On Mathematical Problem Posing*

In mathematics classes at all levels of schooling in all countries of the world, students can be observed solving problems .. The quality and authenticity of these mathematics problems has been the subject of many discussions and debates in recent years. Much of this attention has resulted in a rich, more diverse collection of problems being incorporated into school mathematics curricula. Although the problems themselves have received much scrutiny, less attention has been paid to diversifying the sources for the problems that students are asked to consider in school Students are almost always asked to solve only the problems that have been presented by a teacher or a textbook Students are rarely, if ever, given opportunities to pose in some public way their own mathematics problems. Traditional transmission/reception models of mathematics instruction and learning, which emphasized students passively receiving knowledge as a result of transmission teaching, were compatible with a pedagogy that placed the responsibility for problem posing exclusively in the hands of teachers and textbook authors On the other hand, contemporary constructivist themies of teaching and learning require that we acknowledge the importance of studentgenerated pmblem posing as a component of instructional activity. Problem posing has been identified by some distinguished leaders in mathematics and mathematics education as an important aspect of mathematics education [e.g, Freudenthal, 1973; Polya, 1954].. And problem posing has recently begun to receive increased attention in the literature on curricular and pedagogical innovation in mathematics education. In the United States, fOr example, recent reports, such as the Curriculum and evaluation standards for school mathematics [NCTM, 1989] and the Profe,ional standards for teaching mathematics [NCTM, 1991], have called for an increase in the use of problem-posing activities in the mathematics classroom. Both reports have suggested the inclusion of activities emphasizing studentgenerated problems in addition to having students solve pre-formulated problems, as is clearly illustrated in the following excerpt from the Profes'sional standards for teaching mathematics: