The purpose of the work is to create models of children's construction of the rational numbers of arithmetic that are useful in scientific and educational endeavors. In the context of working with children in logo microworlds in a constructivist teaching experiment, the psychological goal of the investigators is to create a model of children: construction of the operations that generate the rational numbers of arithmetic, along with a scientific language that specifies the mathematics of children in this contentual domain. The educational goal is to elaborate components of possible learning environments for children based on computer microworlds and to include (1) a specification of possible learning periods along with the goals and intentions within these periods, (2) the mathematical concepts and operations of the involved students within the learning period and itineraries of construction, (3) problem situations that served in the establishment and modification of learning environments by the involved students within the learning periods, (4) critical decisions we made in posing the problems to the students, and (5) examples of operating that carried them to new and more powerful concepts. To facilitate this work, the principal investigators will establish a working group on the rational numbers of arithmetic and will conduct a scientific and an educational workshop.
