I can’t give you a magic recipe for academic writing but I can describe my own practice, and share with you examples of what worked for me and what didn’t.
I’ll illustrate this essay with examples from my own papers. So I’m not slagging off anybody’s work but my own. I take responsibility for all errors pointed out below, including when the paper was written in collaboration.
In this essay, we discuss on the technical side:
They say that success is all in the mind. So we’ll also discuss psychosocial aspects of writing a paper, including:
We conclude with some general advice:
Here’s one of the best introductions I’ve ever written (from this paper). Check out the four paragraphs highlighted (click to open in a new window).
The first duty of these first few paragraphs of the introduction is not to excite, fascinate, inspire, and engage the reader—that is their second duty.
What’s most important is that your text should express an idea that the reader can latch on to. The reader will ask himself the following question, and it’s your job to make sure he can answer it:
I read page 1. Do I know what this paper is about, why the author is doing it, and how it makes the world better?
That’s it. Suppose the reader is asked “So what’s this paper about and why should we care?”. If your text enables him to give a pithy and compelling answer to that question, then your introduction has done its job—and you’re half-way to getting published.
This is so important that I want to say it all over again in different words. To illustrate my point let me give you an example of two ways of describing a window:
I see (and write) too many introductions like sample 1 above. It’s not because authors are stupid: if you spent three years inventing something, you tend to lose a more global perspective on your own work.
Note the last bit of sample 2 above. This addresses the “how does it make the world better?” part of Test 1 above. Ideally, you need your introduction to make it clear, in a non-threatening way, how your paper advances the state of the art. Yes, this can be hard. That’s why talking with people is so very important. You learn how to … well … communicate.
Here’s one of my poorer introductions (since the paper is co-authored, let me repeat that I take responsibility for it). See how I failed to express what the paper is about, what makes it special, and so on and so forth.
It got published, but I just got lucky; I didn’t make it easy for the referees.
The good example was published in 2011, the lemon was published in 2005. I’ve got better! I wouldn’t have written the text to the right, today, because I’d have applied the test above and seen that it fails.
I still write lemons. When I find myself writing and re-writing the introduction and I can’t get it right, here are a few possibilities I consider:
Here’s a little trick I like to use. Work your text as much as you like. Make it so it’s really nice. Now read it through again, from start to finish. Read it sequentially. Go on …
Now … you see that paragraph that you skimmed over? You read the first sentence and then jumped to the end or to the start of the next paragraph? That paragraph is weak. Something’s wrong. The sentences are flabby; the ideas aren’t clear.
That’s where you lost the reader. Edit that paragraph down … and start again.
If you can’t be bothered to read your own text, then it’s not good text.
Here’s some original text; see above for what I did with it.
Oooh. That’s frustrating. Solution: The words do exist: you haven’t found them yet. After I wrote my thesis I knew I was on to something, but I didn’t know what it was. Ten years later, I still struggle to express what it is I’m trying to do, but my words have got better as you can see from the examples already cited, even if I still don’t get it a hundred percent every time (and never will).
The more papers I write, the more time and effort I find myself spending on finding words that will motivate the reader. This is not just about the reader, it’s also about you: with the right words, will come new insights into your maths.
There’s nothing wrong with that except it’s just a bit long-winded. Basically, the bit about my thesis and how I’ve laboured for ten years, doesn’t work. I discovered that by using the skim-test—my eye skipped over the text so I knew it needed more work.
What are the main theorems? The main definitions? The main technical lemmas?
What places in the paper do you need the reader to pay particular attention to? Some nice tricks in the proofs? Something that looks like previous work, but actually improves on it or requires a new proof-method?
Do any of the highlights relate in broad terms to previous work by you or by somebody else (this is not a full comparison with related work—just mention the highlights).
Now is not the time for technical details. Just give the reader a framework to help him get into the paper. To the right is part of a page 2. Have a look.
