by Eamon Fulcher
Chapter 7: Problem-solving and creativity
You have a problem when you want to achieve something but do not know how to achieve it. Of course, there may be occasions when you know how to reach the goal, but cannot face doing those things necessary to reach it (such as studying hard to get your degree). In this case the problem would be how to get yourself motivated (and for that you can read the Introduction to this book!). Generally, psychologists are interested in understanding how people reach a goal or achieve a task by thinking about the problem.
In this chapter we discuss good and bad problem-solving strategies, the state space analogy for representing problems and their solutions, how experts acquire their expertise and the issue of creativity.
In this first section you will be studying the various approaches psychologists take in researching this area. You will learn about the state space metaphor that is used to represent a problem and how it might be solved, as well as other ways of representing a problem. You will also learn about the strengths and limitations of this approach.
We solve problems every day of our lives. For example, on a given day you could be faced with a number of problems: a flat tyre on your bicycle, an essay you haven't written needs to be handed in by 6 p.m., your daughter refuses to get dressed for school, your PC fails to recognise that it has a hard disk, or all of these (including the headache!). The solution to each problem may not be obvious, although they are certainly solvable.
Problems have three main features:
An initial state
This is the position or state you are in when you begin working towards a goal or begin a task.
The goal state
This is the position you will be in when the goal is reached or when you have completed the task.
These are the things that prevent you, and get in the way of, reaching your goal.
For example, for the PC problem, the initial state might be turning the PC on, the goal state would be to sit in front of the PC with a word processing package open, and the obstacle might be a blank screen with a flashing cursor that remains this way for several minutes. Of course, another obstacle in this example is the user's lack of knowledge of operating systems and why PCs often ‘hang up' in this way.
In order to solve problems people employ plans or strategies. A plan consists of breaking down the problem into a number of logically arranged sub-goals and solving each in turn. A strategy is a particular method for solving the problem and each problem may be solved by the use of different strategies. For example, to solve the PC problem, one strategy would be to seek help from a technician, and another would be to consult the manual (despite the overwhelming urge to hurl the thing out of the window).
In order to solve a problem successfully, one first needs to understand the problem correctly. When faced with a problem we develop an internal representation of it. This means that we have a certain understanding of the problem, and this understanding will vary in accuracy and complexity. Greeno (1991) suggested that a good internal representation has three features:
The problem is understood with in a specific context and we are aware of the constraints on the problem.
A close correspondence with the true nature of the problem
Sometimes our understanding of the problem is incomplete.
It is built on relevant knowledge
Sometimes people fail to use their background knowledge in understanding a problem.
Very often problems are solved by the use of an analogy or a representation of the problem that makes it easier to understand. Types of analogies are:
Some problems can be represented mathematically. For example, suppose you and a friend purchased some items for a day of studying at home. You purchased 3 bars of chocolate and 1 tin of biscuits and spent £4.60, while your friend spent £4.20 on 2 bars of chocolate and 1 tin of biscuits. A third friend sees the biscuits and wants to buy one too so asks you how much they cost. You solve this by representing the problem mathematically (and solving using simultaneous equations). The major difficulty with using symbols is in their translation. For example, some students are uneasy with simultaneous equations. In one study, Schoenfeld (1985) found that about 30 per cent of students made translation errors when solving similar kinds of algebra problems.
A matrix is a table or chart useful for solving problems that are categorical in nature. An example is the table one uses to solve the popular logic problems found in puzzle books. Schwartz (1971) found that students encouraged to use a matrix to solve a logic problem were more likely to solve it correctly than students who used alternative strategies that employed different types of representation.
One of the most common ways of solving a problem is to draw it as a diagram. This way, a simple form can represent a large amount of information. However, diagrams can suffer from the problem of ‘functional fixedness' (see later section).
Like diagrams, visual images can contain a large amount of information, but they have the advantage over diagrams in that they can be ‘irrational' as Koestler (1964) pointed out. Thus, they encourage problem-solving by breaking out of the boundaries of diagrammatic representations. According to Adeyemo (1994), visual imagery is commonly used when the task is to construct a figure – it is visualised before and during construction.
A problem can be likened to a maze in which there is a start state, a number of paths, several dead ends or cul-de-sacs and a goal (Newell and Simon, 1972). An action (such as turn left, turn right or go back) takes the problem into a different state (e.g. one step closer to the exit), and all of the locations or states of the maze collectively define the state space of the problem. To solve the problem one might use several strategies to determine which move to make and at which point.
