The double nature of didactic models in conceptualizing the evolution of number systems: A mathematical model and a learning tool

Overview & Synopsis

This paper is a small part of an ongoing theoretical study on didactic models as a form of didactic transformations of mathematical notions, concepts and ideas, i.e. of adjusting mathematics for teaching. In what we propose here we argue that regarding this adjustment as mathematical modeling should be inherent to the mathematics teaching: they may enhance the concept development as the ongoing result of students' learning; foster embedding the "big ideas" approach to mathematics learning, and lead to more self-consistently evolving mathematical knowledge. The "big idea" is that numbers are to be studied in the context of number structures, i.e. together with operations defined on them and properties of these operations, and that a familiar number system may serve a model for studying a new one. We illustrate the didactical model's approach at the initial stage of learning fractions.