The reader has landed in this part of the paper and has probably not read the paper linearly. He’s been attracted to this part of the text because the section heading created a visible break in the flow of text, and perhaps because the title of the section interested him.
Just help the reader to get oriented and find his way around. Mention what the main definitions and theorems of this section are. At the moment I often write “The main definition is Definition X, the main results are Proposition Y and Theorem Z, which depend on technical Lemma Q”.
I’ve slipped in a couple of recent examples, just to illustrate what I mean.
I think layout is important.
Aside from the obvious, like choice of colour and improvements in notation, note that in the second paper above I don’t line up the arrows in Figure 2 with the arrows in the examples. They don’t belong together. The examples, however, do belong together, and this is expressed by the alignment. The reader’s eye understands this and is drawn to the three examples as a unit, in a way that does not happen so much in the first paper.
I enjoy the challenge of writing for conferences, but I do prefer journals; there’s more space for manoeuvre and you can communicate with referees via the editor. The only disadvantage is that sometimes, a paper vanishes for 12 months and then comes back summarily rejected.
(That happened to me once: I submitted a survey paper and the journal didn’t publish surveys, but it took the editor a year to tell me. I had enough sense to stay polite, which is good because it turns out that that editor had been bereaved and really wasn’t thinking straight. But the journal should have structures in place to handle that. We’ve got real problems in academia with the way we’ve set things up—some kind of Amazon or eBay like points system for journals, might actually not be such a bad idea. But I digress again.)
In Two-level nominal sets I open with a pithy paragraph (see the highlighted text to the right), but there then follow several pages of discussion—you couldn’t do that in a conference paper.
But look at page 5. I boil the whole discussion down to a single line, in a quoted box so the reader absolutely can’t miss it. Above that line, I pick holes in previous work (including my own) while giving the reader specific and mathematically precise metrics of this paper’s success. Below that line, I sum up exactly what the paper does and the two ideas underlying it.
Though I do say so myself, it’s a beautiful page. Layout, ideas, and text all come together very nicely.
Notice the footnote. I’m breaking one of my own rules: this is a direct critisism of previous work. The only mitigating factor is that I was one of the authors. I felt so strongly about this point that I had to raise it, even though I knew I was running a risk of rejection if the paper hit the wrong referee when he was feeling touchy.
You might also like to take a look at page 6. Because the introduction is so long, I condensed it down for the reader to an executive summary in Subsection 1.4.
I personally don’t have a standard format for my conclusions. I do whatever suits the material and the space available.
Conclusions are retrospective. You’re reminding the reader what you’ve done. So:
My current practice is to write the introduction in the present or future tense, and the conclusions in the past or present tense.
I start with a brief summary of the paper, and then segue into a discussion of significant challenges and difficulties, and/or related work. I conclude with future work. And, I put in precise references so the reader can check up on me.
(Talking about having “shown it sound and complete” without references is OK because there was a section called “soundness” and another called “completeness”. Still, precise references might have been better.)
If there are any likely misunderstandings on technical matters that would have confused the reader in the introduction, now is a good time to head them off (also do this at the point of definition, of course). Often, you only find out about these likely misunderstandings after the paper is refereed. This might cost you a resubmission or two. That’s another reason to prefer journals: you can get into a back-and-forth discussion with the referee which converges quicker to a fixedpoint than a sequence of conference rejections (best, of course, to just get it right first time; but one can only do ones best).
The example to the right is a case in point. Summary, a few pointers into the text, a few nice intuitions, and heading off a potential misapprehension (and yes, that came from a previous version of the paper).
It is often suggested that you should write the introduction, abstract, and title last. This is because you should write the introduction, title, and abstract last. By all means write drafts to get the ideas straight in your head, but don’t be afraid to tear them up repeatedly as the material evolves. The titles of my papers may undergo revision up until the last minute.
What happens with the abstract?
