ACTIVITY 3.5 The Radiation problem

Suppose you are a doctor faced with a patient who has a malignant tumour in his stomach. It is impossible to operate on the patient, but unless the tumour is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumour. If the rays reach the tumour all at once at sufficiently high intensity, the tumour will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumour will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumour either. What type of procedure might be used to destroy the tumour with the rays, and at the same time avoid destroying the healthy tissue?

(Gick & Holyoak, 1980, pp. 307–308) If you don’t know anything about Heisenberg’s Uncertainty Principle then you won’t get the joke—you would be terminally stuck. Similarly if an insight problem requires for its solution knowledge that you do not have, then there is no way you can get out of the impasse.

A third aspect of insight problems is that, as Weisberg (1995) pointed out, you can have an “Aha!” experience on problems that would not normally be classed as insight problems. One might suddenly have an insight into how to solve an algebra problem. Furthermore, an insight may well be completely wrong. Some of these and other points made by Ohlsson will be illustrated with the Radiation problem originally used by Duncker (1945). The version here (Activity 3.5), however, is taken from Gick and Holyoak (1980).

Ohlsson’s theory is based on five problem-solving principles.

1. Reading a problem generates a mental representation of the problem’s givens (the situation described by the problem) and some solution criterion.

2. Based on this mental representation we access a set of mental operators that we think might apply. Associated with the operators is information about pre-requisites and the effects of applying them. The solver can “see” what happens in his or her mind’s eye when an operator is applied.

3. Problem solving is sequential, hence only one operator can be selected and applied at a time from those retrieved from memory. On the basis of some heuristic or plan, an operator is chosen from the ones retrieved from memory. Any operators not retrieved naturally cannot be executed.

4. Retrieving a relevant operator from the vast number in memory is not trivial. It comes about through

spreading activation (bits of information in a semantic network that are related to the current context

are activated, some more strongly than others). Activation spreads from information currently in working memory or in the goal stack. Spreading activation is an unconscious process.

5. The mental representation we form of the problem situation acts as a memory probe for relevant operators in long-term memory. The operators retrieved will have some semantic relationship with the problem situation and goal. Operators that have no such semantic relationship will not be retrieved. Notice that this aspect of the theory resembles Keane’s, in which salient features of the problem activate related operators.

When a problem is unfamiliar we may not interpret it in an optimal way. We therefore encounter an impasse when we generate a representation based on an interpretation that does not allow us to retrieve relevant operators from memory. When solvers hit an impasse, the only way out is to construct a new representation of the problem (we “restructure” the problem). The new representation generates a different spread of activation.

In the Radiation problem any interpretation that involves rays projected at full power is not going to work. Firing the rays down the oesophagus, for example, won’t work as there is no straight path that way to the tumour. Opening the stomach to clear the way for the rays is expressly forbidden. However providing hints

or rephrasing the problem statement can have an effect by acting as different memory probes, allowing different mental operators and hence different solutions to be attempted. Duncker gave two diagrams with the Radiation problem to two groups of people (Figures 3.12a and 3.12b). Figure 3.12b was more effective at generating a solution. He also tried rephrasing the problem statement causing the participants to focus on different aspects of the problem:

1. How could one prevent the rays from injuring…

2. How could one protect the healthy tissues from being injured.

Giving different diagrams and presenting the question in different ways influenced the representation subjects formed of the problem. There are, according to Ohlsson, three ways that one can change an initial representation:

Elaboration. The solver might notice features of the problem that he or she had not noticed before. This

might enrich or extend the representation of the problem. In the Mutilated Chequer Board problem, for example, the solver might notice that the domino has to cover one square of each colour, so if two squares of the same colour are missing then the Chequer Board cannot be entirely covered by dominoes. Elaboration can also come about by retrieving relevant information from long-term memory

Re-encoding. The representation of the problem may be mistaken rather than incomplete. In Duncker’s

Candle Holder problem, the solver has to re-encode the boxes from containers to platforms. Similarly the thinker has to re-encode the pliers in Maier’s Two-string problem as a pendulum weight.

Constraint relaxation. Sometimes the solver may have assumed that there were constraints placed on the

problem that were not in fact there. In the Radiation problem there is nothing to stop you using more than one ray machine or changing the intensity of the rays. To solve it you have to relax any constraints you may have imposed about the number of ray machines at the doctor’s disposal.

