7

Insight

Like intelligence and consciousness, we can usually recognise creativity when we encounter it but the concept is very hard to pin down. It is often associated with insight and, to a lesser extent, analogy. Creative problem solving (CPS) is unlike analytical problem solving in a number of ways. First, it usually involves finding a solution to a problem rather than the solution, so the same issue might produce a range of possible solutions some of which might never have been generated before. Second, it often involves discovering that there is a problem to be solved in the first place. Problem finding is often a first step in creative problem solving. Third, sometimes one is faced with a problem for which there is no known pre-existing structure so you are obliged to try to define the problem in your own way and construct a solution. Fourth, a creative insight would seem to involve thinking of something that has never been thought of before when all we currently know is what we have learned in the past – something previously unknown and entirely novel has emerged from what was previously known. How does that happen?

When we talk of “thinking of something” or “defining a problem” we are referring to how a situation or idea can be mentally represented. When we talk about insight or CPS we are referring to a change in representation such that a previous way of representing something is replaced by a new way of thinking of it. If this re-representation, reformulation and restructuring happens apparently suddenly such that the solution appears before us, then this gives rise to what has been called the “Aha!” or “Eureka!” experience that carries with it an emotional tag. Alternatively, such a re-representation might indicate a new promising search path rather than the solution itself. In both cases we have an example of an insight and this is where we will start.

Insight problems

An insight tends to involve the sudden realisation of a solution or of a straightforward path to a solution. In the middle of a complex mathematical problem a mathematician might suddenly see how to get to the solution. A mechanic trying to fix a problem in a car might suddenly realise what the problem might be. Someone trying to debug computer code might suddenly arrive at a solution after several vain attempts and reaching a mental block – an impasse. A store manager might suddenly think of a simple way of keeping control of stock. However these are not what one would call typical insight problems, so one question to be addressed is what, if anything, is the difference between an insight problem and, say, a well-defined analytical problem.

However, even if there are differences in problem types that doesn’t explain what goes on in the mind of the solver. As with any other problem there is an interaction between the solver and the problem which, in this case, produces an “insight”. In many everyday problems one can search through the problem space until a solution is reached. If you fail to reach a solution then you need to find another path through the space. With insight problems, failure to reach a solution can lead to an impasse because the initial representation you form does not contain the goal. So rather than finding another path through the search space, the solver has to find a different representation of the problem – a different problem space.

Insight problem solving is the Cinderella of problem solving research. While much has been discovered about the cognitive and neurological processes involved in memory, reasoning, attention and perception, and while the studies of judgement, decision making and human problem solving have won Nobel prizes, the study of insight is languishing in the kitchen with only a slim prospect of being invited to the cognitive ball. This is unfortunate as the study of insight as an important facet of creativity and is important if we want to understand its role in scientific discovery as well as literary and artistic accomplishments (Sternberg, 1985).

Gestalt accounts of problem solving

It was the question of how we represent problems and how we might re-represent them that interested the Gestalt psychologists. Understanding how a problem might be solved requires an insight into the problem structure. For Gestalt psychologists, the relationships between elements in the visual field gave rise to “wholes”. In Figure 7.1 the viewer sees a square rather than a series of black dots with chunks cut out, and in Figure 7.2 the “Necker Cube” can be seen in two different ways. Shifting from one view to another involves restructuring the figure.

As with their study of perceptual phenomena, the Gestalt school provided a description of a number of problem solving phenomena and provided a number of useful labels for them. Gestalt psychologists laid great stress on how we “structure” problems and on why we often either fail to solve them or fail to see a simple solution. When we have difficulties solving a problem, insight into its solution can come about by restructuring the problem analogously to perceptually restructuring and ambiguous figure such as Figure 7.2. Nowadays we would talk of having the wrong, or at least an unhelpful, initial representation of a problem requiring a re-representation of it in order to find a solution. Activity 7.1 gives an example of the types of mathematical and geometry problems that Gestalt psychologists were interested in.

One way to approach the question in the activity is to try to divide the number by 9 to see what happens. This would be a perfectly natural approach to what appears to be a division problem. Our past experience of division problems predisposes us to attempt it by applying the procedure we have learned for dealing with division problems. In this case it would be somewhat time-consuming, not to say tedious. However, sometimes a learned procedure is not the easiest way of solving the problem. In Activity 7.1 notice what happens when you subtract 9 from the number. Now can you say whether it is divisible by 9?

