Mathematical papers and the IMRaD-structure: how comparable are they with each other?

Question

I have a background in the social sciences, but I teach academic writing to students from various disciplines (incl. STEM). Since the IMRaD-structure seems universal, I teach them this format. While I do not always understand the content of the seminar papers, I can recognize quite well whether they accomodate to the 'syntax' of the IMRaD-structure.

(That is, the Introduction outlines the topic, contains a literature review, identifies a research gap and puts the research question into that context; the Method section details the approach with which the research question is to be answered step by step; the Result section is strictly factual about what the study found; and the Discussion section puts the result into a wider context, interprets it, admits some limitations, and calls for specific future research venues.)

From time to time, however, I get papers from students of mathematics, and they use a terminology I am unfamiliar with; in fact, I understand nothing. They label their sections 'lemmata' and 'axioms' and 'proof' to derive some equations, but I cannot even gauge whether the sentences are more on the 'method' side or the 'result' side of IMRaD.

Is there any 'easy' way for non-mathematicians to understand mathematical papers? Does anyone of you have experience with teaching academic writing in a way that could also integrate the specific conventions of mathematics, do you have any tips in how to do that, or can anyone suggest good resources to render comparable mathematical papers with that of other disciplines?

Answer 1

There is no fixed format for papers in mathematics, except perhaps that they tend to have an Introduction or an equivalent section. The key thing is mathematical writing is clarity, not conforming to a particular structure, and if you have no background in mathematics, you basically have no way of telling whether or not they are presenting their mathematical argument clearly.

If you really must compare mathematical writing to the Intro-Methods-Results-Discussion format, two notable differences come to mind. Firstly, there is not really any clear distinction between "methods" and "results". Whenever I am forced to make such a distinction (for example, if a grant application requires this), it feels very arbitrary and forced. One could force such a distinction if one really tries, but it does not feel very meaningful to me. Secondly, there is no separate discussion section but rather the discussion takes places throughout the paper. If there is something like a discussion section, it's probably part of the introduction.

By the way, the Intro-Methods-Results-Discussion format is certainly not universal even outside of mathematics. To me, it appears to be more a matter of empirical/experimental vs. non-empirical science. For instance, in my limited experience it does not appear to be the norm in historical linguistics either, despite the fact that it's not at all adjacent to mathematics.

I appreciate that you're trying to do good by the math students who come to your class, but my frank opinion is that it's basically impossible for you to teach them good mathematical writing unless you have a background in mathematics or an adjacent area. As you have correctly recognized, you simply have no standard by which to gauge what good mathematical writing looks like. Unfortunately, there is no simple "trick" or translation into a more familiar format that you can just learn and then math papers will start making sense to you.

My advice is to be honest with the math students who come to your class about the fact that a lot of what you're teaching simply does not apply to their subject and that as a consequence, the course might not be terribly useful for them. (This is why I personally avoided such courses as a Ph.D. student. They are simply not designed for math people, and that's okay.)

Instead of trying to do what you cannot, I would recommend giving your students pointers to some useful literature on mathematical writing. Two absolute classics that every aspiring writer of mathematics needs to read are "How to write mathematics" by Paul Halmos and "Mathematical Writing" by Donald Knuth. There is also this book by Steven Krantz, but I have not read it myself. Encouraging your math students to read this literature will in my opinion be vastly more useful to them than trying to give them your own advice, which might very well turn out to be counterproductive.

Answer 2

Since the IMRaD-structure seems universal

This is simply incorrect, and I'll echo this comment that math isn't the outlier.

Speaking as a mathematician, I find it quite frustrating that I was simply never taught "research methods" or writing for my discipline (beyond the scope of an individual proof). My university's programming for undergraduate researchers had no concept of what math research entails, and I was expected to just "figure it out" by those in my department.

As a faculty member now, I think students would be best served to learn writing from experts in their discipline, and universities should be spending resources on training those faculty rather than expect a single faculty member to be responsible for supporting student research development across all disciplines.

