A recent study on the utility of concrete examples in mathematics education has been gaining significant attention. In a nutshell, the researchers arrived at the seemingly counterintuitive result that students who were taught a particular concept using specific examples (e.g., filling water in a jug) did worse than students who were taught in terms of abstract symbols (such as “meaningless” geometric shapes being used as labels) – when measured on a new problem that tests whether they understood the underlying idea (in this case, that of a group).
Personally, I am not very surprised. I have always felt that the mere use of examples do not achieve much unless the examples are really well chosen. It is nothing short of a fine art to pick a concrete story that actually delivers a nontrivial appreciation for a deep concept. Also, I think that I would have used a different measure of “successful learning” – I’ll come back to this point shortly.
Of course, some concepts just take time and effort to get across. For instance, I have been taught the concept of eigenvalue (and its use) in numerous courses – ranging from my second year undergraduate math course (at which time, I doubt that I understood the concept at all) to a nonlinear functional analysis course I audited during the last year of my PhD studies. And I still arrive at a new(er) appreciation for the concept as I continue to learn, e.g., when I try to read bits and pieces from works such as this.
I guess what I am trying to get at is this – if you teach a student the abstract version of a concept then she learns the rules and consciously thinks about the rules each time a new problem requires their use (as opposed to a slightly more naive version of pattern matching that happens when the same concept is explained using a few limited examples). Eventually, after numerous attempts, the student gains sufficient dexterity with the concept that she begins to ‘understand’. More often, as has been famously noted by Von Neumann, one may just come to terms with a tricky concept before (possibly, eventually and really) understanding. Sometimes, the specific examples are used to short circuit the hard work that it takes to arrive at this state – which is not a great idea. There are surely instances where the use of specific examples yield quicker learning in narrow domains but there must be a careful balance with the general idea if the student is to have any hope of eventually reaching a deeper understanding. In this sense, I would not measure success by speed of learning – but instead I’d like to see how far the student manages to “stretch” based on what she has been taught.
One excellent example of a successful attempt to convey a subject area primarily using concrete examples is a book by Jeff Weeks called The Shape of Space (perhaps this is a nice counterexample to the study mentioned above). It was an absolute pleasure to read Weeks’ explanations of concepts like orientability, curvature and the structure of three-manifolds. He does a careful job of picking examples that are actually proxies for the more abstract concepts – so that a student who first reads his book and then goes on to a more serious course will surely just say, “Oh yeah, that’s just like…”! One example that particularly stands out is his explanation of the difference between one-sidedness and orientability. I can easily imagine someone in graduate school using the same basic explanation, perhaps aided by more fancy symbols. In this sense, I think that Weeks has done a much better job than more glamorous popular science books that just replace the concepts with wishy-washy analogies.
I wonder whether the same study, repeated with Weeks’ examples and corresponding problems, would yield the same results…