Problem Solving
Problem Solving
Information-processing approach
Frequent references to “the problem-solving process,” “the decision-making process,” and “the creative process” may suggest that problem solving can be clearly distinguished from decision making or creative thinking from either, in terms of the processes involved. These phrases can also imply that each involves a single process. On the basis of present evidence, however, it appears that each involves a variety of processes. Moreover, it appears that processes important in problem solving are also often important in decision making or in creative thinking (Guilford 1960; Taylor 1960). Each of these activities may therefore best be defined not in terms of process but in terms of product. For example, “Creativity is that thinking which results in the production of ideas (or other products) that are both novel and worthwhile. Similarly, decision making is that thinking which results in the choice among alternative courses of action; problem solving is that thinking which results in the solution of problems” (Taylor 1965, p. 48). Thus problem solving, decision making, and creativity are all to be regarded as kinds of thinking. The question of the degree to which these and other kinds of thinking—for example, concept attainment—involve the same or different processes is a question that remains to be answered by empirical investigation [seeConcept Formation],
The experimental psychological study of problem solving may be said to have begun shortly before the turn of the century. In 1894 C. Lloyd Morgan introduced the term “trial and error” to describe the process by which his dog learned such tricks as opening a gate by raising the latch with his muzzle. Shortly thereafter, in 1898, Thorndike reported experimental studies of the behavior of cats in various problem boxes—boxes with doors to be opened by turning a door button or pressing a lever or pulling a string attached to the outside bolt. In his book Animal Intelligence (1890—1901) he concluded, on the basis of his experimental results, that animals are incapable of processes such as reasoning or insight, and he supported Morgan’s suggestion that their behavior in solving problems is to be accounted for in terms of trial and error. [SeeMorgan, C. Lloyd; andThorndike.]
Beginning with Hobhouse (1901), however, the objection has been made that the trial and error observed by Thorndike was the result of the kinds of problems that he employed and not a general characteristic of animal behavior in problem solving. In the problems Thorndike presented, the elements are not open to the inspection of the animal at the outset; hence trial-and-error behavior is inevitable. Hobhouse devised a number of problems in which no essential element is concealed. Employing such problems, he obtained results indicating that animals are capable of what he called “practical ideas”—“a combination of efforts to effect a definite change in the perceived object.”
During the four decades that followed the publication of Thorndike’s book, the investigation of problem solving was continued by psychologists in several countries. But much less attention was devoted to this area or to the more general area of thinking than to other areas of psychology, such as perception, learning, motivation, and personality. However, within the past decade and a half an important increase has occurred in the amount of time being devoted to the study of not only problem solving but also the closely related areas of decision making (Edwards 1954; 1961; Taylor 1965) and creative thinking (Taylor 1960).
A general review of published research on problem solving is beyond the scope of this article. Hence, useful supplements to the present discussion may be found in the review by Duncan (1959) or in the articles by Johnson (1950), Taylor and McNemar (1955), and Gagne (1959) that have appeared in the Annual Review of Psychology. Summaries of earlier work on problem solving are available in Osgood (1953, pp. 603637), Woodworth and Schlosberg ([1938] 1960, pp. 814–848), Van De Geer (1957), and Humphrey (1951).
A comprehensive discussion of theories of problem solving is also beyond the scope of the present discussion. No description, for example, will be given of F. C. Bartlett’s treatment of thinking or of the writings of Piaget over a period of more than thirty years (see Piaget 1926; 1947; 1957; Inhelder & Piaget 1955; for a brief summary, see Taylor 1963). Instead, attention will be limited here to three theoretical approaches to problem solving. The first two approaches derive from more extensive work in another area of psychology. As a result, each is treated in articles dealing with other subjects in this encyclopedia. Accordingly, more attention will be given to the third approach. [SeeDevelopmental psychology, article ona theory of development; Bartlett.]
