Within research into the teaching profession, there are two main areas of interest, which have been thoroughly defined and consolidated: teacher's knowledge and teacher's professional development. This paper considers the latter area, and specifically sets out to define the concept of the reflective teacher and to show how we have applied this concept in our research. It aims at defining a more precise notion of teachers' professional development linked to the concept of a reflective teacher, drawing on information from different interdisciplinary sources so as to frame the concept clearly. We discuss the conditions which determine the reflective teacher, and report on how these were put to use in a training course for practising teachers in which participants reflected on challenging episodes from their professional experience. We hope that clarifying this term will enable different educational agents (school teachers, teacher trainers or researchers) to have an informed view that can be interpreted and critiqued, and that it will also be of value to initial teacher training and to in-service courses.

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A concern in Mathematics Education is the professional development of the teacher and to promote effective training programs. The literature provides principles guiding the design of such programs, which were considered for an instruction intended to strengthen the teacher’s practice in relation to the development of mathematical skills. The objective of this work was to study if the designed program was effective, in relation to the impact on the teacher’s teaching. A case study on a group of in-service teachers focused on the ability of mathematical modeling (MM) and their perspectives of this skill was carried out. This was divided into three moments: before, during, and after the program. The findings show that, before the program, teachers conceived modeling from epistemological, pragmatic and conceptual perspectives; during the program, they focused on the pragmatic and educational perspectives; and at the end of the program, the group was stripped of the pragmatic, epistemological and conceptual perspectives to move towards the educational and socio-critical. They were also able to justify the choice or disassociation of one or more of them. The study advances towards the concretion of more specific and robust professional development programs for mathematics teachers.

ResumenParte de un trabajo doctoral, enmarcado dentro de la línea de formación de profesores, es analizar las tareas de enseñanza propuestas por docentes en un curso de formación realizado en Chile, en el año 2012, en donde sus participantes tienen que identificar un problema profesional referido a la enseñanza del álgebra y profundizar sobre el mismo diseñando, implementando y analizando tareas matemáticas para una clase. Luego de presentar algunos descriptores de las tareas, con una metodología cualitativa empleamos el análisis de contenido para describir la evolución de las tareas de enseñanza propuestas por dos profesoras participantes de este curso formativo. El análisis de los datos ponen de manifiesto cómo las docentes afrontan un problema de su práctica, incorporando nuevos elementos de actuación respecto al diseño de tareas, incrementando en sus conocimientos, en específico, en relación con la noción de modelación, avanzando en este proceso en su desarrollo profesional. AbstractPart of a doctoral work, framed within the teacher training online, is to analyze the tasks proposed by teachers teaching in a training course held in Chile in 2012, where participants have to identify a professional problem referred to teaching algebra and deepen on it designing, implementing and analyzing mathematical tasks for a class. After presenting some descriptors of tasks, with a qualitative methodology we use content analysis to describe the evolution of teaching tasks proposed by two teachers participating in this training course. Analysis of the data show how teachers face a problem of practice, incorporating new elements of action regarding the design of tasks, increasing their knowledge, specifically in relation to the notion of modeling, through this process in their professional development.Keywords: Teacher Education. Math Task. Álgebra Problem. IntroducciónEl desarrollo profesional del docente se puede percibir a través de los cambios en su actuación en relación con diversos elementos que involucra su práctica, en palabras de Eraut (1977), implica un proceso natural de crecimiento profesional en que un profesor va adquiriendo, gradualmente, confianza, ganancias de nuevas perspectivas, incrementando en el conocimiento, descubriendo nuevos métodos y asumiendo nuevos roles.En particular, los cambios en la actuación de los docentes pueden verse reflejados en relación con las tareas para el aula diseñadas por ellos para afrontar un su práctica. El diseño de la clase se aborda de una manera práctica delimitando qué se va hacer en la misma, formulando tareas que reflejan las expectativas de enseñanza y aprendizaje. Las tareas para el aula, se han estudiado profusamente en la literatura (DOYLE, 1983; PONTE, 2005;PONTE et al., 1997;RAMOS, 2011;STEIN et al., 2000) e incluso han sido objeto dentro de estudio de casos de la propia práctica profesional del investigador para comprender mejor ciertos aspectos de su trabajo, como lo menciona Ponte (2006).En Chile, entre junio y octubre del año 2012, organizamos e implementamos...