The Process of Mathematical Agreement : Examples from Mathematics History and an Experimental Sequence of Activities

Teaching with Primary Historical Sources: Should it Go Mainstream? Can it? David Pengelley, New Mexico State University, Las Cruces Many are now teaching mathematics directly with primary historical sources, in a variety of courses and levels. How far should this be taken? Should we adapt or redesign standard courses to a completely historical approach, chiefly from primary sources? If so, what are the obstacles to achieving this? Materials? Instructor training and attitudes? Class time? Textbooks? Classroom pedagogy? What should and can we do about such things? We attempt to provide answers to these questions, and illustrate with a sample student project based on Pascal’s Treatise on the Arithmetical Triangle how numerous core course topics can be learned via a primary historical source. Dialogism in Mathematical Writing: Historical, Philosophical and Pedagogical Issues Evelyne Barbin, Université de Nantes, France The notion of dialogism was introduced by the Russian semiotician Mikhail Bakhtin. For him, every sentence or every discourse must be understood as a rejoinder in a dialogue: it is an answer to other sentences, or discourses and it is intended to be received by somebody. My purpose in this paper is to explore the meaning and the implications of this notion for mathematical texts. The consequences for historical works are clear, in the sense that one should pay attention to the nature of texts (letters, papers, books), to the texts known to the authors, and so on. In a philosophical perspective, the notion of dialogism leads to a reflection on mathematical proof. At the classroom level, dialogism is an interesting notion for the reading of ancient texts by pupils and thinking about the persons for whom the pupils write. The Process of Mathematical Agreement: Examples from Mathematics History and an Experimental Sequence of Activities Gustavo Martinez-Sierra and Roco Antonio-Antonio, CICATA-IPN, Mexico This article presents the basic evidence that allow us to give account of a process of production of knowledge, that we have called process of mathematical agreement (Martı̀nez-Sierra, 2005, 2008), that allows to establish a statement at the same time as its truth. The truth of the statement can be interpreted as “agreed truth;” in the sense that it is set from the necessity to make a theoretical corpus. We will present three examples from mathematics history, that show that it is possible the production of the meaning of: 1) the fractional exponents, 2) the square root of negative numbers as precursor on the meaning of complex numbers and 3) the radian and the trigonometric functions. In order to experiment the process of mathematical agreement finally we present the results of an experimental sequence that has the objective to favor the acceptance and the operativeness of the square root of negative numbers.