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The paper is organized into five major sections and a summary. Each of the five major sections is devoted to a different aspect of our analysis of proof skills in geometry and their acquisition. The first section presents an analysis of how a student searches for a proof in geometry after he has acquired his basic skills. This provides a basic framework within which to understand each of the types of learning that we then discuss in the remaining four sections. The second section is concerned with what we call text learning. It is concerned with what the student directly encodes from the text, and more importantly, the processes that use this direct encoding to guide the problem solving. The third section is concerned with a process that we call subsumption, a means by which the student encodes new information into existing knowledge structures. Fourth, we discuss the processes of knowledge compilation by which knowledge is transformed from its initial encoded form, which is declarative, to a more effective procedural form. The final major section discusses how practice serves to tune and optimize the skill so that the proof search is performed more successfully. (Author), Prepared in cooperation with Pittsburgh Univ., PA, Contract N000014-78-C-0022.

In experiments measuring latencies to answer questions about relative locations, Lea (1975) supported a hypothesis that locations are represented in a list ordered according to adjacency relations, and Hintzman, O'Dell, and Arndt (1979) supported a hypothesis that distances and directions between objects are represented. We propose a hypothesis that locations are represented using a spatial schema, and that procedures for answering questions include processes that operate on positions in the spatial schema. We hypothesize further that different performance observed by Hintzman et al and Lea can be interpreted as a difference of retrieval procedures induced by differences in the ways that questions were presented. We conducted an experiment in which different questions were asked in different conditions about locations of objects, where the tasks were comparable in ways that Hintzman et al's task and Lea's task were not. Latencies in the two conditions differed in the same way as the earlier findings, lending support to the hypothesis of different retrieval procedures induced by differences in the question-answering tasks. (Author)

This project examined processes of informal reasoning and relationships between informal and formal reasoning in problem solving. One set of studies focused on relationships between knowledge of problem-solving procedures and knowledge of general conceptual relationships. We have specified ways in which problem-solving performance can be influenced, and problem-solving procedures can be understood, because of their relationships to conceptual knowledge in the form of schemata. The second set of studies was concerned with relationships between general formal principles and cognitive procedures. Contributions included a method for formally analyzing relationships between procedures and constraints, and a theoretical and empirical analysis of understanding the principle of formal deductive consequence in the context of geometry proof problems. (Author)

By 1960 there was a strongly developed theory of learning in which learning was considered as change of behavior. Neobehaviorist theories and then formal stochastic models analyzed processes in which probabilities of responses are altered. In the 1960's we began to analyze learning as discrete change between states of knowledge or stages of processing that differ in qualitative characteristics; stochastic models were used to represent these states and stages. In addition, the processes and structure of human memory were studied in detail. In the 1970's we have developed detailed analyses of the organization of knowledge for understanding language and solving problems, using programming languages as formalisms for representing models that simulate human performance. A prospect for the 1980's is the analysis of learning considered as acquisition of knowledge, in which basic processes will involve modification and combination of cognitive structures; this development is likely to include and profit from analyses of learning tasks used in school instruction. (Author)

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 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 construction 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)

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 processses are also noted.

Analyses are described of knowledge structures used to understand and solve problems in high school geometry and in primary-grade arithmetic word problems. Analysis of geometry problem solving has clarified the nature of strategic knowledge needed by students and raises the question whether more explicit training in strategies would be beneficial in school instruction. Analysis of semantic knowledge needed to understand word problems raises questions about relationships between students' learning of computational procedures and their understanding of general types of quantitative relationships.

This project addressed the question: What knowledge and processes enable individuals to construct and modify representations of novel, nonroutine problems? The theoretical goal was to extend the information-processing theory of problem solving to include processes that have been characterized as restructuring of problems and productive thinking by Gestalt psychologists such as Duncker and Wertheimer. Three lines of research were carried out. One investigated different kinds of knowledge that result in problems that are routine, semiroutine, or nonroutine, according to distinctions in a simulation model that was developed in the project. Experiments distinguished between functional knowledge, involving relations between components of a device, and component knowledge, involving information about behavior of components independent of their interconnections. A second line of work studied processes of generating representations of physics problems, and identified important interactions between everyday experience-based knowledge about motions of objects and knowledge of theoretical principles. The third line of research studied processes of understanding and solution of back-of-the-envelope problems, and identified use of general methods for reasoning about quantities that are combined with domain-specific information. (Author)

We distinguish between routine, semiroutine, and nonroutine problems based on the problem solver's knowledge. Routine problems are solved by applying a known procedure, semiroutine problems require planning that uses functional knowledge, and nonroutine problems require generation of new functional knowledge. We have simulated nonroutine problem solving in which functional knowledge is derived from properties of objects that are available in the situation. In an experiment, we provided knowledge about functional relations of components of a device and found that this facilitated inference of operating procedures, in contrast to knowledge about the states of the individual components, which was ineffective.

An information-theoretical analysis of theories having statistical hypotheses and postulated entities is carried out. The analyses show that it is possible to perform analyses that are relevant to evaluating one's state of knowledge when one has statistical knowledge about a system. It is universally recognized that theoretical entities can serve as an heuristic aid in development of new empirical knowledge about a system, and in some cases can provide a substantial simplification in the representation of knowledge. These facts are clarified and specified to some extent by the present analysis. It has been seen that the information transmitted by dependencies between a set of empirical variables and a set of theoretical variables is generally greater than the information transmitted between set of empirical variables, and this fact clarifies the usefulness of postulated entities in guiding empirical investigations. The role of theoretical entities in relation to simplicity is especially striking in nonstationary systems, particularly when the theoretical system has the Markov property., Sponsored in part by the DARPA.

Measurements of difficulty in two stages of paired-associate memorizing are obtained for four paradigms of negative transfer. Data are presented from a series of different retention tests given after interpolated learning in the four paradigms. Evidence is presented indicating that recall of two responses paired with the same stimulus is independent. An interpretation of the results is given, based on two hypothetical processes. One process in negative transfer is interference with the storage of new items, produced by carryover of the encodings of stimuli and responses from earlier pairings. The other process in transfer, and the main process in forgetting, is interference with the retrieval process, produced mainly by the inclusion of the same stimuli in two retrieval systems. (Author), Presented at meetings of American Psychological Association, Washington, D. C. Sep 69.