Polya's prescription for solving problems consists of four steps that use the 3 R's of problems solving, Request-Response-Result, and a verification of the result.
The solutions below show the incorporation of these steps in forming a logically sound, reliable problem solving process. The solution developments are concept-based instead of the prevailing manipulation-based process.
The first 'Plan ' is the 'Plan ' devised for responding to what the problem asks for, the number of apples Mary has, in this case. The information for formulating the 'Plan ' is in the problem statement.
The second 'Plan ' is the plan devised to respond to what the first 'Plan ' asks for. The information for the second 'Plan ' is contained in the problem statement. It is indented one level since such hierarchical organization is universally used in effective communication.
The third 'Plan ' is the plan devised to respond to what the second 'Plan ' asks for. This is indented to the right since it is a sub problem of the previous problem ( 'Plan '). The process is repeated until there are no more requests. The total 'Plan ' for solving the problem consists of the collection of indented 'Plans '.
With the 'Plan ' complete the third step suggested by Polya, 'Carrying out the plan ' can be done. This consists of successive substitutions.
It is to be noted that each of the 'Plans ' is a new problem independent of the previous 'Plan '. Step 1, 'Understanding the problem,' is applied to the 'Plan ' last written to determine what that 'Plan ' asks for.
In the solution above, the third step, 'Carrying out the plan, ' was delayed till the total 'Plan ' was developed. The problem is "solved" once the total 'Plan ' has been developed. The remainder of the steps, 'Carrying out the plan , ' is not problem solving but rather the use of mathematics to generate the final result. It is a mechanical process. The problem is "solved" once the 'Plan ' is complete. This is quite different than the existing problem solving paradigms. These uniformly focus on 'Carrying out the plan , ' at the expense of devising the 'Plan ' from concepts.
It is characteristic of textbooks and supporting material that only the bottom part of the solution, the left-moving indents here (the manipulation part), is displayed. The problem solving part (the right indents) is not typically shown.
The planning of the solution is the "thinking" part. In general it requires use of knowledge. In simple problems like this one the "knowledge" is contained in the problem statement.
Delaying 'Carrying out the plan ' is not necessary. The display below shows 'Carrying out the plan ' executed as each 'Plan ' is devised.
This arrangement of the solution brings out the Request-Response-Result character of the steps in a problem solution and shows the connection that the solutions presented in these web pages make with Polya's prescription. In the typical problem the unique nested, in parallel/series, structures 'Plan ' developed by this author provide a logically consistent approach to problem solving.
The extension of the four-step process suggested by Polya used in the examples in these web pages applies equally well to more complex problems. The four steps are used recursively, as described above, to generate the solution to a large class of problems. The solution of a physics problem in this manner is shown below.
This solution has the same logical organization as the first one shown. The successive 'Plans ' in this case were devised from knowledge of the subject, resulting in a solution that focuses on concepts as opposed to the usual manipulation-based approach.
Those not possessing the needed knowledge can follow the steps of the solution, being aware of the manner in which using a concept leads to using successive concepts as needed. This is of particular value to those learning the subject.
Textbooks typically display only the bottom part of the solution ('Carrying out the plan '). This is the manipulative part of the solution. It has been observed that students become quite adept at manipulation. Omission of the "thinking" part, 'Devising a plan ' , in examples the students encounter provides no opportunity for the student to develop thinking skills. The 'Devising a plan ' part of the solution depends heavily on understanding concepts. Since coming to an understanding of concepts is a primary goal, the omission of concepts in problem solutions, which are intended to enhance understanding, is particularly distressing.
The solution presentations that students frequently encounter are not the same as the solutions developed by the expert who developed the problem solutions. The expert indeed does the 'Devising a plan ' part of the solution. However, this is frequently done subliminally. The expert may not even be consciously aware of what was done before the first line was written. This effect can be seen quite vividly in a lecture. The lecturer frequently says the words that build the ' planning ' but the first thing written on the board is the manipulative part ('Carrying out the plan ').
The two examples above are particularly simple logical structures consisting of nested 'Plans '. More involved structures are shown elsewhere in these web pages. In all cases, though, Polya's four steps are identifiable, used in combinations which are nested in a parallel and/or series solution structure.
As a consequence all solutions have the same look and feel. The words change, the symbols change, and the concepts change but the problem solving process remains the same.
The reason that the look and feel is the same for all problem solutions is simply that a problem is a request for some result subject to a set of conditions that must be simultaneously satisfied. The ideas used in solving simultaneous equations are then appropriate. A convenient method which handles many problems is elimination by substitution. It is then a mathematical necessity that the look and feel of all problem solutions will be the same.
There are situations in which difficulty is encountered in developing a solution presentation that has the attributes described above. The cause of this is lack of available tools. This leads to the anything goes approach.
The anything goes approach is unfortunately encouraged by NCTM and is the central theme of contemporary (1999) problem solving instruction. This is particularly distressing when used on students in early grades. If builds destructive problem solving habits which leads to the I can't solve word problemssyndrome in later grades where problems are encountered for which the anything goes approach does not work.
The chaotic nature of contemporary K-12 problems solving instruction is wasteful of resources and deprives the student of many opportunities. That this should be the case in view of the extensive understanding of problem solving that exist, as shown in the references of the the previous page, is particularly tragic.