In 500 words, what did you find most interesting or astonishing about Chapter 8? In 500 words, what did you find most in
In 500 words, what did you find most interesting or astonishing about Chapter 8?
In 500 words, what did you find most interesting or astonishing about Chapter 9?
In 500 words, what did you find most interesting or astonishing about Chapter 10?
In 500 words, what did you find most interesting or astonishing about Chapter 11?
Respond with some scholarly references. Use citations, cite your references. Cite every sentence with content from your sources. Put the citation at the end of each sentence.
8
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?
209
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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
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 210
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
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 212
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)
2 1 6
4 8
7 5 3
2 1 6
4 8
7 5 3
(w) (v)
2 6 4
7 5
8 1 3
(x)
2 4
7 6 5
8 1 3
(y)
2 4
7 6 5
8
1 3
(z)
2 4
7 6 5
8
1 3
Goal state
2
7 6 5
8 4
1 3
8 4
6
2 7 5
1 3
8
6
4
2 7 5
1 3
8 4
2 7 5
1 3
6 8 4
2 7 5
6
1 3 8
4
2 7 5
6
1 3
2
7 5
6 4
8 1 3
2 4
8
1 6
7 5
3
2
8
1 6
4
7 5
3
2
1 6
8
4
7 5
3
2 8
7 5
1 4 6
3 2
7 8 5
1 4 6
3 2 4
7 8 5
1 6
3
1 6
2 4
7 8 5
3
2
1 6
4 8
7 5 3
1 6
2 4 8
7 5 3
1 6
2 8
4
7 5 3
2 8
1 4 6
7 5 3
8 4
7 5
2
1 6 3
2
1
4
7 5
6 3
8
4
5
3
7
8 1
2
6
into
2
1
8
4
7 5
3 1 2 3
8 4
7 6 5
6
FIGURE 8.3 The author’s sequence of moves for solving an eight-tile puzzle.
Figure 8.4 is like an upside-down tree with a single trunk and branches leading
out from it. This tree begins with the start state and represents all states reachable
from this state, then all states reachable from those states, and so on. Any
path through such a tree represents a possible sequence of moves that a problem
solver might make. By generating a complete tree, we can also find the shortest
sequence of operators between the start state and the goal state. Figure 8.4 illustrates
some of the problem space. In discussions of such examples, often only a
path through the problem space that leads to the solution is presented (for
instance, see Figure 8.3). Figure 8.4 gives a better idea of the size of the problem
space of possible moves for this kind of problem.
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 213
This search space terminology describes possible steps that the problem
solver might take. It leaves two important questions that we need to answer
before we can explain the behavior of a particular problem solver. First, what
determines the operators available to the problem solver? Second, how does the
problem solver select a particular operator when there are several available? An
answer to the first question determines the search space in which the problem
solver is working. An answer to the second question determines which path the
problem solver takes. We will discuss these questions in the next two sections,
focusing first on the origins of the problem-solving operators and then on the
issue of operator selection.
214 | Problem Solving
2
1 6
8
4
7 5
3
2
1
6
8
4
7 5
3
2
1
6
8
4
7 5
3
2
1
6
8
4
7 5
3
2
1
8 6
4
7 5
3
2
1
6 8 4
7 5
3 2
1
6
8
4
7 5
3 2
1
6
8
7 4
5
3
2
1
6
8
4
7 5
3
2
1
6
8
4
7 5
3
2
1
6 8 4
7 5
3 2
1
6 8 4
7 5
3 2
1
6
8
4
7 5
3
2
1
6
8
4
7
5
3 2
1
6
8
7 4
5
3 2
1
6
8
7 4
5
3
FIGURE 8.4 Part of the search tree, five moves deep, for an eight-tile problem. (After Nilsson, 1971.
Adapted by permission of the publisher. © 1971 by McGraw-Hill.)
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 214
Problem-solving operators generate a space of possible states through which
the problem solver must search to find a path to the goal.
