This book is a thoroughly revised result, updated to mid-1995, of the NATO Advanced Research Workshop on "Intelligent Learning Environments: the case of geometry", held in Grenoble, France, November 13-16, 1989. The main aim of the workshop was to foster exchanges among researchers who were concerned with the design of intelligent learning environments for geometry. The problem of student modelling was chosen as a central theme of the workshop, insofar as geometry cannot be reduced to procedural knowledge and because the significance of its complexity makes it of interest for intelligent tutoring system (ITS) development. The workshop centred around the following themes: modelling the knowledge domain, modelling student knowledge, design ing "didactic interaction", and learner control. This book contains revised versions of the papers presented at the workshop. All of the chapters that follow have been written by participants at the workshop. Each formed the basis for a scheduled presentation and discussion. Many are suggestive of research directions that will be carried out in the future. There are four main issues running through the papers presented in this book: • knowledge about geometry is not knowledge about the real world, and materialization of geometrical objects implies a reification of geometry which is amplified in the case of its implementation in a computer, since objects can be manipulated directly and relations are the results of actions (Laborde, Schumann). This aspect is well exemplified by research projects focusing on the design of geometric microworlds (Guin, Laborde).

Inhalt
A Model of Case-Based Reasoning for Solving Problems of Geometry in a Tutoring System.- Modelling Children's Informal Arithmetic Strategies.- Cognitive Interpretation of Microworld Operations.- Calculus Revisited.- Computer Aided Proofs in School Geometry.- A Cognitive Analysis of Geometry Proof Focused on Intelligent Tutoring Systems.- Modelling Geometrical Knowledge: The Case of the Student.- Intelligent Micro worlds and Learning Environments.- A Constructivist Model for Redesigning Al Tutors in Mathematics.- The Influence of Interactive Tools in Geometry Learning.- Socratic Tutoring with Software: An Example and a Prospectus.- Students' Constructions and Proofs in a Computer Environment Problems and Potentials of a Modelling Experience.- Some Hyperbolic Geometry with CABRI-Géomètre.- Micro-Robots as a Source of Motivation for Geometry.- Complex Factors of Generalization Within a Computerized Microworld: The Case of Geometry.- Contributors and Participants.