The historical and logical movement of algebraic concepts as a principle for the constitution of the teachings object of algebra

Author(s)

Maria Lúcia Panossian

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Faculdade de Educação (FE/SBD)
• Defense date: 2014-02-24

Examining board members

Manoel Oriosvaldo de Moura; Ademir Damazio; Marisa da Silva Dias; Fumikazu Saito; Maria do Carmo de Sousa

• Advisor: Manoel Oriosvaldo de Moura

Abstract

This thesis presents the results of the research carried out to investigate the relationship between the historical and logical movement of algebraic concepts and the teachings object of algebra. Based on categories of dialectical materialism and on the theoretical framework of cultural-historical theory, modes of thinking, language and concept formation were analyzed through algebra history records. This first analysis allowed to highlight and to explain the conceptual connections and to characterize what was considered as the essence of algebra: to establish the relationship between variable magnitudes in a general way. The essence of algebra was considered as a category for another movement of analysis on the constitution of the teachings object of algebra. This object was recognized in curriculum proposals, in teachers speech and in teaching situations. Thus, data for analysis were seized from both: the logical and historical development of algebraic concepts (which present themselves as an object of study in a first moment and subsequently they assumed the role of an instrument and an analytical category), and the process of constructing and developing a course for teachers, which aim was to put in movement the relationships between the historical and logical movement of concepts and the teachings object of algebra. The topics \"sequences\", \"equations\" and \"functions\" that pervade algebra teaching in elementary education were the manifestations that allowed to evidence the analysis of the historical and logical movement of algebraic concepts as a principle for teaching organization. These processes of analysis allowed to recognize the importance of the generalization process for the algebraic knowledge, which led to the development of a \"model of analysis of generalization in teaching situations. From what was recognized as the essence of algebraic knowledge, planning actions for teaching equations established between the researcher and a teacher were also analyzed. In this research, it was recognized that the essence and the conceptual connections of algebraic knowledge revealed in its historical and logical movement are the central elements to be considered for the constitution of the teaching object of algebra. This is the main relation between the study of the historical and logical movement of concepts and the teaching object of algebra. The recognition of this relationship generates implications in the development of curricula, once it presents the fundamentals to review the concepts on the subject of teaching algebra process. Thus, it also generates consequences for the process of teacher training, so that they may consider, in addition to didactic and methodological guidelines, the specificity of the form of knowledge to be taught for students theoretical thinking development. (AU)

• FAPESP's process: 10/15383-4 - Contributions of cultural-historical theory to organize the teaching of algebra
• Grantee: Maria Lucia Panossian
• Support Opportunities: Scholarships in Brazil - Doctorate