•Problem-Solving Operators
Acquisition of Operators
There are at least three ways to acquire new problem-solving operators.We can
acquire new operators by discovery, by being told about them, or by observing
someone else use them.
Problem-Solving Operators | 215
2
1 6
8
4
7 5
2 3
1
6
8
4
7 5
3
2 1
6
8
4
7 5
3 2
1
6
8
7 4
5
3 2
1
6
8 4
7 5
3 2
1
6
8 4
7 5
3
2 1
6
8
4
7 5
3 2
1
6
8
7 4
5
3 1 2
6
8 4
7 5
3
2
1
6
8
4
7 5
3 2
1
6
8 4
7 5
3 2
1
6
8
4
7 5
3
2
1 6
8
4
7 5
3
2 1
6
8
4
7 5
3
2
1
6
8
4
7 5
3 2
6 1
8
7 4
5
3 2
1
6
8
7 4
5
3 1
7 5
2
8 4
6
3 1 2
6
7 8 4
5
3
Goal state
Start state
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 215
Discovery. We might find that a new service station has opened nearby and
so learn by discovery a new operator for repairing our car. Children might
discover that their parents are particularly susceptible to temper tantrums and
so learn a new way to get what they want. We might discover how a new
microwave oven works by playing with it and so learn a new way to prepare
food. Or a scientist might discover a new drug that kills bacteria and so invent a
new way of combating infections. Each of these examples involves a variety of
reasoning processes. These processes will be the topic of Chapter 10.
Although discovery can involve complex reasoning in humans, it is interesting
that it is the only method that most other creatures have to learn new operators,
and they certainly do not engage in complex reasoning. In a famous study
reported in 1898, Thorndike placed cats in “puzzle boxes.” The boxes could be
opened by various nonobvious means. For instance, in one box, if the cat hit
a loop of wire, the door would fall open. The cats, who were hungry, were rewarded
with food when they got out. Initially, a cat would move about randomly,
clawing at the box and behaving ineffectively in other ways until it
happened to hit the unlatching device. After repeated trials in the same puzzle
box, the cats eventually arrived at a point where they would immediately hit
the unlatching device and get out. A controversy exists to this day over whether
the cats ever really “understood” the new operator they had acquired or just
gradually formed a mindless association between being in the box and hitting
the unlatching device. More recently it has been argued that it need not be an
either–or situation. Daw, N.D., Niv, Y., and Dayan, P. (2005) review evidence that
there are two bases for learning such operators from experience—one involves
the basal ganglia (see Figure 1.8), where simple associations are gradually reinforced,
whereas the other involves the prefrontal cortex and a mental model
of how these operators work. It is reasonable to suppose that the second system
becomes more important in mammals with larger prefrontal cortices.
Learning by Being Told or by Example. We can acquire new operators by
being told about them or by observing someone else use them. These are examples
of social learning. The first method is a uniquely human accomplishment because
it depends on language. The second is a capacity thought to be common in primates:
“Monkey see, monkey do.” As we will see, however, the capacity of nonhuman
primates for learning by imitation has often been overestimated.
It might seem that the most efficient way to learn new problem-solving
operators would be simply to be told about them, but seeing an example is often
at least as effective as being told what to do. Table 8.1 shows two forms of
instruction about an algebraic concept, called a pyramid expression, which is
novel to most undergraduates. Students either study part (a), which gives a
semiformal specification of what a pyramid expression is, or they study part (b),
which gives the single example of a pyramid expression. After reading one
instruction or the other, they are asked to evaluate pyramid expressions like
10$2
Which form of instruction do you think would be most useful? Carnegie
Mellon undergraduates show comparable levels of learning from the single
216 | Problem Solving
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 216
example in part (b) to what they learn from the rigorous specification in part (a).
