142 THE TEACHING AND LEARNING OF ALGORITHMS IN SCHOOL MATHEMATICS

DISCUSSIONS of the role of algorithms in the elementary school curriculum give rise to conflicting notions of how algorithms should be approached in the classroom. These views range from encouraging students to invent their own algorithms with minimal guidance to teaching students to perform traditional algorithms. In this paper, we will discuss an approach that eschews both these extremes. This approach values students ’ construction of nonstandard algorithms. However, it also emphasizes the es-sential role of the teacher and of instructional activities in supporting the development of students ’ numerical reasoning. In addition, this approach highlights the importance of discussions in which students justify their algorithms. It therefore treats students’ development of increasingly sophisticated algorithms as a means for conceptual learn-ing. In our view, an approach of this type is consistent with reform recommendations, like those of the National Council of Teachers of Mathematics (1989), that stress the need for students to develop what Skemp (1976) calls a relational understanding rather than merely to memorize the steps of standard procedures. The contrast between thi