The diagram at the left 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.
The fundamental element of a problem solution is the REQUEST-RESPONSE-RESULT structure shown above. This is the familiar hierarchical arrangement commonly used in organizing information. Note that the solution presented above shows exactly what was actually done.
The typical textbook solution presentation omits step 1 above. The thing to be done is not included in the presentation. Thus the reason for obtaining the values chosen in step 2 is not known to the reader. The solution to a textbook problem is only partly shown on the printed page. Much of the conceptual development which the writer did is not shared with the reader.
Answer book solutions tend to focus on steps 3 and 4. Concepts are rarely shown as a basis for problem solutions in textbooks and answer books.
Solutions to more complex problems involve the recursive use of the REQUEST-RESPONSE-RESULT structure. An example is shown below and on other pages of this web.
The starting equation is the answer to the question Where to Start? This equation then tells one What to do Next. This process is applied recursively by reading equations left-to-right, responding to requests made by the equations. This hierarchical use of REQUEST-RESPONSE-RESULT leads to the solution of the problem.
Using different solution paths and making different decisions at branches along the path leads to a variety of solution structures for a given problem. In this way one can explore a problem. It provides a way of finding the most elegant solution. This is the solution we show to others! By exploring a problem, one comes to understand the problem and the concepts and tools used to solve the problem.
Each step of a problem requires a verbal statement of the reason for the step. The more challenging a problem is, the greater is the need for the verbal statement. Verbal statements form probes to locate information in one's knowledge space. Verbal statements inform others and one's "self" of the reasoning involved in solving a problem.
Mathematically, the solution consists of an organized, self-driven approach to solving simultaneous equations. Typical word problems consist of several conditions which must be satisfied simultaneously. For this reason problems in all subjects across the curriculum are solved in the same way. The words change, the symbols change, the ideas change but the problem solving process remains the same.
The diagrams above illustrate an essential feature of a problem solution. This is the indentation of subproblems. The provides a logical structure to the solution. The indentation process is used in a large number of areas for effective communication. Outlines, indexes, tables of contents, bulleted charts, and computer programming are a few example. Indentation serves to break a problem into smaller parts in coherent manner.
There has been a massive amount of research, mainly by the interview method, showing that students use the spaghetti approach to organizing a problem solution. This is a natural consequence of the emphasis on the spaghetti approach in early training in arithmetic and algebra.