Games such as noughts and crosses and chess can also be represented by the state space analogy. The starting state of noughts and crosses is an empty grid, and the first move (say X in the top left-hand corner) takes the game into a new state (or location in the maze analogy). A move by O into the centre square, say, takes the game into another distinct state. Every conceivable (and legal) configuration of X and O on the grid represents the game's state space (and in noughts and crosses there are, believe it or not, several hundred possible states). Similarly, in chess, the start state is the board configuration before a move has been made, and the first stage might be ‘pawn to king 4' . This new state could lead to numerous other states next, depending upon how black makes her first move. In the maze analogy, the state or location created by white's first move has several ‘paths' leading from it based on all of black's options, which I've just counted as being 20 (eight pawns, each with two options, and two knights, each with two options). As the game progresses the number of options increases dramatically (which is one reason why the game is so difficult to master) and it has been estimated that there are more possible game states in chess than there are known atoms in the universe!
In finding solutions to problems that can be represented in this way one uses a ‘search' technique or algorithm. The idea is to search the state space of the problem for routes that lead to one of the goal states. In noughts and crosses, for example, you consider placing your X or O in one of a number of empty squares on the grid and then consider what your opponent would do next (see Figure 7.1).
Try drawing the state space of the first few moves of noughts and crosses. This will help you appreciate the method as well as understand how such an apparently simple game can have a huge state space.
Figure 7.1 An example of the problem space of a game of noughts and crosses.
By expanding the diagram to include O's possible replies to each move, and X's final move,
the diagram can be used to obtain the best result possible for X. Here X has three options,
and only option 2 can guarantee avoiding defeat.
A simple, though extremely long-winded, algorithm is to do an exhaustive search. This means considering every option and every possible consequence of each option. In a game like noughts and crosses an exhaustive search is quite manageable and would help make sure that you did not lose the game. However, in more complex problems, where the state space could be huge, this method is unrealistic. For example, solving the anagram LSSTNEUIAMYOUL using only an exhaustive search would take a very long time since there are more than 87 billion possible arrangements of the 14 letters of the word SIMULTANEOUSLY (Matlin, 2002). Fortunately, problems can be solved using other methods, known as heuristics. A heuristic is a strategy for searching state space in a more meaningful way. For example, to solve the anagram you could try to arrange some of the letters that commonly go together, such as NE and OU and many words end with Y. In addition, we know that the first two letters could be LE, LA, SE, SI, or ME, and so on. We also know that the first two letters cannot be SS, LS, ML, and so on. By using our knowledge of language we can develop useful heuristics for solving this kind of problem.
A useful type of heuristic is the hill-climbing method. To use the maze analogy, we assume that the goal location is at the top of the hill and all we have to do is make sure the next step takes us further up the hill. In order to know how far up the hill we are, we use an evaluation function – some measure of how good each of the possible paths ahead are. For this we need some way of measuring how good a possible state is. In chess, for example, a state can be assigned a value according to a number of criteria, such as whether the next move gets the opponent into check, or whether it removes a player's piece (and how important that piece is), whether it increases control of the centre of the board, and so on.
One problem with the hill-climbing heuristic is that its success depends on the way the evaluation is formulated, and some will be clearly better than others. A second problem often cited against the hill-climbing heuristic is that it could not produce solutions that involve doing something that is temporarily bad but very good in the long-term. It is often the case that you have to take a step back in order to go forward. For example, in chess players often ‘sacrifice' a piece in order to gain advantage over the opponent. In the Tower of Hanoi problem (see Figure 7.2), the solution involves taking a step that appears to be a move away from the goal. The hill-climbing heuristic would not yield such a move since it encourages the selection of moves that produce better board states. It could be argued, though, that this is not necessarily the case since it depends the breadth and depth of the search.
Option 1 Option 2 Option 3
Figure 7.2 The tower of Hanoi problem. The aim is to move the rings of the tower from the first peg to the last peg by moving only one ring at a time and never placing a larger ring on a smaller ring. Heuristics may employ a broad search, which involves considering a large number of immediate options (such as considering the next move of each of your pieces in chess).
Heuristics may also employ a deep search, which involves considering the many implications of a single option (for example, imagining a long sequence of moves in chess – ‘if I move here, and she moves there, then I could move here'). The success of the hill-climbing heuristic can depend on the depth of the search, and if this is large then it could conceivably uncover a sequence of steps where one of the steps results in a ‘bad' state in the short term but a useful solution in the long term.
