Problem Solving

Gestalt approach

Stimulus–response approach

Information-processing approach

BIBLIOGRAPHY

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.]

BIBLIOGRAPHY

Duncan, Carl P. 1959 Recent Research on Human Problem Solving. Psychological Bulletin 56:397–429.

Duncker, Karl (1935) 1945 On Problem-solving. Psychological Monographs, 58, no. 5. → First published as Zur Psychologie des produktiven Denkens.

Edwards, Ward 1954 The Theory of Decision Making. Psychological Bulletin 51:380–417.

Edwards, Ward 1961 Behavioral Decision Theory. Annual Review of Psychology 12:473–498.

Gagne, Robert M. 1959 Problem Solving and Thinking. Annual Review of Psychology 10:147–172.

Groot, Adrianus Dingeman De (1946) 1965 Thought and Choice in Chess. Rev. ed. The Hague: Mouton. → First published in Dutch.

Guilford, J. P. 1960 Basic Conceptual Problems in the Psychology of Thinking. Pages 6–21 in New York Academy of Sciences, Fundamentals of Psychology: The Psychology of Thinking, by Ernest Harms et al. Annals, Vol. 91, Art. 1. New York: The Academy.

Hobhouse, L. T. (1901) 1926 Mind in Evolution. 3d ed. London: Macmillan.

Hull, Clark L. 1932 The Goal Gradient Hypothesis and Maze Learning. Psychological Review 39:25–43.

Hull, Clark L. 1934 The Concept of the Habit-family Hierarchy and Maze Learning. Psychological Review 41:33–54, 134–152.

Humphrey, George 1951 Thinking: An Introduction to Its Experimental Psychology. London: Methuen; New York: Wiley.

Hunt, Earl B. 1962 Concept Learning: An Information Processing Problem. New York: Wiley.

Inhelder, Barbel; and Piaget, Jean (1955) 1958 The Growth of Logical Thinking From Childhood to Adolescence. New York: Basic Books. → First published as De la logique de Venfant a la logique de Vadolescent.

Johnson, Donald M. 1950 Problem olving and Symbolic Processes. Annual Review of Psychology 1:297310.

Katona, George (1940)1949 Organizing and Memorizing. New York: Columbia Univ. Press.

KÖhler, Wolfgang (1917) 1956 The Mentality of Apes. 2d ed., rev. London: Routledge. → First published in German. A paperback edition was published in 1959 by Random House.

KÖhler, Wolfgang (1929) 1947 Gestalt Psychology. Rev. ed. New York: Liveright. → A paperback edition was also published in 1947 by New American Library.

Maier, Norman R. F. 1930–1931 Reasoning in Humans. Journal of Comparative Psychology 10:115-143;12:181–194. → Part 1, “On Direction,” is in Volume 0; Part 2, “The Solution of a Problem and Its Apperance in Consciousness,” is in Volume 12. See especially Part 1.

Maier, Norman R. F. 1940 The Behavior Mechanisms Concerned With Problem Solving. Psychological Review 47:43–58.

Maltzman, Irving 1955 Thinking From a Behavioristic Point of View. Psychological Review 62:275–286.

Miller, George A.; Galanter, E.; and Pribram, K. H. 1960 Plans and the Structure of Behavior. New York: Holt.

Morgan, C. Lloyd (1894) 1906 An Introduction to Comparative Psychology. 2d ed. New York: Scribner; London: Scott.

Newell, Allen; Shaw, J. C; and Simon, Herbert A. 1958a Elements of a Theory of Human Problem Solving. Psychological Review 65:151–166.

Newell, Allen; Shaw, J. C ; and Simon, Herbert A. 1958b Chess-playing Programs and the Problem of Complexity. IBM Journal of Research and Development 2:320–335.

Newell, Allen; Shaw, J. C ; and Simon, Herbert A. 1960 Report on a General Problem-solving Program. Pages 256–264 in International Conference on Information Processing, 1959, Proceedings: Information Processing. Paris: Unesco.

Newell, Allen; Shaw, J. C ; and Simon, Herbert A. 1962 The Processes of Creative Thinking. Pages 63-119 in Howard E. Gruber, Glenn Terrell, and Michael Wertheimer (editors), Contemporary Approaches to Creative Thinking. New York: Atherton.

