In writing, for example, an opening-body-close structure is used in all elements. The opening of an essay is called a thesis sentence.
Similarly, in public speaking, business letters, computer file structures, income tax forms and so forth, hierarchical organization is standard practice for effective communication.
Computer programming is a particularly strong example of the effective use of hierarchical organization. Prior to its introduction in the '70's computer programs could become intractable.
Current problem-solving methods in subject matter courses, such as algebra, physics and chemistry, do not take advantage of the power available in using hierarchical organization.
It would be expected that physics, by its nature, would present subject matter and solve problems using hierarchical organization. The structure of the discipline is very much hierarchical.
However, contemporary organization in presenting physics subject matter and using that knowledge in problem solving is in about the same state that computer programming was 30 years ago.
That this is the case is alarming, in view of the understanding of hierarchical organization reflected in the literature, .
The hierarchical problem solving structure,
is routinely used in every day problems. For example, the sound of a car horn is a Request for action. The Response draws on knowledge space. The sufficiency of the response determines the quality of the Result .
Hierarchical organization is the natural process in writing an essay, planning one's activities, taking notes, studying, outlining and many other activities. Students receive a great deal of training to become proficient in these uses of hierarchical organization. .
There are frequently several requests in a problem. If these requests are sequential, the hierarchical problem-solving structure is typified by the following diagram:
Sequential responses are responses to sequential requests made by the initial request.
It is common for a response to make one or more requests. The basic hierarchical structure is illustrated by the following example.
The logical hierarchical structure is shown at the left and a
particular problem solution at the
right. The first statement is a response to the request made by the problem statement.
In all cases there is an orderly, logic-driven breaking of the problem into its embedded subproblems.
In the example above each response made a single request. For the general case, responses can make multiple sequential requests.
The following diagram shows the type of hierarchical structure that arises in the general problem solution.
Response 1.3, Request 2
Response 2.1, Request 3
Response 3.1, Request 4
Requesting equations are identified by the integers 1, 2, 3 and 4. Responses to the requests by particular requesting equations are 1.1 for the first response to the first request by equation 1, 1.2 for the second, etc. The notation 3.2, for example , means the second response to requests made by requesting equation 3. Results of the embedded subproblems are shown as Result 1, Result 2 and Result 3. The result for subproblem 4 has not been displayed. Obtaining the result of embedded problems is optional. The result for the problem is Result 1 obtained by using the results obtained from the embedded subproblems.
The above material describes the mathematical structure of a problem solution. Of greater importance are the words that reflect the thinking that led to the writing of an equation. The equations are simply a picture of the ideas expressed by the words. The resulting organization of a problem solution is shown schematically below.
The words, which may be a few words, or several sentences or even paragraphs, are indented at the same level as the equation to which they belong. This keeps the thoughts and the mathematical expression of the thoughts together.
In a well developed problem solution, removal of the words would show the coherent mathematial structure of the solution.
There is no need to understand a problem before proceeding to solve it. All that is needed to start a problem solution is to recognize what is asked for. Frequently this is in the sentence preceding a question mark. What is asked for often follows such action words as find, show that, what, how much. This identifies the first request.
All the words that follow such words as if, assuming that, while, and similar constraining words, are of no use in starting a solution and must be ignored in starting a problem solution. These will be asked for when needed as the solution progresses.
Problems are self-solving so far as the logical organization is concerned. The problem solver supplies knowledge and mathematical skills as needed but does not need to figure out how to solve the problem. The equations tell you what to do.
Progress in understanding a problem evolves as the solution evolves much as the meaning of a sentence evolves as it is read.
Frequently a problem, or a sentence, can be understood after it has been processed. Asking "What does the result tell me?" is helpful in developing the understanding.
Solving a problem and obtaining a result does not necessarily guarantee understanding the problem. We encounter many such problems. The problem solving process provides a means of exploring such problems to better define our understanding of the problem.
Copyright: Howard C. McAllister, 1997.