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Students should be given opportunities to formulate problems from given situations and create new problems by modifying the conditions of a given problem. (NCTM 1991, 95)
Children have traditionally solved our problems-problems that we think will be of interest and relevance to them. We need to shift some of this responsibility to students and let them pose problems that they consider to be worthwhile pursuing. Child-generated problem posing is recognized as an important component of children's mathematical activity, if not at the very heart of their learning (Moses, Bjork, and Goldenberg 1993; NCTM 1989, 1991; Silver 1994). Our current vision of curriculum reform is of a balanced variety of rich problem situations, ones that reflect cultural diversity and that engage children in investigating, formulating, representing, and reasoning.
Such reform necessitates a shift in our role from "telling" children what to do to "questioning and listening" (NCTM 1995, 2). This approach will require first, a commitment to creating an environment in which problem posing is a natural process of mathematical learning. Second, it will require us to have strategies for helping children pose meaningful and enticing problems. Before considering these issues, we need to define problem posing and consider some of its merits.
What Is Problem Posing?
Problem posing involves generating new problems and questions to explore about a given situation, as well as reformulating a problem during the course of solving it (Silver 1994). Problem posing can occur before, during, or after solving a given problem. This article is concerned primarily with the "before" and "after" activities; however, this emphasis does not deny the importance of reformulating a problem as one attempts to solve it.
Problem posing and problem solving are obviously closely related. On the one hand, problem posing draws heavily on the processes of problem solving, such as identifying the key elements of a problem and how they relate to one another and to the goal of the problem. On the other hand, problem posing takes children beyond the parameters of the solution process. For example, problem posing can involve children in questioning the origin of the ideas in a problem, or in considering what other problems might arise if the elements of the problem were modified or extended. These ideas...