The transition from the elementary word problems discussed in chapter 4 to the multistep problems discussed in chapter 5 was relatively straightforward. Multistep problems can be solved by formulating a plan for combining the elementary schemas found in chapter 4. But the transition from arithmetic word problems to algebra word problems is more complicated. Although I have used algebra word problems in most of my research on problem solving, I have never been very confident that I could distinguish between algebra word problems and other kinds of problems. One reason is that this distinction seems to depend more on the approach used to solve the problem (whether or not the student uses algebra) than on the problem itself. A better distinction therefore is to distinguish among the different methods that students use to solve problems that we might typically classify as algebra word problems.
According to the NCTM standards, the ability to represent situations with algebraic quantities is a central skill that is a prerequisite to understanding many areas of mathematics. The first section of this chapter looks at what is involved in making this transition from arithmetic to algebra. The transition requires thinking differently about mathematical operations and about the meaning of symbols such as the equals sign. It also requires understanding how letters are used in equations to represent variables. The difficulty of representing relations between variables is illustrated in the extensive research that has been done on variations of the student-professor problem. One consequence of student's difficulty with algebra is that they often attempt to solve word problems by using nonalgebraic approaches. When this is possible, students typically do better by having access to a variety of strategies.