The Role of Problem Solving in School Mathematics

The Role of Problem Solving in School Mathematics -  Stanic and Kilpatrick (1989) identify three general themes that have historically characterized the role of problem solving in school mathematics: problem solving as context, problem solving as skill, and problem solving as art.

Problem solving as context. The authors divide problem solving as a context for doing mathematics into several subcategories. Problem solving has been used as justification for teaching mathematics. To persuade students of the value of mathematics, the content is related to real-world problem-solving experiences. Problem solving also has been used to motivate students, sparking their interest in a specific mathematical topic or algorithm by providing a contextual (real-world) example of its use. Problem solving has been used as recreation, a fun activity often used as a reward or break from routine studies. Problem solving as practice, probably the most widespread use, has been used to reinforce skills and concepts that have been taught directly.

When problem solving is used as context for mathematics, the emphasis is on finding interesting and engaging tasks or problems that help illuminate a mathematical concept or procedure. To use problem solving as context, a teacher might present the concept of fractions, for example, assigning groups of students the problem of dividing two pieces of licorice so that each gets an equal share. By providing this problem-solving context, the teacher’s goals are multiple: to create opportunities for students to make discoveries about fraction concepts using a familiar and desirable medium (motivation); to help make the concepts more concrete (practice); and to offer a rationale for learning about fractions (justification).

Problem solving as a skill. Advocates of this view teach problem solving skills as a separate topic in the curriculum, rather than throughout as a means for developing conceptual understanding and basic skills. They teach students a set of general procedures (or rules of thumb) for solving problems—such as drawing a picture, working backwards, or making a list—and give them practice in using these procedures to solve routine problems. When problem solving is viewed as a collection of skills, however, the skills are often placed in a hierarchy in which students are expected to first master theability to solve routine problems before attempting nonroutine problems. Consequently, nonroutine problem solving is often taught only to advanced students rather than to all students. When defining the learning objectives of a problem-solving activity, teachers will want to be aware of the distinction between teaching problem solving as a separate skill and infusing problem solving throughout the curriculum to develop conceptual understanding as well as basic skills.

Problem solving as art. In his classic book, How To Solve It, George Polya (1945) introduced the idea that problem solving could be taught as a practical art, like playing the piano or swimming. Polya saw problem solving as an act of discovery and introduced the term “modern heuristics” (the art of inquiry and discovery) to describe the abilities needed to successfully investigate new problems. He encouraged presenting mathematics not as a finished set of facts and rules, but as an experimental and inductive science. The aim of teaching problem solving as art is to develop students’ abilities to become skillful and enthusiastic problem solvers; to be independent thinkers who are capable of dealing with open-ended, ill-defined problems.

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