1. In the book, L’arithmétique des écoles (1927), we find the following definition of algebra which illustrates well the role it played in teaching: “Algebra is a science which simplifies problem solving and generalizes solutions by establishing formulas to solve problems of the same type.” This role was put into practice in problems without numeric data and which were explicitly aimed at a general solution as, for example, the following problems: “A sum of money is divided between two children such that one receives twice what the other gets. How much did each receive?” “Two cyclists set off in the same direction with one traveling a certain number of kilometers more than the other. What is the distance between them after a certain number of hours?”
2. For an analysis of problems involving different types of relations between the quantities (transformation problems, rate problems, comparison problems,…), see Bednarz et al., 1992; Bednarz & Janvier, 1994.