Like principles in general, the principles of problem solving have considerable content which is not immediately evident, primarily because of the very simplicity of the statements. The content will become more apparent as you work with the principles.
When first introducing principles in any subject matter new to the reader, it is necessary to use very simple examples, such as our first puzzle. Be assured that these simple examples are solely for the purpose of illustrating principles which are essential to working with more complex situations. Do not fail to use the principles of solving problems just because these simple examples might be solved "in your head." Practice has to be done on the simple cases so the process will come easily on the more challenging ones.
The structure of a more complex problem solution is illustrated by the following schematic diagram.
The banana puzzle also provides an opportunity to set aside some common fallacies about problem solving.
One of these fallacies concerns "given material." The student is often advised to list what is given and what is unknown in a problem. To do that with this puzzle would show the uselessness of that advice. It is time-consuming and is not informative.
We need only seek out what is ASKED FOR, ignoring all else in the beginning step. That step provides a rapid and reliable means of starting and continuing a problem solution.
Another problem solving fallacy is: "Completely understand the problem before trying to solve it." The banana problem and many, many others deny the usefulness of such methods.
"Completely understand the problem before trying to solve it" is quite common in various publications about problem solving. It is easily the most devastating of the problem-solving instructions a novice receives. "Completely understand the problem before trying to solve it" is a way of saying, "Solve the problem before you solve the problem." The short term memory is not up to such a task on any but the most trivial problems.
It is not meaningful to require that a problem be understood before solving it. What does need to be understood clearly at the beginning of any problem is, "What does the problem ASK FOR?"
The understanding of the problem, and the subject matter concepts used in it, develops as the problem is solved. It is the process of solving the problem that brings about the understanding. A problem that is understood is no longer a problem. When you understand it, you have the solution.
In order to solve a problem, a goal must be established, namely: What does the problem ASK FOR?
A problem is not truly understood until a solution is obtained and a conclusion developed. A problem is not solved until one can answer the question,
Although the process of solving a problem is well defined by the principles presented here, and available in the research literature, reliable problem solving is not possible without at least some knowledge of the subject matter of a particular problem. Content and process are so interwoven that they cannot be separated. Having a little knowledge enables one to solve some of the problems, thereby gaining more knowledge and putting more problems within reach.
This is the reason that good problem solutions always lead to better understanding of the subject matter. It is a recursive process, constantly building and reinforcing.
QUESTION: Show that the total energy of a satellite in a circular orbit equals half its potential energy.
COMMENTS
The presentation of this solution emphasizes the use of ideas as opposed to mathematical manipulations.
Ideas are verbalized prior to expressing them in mathematical form. Developing the solution of a problem or subproblem begins with a basic concept responding to the question asked. Verbalization serves to access the knowledge requested by the previous step.
This presentation is markedly different from that used in textbooks. Textbooks quite commonly place the emphasis on mechanisms rather than ideas. The driving concept is rarely shown at the beginning of the solution, but makes its appearance at the end.
The logic of the problem solution is known to the writer, of course, but is not shared with the reader. Novices are then forced to resort to algorithmic methods since they have not been informed of the sequence of ideas actually used by the expert who developed the solution. Textbook presentations are designed to save space and simplify the appearance of the solution by avoiding analytical methods. This deprives the student of an opportunity for a genuine learning experience and development of analytical skills.
The indented organization of subproblems automates the logic of the solution by keeping them in their proper sequence. In addition, the visual arrangement of this presentation conforms to the widely accepted hierarchical structure used in other types of effective communication.
Indentation and verbalization form context-sensitive connections between the steps of solving the problem and the knowledge that supports the solution.
Problem solutions are a very effective communication channel. Optimization of their presentation contributes substantially to the learning process.