Sometimes, you might want your abstract to be short and scary. An example is below right. I sent this in to a conference and I wanted to make sure that it would go to referees who were comfortable with the material in the paper. That material was quite technical, so I needed to make sure it would be picked by a referee who enjoyed a mathematical challenge. Being a conference, space was also at a premium.
Sometimes, you might want your abstract to be long and friendly. The longest abstract I’ve ever written is below left. It was for a survey journal paper; it had already been accepted so I didn’t have to worry about referees. I had plenty of space. The editor rightly asked me to make the abstract more descriptive so I thought: “what the hell, I’ll write an essay”. So I did, and it worked fine.
The bottom line? The abstract is like a flower. Design it to attract the pollinators that you think you need.
Titles should be descriptive. Doh!
At the moment, my titles are long (about two lines). I tend to split them up with a colon. Why? Because in this day and age of Google, papers are not taken off the shelves of libraries; they are indexed by computers. Lists of papers are not compiled and perused; they are auto-generated and skimmed.
The questions I ask myself are:
So the third title above would presumably show up in searches for “nominal meta-variables”, or “nominal terms”, or “nominal sets”, or “sets meta-variables”—to all of which this paper might be relevant. You get the idea.
I try to put as much information in the title as I possibly can. That’s just my current policy based on observing my own search behaviour at the moment. It might change.
After all, academic search tools are improving all the time, albeit from a low baseline. It’s going to get better, right?
There’s a very similar result in the literature (it’s cited in the extract). I know that the expert reader will skim over 4.11 thinking “Oh yes, that’s just like Corollary 5.2 of ”.
It isn’t; the proof is very different and it’s actually a little bit surprising that it works at all. But if I don’t say that, the reader won’t figure it out for himself, will he?
In my early days, I imagined the reader poring over every sentence of my paper, as I had. I’d have left Proposition 4.11 unadorned, to leave the reader the pleasure of figuring out for himself the sublime difference from Corollary 5.2. That flattering illusion didn’t last long: all my papers just got rejected.
This is very easy to get wrong. You spend half a year with your nose jammed up against a theorem, and you lose track of …
If there’s something in your paper that looks simple, but isn’t—then say so. Explain it in the introduction or conclusions and also discuss it at the appropriate point in the paper.
If you don’t, the reader won’t know it’s there.
At some point, you’re going to have to mention previous work. Maybe you wrote that previous work, maybe somebody else did. Here’s what not to do:
Professors X and Y’s work on Widgets was clunky [X&Y2001]. They’ve repeatedly tried to improve [X&Y2003,2004,2007]—and just as repeatedly, they failed. The syntax is obscure, the semantics uninformative, and basically the whole thing is a mess.
My Widgets on the other hand are clean, elegant, and mathematically well-motivated.
There’s lots wrong with this.
Let’s try this again:
Professors X and Y laid the groundwork for this field in [X&Y2001] and developed it further in [X&Y2003,2004,2007].
This paper improves on that state of the art by giving Widgets a new mathematical semantics (Definition Z below; essentially, a translation to Wombats), which can be proved sound and complete (Theorem Q). Note that following X and Y’s approach Theorem Q fails because there is not necessarily a 1-1 correspondence between Widgets and Wombats. Theorem Q works for us, because Definition Z imposes an apparently innocuous but actually very important extra condition (compare Definition Z with Definition P of [X&2001]). This restricts our Widgets just enough to make Theorem Q work. Having said that, the two approaches remain very much in the same spirit."
There. Now you’re happy, the reader’s happy, and Professors X and Y … well, they may or may not be happy, but at least it’s all about the maths and you’ve given the reader honest evidence for your position.
Look at it from the point of view of Professors X and Y (who, perhaps, had good reason for setting things up as they did and see the subsequent developments very differently). God willing you’ll be in their position one day, if you’re not already. It’s no fun seeing some smug little bastard who doesn’t understand what he’s talking about trashing your work in public. This is not just for the humiliation of it, but also for the crushing sense of despair that for all that hard work—people still don’t understand what you wrote. So be nice and try to talk to everybody concerned.
