Conhecimento matemático específico para o ensino na educação básica: a álgebra na escola e na formação do professor - Maria Cristina Costa Ferreira (2014)
There is some consensus around the idea that mathematics teachers must know the subject they teach in a deeper way. However, just knowing more advanced mathematics, not connected to other dimensions of teachers professional knowledge, has been seen as insufficient and even innocuous in terms of adequate teacher preparation to work in a setting as complex as a mathematics school classroom. Thus, the discussion about What mathematics is needed to prepare mathematics school teachers? has been of great interest for researchers and teachers who work in teacher preparation programs. Shulman introduced the notion of pedagogical content knowledge to designate a particular type of professional teaching knowledge, that special amalgam of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding. Drawing on Shulmans idea of Pedagogical Content Knowledge, researchers at the University of Michigan, led by Deborah Ball, developed the concept of Mathematical Knowledge for Teaching, which is a special kind of professional mathematical knowledge needed to carry out the work of teaching mathematics, as Ball herself puts it. In this research we sought to identify elements of mathematical knowledge for teaching related to actual algebras work in school. The observations took place in 8th and 9th grade classrooms, with two different teachers, at a public school in Brazil, from April to August 2012. We focused on specific mathematics knowledge which were effectively mobilized or might have been mobilized by those teachers in their algebras classroom. Along the process of data collection, we were able to notice that several important issues raised in current algebras teaching and learning research literature actually came about in the observed classrooms setting. This shows the convenience and the relevance of analyzing them in terms of specific mathematical knowledge for teaching. In our analysis, two issues stood out, acquiring a prominent position: the justification of results obtained in the generalization process in school algebra classrooms and the process-object duality in the construction of abstract concepts, in particular those associated to algebraic expressions. We were able to highlight tensions between validation processes used according to the structure of mathematics, as conceived by professional mathematicians, and those appropriate to the process of concept acquisition in school mathematics classrooms. The process-object duality has been unfolded and our analysis illuminates the tension between the structural and procedural ways teachers and students signify algebraic expressions. This study identifies important and fundamental mathematical knowledge for teaching which are not mentioned in the recommendations for mathematics teacher preparation programs in Brazil.