It is always possible you tried to solve the problem by the “method of extreme cases”. If you did so you will have noticed that the simple case of 18 is divisible by 9, 108 is divisible by 9, 1008 is divisible by 9, and so on. You may have boldly extrapolated to the number in the Activity and assumed it was divisible by 9 as well. You may even have worked out why.

Another example of this kind of restructuring can often be found on the walls of mathematics classrooms in secondary schools. You sometimes find a poster there describing how 6-year-old Karl Gauss, who later became a prominent mathematician (the gauss, a unit of magnetic flux, is named after him), solved a tedious arithmetic problem very quickly by reconstruing the problem. His teacher, thinking to give himself a few minutes’ peace, had asked the class to add up the numbers 1 + 2 + 3 + 4 and so on up to 100. Hardly had the class begun laboriously to add up all the numbers when Gauss put his hand up with the answer. How had he done it so fast?

Gauss had seen the problem structured in a way similar to Figure 7.3. The figure looks a bit like a rectangle with a side 100 units long cut diagonally in half. All Gauss did was to complete the rectangle by imagining Figure 7.3 duplicated, flipped over and added to itself as in Figure 7.4. You end up with a rectangle 100 × 101 giving an area of 10,100. Adding the series 1 + 2 + 3 + 4 up to 100 is therefore the same as taking half of 10,100; that is, 5,050 (see also Gilhooly, 1996). An alternative representation is an algebraic one: $\frac{n(n+1)}{2}$.

The young Gauss was able to see relationships between the parts of the problem that the other children in the class were unable to see. In other words, he had understood the underlying structure of the problem. Wertheimer (1945) referred to the kind of thinking exhibited by Gauss as productive thinking. As mentioned in Chapter 1 this kind of thinking can be contrasted with reproductive thinking, where the solver attempts to solve the problem according to previously learned methods – in this case by simply adding up the numbers 1 to 100. Reproductive thinking of this latter kind is structurally blind. There need be no real understanding of the underlying structure of the problem.

Set effects

As Wertheimer’s analysis showed, one of Gestalt psychology’s achievements in the study of problem solving was to point out the difficulties people often have in solving problems because of the inappropriate use of prior knowledge. Past experience can sometimes make us psychologically set in our ways. Applying a learned rule or procedure for doing something when there is a simpler way of doing it is therefore called a set effect. The Gestalt term for using a learned method for solving problems, where a simpler method might be quicker and more appropriate, is Einstellung, which can be regarded as “the blinding effects of habit” (Luchins & Luchins, 1950). Learned procedures for doing things are extremely useful most of the time. We wouldn’t get on too well if we didn’t apply the rules we had learned in the past to new occurrences of the same situation. Just how many novel ways are there of making a cup of tea or of riding a motorbike? Do we need to think up novel ways of doing them? Nevertheless, Luchins (1942) argued that a mechanised procedure for solving a particular problem type ceases to be a tool “when … instead of the individual mastering the habit, the habit masters the individual” (p. 93).

Another type of mental set is functional fixedness (or functional fixity), where we are unable to see how we might use an object as a tool to help us solve a problem because that is not the normal function of the object. To get an idea of what functional fixedness means, have a go at Activity 7.2 before reading on.

The point here is that the objects you have found in your pocket all have specific functions, none of which has anything to do with removing corks from bottles. Functional fixedness is being unable to forget for a moment the normal function of an object to be able to use it for a totally novel purpose. In doing Activity 7.2 you may have realised that you can force the cork into the bottle using the pen. You may even have done so in the past. The cork, however, is still in the bottle and tends to get in the way when you are pouring the wine. If you tie a largish knot in the string and push it into the bottle below the cork, you can then pull on the string which causes the knot to pull the cork out of the bottle.

Restructuring, Einstellung and functional fixedness can be illustrated by classic experiments conducted by Maier (1931) and Duncker (1945) as well as the Luchins and Luchins (1959) study described in Information Box 3.1 in Chapter 3. Information Box 7.1 describes Maier’s account of restructuring in an insight problem and Information Box 7.2 describes studies by Duncker into functional fixedness.

While the Gestalt psychologists provided descriptions of the situations in which insight occurred or failed to occur, as well as useful methods for examining problem solving (such as the use of verbal protocols), they were less precise about the processes underlying insight. Explanations such as “short-circuiting” normal problem solving processes don’t tell us how or why such a short circuit takes place. More recently, therefore, information processing explanations have been put forward to explain insightful problem solving.

