The diagram at the left below shows solution paths that result from using a concept-based approach to problem solving. There may be several possible concepts which respond to the question asked. These lead to different paths through solution space and draw on different concepts in knowledge space. A solution path consists of continuous, logically connected steps. This produces a natural, logical evolution of the problem solution.
A particular path may provide alternatives as one proceeds along the solution path. By choosing different paths and branches, the solution, and the concepts that support, it can be explored. There is much, though, that is common to the different paths. By such exploration, an understanding of the problem, its solution and the concepts can evolve. The most elegant solution can be discovered. This is the one we show to others!
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 diagram at the right above is the result of not using a concept-based approach to problem solving. No satisfaction or enjoyment is available from the approach shown in that diagram.
Imagine trying to drive from your home to the mall using the randon process shown in the right diagram. Compare this with the systematic process used in the left diagram.
Introductory physics and algebra textbooks and teaching methods and materials generally present problem solutions in the manner depicted in the right-hand diagram. The resulting lack of logical structure makes the material unduly difficult to learn. The potential of problem solving as a path to understanding concepts is, then, not realized.
The National Council for Teaching of Mathematics provides standards that state the goals for mathematics in the K-12 curriculum.
The material presented here will help those who are interested in making the transition to a coherent, concept-based approach to problem solving as represented by the left diagram and supports the NCTM objectives.