Notas para um estudo da transposição didática da prova matemática
DOI:
https://doi.org/10.23925/1983-3156.2025v27i3p309-354Palavras-chave:
Prova, Demonstração, Transposição didática, Ensino, AprendizagemResumo
Este artigo estuda o longo caminho entre a ausência da demonstração como tal no passado e sua presença contemporânea como um conteúdo a ser ensinado em todas as séries. No entanto, a demonstração teve que passar por um processo de transposição didática para satisfazer uma série de restrições diferentes, de natureza epistêmica, didática, lógica ou matemática. O autor, portanto, segura no seu estudo uma ordem cronológica para delinear as principais características desse processo, com o objetivo de compreender melhor o problema didático que nossa pesquisa atual enfrenta. Balacheff mostra que a prova é tanto um fundamento quanto um organizador do conhecimento. No curso da aprendizagem, ela contribui para reforçar a evolução do conhecimento e fornece ferramentas para sua organização. No ensino, ela legitima novos conhecimentos e constitui um sistema: conhecimento e prova, interligados, fornecem à base de conhecimento de uma estrutura que pode funcionar como precursora da base teórica necessária à matemática. A função de institucionalização das situações de prova coloca a validação explícita sob a arbitragem do professor, que é, em última análise, o garantidor de seu caráter matemático. Essa dimensão social, no sentido de que o funcionamento científico depende de uma organização construída e aceita, está no cerne da dificuldade de ensinar prova em matemática.
Referências
Ausejo, E., & Matos, J. M. (2014). Mathematics Education in Spain and Portugal. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 283‑302). Springer Science & Business Media.
Balacheff, N. (1990). Beyond a psychological approach of the psychology of mathematics education. For The Learning of Mathematics, 10(3), 2‑8.
Balacheff, N. (2022). Penser l’argumentation pour la classe de mathématique. Petit x, 116, 75‑105https://irem.univ-grenoble-alpes.fr/revues/petit-x/consultation/numero-116-petit-x/4-penser-l-argumentation-pour-la-classe-de-mathematique-1114578.kjsp?RH=1661422241433
Balacheff, N. (preprint). Mathematical Argumentation, a Precursor Concept of Mathematical Proof. Proceedings ICME14 Invited Lectures, 17.
Balacheff – Notes for the study of the didactic transposition of proof - 03/03/2023 09:20 26 / 29 Delarivière, S., Frans, J., & Van Kerkhove, B. (2017). Mathematical Explanation: A Contextual Approach. Journal of Indian Council of Philosophical Research, 34(2), 309‑329. https://doi.org/10.1007/s40961-016-0086-2
Ball, D. L. (1993). With an Eye on the Mathematical Horizon : Dilemmas of Teaching Elementary School Mathematics. The Elementary School Journal, 93(4), 373‑397.
http://www.jstor.org/stable/1002018
Barbin, E. (2007). On the argument of simplicity in Elements and schoolbooks of Geometry. Educational Studies in Mathematics, 66(2), 225‑242. https://doi.org/10.1007/s10649-006-9074-9
Barbin, É. (2021). L’écriture de manuels de géométrie pour les Écoles de la Révolution : Ordre des connaissances ou « élémentation ». In A. Le Goff & C. Demeulenaere-Douyère (Éds.), Enseignants et enseignements au cœur de la transmission des savoirs. Éditions du Comité des travaux historiques et scientifiques. https://doi.org/10.4000/books.cths.14562
Barbin, E., & Menghini, M. (2014). History of teaching of geometry. In A. Karp & F. Furinghetti (Éds.), Handbook on the History of Mathematics Education (p. 473‑492). Springer New York.