Information processing approaches to insight

When one encounters a problem, elements of the problem within the task environment trigger retrieval of elements from long-term memory that appear relevant (an unconscious process). These generate an initial representation of the problem that acts to constrain the search through the problem space. If this representation happens to be inappropriate then the solver is likely to reach a dead end – an impasse. Getting round the impasse involves finding a different way of representing the problem (sometimes known as lateral thinking), and if a new representation allows the solver to get to the goal state immediately or very quickly then this constitutes an insight which involves “seeing a problem in a new light, often without awareness of how that new light was switched on” Jung-Beeman et al. (2004, p. 14).

Another aspect of insight problems is that people are usually able to solve them but they don’t realise it. The answer is often obvious once they hear it. Understanding how to solve an insight problem is therefore a bit like getting a joke (Koestler, 1970). Jokes often rely on the listener generating a typical but, in this case wrong, interpretation of a situation (Koestler referred to a “matrix”). One could turn the statement “a man walked into a bar and fell on the floor” into an insight problem by adding the question “Why?” The answer (the punchline) is that it was an iron bar. According to Koestler, a joke, and by extension an insight, occurs when two unrelated matrices come together. The point here is that you could have solved the problem (got the joke) if you had accessed the relevant meaning of bar (a different matrix). It is not that you didn’t know that bar had two meanings; it’s just that you didn’t realise which one was relevant. Thus, insight often occurs in the context of an impasse, which is unmerited in the sense that the thinker is, in fact, competent to solve the problem (Ohlsson, 1992, p. 4). The corollary of this is that if you do not have a particular competence then you cannot have an insight. You are terminally stuck as Ohlsson puts it: the impasse is therefore “warranted” (Ohlsson, 2011). Once again a joke can illustrate this point:

• QUESTION: How many Heisenbergs does it take to change a light bulb?
• ANSWER: If you knew that you wouldn’t know where the light bulb was.

If you don’t know anything about Heisenberg’s Uncertainty Principle, then you won’t get the joke. (Each domain of knowledge tends to have its own in-jokes that only those familiar with the domain are likely to understand.) 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, according to Weisberg (1995), people sometimes solve them without any “Aha!” experience (see later), while on the other hand people sometimes have a sudden mental restructuring during what are normally regarded as non-insight problems. Furthermore, an insight may turn out to be completely wrong.

A fourth aspect of insight problems is that there is no agreed upon way of categorising problems as insight problems, nor is there an agreed way of studying the phenomenon. For example, Evans (2005) discusses the history of insight versus logical reasoning in the Wason decision task (Wason, 1960) where people are shown four cards with a letter or number (e.g., E T 4 7) written on them and given a rule: if there is a vowel on one side of the card, then there is an even number on the other side of the card. They are then asked to decide which card or cards to turn over to determine if the rule is true or false.) It can be argued that the few people who solve the problem must be able to see the underlying structure of the problem in the Gestalt sense and hence have an insight into the nature of the task, although solvers don’t always show the “Aha!” response. Most researchers will, however, agree with the list of general characteristics presented by Batchelder and Alexander (2012):

Another source of disagreement is whether an insight has to follow an impasse. Kounios and Beeman (2014), for example, do not regard insight as necessarily following an impasse. In their view insights can come about when one is not solving a problem, during analytical problem solving when no impasse is reached, or spontaneously, when an idea just occurs to someone more or less out of the blue. In a broad definition, they argue, an insight is any “deep realization, whether sudden or not” (p. 73).

Classifying insight problems

Gilhooly and Murphy (2005) attempted to find out whether insight problems and non-insight problems were indeed distinct classes with both types chosen mainly on the basis of what had been used before (e.g., Weisberg, 1995). They used a set of problems from what Batchelder and Alexander (2012) refer to as a “sizeable folklore of insight problems” (p. 59). The use of the word “folklore” indicates the lack of any kind of scientific rigour in categorising insight problems, a variety of which can be found in Activity 7.3. Using cluster analysis, Gilhooly and Murphy found that different problem types clustered together and tended to form clusters that were either exclusively or predominantly insight or exclusively non-insight, providing support for the view that the two problem types formed distinct general categories. That said, many of the insight problems are distinct from each other in terms of how they can be solved. What is needed to solve the Nine-Dot problem is entirely different from the solution to the matchsticks problems or Duncker’s Candle Holder problem or indeed other problems listed in previous chapters.

