The process of problem solving                          5/1

Continuing

How
to
do it
Having learned how to start a problem solution, you already know how to continue a problem solution. The process is the same, namely, responding to what is asked for.

It is now the equations that do the asking. Work from left to right in the requesting equation, responding to what each term requests.

The first step in continuing the solution to the Brake Liner problem previously started is shown in Figure 5.1.

Figure 5.1
To continue a problem solution it is only necessary to respond to the request(s) made by the equation(s).

In problems encountered in elementary and middle school the answer to what is asked for is usually in the problem statement. As you move on to more advanced work the answer to what the equation asks for is less likely to be in the problem statement. Indeed in subject matter courses like physics or chemistry the answer to what is asked for will rarely be in the problem statement.

Figure 5.2 shows the next step in the solution. Again it is simply a matter of responding to what the current requesting equation asks for.

Figure 5.2
The process of solving a problem continues in the manner illustrated by the previous two figures. There is one more step in completing the solution to the current problem. It is accomplished in the same way. Simply respond to what the last equation asks for. This is shown in Figure 5.3.

Figure 5.3
The last equation in Figure 5.3 does not ask for additional information. This alerts us to the fact that the solution is complete. We obtain the result of solving the problem by successive substitution of the responses into the starting equation. Standard algebraic operations are used in the process of substituting and simplifying to obtain the result. Figure 5.4 animates the process of obtaining the result. The verbal statements have been suppressed in order to better illustrate the process of generating the result of solving the problem.

Figure 5.4
Use
modern
methods
for
problem solving
Problem solving instruction normally emphasizes listing the facts given in the problem statement. This is done in a logically consistent manner in this example. Listing the facts logically automatically generates the solution of the problem.

Developing the solution of a problem is a straight-forward process. Generating the result of solving the problem requires algebraic skills. Such skills are developed by practice.

Just as word processors have greatly increased creativity in writing, the use of symbolic algebra processors for mathematics greatly extends the creative possibilities in problem solving. Because of the tedious, error-prone steps in mathematical operations, curriculum development and research focus on devising means of solving problems that avoid the use of mathematics as a language. The existence of symbolic algebra processors makes this self-defeating process unnecessary.

In applying the Request-Response-Result paradigm to the current problem, the Result was implicit in the individual steps. It is sometimes useful to show the intermediate results. This is particularly useful for the problem solving novice and for those whose mathematical skills are not well developed. The Result for each step of this problem is shown Figure 5.5.

Figure 5.5
Problem solving is a natural process that we use in all of our affairs. We are born problem solvers. The logic of problem solving, Request-Response-Result, is common to developing solutions for all problems. Problems differ only in the tools and resources used to solve them. The resources needed to solve mathematical problems, social problems, relationship problems, political problems, and problems between nations, as examples, differ in important aspects. However, the problem solving process is common to these diverse problems.
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