To solve this problem, people often represent it by a simple tree structure, reasoning that the first round will include 65 matches, the second round 32 matches plus one “bye,” and so forth. However, people using this tree structure approach stumble badly and fail when trying to figure out the general case of n entrants. The problem is that a spatial diagram like a tree structure does not lead to a simple solution; simple solutions arise from considering the verbal fact that in order for there to be a single winner, every other entrant must lose. Given that one match determines one and only one loser, there must be a total of n - 1 matches to determine a champion.
In general, problem solving by college students shows that translating a problem into terms that fit it, often spatial terms, aids greatly in solving the problem. The type of representation that leads to the fastest and best solutions depends on what the learner already knows and on the structure of the problem—that is, on what type of solution can, in principle, be used to solve the problem. Novick and Morse (2000) demonstrated that spatially diagrammed representations can help to solve problems, and Novick and Hurley (2001) showed that some problems are most efficiently solved with tree hierarchy schematics, whereas others are best approached via matrices or networks.
Many problems are more readily solved using spatial representations, but in some cases, trying to use spatial representations can interfere with problem solving. Perceptually rich images can enhance reasoning. Barsalou (1999) showed, however, that perceptually rich images do not necessarily help us to think about things in order to recall and reason about their features. For example, in one experiment, participants were asked to think about a situation and then to list all of the characteristics of the main concept. In one condition, people were asked to think about a grassy field and then to generate a list of all of the features of their concept of “grass.” In the other condition, people were asked to think about sod being rolled and transported on a truck, and then to generate the same type of list. The results were revealing—in the former condition, people generated relatively few features, and very few listed “roots”; in the latter condition, people generated many more features and everyone listed roots, many listed root hairs, and so forth. The difference between the images is the degree of perceptual richness. Indeed, rich perceptual images involve some of the same neural activation patterns as those apparent when someone is actually looking at such a situation instead of just remembering it.