Imagistic Processes in Analogical Reasoning: Conserving Transformations and Dual Simulations

Imagistic Processes in Analogical Reasoning: Conserving Transformations and Dual Simulations John J. Clement (clement@srri.umass.edu) Scientific Reasoning Research Institute College of Natural Sciences and Mathematics and School of Education Lederle GRT 434 University of Massachusetts Amherst, MA 01003 USA Abstract One expert subject proposed the analogy that the wheel acts like a heavy lever perpendicular to the slope, with its fulcrum at the point of contact. Intuitively, the lever would be easier to move by pushing at X, suggesting that the same would be true for the wheel. But in the wheel the point of contact is moving, and ordinarily lever fulcrums do not move. In addition some subjects assume that the fulcrum should instead be at the wheel’s center. Therefore the evaluation of the validity of the analogy relation (shown as the dotted line between A and B in Figure 1) was in question. This is distinguished from the subject’s confidence in his understanding of the analogous case B itself, which was quite high in this case. The classical theory of analogical reasoning focuses on mappings between discrete symbols as the mechanism of analogy evaluation and transfer. This paper introduces several other analogy evaluation strategies discovered in expert reasoning protocols: bridging analogies, conserving transformations, dual simulations used to detect perceptual-motor similarity, and overlay simulations. These findings provide evidence for the hypothesis that certain analogical reasoning processes can be imagery based. Earlier work on higher order reasoning has indicated that expert subjects use various methods to generate analogies spontaneously when solving difficult problems (Clement, 1988), and that evaluating the validity of such analogies is essential to using them (Clement, 1989). That is, even if one has generated a confidently understood analogous case, one must evaluate one’s confidence in the validity of the analogy relation to have confidence in transferring results to the target. The classical theory of analogical reasoning (Gentner, 1983; Holyoke and Thagard, 1989; Forbus, et al, 1997) focuses on mappings between discrete symbols as the mechanism of analogy evaluation and transfer. This paper examines several other analogy evaluation strategies observed in expert think aloud protocols. The data base for the study comes from professors and advanced graduate students in scientific fields who were asked to think aloud about a variety of problems. This paper focuses on two mathematicians solving physics problems they found difficult. By focusing on problems with which they were unfamiliar (i.e., a problem on the frontier of their own personal knowledge). it is plausible that the thought processes analyzed will share some characteristics with hypothesis formation and model construction processes used on the frontiers of science. An example of a problem where analogy evaluation is important is the “Sisyphus problem” in Figure 1A: “You are given the task of rolling a heavy wheel up a hill. Does it take more, less, or the same amount of force to roll the wheel when you push at x, rather than at y? Assume that you apply a force parallel to the slope at one of the two points shown, and that there are no problems with positioning or gripping the wheel. Assume that the wheel can be rolled without slipping by pushing it at either point. ” X Y 1C X Y 1B 1A Figure 1: S2’s lever analogy for the wheel Bridging Analogies One method for evaluating this analogy used by this subject was the bridging analogy shown in Figure 1C of a spoked wheel without a rim. By breaking the problem of confirming a “farther” analogy into the problem of confirming two “closer” analogies. such a bridge can make it easier to develop confidence that the wheel does work like the lever in Figure 1B (a correct analysis). Bridging analogies are defined as occurring when the subject finds or generates an intermediate case which shares features with both the target and source analog. Their value has been documented previously in a number of expert problem contexts and in instructional applications (Clement, 1986). While it can be very helpful to subjects, bridging in itself is an incomplete strategy for analogy evaluation, since each half of the bridge must itself be evaluated. Therefore bridging is most useful in conjunction with other evaluation methods and it adds to, rather than reduces, the number of tasks to be performed.