Engenharia didática: análises preliminares e a priori para a noção dos quaternions de Fibonacci

Referencias

Abramovich, S. (2009). Extending Fibonacci numbers to negative subscripts through problem solving. International Journal of Mathematical Education in Science and Technology. 41(6), 836 – 842. Almouloud, A. S. (2007). Fundamentos da Didática da Matemática. São Paulo: Editora UFPR. Alves, F. R. V. & Borges Neto, H. (2011). A existência da Sequência de Fibonacci no campo dos Inteiros: uma atividade de investigação apoiada nos pressupostos da Sequência Fedathi. Boletim GEPEM. 1(53). 135 – 140 Alves, Francisco. R. V. (2015a). Sobre a evolução histórica do modelo de Fibonacci: a classe das funções hiperbólicas de Fibonacci. Vydya Educação, 35(1), 1 – 15. Alves, F. R. V. (2015b). Sequência Generalizada de Fibonacci – SGF e relações com a razão áurea. Boletim Cearense de Educação e História da Matemática – BOCEHM, 2(6), 1 – 5. Alves, F. R. V. (2016b). Sequência de Pell Generalizada – SGP: aspectos históricos e epistemológicos sobre a evolução de um modelo. Revista THEMA, 13(1), 1 – 25. ALVES, F. R. V. (2016b). Descobrindo definições matemáticas no contexto de investigação histórica: o caso da sequência generalizada de Fibonacci. BOLETIM GEPEM, 69(1), 1 – 5. Alves, Francisco. R. V. (2016c). Engenharia Didática para a generalização da Sequência de Fibonacci: uma experiência num curso de licenciatura. Educação Matemática Pesquisa. 18(1), 61 – 93. Alves, Francisco. R. V. (2017). Fórmula de de Moivre, ou de Binet ou de Lamé: demonstrações e generalidades sobre a sequência generalizada de Fibonacci – SGF. Revista Brasileira de História da Matemática. 17(33), 1 – 16. Artigue, M. (1984). Modélisation et Reproductibilité en Didactiques de Mathématiques. Les Cahiers Rouge des Didactiques des Mathematiques, 8(1), p. 1 – 38. Artigue, M. (1989). Une recherche d´ingènierie didactique sur l´enseignement des Èquations différentielles en premier cycle universitaire. Cahier du Séminaire de Didactique des Maths et de l´Informatique de Grenoble, p. 183-209, IMAG, Grenoble. Artigue, M. (2009). Didactical design in Mathematics Education. Carl Winslow (eds). NORMA08, Copenhaguen: Sense Publishers, Denmark, 7 – 16. Artigue, M. (2012). L´éducation mathématiques comme champ de recherché et champ de pratique: resultats et défis. EM TEIA: Revista de Educação Matemática e Tecnológica Iberoamericana, 3(3), 1 – 18. Ávila, G. (2002). O ensino do Cálculo e da Análise. Matemática Universitária. 33(1), p. 83 – 95. Baez, John. C. (2001). The Octonions. Bullettin American Mathematical Society, 39(1), 145 – 205. Bernard, Beck. (1979). Sur les équations polynomiales dans les quaternions. In. L’Enseignement Mathématique, 25(1), 193 – 201. Brother, Alfred. U. (1965). An introduction to Fibonacci Discovery. Santa Clara: Santa Clara University, California. Brousseau, Alfred. (1971). Linerar recursion and Fibonacci sequences. Santa Clara: Santa Clara University, California. Brousseau, G. (1986). Théorisation des phénomènes d´enseignement des Mathématiques (these de doctorat). Bourdeaux: Université Bourdeaux I, 905f. Brousseau, G. (1988). Le contrat didactique: le milieu. Recherche en Didactiques des Mathematiques, 9(2), p. 309 – 333. Brousseau, G. (1998). Les obstacles épistémologiques, problèmes et ingénierie didactique. G. In: Brousseau, (org.) Théorie des situations didactiques. Grenoble La Pensée Sauvage, p. 115 – 160. Buffet. Cécile. Ouvrier. (2003). Construction de definitions/ construction de concept: vers une situation fondamentale pour la construction de définition