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Universidade Federal Do Rio Grande Do Norte Centro De Ciências Exatas E Da Terra Programa De Pós-Graduação Em Ensino De Ciências Naturais E Matemática

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Universidade Federal Do Rio Grande Do Norte Centro De Ciências Exatas E Da Terra Programa De Pós-Graduação Em Ensino De Ciências Naturais E Matemática UNIVERSIDADE FEDERAL DO RIO GRANDE DO NORTE CENTRO DE CIÊNCIAS EXATAS E DA TERRA PROGRAMA DE PÓS-GRADUAÇÃO EM ENSINO DE CIÊNCIAS NATURAIS E MATEMÁTICA MARLENE GORETE DE ARAÚJO ABU JA'FAR MUHAMMAD IBN MUSA AL-KHWARIZMI: CONTRIBUIÇÕES DA ÁLGEBRA PARA O ENSINO NATAL – RN 2019 MARLENE GORETE DE ARAÚJO ABU JA'FAR MUHAMMAD IBN MUSA AL-KHWARIZMI: CONTRIBUIÇÕES DA ÁLGEBRA PARA O ENSINO Dissertação apresentada ao curso de Pós- Graduação em Ensino de Ciências Naturais e Matemática da Universidade Federal do Rio Grande do Norte, como requisito parcial à obtenção do título de Mestre em Ensino de Ciências Naturais e Matemática. Orientadora: Profª Dra. Bernadete Barbosa Morey. Coorientador: Prof° Fabian Arley Posada Balvin. NATAL – RN 2019 Universidade Federal do Rio Grande do Norte - UFRN Sistema de Bibliotecas - SISBI Catalogação de Publicação na Fonte. UFRN - Biblioteca Central Zila Mamede Araújo, Marlene Gorete de. Abu ja'far muhammad ibn musa al-khwarizmi: contribuições da álgebra para o ensino / Marlene Gorete de Araújo. - 2019. 141f.: il. Dissertação (Mestrado) - Universidade Federal do Rio Grande do Norte, Centro de ciências exatas e da terra, Programa de pós- graduação em ensino de ciências naturais e matemática, Natal, 2019. Orientador: Dra. Bernadete Barbosa Morey. Coorientador: Dr. Fabian Arley Posada Balvin. 1. Ensino de Álgebra - Dissertação. 2. Matemática islâmica medieval - Dissertação. 3. História e Ensino de Matemática - Dissertação. I. Morey, Bernadete Barbosa. II. Balvin, Fabian Arley Posada. III. Título. RN/UF/BCZM CDU 512 Elaborado por Raimundo Muniz de Oliveira - CRB-15/429 MARLENE GORETE DE ARAÚJO A ÁLGEBRA DE ABU JA'FAR MUHAMMAD IBN MUSA AL-KHWARIZMI: CONTRIBUIÇÕES PARA O ENSINO Dissertação apresentada ao curso de Pós- Graduação em Ensino de Ciências Naturais e Matemática da Universidade Federal do Rio Grande do Norte, como requisito parcial à obtenção do título de Mestre em Ensino de Ciências Naturais e Matemática. Orientadora: Profª Dra. Bernadete Barbosa Morey. Coorientador: Prof° Fabian Arley Posada Balvin. Banca Examinadora ___________________________________________________________________ Profª Dra. Bernadete Barbosa Morey (Orientadora) - Presidente ___________________________________________________________________ Profa. Dra. Giselle Costa de Sousa – examinador interno ___________________________________________________________________ Profa. Dra. Ana Carolina Costa Pereira (UECE) – Examinador externo ___________________________________________________________________ Prof. Dr. Fabian Arley Balvin Posada – Co-orientador AGRADECIMENTOS Agradeço a Deus, por todas as vezes que Ele me deu força quando pensei em desistir ou redefinir meus objetivos. A meus pais e demais familiares, pela educação e conselhos que me foram dados. À minha orientadora, professora doutora Bernadete Morey, pela acolhida e ensinamentos. Confesso que sem as orientações que me foram prestadas, não teria conseguido chegar até onde estou. Aqui lhe exprimo a minha gratidão! Ao meu companheiro de todas as horas, Ramon Costa, que divide comigo todas as dificuldades e felicidades da vida acadêmica e pessoal. A todos os meus amigos que entenderam a minha ausência durante esses últimos tempos, ouviram todas as minhas inquietações e me incentivaram a seguir adiante. A todos os amigos do Programa de Pós-graduação em Ensino de Ciências Naturais e Matemática (PPGECNM), que dividiram comigo o peso das disciplinas e da dissertação e as angustias e as alegrias vividas durante esse período. A todos os orientandos da professora Bernadete Morey, por todos os conhecimentos compartilhados em nosso grupo de estudo. A todos os professores do PPGECNM, pelos ensinamentos durante o mestrado. Aos professores que participaram da banca de qualificação e defesa: Dra. Giselle Costa de Sousa, Dra. Ana Carolina Costa Pereira e Dr. Fabian Arley Balvin Posada, cujas contribuições foram primordiais para a conclusão desse trabalho. A todos, mesmo àqueles que por ventura não tenham sido registrados aqui, meus sinceros agradecimentos. RESUMO Reconhecendo que a matemática islâmica medieval produziu diversos nomes de estudiosos de primeira grandeza, a presente pesquisa tem como foco de estudo a obra intitulada Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (Livro de Restauração e Balanceamento), mais conhecida como a Álgebra de al-Khwarizmi, escrita pelo eminente sábio islâmico medieval, Abu Ja'far Muhammad ibn Musa al- Khwarizmi. Desse modo, o objetivo dessa pesquisa foi conhecer as potencialidades histórico-pedagógicas do o Tratado de algébrico de al-Khwarizmi (Álgebra de al- Khwarizmi) tendo