Research in Brief - May 2009 - Volume 109 (5)

Problem Posing and Problem Solving: A Dynamic Connection

Victor Cifarelli & Charlene Sheets

The study of mathematical problem posing has become an important area of investigation by researchers in mathematics education (English, 1997a, 1997b; Kilpatrick, 1987; Silver, 1994; Silver & Cai, 1996). Problem posing, with its emphasis on having students generate and develop their own mathematical problems from particular situations or on the basis of their prior problem solving experiences, may be an important activity that in fact helps students develop as inquiry-based problem solvers.

Proponents of problem posing advocate for the inclusion of problem posing as part of the mathematics curriculum for several reasons. These include the view that problem posing involves the process of reflection to consider imagined mathematical activity (Goldin, 1987; Schoenfeld, 1994; Thompson, 1994), a type of self-generated activity that may stimulate the students' overall abilities to mathematize and develop understanding within new situations. In addition, there is widespread optimism that having students reflect on their mathematics in order to pose new problems may help them become better problem solvers. Hence, posing problems is viewed by many as a useful classroom activity that may help nurture the mathematical thinking, and particularly, the problem solving actions of students.

The exemplary research on problem posing has been reported in a series of studies by English (1997a, 1997b) and Silver and his colleagues (Cai, 1998; Silver, 1994; Silver & Mamona, 1989; Silver & Cai, 1996; Silver & Stein, 1996). For example, English designed a comprehensive framework for developing young children's mathematical problem posing (English, 1997a) and assessed the effectiveness of using problem posing in middle grades classrooms (English, 1997b). In addition, Silver and his colleagues conducted studies of problem posing that encompass a range of important issues related to problem posing, including studies of the problem posing activities of Middle Grades students (Silver & Cai) and in-service teachers (Silver & Mamona), and the effectiveness of problem posing within a contemporary Middle Grades mathematics curriculum (Silver & Stein).

While these studies have undoubtedly added to our knowledge of problem posing as a mathematical activity, the research is less certain concerning the specific roles that problem posing plays in problem solving situations. A particular issue concerns the ways that problem posing and problem solving interact while a student is in the process of solving a problem. In what ways do the solver's initial problem formulations impact on his/her solution activity? Students may view the problem in a particular way that influences the goals he or she sees fit to develop and pursue. For example, he or she may look to break down or reduce the original problem into smaller problems, the solution of which lead to a solution of the original problem. Research studies that examine how the students' personally constructed problems lead to appropriate solution activity would enable mathematics educators at all levels to better understand the diverse reasoning actions of students.

Conversely, in what ways do the solver's results of carried out solution activity help him or her to re-formulate the current problem or pose additional problems? For example, students engaged in the solution of a problem may generate a result that challenges or calls into question their prior goals and actions. In these situations the ways that the student acts to resolve the new question often leads to a reformulation of the original problem which may in turn lead to progress in finding a solution. While studies of college students have documented this recursive property of problem solving, involving successive and on going reformulations of problems (Carlson & Bloom, 2005; Cifarelli & Cai, 2005), studies are needed that examine the development of this process in K-12 settings.

The discussion presented here suggests how problem posing connects naturally to the solver's on going development of solution activity in problem solving situations and thus suggests some important issues for researchers to consider about the role of problem posing in problem solving. Studies are needed that examine these natural connections between the problem posing and solving actions of solvers.

References

Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem-solving framework. Educational Studies in Mathematics, 58(1), 45-75.

Cai, J. (1998). An investigation of U.S. and Chinese students' mathematical problem posing and problem solving. Mathematics Education Research Journal, 10, 37-50.

Cifarelli, V. V., & Cai, J. (2005). The evolution of mathematical explorations in open ended problem solving situations. Journal of Mathematical Behavior, 24, 302-324.

English, L. D. (1997a). Development of fifth grade children's problem posing abilities. Educational Studies in Mathematics, 34, 183-217.

English, L. D. (1997b). Development of seventh grade students' problem posing. In E. Pehkonen (Ed.), Proceedings of the 21st annual conference for the International Group for the Psychology of Mathematics Education, 2, 241-248. Lahti, Finland: University of Helsinki and Lahti Research and Training Center.

Goldin, G. (1987). Cognitive representational systems for mathematical problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. (pp. 125-145). Hillsdale, NJ: Erlbaum.

Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In Alan H. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 123-148). Hillsdale, NJ: Erlbaum.

Schoenfeld, A. H. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.), Mathematical thinking and problem solving (pp. 53-70). Hillsdale, NJ: Erlbaum.

Silver, E. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.

Silver, E. A., & Stein, M. K. (1996). The QUASAR project: The “revolution of the possible” in mathematics instructional reform in urban middle schools. Urban Education, 30, 476-522.

Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27, 521-539.

Silver, E. A., & Mamona, J. (1989). Problem posing by middle school teachers. In C. A. Maher, G. A. Goldin, & R. B. Davis (Eds.), Proceedings of the eleventh annual meeting of the North American Chapter for the International Group for the Psychology of Mathematics Education (p. 263-269). New Brunswick, NJ: Rutgers University.

Thompson, P. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179-234). Albany, NY: SUNY.

Authors' Notes

Dr. Victor V. Cifarelli is an Associate Professor of Mathematics in the Department of Mathematics and Statistics at the University of North Carolina at Charlotte. He earned a BA in Mathematics from the University of Connecticut, an MA in Mathematics from Purdue University, and a PhD in Mathematics Education from Purdue University. His research interests include mathematical reasoning and problem solving, and the mathematical preparation of pre service teachers.

Dr. Charlene Sheets is an Instructor of Mathematics in the Department of Mathematics and Statistics at the University of North Carolina at Charlotte. She earned a BS in Intermediate Education with concentrations in Mathematics, French and Language Arts, and an MA and a PhD in Mathematics Education from the University of Maryland. Her research interests include mathematical reasoning and problem solving, and the mathematical preparation of pre service teachers.