### The Definition of a Problem

Before we discuss problem solving, we first probe into the definition of a problem. Is the following a problem? Keefe has two cats and three dogs. How many pets does he have? For many pre-school-aged children, this could be considered a problem. For the average second-grade student, this problem would be solved within a matter of seconds—no problem. To truly be considered a problem, a mathematical quest must contain some effort or thought on the part of the solver (Brownell, 1942; Polya, 1945). As a teacher you must remember that what is a problem for some students may be a mere exercise for others.

Problem solving involves a variety of skills.

Problem-solving situations call upon children to retrieve previously learned information and apply it in new or varying situations. Knowing the basic arithmetic skills, knowing when to incorporate them into new contexts, and then being able to do so are three distinct skills. Having all three skills makes problem solving easier, but inability in one does not mean that a student does not understand a problem. It may mean that the student’s learning style has not been addressed. Similarly, because students can carry out the operations in isolation does not mean they know when to apply them or how to interpret the numbers involved (Bley & Thornton, 2001, p. 37).

Good problems include modifications that may be made for students with varying skills, abilities, and learning styles. Multiple solutions and multiple methods of solution are encouraged in a classroom that fosters a problem-solving atmosphere. The challenges of creating good problems and maintaining a problem-solving atmosphere in the classroom will be addressed in the next section. In this section we focus on the process and strategies of problem solving.

### Polya’s Four-Step Process

Probably the most famous approach to problem solving is Polya’s four-step process described below (Polya, 1945). Polya identifies the four principles as follows:

- Understand the problem
- Devise a plan
- Carry out the plan
- Look back

The problem-solving process is merely a general guide of how to proceed in solving problems. In many cases, steps of the process will overlap, thus it may not be possible to perform each step of the process in the order given above. These four principles appear in many elementary-level textbook series as early as the kindergarten grade level.

Use of the problem-solving process and specific strategies may be mentioned in a state or district model or framework.

After describing each of the four steps of Polya’s problem-solving process in more detail below, we will discuss a fifth step. This is not suggesting that Polya’s process is incomplete. In fact, the fifth step, extend the problem, is mentioned in Polya’s manuscript as part of the fourth step—look back. These ideas are separated so that the process of extending the problem, especially relevant for teachers, does not become lost in the process of verifying the solution.

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