What is problem solving?
A problem arises when we need to overcome some obstacle in order to get from our current state to a desired state. Problem solving is the process that an organism implements in order to try to get from the current state to the desired state.
An historical review of approaches to problem solving
The behaviourist approach
Behaviourist researchers argued that problem solving was a reproductive process; that is, organisms faced with a problem applied behaviour that had been successful on a previous occasion. Successful behaviour was itself believed to have been arrived at through a process of trial-and-error. In 1911 Edward Thorndike had developed his law of effect after observing cats discover how to escape from the cage into which he had placed them. This greatly influenced the behaviourist view of problem solving:
The Gestalt approach
By contrast, Gestalt psychologists argued that problem solving was a productive process. In particular, in the process of thinking about a problem individuals sometimes "restructured" their representation of the problem, leading to a flash of insight that enabled them to reach a solution. In The Mentality of Apes (1915) Wolfgang Köhler described a series of studies with apes in which the animals appeared to demonstrate insight in problem solving situations. A description of these studies, with photographs, can be found here.
The Gestalt psychologists described several aspects of thought that acted as barriers to successful problem solving. One of these was called the Einstellung effect, now more commonly referred to as mental set or entrenchment. This occurs when a problem solver becomes fixated on applying a strategy that has previously worked, but is less helpful for the current problem. Luchins (1942) reported a study in which people had to use three jugs of differing capacity (measured in cups) to measure out a required amount of water (given by the experimenter). Some people were given a series of "practice" trials prior to attempting the critical problems. These practice problems could be solved by filling Jug B, then tipping water from Jug B into Jug A until it is filled, and then twice using the remainging contents of Jug A to fill Jug C. Expressed as a formula, this is B - A - 2C. However, although this formula could be applied to the subsequent "critical" problems, these also had simpler solutions, such as A - C. People who had experienced the practice problems mostly tried to apply the more complex solution to these later problems, unlike people who had not experienced the earlier problems (who mostly found the simpler solutions).
Another barrier to problem solving is functional fixedness, whereby individuals fail to recognize that objects can be used for a purpose other than that they were designed for. Maier (1930) illustrated this with his two string problem.
For a real life example of overcoming fuctional fixedness, see: Overcoming functional fixedness: Apollo 13
What do you think of Köhler's claim that his apes had demonstrated insight?
What proportion of Maier's participants spontaneously found the solution before getting any kind of hint? What did Maier do that led some people to get the correct solution? (these questions require some research)
The cognitive approach to problem solving
Problem space theory
In 1972, Allen Newell and Herbert Simon published the book Human Problem Solving, in which they outlined their problem space theory of problem solving. In this theory, people solve problems by searching in a problem space. The problem space consists of the initial (current) state, the goal state, and all possible states in between. The actions that people take in order to move from one state to another are known as operators. Consider the eight puzzle. The problem space for the eight puzzle consists of the initial arrangement of tiles, the desired arrangement of tiles (normally 1, 2, 3….8), and all the possible arrangements that can be arrived at in between. However, problem spaces can be very large so the key issue is how people navigate their way through the possibilities, given their limited working memory capacities. In other words, how do they choose operators? For many problems we possess domain knowledge that helps us decide what to do. But for novel problems Newell and Simon proposed that operator selection is guided by cognitive short-cuts, known as heuristics. The simplest heuristic is repeat-state avoidance or backup avoidance1, whereby individuals prefer not to take an action that would take them back to a previous problem state. This is unhelpful when a person has taken an inappropriate action and actually needs to go back a step or more.
Another heuristic is difference reduction, or hill-climbing, whereby people take the action that leads to the biggest similarity between current state and goal state. Before reading further, see if you can solve the following problem:
In the hobbits and orcs problem the task instructions are as follows:
On one side of a river are three hobbits and three orcs. They have a boat on their side that is capable of carrying two creatures at a time across the river. The goal is to transport all six creatures across to the other side of the river. At no point on either side of the river can orcs outnumber hobbits (or the orcs would eat the outnumbered hobbits). The problem, then, is to find a method of transporting all six creatures across the river without the hobbits ever being outnumbered.
The solution to this problem, together with an explanation of how difference reduction is often applied, can be found by clicking here.
A more sophisticated heuristic is means-ends analysis. Like difference reduction, the means-ends analysis heuristic looks for the action that will lead to the greatest reduction in difference between the current state and goal state, but also specifies what to do if that action cannot be taken. Means-ends analysis can be specified as follows2:
- Compare the current state with the goal state. If there is no difference between them, the problem is solved.
- If there is a difference between the current state and the goal state, set a goal to solve that difference. If there is more than one difference, set a goal to solve the largest difference.
- Select an operator that will solve the difference identified in Step 2.
- If the operator can be applied, apply it. If it cannot, set a new goal to reach a state that would allow the application of the operator.
- Return to Step 1 with the new goal set in Step 4.
The application of means-ends analysis can be illustrated with the Tower of Hanoi problem.
In 1957 Newell and Simon developed the General Problem Solver, a computer program that used means-ends analysis to find solutions to a range of well-defined problems - problems that have clear paths (if not easy ones) to a goal state. In their 1972 book on problem solving they reported the verbal protocols of participants engaged in problem solving, which showed a close match between the steps that they took and those taken by the General Problem Solver.
There are three ways in which operators can be acquired:
- Trial-and-error. As noted above, this formed the basis of the behaviourist account of problem solving.
- Direct instruction.
- Analogies. Analogies are examples from one domain (the source), whose elements can be used to aid problem solving in another domain (the target). However, novices often struggle to spot analogies, as described here.