Gestalt approach
Gestalt theory, as embodied in the work of Wertheimer, Kohler, and Koffka, is first of all a theory of perception. The gestalt approach to problem solving gives clear evidence of this fact. In 1951 G. Humphrey remarked, “At the present time, the Gestalt theory of thinking constitutes a programme rather than a fulfilment” (1951, p. 150). This remains true, but it is nevertheless a theory sufficiently developed to demand consideration from one seriously interested in problem solving.
In its approach to thinking, as well as to perception and to other areas, the theory has as its central concept the organized whole, or gestalt. Thus Kohler wrote: “This, indeed, is the most general concept of gestalttheorie: wherever a process dynamically distributes and regulates itself, determined by the actual situation in a whole field, this process is said to follow principles of gestalttheorie” (1929, p. 193). These self-regulating wholes are characteristically extended in time; they are fourdimensional.
The question may be raised as to whether the “whole field” involves experience or underlying brain processes. The answer is both, because according to gestalt theory, isomorphic relations exist between the two such that the same principles apply to both.
When an individual is confronted with a problem, processes occur in the brain (and in consciousness) that by dynamic interaction with each other and with memory traces result initially in “seeing the problem.” Seeing the problem sets up stresses in the psychological field that determine the subsequent course of thinking. These stresses lead to reorganization or restructuring of the field, enabling the individual to perceive the interrelations that form the solution to the problem. Such reorganization tends to occur suddenly, but a problem may require for solution not one but a series of progressive reorganizations, thus giving the appearance of gradual solution.
The first gestalt studies of problem solving were those by Kohler (1917) and were concerned with the behavior of anthropoid apes. Like Hobhouse, Kohler contended that the problems which Thorndike had employed artificially produced trial and error. Kohler presented to his chimpanzees problems in which the elements necessary for solution were available and in which the solution required natural responses. Although he reported considerable behavior that might be described as trial and error, he emphasized that the critical point in the solution of a problem occurred when the animal grasped the appropriate relational pattern–when, for example, he saw a box upon which he could climb as a means of making a workable connection between himself and the fruit suspended beyond his farthest reach.
In discussing these early studies, Kohler accorded a central role to the concept of insight. Employed by gestalt writers in subsequent treatments of thinking—for example, in the account by Wertheimer (1945) of the exceptional course of thought that led Albert Einstein to his theory of relativity—this concept has received varying emphasis and has been used with more than one meaning. Thus, insight has been used to refer to the report by a subject of a particular kind of experience accompanying the solution of a problem. It has also been employed as a descriptive term for a certain kind of behavior—behavior that satisfies a number of criteria, such as a sudden improvement in performance or the ability to repeat a solution immediately. On the other hand, it has also been employed as an explanatory concept, more or less equated with restructuring. This last usage must be objected to, for neither insight nor restructuring is adequate as an explanatory concept, even taken together with the concept of the psychological field. A more adequate account is needed of the processes that result in the occurrence of insight.
Gestalt psychologists, including Duncker (1935), have emphasized a distinction, first made by Selz (1924), between reproductive and productive thinking. Productive thinking is thinking that cannot be accounted for simply on the basis of past learning because it involves something essentially new. The same distinction is emphasized by Maier (1930–1931; 1940), who, although he studied in Berlin, would not be described as a gestalt psychologist. In more recent gestalt writing, Katona (1940) and Wertheimer (1945) have suggested that the mere newness of the result is not sufficient cause for distinguishing reproductive and productive thought. Both emphasize that the way in which learning originally occurs will be important in determining whether subsequent thinking is reproductive or productive. Katona has thus contrasted learning by memorizing with learning by organizing. In the latter, the subject learns to grasp some principle according to which situations are structured; as a result, he is able to recognize subsequent situations not in terms of their superficial features but in terms of their underlying organization, and hence he is able to deal with them productively.
Stimulus–response approach
Whereas gestalt theory is first of all a theory of perception, modern behaviorism, stemming from the work of Hull, is first of all a theory of learning. The central concern of the theory is the process whereby the individual animal or human subject acquires the appropriate response to a stimulus. The concepts developed to account for learning responses to stimuli are employed to explain problem-solving behaviors. Among the important concepts are goal gradient, generalization, and habitfamily hierarchy.
