Visions of Problems and Problems of Vision: Enhancing the Messiness of Mathematics in the World

Problems and problem solving have a long history in mathematics education (Dewey, 1910; National Council of Teachers of Mathematics [NCTM], 1980; Pólya, 1945; Schoenfeld, 1992; Stanic & Kilpatrick, 1988). The Curriculum and Evaluation Standards for School Mathematics asserted, "Problem solving should   be the central focus of the mathematics curriculum" and placed it as Standard 1 (NCTM, 1989, p. 23). The 1990s saw the development of school mathematics   curricula based on various interpretations of these Standards. In most of these curricula, the mathematics emerges from the solution of problems, and there is a   growing body of research evidence supporting the effectiveness of these curricula (Senk & Thompson, 2003). Teaching mathematics through problem solving also continues to be a focus of mathematics educators independent of the curriculum that is used (Schoen & Charles, in press).