Schooler, Ohlsson and Brooks (1993) examined the role of verbalisation on solution success in both insight and non-insight problems. They found that concurrent verbalisations had no effect on participants’ ability to solve non-insight problems but appeared to impede the solutions to insight problems. They argued that this result was due to the fact that insight involves unconscious and therefore unreportable processes that are “overshadowed” by the concurrent verbalisations.

Fleck and Weisberg (2004) however, found no effect of verbalisations on participants’ solutions to Duncker’s Candle Holder problem. They did find evidence of impasses and restructuring, but not all restructuring was due to an impasse. One reason for the difference in outcomes, they argue, may be due to the training given to the participants in the Schooler at al. study; as only one training session was delivered, it is therefore possible that participants occasionally tried to go beyond simply verbalising what is currently in working memory (Ericsson and Simon’s Type I and Type II categories) and tried explaining or justifying. Alternatively, it may be that that different types of insight problems are differentially affected by verbalising.

Gilhooly, Fioratou and Henretty (2010) argued that some of the items used in Schooler et al.’s study were verbal and some were spatial. In two of the experiments the non-insight problems were “predominantly verbal in character while at least two of the three insight problems used […] could be regarded as having a large spatial element” (p. 83). The disruption caused by the concurrent verbalisations could be due to the nature of the spatial coding of the problems requiring a switch to a verbal coding as a result of having to verbalise what they are thinking.

Cunningham, MacGregor, Gibb and Haar (2009) suggest:

Indeed, Chu and MacGregor (2011) suggest that studying insight could be put on a more consistent basis if we use sub-categories of some insight problems. For example, the matchstick arithmetic problems can be varied in terms of the number of transformations that need to be done to find a solution and whether the individual chunks can be readily broken down into smaller elements. In the example IV = III − I, solvers are more likely to try to move one of the upright matches than the V or = since these are more readily seen as a single chunk. Chu and MacGregor (2011) argue that three such categories “promise to provide essentially unbounded sources of relatively homogenous problem [sic]. These include Matchstick Arithmetic explained above (Knoblich, Ohlsson, Haider, & Rhenius, 1999), Compound Remote Associates (CRAs) (Bowden & Jung-Beeman, 2003), and Rebus Puzzles (MacGregor & Cunningham, 2008)” (p. 126). CRAs are groups of usually three words that can be linked by providing a third. Thus age – mile – sand can all be linked by adding the word “stone” to give stone age, milestone, sandstone. Rebus puzzles can vary greatly, so again the experimenter can manipulate the level of complexity by increasing the number of implicit assumptions involved. “You just me” gives you “just between you and me” – a relatively simple rebus; GGES GESG ESGG GSEG involves four anagrams and a word that refers to the fact that the letters are mixed up, hence “scrambled eggs”.

Batchelder and Alexander suggest three other problem types that can be manipulated in the same way: Bongard problems, series completion problems, and self-reference problems. Examples are given in Activity 7.3.

Are the processes involved in Batchelder and Alexander’s list special to insight, or are they part of everyday problem solving – the “business-as-usual” approach? Is insight a “sudden” process below awareness, or do they show the same gradual, incremental and heuristic processes as are found in well-defined or analytical problems?

Insight as something special

In 1981 Weisberg and Alba claimed that there was no evidence for insight (but see later), and therefore there was no reason to suppose that insight problem solving involved different processes from any other type of problem solving. They argued (1981, 1982) that restructuring in Gestalt insight problems comes about through the same type of search through the problem space and a search through memory, as described by Newell and Simon (1972). “Restructuring of a problem comes about as a result of further searches of memory, cued by new information accrued as the subject works through the problem. This is in contrast to the Gestalt view that restructuration is spontaneous” (Weisberg & Alba, 1982, p. 328).

Metcalfe (1986), on the other hand, argued that if the same memorial processes were at work in insightful as in non-insightful problem solving, then one should find that the metacognitive processes would also be the same. Metacognition (also sometimes referred to as “metamemory” or “metaknowledge”) means knowing what you know. If you have played Trivial Pursuit you may well have experienced the tip-of-the-tongue phenomenon where you are sure you know the answer but just can’t quite get it out. It has been shown that people are quite good at estimating how likely they are to find an answer to a question that produces a tip-of-the-tongue state given time or a hint such as the first letter (Cohen, 1996; Lachman, Lachman, & Thronesberry, 1979). In fact you can produce a gradient of feeling-of-knowing (FOK) from “definitely do not know” to “could recall the answer if given more hints and more time”. If, therefore, insight problems involved a search through memory using the current state of the problem as a cue one might reasonably expect that one could estimate one’s FOK (in this case feeling that you know how close you are getting to an answer) just as readily for insight problems as for trivia knowledge or even algebra problems.

