Engenharia Didática de Formação (EDF): sobre o ensino dos Números (Generalizados) de Catalan (NGC)

Referencias

ALMOULOUD, Ag S.. Fundamentos da Didática da Matemática. São Paulo: Editora UFPR, 2007. ALMOULOUD, Ag S.;SILVA, M. J. F. Engenharia didática: evolução e diversidade. Revemat. Florianópolis. v. 7, n. 2, p. 22-52, 2012. ALVES, F. R. V. On teaching of generalized catalan numbers with the maple ́s help. Acta Didactica Naposcencia. v. 11, nº 1, 25 –40, 2018. Disponível em: http://adn.teaching.ro/ ALVES, F, R. V. Didática da Matemática: seus pressupostos de ordem epistemológica, metodológica e cognitive. Interfaces da Educação. v. 7, nº 21, 131 –150. 2016. ALVES, F, R. V.Fórmula de Moivre, ou de Binnet ou de Lamé: demonstrações e generalidades sobre a sequencia generalized de Fibonacci. Revista Brasileira de História da Matemática. v. 17, nº 33, 1 –16,2017a. ALVES, F. R. V. Engenharia Didática sobre a noção de Sequência Generalizada de Jacobsthal. Revista UnionIberoamericana de Matemática. Nº 51, 83 –106. 2017b. ALVES, F. R. V.; CATARINO, M. P. Engenharia didática de Formação (EDF): repercussões para a formação do Professor de Matemática no Brasil. Educação Matematica em Revista –RS. v. 2, nº 18, 121 –137. 2017. Disponível em: http://sbemrs.org/revista/index.php/2011_1/article/view/304 ALLEN, E.. Combinatorial Interpretations Of Generalizations Of Catalan Numbers And Ballot Numbers. (Thesis in Mathematics and Science). Pittsburgh: Pittsburgh University, 2014. ANDREWS, G.. Number Theory. New York: West Washington Square, 1971. ARTIGUE, M.. Ingénierie Didactiques. BRUN, J. (org.). Didactiques de Mathématiques, 243 –264. LAGRANGE J.B. & al. (eds). Jun, Reims, France. 1995. ARTIGUE, M.. Modélisationet Reproductibilité enDidactiques de Mathématiques. Les Cahiers Rouge des Didactiques des Mathematiques, v. 8, nº1, p. 1 –38, 1984. ARTIGUE, M.. Perspectives on Design Research: The Case of Didactical Engineering. In: BIKNER-AHSBAHS, Angelika; KNIPPING, Khristine; PRESMEG, Norma. Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods. New York: Springer, 467 -497, 2015. BARCUTTI, E.; VERRY, M. C. Some more properties of Catalan numbers. Dicrete Mathematics. nº 102, 229 –237, 1991. BERNHART, F. R. Catalan, Motzkin, and Riordan numbers. Discrete Mathematics.v. 4, nº 4, p. 73 –112, 1999.Disponível em: http://users.dimi.uniud.it/~giacomo.dellariccia/Glossary/Catalan/Bernhart1999.pdf BILU, Y. F.;BUGEAUD, Y; MIGNOTTE. M. The problem of Catalan. New York: Springer, 2010. BOBROWSKI, J.. Generalized Catalan Numbers and Some Divisibility Properties. (Master in Science). Nevada: University of Nevada, 2015. Disponível em:https://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=3519&context=thesesdissertations BRASIL JUNIOR, N. G. Bijeções envolvendo os Números de Catalan. (Dissertação de Matemática. Estatistica e Computação). Campinas: UNICAMP, 2014. BROUSSEAU, G. Théorisation des phénomènes d ́enseignement des Mathématiques(these de doctorat). Bourdeaux: Université Bourdeaux I, 1986. 905f. BUESCU, J.. Os incríveis números de Catalan. Ingenium. 91 –95. 2010. Disponível em: http://nda.tecnico.ulisboa.pt/files/sites/40/Os-incriveis-numeros-de-Catalan-ingenium-93-2010.pdf BOYER, C.. A History of Mathematics. London: Wiley and Sons, 1968. BROUSSEAU, G.. Le contrat didactique: le milieau. Recherche en Didactiques des Mathématiques. v. 9, nº 3, 309 –336. 1988. Disponível em: http://guy-brousseau.com/wp-content/uploads/2012/02/contrat-milieu1.pdf CAMPBELL. D.. M. The Computation of Catalan Numbers. Mathematics Magazine, v. 57, No. 4 (Sep., 1984), 195-208. 