Give the values of SOURCE: Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. Used by permission of the National Council of Teachers of Mathematics. |

merical values and use it when *x*=3 (with success rates of 69% and 81% for the two groups of students, respectively). They were less successful, however, when asked to derive from the same table the value of *y* when *x=n* (correct response: *y=n*+7; success rates: 41% and 58%, respectively).

The next three sections of the chapter present representative findings from the large body of research on algebra learning and teaching for the three types of algebraic activity sketched above. Since much of this research has been carried out with students making the transition from arithmetic to algebra, it casts light on the kinds of thinking that students bring with them to algebra from the traditional arithmetic curriculum centered on algorithmic computation that has been predominant in U.S. schools.^{17} Indeed, many studies have been oriented toward either developing approaches to teaching algebra that take this arithmetic thinking into account or, more recently, developing approaches to elementary school mathematics that build foundations of algebraic reasoning earlier.

Much research also has focused on linear relations and linear functions, perhaps because these are considered the easiest and are the first ones encountered by students making the transition from arithmetic to algebra. Although the domain of algebra is far richer than linear relations, much of the research at the cusp of arithmetic and algebra focuses on them.^{18} Some of the newer curriculum programs, however, introduce nonlinear relations along with linear relations in the middle grades. In particular, exponential growth relations (e.g., doubling) have been shown to be an accessible topic for middle school students.^{19}