Problem-Solving
Being able to solve problems is important in our daily live: be it at work, in our family or even (or: especially) when we are on holiday. It reveals a lot about human thinking which is why it has been studied over many years by cognitive psychologists.
There are many different types of problems and this section will look at only a few, and more specifically only at those problems that can be studied scientifically.
From a combination of reading this section & your independent study, you should be able to:
• Define problems and problem-solving
• Have an understanding of the main cognitive approaches to the study of problem-solving
• Describe and evaluate models of problem-solving
• Discuss applications of problem-solving research to everyday problems
What is a Problem? What is Problem-Solving?
Generally speaking, a problem can be seen as a goal that has been blocked for whatever reason (e.g., Newell & Simon, 1972) and problem-solving as the actions and higher-order cognitive processes applied to achieve that initially unattainable goal.
In many ways, problem-solving is the culmination of all the processes that makeup our cognitive arsenal.
(Robinson-Riegler & Robinson-Riegler, 2012, p. 494)
This general characterisation distinguishes problems from momentary ‘glitches’ and it also highlights the fact that problems depend on a specific person lacking the relevant knowledge that is required to achieve the goal state. For example, for many people solving a maths task constitutes a problem but this is not the case for a maths professor with relevant knowledge.
Have another look at the instances of problems you have listed in the first activity: they probably differ in many respects, for example in their seriousness, difficulty, context etc. However, they probably all involve some desired goal and the lack of an obvious way of achieving that goal through lack of knowledge or resources.
Here are some further examples of problems:
1. How can I get a grade A in my exam?
2. How can I get a pay increase?
3. How can I transfer three rings form one peg to another while following the rules?
4. How can I draw three lines through a 3 by 3 array of dots without taking my pencil off the paper?
5. What time should I leave home to be sure of getting to work on time?
6. How do I change a car tyre?
Activity 2: The Tower of Hanoi
Try to solve The Tower of Hanoi Problem:
Figure 1 – The tower of Hanoi Problem
You are presented with three pegs and three rings of differing sizes. Initially all three rings are on peg A and your task is to move all three rings to peg C. However, you may only move one ring at a time, you cannot place any ring on top of a smaller one, and you cannot put the pegs on the floor or anywhere else.
As I suspect you won’t have either rings or pegs to try solving this task, you need to adapt the problem by
drawing three circles on a piece of paper and label them A, B & C. Now place three different sized coins in circle A with the largest at the bottom and the smallest at the top. Now transfer all of the coins to circle C, remembering that you can only move one at a time and that a coin cannot be placed on one smaller than itself.
How many moves did you take?
Try this with other people and observe how people solve the problem.
Describe the initial state of the problem, the goal state and the legal moves (= things that can be done) and restrictions (things that can’t be done)?
What is the minimum number of moves needed to solve the problem? Post your answers to your Discussion Board Group.
A site that allows you to try the Tower of Hanoi problem electronically can be found here: Tower of hanoi DHTML game
If you like, you can try a similar problem: Missionaries & Cannibals
Types of Problems
Several types of problems are commonly distinguished in problem-solving research:
The Tower of Hanoi is a problem that has been used in many laboratory-based studies. Our above list of six example problems contains some other commonly used laboratory problems (How can I draw three lines through a 3 by 3 array of dots without taking my pencil off the paper) but it also gives real world problems (e.g., How can I get a pay increase?) Real world and laboratory problems differ in how well they are defined. Traditionally, problem-solving research has used well defined problems, which means the solver is provided with all the information that is needed to solve the problem. There are four different sorts of information: Information about the initial state of the problem.
• Information about the initial state of the problem
• Information about the goal state
• Information about legal operators (things that can be done)
• Information about operator restrictions (things that can’t be done)
However, problems differ in whether all aspects of the problem are clearly specified including the initial state, strategies for solving them, and desired goals. Have another look at your examples and decide the extent to which they are well-defined or ill-defined.
Well-defined problems are “problems in which the initial state, goal, and methods available for solving them are clearly laid out” (Eysenck & Keane, 2012, p. 641) – the Tower of Hanoi problem is a classic example.
In contrast, for ill-defined problems, these four types of information are not all available.
For example, the problem of how to get a grade A in an exam is ill-defined and can only be vaguely analysed in terms of its initial state: it will vary depending on many aspects such as the actual exam questions, how many exam questions there are, how many need to be answered and the total amount of time for the exam, among others. In terms of goal state again the information is ill-defined; a grade A answer is better than a grade C answer, but what exactly is required for a grade C answer, how much better than a grade C answer is a grade A answer, etc.? In other words, how can you be sure that you have achieved the goal? There are similar difficulties in defining the operators and restrictions, e.g., what did the lecturer say? How much further reading is needed? How should you plan the answer? etc. Typically, in real world problems the solver has to define the problem; the parts of the problem are not made explicit in the instructions. Again, this draws our attention to the fact that solving problems is dependent on the solver’s knowledge and experience.
A further distinction is often made between knowledge-rich problems vs. knowledge-lean problems: Problems differ in the amount of specific knowledge that is required to solve them. Have another look at your examples in Activity 1 and our list above. Eysenck & Keane (2012, p. 365) define knowledge-lean problems as “problems that can be solved without the use of much prior knowledge, with most of the necessary information being provided by the problem statement” whereas knowledge-rich problems “can only be solved through the use of considerable amounts of prior knowledge”.
Most traditional research of problem-solving has used well-defined, knowledge-lean laboratory based problems in which all solvers have the same information and experience. However, research into expertise and problem-solving often uses knowledge-rich problems as we will see later on.
In the next section, we discuss two main approaches to problem-solving research: the Gestalt approach and Information-processing theory. We then discuss the relevance of problem-solving research to everyday problems and can consider some factors that lead to problem-solving success and failure.
The Gestalt Approach
Gestalt psychologists view problem-solving as a requiring the restructuring or reorganisation of problems which lead to insight, the sudden realisation of the solution. This is an early account of human problem-solving: it was popular at the beginning of the 20th century and has been very influential. The main point relevant to problem-solving is the proposal that Gestalt theories of PERCEPTION can be usefully extended to studying problem-solving. Gestaltists initially looked at problem-solving in animals, e.g., apes (arguing against a pure trail-and error account), but we won’t go into this here.
