Chapter 7, page 147
- Understand the problem. Effective problem solvers take time to think about all the given
information and how the information is related until they understand the problem thoroughly. - Develop a plan for solution. When the problem is understood, the problem solver can work out a
plan to solve the problem. - Carry out the plan. Once a plan is worked out, the problem solver carries out the steps in the plan.
- Look back to see what can be learned from this process. An effective problem solver does not
stop once the problem is solved.
Polya’s four strategies are not always carried out in a straightforward sequence. For example, a problem
solver might discover while developing a solution plan or while carrying out the solution steps that he had
made a mistake because of a failure to understand the problem properly. This would lead him to cycle back
and redevelop a new understanding of the problem before continuing again with the other steps.
In addition to Polya’s strategies, psychologists have investigated other strategies for solving problems
in a variety of fields, including but not limited to mathematics. In this section we will discuss four strategies
that can help ineffective problem solvers become more effective. As we will discuss, some of these
strategies are specific ways of implementing Polya’s general strategies.
Representing problems. When trying to understand the problem (Polya’s first strategy), an
important step is construct a representation of the problem. Representing the problem (also called
problem representation) means developing a clear “picture” of what one knows about the problem and
what one needs to find out. The picture can be literally a physical drawing or diagram, or it can be a mental
model of the situation. An example of a student drawing an actual physical diagram is a mathematics
student who draws a diagram that captures all the information given in the problem (see Figure 7.6a). An
example of a student constructing a mental model is a high school economics student solving a test problem
regarding effects of inflation on international balances of payments. The student does not draw a diagram
but spends several minutes carefully reflecting on the relevant factors and how they are interrelated.
Effective problem solvers construct complete, meaningful representations of problems before solving
them; ineffective problem solvers often do not (Novick & Bassok, 2005; Pretz, Naples, & Sternberg,
2003). The following simple example illustrates the difference. First and second graders were given this
word problem to solve: “There are 26 sheep and 10 goats on a ship. How old is the captain?” Of 97
children, 76 answered “36” (Reusser, 1988). Only a few students correctly answered that the question did
not provide any information to answer the question; they had taken the time to envision the situation and
realized that the numbers 26 and 10 were irrelevant to the captain’s age. The students who answered 36
simply added the numbers without ever generating a complete representation of what the problem was
asking. They did not pay attention to what the problem was actually about. Clearly, these students were
making no attempt to understand the problem by constructing a meaningful understanding of it; instead,
they were undoubtedly processing the problem very superficially, without paying attention to what the
problem was actually about.
Three characteristics of good problem representations is that they are (a) complete, (b) include
inferences, and (c) exclude irrelevant information. We briefly discuss each of these below.
One key to good problem representations is to make sure that the problem representations are
complete. For complex problems, the amount of information that should be included in the problem
representation will often be more than working memory can hold; hence, written diagrams or notes are
often necessary to make sure that important information is not forgotten. But whether students draw
diagrams, take notes, or carefully envision the situation in their minds, effective problem solvers take the
time to thoroughly represent what they know about the problem before attempting a solution (Voss, Greene,
Post, & Penner, 1983).