Uma geometria para ensinar: elementos do saber profissional do professor que ensina matemática (1870- 1920)

Abstract

The general objective of this doctoral research consist in characterize a geometry for teaching from Pedagogy manuals directed to the formation of teachers of the first school years in Brazil between 1870 and 1920.To we reach this goal, we rely on the enabled reflections by the following questioning: what geometry was constituted as a working tool by the teacher who taught mathematics in times of intuitive method in Brazil? The reference theoretical-methodological on which we base ourselves for the development of the writing of this text that deals with the knowledge of the teaching professions and teacher training that have been systematized and objectified in each historical-educational period. Based on such references, we agree specifically to the theoretical categories of mathematics to teach and mathematics for teaching, so that our object of study consists of geometry for teaching that we were able to systematize from different Pedagogy manuals, materials that are our sources of research. To develop this systematization, we used the methodological process that indicates how to transform dispersed information into objectified knowledge. From the completion of the stages of recompilation, comparison and systematization, we concluded that all Pedagogy manuals that composed our sources of research present systematizations that guide the pedagogical work of the teacher for teaching geometry, some have greater systematic depth than others in terms of constitution and objectification of a geometry for teaching, but all are specific to form the teacher for teaching this school subject. The elements that we consider as constituents of the geometry that is the teacher's work tool present consensus and converge to the same idea about what the teacher must know for teaching geometry in the first school years. This allowed us to systematize a geometry for teaching characteristic of the institutional training of teachers of the first school years in Brazil between 1870 and 1920, which is based on elements of Euclidean geometry; mobilizes teaching materials such as a collection of solid forms; makes use of the teaching order analytical-synthetic, which means that geometric shapes are taught from the whole to the parts and that, when studied, the parts are moved to the whole, graduating from spatial geometry to flat and then, conversely; the contents are presented to the students from the mobilization of those materials, stimulating the use of the senses to construction of perceptions about the forms; generalization is gradually encouraged. These elements are articulated, configuring principles of the intuitive method elaborated for the teaching of geometry, associating with the pedagogical ideas to which the authors of the manuals say they are affiliated. Therefore, this geometry for teaching is based on the articulation and mutual dependence between the geometry to teach mobilized and the knowledge for teaching geometry, so that the way as happen these joints and mobilizations characterizes this geometry as intuitive, which allows us sustain that between in the 1870s and 1920s, school culture developed and maintained stable in teacher training of first school years a working tool for teachers regarding teaching in geometry, which can be characterized as intuitive geometry for teaching.