SureMath problem solving
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Summary of the Principles of Problem Solving

THE PRIME OBJECTIVE: Produce awareness.

Request-Response-Result Is a natural process that kids use from the day they are born (and quite possibly earlier).

From simple forms this evolves into sequential and nested requests. Responses are sought from their environment and from themselves that will lead to the desired or specified result(s). This produces a hierarchical organization in which each action is a consequence of previous actions and the precursor to subsequent actions.

This natural process is widely used in formal and informal education.

These problem solving plans show the application of the process to solving problems in subject areas that use mathematics for problem solving.

The primary objective is to help the student become aware of the fact that problem solving is not a special area but instead uses the same logical processes with which they are already familiar and use routinely. Indeed, significant mathematics is used without awareness on the part of the user. Helping the student become aware of the mathematical ideas the student already uses in an informal manner helps considerably in establishing formal use of mathematics.

Mathematical problem solving uses some different tools in addition to those used in other problems such as writing a poem or an essay, or riding a bicycle. The logical process is, however, the same. An essay has an opening, body and close. A problem solution has an opening, body and close embodied in the Request-Response-Result paradigm.

About the problem statement. Starting a problem solution:

As problems become more involved the problem statement itself is the primary cause of novice students' difficulty in solving word problems.

The student can become so involved with, and confused by, the various ifs, ands, buts, assume thats, given thats and so forth that there is no hope of solving the problem. The reason for this is quite simple. The human working memory is too small to handle more than a few chunks without strain. When the working memory is overloaded, confusion results.

The solution to this induced confusion is simply to ignore any phrases that start with words like if when reading a problem statement. The item being sought is what the problem asks for quite independent of the constraints contained in the ifs.

The problem statement is simply a source of information. There is no need to unravel, dissect, understand, sort, list, rearrange, reword, interpret, simplify, analyze or take any other action on the problem statement. The initial, and only, action in starting a solution is identifying what is asked for. This is what is meant by understandingthe problem.

Responding to what is asked for:

In general, the response to the request made by the problem requires use of a concept. The concept that the whole is the sum of its parts was used in some problems in the previous lesson plans. The meanings of circumference and area were used in a couple of examples. Identifying appropriate concepts is ordinarily referred to as thinking.

The concept needed may be readily available to the problem solver through common sense and previous experience. Other sources of needed concepts include textbooks, reference books, dictionaries, encyclopedias, publications, the world wide web, teachers, parents and friends.

There are problems in which the needed concept does not exist. Many people are engaged in the research needed to discover and develop these missing concepts.

Use of concepts is quite powerful. For example, the idea that the whole is the sum of its parts is the most widely used concept. Not only is it the basis for solving algebra problems containing words such as coins, lawn mowing, painting, stamps, mixtures, jobs and so forth it is also present in subjects such a physics and chemistry.

Dalton's Law, the law of linear superposition and various conservation laws are all restatements of the idea that the whole is the sum of its parts.

The words change but the idea remains the same.

It is important to help the student become aware of using concepts in early grades. The concepts used there will reappear, albeit using different words, in future learning experiences.

Continuing a problem solution:

A problem solution is continued in the same way it is started except that the equations do the requesting. They will ask for information about their variables. The information needed to respond to these requests may be in the problem statement. You know what you are looking for and can scan through the ifs to locate it. There is no need to fish around.

If the information needed is not present in the problem statement, then it must be obtained from other knowledge sources as described above.

Sometimes the response to what is needed is sufficiently obscured by artful wording of the problem statement that the problem statement is quite puzzling. Such problems are called puzzles. Responding to what is asked for becomes quite important in this case. It provides a means of knowing what to look for.

Help the student become aware of the problem statement as only a source of information like a dictionary or telephone book. It is a place to find information.

Indentation and verbal statements:

The response to a step is indented one step to the right of the requesting equation. This serves to visually display the logical position of the step in the solution.

When responses to all the steps are completed, substitution is made in the requesting equation. The result of this substitution is positioned at the same indent level as the requesting equation. This displays subproblems as distinct entities.

A verbal statement is placed at the same indent level as the equation to which it applies. This verbal statement is a written record of the thoughts that led to writing the equation to which it applies. The words and the equation belong to one another.

Help the student to learn to verbalize. Students tend to be reluctant to state their actions in words. This reluctance stems from not knowing what to say and how to say it. In some instances it is a lack of understanding of what they are doing or lack of familiarity with the needed vocabulary.

Completing a problem solution

Since the responses are substituted into the requesting equation, the last substitution will be at the left margin. This is the result.

A verbal statement following the final result is of particular importance. This statement should answer the question, "What does the result tell me?" In addition to completing the solution, the ending statement serves as a quick check of one's work. For example, having to write that the distance to the moon is two feet would cause the student to reexamine the problem solution.

Help the student understand that an adequate solution presentation does not have to be explained. The explanation should all be recorded in the solution presentation itself. It should not require additional augmentation. The reader of a problem solution should not be required to solve the solution.


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A SureMath solution. Copyright 1998, Howard C. McAllister.