Connecting mathematics problem solving to the real world

Introduction A major argument for including verbal problems in the school curriculum has always been their potential role for the development in students of skills in knowing when and how to use their mathematical knowledge for approaching and solving problems in practical situations. The application of mathematics to solve problem situations in the real world, otherwise termed mathematical modeling, can be usefully thought of as a complex process involving a number of phases: understanding the situation described; constructing a mathematical model that describes the essence of those elements and relations embedded in the situation that are relevant; working through the mathematical model to identify what follows from it; interpreting the outcome of the computational work to arrive at a solution to the practical situation that gave rise to the mathematical model; evaluating that interpreted outcome in relation to the original situation; and communicating the interpreted results. As several authors have stressed, this process of solving mathematical application problems has to be considered as cyclic, rather than as a linear progression from givens to goals (Burkhardt, 1994; Greer, 1997; Lesh & Lamon, 1992). For several years, it has been argued by many mathematics educators that – in contrast to the intention mentioned above the current practice of word problems in school mathematics does not at all foster in students a genuine disposition towards mathematical modeling, i.e. treating the text as a description of some real-world situation to be modeled mathematically. According to these authors, by the end of the elementary school many pupils have constructed a set of beliefs and assumptions about doing mathematical application problems, whereby this activity is reduced to the selection and execution of one or a combination of the four arithmetic operations with the numbers given in the problem, without any serious consideration of possible constraints of the realities of the problem context that may jeopardize the appropriateness of their standard models and solutions (Davis, 1989; Greer, 1997; Freudenthal, 1991; Kilpatrick, 1987; Nesher, 1980; Reusser, 1988; Schoenfeld, 1991; Verschaffel & De Corte, 1997a). However, evidence supporting this claim was, until recently, rather scarce, except for some oft-cited examples of striking evidence of "suspension of sensemaking" by students when confronted with the well-known problem "How old is the captain?" (IREM de Grenoble, 1980) or the buses item from the NAEP in the U.S. (Carpenter, Lindquist, Matthews, & Silver, 1983). This chapter first reviews briefly a series of recent studies that provide robust empirical evidence showing the omnipresence and the strenght of the phenomenon of disconnecting word problem solving from the real world. Next it is argued that major features of the current mathematics classroom practice and culture are largely responsible for this phenomenon. Finally, a radically different approach to the teaching of mathematical problem solving, based on the so-called “modeling perspective”, is proposed, and exemplarily illustrated by a brief review of a design experiment. A much more detailed review and discussion of the theoretical and empirical work summarized in this paper, is given in Verschaffel, Greer, and De Corte (2000).