain of truth to it. Certainly its difficult for me to imagine majors learning how to write mathematics well in any context outside of mathematics.

There is another side of this that is unique. I teach a course in the history of mathematics, and thats a different story. The type of writing that is needed there is much different because it is after all a history class. Im expecting papers to be written as a history student would write them. My general requirements for a paper in that class are the paper should have some historical context and some nontrivial piece of mathematics. For instance, if a student opts to do a biographical piece (and that accounts for most students), then a paper should have both biographical details and some description of the persons mathematical accomplishments. But the writing of that part of the paper is not the same as writing in other mathematics courses; it is probably closer to the sort of writing that a student would do in other history courses. It can be quite difficult for students, because first they have to come to some understanding of the mathematics, and then they have to construct a presentation of it that is suitable for a general audience (such as their fellow students). In a sense, this is popularization, which is one of the most difficult types of writing for any scientist. I suppose composition specialists would refer to this type of writing as repurposing.

Could you say more about how you use writing in your own teaching? What experiences have you had with mathematics students as writers?
In the classroom, I have to focus mostly on the mathematical content. I dont really have the time to address the issue of writing as a process beyond the issues of logic I mentioned earlier. That said, I still pay a great deal of attention to writing, but mostly at the stage of giving feedback. In upper division mathematics courses, I am not grading on writing style but on putting together a coherent argument. An incorrect proof written with passion and conviction and style is still an incorrect proof, and that needs to be pointed out. Sometimes when students attempt to be too writerly, it can get in the way of their present information. Several years ago, I had a student who thought it was cute to write proofs in the form of personal narratives. She would tell fairy tales of herself coming up with the proof. I thought it was funny the first few times, but then I got annoyed and told her to stop. What makes that case interesting is that the student had a clear understanding of the mathematical concepts, but for reasons of her own, decided to present them in a rather strange way.

When writing journal articles or books, how would you describe your own writing process?
It takes me a while to force myself to sit down and write. (I guess I could stand up and write, but thats probably more difficult.) Sitting and thinking about ideas, that is, doing the research, seems easier than putting what I know into coherent form. However, once I get started, writing the paper helps me firm up the ideas in my own mind. Usually the process works like this: I already know the main results, so I start by figuring out what is the best way to organize the paper in terms of sectioning. This isnt detailed outlining (which Ive never found helpful), but just broad organization. What I spend most of the zeroth draft on are the statements of the main results and their proofs, or at least sketches of the proofs, stuck into the various sections with little to no interconnecting text.

The process of writing the main results leads to the first expansions of the paper. Looking at the statements and the proofs, Ill realize what has to come before. Maybe this paragraph of the proof would work better as an independent lemma in an earlier section. Maybe that explanation would be clarified with a new definition. And so on.

I also have to decide how much background knowledge I can assume on the part of the reader. The areas I work in tend to be esoteric, that is, a bit outside of the mathematical mainstream. Thus I often have to include rather basic definitions of terms simply because I cant assume familiarity with those terms among the mathematical community at large. By contrast, a paper closer to the mainstream can start in medias res, where someone who is not an insider has no hope of penetrating even the beginning.

Typically, I go through many revisions of a paper. I dont think of it as entire rewrites but rather tweaking individual parts. Just about the last part I write is the introduction. In addition, most of my work has been collaborative. That leads to a number of unique problems. For example, my coauthor and I might have quite different visions of what the paper is supposed to be about, and those differences might not emerge until we are pretty well into the writing process. I tend to do a lot of rewriting of what my coauthors write. That is not so much a reflection on them as it is an indication of my overall fussiness about how I want the finished product to look.

The hardest part of writing a paper is letting go of it. I have to tell myself that this is as good as it is going to be. Maybe I have other results I was trying to work into it, but I just have to force myself to set them for a future paper. Ill send a paper off for publication when I conclude that I cant sit on it any more. Ive been writing it for months and have to get it out the door. Once I do that, my mind almost immediately starts to move on to other things. I look back at papers of mine that were published several years ago, and I find them almost unrecognizable. The claim that writing is a process of forgetting is, in my experience, absolutely correct. By the way, this can be a problem when I submit to a journal where the refereeing process is slow. Ill get back the referees comments and get ready to make the requested changes, and then find myself struggling to reconstruct what were my thought processes during the writing.

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