The concept of the goal gradient includes, but is not fully represented by, the idea that the closer in distance or time an organism is to the goal, the stronger the organism’s motivation will be. Utilizing the goal-gradient hypothesis, Hull (1932) made a number of predictions, one of which was: Animals will tend to choose the shorter of two alternate paths to the same goal. Thus, for the stimulus-response theorist as for the gestalt theorist, the blocking of the most direct path creates the situation in which problem-solving behavior may be expected to occur.
Of central importance in accounting for the behavior that occurs in solving a problem is the concept of the habit-family hierarchy. A convergent habit-family hierarchy is one in which a number of stimulus situations have been associated with the same response. A divergent hierarchy is one in which a given stimulus situation is associated with a number of different responses (Hull 1934). In the latter, responses form a hierarchy in the sense that they differ in probability of occurrence. The position of a given response in the hierarchy varies with the degree to which it has been previously reinforced in similar situations. Given a new problem, the hierarchy determines which responses will occur and the order of their occurrence. If a response high in the hierarchy is unsuccessful, it will tend to be extinguished; this process will be repeated until some response lower in the hierarchy is successful. The success of that response will reinforce it and increase the probability of its occurrence when that same problem situation recurs.
The concept of generalization is employed to account for the fact that rewarding an organism for making a given response to a particular situation will increase the probability of its making the same response not only to that situation but also to similar situations. This increase in probability varies directly with the similarity of the new to the original stimulus situation. Thus a man who is confronted with the need to tighten a screw but has no screwdriver may readily use a fingernail file, but he may engage in much trial and error before employing a thin coin for the same purpose.
Perhaps even more important than simple stimulus generalization in accounting for human behavior in solving problems is the concept of mediated generalization (Osgood 1953, pp. 359–360, 392–412). Generalization may occur not because of similarity between the two stimulus situations but because of the prior establishment of a common mediating response to the two situations. When the individual acquires a new response to the first situation, that response also becomes associated with stimuli produced by the mediating response. Hence when the second situation occurs, the mediating response produces stimuli that result in the elicitation by the second situation of the response which has previously been given to the first situation.
In order to account for the complex behavior involved in human problem solving, Maltzman (1955) suggested an extension of Hull’s habitfamily hierarchy. He proposed a combination of the concept of a divergent hierarchy with that of a convergent hierarchy in order to produce a compound habit-family hierarchy. In a compound hierarchy a given stimulus situation has the disposition for arousing not only its own family hierarchy but, to a varying degree, the habit-family hierarchies of other stimulus situations. There is a hierarchy of habit-family hierarchies elicitable by that situation. An analogous condition exists for the other stimulus situations. Maltzman (1955) employed this concept of a hierarchy of hierarchies in discussing, for example, the distinction between reproductive and productive thinking. He suggested that in reproductive thinking a series of problems is presented that requires for a solution the elicitation of different responses in the same family hierarchy. With the presentation of successive problems, this habit family becomes dominant in the compound hierarchy. In contrast, in situations requiring productive thinking, a habit family initially low in the compound hierarchy must become dominant before the problem can be solved. Once the appropriate family is dominant, a solution will occur provided that the correct response within that hierarchy in turn becomes dominant.
Information-processing approach
A newer approach to problem solving is that suggested by Newell, Shaw, and Simon (1958a). What they propose essentially is that the problem solver be regarded as an information-processing system. Such a system consists of a number of “memories,” which contain symbolized information and are interconnected by various ordering relations. It also includes precisely specified “primitive information processes,” which operate on the information in the memories, and rules for combining these processes in “programs.”
An investigator employing this approach begins by selecting for study some particular kind of problem solving. He attempts to identify the processes involved. In doing this he may observe and analyze the behavior of individuals engaged in solving such problems, have them “think aloud” while working, examine his own experience, or use any other source of information that may be helpful. When he has formulated tentative hypotheses to describe the processes involved, he undertakes to write the program that employs these processes and simulates the thinking of the human problem solver. [SeeSimulation, article onindividual behavior.]