Osgood, Charles E. (1953) 1959 Method and Theory in Experimental Psychology. New York: Oxford Univ. Press.

Piaget, Jean (1926) 1929 The Child’s Conception of the World. New York: Harcourt. → First published as La representation du monde chez Venfant.

Piaget, Jean (1947) 1950 The Psychology of Intelligence. London: Routledge. → First published as La psychologie de Vintelligence.

Piaget, Jean 1957 Logic and Psychology. New York: Basic Books.

PÓlya, GyÖrgy 1945 How to Solve It. Princeton Univ. Press.

PÓlya, GyÖrgy 1954 Mathematics and Plausible Reasoning. 2 vols. Oxford Univ. Press; Princeton Univ. Press.

Reitman, Walter R. 1965 Cognition and Thought: An Information Processing Approach. New York: Wiley. Selz, Otto 1924 Die Gesetze der produktiven undre produktiven Geistestdtigkeit. Bonn: Cohen.

Shannon, Claude E. 1950 Programming a Computer for Playing Chess. Philosophical Magazine 41:256–275.

Taylor, Donald W. 1960 Thinking and Creativity. Pages 108–127 in New York Academy of Sciences, Fundamentals of Psychology: The Psychology of Thinking, by Ernest Harms et al. Annals, Vol. 91, Art. 1. New York: The Academy.

Taylor, Donald W. 1963 Thinking. Pages 475–493 in Melvin H. Marx (editor), Theories in Contemporary Psychology. New York: Macmillan.

Taylor, Donald W. 1965 Decision Making and Problem Solving. Pages 48–86 in James G. March (editor), Handbook of Organizations. Chicago: Rand McNally.

Taylor, Donald W.; and Mcnemar, Olga W. 1955 Problem Solving and Thinking. Annual Review of Psychology 6:455–482.

Thorndike, Edward L. (1890–1901) 1911 Animal In telligence: Experimental Studies. New York: Macmillan. → See especially Thorndike’s 1898 essay, “Animal Intelligence: An Experimental Study of the Associative Processes in Animals.”

Van De Geer, Johan P. 1957 A Psychological Study of Problem Solving. Haarlem (Netherlands): De Toorts.

Wertheimer, Max (1945) 1961 Productive Thinking. Enl. ed. Edited by Michael Wertheimer. London: Tavistock.

Woodworth, Robert S.; and Schlosberg, Harold (1938) 1960 Experimental Psychology. Rev. ed. New York: Holt. → Woodworth was the sole author of the 1938 edition.

Problem Solving

Family problems come in many sizes and shapes. They range from minor annoyances, such as spats between children, to life-threatening situations such as physical abuse by a parent. They may be brief events that disappear in minutes or recurring disputes that last a lifetime. Whatever their form and duration, problems are distinguished by the presence of negative experiences for some family members. Such experiences provide natural motivation to eliminate the problem. Solving a family problem means finding a way to remove the negative experiences without creating new difficulties.

Humans have a variety of innate capabilities that are used for solving problems (Pinker 1997; Ellis and Seigler 1994). These include the abilities to recognize patterns in human situations, to recall relevant events from the past, to visualize events that may occur in the future, and to weigh the likely consequences of alternate future actions. One particular combination of abilities, rational problem solving, is especially important. It was initially identified in studies of the human thought process (Dewey [1910] 1982) and has been widely applied in work with couples and families (Forgatch and Patterson 1989; Vuchinich 1999). This form of problem solving occurs in a sequence of stages: (1) the problem is clearly defined; (2) several alternative possible solutions are generated; (3) each alternative is evaluated in terms of potential costs and benefits; (4) one alternative is selected as having the best potential to solve the problem; and (5) the solution is applied and adjusted as necessary.

These stages are generally recognized as being logical and based on elements of common sense. Indeed, they may be seen as essential to adaptation in the process of evolution (Pinker 1997; Vuchinich 1999). Using some variation of them provides a way to make changes that are likely to help eliminate the problem. However, individuals do not always use a rational approach to dealing with the difficulties in their lives. Other approaches to problem solving are prevalent and are often linked to couple and family dysfunctions.

