Memory & Cognition 2001, 29 (4), 627-633 Principle explanation and strategic schema abstraction in problem solving ALLAN B. I. BERNARDO De La Salle University, Manila, Philippines This study was undertaken to examine the effects of strategic schema-acquisition tasks (problem comparison or problem construction) and the method of principle explanation (abstract or embedded principle method) on schema acquisition. Ninety-eight subjects studied a set of problems in probability, presented according to either method of principle explanation. Half the subjects in each principle- explanation group were then asked to compare analogous problems, and the rest constructed new analogous problems. To determine whether subjects generalized problem schemas, they were given new analogous problems to solve. The results showed that when the abstract principle method was used, schema acquisition was better in problem comparison; but with the embedded principle method, schema acquisition was better in problem construction. Results were discussed in relation to the im- portance of some fit between the presentation of problem information and the processes that will draw from or build on this information in tasks designed to allow novice problem solvers to acquireadvanced problem representations. In many domains of formal learning, such as mathe- face elements of the problem, and the problem-solving matics, one of the goals of learning and teaching is to de- context play an important role in the development of velop students’ knowledge of the abstract principles and schemas. For example, the reminding model of Ross procedures that they use to solve problems in that do- (1987, 1989;Medin & Ross, 1989) and the multiple-trace main. Research has identifiedanalogicalproblem solving model of Hintzman (1986) assume that content informa- as one of the means by which students can develop such tion plays very important roles in driving the retrieval and abstract knowledge for solving problems. In particular, mapping processes that are integral to analogicalproblem research has shown that through the use of earlier source solving, and that as such, content information acquires problems to solve new analogous problems, problem functionalsignificance in the problem-solvingprocesses. solvers come to develop problem schemas. These models are generally well supported by empiri- Most researchers (for a review, see Reeves & Weis- cal evidence (see Reeves & Weisberg, 1994,for a detailed berg, 1994) agree that abstract principles and procedures review). Numerous studieshave documented the effects of can be derived from solving individual analogous prob- problem content on the retrieval of relevant source prob- lems, and that this abstract information can be represented lems for analogical problem solving (Blessing & Ross, independently of the analogous problems themselves. 1996; Novick & Holyoak, 1991; Reed, Ackinclose, & Moreover, research in analogical problem solving sug- Voss, 1990;Ross, 1989),although contentmay affect some gests that a wide range of problem-specific factors such domains more than others (Bassok & Holyoak, 1989). as the content domain of the problem, superficial or sur- Studies also show that content affects the complex pro- cess of mapping between the source and target analogous problems (Bassok, 1990;Bassok, Chase, & Martin, 1998; This research was supported by a Spencer Postdoctoral Research Fel- Novick & Holyoak, 1991; Ross, 1987, 1989) even with lowship awarded to the author by the National Academy of Education, highly experienced (Blessing & Ross, 1996) and expert United States of America. The paper was completed while the author was a Visiting Scholar at the Faculty of Psychology and Educational problem solvers (Novick, 1988). Similar effects on re- Sciences, Leuven University, Belgium, and with partial support from trieval and mapping of analogous information have been the College of Education Research Fund,De La Salle University,Manila. found for factors other than the immediate content ele- I thank the students, teachers, and administrators of Lourdes School ments of the problem (Bernardo, 1998; Spencer & Weis- and St. Mary’s College, both in Quezon City, for their enthusiastic par- berg, 1986). These findings suggest that aspects of the ticipation in the study. I also acknowledge the valuable assistance pro- vided by Niño Mateo, Alma San Buenaventura, Chinee Tan, Annalyn content of the problem and the problem-solving episode de Guzman, Raffy Banaag, Jezzle Ticao, SylviaSajo, ChristopherCadua, must be retained in the memory representations devel- and Sendy Mangilit in preparing the research materials and gathering oped after analogicalproblem solving.Concreteproblem- the experimental data. Finally, I thank two anonymous reviewers for specific information is retained because it is assumed to their most thoughtful comments on earlier versions of this paper. Corre- spondence should be addressed to the author at the College of Education, be useful for guiding the retrieval and use of abstract in- De La Salle University, 2401 Taft Avenue, Manila, 1004 Philippines formation (Bernardo, 1994; Medin & Ross, 1989; Ross (e-mail: [email protected]). & Kennedy, 1990). 627 Copyright 2001 Psychonomic Society, Inc. 628 BERNARDO The present experiment was carried out in an attempt information, thus facilitating the development of ab- to investigatefurther the effects of more concrete aspects stract problem schemas. of the problem-solvingepisodeby lookingat two factors: Recently,another task that requires the problem solver the manner in which abstract solution principles are pre- to focus on structural information has been studied. sented, and the learning task given to facilitate the strate- Bernardo (2001) studied the effects of analogical prob- gic abstraction of the schema. lem constructionon analogical transfer. This task involves The abstract principles used to solve problems can be asking subjects to construct their own problems after presented in different ways; Ross and Kilbane (1997)have they have been presented exemplars of a problem cate- distinguished two. The first, called the abstract principle gory. A series of experiments showed that this task was method, involves the approach typically used in instruc- effective in facilitating acquisition of schema informa- tion in mathematics. A general principle is given and ex- tion and transfer. The task was assumed to involve the plained,and then a problem is given to illustrate how the reverse of the mapping process that is essential to analog- principle is applied. The second is the embedded princi- ical problem solving. In analogical problem solving, the ple method, whereby a problem illustrating a specific prin- problem solver is required to map out the similaritiesbe- ciple is given and the principles for solving the problem tween the analogoussource and target problems, and this are integrated with the completion of the problem solu- mapping leads to the abstraction of a general principle tion. Hence, there is no explicitpresentationof the princi- (Novick & Holyoak, 1991; Ross & Kennedy, 1990). In ples that one uses to solve the problem. Ross and Kilbane analogicalproblem construction, the problem solver cre- studied how these two modes of presenting principles af- ates a new problem by projecting new problem elements fected the processes of analogicalproblem solving.They that would map with the elementsof the originalproblem. found that problem solvers were likely to be misled by Studying the mode of principle explanation and the non-analogous superficial similarities when the abstract type of schema-acquisition task has both theoretical and principleswere presented explicitly.They concludedthat practical significance. Both of these variables relate to with this method,the problem solver relies heavily on the instructional processes for teaching problem solving in features of the source problem but that these features are formal schools. Ross and Kilbane (1997) have suggested not yet meaningfully linked to the abstract solution infor- that looking at such variables will not only contribute to mation. Thus, the problem solver becomes more vulnera- our understanding of the basic cognitive processes that ble to misleading superficial similarities between the occur in analogical problem solving and schema acqui- source problem and the analogous problems or among sition, but also provide some practical insights into how analagous problems. But with the embedded principle instructionalprocesses can be changed to facilitate better method, the problem solver seems to acquire more con- learning of abstract problem information. textualizedknowledge about the problem solutions.When It was not the objective of the study presented here to solving new analogous problems, the problem solver is ascertain which mode of principleexplanationor strategic less susceptible to misleading superficial similarities, schema acquisition task might be better. Indeed, the no- because the abstract principlesthat need to be applied are tion that various modes of schema acquisitionmay all lead functionallyrelated to other elements of the episodicmem- to the abstraction and generalization of structural problem ory trace for the source problem. information has been suggestedby a number of researchers If there are at least two ways in which abstract solution (e.g., Reeves & Weisberg, 1994; Ross & Kennedy, 1990). principles may be presented, there are several ways by The objective was rather to find out whether the effec-