A Organização do Ensino de Álgebra: Contribuições da Atividade Orientadora de Ensino

ResumoAs constantes dificuldades manifestadas por estudantes em relação à aprendizagem do conhecimento algébrico têm provocado a necessidade de repensar o processo de organização do ensino deste conteúdo. Nesse sentido, grupos de pesquisas vinculados às Instituições de Ensino Superior se debruçaram nesta temática e aqui serão destacadas as pesquisas de membros do Grupo de Estudos e Pesquisa sobre a Atividade Pedagógica (GEPAPe), coordenado pelo professor Dr. Manoel Oriosvaldo de Moura, junto ao departamento de Educação da Universidade de São Paulo (USP). Assim, neste artigo serão analisadas as contribuições da Atividade Orientadora de Ensino para a organização do ensino de álgebra, a partir das produções do GEPAPe. Para o cumprimento deste objetivo foram buscados textos na forma de artigos, de livros, de capítulos de livros e anais de eventos que foram selecionados a partir da plataforma lattes de cada um dos pesquisadores deste grupo. Como resultado foi possível reconhecer que as produções revelam a compreensão sobre a álgebra e seu ensino sustentadas pelo estudo do movimento dialético histórico e lógico visando o desenvolvimento dos nexos conceituais do conhecimento algébrico e o desenvolvimento do pensamento teórico dos estudantes, superando o reconhecimento empírico e aparente do simbolismo algébrico. Tal compreensão sobre a organização do ensino de álgebra se materializa por meio das Situações Desencadeadoras de Aprendizagem (Situação Emergente do Cotidiano; Jogo; História Virtual do Conceito), que são desenvolvidas em espaços de aprendizagem como Clubes de Matemática e Oficinas Pedagógicas de Matemática em diferentes instituições brasileiras. Palavras-chave: AOE. GEPAPe. Situação Desencadeadora de Aprendizagem. AbstractThe constant difficulties expressed by students regarding learning algebraic knowledge have led to the need to rethink the teaching organizing process of this content. Thus, research groups linked to Higher Education Institutions, focused on this theme and the research will be highlighted of the Study and Research Group on Pedagogical Activity members (GEPAPe), coordinated by Professor Dr. Manoel Oriosvaldo de Moura, with the Department of Education at the University of São Paulo (USP). Thus, in this article, the Teaching Guidance Activity contributions to the algebra teaching organization will be analyzed, based on the GEPAPe productions. To fulfill this objective, texts were sought in the form of articles, books, book chapters and event proceedings that were selected from the lattes platform of each of the researchers in this group. As a result, it was possible to recognize that the productions reveal the understanding of algebra and its teaching supported by the study of the historical and logical dialectical movement aiming at the development of the conceptual nexus of algebraic knowledge and the students' theoretical thinking development, surpassing the empirical and apparent recognition of algebraic symbolism. This understanding of the algebra teaching organization is materialized through the Learning Trigger Situations (Emerging Situation of Everyday Life; Game; Virtual History of the Concept), which are developed in learning spaces such as Math Clubs and Mathematics Pedagogical Workshops in different institutions Brazilian companies. Keywords: AOE. GEPAPe. Learning Triggering Situation.

Mathematics in the archives: deconstructive historiography and the shaping of modern geometry (1837–1852)

This essay explores the research practice of French geometer Michel Chasles (1793–1880), from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing.

On the Origin of Symbolic Mathematics and Its Significance for Wittgenstein’s Thought

The main topic of this essay is symbolic mathematics or the method of symbolic construction, which I trace to the end of the sixteenth century when Franciscus Vieta invented the algebraic symbolism and started to use the word ‘symbolic’ in the relevant, non-ontological sense. This approach has played an important role for many of the great inventions in modern mathematics such as the introduction of the decimal place-value system of numeration, Descartes’ analytic geometry, and Leibniz’s infinitesimal calculus. It was also central for the rigorization movement in mathematics in the late nineteenth century, as well as for the mathematics of modern physics in the 20th century.However, the nature of symbolic mathematics has been concealed and confused due to the strong influence of the heritage from the Euclidean and Aristotelian traditions. This essay sheds some light on what has been concealed by approaching some of the crucial issues from a historical perspective. Furthermore, I argue that the conception of modern mathematics as symbolic mathematics was essential to Wittgenstein’s approach to the foundations and nature of mathematics. This connection between Wittgenstein’s thought and symbolic mathematics provides the resources for countering the still prevalent view that he defended an uttrely idiosyncratic conception, disconnected from the progress of serious science. Instead, his project can be seen as clarifying ideas that have been crucial to the development of mathematics since early modernity.

Errores en la traducción de enunciados algebraicos entre los sistemas de representación simbólico y verbal

Este artículo describe parte de una investigación sobre las traducciones de expresiones algebraicas, entre los sistemas de representación simbólico y verbal, que realizan escolares de educación secundaria. El objetivo principal abordado es identificar y clasificar los errores en los que incurren los estudiantes al realizar dichas traducciones. Entre los resultados obtenidos destacamos la influencia de las características del simbolismo algebraico en los errores cometidos en la traducción de expresiones verbales a simbólicas y la mayor facilidad para traducir enunciados del sistema de representación simbólico al verbal.Errors in the Translation of Algebraic Statements between Symbolic and Verbal Representation SystemsThis article describes part of a study on secondary students’ translations between the symbolic and the verbal representation systems. The main objective is to identify and classify the errors they made when performing these translations. Among the obtained results we highlight the influence of the characteristics of algebraic symbolism in the errors made when translating from verbal to symbolic expressions and the larger ease to translate statements from the symbolic to the verbal representation system.Handle: http://hdl.handle.net/10481/36051WOS-ESCINº de citas en WOS (2017): 1 (Citas de 2º orden, 0)

Using Spreadsheets to Promote Algebraic Thinking

For students to find algebra conceptually meaningful, as well as useful in modeling and analyzing real-world problems, they must be able to reflect on, make sense of, and communicate about general numerical procedures (Kieran 1992). Such procedures consist of set sequences of arithmetic operations performed on numbers. Examples include computing an average and performing the standard division algorithm. Thinking about numerical procedures starts in the elementary grades and continues in successive grades until students can eventually express and reflect on the procedures using algebraic symbolism. This article outlines how such thinking can progress to algebraic reasoning and illustrates how computers used to promote this progression.