Η κατανόηση της πυκνής διάταξης των ρητών από μαθητές Β΄ Λυκείου: ένα διδακτικό πείραμα

Abstract

A major source of difficulty in the transition from natural to rational numbers is the interference of natural number knowledge in rational number learning. The framework theory approach to conceptual change assumes that students use their initial theories of numbers as natural numbers to make sense of rational numbers; and that the background assumptions of these theories are not easily revised, an example being the idea that numbers are discrete. Natural numbers are indeed discrete (i.e., given any natural number, it is always possible to define its successor in the natural numbers set). On the contrary, rational numbers are densely ordered (i.e., given any rational number, it is never possible to define its successor). The idea of discreteness stands in the way of students’ understanding of the density of rational numbers. We present a teaching experiment that investigated 15 11^th graders’ understanding of two different aspects of density, namely the fact that there are infinitely many numbers in any rational number interval (“infinity of intermediates”); and that no rational number has a successor (“no successor principle”). We hypothesized that these aspects of density, although mathematically equivalent, are not equally challenging for students. First, students’ initial understandings of density were pre-tested in individual interviews; then each student was exposed to the idea of the arithmetic mean of two numbers that, in principle, could support understanding of both aspects of density; and, finally, each student  revisited the tasks of the pretest and was prompted, when necessary, to use the idea of the arithmetic mean. Most students grasped the “infinity of intermediates” but all continued to assume the successor principle. From the perspective of the framework theory approach to conceptual change, this is a synthetic conception, since students accept the infinity of intermediates while retaining the successor principle.