PAPERBACK
\$34.95

• #### Index 441-454

2 yields the solution Facility with symbolic computation in algebra has an obvious parallel with, and indeed draws upon, procedural fluency in the domain of number. Just as in arithmetic, aspects of conceptual understanding and strategic competence interact with each other and with procedural fluency in transformational activities in algebra.

Lastly, there are the generalizing and justifying activities. These include problem solving, modeling, noting structure, justifying, proving, and predicting. These activities are not exclusive to algebra, but they often use its language and tools. For example, the consecutive numbers problem (show that the sum of two consecutive numbers is always an odd number) illustrates how algebra is used to generalize and justify.5 Arithmetic can be used to generate many instances to show that the sum of two consecutive numbers is odd: 3+4=7, 12+13=25, and so on. But the representational and transformational aspects of algebra make it possible to justify that the sum is always odd. The sum of two consecutive integers can be represented with algebra as x+(x+1), where the key is the recognition that x represents any whole number. This expression can be transformed into the equivalent expression 2x+1, which is the general form of any odd number. This example illustrates the power of algebra, as against arithmetic, as a tool for making generalizations and providing justifications, at least for those learners who understand how statements using variables express generality.

Generalizing and justifying activities typically involve examining and interpreting representations that have previously been generated or manipulated. Such activities can provide insight into, for example, the underlying mathematical structure of a situation, or they can yield answers to specific questions or conjectures. They encourage students to develop an awareness of the role that algebra can play in mathematical thinking. All of the strands of algebraic proficiency come together in these activities, especially adaptive reasoning.

One of the great strengths of algebra is that, for experts, a great deal of its transformational activity can be carried out in what appears to be a rather automated manner. Once a student makes the transformation rules his or her own, the algorithms of algebra can be executed, in a sense, without thinking. The student needs to be thinking, for example, not of what the letters in the expressions refer to or of the operations he or she is carrying out, but only that the actions on the symbolic objects are allowable. In fact, once an expression or equation has been generated (or provided) and the goal is known, it seems to be treated in an almost mindless fashion. But is that possible?

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001