The Gestalt psychologists had been fairly successful in showing that perception frequently involves restructuring the object that is being looked at.
Figure 2 – The Necker Cube
The Necker cube above, for example, illustrates the occurrence of perceptual re-structuring in that the corner marked with a Y sometimes appears to be at the front of the figure and sometimes at the back of the figure (try it for yourselves – it can be difficult to shift one’s perception.). In Gestalt terms, the figure is re-structured to be perceived in one way or the other. In a similar fashion, Gestalt psychologists maintained that one has to ‘re-structure’ a problem in order to gain ‘insight’ into its solution.
The classic example to illustrate these notions was Köhler’s (1927) research on problem-solving in monkeys. In these studies monkeys had to reach bananas that were placed high up on the outside of their cages. Bamboo canes were the only tools that were available. On one occasion Kohler observed a monkey sitting down for a while after a number of failed attempts at using single bamboo canes. The monkey then took two bamboo canes and joined them together to reach the bananas. Kohler heralded this as an example of impasse, followed by problem restructuring and insight. More details about this influential work can be found here: Kohler’s Research on the Mentality of Apes
Another example involving re-structuring is Maier’s (1931) two-string problem (or pendulum problem). In the original version of the problem, participants were brought into a room which had two strings hanging from the ceiling and a number of objects lying around including a set of pliers. Participants were asked to tie the two hanging bits of string together. The room and items were set up in a way which made it impossible to reach one string while holding the other. Thus, participants soon found that when they took hold of one bit of string and went to grab the other, it was out of reach and they couldn’t solve the problem. Some participants, after lots of thinking, would, however, attach the pliers to one of the strings and set it swinging like a pendulum.
Then, whilst holding the other string, it was possible to catch the string on its up-swing and tie the two together. Thus, once they restructured the problem and had the insight to use a pendulum, they easily solved the problem. For many of Maier’s participants such problem re-structuring and insight would only occur after he brushed against a string and set it swinging – albeit apparently accidentally (indeed few participants reported noticing this event). According to Maier, the subtle pendulum hint resulted in a re-organisation or re-structuring of the problem so that a solution could emerge.
he Two-String Problem
Figure 3 – The two-string problem
Source – Eysenck, M. W., & Keane, M. T. 2010. Cognitive Psychology. A student’s handbook. 6th edition. Hove: Psychology Press
When solving-problems in any domain, attempts at finding a solution can get ‘stuck’ and our thoughts might be misdirected or blocked. A lot of research within the Gestalt approach looked at such situations.
Insight problem solving is defined by an ‘Aha’ experience: generally participants initially struggle to solve a problem (such as the two-string problem) and they (usually) reach an impasse until (hopefully) the solution appears in a sudden moment of clarity. The Gestaltists were the first to put forward the idea of insight.
Gestaltists argued that insight involves productive rather than reproductive thinking – we get fixated on the function of everyday objects and fail to see that a pair of pliers can be a pendulum weight. So if prior experience causes people to only think of the typical function of an object to reach a solution, we must reduce the ‘activation’ of thinking about its everyday function, which would then increase the likelihood of solving the problem. The impasse phase presumably results in lowering activation and increases the chance of finding an alternative use for an object.
People’s tendency to view objects narrowly in terms of their usual, everyday function is called functional
fixedness. The phenomenon of functional fixedness – and mental blocks more generally – was illustrated in a nice study by Duncker (1926) using the candle problem.
Figure 4 – Duncker’s candle experiment
Source – Andrade, J. & May, J, 2003, Instant Notes in Cognitive Psychology, London: BIOS Scientific Publishers
Participants were given a candle, a wooden tray of nails, a box of matches and a hammer, all placed on a
table. They were told to try to attach the candle to the wall above the table such that it would not drip on the table. Participants tried to nail the candle directly to the wall or to melt in onto the wall. Few, however, thought of using the wooden box which held the nails as a candle holder that could be tacked to the wall. Duncker argued that participants were fixated on the box’s normal function of holding nails and could not re-conceptualise it in a way that would allow the problem to be solved. (cf. also Maier’s two-string problem – a failure to re-conceptualise the pliers as a pendulum weight).
Thus, problem representation can be seen as the key to solving the problem (Robinson-Riegler & Robinson-Riegler, 2012): you need to go beyond narrowly perceiving the box as a container. Functional fixedness is a common phenomenon and does not seem to be limited to technologically sophisticated cultures where objects might be predominantly viewed in terms of narrowly prescribed functions (German & Barratt, 2005).
Stop & Think
Before reading on, you might want to stop for a minute and generate your own examples of overcoming
functional fixedness: think of a practical problem which you only managed to solve by overcoming the initial tendency to narrowly focus on the usual function of an object.
Activity 3: 9-DOT Problem
Another example of fixity hindering problem solving is the 9-dot problem (another famous experimental task designed by Gestalt psychologists): in front of you are nine dots organised in a three-by-three matrix. You might want to reproduce this onto some paper or print out this page. Your task is to draw four continuous lines connecting all the dots without lifting the pen from the paper.
Click here to see the solution
Evaluation of Gestalt problem-solving theory
In sum, the main assumptions of the Gestalt approach to problem-solving theory are:
• Problem-solving behaviour is both productive and reproductive.
• Productive problem solving is characterised by restructuring of the problem which leads to insight.
• Insight often occurs suddenly and is accompanied by an ‘ah-ha’ experience.
• Reproductive problem-solving involves the use of previous experience. This may hinder successful problem-solving – for example because of the occurrence of ‘functional fixedness’ or ‘fixity’.
Positive aspects of this approach:
• It attempts to explain novel and creative problem-solving (Robinson-Riegler & Robinson-Riegler, 2012).
• It highlighted that problem-solving is often more than re-producing past experience.
• It showed that past experience can lead to failure rather than success.
• The tasks created by the Gestaltists and their theoretical constructs have had an influence on many future generations of problem-solving researchers (see for example modern theories of insight (Knoblich et al., 1991, MacGregor et al., 2001).
