Many teachers agree that solving problems mathematically involves important cognitive dispositions and skills. The rhetoric of problem solving has been so pervasive among mathematics educators over the last two decades, however, that this agreement has taken on different interpretations, and the related arguments have become confused and misleading. First, many teachers do not distinguish between “doing exercises” and “solving problems.” Second, many confuse the idea of structuring mathematics instruction around problem-solving activities with the idea of teaching the strategies of problem solving. Third, some view problem solving as a process or ability, while others view it as content knowledge.
Consider first the issue of doing exercises compared with solving problems. Exercises are typically considered to be tasks that challenge students to apply known procedures to similar situations. In contrast, problem solving requires analysis, heuristics, and reasoning toward self-defined goals (Smith, 1991). It is clear from this distinction that many students are mostly engaged in doing exercises. They are asked only to apply or match given procedures to similar “problems.” True problem-solving activities are rare in most mathematics lessons. If students are to learn how to solve problems, teachers must engage them with real problems on a regular basis.