But, I digress.
So you’ll get referees saying “this is clearly wrong, because Y has properties that X does not” (but you never intended to model all of Y, only the bits you’re interested in). You’ll get referees saying “X just isn’t equal to Y”, or “How dare this stupid author claim that we don’t need Y?”.
So I tend to conclude by saying something like “I’ve just built a model of Y. There are bits missing, and I’m not claiming that X=Y. Nevertheless, by studying X we can better understand Y, because of Theorem Q”.
I don’t mean to be cynical here. We can’t see the future, so try to play to all sides of the audience wherever mathematically possible.
You never know; your next paper might contradict this one. Then you’ll look pretty silly if you said (as I see people writing, without substantiation) “This author is absolutely definitely sure that what he is doing is absolutely right”. Explain your reasons and allow the reader to draw his own conclusions.
As a general rule, my papers are always better for being written with coauthors. Even if the coauthor doesn’t do much, just having somebody to talk to makes a huge difference.
My advice is: find good coauthors and treasure them, and if you can’t, then find bad ones and be grateful for what you’ve got! Don’t worry if you end up doing all the work: what matters is producing the best document you can. Two people are better than one, at this game.
If a coauthor is slow, as is often the case, then chase him up politely. I call this a “standard pester”.
Dear Professor X,
This is a standard pester, just to make sure I stay near the top of your stack. You were going to spend five minutes looking at two weeks of intense effort on my part. Please remember to do this as soon as possible.
One week later:
Dear Professor X,
This is another standard pester, just to make sure I stay near the top of your stack. You were going to spend five minutes looking at two weeks of intense effort on my part that I sent you last week. Do you have any idea when you’ll be able to do this?
Dear Professor X,
This is a standard pester. You were going to spend five minutes looking at two weeks of intense effort on my part that I sent you a month ago. Do you have any idea when you’ll be able to do this? Have you finally keeled over and they can’t resuscitate you this time? [just kidding -mjg]
Remember that the submission deadline is rapidly drawing closer so I really need you to get on this now, or we’ll miss it.
Rules of thumb:
You can phone people up, too. If a couple of e-mails don’t do the trick, pick up the phone.
Sometimes, you just have to deal with difficult characters. I had one coauthor who I knew from experience was going to procrastinate. I set a deadline and when he missed it. I took his name off the paper and told him I’d done it. Tee hee, that made him sit up. He needed publications so that really got his attention.
I had one coauthor who disappeared for a year. No reply to e-mail. No answer to phone. I couldn’t finish the paper without him, so I was just stuck. I bumped into him in a conference and asked “Look, I’m really sorry for whatever I did to offend you, but what did I do or say to drive you away?”. He replied “Oh no, you didn’t offend me. I just got busy.” I finished the paper with him, got it published, and didn’t work with him again. Too much like hard work. But we did finish the paper!
I don’t mind the nonsense above. But I’ll tell you what does annoy me. A coauthor who vanishes, doesn’t respond to e-mail, lets me painstakingly polish a paper to publishable standard all on my own, and then suddenly reappears perhaps a week before the submission deadline and starts tearing into the paper.
I hate that. What a lack of consideration! Polishing a text is hard work: forcing your coauthor to do it twice (once on his own, once with you) means you’re either self-centred or trying to exert control and show who’s boss. Yet even so, when this has happened to me the paper ended up better for having had the second pair of eyes on it. I considered carefully whether to work with the person concerned again (to their subsequent detriment, I like to think), but only after we had finished the paper.