Bartle, R. G. (1995). A brief history of the mathematical literature. Publishing Research Quarterly, 11(2), 3‑13. https://doi.org/10.1007/BF02680421
Bell, A. W. (1976). A study of pupils’ proof-explanations in mathematical situations. Educationa Studies in Mathematics, 7(1‑2), 23‑40. https://doi.org/10.1007/BF00144356
Bieda, K. N., Conner, A., Kosko, K. W., & Staples, M. (Éds.). (2022). Conceptions and Consequences of Mathematical Argumentation, Justification, and Proof. Springer International Publishing. https://doi.org/10.1007/978-3-030-80008-6
Bock, D. D., & Vanpaemel, G. (2015). Modern mathematics at the 1959 OEEC Seminar at Royaumont. In K. Bjarnadottir, F. Furinghetti, J. Prytz, & G. Schubring (Éds.), Proceedings of the Third Conference on the History of Mathematics Education (p. 151‑168). Department of Education, Uppsala University.
Boero, P. (Éd.). (2007). Theorems in school : From history, epistemology and cognition to classroom practice. Sense Publishers.
Brentjes, S. (2014). Teaching the Mathematical Sciences in Islamic Societies Eighth–Seventeenth Centuries. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 85‑107). Springer New York. https://doi.org/10.1007/978-1-4614-9155-2_5
Brentjes, S. (2019). Pourquoi et comment étudier l’histoire de l’enseignement des mathématiques dans les sociétés islamiques entre 750 et 1500. Médiévales, 77, 11‑35.
https://doi.org/10.4000/medievales.10194
Brousseau, G. (1997). Theory of didactical situations in mathematics. Kluwer Academic Publishers. 1997
Chevallard, Y. (1985). La transposition didactique : Du savoir savant au savoir enseigné. La Pensée Sauvage.
Chevallard, Y. (1998). Analyse des pratiques enseignantes et didactique des mathématiques :
L’approche anthropologique. Analyse des pratiques enseignantes et didactique des
mathématiques, 91‑120.
Chevallard, Y., & Bosch, M. (2014). Didactic Transposition in Mathematics Education. In S. Lerman (Éd.), Encyclopedia of Mathematics Education (p. 170‑174). Springer Netherlands. https://doi.org/10.1007/978-94-007-4978-8_48
Chevallard, Y., Bosch, M., & Kim, S. (2015). What is a theory according to the anthropological theory of the didactic? Proceedings of the Ninth Congressof the European Society for Research in Mathematics Education, 2614‑2620.
Clairaut, A. C. (1741). Elemens de géométrie (1753e éd.). David fils. https://gallica.bnf.fr/ark:/12148/bpt6k15218356
Condillac, E. (1746). Essai sur l’origine des connaissances humaines (édition 1798, édition numérique UQAC 2010). Ch. Houel imprimeur. http://classiques.uqac.ca/
Cunning, D. (2015). Analysis versus Synthesis. In L. Nolan (Éd.), The Cambridge Descartes Lexicon (p.7‑12). Cambridge University Press. https://doi.org/10.1017/CBO9780511894695.007
Dechalles, P. (1660). Les elemens d’Euclide (M. Ozanam, Trad.; Nouvelle édition 1720). Claude Jombert.
D’Enfert, R. (2003). Inventer une géométrie pour l’école primaire au XIXe siècle. Tréma, 22, 41‑49. https://doi.org/10.4000/trema.1536
D’Enfert, R., & Gispert, H. (2011). Une réforme à l’épreuve des réalités : Le cas des « mathématiques modernes » au tournant des années 1960-1970. L’État et l’éducation, 1808-2008, 27‑49.
Descartes, R. (1953). Œuvres et lettres. Gallimard.
