The structure of the solution of the Mary's Apples puzzle is present in a
large number of problem solutions. The following presents a particularly
simple instance of this. The same problem solving principles used in
introductory algebra story problems apply to subject matter problems that
use mathematical methods. The need for a unified approach to problem
solving based on fundamental mathematical principles is particularly
apparent in physics.
Problem solutions communicate ideas just as sentences or paragraphs of text do. The ideas conveyed by a sentence evolve as the sentence is read. This is a linear process.
Solving a problem is naturally a linear process just as is the reading of a sentence. The ideas used and contained in a problem solution evolve as the problem is solved in the same sense that ideas evolve as a sentence is read. It is failure to use well known principles of communication in solving problems that causes difficulty in problem solving.
It is common that one hears the words, "I understand the ideas, I just can't solve the problems." This is much like saying, "I understand the words, I just can't write a sentence." In both cases it is failure to know the rules for accomplishing the task. In the case of writing and speaking, a great deal of training is provided in learning the rules for these forms of communication. In the case of problem solving, little, if any, training is provided for learning the rules, even though these rules are the same as those used in writing and speaking. It is not at all surprising that students cannot solve problems. They quite simply have not had an opportunity to learn how to solve problems. Indeed, they are trained to be poor problem solvers by the chaotic problem solving present in physics textbooks.