ORIGINAL ARTICLE Didactic and theoretical-based perspectives in the experimental development of an intelligent tutorial system for the learning of geometry Philippe R. Richard • Josep Maria Fortuny • Michel Gagnon • Nicolas Leduc • Eloi Puertas • Miche `le Tessier-Baillargeon Accepted: 23 February 2011 / Published online: 12 April 2011 Ó FIZ Karlsruhe 2011 Abstract This paper aims at showing the didactic and theoretical-based perspectives in the experimental devel- opment of the geogebraTUTOR system (GGBT) in inter- action with the students. As a research and technological realization developed in a convergent way between math- ematical education and computer science, GGBT is an intelligent tutorial system, which supports the student in the solving of complex problems at a high school level by assuring the management of discursive messages as well as the management of problem situations. By situating the learning model upstream and the diagnostic model down- stream, GGBT proposes to act on the development of mathematical competencies by controlling the acquisition of knowledge in the interaction between the student and the milieu, which allows for the adaptation of the instructional design (learning opportunities) according to the instru- mented actions of the student. The inferential and con- struction graphs, a structured bridge (interface) between the contextualized world of didactical contracts and the formal computer science models, structure GGBT. This way allows for the tutorial action to adjust itself to the com- petential habits conveyed by a certain classroom of stu- dents and to be enriched by the research results in mathematical education. Keywords Mathematical education (didactics of mathematics) Á Intelligent tutorial system Á Mathematical competencies Á Computer science models (informatics) Á Geometry learning Á Dynamic geometry software Á Student–milieu cognitive interactions 1 Preliminary point of view on the tutorial systems for the learning of geometry Geometry at a high school level can be seen as a deductive science allowing the solving of problems in the mathe- matical field as well as a theoretical reference, which ori- ents the wider process of extra mathematical modelling. This permits amongst other things the laying down of problems inspired by what is referred to as the real world or reality. Even if the solving of modelling or proof problems, along side of the curricular obligations, is a mathematical competence to prioritize in the education of young people, it remains difficult to develop in regards of the traditional relationship where the teacher who, alone in front of his class, gives insight into the mathematical reasoning, cal- culations and other problems to which are confronted stu- dents. Also, when it is the instrumental workings of the dynamic geometry tools that liven up his didactic inter- ventions, the teacher may feel at loss when faced with the flow of interactions between each student and the computer device, and this in spite of the fact that these interactions P. R. Richard Á J. M. Fortuny Departament de Dida `ctica de la Matema `tica i de les Cie `ncies Experimentals, Universitat Auto `noma de Barcelona, Barcelona, Spain P. R. Richard (&) Á M. Tessier-Baillargeon De ´partement de didactique, Universite ´ de Montre ´al, Montre ´al, Canada e-mail: philippe.r.richard@umontreal.ca M. Gagnon Á N. Leduc De ´partement de ge ´nie informatique et ge ´nie logiciel, E ´ cole Polytechnique de Montre ´al, Montre ´al, Canada E. Puertas Departament de Matema `tica Aplicada i Ana `lisi, Universitat de Barcelona, Barcelona, Spain 123 ZDM Mathematics Education (2011) 43:425–439 DOI 10.1007/s11858-011-0320-y