Aspects of this approach that have been criticised include:
• It is overly-reliant on the perceptual metaphor coming from Gestalt theories of perception. The main problem here is that perceptual notions of insight and restructuring are too under-specified and vague to capture fully the sophistication and complexity of human problem-solving ability. E.g., what are the exact conditions that lead to re-structuring?
• Definitions of key terms are often not testable and possibly circular.
• Exactly what psychological processes are involved in attaining a state of insight?
The Information Processing Approach
This approach emerged as the dominant paradigm for studying problem-solving in the 1950’s. This approach was heavily influenced by research undertaken by Alan Newell and Herb Simon from Carnegie-Mellon University. Their work from the late 1950’s onwards has contributed enormously to the whole framework of Information Processing Psychology. If necessary, remind yourselves of the main features of the information processing approach by checking relevant sections in any good introductory textbook to Cognitive Psychology.
In the late 50’s, Newell and Simon produced the first computational models of psychological phenomena and they also made landmark discoveries in cognitive psychology and artificial intelligence. Their problem-solving theory – put forward as a whole in a huge volume published in 1972 entitled Human Problem-solving – had a major impact on the cognitive community and still remains at the centre of most contemporary problem-solving research.
Newell and Simon proposed that when a person is confronted with a problem, the first thing they do is to try to construct a mental representation of its relevant features. In this theory, this internal mental representation of a problem is termed a problem space. If its construction has been successful, the problem space will consist of the four key sets of information that we mentioned earlier when we gave you the Tower of Hanoi problem to solve. Here are these four key sets of information again:
1. Information about the problem’s initial state.
2. Information about the problem’s goal state.
3. Information about the problem’s legal operators (i.e., things that you are allowed to do in solving the problem).
4. Information about operator restrictions (i.e., factors which constrain the application of operators).
In this approach, problem-solving is as a step-by-step process from the initial to the goal stage, using the legal operators and observing the operator restrictions. The basic idea is to minimise the distance between the initial and the goal state by breaking the problem down into several smaller sub-goals. So, going back to one of the example problems listed at the beginning of the section: “How can I get a grade A in my exam?” your sub- goals might include:
Sub-goal 1:
Clear diary for two-week period before exam
Subgoal 2:
Check reading list & decide which books to read
Sub-goal 3:
Meet with study group
Sub-goal 4
Find previous year’s exam questions and write an essay under time pressure
As you can easily see, each sub-goal needs to be broken down into further sub-goals.
Problem-solving is essentially a journey through the mentally represented problem space. For any given
problem there are a large number of possible alternative paths from the initial state to the goal state. Thinking about and evaluating all of these alternative paths and all alternative states in order to decide if any of them was the goal state would clearly take an inordinate amount of time. Furthermore, certainly for human problem solvers, things are made worse by the fact that working memory capacity is limited. For example, people often forget what states they have visited before, or they forget the route that they took from them. Whilst trying to solve a problem in such an exhaustive manner (going through all possible paths) would – eventually – lead to the solution, it would be very inefficient and it would take a long time. This method is termed an algorithm, which simply means a problem-solving method that definitely will solve the problem – but it will take a long time and make huge demands on working memory capacity.
Before we discuss an alternative, less exhausting, way of solving problems, let’s illustrate the complexity of the path through a problem-space, even for a well-defined lab problem.
State Action Representations
A well-defined problem can be represented as a series of actions which will transform the initial state to the goal state. A diagram can be drawn in which the possible sequence of actions and their results can be
represented. These are known as State-Action trees. Consider the 8 puzzle (there are also versions with more than 8 tiles, e, g., the 15-puzzle version). Various online versions exist (e.g, Maze Works)
Figure 5 – An example 8 puzzle
The tiles are in a 3X3 frame. One cell is always empty so that adjacent tiles can be slid into the space. You can only slide one tile at a time. The problem here is to move the tiles in the puzzle from their start state to the goal state. The possible sequence of moves to achieve the goal can be represented by a tree diagram:
Figure 6 – First three levels of state action tree for the 8 puzzle.
By the third level of the state action tree, there are five possible states but the number of possible states grows exponentially with each level. In fact there are over 100000 possible states that can be reached from the start and the solver’s task is to discover one of these. This is clearly a challenge! A major characteristic of state action trees is the large number of possible states that can be reached. The difficulty when problem solving is to find a short path to the goal state with little search of the state action tree. This can be achieved in one of two ways: a depth first search or a breadth first search. Depth first searching means following one path until a dead end is reached when no further moves can be made. In fact the 8 puzzle is cleverly constructed such that a dead end cannot be reached. In breadth first searching all states at a given level are examined before moving on to the next level. Although breadth firstsearching causes a much heavier memory load than depth first searching, it will always find the correct answer to a problem – in other words, it is algorithmic.
Having looked at algorithmic methods of problem-solving, let’s turn to the alternative approach: this is, the solver applies a variety of heuristics so that the potentially enormous search space can be rapidly narrowed down. Heuristics are ‘rules of thumb’ that do not guarantee a solution to the problem. However, if they do succeed, they save a lot of time and effort.
One of the most important heuristic methods proposed by Newell and Simon is means-ends analysis (MEA). When using this strategy, the problem solver examines the difference between the initial state and the goal state and identifies a series of sub-goals that can be worked on individually (see our grade A exam – example above). If a sub-goal cannot be met, then that sub-goal is further broken down into smaller sub-goals. The solver then identifies appropriate operators to meet each sub-goal.
Obviously such an approach is not always appropriate. In the 8 puzzle, for example, MEA could not be used because if a sub-goal of moving tile 1 to the correct place was achieved, it may well become corrupted when trying to achieve a second sub-goal. The advantage of MEA however is manifest when there are very many possible alternative actions as MEA supplies rules for selecting actions for further exploration. The usefulness of the approach was demonstrated by the success of Newell and Simon’s (1963) General Problem Solver (GPS) computer programme. MEA is a heuristic for problem-solving and it describes a general method for problem-solving rather than using specific information about the problem domain. MEA allows problems to be solved in a shorter time (than an exhaustive algorithmic approach) as the problem solver does not examine all the possibilities. However, being a heuristic, MEA does not always find the solution. Indeed it was found that MEA could only solve comparatively simple problems in a limited amount of time.