1984. Disponível em: http://www.jstor.org/stable/pdf/2689673.pdf?refreqid=search%3Af334a92a758df808910f31b230e73395 CATALAN, E. Sur les nombre de Segner. Circolo Matematico di Palermo. Itália. 1886. Disponível em: https://orbi.uliege.be/handle/2268/207766 CHEVALLARD, Y. La transposition didactique. Paris: presses universitaires. 1991. CHEVALLARD, Y.;JOHSUA, M. A. Un exemple d'analyse de la transposition didactique. Recherches en didactique des mathématiques. Vol 3.2, 1982. CHUNNING, X. Some interger properties of generalized catalan numbers. Buletin Sinica. v. 22, nº 12, 66 –74. 2016. Disponível em: http://web.math.sinica.edu.tw/math_media/d402/40208.pdf DEVELAY, M.. Pour une épistémologie des savoirs scolaires. Pédagogie Collegialle. v. 7, nº 1, 35 –40. 1993. Disponível em: http://aqpc.qc.ca/sites/default/files/revue/develay_07_1.pdf DOUADY, R.. La ingeniería didáctica y la evolución de su relación con el conocimiento. Gomez, P. (org.) Ingenieria Didactica en Educación Matemática. Bogotá: Grupo Editorial Iberoamericano, 1 –7, 1995. EVES, H.. An introduction to History of Mathematics. New York: Holt, Rinehart and Winston, 1976. GESSEL, I. E. Super ballot numbers. Journal of Symbolic Mathematics. 179 –194.1992. disponívelem: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.4040 GESSEL,I. E.; Xin, G. A combinatorial interpretation of the numbers. 2(3), 1 –20. 2004. Disponível em: https://arxiv.org/pdf/math/0401300.pdf GRAHAM, R, L; KNUTH,D. E.; PASTASHNKI, O. Concrete Mathematics. Second edition. California: Addison & Wesley Publshers, 1988. GRIMALDI, R.. P. Fibonacci and Catalan Numbers. New Jersey: Wiley & Sons Publications, 2012. GUIMARÃES, H. L. A. O passeio de Catalan na Praia e as gramissianas de retas. (dissertação de mestrado). Recife: UFPE, 2012. HURTADO, F.; NOY, M. Graph of triangulations of a convex polygon and tree of triangulations. Computational Geometry, v. 13, 179–188, 1999. Disponível em: https://pdfs.semanticscholar.org/15b6/a9de791dcf14a4f5276577c336dfd799ee0b.pdf HURTADO, F.; NOY, M. Ears of triangulations and Catalan numbers. Discrete Mathematics. nº149, 319 –324, 1996. LARCOMBE. P. L. The 18th Century Chinese descovery of the Catalan numbers. Mathematical spectrum. 32(1), 6 –7, 1999. LAURENTIN, J.. Réflexions sur la triangulation des polygones convexes. Bulletin de la Sabix. 2(2), 141 –150, 2016. KOSHY, T.. SALMASSI. M.. Parity and Primality of Catalan Numbers. The College Mathematics Journal, Vol. 37, No. 1, Jan. 52-53. 2006. KOSHY, T.. The ubiquitous Catalan numbers. Mathematic Teacher. 100(3), 184 –188. 2006. Disponível em: http://lyceeenligne.free.fr/IMG/pdf/Catalan.pdfKOSHY, T.. Some Catalan identities with interesting consequences. The Mathematical Gazette, 96(536), (July 2012), p. 323-328. 2012. Disponível em: http://www.jstor.org/stable/pdf/23248576.pdf?refreqid=excelsior%3A61b9ded96ef2b57e3adaf21f6f15b358 KOSHY, T..Catalan Numbers with application. Oxford: Oxford University Press, 2007. KOSHY,T..; Gao, Zhenguang. Convergence of a Catalan Series. The college mathematics journal.43(2), 141 –146. 2012. Disponível em: http://www.geofhagopian.net/papers/Convergence_Catalan_Series.pdf KOSHY,T.; GAO, Z.. Some divisibility properties of Catalan numbers. The Mathematical Gazette, v. 95 nº 532, 96 –102. 2011. Disponível em:http://www.jstor.org/stable/pdf/23248631.pdf?refreqid=excelsior:46af9b06c52625441d77125d755cfb22 MARGOLINAS, C. Essai de généalogie en didactique des mathématiques. Revue suisse des sciences de l’éducation. 