Answer 3

Most mathematics papers simply do not use the rigid IMRaD structure. In fact, most papers do not adhere to any fixed style at all, but are more like books: They choose whatever subdivision of the material into sections and sub-sections makes sense, and that may (and often will) be different from paper to paper.

Personally, I also think that this makes for better papers when the freedom so gained is wielded by good writers. Structures like IMRaD make sense for inexperienced authors who tend to make a mess otherwise.

Answer 4

Yes, pure math papers are a bit different. There is usually an introduction, which may give some background and also name the problem to be solved. There is the usual conclusion, recapitulating the important bits. But in the middle there are proofs of one or more theorems.

Since proofs can be long and complex, important parts may be broken out separately. Lemmas are such things - smaller scale theorems that have their own proofs but are intended for support of a larger result.

Corollaries are other theorems that, may have a lesser impact (or not) but which follow (fairly) easily from the main theorems.

Axioms, on the other hand are more difficult for the non-mathematician to understand. They are statements, given without proof, that form the basis of what follows. The proved theorems are true, then, only in the context in which the axioms are valid.

One of the most important standard axioms is: Zero is a Natural Number. When that is accepted along with a few other simple things, logic can be used to prove quite complex things. But those things might not be true in areas in which the concept of "Zero", or "Natural Number" make no sense.

I suggest that you build yourself a glossary of things like the above for future reference. One way to extend the glossary is to ask your students to explain organizational terms that you don't understand. They will probably benefit themselves by providing explanations.

A typical math paper might have sections like the following, in this (or similar) order, though there might not be actual section separations.

Introduction and Background
Problem to be Solved
Axioms (the basis)
Lemmas (with proofs)
Theorem(s) (with proof)
Corollaries (proofs if needed, some are obvious the reader)
Conclusion and Significance
Future Work

The lemmas and theorems might be one section or a repeated pair. Some definitions are also likely to be given, perhaps in a separate section or perhaps interspersed with other things so that they appear at point of first use.

One issue is that the methodology is usually just symbolic logic, formalized or not. That is understood to be the case so there is no need for such a section. And it is the major theorem(s) that is the result: statements that we can now take to be true given the axioms with the "evidence" being the proofs.

In some fields the axioms are understood from the field itself and so won't be stated. Important papers in which a new field is introduced (rare) would probably state axioms, likewise papers that deviate from common practice. Simple things may not need lemmas. Not everything has corollaries. And many papers have such a structure even if there are no headings to separate them out. And sometimes the headings are more specific, perhaps the statement of a lemma.

Also note that some other theoretical fields (say, theoretical CS or theoretical physics, even history) will also have paper structures tailored to the field, whereas IMRaD is more useful in the social sciences.

You can't expect to find the above headings/sections in any particular paper. They just represent a natural flow that can be seen if you look for it.

And, finally, some papers in applied math will have a structure something like what you are used to, especially as it may be important to make the methodology explicit.

Answer 5

TL;DR

This answer is a frame challenge. Non-mathematicians should not be teaching mathematical writing.

WAD and WID

By way of background: my professional career took a path through secondary education, as a mathematics instructor (I used to teach high school and middle school). When I was doing this (about 15 years ago, now), there was a major push for "writing across the curriculum" (WAD). The idea was that all instructors were expected to instruct their students on writing. While I think that this is a well-intentioned idea, it was led by English faculty. In my experience, the English faculty were telling other faculty what kind of writing they should be teaching ("five paragraph essays", MLA style, etc), and that the nuances of writing in other fields was lost. This was a bad experience for me, and very much soured me on the idea of WAD (and, in particular, I don't really like the idea of non-mathematicians teaching mathematical writing, non-chemists teaching chemical writing, non-historians teaching historical writing, etc.).