en mathématriques. Grenoble: Université Joseph Fourier. Burton, D. (2007). The History of Mathematics: an introduction, New York: McGraw-Hill Science. Catarino, P. M.; (2015). A note on h(x) − Fibonacci quaternion polynomials. Chaos, Solitons & Fractals, 77(4), 1 – 5. Catarino, P. M. C. (2016). The modified Pell and the Modified k-Pell Quaternions and Octonions, Advances in Applied Clifford Algebra, 26(2), 577 – 590. Catarino, P. M. C. (2019). On k-Pell hybrid numbers, Journal of Discrete Mathematical Sciences and Cryptography, 22(1), 83 – 89. Chevallard. Y. (1991). La Transposition didactique. Paris: La Pensée Sauvage Édition. Conway, John. H. & Smith, Derek. A. (2003). On quaternions and Octonions: their geometry, arithmetic and symmetry. London: A. K. Peters. Davis, P, J. & Hersh, R. (1981). L´Univers Mathématique, Paris: Gauthier & Villars. Douady, Régine. (1984). Jeux de Cadres et dialetique d´outil-objet dans l´enseignement de Mathématiques – une réalisation dans tout cursus primaire (thése d´État). Paris: Université Paris VII. 262f. Douady, Régine. (1995a). La ingeniería didáctica y la evolución de su relación con el conocimiento. Gomez, P. (org.) Ingenieria Didactica en Educación Matemática. Bogotá: Grupo Editorial Iberoamericano, p. 1 – 7. Douady, Régine. (1995b). Nacimiento y desarrollo de la didáctica de las matemáticas en Francia: rol de los IREM. Gomez, P. (org.) Ingenieria Didactica en Educación Matemática. Bogotá: Grupo Editorial Iberoamericano, p. 61 – 97. Douady, Régine. (2008). Géométrie, graphiques, fonctions au collège. Revista Eletrónica de investigación en educación e ciencias. 1(2), p. 1-7. Douady, A. & Douady, R. (1994). Changement de cadre a partir de surface minimale. Le Cahier Rouge, nº 23, 1 – 48. Eves, H. (2004). Introdução à história da matemática; tradução: Hygino H. Domingues. Campinas, SP: Editora da Unicamp. Falcon, S. (2016). On the complex k-Fibonacci numbers. Cogent Mathematics, v. 3, nº 1, 1 – 9. Flaut, C. & Shpakivskyi, V. (2013). Real Matrix representations for the Complex Quaternion. Advanced Applied Clifford Algebra, May, 10(2), 657 – 671. Grenier, D. & Legrand, M. (1986). Un sequence d´enseignement sur l´integrale en DEUG a 1º anée. Le Cahier Rouge, nº 22, 1 – 90. Gulberg, J. (1997). Mathematics: From the Birth of Numbers. New York: W. W. Norton & Company. Halici, Serpil. & Karatas, Adnan. (2016). Some Matrix Representations of Fibonacci Quaternions and Octonions. Advanced Clifford Algebra. April, 12(2), p. 123 – 143. Halici, Serpil. (2015). On Dual Fibonacci Octonions. Advanced Clifford Algebra, April, 25(4), 905 – 914. Halici, Serpil. (2012a). On Complex Fibonacci quaternions. Advanced Clifford Algebra, June, 23(1), 105 – 112. Halici, Serpil. (2012b). On Fibonacci Quaternions. Advanced Clifford Algebra, June, 22(2), 321– 327. Hamilton, William, H. (1853). Lectures on Quaternions. Dublin: The Royal Irish Academy. Hanson, Andrew. J. (2006). Visualizing Quaternions, London: Elsevier. Harman, C. H. (1981). Complex Fibonaci Numebrs, The Fibonacci Quarterly, 19(1), 82 – 87. Hoggat, Vernner, E. (1979), Fibonacci and Lucas Numbers, The Fibonacci Association, Houghton Mifflin Company: USA. Horadam, A. F, (1961). A generalized Fibonacci sequence, A. Math. M. 68(5), 455–459. Horadam. A. F. (1963). Complex Fibonacci Numbers Quaternions, Americal Mathematical Monthly. 