Adopting the MEA heuristic consists of the following steps:
• Note the biggest difference between the current state and the goal state.
• Create a sub-goal to remove this difference.
• Select an operator that can achieve the sub-goal.
• If the operator can be applied immediately, then do so and continue from the newly attained state using further means-ends analysis.
• If the operator cannot be applied immediately, then use further means-ends analysis to remove the blocking conditions.
The Tower of Hanoi problem that you did earlier lends itself to a MEA. Research has shown that the time it takes to make a move correlates positively with the number of sub-goals which need to be set up before a move is made (Smyth et al., 1994).
The original Tower of Hanoi problem as described in this section uses three rings. However, it is possible to extend the problem to any number of rings, with the time needed to solve the problem increasing rapidly! Say you decided to work on a 64-ring problem. Given that if there are n rings, the minimum number of moves is 2n-1, it would take you nearly a trillion years to complete the puzzle provided you make one move per second and don’t make any mistakes. Research using the Tower of Hanoi with more than three rings shows that participants impose a sub-problem structure on the task, i.e., they create sub-goals.
Restricting ourselves to the original three-ring version of the Tower of Hanoi problem, let’s illustrate the search through the mental problem space from the initial state, toward the goal state via a set of intermediate states. The diagram below shows the possible states that intervene between the initial state and the goal state for this problem. These are the states that a problem solver would need to search through:
Figure 7 – Goal States in the Tower of Hanoi Task.
To illustrate the heuristic method of means-ends analysis, let’s assume that the solver is at state 15 of the
Tower Of Hanoi problem. At this point three possible moves can be made (to states 11, 14, 19) – but only one of the three moves will actually take you closer to solving the problem (namely, the move to state 19). Means-ends analysis proposes that one should note the difference between the current state and the goal state (i.e., the difference is that the medium disk is on the second peg instead of the third peg). Second, one should establish a sub-goal to reduce the difference (create a new sub-goal of moving the medium disk from B to C). Third, one needs to select an operator to achieve the sub-goal and apply it (i.e., actually move the small disk from B to C).
Applications of Problem-Solving Strategies
As mentioned earlier, the early information processing approach to problem solving typically studied clearly defined problems in the lab, often called puzzle problems. Consider some differences between such lab-based problems and the problems people face everyday:
Puzzle problems are unfamiliar as there is little prior knowledge or experience that we can bring to bear in trying to solve them. In contrast, because real-life problems are often relatively familiar, there is often a lot of 13 prior knowledge and experience that we can use to tackle them. However, note that prior knowledge may make finding a solution more difficult.
In puzzle tasks, the information required to solve the task is present in the statement of the problem (see legal operators and operator restrictions in the Tower of Hanoi problem). In contrast, in real-world problems the information required to solve the task is often not present. In fact, much of real-world problem solving involves attempts to find the right information to solve the problem (e.g., consider having to buy a house: you have to find out which mortgages are available; what houses are on sale; what deals are on offer etc.). Finding out all relevant information is a significant part of solving the problem.
Finally, the requirements in lab-based puzzle problems are relatively unambiguous: the start state and goal state are clearly specified, and what can and cannot be done is also clearly stated. In contrast, in everyday problems, specifying the nature of the goal state can be of considerable difficulty. For example, problems involving goal states such as “attaining success” or “being happy” are difficult to solve partly because they require you to define what it means to you to be happy or successful (e.g., is local notoriety good enough or do you want world fame?).
Real-life examples of means-ends analysis
Newell and Simon claimed that means-ends analysis is a very powerful and commonly used heuristic in
problem-solving and does not only apply to lab-based problems. A good everyday example of means-ends analysis is deciding to go to see a show in London. Imagine that you had a telephone call at home in the morning: it is your friend inviting you to go to London that same day because she has spare tickets for a show and has invited you to come along. Comparing your current state with your goal state reveals that the biggest difference is one of location (assume you live in Nottingham). Operators such as ‘walk’ or ‘cycle’ can be rejected as unfeasible but ‘catch a train’ seems a good bet. Unfortunately this operator cannot be applied immediately because you’re not at the train station. The new goal is to be at the train station and again the biggest difference between goal state and current state is one of location. Travel operators are again generated and ‘taxi’ is selected. But the taxi driver does not know that they are needed, therefore your new goal is to get hold of a taxi driver. The biggest difference is one of communication. What operator enables communication? Well, a telephone … and so on.
When this mean-ends analysis is complete, a mental plan has been produced consisting of a sequence of
operators that can be applied (in the reverse order in which they were generated) to enable the main goal to be attained.
Activity 4
Describe a concrete example of means-ends analysis in a real-world situation.
Post your answers to your Discussion Board Group
The examples above (and your own example) indicate that heuristics such as means-ends analysis are
powerful strategies even when dealing with everyday-type problems and tasks. Some everyday problems are well-defined and have (relatively) small problem spaces to search through. Smyth et al. (1994) argue that general-purpose heuristics such as means-ends analysis are useful if there is no previous specialised
knowledge which enables us to solve a problem (Smyth et al.,1994). After all, if there were a fool-proof method (= algorithm) for getting that ‘A’- grade in an exam, somebody would have sold it for a lot of money by now!
However, in many real-world problem-solving situations, the problem space is not easily defined. Think of various workplace scenarios, for example detectives in a murder enquiry, medical doctors trying to find effective treatments when dealing with varied and ambiguous features in their clinical practice (Norman, 2005), and many more. The actual formulation and mental representation of the problem, the selection of heuristics, recognising when a goal or sub-goal is reached and many other aspects of problem-solving are not well defined. The mental representation of such problem spaces is affected by the solver’s experience and expertise.
Evaluation of Newell & Simon information-processing theories of problem-solving:
Positive aspects of this approach:
• It emphasised the need to be explicit about the mental operations (and sequences of operations) by which people solve problems, in marked contrast to the Gestaltists’ “sudden insight”.
