Abstract:

Curves represent a rich field of study which remains partly unexplored (Bonanno, et al., 2006, 2010; Costabile, & Serpe, 2003), and that can be discovered and rediscovered at school also with the aid of mathematical machines, in a process which falls between tradition and modernity. Research about the educational use of mathematical instruments is not widespraed (Bartolini Bussi, 2000). That is why the educational potential of traditional instruments in the teaching - learning of geometry is at the origin of a limited number of educational experiments carried out in different countries in the world. The reasons for this are many, and range from the affective, cultural, educational and cognitive aspects for the study of the processes of construction of meanings, and the production of typical mathematical reasoning (Bartolini Bussi, & Maschietto, 2007).

This paper presents an example of spirograph use aimed at the discovery and construction of the deltoid curve. The aim of the spirograph use – first in a real and then in a virtual setting - is to help students to develop the skills required for the use of mathematical instruments, methods and models in different situations. For this reason, the example starts off in the classroom with an empirical approach, that is the exploration of a spirograph, a mechanical instrument, which is based on some mathematical principles. Later we move on to the construction of the meaning of representation, filling the gaps between experience and representation, and then gradually reach the graphic construction of the curve, with the use of the dynamic geometry software GeoGebra. The choice to study and represent graphically the deltoid curve using mathematical equipment - between tradition and modernity - is explained by the need to propose methods and processes alternative to traditional educational courses, and to make students aware of the cultural significance of mathematics, which is a dynamic science combining different branches of knowledge.

An instrument is the result of a process, named instrumental genesis, through which the subject builds a scheme of utilization of the artefact (Bèguin, & Rabardel, 2000).

The use of these artifacts, such as teaching aids, fosters the ability to make conjectures and build even complex demonstrations on their properties and function; the interaction between real and virtual, then, is an added value to teaching as it allows the student not only to design, formalize and display, but also to make active and experimental one of the most beautiful and fascinating topics of geometry.

Keywords:

Dynamic Geometry Software, mathematics education, technology-enhanced learning, deltoid curve, spirograph.