Word Problems

According to the NCTM, the curriculum should be broadened to place more emphasis on conceptual understanding and underrepresented topics in geometry, measurement, and statistics. In addition there should be greater emphasis on interacting with technology to facilitate calculations and graphing, as well as solving more difficult problems. These problems should include word problems with a variety of structures, everyday problems, and problems that take more than a few minutes to solve. Instruction should emphasize the acquisition of knowledge within the context of purposeful activity, rather than first learning computational skills for later application in solving problems.

An example of how a cognitive model can guide instructional choices is illustrated by the work of Fuson and her colleagues at Northwestern University. The curriculum and teaching/learning framework is based on several years of research on how urban Latino children solve addition/subtraction word problems. The framework uses Fuson's conceptual phase model of solving word problems and specifies types of classroom activities for each phase in solving the problem. There is the link between various cognitive phases in solving word problems and classroom activities that can help students become better problem solvers. Although the phases represent stages in solving simple arithmetic problems, they are general enough to capture much of the more complex problem solving that is needed to solve algebra word problems. As students progress through school they move from using objects to using numerical notation to using algebraic notation to represent the problems.

According to a theory proposed by LaBerge and Samuels ( 1974), students would never learn to read unless they could automate some of these components. The reason is that we have a limited amount of cognitive capacity for performing tasks ( Kahneman, 1973) and this capacity would become quickly overloaded if each component skill required substantial mental effort. For example, a child who is investing all his mental effort in recognizing individual words has little mental effort left over to integrate the meanings of the words. The proposed solution is to practice performing the component skills until they can be performed fairly automatically, requiring little cognitive effort. The basic skills for solving word problems-understanding the text, determining temporal and spatial relations, eliminating irrelevant information, identifying the unknown variable, selecting the correct arithmetic operations, determining what to equate in an equation-need to be practiced enough to prevent this mental overload.

It is now time to begin applying some of the general ideas about procedural and conceptual knowledge to word problems. We begin with arithmetic problems that children encounter in the lower elementary grades. These problems are interesting for two reasons. The first is that they provide a valuable source of information regarding the kinds of difficulties young children encounter when they begin working on mathematical problems. For instance, is successful performance at this age constrained by a lack of ability to do addition and subtraction or is successful performance constrained by an inability to understand the problems? The second reason these problems are useful is that it can later be seen whether the analysis of more complex problems builds on the same theoretical ideas used to analyze simple problems. This issue is on multistep arithmetic problems and on algebra word problems.