Posted by: Jeremy Fox | May 22, 2012

On confusing specific examples and general principles

Here’s something I struggle with in my teaching and writing (blogging as well as papers). How do you keep your audience from mistaking specific examples for general principles, and vice-versa?

For instance (to pick a specific example!), “density dependence” is a general principle. It just means that per-capita growth rate varies with density (in any fashion, for any reason). The logistic equation is the first specific example of density dependence taught to undergrads, because it’s the simplest example. Which causes many students to answer exam questions as if they think logistic growth is one and the same thing as density dependence.

The obvious way to deal with this is to use multiple examples of the general principle. Which I try to do, of course. But just using, say, two examples rather than one doesn’t magically allow your audience to extract the general principle and distinguish it from example-specific details. If your two chosen examples share other features besides being two examples of the same general principle, your audience may well latch onto those other features as being somehow crucial. I have had this happen in my teaching, using multiple examples of density dependent models with carrying capacity parameters, thereby giving some students the mistaken impression (despite my best efforts to prevent this) that “density dependence equals carrying capacity”. But if your two chosen examples are as different as you can make them, your audience may have trouble seeing that they have anything in common at all. And if you try to get around these problems by using more than two examples, well, how many examples can you possibly provide, given that you only have so much time, or so many words, to work with?

As I said, this problem doesn’t just crop up when teaching undergrads; it’s not a problem specific to that audience. For instance, consider the “storage effect“, a very general type of coexistence mechanism. A colleague of mine has been arguing to me, correctly I think, that Peter Chesson and others interested in the “storage effect” haven’t always explained it in the best way. Part of the problem may be their reliance on an overly-limited range of examples. Briefly, the storage effect is a coexistence mechanism that can operate when environmental conditions and species’ densities fluctuate over time, in such a way that the strength of competition a given species experiences covaries in an appropriate way with environmental conditions. One way for the appropriate covariances to arise is if the competing species have stage-structured life histories with a long-lived, difficult-to-kill life history stage.* The examples that always get used include annual plants with seed banks, zooplankton with resting eggs, or coral reef fish and tropical trees with long-lived adults. It’s completely understandable why such examples are emphasized. The example of coral reef fish is what originally inspired Chesson and Warner (1981) to come up with the “lottery model” of coexistence, with Peter Chesson later generalizing the “lottery” mechanism to the storage effect. The storage effect is easy to demonstrate and illustrate using the sorts of mathematical models appropriate to species with such life histories.  Such life histories are common in nature, and all the best empirical examples of the storage effect involve species with such life histories. Such life histories even give the storage effect its name. When species with such life histories coexist via the storage effect, one can view each species as increasing when environmental conditions favor it, and then “storing” the gains in a long-lived, hard-to-kill life history stage, preventing it from being driven extinct even if conditions mostly disfavor it.

All of which has given many people the impression that those sorts of stage-structured life histories are essential to the storage effect. Which they’re not. The only life history feature you need is overlapping generations (Ellner and Hairston Jr. 1994). So even organisms that just reproduce and die continuously, with no stage structure at all, can exhibit a storage effect, as illustrated by the “flip-flop competition” model of Klausmeier (2010).**

Sometimes you can avoid the problem of confusing specific examples and general principles by avoiding general principles entirely. That is, if there’s a specific example or application of a general principle which is familiar to your audience, you can sometimes explain a new example or application by reference to the familiar example rather than to the general principle. I do this when I’m explaining the application of the Price equation to ecology (e.g., Fox 2010 Oikos, Fox and Kerr 2012 Oikos). The Price equation is an extremely general and abstract mathematical formalism with very broad applicability. But my audience is very familiar with one specific application: evolution by natural selection. That is, my audience is familiar with evolution by natural selection, even though they (mostly) don’t have any previous familiarity with the Price equation, or with the notion that evolution by natural selection is merely one specific example of the more general principle of “selection” (Price 1995). So rather than trying to explain the general, abstract principles and how they apply to whatever specific bit of ecology I’m talking about, I start by making an analogy between evolution by natural selection and the specific bit of ecology I’m talking about. I pitch what I’m doing not as applying an abstract, general principle in a specific ecological context, but as taking an idea that (as far as most of my audience knows) is specific to evolution, and transferring that familiar “evolutionary” insight to ecology. But if your goal is to convey the general principle, you obviously can’t get away with just talking about examples, because it wouldn’t be clear what they’re examples of?

