1. Contribution of a computer algebra system (CAS) in the solving of problems in geometry with the help of an emerging tutorial system Philippe R. Richard Université de Montréal Universitat Autònoma de Barcelona

2. The tutorial system Technological research project + realization

3. About geogebraTUTOR Development of geometrical competencies using an Intelligent Tutorial System (ITS) Main objective of GGBT How GGBT could help the student to solve complex problems using a CAS with a DGS Aim of our presentation Technological research project + realization

4. Geometrical competencies Figural to model conjecture define Operational instrumentation instrumentalization Deductive argumentation reasoning Visual observation exploration Technological research project + realization

5. Complex problems Heuristic requirement Multiple problem- solving and posing Cognitive requirement Network of mathematical concepts and processes Discursive requirement Argumentative approach, multistep reasoning or non-routine calculations Competential requirement Connexion and reflection competency clusters (PISA, 2009) Technological research project + realization

6. Student’s interface Technological research project + realization

7. Possible interface with CAS Technological research project + realization

8. Some examples a priori Treatment with and without CAS

9. Examples of treatment In synthetic geometry (without CAS) – Problem and real solution – Instrumented figural solution In analytic geometry (without CAS) – With the trace and the worksheet – With a construction and the locus In analytic geometry (with CAS)

10. From a modelling problem On a rectangular ground planned for the installation of a swimming pool, a city decided to divide it in two parts, according to a diagonal, in order to build also a small community centre. If the swimming pool must remain rectangular, where should it be built so that its surface is maximum?

11. A real solution (15 years old) Synthetic geometry

12. Paper and pencil with DGS Synthetic geometry

13. Instrumented figural solution Synthetic geometry

14. With the trace and the worksheet Analytic geometry

15. With a construction and the locus Analytic geometry

16. Traditional modelling solution Analytic geometry

17. Some technical problems 1.Is the traditional solution adapted to the use of a CAS? 2.If the use of a CAS creates a new space for modelling and solving geometrical problems, how we can use it in a classroom? 3.And, if the use needs to be implemented in an ITS, how can we use it to support the development of geometrical competencies? Analytic geometry

18. Instrumented solutions with CAS Example of a solving tree, planned to be implemented in an intelligent tutorial system, that uses strongly the computer algebra system Analytic geometry

21. To conclude Very briefly

22. Influence on geometrical competences Figural to model conjecture define Operational instrumentation instrumentalization Deductive argumentation reasoning Visual observation exploration Technological research project + realization

23. For the research and development Other ideas in the integrating equation CAS + DGS + ITS ? Technological research project + realization

24. Děkuji ! michel.gagnon@polymtl.ca nicolas.leduc@polymtl.ca philippe.r.richard@umontreal.ca michele.tessier-baillargeon@umontreal.ca