A well-known heuristic is means-ends analysis. It involves breaking a problem down into a number of smaller sub-problems. If a number of ends (sub-problems) can be identified then all one has to do is to identify the means of solving each sub-problem. It seems that we use this heuristic for most everyday problems we encounter. Greeno (1974) has found that people do tend to use the means-ends analysis when solving problems, and Ward and Allport (1997) have observed that working memory is very active when people try to organise sub-problems.
Virtually all of the problem-solving strategies noted above involve what are called well defined problems in which the goal is clear and there are well-understood paths or routes one can take. However, many problems do not have these characteristics and are ill defined (Reitman, 1965). Types of ill-defined problems include how to get a university degree with minimum effort, how to stop the neighbour's dog from barking, how to get more money from those sponsoring our studies (usually our parents), how to get a date, and so on. While many of the methods described above may be suitable for well-defined problems such as the game of chess, the Tower of Hanoi problem or anagrams, it is not clear how they could be used to solve ill-defined problems such as those just listed. Of course, there are those who would argue against this, saying that the aim would be to convert an ill-defined problem into a series of smaller well-defined problems (Simon and Chase, 1973). However, since most of the research in this area has been on what could be called ‘toy problems', it is not clear how these methods would ‘scale up' to everyday, real life problems.
In this section you will study the concept of the ‘expert', how their problem-solving strategies might differ from those of the novice, and how experts acquire their expertise. A common view of experts is that they either have superior memory or that they have superior reasoning skills. However, the view that they have some super general ability is not necessarily the case. Rather, experts have a significant amount of domain-specific knowledge and ability. This conclusion is brought about by the extensive research on expertise, and in particular research that compares the expert with the novice.
Much of this research has focused on masters versus beginners at chess. We know from the previous section that chess is a difficult game to learn because of the huge number of possible game states. One might assume that chess experts search deeper and broader when analysing their move, and that they can hold a large amount of information in their superior short-term memories. However, Chase and Simon (1973) have refuted this idea and show that the main difference between the expert and novice chess player is that the expert is able to chunk information in a more efficient way than the novice.
Three types of chess player were shown mid-games from chess that contained around 24 to 26 pieces for five seconds. The players were then asked to recall the position of each piece by reconstructing it using another chess set. The master chess player recreated 16 pieces successfully, the intermediate player recalled eight successfully, while the novice recreated only four successfully. However, when the same types of player were shown random board configurations (24 to 26 pieces arranged in random positions on the board) there were no differences in the number of successful reconstructions between them (each recalled about two or three). The reason why the expert's recall of random board configurations was so poor was that such a configuration is unlikely to be a typical board state and hence its meaning for the chess expert was no greater than for that of the novice. This study shows that experts do not have superior memories but rather chunk information in meaningful ways.
Similar studies have been carried out in other domains with similar results. For example, expert hockey players could remember more player positions that could non-experts when briefly shown a photograph from a game. In addition, expert volleyball players, but not non-experts, are especially good at identifying the location of the ball when shown a photograph from a game (see Allard and Starkes, 1991).
Experts appear to have developed more advanced problem schemas, which are knowledge structures for understanding problems within a specific domain. Chi et al. (1981) studied expert versus novice physicists on a series of physics problems. They found that while the novice physicists grouped information about the problem in terms of their structural characteristics, expert physicists categorised information on the basis of the laws of physics (such as Newton's Third Law). Through extensive experience, experts have gained knowledge of specific configurations of information that they can apply to problems. Lesgold et al. (1988) examined expert versus novice radiologists in their ability to detect disorders in X-ray films. Interestingly, experts were not only able to spot an abnormality more quickly but could also entertain several plausible diagnoses than could the novices. Novices tended to work backwards from a possible hypothesis.
If we want to know how experts acquired their expertise then the simple answer is that they practised a lot. Simon (1980) estimates that it takes about ten years (or 10,000 hours) to acquire expertise in one domain. He argues that practice leads to ‘automatic' actions in response to a problem. Problem-solving can promote learning. However, the use of some heuristics, such as means-ends analysis, can hinder learning rather than help it (Sweller and Chandler, 1994). This is because experts themselves use schema-driven problem-solving methods and not means-ends analysis. Undirected problem-solving strategies can help the learner. In one example, Owen and Sweller (1985) gave a series of trigonometry problems and compared students who were instructed to calculate particular angles and sides (directed instructions) and others who were told to calculate as many angles and as sides as they could (undirected instructions). Greater learning occurred with the undirected instructions than with directed instructions.