Emotion and Problem Solving

To understand family efforts at problem solving it is first necessary to address the basic nature of problems and how they arise in family life. Since John Dewey's early formulations (1938) it has been acknowledged that the essence of problems is blocked goal attainment (Tallman 1988). Goals are physical or psychological states that individuals or groups seek. These include such things as sexual gratification, a sense of self-esteem, a full stomach, parental approval, or religious salvation. When attainment of such goals is blocked, individuals sense some form of frustration. At a certain level, such frustration creates a negative experience for the individual. Those experiences can be transformed into a perceived problem. When that happens, the individual feels dissatisfied and attempts to remove the blockage and reach the goal. If the individual finds a way to reach the goal, the problem is solved. Because of the negative affect in this process, emotional regulation is a key element in how couples and families try to solve their problems. Certain types of emotional regulation can short-circuit problem solving.

One of these types of emotional regulation is denial. Denial is a normal defense mechanism that allows the individual to avoid the pain of facing negative experiences. The negative emotion is regulated by denying its existence. With denial, an individual or family has negative affective experiences but tries to ignore them and takes no action to eliminate them. For example, a wife who is physically abused by her husband may think and act as though nothing is really wrong with her marriage. Although it may be psychologically useful in certain contexts, denial is a hallmark of a variety of couple and family dysfunctions. Denial stops any problem solving before it can even get started. From the individual's viewpoint, there is no problem. When denial is present, problem solving can only occur after there is some acknowledgement that a problem exists and something needs to be done about it. This may require some form of confrontation in which family members or professionals assertively display the problem or create consequences for not addressing the issues.

A second common type of emotional regulation is conflict engagement (Kurdek 1995). Here the frustration of blocked goal attainment is transformed into anger and aggression, usually against someone in the family. This represents a low level of emotional regulation. Someone is often blamed for the difficulties and becomes the target for verbal or physical aggression. Although a perceived problem may be identified, the impulsive expression of anger and the aroused emotional state forestalls any movement toward a meaningful solution. As a result, the conditions that created the underlying problem are not improved. Thus, negative experiences continue to accumulate and ultimately lead to more aggressive outbursts. This pattern can become dangerous as verbal aggression worsens and is supplemented by physical aggression. In such cases it is typically necessary to improve anger management skills of some family members before problem solving can begin.

Individuals have their own styles of responding to problems. But solving problems in couples and families must take individual styles into account. One particularly damaging combination of styles has been found to be prevalent in U.S. couples. It is known as the demand/withdraw pattern (Heavey, Layne, and Christensen 1993; Gottman 1995). In such cases one partner (the demander), often the wife, pursues discussion of a problem sometimes by demanding or complaining. The other partner (the withdrawer) responds to this by withdrawing and refusing to talk about it. This increases the intensity of the first partner's demands, which leads to further withdrawal by the other. The result, of course, is that nothing is solved, one partner is increasingly frustrated and the other is pushed further into a noncommunicative state. This pattern can also take place between parents and their children. Typically it is the parent who demands and the child who withdraws. The demand/withdraw pattern can be overcome by persuading each party to use different strategies when problems arise (Gottman 1995). For example, the demander may learn to initiate discussions more subtly. The withdrawer may learn to acknowledge the other's concerns and communicate more openly.

Who Defines Couple and Family Problems?

The demand/withdraw pattern highlights the fundamental importance of how a family problem is defined. Is something a problem if one partner perceives it to be a problem but the other does not? In the family context, Does any family member have the right to declare that a problem exists and that changes need to be made to solve it? Ideally if anyone in a couple or family senses that a problem exists, then the others would consider it a problem and seek a solution. This ideal is the basis for models of open communication, regular family meetings, and similar practices (Vuchinich 1999). In reality, power dynamics in couples and families often determine what is defined as a problem. Those with the most power decide whether an issue is a problem or not, and what can be done about it. In healthy couples and families those with power are sensitive to the needs of all members. They acknowledge legitimate problems and seek appropriate solutions. However, in some families, power is used to suppress dealing with important problems based on negative experiences of weaker family members. This is part of a more general pattern of dominance.

The question of whether something is a problem can extend beyond the boundaries of the family. If a couple seeks counseling for marital problems then a professional is involved in determining what problems exist. Resistance to therapeutic efforts is often based on an unwillingness to accept the therapist's definition of problems. Success in therapy can be a result of reformulating the problems that cannot be solved into problems that can be solved. Defining problems is also a common issue in social services work. A family may not view the father's physical beatings of their young son as a problem. But evidence of broken bones and psychological symptoms makes those beatings a problem as defined by the medical, social work, and legal professionals. Solutions to such problems may need to be imposed from outside the family.