Sometimes, examples can be the superior means of instruction. For instance,
Reed and Bolstad (1991) had participants learn to solve problems such as the
following:
An expert can complete a technical task in five hours, but a novice requires
seven hours to do the same task. When they work together, the novice works
two hours more than the expert. How long does the expert work? (p. 765)
Participants received instruction in how to use the following equation to solve
the problem:
rate1 _ time1 _ rate2 _ time2 _ tasks
The participants needed to acquire problem-solving operators for assigning
values to the terms in this equation. The participants either received abstract
instruction about how to make these assignments or saw a simple example of
how the assignments were made. There was also a condition in which participants
saw both the abstract instruction and the example. Participants given the
abstract instruction were able to solve only 13% of a set of later problems; participants
given an example solved 28% of the problems; and participants given
both instruction and an example were able to solve 40%.
Why would giving examples be better for learning problem-solving operators
than telling someone what to do directly? The problem with direct instruction
is that it can often be difficult to understand what such quantities as rate1
refer to. This information can be clearer in the context of an example. On the
other hand, it can be difficult to see how to extend an example solution from
one problem to another problem. Thus, experiments like Reed and Bolstad’s
indicate that the best learning occurs when participants have access to both
methods. Similar results have been obtained by Fong, Krantz, and Nisbett
(1986) in the domain of statistics and by Cheng, Holyoak, Nisbett, and Oliver
(1986) in the domain of logic.
Problem-Solving Operators | 217
TABLE 8.1
Instruction for Pyramid Problems
(a) Direct Specification
N$M is a pyramid expression for designating repeated addition where each
term in the sum is one less than the previous.
N, the base, is first term in the sum.
M, the height, is number of terms you add to the base.
(b) Just an Example
7$3 is an example of a pyramid expression.
7$3 _ 7 _ 6 _ 5 _ 4 _ 22
7 is the 3 is the
base height
M
a
a
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 217
Problem-solving operators can be acquired by discovery, by modeling example
problem solutions, or by direct instruction.
Analogy and Imitation
Analogy is the process by which a problem solver extracts the operators used
to solve one problem and maps them onto a solution for another problem.
Sometimes, the analogy process can be straightforward. For instance, a student
may take the structure of an example worked out in a section of a mathematics
text and map it into the solution for a problem in the exercises at the end
of the section. At other times, the transformations can be more complex.
Rutherford, for example, used the solar system as a model for the structure of
the atom, in which electrons revolve around the nucleus of the atom in the
same way as the planets revolve around the sun (Koestler, 1964; Gentner,
1983—see Table 8.2). Although this is a particularly famous example of an
analogy, scientists and engineers use such analogies, if often more mundane,
with great frequency. For instance, Christensen and Schunn (2007) found engineers
making 102 analogies in 9 hours of problem solving (see also Dunbar &
Blanchette, 2001).
An example of the power of analogy in problem solving is provided in an
experiment of Gick and Holyoak (1980). They presented their participants with
the following problem, which is adapted from Duncker (1945):
Suppose you are a doctor faced with a patient who has a malignant tumor in
his stomach. It is impossible to operate on the patient, but unless the tumor is
destroyed, the patient will die. There is a kind of ray that can be used to
destroy the tumor. If the rays reach the tumor all at once at a sufficiently high
intensity, the tumor will be destroyed. Unfortunately, at this intensity the
healthy tissue that the rays pass through on the way to the tumor will also be
destroyed. At lower intensities the rays are harmless to healthy tissue, but they
will not affect the tumor either. What type of procedure might be used to
destroy the tumor with the rays, and at the same time avoid destroying the
healthy tissue? (pp. 307–308)
218 | Problem Solving
TABLE 8.2
The Solar System–Atom Analogy
Base Domain: Solar System Target Domain: Atom
The sun attracts the planets. The nucleus attracts the electrons.
The sun is larger than the planets. The nucleus is larger than the electrons.
The planets revolve around the sun. The electrons revolve around the nucleus.
The planets revolve around the sun The electrons revolve around the nucleus
because of the attraction and because of the attraction and weight
weight difference. difference.
The planet Earth has life on it. No transfer.
After Gentner (1983). Adapted by permission of the publisher. © 1983 by LEA, Ltd.
