To Be Presented in the 11th Asian Technology Conference in Mathematics
December 12-16, 2006, Hong Kong SAR, China

Instrumentation of the Drag Mode in the Introduction to Deductive Reasoning with Cabri-geometry

Sophie Soury-Lavergne
IUFM of Grenoble & University Joseph Fourier
France

Abstract

The question addressed in this paper is how 11 years old pupils can use dynamic geometry to change the way they identify the geometrical properties of a figure. When they enter in the secondary level school, they are supposed to gradually surrender perceptive identification and begin to use new means to establish the validity of a geometrical property. The final aim is that they state it as a hypothesis or deductible from the hypothesis. Actually, the drag mode in Cabri-geometry allows the user to distinguish the properties of geometrical figure (hypothesis and their consequences) from the ones of a specific diagram. The first ones are preserved while dragging and not the second ones. Thus, Cabri-geometry is a relevant tool for pupils at the very beginning of the learning of deductive reasoning. Nevertheless, the potential of Cabri-geometry aks for a right use of the drag mode, which is far to be obvious. All along the 20 years of use of Cabri-geometry, it has been observed that beginners are reluctant to move their constructions. Even when they finally start to drag elements, they face many difficulties:

  • they only drag partially the diagram (just one point or very little movements around the initial position),
  • they interpret the retroactions on a spacio-graphic level (i.e. the diagram moves up or down or becomes flat etcíK),
  • they are not able to perceptively recognize geometrical properties in some particular positions,
  • they want to fix the moving points instead of analysing the diagram on a geometrical point of view.
Within the instrumental approach, we focus our analysis on the instrumental genesis that turns the dragging into a pupilíŽs instrument which makes them able to state about the validity of a geometrical property. We have identified instrumental schemes involved in the use of the artefact "dragging functionality":
  • to choose the points to drag (different category of points)
  • to identify, by the mean of perception, the geometrical property to be controlled
  • to decide if a geometrical property is conserved or not, always by the mean of perception,
  • to conclude about the property and the figure (is it a counter-example?...).
We have elaborated activities in the context of a French project MAGI (better learning geometry with computers), in order to sustain this process of instrumental genesis of the drag instrument. We have tested those activities during 2 years, with a teacher in 3 different classes. Besides the activities, we also brought out elements, which help the process:
  • to make explicit the different category of points and their consequences on drag.
  • to produce instructions for individual use of Cabri which ask to specify the points to be dragged and those to observe,
  • to produce instructions for the use of Cabri during debates with the whole class; only one pupil manipulates the software and the others have to propose which points to drag and what to look for during dragging.

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