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Discussed is one "quite general" attribute that can differentiate problem representations: the kinds of entities that are included -- the cognitive objects that the system can reason about in a relatively direct way, and that are included continuously in the representation. The ontology of a domain is significant for four reasons. First, ontology is a significant factor in forming analogies between domains, described in terms of two examples involving problem-solving procedures between domains: geometric proofs and subtraction procedures. Second, these entities provide arguments on which general reasoning procedures can operate directly; this is explored through physics problems; distance, time, and velocity; and sound transmission. That conceptual entities can enable more efficient computation is presented in terms of monster problems, isomorphic to the Tower of Hanoi problem. Finally, the ontology of a problem domain has important effects on goal definition and planning, illustrated by studies of binomial probability. (MNS)

In the pair of experiments reported here the authors investigated the relationship between meaningfulness of problem statements and subjects' use of these statements in problem-solving tasks. Subjects (96 university students) were required to memorize meaningful formulae such as "volume = area x height" or corresponding symbolic formulae such as "v = a x h." Formulae were memorized in three sets of three. Some subjects were tested on one formula from each set, while others were tested on an entire three-formula set. In the first experiment subjects were asked to compute values using the formulae, or were asked unanswerable (incomplete or inconsistent) questions about the formulae. In the second experiment subjects were asked to judge computability of a quantity given several others. In both experiments analysis of variance revealed a three-way interaction between meaningfulness, problem type, and grouping. Symbolically stated problems involving more than one formula took much longer when the formulae came from different sets; for meaningful problems no such difference occurred. The authors conclude that subjects reorganize meaningful material for themselves, but use the instructor's organization for symbolic material. (SD)

Hierarchically organized knowledge about actions has been postulated to explain planning in problem solving. Perdix, a simulation of problem solving in geometry with schematic planning knowledge, is described. Perdix' planning knowledge enables it to augment the problem space it is given by constructing auxiliary lines. The planning system also provides a mechanism that can result in problem-solving set. Results of three experiments involving a set and constructions seem consistent with the kinds of knowledge structures hypothesized in the model. Protocols given during solution of 11 geometry problems showed general agreement with the explanation of constructions and set based on planning knowledge, but also indicated processes of human problem solving not represented in the model. Finally, the explanation of constructions is discussed in relation to the general question of ill-structured problems and creativity, and the explanation of set is discussed in relation to other phenomena in the problem-solving literature, including functional fixedness. (Author)

Thinking-aloud protocols of human problem solvers working on geometry problems are presented and discussed. Protocols were obtained from six individuals working on nine different problems in which constructions were used. Nineteen protocols are presented with annotation and discussion, and other protocols are summarized. The primary purpose of the report is to provide documentary evidence relevant to a model of planning knowledge that provides an explanation of constructions. The protocols are consistent with the general features of the model, but also show ways in which human problem solving involves processes more complex than those in the model. Illustrations of interaction between formal and informal reasoning processes are also noted. (Author)

Problems were presented to 13 university students in which letters were to be added or subtracted (e.g., B + D = ? or F - ? = D). After each problem, each subject gave a retrospective protocol indicating the way in which the problem was solved. Models of performance by each subject in each experimental session shared major properties; choices by all subjects depended on a few features of the problems. Individual differences consisted in the features that were included in a subject's choice procedure and the criteria applied in choices based on the features that were used. (Author)

The relation of knowledge for solving problems to understanding of general principles in a subject-matter domain is discussed. Theoretical representations, called "conceptual competence," are presented to represent principles that are believed to be understood implicitly by individuals. Implications for problem-solving procedures can be derived from these principles. In a task for which process models have been developed, the process models are related to the formulation of general principles that are reflected in the process. (Author/PN)

Research was conducted investigating properties of skill in learning, in the domain of elementary algebra. Thinking-aloud protocols indicate that early knowledge of the subjects studied was fragmentary, rather than involving systematically flawed procedures. Computational models, developed to simulate observed errors, focused on the role of structural representations in facilitating reliable performance. Connectionist models for recognizing structural features were investigated, leading to the conclusion that the cognitive system probably requires knowledge functionally equivalent to grammatical rules. Data from information processing experiments indicated that: (1) judgments about the application of an algebraic operator are influenced by low-level features recognized before a completely parsed representation is formed; and (2) recognition of individual characters in expressions is not facilitated by syntactically correct contexts, as it is by lexical contexts in letter recognition, but information about the algebraic categories of characters is obtained early in processing from the syntactic context. The conclusion is made that training in basic symbolic skill might be more effective if more attention were given to teaching the structure of information of the domain, including general features of the information presented in problems as well as general constraints and goals of the procedures to be acquired. (Author/LMO)

In the research described in this document, the goal was to develop definite theoretical characterizations of understanding in a specific domain of problems. Some performance that provides persuasive evidence of understanding was chosen, and hypotheses about cognitive structures and processes that cause the performance to occur were developed. Performance was analyzed on two tasks in high school geometry, developing proofs and checking proofs. For developing proofs, thinking-aloud protocols were obtained from six students who were interviewed approximately once a week during the year they studied geometry. Proof-checking problems were given to 15 different students during interviews in November and May, and a group of 15 college students also were given instruction on proof checking to ascertain whether it could be acquired. Two characterizations of knowledge that can be adopted as objectives of instruction resulted, one on structural understanding and one in understanding the principle of deductive consequence. (MNS)

Research in this project studied cognitive processes involved in understanding and solving problems used in instruction in the domain of mathematics, and explored implications of these cognitive analyses for the design of instruction. Three general issues were addressed: knowledge required for understanding problems, knowledge of the conditions for applying problem-solving operators, and strategic knowledge for problem solving. Computational systems were developed in which graphical displays present explicit information to students that allows them to reflect on processes that are usually implicit, construct representations of problem information, specify assignments of variables, and learn from strategic errors. Research is described that provided analyses of cognitive processes on which instructional designs were based, and preliminary empirical results from use of the systems are presented. (Author/JM)

The uses of verbal rules in solving arithmetic problems are presented. Thirty-six problems were used in individual testing of five students who were enrolled in a remedial mathematics course at a community college. Two or three weeks prior to the individual exams, the students had been tested in their class on the material. The study indicated that verbal rules provide a plan for action in verbal-retrievable form, but students must add a non-trivial amount of knowledge to make the rules effective. Five principles developed to explain the status and use of verbal rules in problem solving are presented and discussed in the context of the study. It is suggested that the next area of study be the process of transition from rules to procedures that occurs within the student's mind. (MP)