SureMath Logo
Home Page How to solve problems



Basic principles of effective communication

A problem solution is a story that is meant to tell how the answer to the question posed evolves from concepts in an orderly, logically coherent manner. A story well told engages the reader.

The presentation of a problem solution is a form of communication. As such it is most satisfactorily executed by using the well known and widely used principles of effective communication.

A sentence is processed linearly and achieves the communication of ideas that are far from linear. A simple model of a sentence is, "The quick brown fox jumps over the lazy dog." This is necessarily processed linearly. In so doing the evolution of the ideas the sentence communicates is readily seen. The focal element of a sentence is the verb, the action. The other elements of the sentence, collectively referred to as the subject and object, attach to the verb in much the way that spokes attach to the hub of a wheel. These can be replaced so that the hub, the verb, can be used to focus a huge range of ideas. Tinker toys provide a means of illustrating this. Understanding sentences and combinations of sentences is a specialty of the field of study called linguistics. Semantics provides the rules for combining the elements of communication in a wide variety of applications.

These principles are taught in such courses as English composition and public speaking. The principles are encountered daily in the form of organized tables of contents of books, well written business letters, essays, reports, indexes, hierarchical directories, effective advertising, computer programming, and many other forms of communication.

A common property of any effective communication process is the existence of an opening, body and close. In composition, emphasis is placed on the importance of the thesis sentence (an opening) which informs the reader of the intent of the essay. The presentation is divided into ordered sections and paragraphs. Each component has an opening, body and close.

Application to problem solving

With these ideas in mind, it is illuminating to examine the problem solutions presented in typical introductory physics books and physics solution manuals. Some examples are shown below. These demonstrate the lack, sometimes total lack, of recognition of any of the principles of effective communication. It is indeed surprising that in introductory physics, of all subjects, problem statements and problem solution presentations are as chaotic as they are.

The words "one must first know" used in the first sentence of the Answer Book Solution are quite common in solution presentations in textbooks. The reader has not been informed of the reason "one must first know." To solve this problem the first thing to be known is Newton's law of motion. The writer "knew" this and anyone well versed in the subject matter "knows" this. The student (by definition) is neither an author nor an expert.

The Answer Book Solution would be semantically correct if the first sentence were replaced with "Newton's law of motion states that force acting on an object and the resulting acceleration of the object are proportional. It is therefore necessary to determine the acceleration in order to determine the force." This serves to establish the theme.

In the solution shown above, the concept, Newton's law of motion, appears at the end of the presentation. This solution is presented in the opposite order in which the writer mentally developed the solution.

Even a cursory examination of the presentation of problem solutions in the typical introductory textbook using mathematical methods shows a distinct lack of use of even the most elementary forms of effective communication. The problem statements and the solution presentations are normally inverted; that is, they are presented in a sequence opposite to that used by the expert when solving the problem. The thought process, the use of the ideas, is retained in the author's mind and not presented on the printed page. This is most dramatically seen in the typical multiple part problem.

The negative effect of this on the learning of a mathematically based subject by a novice is dramatic and tragic. Only the most adept can unravel the situation. This is shown quite strikingly by the work done by Sheila Tobias, in They're Not Dumb, They're Different. The subjects of that study were trained in the "normal" methods of communication. They function from a "common sense" base. Confronted with the inverted approach to problem solving that is present in the typical introductory physics text, they are understandably confused.

The following solution starts with the concept, the idea the problem is intended to embed. This beginning establishes the theme.

The concept-based solution starts explicitly with the basic concept that drives the solution. In contrast, the Answer Book solution infers this concept. The inference process is available only to those who already know the subject. Students, by definition, do not already know the subject. It is noted that the inference process is significantly nonlinear and is not actually the process the expert uses. Instead, it is an interpretation of the process. The expert actually uses a linear process but displays it in a nonlinear form. Those who study the actions of expert problem solvers see the inverted, non-linear result of the experts work and conclude that is the way the expert solved the problem. Since the expert did not in fact solve the problem in the same way the expert presented the solution the results of such research are without meaning.

The concept-based solution is developed step-by-logical-step, each step pointing to what to do next. Simple principles of effective communication are used in the solution presentation. For example, subproblems are indented. The same process is used in many forms of communication to display subtopics of the main theme.

White space in the solution presentation makes it possible for the eye and mind to follow the solution easily. White space is user-friendly. White space is a recognized tool in effective written communication. In verbal communication, white space is known as a pause.

The solution presentations in these web pages appear to be longer and more time consuming to execute than the familiar abbreviated solutions. This is a result of the solution process being completely displayed rather than retaining part of the solution process in the problem solver's mind, where it is unavailable to the reader.

