Another strong body of evidence from the Gestaltists that portrays the importance of representation comes from research examining the effects of poor representation (Green and Gilhooly, 2005). Much in the same way that reconstructing a problem can lead to easier solutions being generated, poor representations can actually prevent easier solutions from being found. The 9-dot problem illustrates a finding called the ‘set’ effect (Green and Gilhooly, 2005). This demonstrates how people’s perceptions limit their capabilities of generating relatively simple solutions, in this case, stepping outside the perceived square represented by the dots in order to draw four straight lines that run through each dot at least once.
Another effect found to hinder effective problem solving from constrained representations is ‘functional fixity’ (Green and Gilhooly, 2005). This term relates to the additional difficulty individuals experience when solving problems as a result of being less able use objects in new ways. This was supported by many studies including Dunker’s (1945, cited in Green and Gilhooly 2005) findings which revealed that the solving rate for the ‘box’ problem was higher for people who were not fixed in their view of how the apparatus could be used to solve the puzzle, in other words, individuals who were not constrained by their representations. Elaboration, re-encoding and constraint relaxation are terms suggested by Ohlsson (1992, cited in Green and Gilhooly, 2005) which define the methods so far explored for how individuals accomplish re-representation or restructuring problems which lead to easier solutions.
Linked to representation, but more from the ‘problems’ side than the solver side of the equation presented at the beginning, is the way that problems are structured. This again is interlinked with the characteristics of the solver such as the solving techniques and strategies that make the process easier. Knowledge lean problems broadly fall into one of two categories; well structured or ill structured problems. On these bases, solvers are able to utilise various methods in order to help them accomplish the solutions more effectively. For example, to solve the Tower of Hanoi puzzle (a well structured problem) individuals may employ state-space analysis which is a formal abstract way of representing the underlying structure of problems (Green and Gilhooly, 2005). One difficulty with applying this framework however is that unlike computers, the number of steps that we are able to pre-calculate is limited by the capacity of working memory. The personal representation of the state-space is called ‘problem space’ and another method often used to achieve the end goal is means-end analysis as previously mentioned. This uses a backwards search procedure which systematically looks to reduce the differences between the current state and the goal state. Means-end analysis is widely used in ill structured problem solving which incidentally reflects a large proportion of how our everyday problems are structured. There is however some controversial evidence about the practicality of means-end analysis. Whilst this method seems to be at the core of novices strategies to make problem solving easier, it limits them from using a working forward strategy that experts seem to use (Green and Gilhooly, 2005). The working forward strategy employed by experts appears to be out of reach to novices as they lack domain knowledge. Nevertheless, Sweller, Mawer and Ward (1983, cited in Green and Gilhooly, 2005) argue that this very strategy of working backwards, which is so widely employed actually hinders the efficiency with which we solve problems due to the high processing demands it places of the cognitive system. Thus whilst it would seem unreasonable to suggest that individuals should gain a more extensive knowledge base in all areas so that they can come to solve problems more easily, individuals should attempt to close the gaps between the less useful strategies such as backwards processing (means-end analysis) and more practical strategies (working forward) by implementing other ways around it such as making more use of external representations (which reduce cognitive load) and de-emphasising the end goal (which reduces the bias for adopting a backwards process)(Green and Gilhooly, 2005).
Analogies are another method that have been found to help with problem solving. Citing Gick and Holyoak’s (1980) findings Green and Gilhooly (2005) demonstrate how participants who were presented with the x-ray problem alongside the analogy of a General dividing his forces to attack a castle from all sides were better able to solve the problem than those not presented with the analogy. Furthermore Needham and Begg (1991, cited in Eysneck and Keane, 2005) noted that if analogical reasoning is partnered with understanding the easiness with which problems are resolved is increased. Other studies such as Keane (1988, cited in Green and Gilhooly, 2005) also went on to establish that the closer the surface features of the analogy are to the target problem the more likely they were to facilitate resolution. The strategies presented here once again all interrelated to the notion of representation and how by using good and efficient analogies, problems can be solved relatively easily.
Within ‘complex’ problem solving, the factors that lead to how easily solvable problems are, is largely dependent on who is solving them, thus it is worth having a look at expert problem solving. As with knowledge lean problems, experts who solve knowledge rich problems benefit immensely from having excellent representational skills. The ‘superiority’ of knowledge that experts are believed to have has actually been found to somewhat reside in the way that they represent the problems. Chi, Glaser and Rees (1982, cited in Green and Gilhooly, 2005) found that experts come to generate solutions more efficiently due to their better organisation of grouping problems together according to ‘deep structures’ as oppose to ‘surface structures’ as novices do. This is supported by the notion of ‘chunking’ that had been found to be an immensely practical method for extracting information from the external environment and representing it in internally despite cognitive limitations (Gobert, Lane, Croker, Cheng, Jones, Oliver and Pine, 2001).
A highly influential factor that makes experts who they are is practice. Thus whilst not directly making a problem easier to solve, it is nevertheless a factor that will make problems gradually become easier to solve and thus worth considering. Findings that have been drawn from examining the differences between good and poor learners can offers some insight into ways in which problems can come to be moderately easily solved. Thorndyke and Stasz (1980) and Green and Gilhooly (1990) produced findings that suggest that making effective use of metacognitive processes and strategies enhance learning and thus could be applied to problem solving (cited Green and Gilhooly, 2005).
To conclude, in order to understand how problems can be relatively easily solved, one needs to consider the intricate interactions that exist between the nature of the problem solving tasks and the experience of the problem solver. Whilst acknowledging theoretical limitations, evidence generated from both simple and complex problem solving has lead to a better understanding of how some problems come to be relatively easily solved given their structure. This in turn, combined with the solvers experience also dictates the strategies with which tasks are approached. One answer that seems to apply throughout the equation set out at the beginning of what makes problem solving relatively easy is, representation. This seems a fundamental theme that runs throughout this area of research and one that is suspected to hold many of the answers that link all the variables of problem solving together.
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References
Eysenck, M.W. & Keane, M.T. (2006), ‘Problem Solving and Expertise, Cognitive Psychology, A Students Handbook, Psychology Press, Hove and New York
Gobet, F., Lane.,P.C.R., Croker, S., Cheng, P.C.-H., Jones, G., Oliver, I. and Pine, J.M. (2001) ‘Chunking Mechanisms in human learning’, Trends in Cognitive Sciences, vol.5. no.6, pp.236-43.
Green, A.J.K., and Gilhooly, K. (2005), ‘Problem Solving’ in Braisby, N. and Gellatly, A. (eds), Cognitive Psychology, Oxford University Press, Oxford