Thus my first visceral, gut reaction is to say that if you are not a mathematician (or, at least, if you haven't taken one or two upper-division or graduate-level courses in mathematics in which you have had to read a couple of papers), you should not be trying to teach mathematical writing (ditto for other fields, but I am less defensive there out of a lack of experience). Instead, non-specialized writing classes should focus on those things which really are universal: basic spelling and grammar, tone (professional vs casual), structure (words make sentences, sentences make paragraphs, paragraphs make sections, etc), plagiarism (what is it?!), proper citation (teach students to use citation managers like EndNote, rather than obsessing over a particular citation style), how to read and use a style guide, and so on.

It should be the responsibility of experts in a field to teach appropriate writing in that field (I have heard this called "writing in the discipline" (WID)). In my experience, social scientists are very good at this—a typical anthropology or psychology class will involve a fair amount of writing, and that writing is typically judged within the framework of appropriate writing for the field. Natural / physical scientists are okay—some of the conventions are conveyed via lab reports (I remember being scolded in a college chemistry class for not using the passive voice, for example), but this is probably an area where more could be done.

Those of us in mathematics are terrible at teaching our students to write, and this is (in my opinion) a fairly important problem. I think that part of the difficulty is that typing mathematics is somewhat arcane. One either has to learn LaTeX or something similar, which is seen as a big lift; or one has to tediously muck about with the equation editors in Word, or Google Docs, or whatever other word processor a person is using. I believe that these problems can be overcome, but it is the responsibility of mathematics departments to do this.

Some discussion of mathematical writing

For what it is worth, mathematical has a number of unique quirks (I assume that every field has quirks, but I am most familiar with mathematics), and these quirks are not just limited to the overall structure of the paper. Here are a few which stand out quite a lot to me (this list is by no means exhaustive).

• Structure: While mathematical papers do not stick to any kind of strict structure, there are a few general formats that I have seen in the wild. My observations on these are as follows:

  • Most authors will begin with a section of background. The goal of this section is to introduce notation, set the paper in a broader mathematical context (e.g. discuss past work, and how the current work relates to it), and to highlight the novelty or interest in the paper.
  • Many authors will then proceed directly into their results. This "Results" section of the paper may be divided into several sub-sections, if the authors have several distinct ideas to present, or it may consist of a series of propositions and lemmata which lead up to a main result. In either case, the expectation is that this section will roughly follow a "theorem -> proof -> discussion" format (repeated as many times as it takes to get to the main result(s)).
  • Other authors will state the main result(s) right after the introduction, then use much of the remainder of the paper to state-and-proof the theory which leads to that result, culminating in a proof of the main results.
  • Personally, I like to see examples, and will tend to put a section of examples near the end of a paper. I've seen other authors do something similar, though whether or not this is appropriate will depend a lot on the kinds of results being proved in the paper.
  • Many authors will conclude with a section on future research directions, but this is not necessarily required.

• Typesetting: There is a lot of notation in mathematics (which, again, is what makes it hard to teach mathematical writing). Because of this, the presentation of notation is important—sometimes, it is appropriate to use inline expressions (expressions which fall into the main flow of the text). At other times, displayed expressions are useful (expressions which are set off from the rest of the text and "displayed" in the center of the line).

  Moreover, the choice of fonts, capitalization, etc. is typically important. Variables are italicized, vectors are (often) bolded, certain objects have standard notation (blackboard bold letters for the natural and real numbers; "mathfrak" letters for Lie... things; etc), and so on. From experience, teaching students that A is not the same thing as a is a challenge.

• Grammar and Punctuation: Mathematical expressions should be treated as part of the text. If a sentence ends with an expression, there should be a period after the expression. Mathematicians also handle quotation marks differently from other fields: punctuation goes outside of the quoted material, unless the punctuation is part of the quoted material. There are lots of little pitfalls like this, hence it is helpful to have a style guide handy when writing.

• Voice: Mathematics is usually written in the first-person plural (the "mathematical 'we'"), or in the imperative. This differs from the sciences (which I understand are meant to be written in the passive voice) and the social sciences (where things are often either much more narrative and third-persony, or written from the first-person singular).