In our presentation we will report about the development process of the university mathematics study module (25 ECTS) called Didactic Mathematics and some curricular tensions we have met in the development process of its curriculum. The study module of the Didactic Mathematics consists of five first university level mathematics courses taught to prospective math teachers either at elementary or secondary level. The basic problem of developing the curriculum and teaching practices of Didactic Mathematics has been to analyze what is meant with the so-called didactization of university level mathematics courses. In conceptualizing the essence of the didactization process of the courses, especially the studies of Abramovich & Brouwer (2003) and Stylianides & Stylianides (2009) have offered us the good terms of reference.

Topics we have especially taken account when choosing problems to the lessons and which we have discussed a lot with students have been both the impact which the development of computing technology has had and still has both to school and university level mathematics and the tensions experienced between concretization and abstraction, between perceiving and proving and between the communicative power of ideas and the mathematical rigor. A necessity we especially have to take account is that after passing the pedagogically oriented courses of Didactic Mathematics students continue to study ‘normal’ courses of university mathematics. That forms a kind of counterforce to the goal of the didactization.

For prospective teachers there is always a kind of controversial situation: as future teachers they should learn to evaluate the mathematics taught at university courses from the point of view of school mathematics but as university students they have to learn to think the way university mathematicians think. Stylianides & Stylianides (2010) conceptualized mathematics tasks as a special kind they called Pedagogy-Related mathematics tasks (P-R mathematics tasks). These tasks are intended to provoke activity that can support the development of mathematical knowledge for teaching (Mft) as Ball & Bass (2000, 2003) call the type of mathematical knowledge. The essential part of our development process of Didactic Mathematics has been producing such P-R mathematics tasks which emphasize the learning trajectories from elementary to university mathematics and strengthen student’s learning to learn – skills like abstraction, defining and argumentation as learners of the university level mathematics.

Abramovich, S. & Brouwer, P. (2003). Revealing hidden mathematics curriculum to pre-teachers using technology: the case of partitions. International Journal of Mathematics Education in Science and Technology 34(1), 81–94. Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 83–104). Westport, CT: Ablex Publishing. Ball, D.L., & Bass, H. (2003). Making mathematics reasonable in school. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 27-44). Reston, VA: National Council of Teachers of Mathematics. Mishra, P., & Koehler, M. J. (2006). Technological Pedagogical Content Knowledge: A Framework for Teacher Knowledge. Teachers College Record 108(6), 1017–1054 Poranen, J. & Haukkanen, P. (2012). Didactic Number Theory and Group Theory for School Teachers. Open Mathematical Education Notes, Vol. 2 (2012). Silfverberg, H. (2004). DGS and CAS as tools supplementing each other in an inquiry task "Locus curves" In J., Boehm (Ed.) Proceedings TIME-2004, 14-17 July 2004, Montreal, Canada. Simonson, M. (2006). Design-Based Research. Applications for Distance Education. The Quarterly Review of Distance Education, Volume 7(1): vii–viii. Stylianides, G.J. & Stylianides, A.J. (2010). Mathematics for teaching: A form of applied mathematics. Teaching and Teacher Education 26, 161–172. Wang, F., & Hannafin, M. (2005). Design-based research and technology-enhanced learning environments. Educational Technology Research and Development, 53(4), 5–23.