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 218
This is a very difficult problem, and few people are able to solve it. However,
Gick and Holyoak presented their participants with the following story:
A small country was ruled from a strong fortress by a dictator. The fortress was
situated in the middle of the country, surrounded by farms and villages. Many
roads led to the fortress through the countryside. A rebel general vowed to capture
the fortress. The general knew that an attack by his entire army would
capture the fortress. He gathered his army at the head of one of the roads,
ready to launch a full-scale direct attack. However, the general then learned
that the dictator had planted mines on each of the roads. The mines were set so
that small bodies of men could pass over them safely, since the dictator needed
to move his troops and workers to and from the fortress. However, any large
force would detonate the mines. Not only would this blow up the road, but it
would also destroy many neighboring villages. It therefore seemed impossible
to capture the fortress. However, the general devised a simple plan. He divided
his army into small groups and dispatched each group to the head of a different
road.When all was ready he gave the signal and each group marched down
a different road. Each group continued down its road to the fortress so that the
entire army arrived together at the fortress at the same time. In this way, the
general captured the fortress and overthrew the dictator. (p. 351)
Told to use this story as the model for a solution, most participants were able
to develop an analogous operation to solve the tumor problem.
An interesting example of a solution by analogy that did not quite work is a
geometry problem encountered by one student. Figure 8.5a illustrates the steps
of a solution that the text gave as an example, and Figure 8.5b illustrates the
student’s attempts to use that example proof to guide his solution to a homework
problem. In Figure 8.5a, two segments of a line are given as equal length,
and the goal is to prove that two larger segments have equal length. In Figure 8.5b,
the student is given two line segments with AB longer than CD, and his task is to
prove the same inequality for two larger segments, AC and BD.
Our participant noted the obvious similarity between the two problems and
proceeded to develop the apparent analogy. He thought he could
simply substitute points on one line for points on another, and
inequality for equality. That is, he tried to substitute A for R, B
for O, C for N, D for Y, and _ for _.With these substitutions, he
got the first line correct: Analogous to RO _ NY, he wrote AB _
CD. Then he had to write something analogous to ON _ ON, so
he wrote BC _ BC! This example illustrates how analogy can be
used to create operators for problem solving and also shows that
it requires a little sophistication to use analogy correctly.
Another difficulty with analogy is finding the appropriate
examples from which to analogize operators. Often, participants
do not notice when an analogy is possible. Gick and
Holyoak (1980) did an experiment in which they read participants
the story about the general and the dictator and then gave
them Duncker’s (1945) ray problem (both shown earlier in this
section). Very few participants spontaneously noticed the relevance
of the first story to solving the second. To achieve success,
Problem-Solving Operators | 219
R
(a)
O
N
Y
Given: RO = NY, RONY
Prove: RN = OY
RO = NY
ON = ON
RO + ON = ON + NY
RONY
RO + NY = RN
ON + NY = OY
RN = OY
A
(b)
B
C
D
AB > CD
BC > BC
!!!
Given: AB > CD, ABCD
Prove: AC > BD
FIGURE 8.5 (a) A worked-out
proof problem given in a geometry
text. (b) One student’s attempt
to use the structure of this
problem’s solution to guide his
solution of a similar problem.
This example illustrates how
analogy can be used (and
misused) for problem solving.
Anderson7e_Chapter_08.qxd 8/20/09 9:48 AM Page 219
participants had to be explicitly told to use the general and dictator story as an
analogy for solving the ray problem.
When participants do spontaneously use previous examples to solve a problem,
they are often guided by superficial similarities in their choice of examples.
For instance, B. H. Ross (1984, 1987) taught participants several methods for
solving probability problems. These methods were taught by reference to specific
examples, such as finding the probability that a pair of tossed dice will sum
to 7. Participants were then tested with new problems that were superficially
similar to prior examples. The similarity was superficial because both the
example and the problem involved the same content (e.g., dice) but not necessarily
the same principle of probability. Participants tried to solve the new
problem by using the operators illustrated in the superficially similar prior
example. When that example illustrated th
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