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In this article, we elaborate methodologies to study the argumentation speech of a teacher involved in argumentative activities. The standard tool of analysis of teachers’ argumentation concerning pedagogical matters is Toulmin’s model. The theory of argumentation schemes offers an alternative perspective on the analysis of arguments. We propose an integrated way of analysis employing Toulmin’s model and argumentation scheme, based on Walton’s taxonomy. We examine the change of pedagogical argumentation of a teacher that participated in a graduate course that was based on the analysis of hypothetical classroom scenarios on the teaching of calculus in high school. By exhibiting our methodological analysis in a particular case, we show that by adopting two different analytical perspectives, we gain a deeper understanding of the structure and quality of teachers’ argumentation on pedagogical matters. The integration of the two qualitative methodologies could help identify several aspects of possible construction of knowledge in argumentative activities. © 2016, Springer Science+Business Media Dordrecht.

Visualization of mathematical relationships enables students to formulate conjectures as well as to search for mathematical arguments to support these conjectures. In this project students are asked to discover the sufficient and necessary condition so that two circles form the circumscribed and inscribed circle of a triangle and investigate how this condition effects the type of triangle in general and its perimeter in particular. Its open-ended form of the task is a departure from the usual phrasing of textbook's exercises "show that⋯". © 2009 Springer Science+Business Media B.V.