A theory of skill acquisition: Anderson (1987)
Anderson has presented a theory of how skills are acquired and has implemented the theory as a computer program. In the theory skill acquisition occurs through a series of stages:
The learner uses declarative knowledge. When we first learn something we adhere rigidly to the instructions we are given and we solve new problems by going over our knowledge we have acquired so far. That knowledge is declarative in the sense that it is of the ‘facts and figures' variety. In learning to play a new piece on the piano, the learner first acquires knowledge of the notes to be played and by which fingers, and continually rehearses this knowledge through practice.
The learner develops procedural knowledge. Based on procedural knowledge the learner can build up ‘procedures' or knowledge of how and when to do some action. Anderson describes procedural knowledge as a collection of productions, which are rules in the IF-THEN form. Programmers will be familiar with IF-THEN lines in programming code. The general case is IF (some condition is met) THEN (take some specific action), e.g. IF (it is raining) THEN (take an umbrella). For our pianist, this stage might refer to the development knowledge of how to play a bar or a segment of the piece as one ‘chunk'. So, when the learner wishes to play that part of the piece they play it automatically rather than by consulting the manuscript (and it is almost as though the fingers have a mind of their own).
The learner refines and enlarges procedural knowledge. Through extensive practice and experience the learner can conceptualise knowledge in larger chunks. It is the stage where our piano player has learned the entire piece as a small number of units. The theory has made predictions about the speed of learning that have been supported empirically (Anderson, 1993; Singley and Anderson 1989). It has also formed the basis of a computer-based tutoring system (Anderson et al., 1995). Although means-ends analysis can sometimes hinder learning, there may be other heuristics that can be taught in helping people acquire expertise. Schoenfeld (1985) has shown that teaching heuristics does not improve learning and argues that although heuristics may describe some aspects of what experts do, they are not informative enough in letting the learner know exactly what to do. Parents often make similar mistakes with their children. For example, they may ask them to take care when playing on a climbing frame. However, the child may not have a good understanding of what the dangers are and the sorts of behaviours that can put themin danger. Rather, the learner needs to know how to implement heuristics (in the language of Anderson 's theory, they need to be taught procedural knowledge).Creativity
Most of the time, the major factor that determines whether someone will solve a problem or not is the extent of their knowledge. However, some problems are difficult to solve because prior knowledge ‘gets in the way' of a solution. In this section you will be studying creativity and creative solutions. We will be asking whether there are special cognitive processes involved in creativity or whether more routine cognitive processes can be used to explain it.
One of the main obstacles to problem-solving is known as functional fixedness. It refers to the way we tend to think of an object only in terms of its common everyday function, rather than how it may be used differently in new contexts. Consider the following problem devised by Duncker (1945). The task is to fix a candle to a wall. The only objects available are: a handful of drawing pins and a box of matches (and a candle, of course). The pins are too small to go right through the candle. Very few people solve this problem without any hints. Try this yourself before reading on.
The solution becomes obvious when you are told to think of the match box as a candle holder. All you then do is fix the match box to the wall with the pins, melt some wax into the base of the match box, and place the candle in the match box on top of the melted wax. (Alternatively, you could also push a pin through the base of the box into the underside of the candle.) The difficulty with this problem lies in seeing the match box as a useful object for solving the problem.
A related idea is that we tend to see familiar patterns when we look at something and our problem-solving is over-influenced by prior knowledge and prior conceptions.
Lateral thinking is the term used when we try to break out of seeing something in a conventional way. Consider the nine-dot problem in Figure 7.3. You are to connect the nine dots using only four straight lines and the pen must not leave the paper. Have a go at this problem before you read on.
Figure 7.3 The nine-dot problem
The solution involves ‘thinking outside the square' (see Figure 7.4). Most people attempt the problem by drawing lines within the square itself and this may be because the dots themselves form imaginary lines. Hence, prior knowledge of visual patterns can constrain our thinking.
Generally, it could be said that functional fixedness and ‘thinking outside the square' involve some aspect of creative thought, and psychologists are becoming increasingly more interested in the study of creativity.
Creative thinking is often characterised by the difference between convergent and divergent thinking ( Guilford , 1959). Convergent thinking typically occurs when there is one solution and the problem-solver gathers information and develops a single overall plan to home in on the solution. In divergent thinking, the problem-solver gathers a variety of information and develops numerous strategies to find one of many possible solutions. Divergent thinking expands on the number of opportunities to find a solution.
