People have been bugging me to blog about this, and “Give the people what they want” is my motto*, so here goes.
Writing in a forthcoming PNAS article, Fawcett and Higginson report the results of a citation analysis showing that, the fewer equations included in the main text of a ecology or evolution paper, the less likely it is to be cited by the most rigorous papers, although the more likely it is to be cited in the least-rigorous papers. Tellingly, equations in appendices have no effect on citation rates in the least-rigorous papers, indicating that many readers will happily ignore equations when given the least excuse to do so. Fawcett and Higginson take this to indicate a worrying preference for imprecise verbal arguments and handwaving on the part of too many ecologists and evolutionary biologists. They suggest that equations always be presented in the main text so as to force readers to pay attention to the core content of the paper.
Well, that’s what they could’ve said, but they didn’t.
Actually, what they said was that, the more equations included in the main text of an ecology or evolution paper, the less likely it is to be cited in non-theoretical papers, but the more likely it is to be cited in theoretical papers. The former effect outweighs the latter, presumably because non-theoretical papers outnumber theoretical ones, so the more equations in the main text, the less a paper is cited. Equations in appendices don’t affect citation rates. Fawcett and Higginson suggest that theoretical papers need to place more emphasis on words explaining the math, and less emphasis on showing the math itself, in order to increase their impact with non-theoreticians. They also suggest more mathematical training in graduate school, but note that this is a long-term proposal that’s unlikely to happen, for all sorts of reasons.
I actually think Fawcett and Higginson’s paper is fine as far as it goes, I just don’t think it gets at the ultimate issue. Their basic empirical results are hardly surprising, of course. Theoretical work in any area of biology is at least partly, and often almost entirely, its own subfield. That’s why theory journals exist. Theoretical papers have equations in the main text because they’re aimed at theoreticians. Such papers can’t bury all the equations in appendices any more than empirical papers can bury the methods, statistical analyses, and results in appendices. Theoreticians who write equation-heavy papers are not poor communicators (at least not necessarily), they’re good communicators for the audience they’re trying to communicate with. (Plus, even theory papers typically do bury a lot of the equations in appendices—it’s just that, even after doing that, they’re still left with numerous equations in the main text!)
And if you say, “So how come theoreticians write for that narrow audience, instead of making more effort to communicate with the non-theoreticians who want to test their theories?”, my response is, “How come non-theoreticians don’t learn more math?” Understanding is a two-way street. And understanding, not citation, or “impact”, or even “communication”, is the ultimate issue here. Yes, absolutely, theoreticians vary in how good they are at helping readers understand their work.** But exactly the same thing is true of non-theoreticians! Any scientific paper, theoretical or not, necessarily assumes a lot of background knowledge on the part of the reader. Scientific papers are written for people with Ph.D.’s, not undergraduates. Many Ph.D. ecologists’ and evolutionary biologists’ last math class was a now-forgotten first-year undergraduate calculus course. Why should theoreticians be under a special obligation to write their papers at that level? Why are non-theoreticians not under a similar obligation to write their papers for an audience whose last courses in natural history, field methods, zoology, botany, statistics, etc. were early in their undergraduate careers?
Far more important than the issue of who cites what, as far as I’m concerned, is who understands what. The sad history of early attempts to “test” MacArthur’s competition models is a case in point. MacArthur’s papers were fairly light on equations (as far as I recall…), and in any case were hugely influential with non-theoreticians, right down to the present day. Infamously, they were also widely misunderstood by non-theoreticians, and I doubt that was because MacArthur was a particularly bad explainer. Further, getting non-mathematical readers to understand math is not as simple as burying the math in appendices so that you can fit more words into the main text. As Caroline Tucker at The EEB and Flow notes, all that does is give readers an excuse to ignore the math entirely. Non-theoreticians may well cite such papers more often—but do they understand such theoretical papers any better than papers with more equations in the main text? Or do they merely think they understand such papers better?
I admit that I’ve become more pessimistic over time on the possibility of really bridging the gap in understanding between theoreticians and non-theoreticians. That’s in part because of my own increasingly-lengthy experience trying to explain what’s wrong with the intermediate disturbance hypothesis to non-theoreticians reading my blogs. It’s been really, really difficult, and not because my silly zombie jokes and other attention-grabbing rhetoric have gotten in the way. Nor has it been because I’ve used equations, because I haven’t used any. And I’ve still ended up with massive comment threads packed with questions from smart but non-theoretical readers who’ve misunderstood things. Now, part of the difficulty is because I’m trying to unteach an established idea, rather than teach something about which readers have no previous knowledge. But I don’t think that’s all of it, or perhaps even most of it. I think it comes down to non-theoretical readers just having a hard time “getting” how even very simple mathematical models work.
And as my own extensive experience teaching introductory mathematical modeling to ecology undergrads has taught me, the only way most non-mathematicians can really learn to “get” mathematical models is by actually doing math. Figuring out for themselves the equations that correspond to some specified biology. Plotting the shapes of the functions that comprise the model. Actually doing the algebra to solve for the equilibria. Solving for the isoclines and plotting them. Coding up the model and simulating it for different parameter values and initial conditions. Etc. Non-theoreticians routinely scoff at the notion that anyone without extensive field experience can ever really “get” how wild nature works. Well, if you don’t have extensive “field experience” with math, why would you ever think that you could “get” mathematical models just by reading words?
I emphasize that I don’t think everyone should learn more math and less about everything else. I’m not privileging mathematical over non-mathematical knowledge here, and I’m not criticizing my many non-theoretical colleagues for not knowing math. I’m just saying that, if few people cite mathy papers, that’s mostly not because of how those papers are written. It’s mostly because few people understand math. Whether and how that could be changed, and what the costs as well as benefits would be, is a whole ‘nother conversation.
UPDATE: Fawcett and Higginson conclude their paper with the famous line from Steven Hawking about how he only included one equation in A Brief History of Time because he was told that every equation would reduce sales by 50%. Which seems like a poor example to reinforce the message that words can substitute for math. Yes, the book became a bestseller–which was infamous for not actually being read or understood by the vast majority of people who bought it.
UPDATE #2: As noted in the comments, I could’ve just linked to Florian Hartig’s excellent discussion, much of which I unwittingly recapitulated. And Tim Fawcett himself pops up in the comments on Florian’s post.
*”…so long as what they want is zombie jokes” is the second half of my motto.
**Steve Ellner, perhaps the best theory-explainer in all of ecology and evolution, has good advice on how to write theoretical papers that non-theoreticians will cite. Notably, his advice is not simply “bury all the equations in appendices”.