From test and interview data obtained during an investigation into Grade 10 students' conceptions of algebraic generalisation and justification, we have extracted evidence of the extent to which these students have coordinated the "worlds" of arithmetic and algebra, and can move freely between them. The data show more dissociation than we expected, even among students who were successful at standard algebraic tasks. Conceiving algebra as "generalised arithmetic" may obscure the many genuine obstacles that the learner has to overcome in moving from fluent performance in arithmetic to fluent performance in algebra while achieving and maintaining a smooth coordination of both modes of action.
Historically, algebra grew out of arithmetic and it ought so to grow afresh for each individual. (Mathematical Association, The Teaching of Algebra in Schools, p. 5) Our investigations into the algebraic thinking of high school students show that the connection between the algebra and arithmetic in their minds is not always as direct and transparent as the quoted precept might suggest. Introducing algebra to beginning students as "generalised arithmetic" may be a sensible strategy but there are distinct pedagogical difficulties to be faced if it is adopted. Our account of some of the misconceptions that can occur at the arithmetic/algebra interface may give some pointers to the character of these pedagogical difficulties.
For this article we have drawn on the results of an open-ended test administered to 350 Grade 10 students (ages 15-16), on interviews with 25 of these students, as well as some interviews from a pilot project undertaken two years ago. Our main study was designed to examine students' conceptions of generalisation and justification. Preliminary findings have been reported in Lee and Wheeler (1986), and a fuller report is available in Lee and Wheeler (1987). Here, in this article, we focus on what our data tell us about the coordination of arithmetic and algebra in the students' minds as this is indicated by the extent to which they seem able to move into and out of each of these worlds at will. In general we find serious lapses in their coordination of these two worlds though of course not all students exhibit or express the same degree of dissociation between them.