Torrance and Witt (1966) developed a series of tasks to measure creativity which included such tasks as thinking of as many different uses of an everyday object as possible, such as a house brick. Another example, is to draw as many objects as you can that have a circular feature. Creativity is measured by:
fluency – the number of objects drawn;
flexibility – the number of distinct objects drawn (for example, if participants drew two types of door then this would score 2 for fluency but only 1 for flexibility);
originality – the infrequency of the object in terms of how different it is from what other people draw.
Creativity measured in this way focuses on the product of the creative process, and the products must be both original and relevant in order to be considered creative. However, psychologists are more interested in the process of creativity (i.e. creative thought) rather than just the product.
Studying creativity is hampered by the problem of defining it and knowing when it has occurred. One of the earliest definitions was provided by Wallas (1926), who proposed four stages of the creative process:
Before creativity occurs, the individual is carrying out some task or trying to achieve some goal and encounters a difficult problem.
After some time the problem-solver leaves the problem and does something else.
The solution suddenly occurs to the problem-solver in a flash of insight.
As not all insights turn out to be useful, the possible solution is checked out and verified.
There are many examples from great thinkers who seem to have arrived at a creative solution to a problem through something like Wallas' four stages. However, Weisberg (1986) has studied these well-known cases and has tried to debunk the idea of these four stages. First,most solutions are identified during the problem-solving process itself and not during a period of rest from the problem. Second, Wallas' ideas may describe the phenomenology of the creative process and what it feels like but do not describe the process. For example, it is not clear what is meant by illumination and, more importantly, how the process of illumination occurs.
Incubation and illumination are dependent upon the notion of unconscious problem solving and occur when we are consciously engaged in a different task away from the problem. However, despite numerous attempts to find empirical support for unconscious problem-solving, very little has been obtained. Smith and Blankenship (1991) argue that in such cases the problem-solver becomes fixated on an inappropriate strategy and when the problem-solver returns to the problem after a break from it, alternative strategies can be considered. Fixating on a single strategy can block off relevant knowledge that could be useful for solving the problem. Smith and Blankenship (1991) provided support for this idea by giving some participants misleading information in an attempt to simulate fixation on an inappropriate strategy. The findings were more consistent with blocking of relevant information than unconscious problem-solving.
It may be that creativity, rather than occurring as a sudden insight, occurs through a series of small incremental steps, during which previous methods and ideas become modified and elaborated (Weisberg, 1986). Weber and Dixon (1989) examined the historical development of a number of inventions. In many cases, inventions are merely modifications and refinements of already existing ideas and do not just suddenly appear.
Simon (1995) argues that ideas that are considered as creative seem to be brought about not just by finding an original or new solution but by reformulating the problem. Understanding how the problem emerged can lead to a potential solution. For example, suppose you want to stop some children from behaving badly in a classroom. Your belief about what causes their unruly behaviour will determine what solution you choose to employ. If you think it is because of a lack of discipline you might want the teachers to be stricter. If you think it is because the children are seeking attention from the teacher and from other children you might want them taught separately. Your solution is dependent upon how you view the problem. Dunbar (2000) examined the reasoning of molecular biologists over a one-year period. He claims that problem reformulation took place regularly within the research teams, often promoted by tough questioning in lab meetings.
Figure 7.4 The solution to the nine-dot problem
According to Ward et al. (1999) the mental processes in what many people would regard as creative thought are no different to those of non-creative thought. They argue that creativity is simply the result of the combined activity of a number of mental processes.
Ward (1994) asked children to imagine that they had visited another planet that had alien life forms. They were then asked to draw the creatures. One could argue that the drawings that resulted were extremely creative and unusual. However, an analysis of the features that the children used in their drawings casts doubt on this assertion. For example, 89 per cent of the aliens were symmetric and 92 per cent had obvious sensory organs. Hence, the features of the aliens were much like creatures found on earth. Ward (1994) argues that their thinking was based on existing knowledge and non-creative thought processes.ction 4
Typical Exam Questions
1. How do people solve problems?
2. How do people acquire expertise?
3. Is creativity a useful concept for explaining how people solve problems?
Hayes, J. R. (1989) The Complete Problem Solver. Hillsdale , NJ : Erlbaum & Associates.
Robertson, S. I. (2001) Problem Solving. Hove: Psychology Press.
This book was first published in 2003 by Crucial, a division of Learning Matters Ltd [ISBN 1 903337 13 5] © 2003 Eamon Fulcher; © 2009 GEFT Consultance Services (geft.co.uk).
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission in writing from Geft Consultancy Services, who may be contacted via www.geft.co.uk.