The extension of the natural-number domain to the integers in the transition from arithmetic to algebra

The work described in this article was based on a historical perspective in order to analyze the extension of the natural-number domain to the integers in students' transition from arithmetic to algebra in the context of word problems. The initial historical analysis showed the need to consider mutual interrelationships between the algebraic language and the methods of solving word problems and linear equations, for the understanding of the evolution of negative numbers. Four levels of acceptance of these numbers (subtrahend, relative number, isolated number and formal negative number) were extracted from the historical texts. The empirical analysis showed that the first three levels were observed among 35 students of 12–13 years old as well.This revised version was published online in September 2005 with corrections to the Cover Date.