como suporte teórico-metodológico a Teoria da Objetivação. Essa é uma pesquisa histórica bibliográfica e documental, de natureza qualitativa, com algumas pinceladas da pesquisa-ação. No decorrer da pesquisa foi elaborado um material instrucional sob a forma de e-book que foi pensado para satisfazer a exigência de produto educacional no curso de Mestrado Profissional. Na presente pesquisa, o e-book desempenha o papel de material didático de auxílio aos professores de matemática que queiram se familiarizar com a obra Álgebra de al- Khwarizmi. Na apresentação da Álgebra de al-Khwarizmi aos professores, optamos por privilegiar os aspectos formais, externos e físicos da obra, por pensar que a familiaridade com tais aspectos da obra citada, pode potencializar, por parte dos professores, futuramente, uma imersão mais profunda no conteúdo histórico- matemático da obra. Desse modo, concluímos que a Álgebra de al-Khwarizmi é um tratado consideravelmente importante para o desenvolvimento da álgebra. Além disso, esse tratado apresenta um grande potencial para a formação docente, não somente quanto aos aspectos matemáticos, mas também quanto aos aspectos históricos, sociais e políticos. Palavras-chave: Ensino de Álgebra. Matemática islâmica medieval. História e Ensino de Matemática. ABSTRACT Recognizing that medieval Islamic mathematics has produced several names of first- rate scholars, this research focuses on the well-known work entitled Al-Kitāb al- mukhtaṣar fī ḥisāb al-jabr wal-muqābala (Book of Restoration and Balancing) like al- Khwarizmi Algebra, written by the eminent medieval Islamic sage, Abu Ja'far Muhammad ibn Musa al-Khwarizmi. Thus, the objective of this research was to know the historical-pedagogical potentialities of al-Khwarizmi's Algebraic Treaty (Algebra de al-Khwarizmi) having as theoretical and methodological support the Objection Theory. This is a qualitative bibliographical and documentary historical research, with some brushstrokes of action research. In the course of the research an instructional material was prepared in the form of e-book that was designed to satisfy the educational product requirement in the Professional Master course. In this research, the ebook plays the role of teaching aids to mathematics teachers who want to become familiar with al- Khwarizmi's Algebra. In the presentation of al-Khwarizmi's Algebra to the teachers, we chose to privilege the formal, external and physical aspects of the work, thinking that the familiarity with such aspects of the mentioned work may, in the future, enhance the teachers' deeper immersion. in the historical-mathematical content of the work. Thus we conclude that al-Khwarizmi Algebra is a considerably important treatise for the development of algebra. Moreover, this treatise has great potential for teacher education, not only in mathematical aspects, but also in historical, social and political aspects. Keywords: Algebra teaching. Medieval Islamic Mathematics. History and Teaching of Mathematics. LISTA DE FIGURAS Figura 1 - Península Arábica ..................................................................................... 24 Figura 2 - Selo emitido pela União Soviética em 6 de setembro de 1983 ................. 41 Figura 3 - Página do Livro de Restauração e Balanceamento .................................. 45 Figura 4 - Mapa atribuído aos geógrafos do califa al-Mamun ................................... 47 Figura 5 - Capa do El libro del Álgebra: Mohammed ibn-Musa al-Jwarizmi .............. 55 Figura 6 - Tratado algébrico de al-Khwarizmi em árabe e em espanhol ................... 56 Figura 7 – Momento de discussão da tarefa 1 .......................................................... 93 Figura 8 – Utilização de mapa para apresentação do mundo islâmico medieval ...... 94 Figura 9 – Conversa sobre a Álgebra de al-Khwarizmi ............................................. 96 Figura 10 – Sequências de imagens que apresentam o labor conjunto entre participantes .............................................................................................................. 98 LISTA DE MAPAS Mapa 1 - Conquistas do Islã até 632 ......................................................................... 27 Mapa 2 - Expansão do Islã até 750 ........................................................................... 28 Mapa 3 - Expansão do Islã durante 750 a 1700 ........................................................ 30 Mapa 4 - Khwarezm, localizado em Uzbequistão ...................................................... 42 LISTA DE QUADROS Quadro 1 – Percurso Metodológico da pesquisa ....................................................... 21 Quadro 2 - Síntese das conquistas do mundo islâmico ............................................ 31 Quadro 3 - Sumário do Tratado algébrico de al-Khwarizmi ....................................... 59 Quadro 4 - Os seis tipos de equações polinomiais propostas por al-Khwarizmi ....... 64 Quadro
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