Viewed as a theory of behavior, a program is highly specific in that it represents only the behavior of one individual in one set of situations. If either the individual or the class of situations is changed, the program must be changed. There will, however, be important similarities among the programs that represent the behavior of the same individual in different situations or among those that represent the behavior of different individuals in the same situations. On the basis of these similarities a more general theory of the behavior under study may be developed.
Such programs are written in one of the so-called information-processing languages—precise languages that have been created for this kind of use. Because the language employed is precise, it is possible to run the program on any modern highspeed computer by supplying the computer with the appropriate “interpretive deck.” In this way one may determine what behavior any given program will produce. It should be emphasized that this approach to thinking does not depend upon any crude analogy between the structure of the computer and the structure of the brain. The function of the computer is simply to enable the investigator to determine quickly whether the consequences of a program are in fact consistent with the observed human behavior that it is supposed to explain. If a program is to be more than an ex post facto explanation, it must not only simulate adequately the behavior it was written to account for but also predict the effect of changing conditions on behavior.
An important distinction is that between algorithmic and heuristic processes in problem solving. An algorithm is a process for solving a problem that guarantees solution in a finite number of steps if the problem has at least one solution. An example of a very simple algorithm would be that for obtaining temperature on the Fahrenheit scale when the value for the centigrade scale is known: Multiply the known value by 1.8 and add 32.
A heuristic process may aid in solution but offers no guarantee of doing so. In his little book How to Solve It, the mathematician Poly a (1945) described a variety of heuristic processes useful at the level of high school mathematics. In Mathematics and Plausible Reasoning (1954) he dealt in more detail with the role of heuristic procedures. Some heuristics are applicable only to a particular class of problems. Thus, the following heuristic would be appropriate for use only in discovering proofs for theorems in geometry: If the figure has one axis of symmetry and it is not drawn, then draw it. But other heuristics may aid in solving quite varied kinds of problems. One example of a more generally useful heuristic is what is called “means-end” analysis: Compare what you have with what you wish to obtain; identify a difference between the two; find and carry out an operation that may reduce the difference; repeat this procedure until the problem is solved. Another generally useful heuristic is the one called “makea-plan”: Find a problem that is similar to, but simpler than, the one you are attempting to solve; solve the simpler problem; use the procedure successful in solving the simpler problem as a plan for solving the more complex problem.
Within recent years several programs have been written employing heuristic procedures and simulating various kinds of human problem solving. One of the first was the Logic Theorist by Newell, Shaw, and Simon (1958a). In writing this program they undertook to identify the heuristics involved in discovering proofs for theorems in symbolic logic. When completed, the program produced impressive results in a number of experiments carried out by running it under various conditions on a high-speed digital computer. Principia mathematica by Whitehead and Russell is a classic of modern symbolic logic. In one experiment the Logic Theorist, employing the same axioms, definitions, and rules used in the Principia, was presented with the task of constructing in turn a valid proof of each of the first 52 theorems in Chapter 2 of the Principia in the order in which they appear there. Whenever a theorem was proved, it was stored in memory and was available, together with the original axioms, for use in proving subsequent theorems. Under these conditions the Logic Theorist discovered proofs for 38 of the 52 theorems and did so in times ranging from less than a minute to more than 15 minutes.
In a sense, the Logic Theorist has been superseded by the General Problem Solver, another program by Newell, Shaw, and Simon (1960). Construction of this program began with the study of “thinking aloud” protocols produced by individuals engaged in solving a type of problem that also involved symbolic logic but did not involve discovering proofs for theorems. Although the original objective was to simulate the behavior involved in solving the particular type of problem chosen for study, the evidence clearly indicates that the program constructed can also do trigonometric identities and formal integration, solve algebraic equations, and may be extended to an even wider range of tasks. The program is called general for this last reason, not because it can solve problems in general. Of marked interest is the fact that the General Problem Solver employs only two principal heuristics, the means-end heuristic and the makea-plan heuristic. In the past such procedures, if considered at all, were often regarded as ambiguous and trivial. One major result to date of the use of simulation has been the rigorous demonstration of the potential power of heuristic processes in solving problems.