It is often said that admitting that a problem exists is half of the solution, but that may be an understatement. If a real problem is not acknowledged, there may be little chance for a solution. But family members who sense a problem tend to take a "wait and see" approach. There is a stigma attached to couple and family problems in U.S. culture. Bringing up problems risks a negative reaction from loved ones, or even making matters worse. Indeed, minor difficulties do disappear without professional help. So there are some reasons why addressing problems is avoided. The average troubled couple waits six years before seeking marital counseling (Gottman 1999). But as a consequence, problems are usually well developed by the time any action is taken. This is not inevitable. Couples who regularly "check in" with how each other are doing can resolve problems quickly. Weekly family meetings serve the same purpose for family groups (Forgatch and Patterson 1989). These practices acknowledge that minor problems are normal. They can draw families closer together by opening lines of communication with an orientation toward helping all family members solve problems that concern them.

The Problem-Solving Process in Couples and Families

Once a problem is defined, elements of the rational process can be used to seek solutions. However, the rhetoric of problem solving can be somewhat deceptive when applied to couples and families. It is important to recognize that problems in couples and families are not like math problems. There is no single correct solution. There are many different solutions that might help eliminate the negative experiences at the core of the problem. One solution might solve only part of the problem, and components may be needed. Moreover, it may not be clear in advance whether a given solution will be helpful or not. It may seem like a good idea at the time; it may not work. Furthermore, individuals, couples, and families change over time. Such changes mean that solutions that worked at one point in time may not work at a later point.

Thus, effective family problem solving is an ongoing process that involves more than pure logic and reasoning. Couples and families must have certain minimal levels of communication skills and cohesion. They must be willing to change some of their behaviors for the well-being of others. Possible solutions must be tried out to see what happens. The results must be evaluated to see how well a given solution worked and whether other parts of solutions are needed. Family members must have enough patience and willingness to persevere through negotiations that, at times, may seem tedious or unpleasant. These are endured because of the rewards associated with living in a healthy functioning family that solves its problems.

Some family problems may not have a complete solution. For example, a family in poverty may have many problems associated with not having enough resources to meet basic family needs. Or a parent may acquire a serious disability that prevents them from fulfilling their roles as spouse and parent. In such circumstances family members may have to accept that some of their goals can not be attained. It is important to acknowledge that some problems have no solution and further efforts to solve them is counterproductive (Gottman 1999). Problem solving can still be used to find ways of making the best of the situation.

The involvement of issues such as emotional regulation and power dynamics complicates the problem solving process. However, the rational model remains at the core. Teaching couples, parents, and children to use it has proven beneficial in both prevention and therapeutic applications (Vuchinich 1999). Specific issues emerge within each stage when working with couples and families. First, constructing a clear definition of a problem is often difficult. Yet this is a crucial step in starting the process. Problem definitions are often expressed initially as complaints, and complaints are often met by immediate denials or countercomplaints. In the definition stage it is important to avoid such instantaneous negative reactions that engender conflict. Complaints deserve a fair hearing and some displays of empathy. One family member may have to facilitate this and later stages.

Once a problem is defined, possible solutions are suggested. Again, there is a tendency for proposals to be met with immediate negative response. But this stage should follow a brainstorming session in which various proposals are solicited, one after another, with neither criticism nor approval. Unrealistic or humorous proposals are allowed. This format promotes novel or creative approaches to the problem and participation from everyone.

The evaluation stage is facilitated by considering the potential consequences of each realistic proposed solution. By discussing the pros and cons of each proposed solution, family members can project what implications it would have for each of them. There are still disagreements here, but they should focus more on specific details of a solution rather than direct interpersonal conflict. In some cases it is useful to have someone write down each solution and its pros and cons. In this format the most severe objections of some family members get aired and acknowledged. It usually becomes apparent that only one or two solutions have a realistic chance of working. Ideally, the final selection of one solution is a consensual decision. This is not always possible. In such cases family cohesion and commitment to solving the family problems provide the motivation for everyone to try one solution, even if it wasn't everyone's first choice. Social skills can be especially valuable in this phase in reassuring everyone that their interests will be taken into account and the solution will not exploit them.

