A Grossone-Based Numerical Model for Computations with Infinity: A Case Study in an Italian High School

The knowledge and understanding of abstract concepts systematically occur in the studies of mathematics. The epistemological approach of these concepts gradually becomes of higher importance as the level of abstraction and the risk of developing a “primitive concept” which is different from the knowledge of the topic itself increase. A typical case relates to the concepts of infinity and infinitesimal. The basic idea is to overturn the normal “concept-model” approach: no longer a concept which has to be studied and modeled in a further moment but rather a model that can be manipulated (from the calculation point of view) and that has to be associated to a concept that is compatible with the calculus properties of the selected model. In this paper the authors want to prove the usefulness of this new approach in the study of infinite quantities and of the infinitesimal calculus. To do this, they expose results of an experiment being a test proposed to students of a high school. The aim of the test is to demonstrate that this new solution could be useful in order to enforce ideas and acknowledgment about infinitesimal calculus. In order to do that, the authors propose a test to their students a first time without giving any theoretical information but only using an arithmetic/algebraic model. In a second moment, after some lectures, the students repeat the test showing that new better results come out. The reason is that after lessons, students could join new basic ideas or primitive concepts to their calculus abilities. By such doing they do not use a traditional “concept–model” but a new “model–concept” solution.