International Journal for Technology in Mathematics Education, Volume 13, No 1 Using Cabri3D Diagrams For Teaching Geometry By Giuseppe Accascina 1 , Enrico Rogora 2 1 Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza” accascina@dmmm.uniroma1.it 2 Dipartimento di Matematica, Università di Roma “La Sapienza” rogora@mat.uniroma1.it Received: 4 October 2005 Revised: 29 March 2006 Cabri3D is a potentially very useful software for learning and teaching 3D geometry. The dynamic nature of the digital diagrams produced with it provides a useful aid for helping students to better develop concept images of geometric concepts. However, since any Cabri3D diagram represents three-dimensional objects on the two dimensional screen of a computer, some care is needed in order to avoid serious misconceptions which can arise from its use, in particular those due to the fact that projections do not preserve, in general, angles and distances. In this paper, after comparing digital diagrams (i.e. diagrams on a computer screen) with the more usual diagrams and models, we illustrate an experience around the use of Cabri3D with prospective high school teachers, aimed at clarifying which misconceptions may arise while interpreting a Cabri3D diagram. 1 INTRODUCTION Three-dimensional Euclidean geometry is not a popular subject nowadays. One of the main reasons for this is that diagrams representing three-dimensional geometric objects are difficult to interpret. “The survey by the French Ministry of Education shows that the fifteen-year-old students’ most repulsive subjects in mathematics were spatial geometry and statistics. Only ten percent of teachers taught spatial geometry. They said that they did not have enough time to teach it, but the real reason is that the students cannot see in 3D. We mean this, as the students cannot picture spatial situation of a teacher blackboard figure” (Bako, 2003, p. 1). Insufficient attention is paid to the visualisation capabilities of the students. “It seems that there is a hidden naïve assumption that somehow students do have visual thinking abilities and that they apply visual reasoning when they have to” (Hershkowitz, Parzysz, and Van Dormolen, 1996, p 166). This assumption is not only naïve but also dangerous, especially in solid geometry, where it is quite easy to pick vague and distorted ideas about geometric objects and relations and it is quite hard to get rid of them. For helping the development of good concept images of three-dimensional geometric objects, educators have some possible aids: models, manipulatives and diagrams. The recent availability of 3D dynamic software, like Cabri3D (Bainville and Laborde, 2004) gives a potentially important new tool for developing visual education for solid geometry. In this paper we consider some teaching possibilities with respect to Cabri3D and compare them with the use of models and diagrams. We shall particularly be interested in misconceptions which may arise from interpreting Cabri3D diagrams. In section 5 we give a precise statement of the claims we tested. Glossary of terms For the sake of clarity, we briefly recall the meaning we assign to some crucial terms. • Models: concrete objects representing concrete instances of geometric objects like wood polyhedra, plastic triangles and so on. • Manipulatives: set of concrete objects over which one can perform manual activity in order to represent geometric objects, constructions and relations. For example: Geoboard, Mira, Polymorf, Polydron, the look-through- window. • Diagrams: drawings representing geometric objects and relations on a sheet of paper. Diagrams may be further subdivided into: • Sketch diagrams, i.e. outline drawings representing some geometric features of two or three-dimensional geometric objects, possibly with the aid of graphic conventions. • Euclidean 2D diagrams, i.e. diagrams built by means of ruler and compass, which faithfully represent one particular instance of some geometric 2D configuration; • Euclidean 3D diagrams i.e. diagrams built by means of ruler and compass, which represent one particular instance of some geometric 3D configuration by means of the methods of descriptive geometry. These in general are not a faithful representation although some methods of projection can faithfully preserve some of the properties of the original 3D configuration, like parallelism.