The example below shows that the problem solving process is the same for puzzles (story problems) and for problems based on subject matter. Using concept-driven problem solving assures reliable use of ideas and mathematics.
Simple problems are the first ones encountered and can be solved in the head and with a little work on scratch paper. As course level advances this method becomes more and more frustrating. This is particularly damaging to the more capable students. For these, this type of problem is trivial and they are encouraged to handle it in a trivial way. They do not have the opportunity to develop reliable problem solving skills. Students are trained to be poor problem solvers.
To change this it is necessay to introduce grammar into problem solving. Problems cannot be solved reliably by knowing ideas only. The composition of the solution is equally important. To not use grammar in problem solving is much the same as trying to write a sentence knowing only words but no grammar. The statement, "If they only understood the ideas, they could solve the problems," is the like saying, "If they only knew the words, they could write a sentence."
The perception that "If they only understood the ideas, they could solve the problems" is true has given rise to substantial distortion of the teaching/learning process. A considerable amount of the teaching revolution is based on the premise that "If they only understood the ideas, they could solve the problems" is a true statement. Unfortunately this statement is not true. The statement by students, "I understand the ideas, I just can't solve the problems! " is rather universal. In response to this the threorists in education put ever more stress on teaching ideas. This internal contradiction in educational reform has led to unfortunate results.