Metcalfe (1986) found that there was a positive correlation between subjects’ estimates of FOK for trivia questions but a zero correlation for insight problems. Furthermore, as subjects solved algebra problems, deductive reasoning problems or the Tower of Hanoi problem they were able to produce warmth ratings as they got closer to a solution – the closer they were to a solution the warmer they were (the more confident they were that they were close to a solution) (Metcalfe & Wiebe, 1987). Indeed, these types of problems showed gradual increases in the subjects’ warmth ratings from 1 (cold) to 7 (very warm) every 15 seconds. For insight problems, on the other hand, there were hardly any increases in feelings of warmth until immediately before a solution was found. Metcalfe and Wiebe argued that their study shows an empirically demonstrable distinction between problems that people thought were insight problems and those that are generally considered not to require insight such as algebra or multistep problems, and that such warmth protocols might be used to diagnose problem types. They concluded that their findings “indicate in a straightforward manner that insight problems are, at least subjectively, solved by a sudden flash of illumination; non-insight problems are solved more incrementally” (p. 243).

Insight as “business as usual”

Despite Metcalfe’s conclusion that insight and non-insight problems involve different processes, there have been several attempts at explaining Gestalt insight problems in classical information processing terms. Most of the accounts have a lot in common since they appeal to a number of cognitive processes usually involved in other forms of problem solving, such as retrieval of information from long-term memory, search through a problem space, search for relevant operators from memory, problem understanding, and so on.

Kaplan and Simon’s account of insight

The title of Kaplan and Simon’s (1990) paper In Search of Insight is intended to show that search is part of insightful problem solving. The difference is that, rather than developing a representation of a problem and then searching through that representation (the problem space), the solver, having reached an impasse, has to search for the appropriate representation among the space of potential problem spaces. Kaplan and Simon use the metaphor of searching for a diamond in a darkened room to illustrate insight. At first you might grope blindly on your hands and knees as you search for the diamond. After a while, though, you may feel that this is getting you nowhere fast. You therefore look for a different way of trying to find the diamond and decide to start searching for a light switch instead. If you find one and turn it on the diamond can be seen almost at once. Now try Activity 7.4.

Kaplan and Simon used variants of the Mutilated Chequerboard to examine the process of search in insightful problem solving. If you read the statement of the problem in Activity 7.5, the most obvious apparent solution method is to try covering the squares with dominoes. This method of searching for a solution is equivalent to groping in the dark for the diamond. The reason why this strategy is likely to fail is that the search space is too big. Kaplan and Simon constructed a computer program that used “covering heuristics” and found that the program required 758,148 domino placements – probably not something human beings are normally prepared to attempt (although one graduate student spent over 18 hours trying to solve it this way and failed). Kaplan and Simon argued that the problem was hard because there were not enough constraints on the problem – there are too many possible paths to search. The only way to find a solution is to stop searching through the initial representation (the problem space of covering squares with dominoes) and search for a representation that provides more constraints. Activity 7.6 provides an analogical solution.

A problem constraint allows you to prune the search tree. Figures 7.14 and 7.15 make this idea a little clearer.

Figure 7.16 depicts this switch from a search through a single representation to a search through a meta-representation (the problem space of problem spaces). Figure 7.16 also illustrates some of the problem spaces used by subjects and identified from think-aloud protocols. All subjects at first searched for a solution by covering squares with dominoes. When this failed to work they eventually switched to another representation of the problem. Some attempted to search for a mathematical solution; some attempted to manipulate the board by, for example, dividing it into separate areas; some sought an analogy or another similar problem. Eventually all tried to find a solution based on parity (that is, a solution based on the fact that the dominoes had to cover two squares of different colours).

To sum up, Kaplan and Simon argued that insight is not a special type of problem solving phenomenon, but involves the same processes as other forms of problem solving. “The same processes that are ordinarily used to search within problem space can be used to search for a problem space (representation)” (Kaplan & Simon, 1990, p. 376). Subjects’ difficulty in solving the problem was mainly due to an inappropriate and under-constrained representation.

