School algebra usually begins in the eighth or ninth grade when students are 14 to 15 years of age. The teaching of equations, functions, and the manipulation of literal expressions and equations traditionally signals the start of algebra. But algebra can also be said to begin when students are taught to extract pertinent relations from problem situations and to express those relations using algebraic symbolism. Both of these conceptions of algebra carry with them strands of continuity and discontinuity with respect to the prior arithmetical learning of students. However, even the seeming continuities require a shift in interpretation from what was considered appropriate in arithmetic.

This chapter begins with a discussion of what algebra is, how it has developed historically, and what demands it makes on the learner from an epistemological perspective. This discussion, which includes a description of the continuities and discontinuities of algebra with respect to arithmetic, forms the backdrop for the second part of the chapter, which presents research findings from several cognitive studies in the learning and teaching of algebra. It should be noted that much of the research cited in this chapter considers the current algebra curriculum as a given. Because recent advances in technology force us to reconsider this given, a section describing the algebra research that has been carried out in computer environments is also included. The chapter concludes with some suggestions for further research.