Problem Posing and Solving with Mathematical Modeling

In recent years, we have introduced elementary school children to the powerful world of mathematical modeling. Models are used to interpret real-world situations in a mathematical format. For example, graphs and tables model complex relationships among various phenomena. Traditionally, mathematical modeling has been a topic for the secondary school only. We choose to reverse this trend because we could envision how such rich learning experiences could promote younger children's problem-solving and problem-posing abilities. Children need to develop these abilities to function effectively in a world that is demanding more flexible, creative, and future-oriented mathematical thinkers and problem solvers. Our modeling activities involve children in challenging but meaningful problem-solving situations that allow for multiple solution approaches and call for multifaceted products. Because modeling is an inherently social experience, designed for small-group work, children develop effective communication and teamwork skills. Furthermore, the activities can be solved at different levels of sophistication, enabling all children to access the powerful mathematical ideas embedded in the problem. In contrast with many classroom mathematics problems, modeling activities promote problem posing as well as problem solving primarily because they evoke repeated asking of questions and posing of conjectures (Brown and Walter 2005). As in real-life situations, modeling activities often comprise information that might be incomplete, ambiguous, or undefined, with too much or too little data, and they might involve visual representations that require interpretation. Numerous questions and problems arise as children try to make sense of this information, elicit and work with the embedded mathematical ideas, and modify and refine their models. In this article we explore mathematical modeling as both problem posing and problem solving from two perspectives - that of the child and that of the teacher. We first provide background information on mathematical models and modeling. In doing so we consider a modeling problem related to Australia's cyclone disasters (see fig. 1). We recently designed and implemented this problem in four fourth-grade classrooms (9-year-olds) and in one combined fourth-fifth-grade class (9- and 10- year-olds). [Note: Australia's cyclones are the same weather phenomenon as North American hurricanes.-Ed.