70(2), 289 – 291. Horadam, A. F. (1967). Special Properties of the sequence Wn (a, b; p, q) . The Fibonacci Quaterly, 5(5), December, 424 – 435. Horadam, A. F. (1974). Oresme Numbers, The Fibonacci Quarterly, 12(3), 267 – 271. Iakin, A. L. (1977). Generalized Quaternions with Quaternion Components. The Fibonacci Quarterly, 15(4), December, 225 – 230. Huntley, H.E. The divine proportion: a study in mathematical beauty. New York: Dover Publications, 1970. Iakin, A. L. (1981). Extended Binet Forms for Generalized Quaternions of Higher Order. The Fibonacci Quarterly, 19(5), December, 410 – 414. Jeannin, André, R. (1991). Generalized Complex and Lucas functions. The Fibonacci Quarterly, 29(1), February, 13 – 19. Laborde, C. (1997). Affronter la complexité des situations didátiques d´apprentissage des mathématiques en classe: défis et tentatives. DIDASKALIA, 10(1), p. 97 – 112. Killgore, Peter. L. (2015). The Geometry of the Octonionic Multiplication Table. Baccalaureate of Science in Mathematics, Oregon: Oregon State University. Koshy. T. (2007). Elementary Number Theory and Applications, second edition, Boston: Elselvier. Krantz, S. G. (2006). An episodic history of mathematics. Mathematical culture through problem solving, New York: Mathematical Association of America. Margolinas, C. (1995). D´evolution et institutionnalisation: deux aspects antagonistes du rôle du maître. Didactique des disciplines scientifiques et formation des enseignants, Paris: Maison Édition, p. 342-347. Nurkan, Semra. K. & Güven, I. Arlans. (2015). On dual Fibonacci quaternions. In: Advanced Applied Clifford Algebra, 25(2), 403 – 414. Oliveira, R. R. (2018). Engenharia didática com o tema: relações bidimensionais, tridimenssionais e n-dimensiionais do modelo de Fibonacci. (Dissertação de mestrado em Mestrado em Ensino de Ciências e Matemática, PGECM) - Instituto Federal de Educação Ciencia e Tecnologia – IFCE. Polatli, E. & Kesim, S. (2015). A note on Catalan´s identity for the k-fibonacci Quaternions. Journal of Integer Sequences, 18(1), 1 – 4. Ramirez, J, L. (2015). Some Combinatorial Properties of the k-Fibonacci and the k-Lucas Quaternions. Versita, 23(2), 201 – 212. Robinet, J. (1983). De L´ingenierie Didactiques. Les Cahiers Blancs. 1(1), p. 1 – 11. Seródio, Rogério. (2008). Construction of Octonionic Polynomials. Advanced Clifford Algebra, 20(1), 155 – 178. Singh, P. (1985). The So-called Fibonacci Numbers in Ancient and Medieval India, Historia Mathematica, 12(1), 229 – 244. Singh, Mamta. et all. (2014). Generalized Fibonacci-Lucas Polynomails. International Journal Advanced Mathematics Sciences, 2(1), 81 – 87. Swamy, M. N. S. (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 31(4), November, 315 – 325. Soykan, Y. (2019). Tibonacci and Tribonacci – Lucas Sedenions, Mathematics, 7(1), 1 – 19. Disponível em: https://www. mdpi.com/2227-7390/7/1/74. Tasci, Dursun & Yalcin, Feyza. (2013). Complex Fibonacci p-numbers, Communications in Mathematics and Applications, 4(3), 213 – 218. Thongmoon, M. (2009). New Identities for the Even and Odd Fibonacci and Lucas numbers, International Journal Contemporary Mathematical Sciences. 4(14), 671 – 676. Vernner, E. & Hoggat, Jr. (1969). Fibonacci and Lucas Numbers. Santa Clara: Santa Clara University. Waddill, M. E; Sacks, L. (1967). Another Generalized Fibonacci Sequence. The Fibonacci Quartely. 5(3), October, 209 – 222. Walton, J. E. & Horadam, A. F. (1974). Some aspects of Generalized Fibonacci Numbers, The Fibonacci Quarterly, 31(4), October, 241 – 250. Wang, H. & Ren, G. (2014). Octonion Analysis of Several Variables. Communications in Mathematics and Statistics. June, 164 – 185. Whittaker, E. T. (1945). The Sequence of Ideas in the Discovery of Quaternions. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Science. v. 50, 93 – 98.