General Kinds of Problems

Choosing a Problem. In 1961, George Polya, considered as the “Father of Modern Problem Solving”, distinguished the difference between immediate application of known material and a real problem. According to him, “To have a problem means: to search consciously for some action appropriate to attain a clearly conceived, but not immediately attainable, aim. ” (p. 117). In 1994, Swenson refined this definition by including “finding solutions for difficulties which: (1) are seen and felt by the learner; (2) he cares about solving; and (3) seem to him to be solvable” (p. 401).

Children will understand mathematics more if they practice solving their own problems instead of merely remembering the teacher’s solutions to a problem. It is important to take note of the children’s natural interest in coming up with good problems which can help the children to learn problem solving. . Word problems have two classifications — closed (only one solution is possible) or open (many solutions are possible). Smith (2006) gives an example for each: Closed: Juanita receives an allowance of $2. 00 each week. What are the fewest numbers of coins she could get?

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Open: Juanita receives an allowance of $2. 00 each week. How many different ways can she get her $2. 00, using coins and dollar bills? In closed problems, only one number sentence is required to solve the problem. Open problems, on the other hand, do not only display solutions using equations. Charts, tables and graphs can be used to show the solution. However different they may be, both provide valuable learning opportunities (Smith, 2006). Problems can also be classified using a variety of methods. It can be mental math, paper-and-pencil math, or calculator math.

Mental math can be used in solving simple operations while calculator math is used in solving more complex operations. A calculator helps the student to focus on the problem as a whole rather than to just remember facts. Using paper and pencil math limits the size of the numbers and the complexity of operations. Children benefit from the same resources as adults do. Teachers must be careful to choose problems that do not promote stereotypes. The teacher must be sensitive to the cultural practices of each family to avoid perceptions of discrimination.

Posing a Problem. After choosing the problem, teachers should now decide on how to propose a problem. Cognitively Guided Instruction suggested that the teacher should read the problem aloud as many times as needed while the children listen even though they were provided a sheet in which the problem was written (Carpenter, Fennema, Franke, Levi, & Empson, 1999; Peterson, Fennema, & Carpenter, 1989). Manipulative objects, such as counters or Unifix cubes, should be present and accessible for use to each child.

In this method, the children solve the problem in whatever way they wish and discuss their thinking. In 1992, Hembree studied the use of pictures in problem solving. The problems chosen by teachers are more often challenges the minds of young learners. A poorly constructed story, or one using mixed order, may only confuse the students. Children want to succeed and share their solutions. Elements that only create confusion take away the children’s enjoyment in solving problems (Smith, 2006). A Positive Environment.

Aside from choosing an interesting and challenging problem, a positive environment also adds in the students’ understanding of math. Many elements of a positive environment are missing in today’s classrooms. Ideally, problem solving should happen on a daily basis, not just at the end of a chapter. The teacher should allow plenty of time. Some students may solve the problem in a day while some need many days before they come up with a solution. The teacher should, ideally, accept any solution however unusual it may be provided that the solutions are valid and give the correct answer.

The teacher should encourage a variety of approaches because there is no one “right way” (Franke & Carey, 1997). In mathematics, what is more important is the process, not the answer. In grading, the teacher should consider the process as well as the product. Discussion within cooperative groups may lessen the risk of having an incorrect solution. However, an individual must sometimes solve the problem alone to exercise his thinking. Problem solving and thinking is a very personal experience, and “no one should be deprived of this valuable opportunity” (Smith, 2006).