The solutions indeed take more time to develop. It is much as writing an essay. If one ignores spelling, punctuation, logical organization and grammar in writing an essay, the essay can be written quite rapidly. However, such an essay would not be successful. It will fail to communicate. The same thing is true of problem solving. If they are solved and displayed with omitted steps, lack logical organization and do not use the grammar of problem solving, the problem solution will not be successful. It will fail to communicate.

There are many excellent problems designed to develop thinking processes and enhance the understanding of physical concepts. (See, for example, Homework and Test Questions for Introductory Physics Teaching, Arnold Arons, Wiley, NY, 1994). As the Answer Book Solution above shows, the thinking process is not typically revealed to the student. It is retained in the writer's mind.

The student is not shown the tool of problem solving which is most essential to developing and exploring a problem and its solution - the thinking process. They necessarily revert to the abbreviated, inverted models available in textbooks and other materials. These models do not show expert thinking processes. The student is forced into rote processes. An appreciation of the ideas involved cannot be attained by a rote process.

The contribution to understanding concepts inherent in well designed problems is lost quite simply by failing to follow basic rules of effective communication.

A number of problem solving strategies and tactics are commonly suggested to the student. One of these is, "First, understand the problem." This is quite impossible in any but the most trivial situations. The purpose of solving a problem is to develop an understanding of the problem. The understanding evolves in the process of solving it. To say, " First, understand the problem" is to say, "Solve the problem before solving it." Returning to "The quick brown fox jumps over the lazy dog," one sees that the ideas communicated by the sentence evolve as the sentence is read. The sentence is, quite simply, processed linearly from left to right. There is no "pre-analysis" done to figure out what the sentence is going to say.

A second common advice given to students is to list what is given and what is unknown and from that list deduce what equation is to be used. Rather elaborate arrays of boxes preceding the solution are common. The approach is tragically remote from developing the use of ideas but instead embeds an algorithmic method of very limited usefulness. In "The quick brown jumps over the lazy dog" one can readily list the nouns, verbs, adjectives and so forth. This is not a useful process in arriving at the ideas that the sentence communicates. Similarly, that process is not useful in problem solving.

In reading sentences, verbs form the hub. In problem solving it is necessary to identify the action verb. These are such verbs as "find," "determine," "show," and the like. These form the hub to which the other elements of the problem are attached as spokes in a wheel. One of the spokes will answer the question, "find what?" Other spokes (often connected with words such as "like," "if," "assume that," "ignore," etc.) delimit the problem. Spokes are replaceable, giving rise to a large array of problems. Advising the student to list the spokes is counter productive. The hub will ask for spokes as needed.

The suggestion to "sketch" a picture of the problem is common. The word "sketch" would be better replaced by the word "draw." The expert indeed draws a picture. Much of this drawing is in the expert's mind. The expert may sketch reminders but a fully drawn diagram evolves in the expert's mind. The novice's sketch is typically so incomplete as to be useless.

Modern technology provides readily available tools for drawing easily. A well drawn diagram communicates a vast amount of information about the ideas involved in a particular situation. The diagram cannot be executed if the ideas are not known. A "sketch" can be made even though a real grasp of the ideas is not possessed by the sketcher.

Just as understanding a problem evolves as the problem solution progresses, the diagram evolves as the problem solution requests additional information. Frequently, students will develop a diagram before anything else is placed on the paper. This is a rote process, not a thinking process. This preemptive drawing is not driven by the demands of the solution steps. The steps in a problem solution will demand additions or modifications to the diagram as the solution evolves.

The diagram cannot logically be drawn in its entirety before the solution is commenced. A particular example of preemptive drawing is the instruction to show all the forces acting in a particular problem. The "presolving" so introduced defeats the aim of understanding ideas. In the case of two connected masses, for example, only the forces acting on one of the masses are necessary to begin the solution. The solution will, at the appropriate point, ask for inclusion of the details of the effects of the second mass on the first mass. There are two free body diagrams involved and one is developed before the other. Problem solving is a linear process.

Rapidly developing technology confronts an ever-growing population with the need to function in more sophisticated areas of thought and application. A part of this is the use of mathematics as a communication tool. Problem solving using mathematical methods has strong training and application attributes. In order to meet this challenge it is necessary to modernize the teaching and use of problem solving as a tool to understanding and applying ideas. Teaching students to be effective problem solvers, instead of training them to be poor problem solvers, will contribute much to making physical and mathematical ideas available to a broader population. This can be accomplished by bringing common sense to problem solving.

Very few, if any, of those who have viewed this page have even the foggiest idea of what is said here.

Return to Cognitive Research


EQUATIONS TALK

Home Page How to solve problems
The modern problem solutions shown were developed using SureMath.
Copyright 1995. Howard C. McAllister