The Complete Problem Solver

John R. Hayes

• Publisher: Routledge
• Year: 2013
• ISBN:
  9781136465208
• Edition: 2
• Topic: Didattica
• Subtopic: Didattica generale

This unique volume returns in its second edition, revised and updated with the latest advances in problem solving research. It is designed to provide readers with skills that will make them better problem solvers and to give up-to-date information about the psychology of problem solving. Professor Hayes provides students and professionals with practical, tested methods of defining, representing, and solving problems. Each discussion of the important aspects of human problem solving is supported by the most current research on the psychology problem solving. The Complete Problem Solver, Second Edition features:
*Valuable learning strategies;
*Decision making methods;
*Discussions of the nature of creativity and invention, and
*A new chapter on writing. The Complete Problem Solver utilizes numerous examples, diagrams, illustrations, and charts to help any readerbecome better at problem solving. See the order form for the answer to the problem below.

I
Problem Solving Theory and Practice

1
Understanding Problems: The Process of Representation

Usually when we solve a problem, we put most of our attention on the problem and very little attention on ourselves—that is, on what we are doing to solve the problem. If we did attend to our own actions, we might notice that they often occur in a characteristic sequence:

1. Finding the Problem: recognizing that there is a problem to be solved.
2. Representing the Problem: understanding the nature of the gap to be crossed.
3. Planning the Solution: choosing a method for crossing the gap.
4. Carrying Out the Plan
5. Evaluating the Solution: asking "How good is the result?" once the plan is carried out.
6. Consolidating Gains: learning from the experience of solving.

This sequence of actions is illustrated in the following problem.

In easy problems, we may go through these actions in order and without any difficulties. In hard problems, though, we may have to do a great deal of backtracking. For example, when we evaluate what we have done, we may decide that our solution is terrible, e.g., "Asbestos bread will not solve the burned toast problem!!" and go back to planning. Or while trying to execute a solution, we may discover something about the problem which will lead us to represent it in an entirely new way—"Oh, now I see what kind of a problem it is!" Retracing of this sort is characteristic of problems that are called "ill-defined." We will discuss these in much more detail later.

Our success as problem solvers depends on the effectiveness with which we can carry out each of the six actions just described. In this chapter, we will examine the nature of problem representations and the processes people use to form them. In addition, we will describe techniques for improving representations so that they make problem solving easier. In the next chapter we will discuss planning, executing, evaluating, and consolidating. We will delay the discussion of problem finding until the final section of the book because this topic is so closely related to the topic of creativity.

How Do People Understand Problems?

Suppose we were to spy on people as they were trying to understand a new problem, such as the Monster Problem below.

Monster Problem #1

Three five-handed extra-terrestrial monsters were holding three crystal globes. Because of the quantum-mechanical peculiarities of their neighborhood, both monsters and globes come in exactly three sizes with no others permitted; small, medium, and large. The medium-sized monster was holding the small globe; the small monster was holding the large globe; and the large monster was holding the medium-sized globe. Since this situation offended their keenly developed sense of symmetry, they proceeded to transfer globes from one monster to another so that each monster would have a globe proportionate to its own size.

Monster etiquette complicated the solution of the problem since it requires: 1. that only one globe may be transferred at a time, 2. that if a monster is holding two globes, only the larger of the two may be transferred, and 3. that a globe may not be transferred to a monster who is holding a larger globe.

By what sequence of transfers could the monsters have solved this problem?

We might see people reading the problem over several tunes and pausing over the hard parts. We might see them drawing sketches or writing symbols on paper, and we might hear them mutter to themselves, something like: "Let's see ... If a monster is holding two globes . . . What does this mean? . . ." If we were to ask people to "think aloud" as they worked on the problem, we would find that their reading, sketching, and muttering reflected a whirlwind of internal activities—imaging, inferencing, decision making, and retrieving of knowledge from memory—activities which are directed toward "understanding the problem." If we look in more detail, we would find that people are selecting information and imaging objects and relations in the problem. For example, after reading the first line of the Monster Problem, a person might form a visual image of three blobs, each touching a circle. The imagined blobs and circles, of course, correspond to the monsters and the globes, and touching in the image corresponds to the relation of holding. The images usually reflect some selection of information, e.g., the blobs may have no hands, or the circles may give no indication that the globes are crystaline.

