Furthermore, skill in algebraic symbol manipulation was not a prerequisite for the students’ success in problem solving, and as the researchers emphasized, “when those students had access to the kind of technological tools that are becoming standard mathematical tools, they could overcome limited personal calculation skills.”^{65}

Although researchers have made notable advances in finding ways to make representing and interpreting algebraic expressions and equations more meaningful for students with the help of computer and calculator technology, similar efforts in the realm of transforming expressions and equations have been less abundant. As inexpensive symbol manipulators continue to become available for the algebra classroom, it may be feasible to develop and evaluate programs that incorporate their use. At present, despite the occasional use of calculator- and computer-supported approaches to the transformational activities of algebra, the traditional rule-based methods for developing manipulative skills tend to dominate. However, few people at any level in education are satisfied that the traditional approach leads to sufficient proficiency in algebra for most students.

**Generalizing and Justifying Activities of Algebra**

In this section, we consider activities such as solving problems, modeling situations, noting mathematical structure, justifying, proving, and predicting. None of these activities is exclusive to algebra, but in all of them algebra is often used as a tool. Several of these activities require a certain level of skill in representing and transforming algebraic expressions, as well as in adaptive reasoning. Two problems from the research literature help illustrate the issues (see Box 8–8).

**Justifying Generalizations**

Students given Problem A in Box 8–8 tended to give a strictly numerical justification in Part 1. The explicit demand of Part 2 to use algebra, however, requires translating the nonspecific number and the sequence of operations into algebraic notation and then manipulating that notation to obtain an expression that can be interpreted in terms of the problem’s conditions. If *x* is the number, that translation yields