A logically connected sequence of steps defines a valid solution path. A solution path may branch along the way to provide an alternative route. There may be more than one concept that responds to what is ASKED FOR at the beginning of the solution or at some later point in the solution. The choice of which branch to follow is most efficiently made by flipping a coin.
This diagram illustrates the idea that problems often have more than one solution path.
All connected solution paths are valid. There is no such thing as the best way, the right way, or the correct way. The various paths and branches are different ways of exploring the problem. They are different ways of coming to understand the problem. Different paths provide different insights.
It is much the same as traveling from one place to another. One can choose different modes of transportation and different routes. So explore the less traveled paths! See things that others have not seen! Have experiences that others have not had!
There usually is a most elegant way to solve a problem. This is the one we enjoy showing to others! It is discovered by exploring.
The diagram below shows what happens when
After many fitful fragments, still no solution! A great deal of time and effort can be expended uselessly if work is disjointed and illogical. The hit-or-miss method does not work!
With no control, no sense of direction, there can be no reliable means of reaching the solution.
No understanding of the problem, no understanding of the subject matter, no worthwhile results can be obtained in this way. The method does not build confidence, or skills, or knowledge.
Following is an example of five paths which could be followed to solve a particular problem. Information obtained by choosing the various branches could serve special purposes for the user.
The problem: Water is flowing into a conical vessel at the rate r. The vessel has the shape of a right circular cone with horizontal base, the vertex pointing downward. Radius of the base is a, altitude of the cone is b. Determine the rate at which the surface of the water is rising when the depth of the water is y. (Scroll down to the lower left-hand corner for a diagram of the problem.)
The problem is treated in How to Solve It , G. Polya, Princeton University Press, 1945. Polya's model for problem solving is used for the the problem solving presentations made in these web page.
The solution development was done with SureMath
