Problem Solving,

Problem Solving

Human ability to solve novel problems greatly surpasses that of any other species,

and this ability depends on the advanced evolution of the prefrontal cortex in

humans. We have already noted the role of the prefrontal cortex in a number of

higher-level cognitive functions: language, imagery, and memory. It is generally

thought that the prefrontal cortex performs more than these specific functions, however,

and plays a major role in the overall organization of behavior. The regions of the

prefrontal cortex that we have discussed so far tend to be ventral (toward the bottom)

and posterior (toward the back), and many of these regions are left lateralized.

In contrast, dorsal (toward the top), anterior (toward the front), and right-hemisphere

prefrontal structures tend to be more involved in the organization of behavior. These

are the prefrontal regions that have expanded the most in the human brain.

Goel and Grafman (2000) describe a patient, PF, who suffered damage to his

right anterior prefrontal cortex as the result of a stroke. Like many patients with damage

to the prefrontal cortex, PF appears normal and even intelligent, and he scored in

the superior range on an intelligence test. In fact, he performed well on most tests,

although he did have difficulty with the Tower of Hanoi problem described later in this

chapter. Nonetheless, for all these surface appearances of normality, there were profound

intellectual deficits. He had been a successful architect before his stroke but

was forced to retire due to loss of the ability to design. He was able to get some work

as a draftsman. Goel and Grafman gave PF a problem that involved redesigning their

laboratory space. Although he was able to speak coherently about the problem, he

was unable to make any real progress on the solution. A comparably trained architect

without brain damage achieved a good solution in a couple of hours. It seems that the

stroke affected only PF’s most highly developed intellectual abilities.

This chapter and Chapter 9 will look at what we know about human problem

solving. In this chapter, we will answer the following questions: • What does it mean to characterize human problem solving as a search of a

problem space? • How do humans learn methods, called operators, for searching the problem

space?

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210 | Problem Solving

• How do humans select among different operators for searching a problem

space? • How can past experience affect the availability of different operators and the

success of problem-solving efforts?

The Nature of Problem Solving

A Comparative Perspective on Problem Solving

Figure 8.1 shows the relative sizes of the prefrontal cortex in various mammals

and illustrates the dramatic increase in humans. This increase supports the

advanced problem solving that only humans are capable of. Nonetheless, one

can find instances of interesting problem solving in other species, particularly

in the higher apes such as chimpanzees. The study of problem solving in other

species offers perspective on our own abilities. Köhler (1927) performed some

of the classic studies on chimpanzee problem solving. Köhler was a famous

German gestalt psychologist who came to America in the 1930s. During World

War I, he found himself trapped on Tenerife in the Canary Islands. On the

island, he found a colony of captive chimpanzees, which he studied, taking

particular interest in the problem-solving behavior of the animals. His best

participant was a chimpanzee named Sultan. One problem posed to Sultan was

FIGURE 8.1 The relative proportions of the frontal lobe given over to the prefrontal cortex in

six mammals. Note that these brains are not drawn to scale and that the human brain is really

much larger in absolute size. (After Fuster, 1989. Adapted by permission of the publisher. © 1989 by Raven Press.)

Squirrel monkey Cat Rhesus monkey

Dog Chimpanzee Human

Brain Structures

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The Nature of Problem Solving | 211

to get some bananas that were outside his cage. Sultan had no difficulty when

he was given a stick that could reach the bananas; he simply used the stick to

pull the bananas into the cage. The problem became harder when Sultan was

provided with two poles, neither of which could reach the food. After unsuccessfully

trying to use the poles to get to the food, the frustrated ape sulked in

his cage. Suddenly, he went over to the poles and put one inside the other, creating

a pole long enough to reach the bananas (Figure 8.2). Clearly, Sultan had

creatively solved the problem.

What are the essential features that qualify this episode as an instance of

problem solving? There seem to be three:

1. Goal directedness. The behavior is clearly organized toward a goal—in

this case, getting the food.

