1. Critical analysis of the evolution of ideas and methodologies in mathematics teaching, with particular regard to the role of the teacher. 2. The mathematics curriculum in compulsory education and in the various addresses of secondary schools (high schools, technical institutes and professional institutes) in an international framework 3. Educational design and teaching methods of mathematics: programming and rhythm, principles and methods for activity building, class management. 4. Troubleshooting. Logic, intuition and history in the teaching of mathematics.