Representational change theory (redistribution theory)

Ohlsson (e.g., 1984, 1992, 2011) developed a theory of insight (usually referred to as Representational Change Theory (RCT) but recently renamed as “Redistribution Theory”). This theory took the views of the Gestalt psychologists that insight was a special process and tried to explain them in terms of information processing theory, particularly the theories of Newell and Simon (1972). He took the Gestalt notion of restructuring and combined it with Newell and Simon’s view of problem solving as a search through a problem space. It is not therefore the case that Ohlsson views insight and restructuring as entirely different from everyday problem solving, but rather that some assumptions must be made to explain impasses and subsequent restructuring.

The initial representation of the problem is based on previous knowledge and experience. As in the Gestalt tradition, perceptual processes group elements of the problem into coherent chunks. Elements or concepts that appear relevant to the problem are retrieved from long-term memory unconsciously through spreading activation. The goal representation thus formed constrains the search through the problem space. However, if the search space does not include a path to the solution then the solver will reach an impasse. The solver will fail to find a solution if he does not, in fact, have the knowledge to find a solution. In this case the impasse is warranted. If, however, the solver is capable of solving the problem but doesn’t know it then the impasse is unwarranted. According to Ohlsson (2011), during the impasse phase the solver will encounter negative feedback and so will try to switch attention (or activation) from the initial cognitive structure that proves unsuccessful in generating a solution two other elements of the structure where an increase in activation of those elements may be enough to produce a revised problem space. Information Box 7.3 presents a summary of Ohlsson’s theory.

There are several reasons why a problem might lead to an impasse. Kershaw and Ohlsson (2004) argue that

Knoblich et al. (1999) provide some evidence of mechanisms that allow a failed or over-constrained representation to be changed. These are chunk decomposition and constraint relaxation. In the matchstick problem mentioned earlier the Roman number VI can be “decomposed” into V and I whereas “+, =, X” are seen initially as chunks that cannot be decomposed. This is an unnecessary constraint, which, when relaxed, can allow a solver to see more readily how the solution might be found. These processes are not necessarily under conscious control but they can bring about a re-representation or restructuring of the problem. As a result the solver may either:

• See the solution immediately (i.e., there is no need for search beyond the limits of working memory) – this is the “Aha!” experience that Ohlsson (2011) refers to as full insight.
• Find a new promising search space involving further analytical problem solving – which Ohlsson refers to as partial insight.
• Find a new promising search space, engage in analytical problem solving, but still fail as the new representation was also ultimately unhelpful – which Ohlsson refers to as false insight.

There have been several attempts to get solvers to avoid impasses by training (e.g., Chronicle, Ormerod, & MacGregor, 2001; Kershaw & Ohlsson, 2004; Weisberg & Alba, 1981), usually in a particular problem such as the Nine-Dot problem (problem 1 in Activity 7.3). A few have looked at the impact of training on a more general category of problems such as verbal insight problems (e.g., Ansburg & Dominowski, 2000; Chrysikou, 2006; Patrick & Ahmed, 2014; Patrick, Ahmed, Smy, Seeby, & Sambrooks, 2014). Patrick et al. (2014) found that they could enhance the process of representational change through training which involved identifying inconsistencies between the solver’s interpretation of a problem and the question statement. Solvers were able to avoid the negative feedback they would normally encounter due to reaching an impasse.

Criterion for satisfactory progress theory

MacGregor, Ormerod, and Chronicle (2001) developed the Criterion for Satisfactory Progress Theory (CSPT) based on Newell and Simon’s (1972) view of problem solving as heuristic search through a problem space. They argued that solving insight problems such as the Nine-Dot problem involves an interplay between a hill climbing heuristic (a maximisation heuristic) and a progress monitoring heuristic. When trying to solve an insight problem the solver will try to choose actions that appear to take him closer to the goal. At the same time the solver is monitoring his progress and selecting moves that meet some criterion for progress. As the solver works through the problem, she gets feedback from the state of the problem and if there is “criterion failure” then the solver has reached an impasse. Solving continues as the solver looks for other as yet unexplored “promising states” that can be evaluated and expanded. For example, the solver may try what appear to be “non-maximal” moves such as drawing lines outside the Gestalt square formed by the nine dots. If an impasse is reached there, then the process is repeated until a new search strategy reveals a new solution path. The success of this process depends essentially on the individual’s working memory, as the probability of reaching an impasse will depend on their ability to look ahead (Ash & Wiley, 2006; Fleck, 2008; Murray & Byrne, 2013). People with a high look-ahead capacity will reach an insight earlier than those who have a lower capacity assuming the heuristics successfully guide their search.