• It provides a normative theory of problem-solving: the notion of problem space allows us to specify the idealised structure of a problem and the ideal solution path. This provides a clear basis from which to empirically study how and why people’s actual problem-solving behaviour deviates from the ideal.
• Theories are explicit about the kinds of heuristic strategies that people adopt to tackle problems and therefore make strong predictions about what they will do, when they will succeed and when they will fail.
• Many of these predictions are borne out in early problem research.
Aspects of this approach that have been criticised include:
• Early problem-solving research focused on a very narrow class of problems (i.e., puzzle-like lab problems).
• For many problems (in the real world), there is no easily defined problem space. How and when heuristics are applied in complex real-world problem-solving is a matter for further on-going research.
Problem-Solving Success (and Failure)
Finally, it’s worth pointing out that a wide range of research has looked into successful problem-solving, with obvious potential impact for everyday problem-solving. This brief section aims to give you a flavour of the many aspects such research investigated.
Previous experience and knowledge can be assumed to make solving of a current problem faster and
easier. Research into skilled performance provides evidence for this (for example research into expert chess players, see Charness et al., 2001). However, past experience sometimes disrupts and slows down current problem-solving, as we have discussed in the context of functional fixedness. Clearly, we need a better understanding of when previous knowledge and experience help and when they are detrimental. Eysenck & Keane (2010) and Robinson-Riegler & Robinson-Riegler (2012) provide good discussions of this research.
Problem representation is a critical component of successful problem-solving, as Robinson-Riegler &
Robinson-Riegler (2012) point out. This applies to all aspects of the problem space, i.e., the mental
representation of the initial and goal states and of the legal operators. Failure in problem representation might result from many factors, such as insufficient attention being directed to certain problem elements, or lack of understanding of problem elements, or, again, functional fixedness, i.e., an overly rigid representation of problem elements due to previous experience. Individual differences in problem representation might be at play, such as anxiety due to stereotype threat: the expectation that a negative stereotype (‘girls are bad at maths’) is going to be used to judge one’s performance can adversely affect problem-solving (Robinson-Riegler & Robinson-Riegler, 2012).
The format of presenting problems can have effects on the solutions being produced (see framing effects in decision-making, Eysenck & Keane, 2010).
Human reasoning is subject to some well researched biases. Bridger (2009) discusses confirmation bias in the context of problem-solving: the tendency to seek out confirming evidence over evidence that might refute a given assumption or hypothesis. Bridger gives the example of a computer programmer who is trying to debug a computer program. She is continuously looking for a syntax error and is unable to shift her attention to search for faults in the structure of the program. In this sense, there is a problem with the mental representation of the initial state of the problem and a reluctance to re-define the initial state, with obvious detrimental consequences for selecting appropriate operators to solve the problem.
A simple way of promoting successful problem-solving might be to introduce a period of incubation and even sleep. Incubation (i.e., ignoring a problem for some time) might help people overcome a ‘mental block’ when solving a problem. Different accounts have been proposed to account for such effects (including the assumption that the “subconscious mind” carries on working on the problem while the conscious mind is dealing with other things) but the evidence is not fully conclusive. A recent meta-analysis concludes that “conditions under which incubation can be used as a practical technique for enhancing problem-solving must be designed with care” (Sio & Ormerod, 2009, p. 94), so don’t rush into adopting this strategy blindly! Eysenck & Keane (2010) concluded that incubation effects were stronger for creative problems with multiple solutions and when there was a long preparation period prior to incubation.
Conclusion
In this section, we introduced different types of problems. We considered two main approaches to problem-solving research: the Gestalt approach and information-processing theory. Gestalt psychologists extended theories of perception to problem-solving. They assumed that problem-solving involves insight into the structure of a problem and re-structuring it. They found that prior experience may hinder performance.
Information-processing theory of problem-solving tended to focus on well-defined and knowledge-lean
problems whereas much of real life seems to involve tackling ill-defined and semantically rich problems. Early information-processing theory saw problem-solving as a mental search through problem space. It was suggested that problem-solving involves the use of heuristics. Whilst early problem solving theory had the potential to be applicable to characterising problem-solving with real-world problems, it was only from the 1970’s onwards that research actually got underway to do this. Problem-solving research is relevant to many everyday problems and can help us better understand the factors that lead to problem-solving success and failure. We discussed some such factors, namely previous experience and knowledge, problems of Problem representation, the format of presenting problems, confirmation bias and incubation.
Activity 5
Verbal Protocol
Do this activity AFTER reading this section and other materials:
Choose some everyday problem (such as planning a birthday party) and spend about 15 minutes thinking about solving it. At the same time, think aloud for the full 15 minutes:
Collect your own verbal protocol (you could do an audio recording, or make detailed notes).
Observe what processes your mind is going through while you are talking about how to solve the problem.
Reflect on your own protocol and describe how it demonstrates the problem-solving principles we’ve discussed in this section.
Quiz
Once you have worked through this section and read around this topic, test your knowledge with this quiz on problem-solving. If you are having problems answering any questions, then go and check your notes or your textbooks for the correct answer and have a think about why you got the answer wrong. Re-read the appropriate section either from your notes or your textbook/s and have another go.
1. Gestalt theory states that productive problem solving means:
a. moving from a starting position to a goal position
b. restructuring a problem in order to think about it in a different way
c. not being able to access the goal state
d. solving a problem in the easiest way
2. What can often hinder our ability for insight?
a. closing our eyes
b. restructuring the problem
c. previous knowledge that is relevant to the problem
d. nothing, insight is a natural human process which is not hindered by anything
3. What is functional fixedness?
a. only being able to think of the normal uses of objects
b. only being able to concentrate on one part of the problem at a time
c. being able to use objects in different ways
d. focusing on the goal of the problem
4. What is a heuristic?
a. when you reach a dead-end in the problem and have to go backwards
b. the process of moving from the initial state to the goal state
c. a problem which has an infinitely large problem space
d. an intuitive guide to aid the selection of an operator
5. Means-ends analysis is an example of:
a. an algorithm
b. exhaustive search
c. a heuristic
d. trial-and-error learning
6. A person’s representation of problem information is referred to as their:
a. constraint space
b. problem space
c. state space
d. operator space
7. In Duncker’s candle problem experiment, people’s problemsolving success is hampered by:
a. productive thinking
b. divergent thinking
c. reproductive thinking
d. convergent thinking
8. What is one of the main problems with means-ends analysis?
a. it takes a long time to find a solution
b. it is not guaranteed to find a solution
c. it doesn’t take the goal state into account
d. it doesn’t take the initial state into account
References
Duncker, K. (1926). A qualitative (experimental and theoretical) study of productive thinking (solving of
comprehensible problems). Journal of Genetic Psychology, 68, 97-116.
