| An education system adjusted to all its pupils, in line with the present reform of the education system of Quebec has led us in this project, to examine how students with learning problems deal with numbers and measurements in mathematics. In the present study, our purpose is to double-check many of the parameters defined in the work of Brousseau (1987, 1992). La theorie des champs conceptuels (TCC) of Vergnaud (1991), The Theory of the Conceptual Fields (TCF) of Vergnaud (1991), applied to the additives structures, was particularly useful in our analysis of the facts and the interpretation of their representations. In this work, the methodology we adopted is the Didactic Engineering, which allows a better understanding in articulating the contents to each. Using Theory of Didactic Situations in Mathematics (TSDM) (La theorie des situations didactiques en mathematiques), we examined the didactic approaches the teachers have in their relationship with their students. The data for our study, which is of the exploratory and qualitative type, was collected with twenty six students of the second cycle of the primary school. That data was analysed in conformity with a methodology of content analysis.;The study of the interactions between teachers and students revealed that only teachers used words in the process, where they used the approach of the control of the actions (approche du controle des actes), or the approach of control of the meaning (approche du controle du sens), or both strategies to help students with problems.;Depending on the type of problem encountered during these activities of measurements of length and masses, the students had recourse to numerous experiments such as manipulation of the standard measure(s). They proceeded by superimposing, by successive deferments, by folding, by cutting when the standard was exceeding in size, or by reduction or addition of some amount of sand to bring into balance the scale. We noticed also that despite the fact that certain students used their fingers to have a global idea of the external measures of the quantities, many of those same students had recourse to a diversity of other procedures during the same test.;The result presented here support the hypothesis that says that the concepts of size and measurement get more meaning in a specific context, where they relate to real situations lived by the students, as well as by direct comparisons. They reinforce and establish links between the so-called sizes, their properties and the numeric knowledge.;The examination of the student's behaviour revealed two attitudes. Almost all the students used the first attitude, which is of the procedural type. It consisted in using counting systems more or less sophisticated from the planning to the following actions involved. The second attitude implied memorizing for the long term, the result associated with a specific couple of actions and the control of their execution. The observation of the students' attitudes reveals that the errors they made are related to a semantic disruption in their interpretation of the varied tips and strategies the teachers tried to help them with to solve the different problems. Thus, it appeared to us that the difficulties at the conceptual and symbolization levels were more important when the exchange activity involved their competence to evaluate an activity related to the understanding of the task to achieve. In other terms, they had more difficulty with the tasks where they had to establish by themselves to links between the variables, and simulate the actions involved by those tasks. Consequently, the tasks involving exchange operations happened to be more difficult to translate into actions, and were clearly more problematic than the other tasks.;Key words: Didactic of Mathematics, Theory of Conceptual Fields, Additive Structures, Theory of Didactic Situations in mathematics, Didactic Engineering, Importance in Size, Learning Problems, Students Representation, Students behavior, Procedures. |