7(3), p. 343 –360, 2005. MARGOLINAS, C..Situations, savoirs et connaissances... comme lieux de rencontre? Formation et pratiques d’enseignement en questions. nº 19, 31 –39, 2015. Disponível em:http://www.revuedeshep.ch/site-fpeq-n/Site_FPEQ/19_files/2015-Margolinas-FPEQ-19.pdf MARGOLINAS, C.. Essai de généalogie en didactique des mathématiques. Revue suisse des sciences de l’éducation. v. 27, nº 3, 343 –360, 2005. Disponível em: http://www.pedocs.de/volltexte/2011/4128/pdf/SZBW_2005_H3_S343_Margolinas_D_A.pdf MARGOLINAS, C.; DRIJVERS, P. Didactical engineering in France; an insider’s and an outsider’s view on its foundations, its practiceand its impact. ZDM Mathematics Education, nº 47, 893 –903, 2015. Disponívelem:https://link.springer.com/article/10.1007/s11858-015-0698-z MAYS, M.E.; WOJCIECHOWSKI, J. A determinant property of Catalan numbers. Discrete Math. 21, no. 1-3, 125–133, 2000. MCCAMMOND, J. Noncrossing partitions in surprising locations. American Mathematical Monthly, nº 113, 598 –610, 2006. PASTRÉ, P.. L ́analyse du travail en Didactique professionnelle. Revue Française de Pédagogie, v. 3, nº 138, 9 –17, 2002. PASTRÉ, P. La Didactique Professionelle. Education, Sciences & Society, v. 2, nº 1, 83 –95. 2011. Disponível em: https://riviste.unimc.it/index.php/es_s/article/view/136/65 PERRIN-GLORIAN, M-J; BELLEMAIN, P. M. B. L’ingenierie didactique entre recherche et ressource pour l’enseignement et la formation des maitres. Anais do I Simpósio Latino-Americano de Didática da Matemática-LADIMA. 1 –15, 2016. PERRIN-GLORIAN, M. J. (2011). L’ingénierie didactique à l’interface de la recherche avec l’enseignement. Vers une ingénierie didactique de deuxième génération? In: MARGOLINAS, Claire; DRIJVERS, P. Didactical engineering in France; an insider’s and an outsider’s view on its foundations, its practice and its impact. ZDM Mathematics Education. 47(4), p. 893–903, 2015. REGEV, A. A proof of catalan’s convolution formula. INTEGERS, v. 12. nº 1, 2011.Disponível em: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/m29/m29.pdf RIBENBOIN, P. A conjectura de Catalan. Matemática Universitária. nº 21, Junho, 3 –16.1996. Disponível em: http://rmu.sbm.org.br/Conteudo/n20_n21/n20_n21_Artigo01.pd ROMAN. S. An Introduction to Catalan Numbers. London: Birkhauser, 2015. ROQUE, T. História da Matemática: uma visãocrítica, desfazendo mitos e lendas.Rio de Janeiro: ZAHAR editora, 2012. RUBENSTEIN, D. Catalan Numbers Revisited. Journal of combinatorial theory. 486 –490, 1993. SHAPIRO, L. W. A Catalan triangle. Discrete Mathematics. nº 14, 83–90, 1976. STANLEY, R.P. The Catalan numbers. Cambridge: Cambridge University Press, 2015. STOJADINOVIĆ, T. On Catalan Numbers. Teaching of Mathematics, 18(1), 16-24. 2015. Available in: http://elib.mi.sanu.ac.rs/files/journals/tm/34/tmn34p16-24.pdf TRIVEDI, M. B.; JHA, P. Approaches differ: Catalan numbers. International Journal of Advanced Science and Research. 2(6), 82 –89. 2017. Available in: http://www.allsciencejournal.com/archives/2017/vol2/issue6/2-6-21 SARACEVI, M. H.; STANIMIROVI, S, P.; KRTOLICA, S; MASOVI. C. Construction and Notation of Convex Polygon Triangulation Based on Ballot Problem. Romanian Journal of information science and technology, v. 17, nº 3, 237–251, 2014. Disponível em: http://www.romjist.ro/content/pdf/03-msaracevic2014.pdf VARADARAJAN, V. S. Euler through time: a new look at old themes, AMS, Providence, RI, 2006. VAREY, A.Catalan Numbers: New and Old(Master of Science and Mathematics). Wisconsin-Milwaukee: Wisconsin-Milwaukee University. 2011. VON SZILY. Über die quadratsummen der binomial coëfficienten. Math Nat Berlim. 12(1), 84 –91, 1894.