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Problem solving

A great deal of the art of problem solving is to understand the kind of question that is posed and the kind of answer that is demanded. It is for this reason that psychologists prefer problems with unique solutions, and that they try to ensure that individuals understand what they have to solve.

There are several theoretical points of view about problem solving, but none is really complete because each tends to be restricted to different problem domains, and there is little definitive agreement about what constitutes a problem. The Gestalt theorists (e.g. Wertheimer 1969) believed that a problem occurs because of the way in which a situation is initially perceived, and that its solution emerges suddenly from reorganizing it in such a way that its real structure becomes apparent. On the other hand, many contemporary psychologists have been impressed by ideas borrowed from research on artificial intelligence. They conceive of the mind as analogous to a computer program which operates in discrete steps ('information processing') to reduce the difference between existing states and 'subgoals'. Their pioneering efforts were devoted mainly to a small number of computable games and puzzles, and they were not deterred by the fact that early computer programs played poor chess. See Newell and Simon (1972), and, for criticism, Dreyfus (1972) and Weizenbaum (1976); and computer chess for recent improvements. For an account of the fundamental difficulties of computer chess, see Hartson and Wason (1983: ch. 6). My own interest has been to devise problems in which the initial response may ensnare the capacity to see the point.

The difficulty of writing about problem solving is that one may either insult the reader's intelligence, or create states of frustration. Hence I shall not report the solution to my first problem. Instead I shall take the reader by the hand (if he will pardon the condescension) and ask him to solve with me two related, simplified problems. These may alter the way in which he conceived the original problem. Of course, he may find the first problem trivial, but then I hope his boredom will be alleviated by the knowledge that others find it rather puzzling. I could say a lot more at this point, but that would be to lay all my cards on the table. Consider the first problem.Problem 1I formulated this problem in 1966, and the present version was devised for the 1977 Science Museum Explorations Exhibition in London. Earlier versions contained some confusing features. The problem is generally known as 'the selection task'.

You are shown a panel of four cards, A, B, C, D (Fig. 1), together with the following instructions:

Which of the hidden parts of these cards do you need to see in order to answer the following question decisively? For these cards is it true that if there is a circle on the left there is a circle on the right? You have only one opportunity to make this decision; you must not assume that you can inspect cards one at a time. Name those cards which it is absolutely necessary to see.

Please record your solution, and then consider the next problem.Problem 2This problem is based on Johnson-Laird and Wason (1970). A more recent, intensive investigation of the issues may be found in Wason and Green (1984). In front of you are two boxes, one labelled 'white' and the other labelled 'black' (Fig. 2). There are fifteen white shapes in the white box and fifteen black shapes in the black box, and the only shapes are triangles and circles. Your problem is to prove the following sentence true, as economically as possible, by requesting to inspect shapes from either box: If they are triangles, then they are black.

The students who were tested in this experiment tended to ask first of all for a black shape — they were handed a black triangle. The task turned out to be fairly easy; on average only six black shapes were requested. Of course, when individuals asked for a white shape they were always handed a white circle. Somebody in my class said recently: 'The best strategy is to alternate your choices between the two boxes.' This would have been a perverse strategy, especially if one were to apply it consistently by exhausting the contents of both boxes. In fact, insight came rapidly, and all the individuals exhausted the supply of fifteen white circles, and requested no more than nine black shapes. Moreover, they tended to do so with a broad grin, as if they had penetrated a secret, or seen the point of a joke. In order to prove the truth of the sentence 'If they are triangles, then they are black', it is merely necessary to establish the absence of a white triangle. The contents of the black box are gratuitous.

What is the connection between problems 1 and 2? In the first place, problem 2 is concerned only with half the amount of information in problem 1. In problem 2 no decision has to be made about 'triangles' and 'circles' which corresponds to the presence and absence of a 'circle on the left'. Secondly, problem 1 involves a single and ultimate decision for its solution, but problem 2 involves a series of decisions so that an earlier error can be corrected. Thirdly, problem 2 involves a number of concrete objects rather than the consideration of symbols positioned on cards.

Are you still satisfied with your solution to problem 1?Problem 3This problem is based on Wason and Shapiro (1971). There are four cards on the table in front of you, showing (respectively) 'Manchester', 'Leeds', 'Train', 'Car' (Fig. 3). The students who were tested in this experiment had first of all examined a larger set of cards (from which these four had been selected), each of which had on one side a town (e.g. Chicago), and on the other side a mode of transport (e.g. aeroplane). They had been asked to satisfy themselves that this condition obtained on every card. The four cards were then placed on the table, and the individuals were instructed to imagine that each represented a journey made by the experimenter. They were then presented with the experimenter's claim about her journeys: Every time I go to Manchester I travel by train.

The problem is to state which cards need to be turned over in order to determine whether this claim is true or false. The solution is 'Manchester' and 'Car' because only 'Manchester' associated with a transport other than 'Train', or 'Car' associated with 'Manchester', would disprove the claim. This thematic problem proved much easier than a standard, abstract version which was structurally equivalent to problem 1. However, later attempts to replicate this effect have not been at all clear. For a general discussion, see Griggs (1983).

What is the connection between problems 1 and 3? In both a single decision has to be made about four cards, so in this sense they are both unlike problem 2. But in problem 3 the cards are not simply cards. They represent four different journeys and their two sides are connected intrinsically in this respect. This means that the material of the problem is intimately related to experience, and the solution can be guided by it. For a more detailed account of these, and similar, experiments see Wason and Johnson-Laird (1972).

