Two teams that cancel: a didactic analysis of a teaching progression for signed numbers from Grade 5 to Grade 7

Abstract

Signed numbers make learners revise what addition, subtraction and the minus sign meant for whole numbers. This paper presents the design rationale and a critical analysis of a three-level source progression for signed numbers (Grades 5-7, nominal ages 10-13), from which French-curriculum and International Baccalaureate chapters published at https://commeunjeu.org are assembled. One representation is presented first and taken up throughout: an integer is drawn as a collection of positive and negative units, a positive and a negative unit forming a zero pair that leaves nothing. An integer therefore has many drawings, which makes visible the construction of the integers as classes of pairs of natural numbers. The design hypothesis is that the gesture pupils perform with the units, inserting a zero in the form a + (-a), is shared by the subtraction of a negative number, by the worked examples that accompany the 2026 French programme, and by the derivation of the sign of a product of two negatives through distributivity; the progression does not model that product, it derives it from the zero pair, a derivation that holds only if distributivity is kept. The number line supports position, order and distance; contexts provide applications. The choices are related to the epistemological obstacles described by Glaeser, to studies of counter and number-line models, including a classroom comparison in which a path model gave better results than chips, immediately and five weeks later, and to the French and IB curricula. A block-level audit of the dated source documents found the representation absent from the first level, no exercise asking pupils to draw, and the decisive subtraction never drawn; the revisions made after the audit are recorded. No empirical evaluation is reported; hypotheses for one are stated.