1. Is creative and critical mathematics considered at the entry to tertiary level? Carmen Sessa Universidad de Buenos Aires, Argentina Analia Bergé, Cégep de Rimouski, Québec, Canada

2. The role of mathematics in the education of critical thinkers and citizens. This includes building self- control and autonomy. The place of learners as producers of mathematical knowledge (and not only as reproducers)

3. The school should be the place where young students construct tools intended to understand the world, to take position in it, to belong critically to it, and eventually, to transform it. Thinking about such a formation in the domain of mathematics, makes us privilege activities that give raise to processes of mathematical production by students. Activities such as producing conjectures, experimenting with particular cases, proposing alternative solutions for a problem, and discussing them in the classroom.

4. We also assign importance to the work on the techniques, as a relevant element of the construction of effective tools for the resolution of problems. We think about a kind of technical work whose basis students have access to, which allows discussion in class.

5. An alive and productive class is not exclusive to Mathematics, and not new in Math Education Guy Brousseau, Paul Cobb, Yves Chevallard, Ole Skovmose, In our country: Patricia Sadovsky

6. Secondary level education reformed programs (Quebec, Canada and Buenos Aires, Argentine) emphasize – the use of mathematical models in representing and solving problems, –the use of mathematics as a tool for interpreting, analyzing and understanding the reality. Both describe the activity of the pupils in terms of – exploring, –observing, –deducing, –producing conjectures and validating or invalidating them, and that with the aid, at several times, of technological tools.

7. From the Québec Education Program: Students become active learners when they take part in activities involving reflection, hands-on tasks and exploration in order to construct knowledge or when they have discussions that allow them to express their opinions, justify their choices, compare results and draw conclusion”. From an Argentinean curricular document: Along this process, the fact that the student constructs (mathematical) tools in order to control the correctness of her/his production is of main importance. It is essential that s/he could justify the decisions s/he takes and be certain of her/his work above the evaluations of the teacher. For this to be possible, it is necessary that the teacher helps the student to maintain his work without replacing him in his task of production.” (our translation)

8. New programs suppose teachers That could tolerate the typical uncertainty due to the fact of facing a productive class. That could accompany students in their processes of study and production. That would need a group of reference (colleagues) with which to discuss his teaching projects and the facts that take place in his class.

9. At the entry to tertiary level, mathematics exams consist mostly of routine technical exercises, at the same jurisdictions. Rupture regarding the purposes of secondary level. Why are these aspects not considered at the entry to tertiary level?

10. (Conjecture) Implicit precept: it is necessary to learn the techniques and rules before, to be able to apply them in problems after. A confronting idea: technical aspects can be learned imbedded in situations that nourish them of sense.

11. Changing the focus at the entry to tertiary level necessarily leads to think differently the evaluations. Can we think the access to tertiary education from other suppositions?

12. When working with teachers in Argentine on the new programs, several of them express that they feel pressed for obtaining the success of their students in the next cycle. Then, they center their teaching in the more technical aspects, making it hard to the pupils to find sense in this practice.

13. Questions for discussion Is there a tension between the role of mathematics in the general high school curriculum and the aspects that are considered at the entry to tertiary level? ¿If yes, what are the consequences of this tension? Why the entry to tertiary level is focused almost only on technical aspects? Can we think the access to tertiary education from other suppositions?