L'expression des nombres rationnels et leur enseignement initial.
ADJIAGE Robert, 23 novembre 1999

Rational numbers today are the object of a variety of teaching models which generally distinguish two learning phases : a longer phase to treat rhetorically a specific class of problems in order to permit the pupil to conceptualise them, a shorter phase to assimilate the symbols needed to communicate and process them. Significant progress has already been made thanks to this approach. Pupils still have trouble, however, discriminating between the various systems expressing numbers, particularly the notation of fractions and decimals. Another major problem is recognising in the symbolically expressed mathematical objects, those discussed during the conceptualisation phase.
In order to overcome these difficulties, we propose an introduction to rational numbers using graduated lines as the main system of expression. A certain number of parameters were retained in the development of this teaching aid, used in connection with an interactive computer program. The first of these is that pupils are familiar with the graduated lines (e.g. rulers) and will use it over a number of years. It is a tool which can easily be transformed into the semiotic register which discourages routine thinking in relation to rationals. Understanding this first semiotic register facilitates the comprehension of the fractional and decimal registers and allows the pupil to verify his calculations. Moving back and forth between the three leads to better conceptualisation of the rational number.
We were able to confirm and refine our hypotheses through observation of a class having used this approach. Seven distinct skills necessary in mastering rationals were evident and linked to the acquisition of the register of graduated lines. This amply justifies the investment. Our observation confirmed the key role played by the common linguistic register and its articulation with the specific symbolic registers.