I’ve twice got involved in coauthors so utterly hopeless that they just were a waste of time from start to finish. One was an idiot, the other couldn’t be arsed. If this happens to you, and it will eventually, then I suggest that you leave their name on that paper (if it ever gets published). If you don’t, then you’ll spend the rest of your life looking over your shoulder, terrified for the comeback: “Did this paper get rejected because X was on the programme committee and he hates me?”. Grin, be polite, chalk it up to experience, perhaps hold back a bit on the results, and later on write another and a better paper without them.
If you are reading this guide, then probably you do.
Whoever is the junior partner in a collaboration will typically do most of the legwork (writing out / checking details of theorems, trying out design options, etc).
It’s not a hierarchical thing: it’s a matter of who is a) most desperate, b) least productive, and c) has the fewest distractions. Junior people typically satisfy a, b, and c, whereas senior people typically don’t. Junior people also typically need to work ten times as hard to get the same results (if they’re lucky) because they lack overview, technique, and experience, so a senior partner can pay for himself many times by providing ideas, guidance, advice, and knowledge of how to present the material to a broader audience.
Everybody brings expertise to the partnership. Everybody is equal. But, if you’re junior and you really want that paper, then be prepared to sweat for it. Conversely if you are junior and your senior co-author is putting in most of the hours, then ask yourself what is in it for him and … will he work with you next time? Just a thought.
Productivity grows with practice. Personally, I relish the opportunity to do most of the work. It gives me a deeper understanding of the underlying maths, which I can then bring to the next paper. So if your coauthor leaves you in the lurch then remember … there are no problems, only opportunities!
My advice: alphabetic order. No egos = no complications.
You think that your contribution was most valuable? What if your coauthor disagrees (bear in mind that he might not say this to you, and you might suffer for it only later)? Do you argue over this? No. Keep it simple. If it wasn’t worth changing your name to “Aarnon, Aliyah”, then that’s your fault and it’s not worth losing a collaboration over it now.
Conversely, if your coauthor is determined to have a particular order, let him have it. Avoid direct conflict. What matters most is who develops the material later; the issue here is not position but velocity. If such a disagreement does arise, then your coauthor is insecure about his contribution—and that suggests that you can overtake him later. Anyway if you’re arguing about priority then there are probably deeper problems with your relationship. Weigh up whether you want or need to write another paper with him. Give him what he wants today, and get out.
We project ourselves onto other people; I tend to assume that you’re just like me—and you tend to assume that I am just like you. It’s only human.
When people forget this they can get very annoyed. “Why don’t they understand? The only possibilities are idiocy or deliberate malice!” Wars have been fought, and many books written. Let me try to discuss briefly how this phenomenon affects academic communication.
Your reader is not equal to you.
As an author of an academic paper you are somebody who loves the material, or at least has laboured intensely to create it. You will be tempted to tell your reader how hard it was to do this; all the false leads; also all the limitations. You will tend to assume that:
At least, you will tend to assume this if you are just like me.
But no. Even for close colleagues, this is usually not the case. What was obvious for you may be a revalation to the reader if you are lucky, or perverse if you are not. Significant distinctions to you may (rightly or wrongly) seem minor details to the reader—conversely, the reader may be fascinated by what seems a minor curiosity to you. As for awareness of research context, your other papers, and the underlying motivations—only if you tell him, and if your reader has any prior awareness of these things it may be very different from yours.
Most likely, your reader just wants to get the idea: “what is in this maths for me?”. Or, your reader will be after a specific piece of information: “does he encounter problem X which I’m familiar with, and if so how does he solve it?”; “what’s his syntax?”; “does he use Widgets or Wombats?”; “can he model Y, Z, and U?”. Once these questions are answered, if you’re lucky, some very few readers may go beyond that.
Your reader may be a future version of you, by the way. It’s amazing how quickly one can forget the minutiae of the material, so by writing a good, clean document you are also helping your future self.
Let me give you a real-life example. A PhD student asked to give a 30 minute lunchtime seminar spent most of it describing the technical limitations of his work. At the 20 minute mark I interrupted him: “But the overview of your work is this, correct?” (I had extracted it from the 20 minutes of technical data; it took me about 30 seconds to sum up). “Yes”, he replied, “that’s right”. “That’s an incredible achievement, you know”, I said encouragingly. He was stunned and silent.