German, T.P., & Barratt, H.C. (2005). Functional fixedness in a technologically sparse culture. Psychological Science, 16, 1-5.
Köhler, W. (1927). The mentality of apes (2nd Ed.). New York: Harcourt Brace.
Knoblich, G., Ohlsson, S., Haider, H. & Rhenius, D. (1999). Constraint Relaxation and Chunk Decomposition in Insight Problem Solving, Journal of Experimental Psychology: Learning, Memory and Cognition, 25 (6), 1534-
1555
Locke, E.A. (2009). It’s time we brought introspection out of the closet. Psychological Science, 4, 24-25.
MacGregor, J. N., Ormerod, T. C., & Chronicle, E. P. (2001). Information-processing and insight: A process model of performance on the nine-dot and related problems. Journal of Experimental Psychology: Learning, Memory and Cognition, 27, 176-201.
Maier, N.R.F. (1931). Reasoning in humans II: The solution of a problem and its appearance in consciousness. Journal of Comparative Psychology, 12, 181-194.
Newell, A. & Simon, H.A. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall.
Norman, J. (2005). From theory to application and back again: Implications of research on medical expertise for psychological theory. Canadian Journal of Experimental Psychology, 59(1), 35-40.
Module Reference List
(Please note that this is not a reading list for you, and some, but not all of these references are available in the library and this list provided for information purposes and to help you with literature searching.)
Allport, D.A., Antonis, B., & Reynolds, P. (1972). On the division of attention: A disproof of the single channel hypothesis. Quarterly Journal of Experimental Psychology, 24, 225-235.
Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.
Anderson,S.J., Yamagishi,N. & Karavia,V. (2002) Attentional Processes Link Perception and Action.Proceedings of the Royal Society, London: B, 269, 1225-1232.
Baddeley, A.D. (1997) Human memory: Theory and Practice (Revised Edition). Hove: Psychology Press.
Baddeley, Alan & Warrington, E K.(1970. Amnesia and the distinction between long- and short-term memory. Journal of Verbal Learning and Learning Behavior, 9. 176-189.
Baddeley, A. D., Thomson, N., & Buchanan, M. (1975). Word length and the structure of short-term memory. Journal of Verbal Learning and Verbal Behavior, 14, 575-589.
Behrmann,M. & Tipper,S.P. (1994) Object-based attentional mechanisms: Evidence from patients with unilateral neglect. In Umilita,C. & Moscovitch,M. (Eds.), Attention and Performance XV, Conscious and Nonconscious Information Processing. MIT Press, Cambridge.
Bisiach,E. & Luzatti,C. (1978) Unilateral neglect of representational space. Cortex, 14, 128-133.
Broadbent, D.E. (1958). Perception and communication. Oxford: Pergamon.
Cohen N. J., and Squire L. R. (1980) Preserved learning and retention of pattern-analyzing skill in amnesia: dissociation of knowing how and knowing that. Science. 210(4466):207-210.
Cherry, E.C. (1953). Some experiments on the recognition of speech with one and two ears. Journal of the Acoustical Society of America, 25, 975-979.
Deutsch, J.A., & Deutsch, D. (1963). Attention: Some theoretical considerations. Psychological Review, 93, 283-321.
Duncan,J. (1984) Selective attention and the organisation of visual information. Journal of Experimental Psychology: General,113, 501-517.
Eriksen,C.W. & Murphy,T.D. (1987) Movement of attentional focus across the visual field: A critical look at the evidence. Perception & Psychophysics, 42, 299-305.
Eysenck, M. W., & Keane, M. T. (2000). Cognitive Psychology. A student’s handbook. 4th edition. Hove: Psychology Press.
Gibson,B. & Egeth,H. (1994) Inhibition of return to object-based and environment-based locations.Perception & Psychophysics, 55, 323-339.
Glenberg, A., Smith, S. M., and Green, C. (1977). Type I rehearsal: maintenance and more. Journal of Verbal Learning and Verbal Behavior 16, 339-352.
Graf & Schachter, 1985
James,W. (1890) The Principles of Psychology. Holt: New York
Lavie,N. (1995) Perceptual load as a necessary condition for selective attention. Journal of Experimental Psychology: Human Perception & Performance,21, 451-468.
Leslie,A.M., Xu,F., Tremoulet,P.D. & Scholl,B. (1998) Indexing and the object concept: developing ‘what’ and ‘where’ systems. Trends in Cognitive Sciences, 2, 10-18.
Linnell,K.J., Humphreys,G.W., McIntyre., Laitinene,S. & Wing,A. (2005) Action modulates object-based selection. Visual Research, 17, 2268-2286
MacLeod, C. M. (1991). Half a century of research on the Stroop effect: An integrative review. Psychological Bulletin, 109, 163-203.
Matthews, G., Davies, R.D., Westerman, S.J., & Stammers, R.B. (2000). Human Performance: Cognition, stress and individual differences. Hove: Psychology Press.
Milner, B. (1966). Amnesia following operation on the temporal lobes. In C.W.M. Whitty & O. L. Zangwill (eds), Amnesia. London: Butterworth.
Norman, D.A., & Shallice, T. (1986). Attention to action: Willed and automatic control of behaviour. In R.J. Davidson, G.E. Schwartz, & D. Shapiro (Eds.), The design of everyday things. New York: Doubleday.
Paivio, 1969
Paivio, 1972
Posner,M.I. (1980) Orienting of Attention. Quarterly Journal of Experimental Psychology. 32, 3-25.
Reason, J. (2000). Human error: Models and management. British Medical Journal, 320, 768-770.
