This poster reports on a teaching experience, developed in the context of a Mathematics in-service teacher education program (Canavarro & Rocha, 2008; Darling-Hammond, Wei, Andree, Richardson & Orphanos, 2009), aiming to know how teachers integrate investigation tasks in the curriculum development and how they reflect upon their teaching practices.

The dynamics of the Mathematics class is highly influenced by the proposed tasks, because they can provide different ways to understand or do Mathematics. Problems and investigations appear as expressions of a non-routine work, referring to complex mathematical processes. Although the two concepts are considered very close, they can fulfil different functions (Chamoso & Rawson, 2001). "Problem solving" refers to a convergent activity in which one tries to achieve a solution to a problem, using appropriate strategies and techniques, while "investigation" is seen as a more divergent activity that encourages someone to be curious, to search for alternative strategies, to consider what would happen if certain conditions change or to generalize the situation.

Obviously that a situation, in the abstract, can have the characteristics of an investigation (Ponte, Oliveira, Cunha & Segurado, 1998) and its exploitation in the classroom can twist out, in practice, these characteristics. Thus the teacher's role in the class organization and management is determinant, in order to lead students to conjecture, explore examples and counterexamples and justify the conjectures based on mathematical arguments.  Assigning a central role to the argumentation in the classroom means empowering all students to present and explain their reasonings, but also to understand and accept the argumentation of the others (Boavida, 2005).

Under these conditions, a Mathematics teacher is confronted with an increasing demand in his work of preparation and conducting classes (and, mainly, the proposed tasks and their exploration) and the consequent decision-making. Being a teacher requires a very diversified and complex professional knowledge, especially the one linked to the teachers' characteristic knowledge, the didactic knowledge (Pires, 2006). This knowledge is related, for example, to the integration of Mathematics in the school reality or to the instructional process — planning, conducting and evaluation of the teaching procedures. As Shulman says (1986, p. 9), “it is pedagogical knowledge, which goes beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teaching”.

But the teacher's knowledge, in particular the didactic one, develops essentially through reflection on his practice and interaction with other teachers. This practical knowledge, resulting from the integration of both theoretical and experiential knowledges, has a contextualised nature, is modeled by the teacher's intentions and own conceptions and it is knowledge in action (Elbaz, 1983; Schön, 1992). For Hiebert, Gallimore and Stigler (2002), this is a very useful knowledge because, while connected to the practice, it evolves in response to specific problems of that practice.

Boavida, A.M. (2005). A argumentação na aula de matemática: Olhares sobre o trabalho do professor. In J. Brocardo, F. Mendes & A. Boavida (Orgs.), XVI SIEM – Atas (pp. 13-43). Setúbal: APM. Bogdan, R., & Biklen, S. (1994). Investigação qualitativa em educação. Porto: Porto Editora. Bolívar, A., Domingo, J., & Fernández, M. (2001). La investigación biográfico-narrativa en educación: Enfoque y metodología. Madrid: La Muralla. Canavarro, A.P., & Rocha, I. (2008). Professional development of mathematics teachers: Challenges from a national in-service teacher education program in Portugal. In ICME11, Mexico. Retreived January 20, 2012, from http://dg.icme11.org/document/get/147. Chamoso, J., & Rawson, W. (2001). En la búsqueda de lo importante en el aula de matemáticas. SUMA, 36, pp. 33-41. Elbaz, F. (1983). Teacher thinking: A study of practical knowledge. London: Croom Helm. Darling-Hammond, L., Wei, R., Andree, A., Richardson, N., & Orphanos, S. (2009). Professional learning in the learning profession: A status report on teacher development in the United States and abroad. Stanford: NSDC. Retreived January 20, 2012, from http://www.srnleads.org/resources/publications/pdf/nsdc_profdev_short_report.pdf . Hiebert, J., Gallimore, R., & Stigler, J. (2002). A knowledge base for the teaching profession: What would it look like and how can we get one?. Educational Researcher, 31(5), pp. 3-15. NCTM (2000). Principles and standards for school mathematics. Reston, VA. Pires, M.V. (2006). A construção do conhecimento profissional: Um estudo com três professores. In Atas do XVII SIEM [CD-ROM]. Setúbal: APM. Ponte, J.P., Oliveira, H., Cunha, H., & Segurado, I. (1998). Histórias de investigações matemáticas. Lisboa: IIE. Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), pp. 4-14. Schön, D. (1992). La formación de profesionales reflexivos: Hacia un nuevo diseño de la enseñanza y aprendizaje en las profesiones. Madrid: Paidós e MEC.