Acculturation institutionnelle du chercheur, de l’enseignant et des élèves de 1re 
secondaire présentant des difficultés d’apprentissage dans la conception et  la gestion de situations-problèmes impliquant des nombres rationnels

Our research looks at how grade seven students with learning disabilities deal  with rational numbers. Several studies have shown the biggest hurdle facing teachers and  researchers is to avoid getting into the vicious cycle of minimizing the issues related to learning rational numbers and learning opportunities of students with a learning disability, which results in students not having the chance to test their knowledge, not taking the risk of becoming involved in their knowledge construction process and not appreciating the impact of their cognitive engagements.    In order to address this challenge, we integrated complex situations. Taking an  acculturation approach, we found the ecological model to be a fitting tool to consider a “dé-transposition/ré-transportation didactique” (Antibi and Brousseau, 2000) and to rebuild a hope-filled memory (Brousseau et Centeno, 1998).    The aim of our research was to: 1) characterize changes in the institutional  acculturation processes of the teacher, the researcher and the students as well as their  effects on developing and managing teaching situations and 2) identify changes in  students’ knowledge of, habits related to and relationship with rational numbers.    Our own six-month classroom integration allowed us to appreciate the effects of  the acculturation process. We noted significant changes in the topogenesis and  chronogenesis of knowledge (Mercier, 1995). Initially, the teacher used the following  approach: she presented the material and the steps the students would need to do; the  students took notes they would then refer to when doing exercises and solving problems.  The teacher would subsequently modify this approach by presenting problems that would  allow students to bring together the different knowledge they had acquired, thus  reconstructing the knowledge they could refer to when making course notes. The students could refer to these notes when completing subsequent exercises. We were also able to observe the effects of integrating various representations of rational numbers on the advancement of didactical timelines (Mercier, 1995) as well as the changes in students’ relationship with and habits related to rational numbers (Bourdieu, 1980). These changes manifested themselves in several ways: a) a noticeable effort during complex situations; b) the use of mathematical methods that took into account the numerical data sets and the  relationship between these sets of data; c) the appearance of “unusual” behaviours [e.g.  coordination of different semiotic registers, using additive/multiplicative operations and non-conventional writing]. Similar behaviours have been observed in several studies conducted on students who did not show signs of learning disabilities (Moss et Case, 1999). Our findings support the obvious importance of considering students with learning disabilities as being mathematically competent, as highlighted by Empson (2003) and Houssart (2002). We feel it is important to note that work done on the representation of rational numbers provided a particularly fertile niche for work on rational numbers, an essential element of knowledge, that would allow students to experience a more harmonious learning process.

Keyphrases: mathematics education, rational numbers, learning disabilities, ecological approach, problem solving