Jones (2003) generated “pertinent” predictions for both the RCT and for the CSPT using the Car Park problem (Figures 7.17 and 7.18). For RCT he predicted that solvers would reach an impasse before moving the taxi and that non-solvers would need a hint before moving the taxi. Based on those who successfully solve the problem, this was the case with a greater time spent on fixations on the problem immediately before moving the taxi. The case for hints for non-solvers is difficult to assess as there were very few who used the hint. For CSPT he predicted

The results showed that the number of impasses did not conform to what CSPT would predict although there was more support based on the time taken on impasses. A direct comparison between the two theories was devised involving a condition in which the puzzle was rotated 90° (with the exit at the side) following some simple practice problems (with the exit at the bottom). For the CSPT this should make no difference as the rotation should not impact on the difference reduction heuristics assumed to be used, but one would expect a difference for the RCT as the rotation triggers a re-representation of the problem. The results showed a significant difference between the two orientations which favoured the RCT interpretation of insight.

Murray and Byrne (2013) presented both single-step problems (e.g., ping-pong in Activity 7.3) and multistep problems (e.g., the Nine-Dot problem in Activity 7.3). They found that single-step problems can be solved immediately and relatively easily giving the typical “Aha!” experience. Multistep problems involve a number of steps before a solution can be verified once a new representation has been found. The latter are more amenable to the processes assumed in CSPT.

While there appears to be some evidence favouring each of the theories, there has been a greater research focus on representational change. For example, Öllinger, Jones, Faber and Knoblich (2013) found more evidence for RCT than for CSPT. CSPT proposes that there are mechanisms for inducing a change in the problem space when an impasse is reached, leading to the solver finding a new solution path. It is not obvious that this is substantially different from re-representing the problem. RCT and CSPT focus on different aspects of insight problem solving behaviour, with RCT dealing with largely unconscious processes (memory retrieval, spreading activation) that take place prior to an initial and a new representation. CSPT concentrates on the conscious process of heuristic search common to other forms of problem solving. Hence Jones (2003) and Ohlsson (2011) among others argue that there is evidence for both theories and that both should be considered together, with Öllinger, Jones and Knoblich (2006) arguing that “representational change is the door opener that ensures that the appropriate heuristics can be applied to the proper problem representation” (p. 252).

Figure 7.19 presents a summary of the influences and some of the underlying processes involved in both RCT and CSPT.

Dual process approaches

Gilhooly and Murphy (2005) looked at the possibility that different processing systems were at play in insight and non-insight problems. The literature on reasoning and decision making postulates two forms of cognitive processing: System 1 and System 2. The first is presumed to include innate, intuitive processes as well as decisions based on basic associative learning or implicit learning mechanisms. The second involves conscious and deliberative thought. Evans (e.g., 2011, 2012) prefers the terms Type 1 and Type 2 since, among other things, there are probably more than two systems involved. He categorised them (2011) thus:

It could be argued that insight problems following an impasse are predominantly Type 1, and non-insight problems predominantly Type 2. Representational change is deemed to occur through unconscious processes such as spreading activation due to switching attention to a previously disregarded problem element, hence insight may involve the processes associated with Type 1. Type 2 is conscious and depends on working memory capacity, and so is associated with individual differences in such executively demanding processes as look-ahead and heuristic processing (Evans, 2005). Analytical problems involving search therefore presumably involve mainly Type 2 processes. If, however, insight problems are no different from non-insight problems in that they use the same cognitive processes, then Kaplan and Simon’s view that insight involves a deliberate search through a problem space or a problem space of problem spaces then Type 2 processes would be involved.

Gilhooly and Murphy found that System 2 (Type 2) processes were involved in both insight and non-insight problems. For example, scores on Raven’s Matrices, used to assess reasoning ability and non-verbal IQ, were correlated with performance on non-insight tasks. The Figural Fluency test, used to assess the ability to generate novel figures using configurations of five dots, is a measure of divergent thinking and was assumed to involve the Type 2 executive process of switching (changing strategy) and inhibition (avoiding misleading strategies). Gilhooly and Murphy found that the scores for the Figural Fluency test predicted performance on insight tasks. They argued that Type 1 processes would determine the initial problem representation, and when an impasse is reached Type 2 processes override Type 1 so that an alternative representation can be found. In summary, Type 2 processes seem to be involved in both insight and non-insight problems, but the particular forms they take “[support] the notion that there are some common processes underlying performance on insight tasks distinct from those underlying non-insight tasks” (p. 298).