Once a solution is chosen, a detailed implementation plan is needed to specify exactly who will do what and when they will do it. Following through with a solution may be difficult. Talk is one thing—action is another. It is essential to plan for meetings or discussions to assess how well the solution worked. Typically an initial solution is only partially implemented and is only partially effective. Later meetings are used to revisit the solution and consider adjustments that will improve it. When family members begin seeing the benefits of solving their problems, their motivation for participating in problem solving activities increases.

The extent to which formal meetings are needed for problem solving varies from family to family. Family members do need to communicate about perceived problems in some way. Where and how often they do it depends on their communication patterns. The family meeting provides an effective structured format. It is important that such meetings do not degenerate into mere "gripe sessions," and some planning and facilitation may be necessary. A family meeting should include other activities besides problem solving. This can include such activities as sharing recent positive experiences, sharing news or feelings about extended family members, playing games, or eating snacks. This involves setting aside some time for the couple or family to be together and affirm their positive bonds. This can occur before or after the problem solving and helps integrate it into other aspects of family life.

See also:Communication: Couple Relationships; Communication: Family Relationships; Conflict: Couple Relationships; Conflict: Family Relationships; Conflict: Parent-Child Relationships; Decision Making; Family Life Education; Nagging and Complaining; Power: Family Relationships; Power: Marital Relationships; Resource Management; Therapy: Couple Relationships

Bibliography

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sam vuchinich

Problem Solving, Multiple Approaches to

Most of the mathematical problems that one encounters have already been solved and documented. Thus, the most important problem-solving tools are references. A reference can be anything—a book, a person, or a past experience—that aids in understanding and solving a problem. References enable one to solve problems independently when no other source of help is available.

Basic Strategies

The following are some general problem-solving strategies: break the problem into smaller parts, find a new perspective, work backward, guess. In all of these strategies, simple logic is used.

Logic. Mathematics is internally consistent. If at any point a false statement is generated, like 0 = 1, it is at once apparent that an error or a false assumption has been made. A false statement can be generated intentionally, as in a proof by contradiction.

Mutually exclusive dichotomies and trichotomies categorize the universe. Perhaps the most well-known dichotomy is true versus false. If x is true, then x is not false, and vice versa. The Trichotomy Axiom is a mathematical truism: If a and b are numbers, then either a = b, a > b, or a < b. Dividing all possible solutions into mutually exclusive categories can quickly eliminate incorrect solutions.

The Whole Is Equal to the Sum of Its Parts. When one is overwhelmed by a problem, it is a good strategy to break the problem down into smaller parts. As a general rule, the whole is equal to the sum of its parts, so, if all the pieces of a problem are solved, then the entire problem is solved.

Sometimes, the whole is greater than the sum of its parts. For example, "proper completed need in parts, may be the to assembled order once," makes no sense, but "once completed, the parts may need to be assembled in the proper order," does. After solving all of the pieces of a big problem, it must be determined that the pieces fit together into a form that makes sense.

Change of Perspective. Mathematical equations and concepts often have different forms that may be better suited to specific situations, and the internal consistency of mathematics guarantees that different forms of the same thing are equally valid. Here are 3 ways to represent 8: 2³, , .

Manipulatives and pictures can also provide a different perspective. When large amounts of information are presented, a visual or tangible connection can aid organization and understanding. For example:

Work Backward. In mathematical proofs in which the solution is known and the problem is proving the solution, working backward determines prerequisites for the solution. A series of questions may need to be asked of the task: What must be true (or false) in order that the solution is true (or false)? Do these prerequisites have prerequisites? And so on. In this manner, one continues working backward until the correct problem–solving strategy is determined.

When in Doubt… If one is confronted with a completely unfamiliar problem, guessing may well be the best strategy. Trial and error are an essential part of science and often the only way to proceed when charting new territory. A methodical approach, including meticulous recording of data and a careful search for patterns, makes guesses more informative and more accurate.

see also Proof.

Stanislaus Noel Ting

Bibliography

Challenging Mathematical Problems with Elementary Solutions. San Francisco: Holden Day, 1967.

Polya, George. Mathematics and Plausible Reasoning. Princeton, NJ: Princeton University Press, 1954.

Wickelgren, Wayne A. How to Solve Mathematical Problems. New York: Dover, 1995.