2. Subgoal decomposition. If Sultan could have obtained the food simply

by reaching for it, the behavior would have been problem solving, but

only in the most trivial sense. The essence of the problem solution is that

the ape had to decompose the original goal into subtasks, or subgoals,

such as getting the poles and putting them together.

3. Operator application. Decomposing the overall goal into subgoals is

useful because the ape knows operators that can help him achieve these

subgoals. The term operator refers to an action that will transform the

problem state into another problem state. The solution of the overall

problem is a sequence of these known operators.

Problem solving is goal-directed behavior that often involves setting subgoals

to enable the application of operators.

FIGURE 8.2 Köhler’s ape,

Sultan, solved the two-stick

problem by joining two short

sticks to form a pole long

enough to reach the food

outside his cage. (From Köhler, 1956.

Reprinted by permission of the publisher.

© 1956 by Routledge & Kegan Paul.)

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The Problem-Solving Process: Problem Space and Search

Often, problem solving is described in terms of searching a problem space,

which consists of various states of the problem. A state is a representation of the

problem in some degree of solution. The initial situation of the problem is

referred to as the start state; the situations on the way to the goal, as intermediate

states; and the goal, as the goal state. Beginning from the start state, there are

many ways the problem solver can choose to change the state. Sultan could reach

for a stick, stand on his head, sulk, or try other approaches. Suppose he reaches

for a stick. Now he has entered a new state. He can transform it into another

state—for example, by letting go of the stick (thereby returning to the earlier

state), reaching for the food with the stick, throwing the stick at the food,

or reaching for the other stick. Suppose he reaches for the other stick. Again, he

has created a new state. From this state, Sultan can choose to try, say, walking on

the sticks, putting them together, or eating them. Suppose he chooses to put the

sticks together. He can then choose to reach for the food, throw the sticks away,

or separate them. If he reaches for the food, he will achieve the goal state.

The various states that the problem solver can achieve define a problem

space, also called a state space. Problem-solving operators can be thought of as

ways to change one state in the problem space into another. The challenge is to

find some possible sequence of operators in the problem space that leads from

the start state to the goal state.We can think of the problem space as a maze of

states and of the operators as paths for moving among them. In this model, the

solution to a problem is achieved through search; that is, the problem solver

must find an appropriate path through a maze of states. This conception of

problem solving as a search through a state space was developed by Allen

Newell and Herbert Simon, who were dominant figures in cognitive science

throughout their careers, and it has become the major problem-solving approach,

in both cognitive psychology and AI.

A problem space characterization consists of a set of states and operators

for moving among the states. A good example of problem-space characterization

is the eight-tile puzzle, which consists of eight numbered, movable tiles set

in a 3 _ 3 frame. One cell of the frame is always empty, making it possible to

move an adjacent tile into the empty cell and thereby to “move” the empty cell

as well. The goal is to achieve a particular configuration of tiles, starting from

a different configuration. For instance, a problem might be to transform

212 | Problem Solving

The possible states of this problem are represented as configurations of tiles in

the eight-tile puzzle. So, the first configuration shown is the start state, and the second

is the goal state. The operators that change the states are movements of tiles

into empty spaces. Figure 8.3 reproduces an attempt of mine to solve this problem.

into

2 1 6

4 8

7 5 3

1 2 3

8 4

7 6 5

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My solution involved 26 moves, each move being an operator that changed the

state of the problem. This sequence of operators is considerably longer than necessary.

Try to find a shorter sequence of moves. (The shortest sequence possible is

given in the appendix at the end of the chapter, in Figure A8.1.)

Often, discussions of problem solving involve the use of search graphs or

search trees. Figure 8.4 gives a partial search tree for the following, simpler

eight-tile problem:

The Nature of Problem Solving | 213

(a) (b) (c) (d) (e) (f) (g)

(o) (p) (q) (r) (s) (t) (u)

(n) (m) (l) (k) ( j) (i) (h)

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