Reason, J. (1990). Human error. Cambridge: CUP.
Rundus, D (1971). “An analysis of rehearsal processes in free recall”. Journal of Experimental Psychology(89): 63–77.
Schneider, W., & Shiffrin, R.M. (1977). Controlled and automatic human information processing: Detection, search, and attention. Psychological Review, 84, 1-66.
Shallice, T. and Warrington, K.(1970)’Independent functioning of verbal memory stores: A neuropsychological study’, The Quarterly Journal of Experimental Psychology,22:2,261-273.
Shiffrin, R. & Nosofsky, R. (1994). Seven plus or minus two: a commentary on capacity limitations.Psychology Review, 101(2):357-61.
Smyth, M. M., Collins, A. F., Morris, P. E., & Levy, P. (1994). Cognition in action. 2nd edition. Hove: Psychology Press.
Stroop, J.R. (1935). Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18, 643-662.
Treisman, A.M. (1960). Contextual cues in selective listening. Quarterly Journal of Experimental Psychology, 12, 242-248.
Tulving E. (1972). Episodic and semantic memory. In: Tulving E, Donaldson W, editors. Organization of memory. Academic Press; New York: pp. 381-403.
Von Wright, J.M., Anderson, K., & Stenman, U. (1975). Generalisation of conditioned G.S.R.s in dichotic listening. In P.M.A. Rabbit & S. Dornic (Eds.) Attention and Performance, Vol. V. London: Academic Press
Language and Grammar Preparation
Introduction
The purpose of this unit is to provide you with a number of simple exercises to to help you become confident about English grammar and the basic terminology people use to describe sentences. This is important because psychological research into children’s language requires a basic understanding of (some) linguistic terminology.
It is not necessary that you complete all of this material in your first week and in fact you might find it more useful to revisit the material as you work through the module. Because we do not expect you to complete this material at once there are no discussion board activities for this unit.
Having completed the work in this section you should:
• Have a basic understanding of grammatical terminology and grammar.
• Understand the importance of formal features to describe word classes.
• Be able to identify phrases and their parts.
• Appreciate the generative nature of grammar
What is Grammar?
As competent users of English, you actually know English grammar very well BUT this knowledge is not necessarily explicit and it is not easily consciously accessible. For example, you would probably all agree that ‘I bought some chocolate’ is a syntactically well-formed sentence (which does not mean, by the way, that it is a truthful or even plausible sentence!) and that ‘Some I chocolate bought’ is not a syntactically well-formed sentence. How do you know this? You know it not because anybody told you so, or because you have never heard a sentence like ‘Some I chocolate bought’ ever before (after all, there are MANY sentences you’ve never heard before but which you can easily recognise as syntactically well-formed) – so how do you know? I’d argue it’s because you have a wealth of implicit grammatical knowledge which tells you exactly which parts of a sentence can be put in which order. What you may need to brush up on, though, is some linguistic terminology so you can easily understand what psycholinguists and psychologists are actually studying when they try to find out how people use language. Most examples in this section were taken from UCL’s website ‘The Internet Grammar of English’ and I highly recommend that you do many more of the exercises on their website.
Word Classes and Formal Features
Since words are fundamental units in every sentence (not just in English), we will begin by looking at these. The words of a language can be categorised into seven MAJOR word classes:
Verb: be, drive, grow, sing, think
Noun: brother, car, David, house, London
Determiner: a, an, my, some, the
Adjective: big, foolish, happy, talented, tidy
Adverb: happily, recently, soon, then, there
Preposition: at, in, of, over, with
Conjunction: and, because, but, if, or
Feel free to add more examples for each of these word classes!
So how do you know if a word belongs to the ‘noun’ class, or the ‘verb’ class, etc?
The best way to define word classes is NOT by referring to their meaning – it is more useful to consider them from the point of view of their formal characteristics or their formal features.
Consider nouns: Nouns are commonly described as “naming” words, that is, as the names of “people, places, or things”. However, many nouns also denote abstract and intangible concepts rather than actual ‘things’ with names (e.g., evolution, politics, hope). Because of the enormous diversity of reference, it is not very useful to study nouns solely in terms of their meaning. It is better to consider them from the point of view of their formal characteristics.
By “formal features” I mean characteristics that are true of all nouns (or all verbs, etc.).
Simply by virtue of being a noun, certain features are true of each particular noun. Please note that while the particular formal features I’ll be discussing apply to English.If you know other languages, you could try and describe some of these formal features yourself!
Formal Features of (English) Nouns
Here’s a brief list of the formal features of (English) nouns:
1. Many nouns can be recognised by their endings. Typical noun endings include:
-er/-or actor, painter, plumber, writer
-ism criticism, egotism, magnetism, vandalism
-ist artist, capitalist, journalist, scientist
-ment arrangement, development, establishment, government
-tion foundation, organisation, recognition, supposition
2. Most nouns have distinctive SINGULAR and PLURAL forms. In English, the plural of regular nouns is formed by adding -s to the singular noun (but note that there are many irregular nouns which do not form the plural in this way, see e.g., men, children, sheep).
3. Nouns often have DETERMINERS in front of them (the car; an artist; my book; these students).
4. Nouns may take an -‘s (“apostrophe s”) or GENITIVE MARKER to indicate possession (the boy’s pen; my girlfriend’s brother; John’s house).
Activity
How many nouns?
If you’re quite happy with what you’ve read so far, you should be able to do this little exercise: work out how many nouns there are in the following paragraph:
The major thoroughfares were already lit by the new gas, but this was not the bright and even glare of the late Victorian period: the light flared and diminished, casting a flickering light across the streets and lending to the houses and pedestrians a faintly unreal or even theatrical quality.
Visit UCL’s website for more exercises: The Internet Grammar of English
Formal Features of (English) Verbs
Let’s now have a quick look at the formal features of (English) verbs:
1. The base form has TYPICAL ENDINGS:
-ate: concentrate, demonstrate, illustrate
-ify: clarify, dignify, magnify
-ise/-ize: baptize, conceptualize, realise
2. Verbs have inflections to indicate TENSE:
-s inflection for PRESENT TENSE
-ed inflection for PAST TENSE (for regular verbs)
3. Verbs have inflections to indicate PERSON in the present tense:
-s inflection: to indicate 3. person singular > HE/SHE/IT /the man/Jane laughs.
no inflection: to indicate all other persons > I / you/ we/ they laugh.