Fehr, H. (1911). Compte-rendu du congrès de Milan. A1- La rigueur dans l’enseignement mathématique des écoles moyennes. IV - Deuxième séance. L’enseignement mathématique, 13, 461‑468. https://dx.doi.org/10.5169/seals-13544
Fehr, H. F. (1908). Rapport préliminaire sur l’organisation de la commission Internationale de l’enseignement mathématique et le plan général de ses travaux (p. 9). https://www.icmihistory.unito.it/documents/RapportPreliminaire.pdf
France> 10. 30 octobre 1794 (9 brumaire an III). Décret relatif à l’établissement des écoles normales. (1992). In L’enseignement du Français à l’école primaire – Textes officiels.: Vol. Tome 1 : 1791-1879 (p. 50‑51). Institut national de recherche pédagogique. https://www.persee.fr/doc/inrp_0000-0000_1992_ant_5_1_1722
Furinghetti, F., & Giacardi, L. (2008). The First Century of the International Commission on
Mathematical Instruction (1908-2008)—History of ICMI. https://www.icmihistory.unito.it/timeline.php
Garden, R. A., Lie, S., Robitaille, D. F., Angell, C., Martin, M. O., Mullis, I. V. S., Foy, P., & Arora, A. (2008). TIMSS Advanced 2008 assessment frameworks. International Association for the Evaluation of Educational Achievement. Herengracht 487, Amsterdam, 1017 BT, The
Netherlands. Tel: +31-20-625-3625; Fax: +31-20-420-7136; e-mail: department@iea.nl; Web
site: http://www.iea.nl. https://timssandpirls.bc.edu/timss_advanced/frameworks.html
Gispert, H. (2002). Pourquoi, pour qui enseigner les mathématiques ? Une mise en perspective historique de l’évolution des programmes de mathématiques dans la société française auXXe siècle. Zentralblatt für Didaktik der Mathematik, 34(4), 158‑163. https://doi.org/10.1007/BF02655809
Gispert, H. (2014). Mathematics Education in France: 1800–1980. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 229‑240). Springer New York. https://doi.org/10.1007/978-1-4614-9155-2_11
Glaeser, G. (1983). A propos de la pédagogie de Clairaut vers une nouvelle orientation dans l’histoire de l’éducation—Revue RDM. Recherches en didactique des mathématiques, 4(3), 332‑344. https://revue-rdm.com/1983/a-propos-de-la-pedagogie-de/
González, G., & Herbst, P. G. (2006). Competing Arguments for the Geometry Course: Why Were American High School Students Supposed to Study Geometry in the Twentieth Century? The International Journal for the History of Mathematics Education, 1(1), 7‑33.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.404.6320&rep=rep1&type=pdf#p
age=11
Goodstein, R. L. (1962). Reviewed Work(s): New Thinking in School Mathematics: Synopses for Modern Secondary School Mathematics. The Mathematical Gazette, 46(355), 69‑72.
Hanna, G. (2000). Proof, Explanation and Exploration: An Overview. Educational Studies in
Mathematics, 44, 5‑23.
Hanna, G., & de Villiers, M. (Éds.). (2012). Proof and proving in mathematics education: The 19th ICMI study (corrected edition 2021). Springer.
Herbst, P. (1999). On proof, the logic of practice of geometry teaching and the two-column proof format [Webzine]. Lettre de la Preuve. http://www.lettredelapreuve.org/OldPreuve/Newsletter/990102Theme/990102ThemeUK.html
Herbst, P. G. (2002a). Engaging Students in Proving: A Double Bind on the Teacher. Journal for Research in Mathematics Education, 33(3), 176. https://doi.org/10.2307/749724
Herbst, P. G. (2002b). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283‑312.
Balacheff – Notes for the study of the didactic transposition of proof - 03/03/2023 09:20 27 / 29 Hill, S. (1976). Issues from the NACOME Report. The Mathematics Teacher, 69(6), 440‑446. http://www.jstor.org/stable/27960539
Houzel, C. (1979). Histoire des mathématiques et enseignement des mathématiques. Bulletin Inter-IREM, 18, 3‑6.