My point is not about PhD students giving bad talks, which is old news. My point is that this PhD student had spent 20 minutes talking to himself. He was a modest, sweet man, very involved in his work, aware of his own shortcomings, and deeply concerned about getting stuff just right—so he projected this onto his audience and spoke accordingly. I intruded with a completely different point of view, which he had not considered, and he was surprised.
Your readers are a strange, amazing, alien species. Get out there and get to know them.
One paragraph = one idea. Make sure that idea is clear from the first couple of sentences of the paragraph. Do not introduce the central idea of a paragraph sandwiched in the middle of the text.
It’s very simple. For every paragraph in your paper ask yourself: what is the point of this paragraph and is that point clear? If not, then why is that text sitting there?
If you find yourself writing a really long paragraph that does not naturally split in two, then odds are you’re really writing an itemised list.
I could go online and find you a dozen great examples of how not to do it, but I promised to slag off only my own work so I’ll try to rewrite the text above as I wouldn’t normally do it. This goes against every fibre of my being (cracks knuckles)…
Each sentence in the excerpt above is reasonable, but they are assembled into a bad paragraph. I take three sentences to get to the point, and than ramble on. You can easily lose your reader with this, before you even get to the hard maths.
So you’ve written some text. Now compact it. The fewer words you use, the clearer your point will be. So:
Technical text is not necessarily like normal prose. You need to express yourself plainly and concisely—like in journalism.
Here’s that paragraph again, after my first edit but before I came back to it the following morning:
The goals when editing a technical text are not necessarily those of editing normal prose. Technical writing should express itself as plainly and as concisely as possible. In that sense I think it has a lot in common with journalism.
English is a superb technical language. I feel privileged to be a native speaker.
English is a superb technical and journalistic language. I feel privileged to be a native speaker and have immediate access to this wonderful tool. I don’t think non-native English speakers really appreciate just how good it is for that kind of thing.
In many continental languages, good style is complex sentences (lots of “the former” and “the latter” and paragraph-long sentences). I suspect it’s descended from Latin. I tell foreigners this is not good technical or journalistic English and am universally ignored. I try to be understanding: their whole culture is at odds with me.
In continental languages—especially Italian, funnily enough—a sign of erudition is construction of complex sentences (lots of “the former” and “the latter” and paragraph-long sentences). It’s kind of an imitation of German, though I suspect it’s descended from Latin. Anyway, it doesn’t suit technical writing. I tell this to foreigners and am universally ignored, which is why I feel entirely free to write this here. There’s a certain freedom in knowing that those who most need my advice, will ignore it. I try to be understanding: their entire culture is at odds with me.
As referee or editor I regularly get rambling, poorly-edited papers. I’m likely to bin them, and it wouldn’t help even if they were published—they wouldn’t be read (relative to the potential audience for lively, direct, and straightforward text).
Not much change from the original:
Let me just say this: as a referee or editor I regularly get rambling and poorly-edited papers. I’m likely to bin them. And it wouldn’t matter if they were published—they wouldn’t be read (relative to the potential audience for lively, direct, and straightforward text).
My idol here is Mark Twain. I haven’t read his fiction but I adore his essays. And of course there’s that great essayist, George Orwell. You get the idea.
A few random things that dropped out of my fortune cookies:
Let’s just remember what this is all about: publishing a clear text that people can read, will want to read, and will be better for having read.
Your papers won’t go stale, they won’t corrode, and thanks to the internet they’ll stay online until the end of civilisation—and, hopefully, until long after you and I are gone.
Get it right, and through your papers you might have the opportunity and privilege of being a colleague, a guide, and perhaps even a friend and inspiration to people you’ve never met.
That’s worth a bit of work.