Any ideas on how to deal with this problem? Does the order in which you introduce general principles and specific examples matter? (I feel like it doesn’t, or shouldn’t, but I’m not sure) There must be papers on this in educational psychology, but I don’t know that literature at all…

*You also need other ingredients. For instance, species can’t all respond in exactly the same way to environmental fluctuations.

**Klausmeier (2010) doesn’t actually say that the storage effect is what generates coexistence in this model, but Chris Klausmeier and I have figured out that that’s what’s going on. See this old post for some discussion. I didn’t walk through the details then and I’m not going to now, because I doubt most readers would be interested. But trust us, it’s a storage effect.

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Responses

  1. I think this is really just a specific instance of the more general phenomenon of human conceptualization and thought process issues. :)

    Perhaps an occasional interjection of “you numbskulls really aren’t getting the bigger picture are you”? would light a fire?

    No but seriously, I think sometimes you just have to come right out and state that you are illustrating a larger concept using some specific examples, some of which illustrate that concept better than others. In your dichotomy of choices presented in the third paragraph, I think using as widely disparate examples as possible is the more effective route, even if those examples may differ wildly in other, non-relevant-to-the-main-point ways, because I think it’s generally harder to see commonalities in such things for a lot of people. But more generally (:)), I think the issue of seeing both similarities and differences in objects, processes, concepts, theories etc., is one of the most important things that can be instilled during the educational process. Everything else, to use (Mayr’s? Dobzhansky’s?) phrase is just “stamp collecting” (although I really think that “just” should be removed from that quip, because that collecting is itself an integral part of the process; you can’t theorize if you don’t have some–usually a *lot* of basic facts in place).

    • Well, wrong as usual on the quote. It was, according to Gaither’s Dictionary of Scientific Quotations, George Gaylord Simpson:
      “Biology starts with biochemistry and goes on to neurophysiology
      and genetics. All else is stamp-collecting. (“This View of Life”, 1964)

      Which is kind of interesting because Simpson was a paleontologist, about whom physicist Luis Alvarez is reported to have said “Paleontologists…they’re really not very good scientists.
      They’re more like stamp collectors.”

      Anyway, just remember, biology starts with biochemistry. :)

    • “I think sometimes you just have to come right out and state that you are illustrating a larger concept using some specific examples”

      Oh, I do. Doesn’t seem to prevent the sort of mistakes I talk about in the post, at least not in my undergraduate teaching.

      Yes, the larger issues here are recognizing similarities and differences, and distinguishing relevant similarities and differences from irrelevant ones.

      • Hmmm.

        The numbskull quote it is then!

      • Um, no. I know you’re kidding, but no.

      • If you’re telling them straight up what your objective with the examples is, and giving good ones, and they still can’t get it, then well, I mean you can’t change peoples’ fundamental thought processes and ability to make connections, that’s beyond your control.
        And yes I was kidding.

      • In order to teach well while saying sane, I find I have to hold two almost-contradictory beliefs at once: that every student who puts in an honest effort *can* get it, if only I find the right way to help them get it, and that students who put in an honest effort but yet don’t get it aren’t a sign of utter failure on my part.

  2. Sometimes the confusion isn’t of your making. My students also confuse density-dependence with a carrying capacity but I find that they walk into my class with that misconception. They learn that before I can get to them. So in addition to the difficulties of abstracting general principals from specific examples, add the difficulty of unlearning a misconception.

    I’m pretty sure there’s theory of learning/pedagogy research on both of these, but I can’t point you to any resources.

    • Good point. Same thing comes up in other contexts. When I coach baseball, the hardest thing by far is unteaching kids’ bad habits and techniques that have gone uncorrected for years.

  3. “If the reader is to grasp what the writer means,
    the writer must understand what the reader needs”.
    This paper is a bit basic (http://www-stat.wharton.upenn.edu/~buja/sci.html), but it may be a good point to start. When there are so many students that confuse such a type of concepts (or others in Ecology), there may be also some problem with the way with which the concept is taught or shown to the students (communication problem, use of language, use of words, examples…).

    In my case, I have much more clear now what the storage effect is about,
    so thanks!!


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