Still happy about the solution to problem 1? If it did cause any difficulties, it seems fairly likely that any error will now have been corrected because that particular problem has been broken down into two much simpler ones, each of which eases the original burden of thought. But suppose, just for the sake of argument, that the solution to problem 1 is still wrong — for instance, it might be cards A and C. In the original experiments based on it I devised therapies which induced contradictions between the first attempted solution and a subsequent evaluation of the material. The card, corresponding to A, which everybody had (rightly) selected, would have been revealed thus:

'What does this tell you about the answer to the question?' ('For these cards is it true that if there is a circle on the left there is a circle on the right?') Everybody said that this told them the answer is 'yes'.

Then the card, corresponding to D, which nearly everyone had (wrongly) omitted, would have been revealed thus:

Conflict: a card which had been chosen allowed the answer 'yes', but now a card which had been ignored indubitably forces the answer 'no'. The majority of individuals remained unmoved — they refused at this point to incorporate D into their solution. When prompted, they made remarks like, 'It's got nothing to do with it', and 'It doesn't matter'. All the available evidence is present, but the correction tends not to be made.

We went on informally to discuss the potential consequences of B and C which were never fully revealed. B had nearly always been omitted (rightly) and C selected (wrongly).

'Can you say anything about the answer from this card [B]?':

'It's got nothing to do with it because there's no circle [on the left].'

'Can you say anything about the answer from this card [C]?':

'There has to be a circle under it for the answer to be "yes".'

'What if there is no circle?'

'Then the answer would be "no".'

What could be the selfsame card has a different meaning according to whether it had been selected initially. The individual is confronted with the possibility of both cards being like this:

But only when this contingency derives from C is it assumed (wrongly) to be informative. It is evidently the individual's intention to select a card which confers meaning on it.

Reality, for the individuals who made these kinds of error, is determined by their own thought. That is not, perhaps, surprising. What is very surprising is that this reality is so recalcitrant to correction. See Wason (1977) for further discussion. For a recent and much more comprehensive account of the issues raised by this problem, see Wason (1983). It is as if the attention mobilized in the initial decision is divided from their subsequent attention to facts, or possibilities. The solution, cards A and D, is systematically evaded in ways which are not yet properly understood. There is even the finding that the origin of the difficulty might arise through differences in the functioning of the two hemispheres of the normal brain. When corrective feedback is induced in the left ('analytic') hemisphere it is more effective than when it is induced in the right ('synthetic') hemisphere.

Our basic paradigm (problem 1) has the enormous advantage of being artificial and novel; in these studies we are not interested in everyday thought, but in the kind of thinking which occurs when there is minimal meaning in the things around us. On a much smaller scale, what do our students' remarks remind us of in real life? They are like saying 'Of course, the earth is flat', 'Of course, we are descended from Adam and Eve', 'Of course, space has nothing to do with time'. The old ways of seeing things now look like absurd prejudices, but our highly intelligent student volunteers display analogous miniature prejudices when their premature conclusions are challenged by the facts. As Kuhn (1962) has shown, the old paradigms do not yield in the face of a few counter-examples. In the same way, our volunteers do not often accommodate their thought to new observations, even those governed by logical necessity, in a deceptive problem situation. They will frequently deny the facts, or contradict themselves, rather than shift their frame of reference.

Other treatments and interpretations of problem solving could have been cited. For instance, most problems studied by psychologists create a sense of perplexity rather than a specious answer. But the present interpretation, in terms of the development of dogma and its resistance to truth, reveals the interest and excitement generated by research in this area.

Fig. 1. Problem 1: the Science Museum Problem.

Fig. 2. Problem 2.

Fig. 3. Problem 3.

(Published 1987)

— Peter Cathcart Wason

Bibliography

Dreyfus, H. L. (1972). What Computers Can't Do.
Griggs, R. A. (1983). 'The role of problem content in the selection task and in the thog problem'. In Evans, J. St B. T. (ed.), Thinking and Reasoning: Psychological Approaches.
Hartston, W. R., and Wason, P. C. (1983). The Psychology of Chess.
Johnson-Laird, P. N., and Wason, P. C. (1970). 'Insight into a logical relation'. Quarterly Journal of Experimental Psychology, 22.
Kuhn, T. S. (1962). The Structure of Scientific Revolutions.
Newell, A., and Simon, H. A. (1972). Human Problem Solving.
Wason, P. C. (1977). 'Self-contradictions'. In Johnson-Laird, P. N., and Wason, P. C. (eds.), Thinking: Readings in Cognitive Science.
— — (1983). 'Realism and rationality in the selection task'. In Evans, J. St B. T. (ed.), Thinking and Reasoning: Psychological Approaches.
— — and Green, D. W. (1984). 'Reasoning and mental representation'. Quarterly Journal of Experimental Psychology, 36.
— — and Johnson-Laird, P. N. (1972). Psychology of Reasoning: Structure and Content.
— — and Shapiro, D. (1971). 'Natural and contrived experience in a reasoning problem'. Quarterly Journal of Experimental Psychology, 23.
Weizenbaum, J. (1976). Computer Power and Human Reason.
Wertheimer, M. (1969). Productive Thinking.