Problems: Definition, Types, and Evidence

Synonyms

Problem solving

Definition

A distinction can be made between “task” and “problem.” Generally, a task is a well-defined piece of work that is usually imposed by another person and may be burdensome. A problem is generally considered to be a task, a situation, or person which is difficult to deal with or control due to complexity and intransparency. In everyday language, a problem is a question proposed for solution, a matter stated for examination or proof. In each case, a problem is considered to be a matter which is difficult to solve or settle, a doubtful case, or a complex task involving doubt and uncertainty.

Theoretical Background

The nature of human problem solving has been studied by psychologists over the past hundred years. Beginning with the early experimental work of the Gestalt psychologists in Germany, and continuing through the 1960s and early 1970s, research on problem solving typically operated with relatively simple laboratory problems, such as Duncker’s famous “X-ray” problem and Ewert and Lambert’s “disk” problem (later known as “Tower of Hanoi”). Various factors account for the choice of simple problems: They have clearly defined optimal solutions, they are solvable within a relatively short time frame, researchers can trace learners’ problem-solving steps, and so on. Furthermore, it can be argued that that simple problems, such as the Tower of Hanoi problem, capture the main properties of “real world” problems, and that the cognitive processes underlying attempts to solve simple problems are representative of “real world” problems.

Thus, researchers used simple problems for reasons of convenience and thought it would be possible to generalize their findings to explain how people solve more complex problems. Perhaps the best-known and most impressive example of this line of research is the work by Newell and Simon (1972). Whereas Gestalt psychologists maintained that problem solving is based on “restructuring” a problem in order to gain “insight” into its solution, cognitive psychologists agreed on the point that problem solving should be considered as information processing.

Cognitive psychologists propose that the first thing a person does when confronted with a problem is to construct a mental representation of its relevant features. This internal representation of a problem is termed a problem space. When its construction has been successful the problem space consists of information about the initial and goal state of the problem as well as information about the operators which can be applied to solve it. Generally, a problem occurs if a person does not know how to proceed from a given state to a desired goal state. Thus, a problem is described by three components: (1) a given initial state s_α; (2) a desired final state s_ω; and (3) a barrier which hinders the solution of the problem, that is, to come from s_α to s_ω. A helpful classification of problems and the barriers involved in them has been provided by Dörner (1976), who argues that the type of a problem depends on the transparency of the goal criteria and how familiar the means of solving it are (see Table 1).

Problems: Definition, Types, and Evidence. Table 1 Classification of problems in accordance with both the clarity of objectives and certainty of resources (Dörner 1976)

In the case of a problem with an interpolation barrier, both s_α and s_ω are known – for example, if you want to travel from New York City to Sydney. The problem consists in the interpolation, that is, the effective order of the necessary transformations of states of time and space. The solution requires the correct combination or order of known operations. In the case of a problem with a synthetic barrier the set of operations aiming at the transformation from s_α and s_ω is not closed. That means that the individual knows after several problem-solving trials that the available means and operations are insufficient. A good example is the task of producing gold from straw: s_α and s_ω are known, but both the effective combination of operations and the necessary operations themselves are unknown. Therefore, the problem consists in finding the effective operations and their correct combination. Accordingly, the major task consists in synthesizing an inventory of effective operations. With reference to our example, we know that such an inventory does not exist because we can produce all sorts of things from straw but not gold. In the case of problems with a dialectic barrier the problem solver knows that a given situation s_α must be changed, but only the global criteria for the desired change are known. For example, a young lady wants to have an apartment which is more attractive than her current one, but she does not know how this can be achieved (combination of colors, style of furniture, etc.). Although it may be easy to find comparative criteria, we can assume that the subjectively satisfying solution to this problem can be found in a dialectic process. Accordingly, a first sketch will be evaluated with regard to both external consistency (e.g., concerning the requirements of the environment) and internal consistency. This sketch must probably be modified or revised and will then be evaluated again, and so on. Another example for a dialectic process of problem solving is the production of a master’s thesis.

The type of barrier evidently depends on the prior knowledge and the applicable skills of the problem solver. If, for example, an individual does not know anything about chemistry then the production of ammonia will be a problem with a synthetic barrier, whereas it will only be a problem with an interpolation barrier for a chemist. Moreover, a complex problem may contain not only one barrier but possibly all kinds of barriers. The experience of a barrier motivates problem solvers to varying degrees to grapple with a problem and leads them to test different solutions.

Problems also vary in terms of how structured they are. Jonassen (1997) classifies problems on a continuum from well structured to ill structured (see also the entry on Problem Typology). This differentiation corresponds to the distinction between well-defined and ill-defined problems, which has its origins in the specification of components of a problem space. Well-structured problems have a well-defined initial state, a known goal state or solution, and a constrained set of known procedures for solving a class of problems. In other words, they require the application of a limited and known number of concepts, rules, and principles (e.g., means-ends analysis) studied within a restricted domain. In contrast, the solutions to ill-structured problems are neither predictable nor convergent because they often possess aspects that are unknown. Additionally, they possess multiple solutions or solution methods or often no solutions at all.

