MEANING OF PROBLEM

Research has demonstrated the effects of knowledge in general on problem solving, as well as its effect on domain-specific expertise. Most of this research has focused on problem representation and can also be applied to our understanding of problem definition. One source of evidence of the effect of knowledge on problem definition and representation stems from early research on the solution of well-defined problems.

Early problem-solving research sought to describe the problem-solving process as a set of steps in higher order, isomorphic problem spaces (e.g., Newell & Simon, 1972). Such research on problem solving and the concept of “problem space” grew from Newell and Simon’s (1972) work on the General Problem Solver, or GPS, a model of human problem-solving processes. This model defined a problem as composed of a problem space, a starting state, a goal state, rules of transition, and heuristics. The problem space refers to all the possible states a problem could be in, such as during a bridge or checkers game. The starting state refers to the initial state of the problem. The goal state is the state to be reached by the system. Rules of transition refer to those functions that move the system from one state to another. Finally, heuristics are defined as rules that determine which moves are to be made in the problem space, as opposed to a random walk. Essentially, the GPS employs means-end analysis, a process that compares the starting state of a problem with the goal state and attempts to minimize the differences between the two. These components are well suited for solving well-defined problems where the space and transitions between states are unambiguous. However, the model offers no solution whatsoever for dealing with ill-defined problems. Nevertheless, the idea of a problem space has become a widely used and effective way of formalizing well-defined problems.

Recall the Tower of Hanoi and Monsters and Globes problems mentioned previously. According to the GPS, isomorphic problems should theoretically be solved similarly regardless of the way the information in the problem is represented. However, this model has been called into question by further studies of problem-solving performance on problems identified to be isomorphic to the Tower of Hanoi problem. Although these problems share with the Tower of Hanoi problem an identical problem space and 12 Pretz, Naples, and Sternberg solution structure, it is clear that the constituents chosen to represent the surface structure of each problem do have an effect (sometimes negative) on the mental representation of the problem space. One source of such evidence comes from a study that used isomorphs of the Tower of Hanoi problem involving acrobats of differing sizes (Kotovsky et al., 1985). Consider one such isomorph:

Three circus acrobats developed an amazing routine in which they jumped to and from each other’s shoulders to form human towers. The routine was quite spectacular because it was performed atop three very tall flagpoles. It was made even more impressive because the acrobats were very different in size: The large acrobat weighed 700 pounds; the medium acrobat 200 pounds; and the small acrobat, a mere 40 pounds. These differences forced them to follow these safety rules.

1. Only one acrobat may jump at a time.
2. Whenever two acrobats are standing on the same flagpole one must be standing on the shoulders of the other.
3. An acrobat may not jump if someone is standing on his shoulders.
4. A bigger acrobat may not stand on the shoulders of a smaller

At the beginning of their act, the medium acrobat was on the left, the large acrobat in the middle, and the small acrobat was on the right. At the end of the act they were arranged small, medium, and large from left to right. How did they manage to do this while obeying the safety rules? ∗

For the Reverse Acrobat problem this rule was reversed so that the smaller acrobat could not stand on the larger one; thus, the large ones had freedom of movement in that version. (Kotovsky et al., 1985, p. 262)

In the reversal of the situation where the large acrobats were standing on the smaller acrobats, participants took significantly more time to solve the problems. When an individual’s expectations about a problem are violated (i.e., smaller acrobats should stand on top of larger acrobats), it requires more time successfully to build and navigate a solution to the problem. Alternatively, performance was facilitated when the information presented was in synchrony with the individual’s knowledge, or in a form that did not lead to inadequate representations. Clement and Richard (1997) again used the Tower of Hanoi framework to examine problem solving, coming to the conclusion that the most difficult versions of the problem were those that required an individual to abandon their initial point of view in favor of a new, more appropriate one.


These findings pose a challenge to the idea that an individual’s representation of a problem is based solely on structure, as implied by the GPS model. Even when the structure of two problem spaces is identical, the solution of those problems will depend on dissimilarities in surface elements and modalities of thought (Kotovsky et al., 1985; Simon & Newell, 1971). Simply put, these results show that one does not enter a problem as a blank slate. Prior knowledge provides a tool to structure the information in the problem, allowing the individual to apply a familiar scaffold to the information, regardless of how helpful or harmful it might be. Prior knowledge mediates an individual’s ability to represent the problem in the
most efficient fashion.

There is also evidence to suggest a developmental trend in the ability to use knowledge, a skill that affects problem definition. Siegler (1978) found that older children outperform younger children on a balance-scale task because of their attention to all the relevant information about the problem. Older children realize that it is necessary to encode information about multiple dimensions of the task, but younger children do not without prompts to do so. Thus, to the extent that problem definition relies on the knowledge that multiple sources of information need to be attended toandencoded, the skill of defining problems will also increase with age.

Related Articles :