Making sense of the minus sign or becoming flexible in ‘negativity’

This article focuses on the kinds of conceptual changes that occur when students have to deal with negative numbers in elementary algebraic operations. Interviews were carried out with twelve 8th-grade level students who were selected on the basis of their results in a test where they were required to reduce polynomials. The questions applied to their strategies and to the meaning they gave to the minus sign. The analysis of the students’ oral and written discourse attests to the presence of two major kinds of conceptual change: the first one results from students’ attempts to reconcile their arithmetical presuppositions about natural numbers and the algebraic rules required to operate with negatives. The nature of the second kind of conceptual change relates to the minus sign and develops through an enlarged understanding and a flexible use of what we called ‘negativity’. We argue that these two kinds of conceptual change cannot fully occur without the students developing a meta-conceptual awareness of their symbolizing activities.