To understand a problem, then, the problem solver creates (imagines) objects and relations in his head which correspond to objects and relations in the externally presented problem. These internal objects and relations are the problem solver's internal representation of the problem. Different people may create different internal representations of the same problem.

Frequently, problem solvers will make an external representation of some parts of the problem. They do this by drawing sketches and diagrams or by writing down symbols or equations which correspond to parts of the internal representation. Such external representations can be enormously helpful in solving problems.

The Relation of Internal and External Representations

Sometimes we can solve a problem using only an internal representation. For example, most of us can multiply 17 by 23 entirely in our heads and, with a little effort, get the right answer. Many problems, however, are very difficult to solve without the aid of an external representation. The Monster Problem and the Driver's License Problem in the Introduction are examples of such problems. While it is possible to solve the Monster Problem entirely mentally, it is very difficult to keep track of where you are in this problem without an external representation. You find yourself asking questions like, "Did I give the small globe to the big monster or didn't I?" In the Driver's License Problem, if you don't invent and write down a good algebraic notation, you are very likely to confuse such things as Tom's age now with his age at an earlier time.

External representations, then, are often very helpful m solving difficult problems. We should note, though, that external representations can't help us at all unless we also have an internal representation of the problem. Imagine that we are playing chess. In front of us the chess board and pieces provide a very useful external representation of the chess game. But when we make a move, we typically try it in our heads before making it on the board. Planning is done internally. Further, we couldn't make moves either in our heads or on the board if we didn't have an internal representation of how each piece moves. In short, intelligent play would be impossible without an internal representation.

In summary:

1. An internal representation is essential for intelligent problem solving. Internal representations are the medium in which we think, in the same way that words are the medium in which we talk. Without internal representations, we can't think through the solution of a problem, just as without words we can't speak.
2. Sometimes an internal representation is sufficient for solving. If we were very skillful, we could play "blindfold chess," that is, we could play using only our internal representation, but it wouldn't be easy.
3. For many problems, an external representation is very helpful. We will explore how external representations can help later in this chapter.

What Do We Need to Represent in an Internal Representation?

Consider the Monster Problem discussed previously. If we are to solve this problem, there are four problem parts that we need to include in our internal representation:

1. The Goal—where we want the globes to be when we are done.
2. The Initial State—that is, which monsters have which globes at the beginning of the problem.
3. The Operators—the actions that change one problem state into another— in this case, passing globes back and forth; and
4. The Restrictions on the Operators—Monster Problem rules 1,2, and 3.

Here is another problem:

However, the white dots can only move to the left and the black dots to the right.

Restriction

Try to identify the goal, the initial state, the operators, and the restrictions in the following problem:

A farmer traveling to market took three possessions with him; his dog, a chicken, and a sack of grain. On his way, he came to a river which he had to cross. Unfortunately, the only available transportation was an old abandoned boat that would hold only himself and one of his possessions. Taking his possessions across one at a time posed a problem, however. If he left his very reliable dog with the chicken, the dog would very reliably eat the chicken. If he left the chicken with the grain, the chicken would eat the grain and then burst, improving neither of them.

How did the farmer manage to get all his possessions safely across the river?

While all four problem parts are essential in these two problems, this isn't always the case. All problems involve at least a goal, but many problems omit one or more of the other three parts. Suppose a friend says to us, "Get to my house at 10 o'clock." That statement specifies the goal you are to accomplish, but nothing else. It doesn't specify where you should start—north, south, east, or west. There is no special initial state. Further, it doesn't matter how you get there—you can walk, hop, skate, unicycle, take a cab, a helicopter, a large bird, anything—it doesn't matter—no operator is specified. Further still, no restrictions were specified—e.g., "If you hop, use only the left foot," or "If you come by bird, don't use a sparrow." Some other problem statements which specify only a goal are: "Be a success, my child," and, "Prove your point."

Some problems specify just initial state...