1. Desarrollo de recubrimientos curriculares y gestión de itinerarios adaptados al aprendizaje instrumentado con el geogebraTUTOR Philippe R. Richard Université de Montréal Canadá Josep M. Fortuny Universitat Autònoma de Barcelona España Pedro Cobo IES Font i Quer España

2. What geogebraTUTOR does? Interface and system

3. Interface of the student Tutor agent helps student to solve the problem

4. Interface of the teacher Problems’ files Strategies’ files Messages’ files Setting up local and global solving process

5. What geogebraTUTOR hides? Spaces, tags, trees and forests

6. Basic space of the problem Contains all cognitive and heuristic states

7. Discursive messages Messages are based on a set of possible inferences

8. Inferential graph Tutor seeks coherence of inferences and strategy 1  2  3

9. Tags and «ID card» The system retains tags from the solving process Characterization of each problem uses a set of tags

10. Trees and forests Each sub-problem is a cognitive message

12. Simple: Si te parece puedes descomponer la figura en otras Intermedio: Puedes descomponer el triángulo ABC en otros triángulos Detallado: Intenta descomponer el triángulo ABC en otros triángulos uniendo vértices o trazando paralelas Ejemplos de mensajes cognitivos heurísticos en la fase de planificación/ejecución

13. Trees and forests (...) Set of root problems – itinerary of learning Set of root problems – itinerary of learning 1 System generates trees using tags of «ID cards» 2 Restructuring trees using retained tags in solving 3 Forests fit to the learning process of each student 4

14. Theory and research Foundations of geogebraTUTOR

15. Context New curriculum paradigm in mathematics learning (from objectives to competencies). Importance of social debate in the building of mathematical knowledge. Pedagogical agents with an artificial tutorial system can convey argumentative strategies to improve mathematical competencies and knowledge.

16. Hyperspace of four reference axis Theoretical background Epistemological Dialectical of proofs and refutations (Lakatos, 1984) Semiotic Registers, models of representation (Duval, 1995; Richard, 2004) Situational Theory of didactical situations (Brousseau, 1998) Instrumental Cognitive approach of the contemporary instruments (Rabardel, 1995)

17. Future of geogebraTUTOR in research Four objectives in research projects (due very soon) Instructional model To analyze the didactic effectiveness of the tutorial system Interpretation and theorization To interpret and theorize on the representational, epistemological, didactic and instrumental characteristics of the system «student-milieu» Evaluation and control To evaluate the constancy of the system «student-milieu» in interaction Implication in vocational training To test the empathy in problem solving and the effects of tutoring actions

18. Any questions? To have inside information outside

19. To know more about evolution Richard & Fortuny (2007). Amélioration des compétences argumentatives à l’aide d’un système tutoriel en classe de mathématique au secondaire. Annales de didactique et de sciences cognitives, 12, 83-116. Strasbourg: Université Louis Pasteur. Cobo, Fortuny, Puertas & Richard (2007). AgentGeom: a multiagent system for pedagogical support in a geometric proof problem. International Journal of Computers for Mathematical Learning, 12, 57-79. Springer Science+Business Media. Proceedings of World Conference E-Learn 2004, 2005, 2007, ED-Media 2005, 2006 and SITE 2007.

20. ¡Gracias! philippe.r.richard@umontreal.ca JosepMaria.Fortuny@uab.cat pcobo@xtec.cat