Activity
How many verbs?
You should now be ready for your next exercise: work out how many verbs there are in the following paragraph.
Her pace slowed and an ache spread from between her shoulders. Vapours swirled and banked; the light of on-coming headlights drained out of the car. […] Sodium 4 street lamps burned phosphorescent holes in the fog, but as she turned off Main Street to the cottage she noticed the one which illuminated the alley was out.
The Generative Nature of Grammar
Knowledge of grammar obviously involves more than knowing what makes up the basic word classes of a language: as a competent speaker of English (for example), you have an implicit understanding of the rules governing the permissible word order in well-formed sentences (see the chocolate example earlier on). In other words, you know the syntax of English. Syntactic rules do not operate on individual words but on so-called phrases, like noun phrases or verb phrases. ‘Phrases’ are the building blocks of sentences and, together with the syntactic rules, allow you to create (and understand) any number of sentences in
your language: this is what is meant by the “generative nature” of language: knowledge of a finite number of syntactic rules operating on the basic building blocks (phrases) allows you to generate or to understand an infinite number of novel sentences.
So what are phrases? The simplest type of noun phrase is a noun, and the simplest type of verb phrase is a verb. I’ll show you examples in a minute. You will see that any syntactic rule that applies to a noun, also applies to a noun phrase: nouns are simply special cases of noun phrases (and ditto for verbs & verb phrases, etc.).
Noun Phrases (NPs)
Noun phrases (NPs) have three parts which are called the head, the pre-head string, and the post-head string.
In a noun phrase, the head is a noun or a pronoun (hence the name noun phrase!):
Head = noun or pronoun (in bold):
[Children] should watch less TV.
[He] likes coffee.
The waitress gave [me] the wrong dessert.
[This] is my car.
You might have noticed the square brackets, as in [children] – this is a common convention to indicate a noun phrase – you’ll see the brackets in the following examples as well. In the examples above, the pre-head string and the post-head string are empty.
The pre-Head string of an NP (= the parts of an NP preceding the head noun) is usually a determiner or an adjective phrases:
pre-Head string: (in bold):
[Happy children] often sing.
She gave [the new waiter] a big tip.5
[The car] did not start.
Note that pronouns do not take determiners or adjectives, so there will be no pre-Head string for them.
And finally, the post-Head string of an NP can be indefinitely long, for example by adding many so-called relative clauses that provide more information about the noun in question. We will come back to relative clauses when we discuss human sentence processing:
post-Head string: (in bold)
He saw [the dog that chased the cat that killed the mouse that ate the cheese that was made from the milk that came from the cow that…].
Do you see what I mean about the generative nature of grammar? This NP could go on forever, at least in principle, if not in practice.
You should be able to see that you can put an NP into certain positions within a sentence regardless of whether the NP just consists of the head, or the head plus pre-head string, or the head plus post-head string, or even the head plus pre-head string plus post-head string! I didn’t give you an example for the latter but you should be able to generate one (or many!!) yourself.
Activity
Parts of the NP
You should now be able to identify the three parts of the NP in this sentence: what is the head, the pre-head string, and the post-head string?
The small children in class 5 fell asleep.
Verb Phrases (VPs)
Like NPs, Verb Phrases (VPs) also consist of three parts, namely the head, the pre-head string, and the post-head string. As you can probably guess by now, the head of a VP is always a verb:
the Head: (in bold)
He [composed] an aria.
David [sneezed].
The pre-Head string is often empty. If there is a pre-head string, it will be a `negative’ word such as not [1] or never [2], or an adverb phrase [3]:
pre-Head string: (in bold)
[1] He [did not compose an aria].
[2] He [never composed an aria].
[3] Jane [deliberately broke the window].
Whether or not there is a post-Head string in a VP depends on the verb: for many English verbs, a post-head string is obligatory (i.e., HAS to be there). These verbs are called TRANSITIVE verbs and what follows the verb is called a direct object:
post-Head string: (in bold) Transitive Verbs
My son [made acake] — (compare: *My son made.)
We [keep pigeons] — (compare: *We keep.)
I [recommend the fish] — (compare: *I recommend.)
Note that the asterisk indicates an ungrammatical sentence (e.g., *My son made.)
In contrast to transitive verbs, some verbs are INTRANSITIVE, i.e., they are verbs that are never followed by a direct object:
post-Head string: (in bold) Intransitive Verbs
Susan [smiled].
The professor [yawned].
As most verbs in English can be both transitive and intransitive, it is perhaps more accurate to refer to transitive and intransitive uses of a verb:
Example:
Intransitive use: David smokes.
Transitive use: David smokes cigars.
Whether or not an English verb is transitive, or intransitive or both is an idiosyncratic feature of each verb: there are no rules that would tell you the answer. Psycholinguists assume that such idiosyncratic features are stored in the mental lexicon as part of the entry for that verb. Furthermore, the post-head string of some verbs is not a direct object but consists of some other ‘bits’, for example a clause (see A below), a direct object AND a prepositional object (see B below), or indirect object AND a direct object (see C below):
post-Head strings: (in bold)
A: He [decided to go to war].
B: Sue [put the book on the table].
C: I gave [the child some chocolate.]
Activity
Parts of the Verb Phrase
You should now be able to identify the three parts of the VP in this sentence: what is the head, the pre-head string, and the post-head string?
The Prime Minister reluctantly showed the journalists the incriminating photos.
Conclusion
Having gained (or consolidated) a basic understanding of word classes, phrases and their structure, and appreciated the generative nature of grammar, you are now in a better position to describe people’s language and to understand psychological research into how human beings use their knowledge of grammar when they speak, listen, talk, read or write.
Further Resources
BBC’s website on how to improve your language skills: http://www.bbc.co.uk/skillswise/words/grammar/
Internet Grammar by University College London (UCL). http://www.ucl.ac.uk/internet-grammar/