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Working Definition of Problem Solving

According to Mayer (1983), any definition of problem solving should consist of three ideas. First, the problem is presently in some state. Second, it is desired that the problem be in another state. And third, there is no direct, obvious way to accomplish the change.
Reitman (1965) has analyzed four categories of problems according to how well the given and goal states are presented. First, there is the well defined given state and the well defined goal state: “How can you turn a sow’s ear into a silk purse?” Although the given state and the goal state are clearly stated, the problem seriously lacks viable ways of solving it. Second, there may be a well defined given state and a poorly defined goal state: “How can you redesign a Rolls Royce to get better gas mileage?” The given state, the car, is clearly stated, but what does “better gas mileage” mean exactly? Third, the problem may be presented such that there is a poorly defined given state and a well defined goal state: “Explain the mechanism responsible for creativity.” The goal “creativity” is clear but the initial state that causes this goal is not. As you may expect, the fourth category involves a poorly defined given state and a poorly defined goal state. To use Reitman’s example: “What is red and goes put—put?” The answer is an outboard apple.
Greeno (1978) has suggested a three-part typology of problems. First, we have the problems of inducing structure, in which several instances are given and the problem solver must discover the rule or pattern involved. Second, there are the problems of transformation. Here an initial state is given and the problem solver must find a sequence of operation that will produce the goal state. Lastly, there are problems of arrangement. In this instance, all of the elements are given and the problem solver must arrange them in such a way that the problem is solved. Greeno points out that not all problems will neatly fall into one of these three classifications; some problems consist of a combination of classifications.
In covering the steps and stages utilized in problem solving strategies, it is best to start with Wallas (1926), since he was the first to purpose such. Wallas suggested that the stages of problem solving consist of four major phases:

1.  Preparation: gathering of information and preliminary attempts at a solution. 
2.  Incubation: putting the problem aside to work on other activities or to sleep. 
3.  Illumination: appearance of the key to the solution (often referred to as the “aha” experience). 
4.  Verification: checking the solution to make sure that it is valid.

Polya (1957) has introduced a series of steps based on his observations as a teacher. Polya’s four steps are:

1.  Understanding the problem: here the solver gathers information. 
2.  Devising a plan: when this phase is reached the problem solver tries to use past experience to find a method of solution. 
3.  Carrying out the plan: the problem solver tries out the plan of solution. 
4.  Looking back: during this final phase the problem solver tries to check the result by using another method or by seeing how it all fits together.

Obviously, there are similarities between the works of Wallas and Polya. Dunker (1945), was the first to successfully attempt to study the stages of problem solving empirically. Through his work he noted several basic phenomena characteristic to problem solving. With the first phenomenon, functional solution or value, the elements of the problem must be seen in terms of their general or functional usefulness in the problem, and the general or functional solution must precede specific solutions. The second phenomenon is reformulating or re-centering, which involves successive stages of restructuring the problem with each new partial solution creating a new, more specific problem. The last two observed phenomenon involve suggestions from above and suggestions from below. In suggesting from above, the individual tries reformulating the goal to make it closer to the givens. When suggesting from below, the individual reformulates the givens so that they more closely relate to the goal.

 Newell and Simon (1972) infer several characteristics about the problem solver’s behavior. First, the behavior will be segmented and under the control of a problem solving method. Second, each method will be ordered within itself; but there will be discontinuities as activity shifts from one method to another. Since a goal may be attached by a sequence of methods, a single segment of behavior addressed entirely to a single goal may encompass several divergent sequences of behavior.

Newell and Simon (1972) also suggest that problem solving is interactive in nature, that is, it consists of repeated loops around a circuit: select a goal, select a method, evaluate the results, select another goal, and so forth. The generation of subgoals implies that behavior can also be recursive: pending goals may be held in suspension while new goals are attended to, then the previous goals may be revoked when the new goal is realized.

 A more elaborate overall organization of the problem solving process is another model proposed by Newell and Simon (1972). Initially there is a process called the input translation, during which an internal representation of the external environment is produced inside the problem solver and the problem space is selected. Once the problem is represented internally, the system responds by selecting a particular problem solving method. A method is defined as a process that bears some rational relation towards attaining a problem. Next the selected method is applied. It then begins to control the behavior, both internal and external of the problem solver. During this phase, the outcome of processes incorporated in the method or more general processes that monitor its application, may be halted. Once a method is terminated, three options are available to the problem solver: (1) another method may be attempted, (2) a different internal representation may be selected and the problem reformulated, or (3) the attempt to solve the problem may be abandoned. During its operation, a method may produce new problems, or subgoals, and the problem solver may elect to attempt one of these. The problem solver may also have the option of setting aside new subgoals and continuing instead with another branch of the original method. It should be added that the continuous influx of new information from the external environment may offer new solution possibilities or demands that cause the problem solver to interrupt his current activities to try different ones.

This model calls for an IPS that can manipulate symbols, switch method and representation, and make decisions as the scheme requires.

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