O estudo dos quaternions iluminando questões sobre a natureza da matemática na formação de professores de matemática

Abstract

This research was developed within the scope of Mathematics Education (ME). We address the inclusion of the History of Mathematics (HM) in Teacher Training (TT). According to Brito (2007), HM can contribute to TT, filling the gap between specific training, pedagogical training and teaching practice. Morey (2013) proposes the use of the historical (original) source as a strategy to carry out this insertion. Starting from an exemplary case study, we seek to show possibilities and contributions of HM in TT. Given this, we assumed the research questions: What contributions can the history of quaternions provide to the training of Mathematics teachers? What philosophical, conceptual and contextual aspects of Mathematics does this case study reveal? The general objective of this research is to investigate the history of quaternions from philosophical, conceptual and contextual perspectives to clarify how to construct a narrative that makes contributions to the training of Mathematics teachers. For that, in a qualitative methodological approach, we adopted as a procedure an analysis bibliography of primary and secondary sources related to quaternions. The historical study of this research is based mainly on the works of Hamilton (1866), Graves (1882, 1885, 1889) and Hankins (1980). We created a historical narrative highlighting elements that can contribute to the training of Mathematics teachers. In this narrative, we highlight the contextual, philosophical and conceptual aspects that were revealed when we began to understand quaternions as a product of human action and as part of an unfinished Mathematics, whose development is non-linear, non-continuous and non-progressive. We consider the contextual and temporal contingencies pertinent to the historical development of quaternions and Fried's (2001) ideas regarding the double synchronic and diachronic vision that highlights Mathematics, according to Fried (2008), as a set of signs and a product of human action. The justification that encouraged this research is based on Fried's double vision, which reveals the nature of quaternions and their possible implications for the training of Mathematics teachers. We understand that, to understand quaternions in their entirety, it is necessary to study them synchronously and diachronically. In this way, we understand that quaternions are made up of human actions and sign systems. The human aspects become explicit when Hamilton (1805-1865) proposed algebra as a science of pure time assuming intuitive assumptions. The formalization of quaternions as a system of signs occurred through their algebraic constitution (imaginary components and operations). From the perspective of Radford (2021), we held a mini-course in which we discussed the historical narrative of quaternions with Mathematics students at UEPB. Through the multimodal analysis of the short course, we observed that historical-cultural knowledge was materialized through semiotic modes (visual, verbal and gestural) through the mobilization of active listening, reading and interpretation skills. We noticed that some students expanded their repertoire of historical-cultural knowledge, mainly through the development of Fried's conception of the mathematical nature of quaternions. Finally, we understand that the discussion of the historical narrative of quaternions in TT enabled the student to understand the contextual, philosophical and conceptual aspects of quaternionic historical development, providing this future teacher with the opportunity to develop a conception of the nature of Mathematics that is based on Fried's double vision.