Høyrup, J. (2014). Mathematics Education in the European Middle Ages. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 109‑124). Springer New York. https://doi.org/10.1007/978-1-4614-9155-2_6
Jones, K., & Herbst, P. (2012). Proof, Proving, and Teacher-Student Interaction: Theories and
Contexts. In G. Hanna & M. de Villiers (Éds.), Proof and Proving in Mathematics Education
(Vol. 15, p. 261‑277). Springer Netherlands. https://doi.org/10.1007/978-94-007-2129-6_11
Kang, W., & Kilpatrick, J. (1992). Didactic Transposition in Mathematics Textbooks. For the Learning of Mathematics, 12(1), 6.
Karp, A. (2014). Mathematics education in Russia. In A. Karp & G. Schubring (Éds.), Handbook of the History of Mathematics Education (p. 303‑322). Springer New York.
Karp, A., & Schubring, G. (Éds.). (2014). Handbook on the History of Mathematics Education. Springer New York. https://doi.org/10.1007/978-1-4614-9155-2
Kilpatrick, J. (2014). Mathematics Education in the United States and Canada. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 323‑334). Springer Science & Business Media.
Kline, M. (1976). NACOME: Implications for Curriculum Design. The Mathematics Teacher, 69(6), 449‑454. http://www.jstor.org/stable/27960539 Kuntzmann, J. (1976). Évolution et étude critique des enseignements de mathématique. CEDIC-
Nathan.
Lacroix, S. F. (1799). Élémens de géométrie, à l’usage de l’école centrale des quatre nations (1804e éd.). Courcier, imprimeur libraire pour les mathématiques. https://gallica.bnf.fr/ark:/12148/bpt6k147494xLakanal, J. (1795). Rapport fait au Conseil des Cinq-cents, par Lakanal, un de ses membres, sur les livres élémentaires présentés au concours ouvert par la loi du 9 pluviôse, an II: séance du 14 brumaire, an IV ([Reprod.]) Corps législatif, Conseil des Cinq-cents. 43. https://gallica.bnf.fr/ark:/12148/bpt6k489424
Lampert, M. (1990). When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching. American Educational Research Journal, 27(1), 29‑63.
Lee, P. Y. (2014). Mathematics Education in Southeast Asia. In A. Karp & G. Schubring (Éds.), Handbook on the History of Mathematics Education (p. 384‑388). Springer Science &
Business Media.
Legendre, A.-M. (1752-1833) A. du texte. (1794). Éléments de géométrie, avec des notes. Par Adrien-Marie Legendre. https://gallica.bnf.fr/ark:/12148/bpt6k1521831j
Loget, F. (2004). Héritage et réforme du quadrivium au XVIe siècle. La Pensée numérique, 211‑230.
Mariotti, M. A. (2006). Proof and proving in mathematics education. In Á. Gutiérrez & P. Boero (Éds.), Handbook of Research on the Psychology of Mathematics Education (p. 173‑204). Sense Publishers. http://math.unipa.it/~grim/YESS-5/PMEbook_MariottiNew.pdf
Mullis, I. V. S., Ed, Martin, M. O., Ed, Boston College, T. & P. I. S. C., & International Association for the Evaluation of Educational Achievement (IEA) (Netherlands). (2017).
TIMSS 2019 Assessment Frameworks. International Association for the Evaluation of Educational Achievement. Herengracht 487, Amsterdam, 1017 BT, The Netherlands. Tel: +31-20-625-3625; Fax: +31-20-420-7136; e-mail: department@iea.nl; Web site: http://www.iea.nl. http://timssandpirls.bc.edu/timss2019/frameworks/
Mullis, I. V. S., International Association for the Evaluation of Educational Achievement, & TIMSS (Éds.). (2007). TIMSS 2007 assessment frameworks. TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College. https://timssandpirls.bc.edu/TIMSS2007/frameworks.html
Mullis, I. V. S., & Martin, M. O. (2014). TIMSS advanced 2015 assessment framework. TIMSS & PIRLS International Study Center.
