The animation below shows the process involved in developing steps in problem solving.

Start a problem solution by responding to what is asked for. There may be more than one appropriate response. The choice of response is guided by:

• Resources available.
• Knowledge.
• Convenience.
The starting equation requests additional information and/or procedures. There may be more than one appropriate response. The choice of response is guided by:
• Resources available.
• Knowledge.
• Given conditions.
• Convenience.
At each node (a solution state) in the solution there will be requests for additional information and/or procedures. There may be more than one appropriate response. The choice of response is guided by:
• Resources available.
• Knowledge.
• Given conditions.
• Convenience.
The problem solving process consists of recursive use of:
• Request
•     Response
• Result
At any of the nodes any of the several choices can be used. All are valid. Using different choices gives rise to different solution paths. In this way a problem can be explored and greater understanding of the problem can be obtained.

It is much like traveling from one place to another. One can choose different routes and different modes of transportation. In this way one can have experiences that others have not had and see things others have not seen.

The understanding of a problem evolves as the solution evolves. Developing the solution using different solution paths deepens the understanding of the problem and its solution.

It is not logically possible to understand a problem before solving the problem. An understood problem is a solved problem. The plan for solving a problem cannot be made before solving the problem. The plan evolves as the problem is solved.

A problem is understood by solving it, not by pondering it.
In contrast, contemporary (1999) teaching materials on problem solving unfortunately emphasize the pondering approach.

The basic problem solving process is a linear, hierarchical process. Each step is a result of the previous step and a precursor to the next step. The application of this process is straightforward in a wealth of problems commonly encountered. Problems exist for which a solution cannot be obtained. Such problems are a consequence of lack of the knowledge and/or resources needed to solve the problem. Problems that have been unsolvable in the past have become solvable as additional knowledge and/or resources have become available. Many unsolved problems remain. Continuing solutions of such problems await additional knowledge and/or resources.

Problems exist in which constraints, the conditions to be satisfied, change during the problem solving process. The problem solving process then becomes nonlinear. In such situations the solution is frequently approximated and the solution changed as the constraints change. More complex problems arise when some of the constraints are inconsistent, even mutually contradictory. Solutions in such cases fall short of being ideal. Improvement in a solution involves the problem of modifying the constraints. Relationships between nations, social problems and political problems are examples of complex problems. Extensive resources have been created to approximate solutions of such problems. Approximate solutions have varying degrees of satisfaction.

The problem solving process provides operational definitions of the such words and phrases as brainstorming, lateral thinking and problem solving.

For definitive reference material on problem solving see Concepts.