Problems vary in complexity. The complexity of a problem is determined by the number of issues, functions, or variables it involves; the degree of connectivity among these variables; the type of functional relationships between these properties; and the stability of the properties of the problem over time (cf. Funke 1992). Simple problems, like textbook problems, are composed of few variables, while ill-structured problems may include many factors or variables that may interact in unpredictable ways. For example, international political problems are complex and unpredictable. Finally, problems vary in their stability or dynamicity. More complex problems tend to be dynamic; that is, the task environment and its factors change over time. When the conditions of a problem change, people must continuously adapt their understanding of the problem while searching for new solutions, because the old solutions may no longer be viable. For example, investing in the stock market is often difficult because market conditions (such as demand, interest rates, or confidence) tend to change, often dramatically, over short periods of time. Static problems are those in which the factors are stable over time. Clearly, ill-structured problems tend to be more dynamic, whereas well-structured problems tend to be fairly stable.

Important Scientific Research and Open Questions

Although cognitive psychologists on both sides of the Atlantic generally agree on the point that problem solving should be considered as information processing, different lines of research have emerged in North America and in Europe. Initiated by the work of Herbert Simon, researchers in North America began to investigate problem solving separately in different natural knowledge domains – such as physics, writing, or chess playing – thus relinquishing their attempts to extract a unique and comprehensive theory of problem solving. The North American line focused on the investigation of problem solving within specific domains such as reading, calculating, political decision making, and personal problem solving (cf. Funke and Frensch 1995). Newell, Shaw, and Simon (1959) introduced the General Problem Solver (GPS), which simulates human problem-solving behavior. This computer program was proposed to provide an essential set of processes to solve a variety of different problems. Accordingly, the GPS solves distinctive formally described problems or tasks by itself and with a specific analogy to human problem-solving performance, which presupposes the sequential transformation of knowledge structures. During a problem-solving process, mental operators generate the shift from an initial knowledge state to the desired final state. An example of such serial information processing is the aforementioned Tower of Hanoi problem. Although the GPS was expected to be a general problem solver, it clearly was limited to “well-defined” problems, such as word puzzles, chess, or the proving of theorems in logic. Nevertheless, the GPS provided a basis for a wide range of common problems in different domains. Whereas the GPS was concerned with solving any given problem in any domain, expert systems (ES) were domain specific to a high degree. ES were developed to aid in decision making and to present results in a well-founded manner to the expert who makes the final decision. But human decision making is not based on individual constituents, even if they are well founded. Due to a lack of systematic empirical research, the effectiveness of ES could not be clarified.

While the North American line of research focused successfully on the implementation of problem solving in computer systems, the European line focused on the simulation of complex environments to empower human problem solving and decision making within complex domains. Two approaches surfaced, one initiated by Donald Broadbent (1977) in the United Kingdom and the other by Dietrich Dörner in Germany (Dörner 1976). The two approaches have a common emphasis on relatively complex, semantically rich, computerized laboratory tasks constructed to resemble real-life problems. The approaches differ somewhat, however, in their theoretical goals and methodology. The tradition initiated by Broadbent emphasizes the distinction between cognitive problem-solving processes that operate under awareness versus outside of awareness and typically employs mathematically well-defined computerized systems. The tradition initiated by Dörner, on the other hand, is interested in the interplay of the cognitive, motivational, and social components of problem solving and utilizes very complex computerized scenarios (e.g., the Lohhausen project by Dörner et al. 1983).

Both approaches focused on laboratory problems with complex structures that were computerized and analogous to real-life situations. Broadbent’s experimental research emphasized the distinction between cognitive problem-solving processes in explicit and implicit modes (Berry and Broadbent 1995). These experimental approaches helped to categorize expert problem solving further, thus strengthening the understanding that there is nothing like one single problem-solving skill or deterministic algorithm which accurately describes human problem solving and that each of the categories comes with different sets of knowledge and skills. On the other hand, Dörner developed complex computer environments with more than 2,000 variables (Dörner et al. 1983). Several experimental studies with the Lohhausen scenario revealed typical errors which occur when one works with complex systems (Dörner 1989). However, the computational models were no longer used to simulate (or imitate) the problem-solving process but to stimulate them. Instead of trying to compute the problem-solving process (as in GPS) or support the decisions, Dörner developed research instruments for a better understanding of problem solving and at the same time provided environments to train problem solving skills. On the computational level, the environments of course still have to be deterministic in order to be implemented. But due to the many variables it was not possible for subjects to understand all of their effects. Having models which are fully available to the researcher (or to the instructor) and yet unable to be disclosed to the learners led to a better understanding of the problem-solving process. These insights still provide us with opportunities to train systematically human complex problem solving today.

Cross-References

Complex Problem Solving

Problem Typology

Problem-Based Learning