We interrupt this program to bring
you this brief message about problem solving.
About understanding the problem
Only the most trivial problems can be understood in a literal sense.
Understanding a problem evolves as the problem is solved. Once the result
is obtained the problem may be understood. This problem illustrates the
uselessness of starting by listing the unknowns and givens, drawing a
picture, making tables and so forth. These only consume time and energy
without adding usefully to developing the solution of the problem. These
tools may be useful at some point in the solution. The solution steps
will tell you when and where to use them.
To solve a problem, simply respond to what the problem asks for in a
manner as simple and direct as possible. This response will tell you
what to do next. Repeating the REQUEST-RESPONSE process leads you through
the problem solution.
Once there are no more requests, you can generate the RESULT. Generating
the result is a mechanical process involving mathematical operations and
can well be tedious and error-prone.
About the result
Obtaining the result and a numerical value, if requested, does not
necessarily mean the problem has completely solved and understood.
Virtually all problems one encounters in school work and real life are
open ended. There can be several solution paths for a problem. These
yield additional understanding of the problem. Tables and graphs of the
result can be made revealing additional understanding. The result can be
evaluated in limiting cases. This often shows whether the result obtained
is in accord with common sense. Any problem can be explored in many ways.
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