dynamic geometry environments

Colette Laborde

Colette.Laborde@imag.fr

University Joseph Fourier and Teacher Education Institute, Grenoble

Abstract

Variation is the essence of dynamic geometry environments. This talk aims at discussing two paradigms of use of

variation in dynamic geometry environments: robust and soft constructions.Robust constructions are constructions for

which the drag mode preserves their properties. Such constructions must be constructed by using the geometrical

objects and relationships characterizing the construction to obtain. In such constructions variation is used as a

verification means. In soft constructions, variation is part of the construction itself and a property becomes visible only

whenanother one is satisfied. By means of several examples based on Cabri Geometry II and Cabri 3D, it will be

shown how the soft paradigm can contribute to the learning. On the one hand, soft constructions can be part of the

« private » side of the work of the students and help them identify dependency relationships between properties, on the

other hand they can be used in mathematics teaching tointroduce students to better understand the functioning of

fundamental notions such as those of implication, valid property, hypothesis and conclusion

1. Variable and variation in mathematics

The notions of variable and variation are often attached to algebra and in particular to the notion of

function. However these notions go beyond these topics and are essential in mathematics. The

dualityvariation/invariant permeates all mathematics, including geometry. A theorem like “Any

isosceles triangle has two congruent angles” expresses a relational invariant between the sides of a

triangle varying in the set of the isosceles triangles, even if the varying nature of the triangle is

expressed by the subtle mark “any”. A geometric property is an invariant satisfied by a variable

object assoon as this object varies in a set of objects satisfying some common conditions. The

variability of geometric objects is generally invisible because the formulation of a geometric

property is most of the time expressed as dealing with a single static object, the quantifiers being

implicit, especially in the secondary school. This is not without causing troubles for the students

who do notperceive the generality of theorems or properties.

Dynamic geometry exteriorizes the duality invariant/variable in a tangible way by means of motion

in the space of the plane. When a figure is constructed in order to satisfy a set of conditions,

properties that derive from them are preserved in the dragging of an element of a figure. Those

properties remaining invariant in the drag mode emergefrom the contrast with the changing

properties of the figure in the drag mode, as Mason & Heal (1995, p.301) wrote:

“Being able to move screen objects around in space (and so over time) can add significantly to the

user’s sense of the underlying concept as an object not just in itself but a something invariant amidst

change”

Geometric properties are perceived in a dynamic geometry environmentas invariant in the variation

of the figure, exactly in the same way as an algebraic identity such as (x + 1)2 = x2 + 2x + 1 can

only be perceived in the variations of x. One could say that a theorem in geometry is of same nature

as an algebraic identity but from the converse point of view, an algebraic identity can be viewed as

a theorem. Although dynamic geometry reveals this deep unity ofmathematics as finally a science

dealing with variable objects, the attention to variation as the essence of mathematics is not new.

Let us quote the French geometer Monge (1792) writing in the “Leçons données à l’Ecole Normale

de l’an III”:

“Il faut que l’élève se mette en état, d’une part de pouvoir écrire tous les mouvements qu’il peut

concevoir dans l’espace, et de l’autre, de se...