International Approaches to Professional Development for Mathematics Teachers

Chapter 9: Formative Processes: Sharing Learning Experiences in Geometry

128 Chapter 9 Formative Processes: Sharing Learning Experiences in Geometry Adair Mendes Nacarato and Regina Célia Grando Introduction This chapter is related to research developed from 2003 to 2007 in a collaborative group constituted by graduate students (pre-service mathematics teachers) and postgraduate students, in-service elementary school teachers , and university professors. The group’s object of study was geometry and its teaching; it was this focus that motivated the name of the group: Collaborative Group in Geometry (Grucogeo). The group met weekly during two-hour sessions in the university. It encouraged, in each semester, voluntary participation, with possible entries and exits of participants, having on average fifteen participants. In this space, five researches were developed: one coordinated by the authors of this chapter (with external financing from MCT/CNPq1 ), two scientific initiation researches, and two master’s degrees in education researches. The look at geometry and its teaching was diversified: exploration of investigative tasks with and without the use of software; experimentation with the same task with different media (in Grucogeo or with elementary school pupils); exploration of the potentialities and limits of each medium (folding, constructions with ruler and compass, use of manipulable materials and computer environments); and discussion and analysis of the argumentation , validation, and proof processes in geometry. The proposal of the work group centred on the elaboration, development , evaluation, and analysis of activities for elementary education. The Formative Processes || 129 group focused on some activities initiated by participants (in-service teachers and university educators), some of them being developed by the university educators. These activities were re-elaborated in a way that could be used with pupils in the school classrooms of the in-service teachers. Sometimes pupils worked in the institutional space of the university. For the classroom activities, participants were divided into subgroups in such a way that, in each subgroup, there was at least one elementary teacher to develop the lesson with pupils along with graduate students to register the process and management of the activity. The data (productions of the pupils, daily field diaries of the graduate students/researchers, and narratives of the teachers) were discussed, analyzed, synthesized, and systematized in Grucogeo. Underlying Framework This work proposal was based on the use of formative strategies (Alarcão, 2003, p. 58), whose “objective [is] to make the teachers more competent in the analysis of the questions that surround their routine, allowing them to act, helping them not only to solve immediate problems but also to situate them in a broader horizon that gives a perspective to their action and to the function of school in the society where we live.” Among the strategies suggested by Alarcão (2003, p. 51), in Grucogeo we used “the questioning of other educative actors,” “the confrontation of opinions and approaches,” “the discussion group or study circles,” and “the pedagogical questions.” There was in the group a culture of questioning and problematization of the questions relative to geometry and its teaching. We also adopted the strategy of production of oral and written narratives about daily practices of the teachers (Alarcão, 2003). Cunha (1997, p. 186), referring to the narratives as formative processes, affirms that, “at the same time that the subject organizes the ideas for the—written or oral—narrative, he or she reconstructs his or her experience in a reflective way and, therefore, makes an auto-analysis that creates new bases of understanding for his or her own practice. The narrative provokes changes in the way people understand themselves and others.” We also started from the supposition that the heterogeneity of the group—composed of in-service teachers and graduate students (pre-service teachers)—and the formative strategies would contribute to the construction of a repertoire of knowledge and professional development of teachers. 130 || Professional Development of Mathematics Teachers Although future teachers have left school recently, in the initial training process, they need to transform themselves from having a student’s posture to having a teacher’s posture. This transformation implies a change in the way of looking at school and classroom culture. From this perspective, inservice teachers assume the role of educators and co-constructors of future teachers’ professional identity, in a process of mutual involvement, using “guided participation” (Rogoff, 1998). Rogoff considers that this participation can be “tacit or explicit, face-to-face or distant, involved in joint efforts with people. ... It includes deliberate attempts to teach, and incidental commentaries...