The objection may be made that in view of the high speed and accuracy with which a computer works, it is hardly surprising that the Logic Theorist is successful in discovering proofs for a variety of theorems or that the General Problem Solver can find solutions for other problems in logic or mathematics. This objection may arise from the incorrect belief that what is occurring is either an exhaustive search or a random search of all possible alternatives. For simple problems solutions may well be attainable by exhaustive or random search. But such searching is simply not feasible in solving problems of great complexity. The following analysis may help clarify this.
A useful abstract model of many kinds of problem-solving activity is provided by the maze (Newell et al. 1962). The task of the thinker in discovering the proof of a theorem, for example, may be regarded as that of finding one of the paths through a very complex maze that begins with one or more of the axioms and ends in the specified theorem. The maze consists of all possible paths that might have been taken. Using the maze as a model, one may make some estimate of the difficulty of a particular class of problems by asking the essential question: How large is the maze that must be searched? The task of the thinker is to find a way through the maze the first time—not, as in studies of animal learning, to learn to run the maze without error.
Estimates have been made of the size of the space of possible solutions for problems handled by the Logic Theorist. If no limit is imposed upon the length or other characteristics of sequences of symbolic expressions generated in the attempt to discover proofs, then the size of the maze is literally infinite. If, however, the space is restricted, “for example, to proofs consisting of sequences of not more than twenty logic expressions, with each expression not more than twenty-three symbols in length and involving only the variables p, q, r, s, and t and the connectives ‘or’ and Implies/ the number of possible proofs meeting these restrictions is about 10235—one followed by 235 zeros!” (Newell et al. 1962, p. 73). Even with the speed of a computer, either the random or the exhaustive search of such a maze is clearly out of the question.
For those classes of problems for which efficient algorithms have been discovered, such procedures are to be preferred. They guarantee solution if the problems have solutions. But for many kinds of problems algorithms are unknown. Moreover, for some types of problems for which algorithms are known, their use is prohibited by the enormous amount of time that would be required to carry them out.
There is, for example, an algorithm for playing chess: Consider all possible continuations of the game from the existing position to termination and then select one move that will lead to checkmate of the opposing king. The mathematician C. E. Shannon (1950) has estimated that if this procedure were employed, it is unlikely that a single game would be completed within a lifetime, even if the players worked at the speed of the fastest electronic computer. The use of the algorithm is simply not feasible. Instead, those who play chess employ heuristic procedures which simplify the task of searching the maze—heuristics that include, for example: “Protect your king,” “Develop your pawns,” “Try to control the center of the board,” etc. (see Groot 1946). Beginning in about 1950, a number of attempts have been made to program a computer to play chess. A review of these attempts reveals a clear trend towardthe use in these programs of increasingly selective heuristics (Newell et al. 1958b).
In the relatively short time that has elapsed since the initial proposal of theinformation-processing approach, heuristic programs have been written for a wide variety of tass. Although not all were written with the primary purose of simulation of human thinking, it may be of interest that heuristic programs now exist for such varied tasks as attaining concepts, discovering proofs for theorems in plane geometry, composingmusic, dealing with the trust investment problemof portfolio selection, and balancing an assembl line. A more complete introduction to this approach to thinking is provided in Miller, Galantr, and Pribram (1960), Hunt (1962), Reitman (1965), and Taylor (1965).
Donald W. Taylor
[Directly related are the entriesCreativity, article onpsychological aspects; Decision making; Reasoning and logic; Thinking. Other relevant material may be found inGestalt theory; Information theory; Learning; and in the biographies ofHull; Koffka; KÖhler; Watson; Wertheimer.]
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