Learning subgoals and methods for solving probability problems

Abstract

We hypothesize that typical example problems used in quantitative domains such as algebra and probability can be represented in terms of subgoals and methods that these problems teach learners. The "quality" ofthese subgoals and methods can vary, depending on the features ofthe examples. In addition, the likelihood of these subgoals' being recognized in novel problems and the likelihood of learners' being able to modify an old method for a new problem may be functions of the training examples learners study. In Experiment 1, subjects who studied examples predicted to teach certain subgoals were often able to recognize those subgoals in nonisomorphie transfer problems. Subjects who studied examples demonstrating two methods rather than one exhibited no advantages in transfer. Experiment 2 demonstrated that ifthe conditions for applying a method are highlighted in examples, learners are more likely to appropriately adapt that method in a novel problem, perhaps because they recognize that the conditions do not fully match those required for any of the old methods. Overall, the results indicate that the subgoal/method representational scheme may be useful in predicting transfer performance.A consistent finding in the problem-solving literature is that learners are often unable to make appropriate use of prior information to solve new problems if the new problems differ from training examples in more than minor ways (e.g., Anderson, Farrell, & Sauers, 1984;Gick & Holyoak, 1980Reed, Dempster, & Ettinger, 1985; Spencer & Weisberg, 1986). Learners often seem to acquire primarily superficial knowledge from training examples, consisting of little more than aseries of memorized steps. For instance, some of the examples used in the present experiments involve finding the average number of times an event OCCUfS per trial (such as the average number of errors per game made by the Detroit Tigers' infield) given the frequencies of the various subevents (e.g., the number of games in which the infield made°errors, the number of games in which they made exactly I error, and so on). Learners are good at memorizing, from examples, the steps of multiplying each subevent (0, I, etc.) by its observed frequency, summing the results, and dividing by the number of trials in order to achieve the subgoal of finding the average frequency. However, they rarely notice that the steps of multiplying each subevent by its frequency and then summing them could be viewed as a way of finding the total frequency of the event. As a result, these learners are usually unable to find the average frequency of an event when a problem provides the total frequency of that event directly, rather than requiring that it be derived from the frequencies of the various subevents.In the present paper, we will argue that the knowledge people gain from examples in domains that emphasize solving problems, such as probability , can be represented by subgoals and methods. Furthermore, manipulations of examples can affect which subgoals and methods a learner acquires, th...

Object encoding, goal similarity, and analogical transfer

“…The second possible explanation is that the modified final goal furnishes an implicit cue for constructing the subgoal. In other words, the fact that the rectangles appear together in the final goal cues moving them together to form the subgoal without any explicit thought on the matter necessarily taking place (Catrambone, 1994(Catrambone, , 1995Catrambone & Holyoak, 1990). Our results support either one of these interpretations.…”

“…What we would like to underline is that what was studied in this research was more the means (goal structure) or procedure that allowed the goal to be reached than the goal state itself. The same is true for the studies of Catrambone (1994Catrambone ( , 1995, Catrambone and Holyoak (1990), and Ross (1987Ross ( , 1989, which focused more on procedure than on the final goal. In these studies, the negative role ofgoal dissimilarity was studied through a dissimilarity of procedures.…”

Improving examples to improve transfer to novel problems

“…A number of studies have shown that manipulations of examples have a powerful and systematic effect on performance on novel problems (e.g., Catrambone & Holyoak, 1990;LeFevre & Dixon, 1986;Pirolli & Anderson, 1985;Reder, Charney, & Morgan, 1986;Ross, 1984). Given the central role that examples play in prob-IMPROVING EXAMPLES 607 lem solving, and given the assumption that people learn subgoals from the examples, it is important to investigate the conditions that influence subgoallearning.…”

“…The term subgoal is used here to represent the task structure to be learned for solving problems in a particular domain (e.g., Catrambone & Holyoak, 1990;Dixon, 1987;Eylon & Reif, 1984). A subgoal groups a set of steps under a meaningful task or purpose (e.g., Anzai & Simon, 1979;Chi & VanLehn, 1991).…”

Prior experience and complex procedures

“…When people solve complex problems, it is common to generate sub goals for achieving major aspects ofthe task and to attempt to solve each of the subgoals as separate problems (see, e.g., Newell & Simon, 1972). For example, when attempting to calculate the probability of an event in a probability problem, subjects may generate the subgoal offirst finding the overall frequency of the event and then after that subgoal has been achieved, divide by the total number of trials to solve the original problem (Catrambone & Holyoak, 1990). Catrambone (1994Catrambone ( , 1995 has argued that the appropriate use of subgoals in problems like this promotes transfer: Learners are able to generalize to novel tasks more readily because they can find new solution methods to achieve the same sequence of subgoals.…”