Mullis, I. V. S., Martin, M. O., Ruddock, G., O’Sullivan, C. Y., & Preuschoff, C. (2009). TIMSS 2011 assessment frameworks. TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College.
Nabonnand, P. (2007). Les réformes de l’enseignement des mathématiques au début du XXe siècle. Une dynamique à l’échelle international. In H. Gispert, N. Hulin, & C. Robic (Éds.), Sciences et enseignement. L’exemple de la grande réforme des programmes du lycée au début du XXe siècle (p. 293‑314). INRP & Vuibert.
NCTM. (2000). Principles and standards for school mathematics. NCTM. https://www.nctm.org/uploadedFiles/Standards_and_Positions/PSSM_ExecutiveSummary.p
df Netz, R. (1999). The shaping of deduction in Greek Mathematics. Cambridge University Press.
NGA Center, & CCSSO. (2010). Common core state standard for mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C. http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf
O’Connor, K. M., Mullis, I. V. S., Garden, R. A., Martin, M. O., & Gregory, K. D. (2003). TIMSS assessment frameworks and specifications 2003 (2nd ed). International Study Center.
https://timssandpirls.bc.edu/timss2003i/frameworksD.html OEEC. (1961). New Thinking in School Mathematics. Organisation for European Economic Cooperation.
O’Reilly, M. F. (1902). Plane and Solid Geometry. By Arthur Schultze, Ph.D., and F. L. Sevenoak, A.M., M.D. The Macmillan Company, New York. Science, 15(375), 384‑386.
https://doi.org/10.1126/science.15.375.384
Pehkonen, E. (1997). Proceedings of the 21st Conference of the International Group for the
Psychology of Mathematics Education. University of Helsinki. https://files.eric.ed.gov/fulltext/ED416082.pdf
Piaget, J. (1973). Remarques sur l’éduction mathématique. Math-école, 12(58), 1‑7. https://www.fondationjeanpiaget.ch/fjp/site/crypt/index.php?DOCID=865 Reid, D. A., &
Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching.
Sense Publishers. Schubring, G. (1987). On the Methodology of Analysing Historical Textbooks: Lacroix as Textbook Author. For the Learning of Mathematics, 7(3), 41‑50.
Schubring, G. (2007). La diffusion internationale de la géométrie de Legendre—Différentes visions des mathématiques. Revue française d’éducation comparée, 2, 31‑55.
Schubring, G. (2015). From the few to the many: On the emergence of mathematics for all.
Recherches En Didactique Des Mathématiques, 35(2), 221‑260.
Silver, E. A. (2000). Spotlight on the standards: Improving Mathematics Teaching and Learning: How Can Principles and Standards Help? Mathematics teaching in the Middle School, 6(1), 20‑23. http://www.jstor.org/stable/41182261
Sinclair, N. (2006). The History of the Geometry Curriculum in the United States. Information Age Pub.
Stylianides, A. J. (2007). Proof and Proving in School Mathematics. Journal for Research in
Mathematics, 38(3), 289‑321. Stylianides, A. J., & Harel, G. (Éds.). (2018). Advances in Mathematics Education Research on Proof and Proving. Springer International Publishing. https://doi.org/10.1007/978-3-319-70996-3
Tall, D., Yevdokimov, O., Koichu, B., Whiteley, W., Kondratieva, M., & Cheng, Y.-H. (2012). Cognitive Development of Proof. In G. Hanna & M. de Villiers (Éds.), Proof and Proving in Mathematics Education (Vol. 15, p. 13‑49). Springer Netherlands. https://doi.org/10.1007/978-94-007-2129-6_2
Wentworth, G. A. (1877). Elements of geometry. (1881e éd.). Ginn and Heath.https://hdl.handle.net/2027/hvd.32044097014377
Yackel, E., & Cobb, P. (1996). Sociomathematical Norms, Argumentation, and Autonomy in
Mathematics. Journal for Research in Mathematics Education, 27(4), 458‑477.
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