Fleck (2008) also found that verbal short-term memory predicted success in insight problem solving, although she also found that working memory capacity was more related to solving analytic problems than insight problems. It appears therefore that the role of working memory and Type 2 processes can vary from one insight problem to another.

Summary

1. 1  The way we represent problems when we encounter them has a powerful influence on our ability to solve them. Sometimes changing one’s representation of a task or situation can lead to a sudden realisation of how to solve the problem. This is an insight.
2. 2  Gestalt psychologists were interested in how we represent and “restructure” problems. They viewed thinking as often either reproductive, whereby we use previously learned procedures without taking too much account of the problem structure; or productive, where thinking is based on a deep understanding of a problem’s structure and is not structurally blind.
3. 3  They were also interested in the failures of thinking due to:
   • Functional fixedness, when we fail to notice that an object can have more than one use;
   • The effects of set, when we apply previously learned procedures when a simpler procedure would work.
4. 4  Insight problems would appear to pose problems for information processing theories of problem solving since it does not involve sequential, conscious, heuristic search. Consequently some researchers have viewed insight as a special case of problem solving. Others have tried to fit insight into traditional information processing accounts – the so-called business-as-usual approach.
5. 5  Kaplan and Simon saw insight as a search for a representation rather than a search in a representation.
6. 6  Ohlsson’s Representational Change Theory (RCT) saw insight as being due to forming an initial representation of a problem, based on salient features of the problem and what is accessed readily form long-term memory, that did not contain a path to the solution. This leads to an impasse. It is possible to get out of an impasse by such means as:
   • Re-encoding: focussing on a different aspects or elements of the problem – activation is subtracted from those elements of the initial cognitive structure found to be unsuccessful (produce negative feedback) and then redistributed across other elements;
   • Constraint relaxation: relaxing constraints that we have inadvertently placed on the problem;
   • Elaboration: addition of new information;
   • Chunk decomposition: breaking elements of a problem down into “sub-elements” (this depends on the nature of the problem);
   • Accessing one operator at a time based on our initial interpretation of a problem. Retrieving operators is an unconscious process involving spreading activation (and hence also depends on the organisation of semantic memory).
7. 7  MacGregor et al. (2001) developed the Criterion for Satisfactory Progress Theory (CSPT) which involves:
   • A search through a problem space using hill climbing and progress monitoring heuristics;
   • Criterion failure forcing the solver to look for other “promising states” to expand and evaluate;
   • Success very often depends on the ability to look ahead – an aspect of working memory capacity.
8. 8  Aspects of both RCT and CSPT would seem to play a part in insight, with RCT focussing on unconscious processes such as cued memory retrieval and CSPT focussing on conscious search processes via heuristics.
9. 9  Gilhooly and Murphy found evidence for both conscious and unconscious processes in solving insight problems from a dual–process paradigm. The degree to which Type 1 and Type 2 processes are used seems to depend on the nature of the insight problem.

Answers to insight problems

1.  1 Nine-Dot problem.
2.  2 Set fire to the £50 note. The note is thereby removed.
3.  3 Triangle problem.
4.  4 Throw the ball straight up in the air.
5.  5 They are triplets.

1.  6 Take one coin from the first bag, two from the second, three from the third, and so on. Once you have done that, weigh all those coins. If all the coins weighed 16 ounces you would have 880 ounces [16 × (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10)]. Any amount over 880 ounces will determine which bag contains the 17 ounces (2 ounces over = bag 2, if it is 7 ounces over = bag 7, etc.).

1.  7 Necklace problem.
2.  8 Turn the first switch on and leave it on for a few seconds. Switch it off and turn the second switch on and go in to the room. If the light is on then it is the second switch that operates it. If the light bulb is off and is warm then the first switch is the right one. If the light is off and the light bulb is cold then it is the third switch that works the light bulb.

1.  9 The prisoner unravels the rope along its length and then ties them together. It now reaches to the ground.
2. 10 The correct answer is 402 pieces of paper. In books, pieces of paper are numbered on both sides. Furthermore, examining the way the books are stacked in [the figure], the worm only eats one piece of paper in the first and tenth volumes; whereas it eats 50 pieces of paper in the other eight volumes (Batchelder & Alexander, 2012, p. 98).
3. 11 The triangles on the left are all isosceles. Those on the right aren’t.