Completing the solution

If an impasse is successfully broken by forming a re-representation of the problem, a new set of operators becomes available. This leads to either a partial insight or full insight. A partial insight occurs when a new path through the problem can suddenly be seen. A full insight is where the answer is suddenly staring you in

Figure 3.12. Diagrams used by Duncker in describing the Radiation problem (adapted from Duncker, 1945).

the face. The latter is hard to explain within an information-processing framework. Ohlsson suggests that a solution is constructed at the moment of insight. This process is fast, hence it seems immediate (the “Aha!” experience).

How such a construction process might come about can be seen from a new view of a well-known insight problem. The Nine Dots problem is an example of a problem where the difficulty seems to be in overcoming self-imposed constraints (Figure 3.13 in Activity 3.6).

ACTIVITY 3.6

Draw four straight lines that connect all nine dots without taking the pencil from the paper. There is a hint in the text (answer on page 235).

A Gestalt explanation of the Nine Dots problem is that there is a perceptual constraint—subjects assume they have to stay within the square formed by the nine dots (Scheerer, 1963). However, there has been some argument over the exact nature of the constraints involved in this problem. Weisberg and Alba (1981) found that only 20% of their subjects produced a correct solution after being given a hint to go outside the dots. As so few people solved it despite the constraint being presumably “relaxed”, Weisberg and Alba argued that the difficulty could not be due to a perceptual constraint. Lung and Dominowski (1985) argued that the constraint was that people felt that lines had to begin and end on dots.

In a series of studies, MacGregor, Ormerod, and Chronicle (in press; Chronicle, Ormerod, & MacGregor, in press; Ormerod, Chronicle, & MacGregor, 1997) argue that the self-imposed constraints are due to the information-processing demands of the problem. For example, Chronicle et al. found that, despite being given visual cues to guide them towards constraint relaxation, participants were still unlikely to solve the problem. MacGregor et al. therefore propose a model of performance on the Nine Dots problem based on two general problem-solving heuristics—a type of means-ends analysis, and progress monitoring based on some criterion. The Nine Dots problem is an abstractly defined problem, so there is no explicit goal state— no “end”, as it were, that can readily be reached by difference reduction. Nevertheless, MacGregor et al. argue that, in abstractly defined problems, people try to use operators that are “locally rational”. Such an operator would allow a solver to reduce the difference between where they are in a problem and some “local sub-goal state”. Specifically, people try to cancel as many dots as possible in one move. This means- end strategy MacGregor et al. refer to as a “maximisation heuristic”. The second strategy is to monitor progress through a problem. One way to do this is to look one or more moves ahead. Unfortunately, capacity limitations restrict our ability to do this. Another way to monitor progress is to evaluate prospective moves based on some kind of criterion: in this case, cancel an average of 2.25 dots per move.

MacGregor et al. argue that fixating on the square or failing to consider non-dot turning points are the

results of applying the local means-end heuristic to the basic Nine Dots problem. It also explains the finding Figure 3.13.

The Nine Dots problem.

shown in Figure 3.14. In the Figure the first line provided in (a) was less likely to lead to a correct solution than the diagonal line in (b), despite the fact that the horizontal line seems to indicate a relaxation of a constraint that says you should stay within the square.

The model also predicts that experience of criterion failure in problems of this kind—possibly brought about by various manipulations—may cause the problem space to be expanded and new operators sought that may in turn lead to an insightful solution (e.g., drawing a line outside the dot array). Such a move may produce a “promising state” that may in turn lead to a re-conceptualisation of the problem.

MacGregor et al. seem to have succeeded in building a detailed process model of success and failure on abstractly defined problems such as the Nine Dots problem. A traditional insight problem has yielded to an information-processing account that can predict success and failure on the problem and its variants. Not only that but their model also shows “how insight may be achieved incrementally through experience of one or many partial solutions”.

THE RELATIONSHIP BETWEEN INSIGHT PROBLEMS AND OTHER PROBLEMS

The necessity to re-represent problems is not confined to those normally referred to as insight problems. All so-called “word” problems in algebra and physics

ACTIVITY 3.7