Revista Integración, temas de matemáticas.
Vol. 31 Núm. 2 (2013): Revista Integración, temas de matemáticas
Artículo Original

Acerca de la enseñanza y el aprendizaje de la demostración en matemáticas

Jorge Fiallo
Universidad Industrial de Santander
Leonor Camargo
Universidad Pedagógica Nacional
Ángel Gutiérrez
Universidad de Valencia

Publicado 2013-12-17

Palabras clave

  • Síntesis de publicaciones,
  • demostración matemática,
  • aprendizaje de la demostración,
  • educación matemática

Cómo citar

Fiallo, J., Camargo, L., & Gutiérrez, Ángel. (2013). Acerca de la enseñanza y el aprendizaje de la demostración en matemáticas. Revista integración, Temas De matemáticas, 31(2), 181–205. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/3756

Resumen

En el presente documento realizamos una recopilación bibliográfica de las principales investigaciones acerca de la enseñanza y el aprendizaje de la demostración, con el ánimo de aportar fuentes de consulta a la comunidad de educadores en matemáticas interesados en el tema. Planteamos una estructura organizativa que incluye las siguientes líneas de investigación: Consideraciones histórico-epistemológicas, La demostración en el currículo, Concepciones y dificultades de los estudiantes al demostrar, Relaciones entre argumentación y demostración y Propuestas didácticas para la enseñanza de la demostración.



Descargas

Los datos de descargas todavía no están disponibles.

Referencias

[1] Alibert D., Thomas M., “Research on mathematical proof”. En D. Tall (Ed.), Advanced mathematical thinking (pp. 215–229). Dordrecht, Los Países Bajos, Kluwer, 1991.

[2] Antonini S., Mariotti M.A., “Indirect proof: An interpreting model”, Proceedings of the 5th Conference of the European Society for Research in Mathematics Education (CERME5), (2007), 541–550.

[3] Antonini S., Mariotti M.A., “Indirect proof: what is specific to this way of proving?” ZDM the International Journal on Mathematics Education 40 (2008), 401–412.

[4] Arsac G., “L’origine de la démonstration: Essai d’épistémologie didactique”, Recherches en Didactique des Mathématiques 8 (1987), no. 3, 267–312.

[5] Arsac G., “Origin of mathematical proof”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 27–42). Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[6] Arzarello F. “The proof in the 20th century: From Hilbert to automatic theorem proving”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 43–64). Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[7] Arzarello F., Micheletti C., Olivero F., Robutti O., “A model for analiysing the transition to formal proofs in geometry”, Proceedings of the 22th PME International Conference 2 (1998), 24–31.

[8] Arzarello F., Olivero F., Paola D., Robutti O., “A cognitive analysis of dragging practises in Cabri enviroments”, Zentralblatt fur Didaktik der Mathematik 34 (2002), no. 3, 66–72.

[9] Arzarello F., Olivero F., Paola D., Robutti O., “The transition to formal proof in geometry”.En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 305–323). Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[10] Arzarello F., Sabena C., “Semiotic and theoretic control in argumentation and proof activities”, Educational Studies in Mathematics 77 (2011), no. 2-3, 189–206.

[11] Back R.J., Wright J.V., “A method for teaching rigorous mathematical reasoning”, Proceedings of ICTM14, (1999), 9–13.

[12] Balacheff N., Une étude des processus de preuve en mathématique chez des élèves de collège, Tesis doctoral, Grenoble, Francia, 1988. [Traducción al español: Balacheff N., Procesos de prueba en los alumnos de matemáticas, Bogotá, Colombia: una empresa docente, 2000].

[13] Balacheff N., “Aspects of proof in pupils’ practice of school mathematics”. En D. Pimm (Ed.), Mathematics, teachers and children (pp. 216–235), Londres, Hodder & Stoughton, 1988.

[14] Balacheff N., “Conception, connaissance et concept”. En D. Grenier (Ed.), Didactique et technologies cognitives en mathématiques, (pp. 219–244), Grenoble (Francia), UniversitéJoseph Fourier, séminaires 1994-1995.

[15] Balacheff N. “The role of the researcher’s epistemology in mathematics education: an essay on the case of proof”, ZDM the International Journal on Mathematics Education 40 (2008), 501–512.

[16] Balacheff N., Margolinas C., “cKc/ modèle de connaissances pour le calcul de situations didactiques”. En A. Mercier, C. Margolinas (Eds.), Balises pour la didactique des mathé-matiques (pp. 75–106), Francia, La Pensée Sauvage, 2005.

[17] Bartolini Bussi M., “Experimental mathematics and the teaching and learning of proof”, Proceedings of the 6th Conference of the European Society for Research in Mathematics Education (CERME6), (2010), 221–230.

[18] Bartolini Bussi, M., Boero P., Ferri F., Garuti R., Mariotti M.A., “Approaching and developing the culture of geometry theorems in school: A theoretical framework”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 211–217), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[19] Bartolini Bussi M., Boni N., Ferri F., “Construction problems in primary school: A case from the geometry of circle”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 219–247), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[20] Battista M.T., Clements D.H., “Geometry and proof”, The Mathematics Teacher 88 (1995), no. 1, 48–54.

[21] Bell A.W., “A study of pupil’s proof-explanation in mathematical situation”, Educational Studies in Mathematics 7 (1976), no. 1, 23–40.

[22] Blanton M.L., Stylianou, D.A., “Exploring sociocultural aspects of undergraduate students’ transition to mathematical proof”, Proceedings of the 24th Annual Meeting of the North American Chapter of the PME International Group 4 (2002), 1673–1680.

[23] Boero P., Consogno V., Guala E., Gazzolo T., “Research for innovation : A teaching sequence on the argumentative approach to probabilistic thinking in Grades I-V and some related basic research results”, Recherches en Didactique des Mathématiques 29(I) (2009), 59–96.

[24] Boero P., Theorems in school: From history, epistemology and cognition to classroom practice, Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[25] Boero P., Garuti R., Lemut E., “Approaching theorems in grade VIII: Some mental processes underlying producing and proving conjectures, and conditions suitable to enhance them”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 249–264), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[26] Boero P., Garuti R., Lemut E., Mariotti A., “Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems”, Proceedings of the 20th PME International Conference 2 (1996), 113–120.

[27] Camargo L., Descripción y análisis de un caso de enseñanza y aprendizaje de la demostración en una comunidad de práctica de futuros profesores de matemáticas de educación secundaria, Tesis doctoral, Universidad de Valencia, Valencia, España, 2010.

[28] Castagnola E., Tortora R., “Some remarks on the theorem about the infinity of prime numbers”, Proceeding of the 5th Conference of the European Society for Research in Mathematics Education (CERME5), (2007), 581–590.

[29] Clark P., The emergence of a classroom community of practice in a mathematical structures course, Tesis doctoral, Department of Philosophy, Arizona State University, 2005.

[30] Cobb P., Yackel E., Woods T., “A constructivist alternative to the representational view of mind in mathematics education”, Journal for Research in Mathematics Education 23 (1992), 2–33.

[31] De Villiers M., “El papel y la función de la demostración en matemáticas”, Epsilon 26 (1993), 15–29.

[32] Douek N., “Some remarks about argumentation and mathematical proof and their educational implications”, Proceeding of the 1st Conference of the European Society for Research in Mathematics Education (CERME1), (1998), 125–139.

[33] Douek N., “Some remarks about argumentation and proof”, En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 163–181), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[34] Duval R., “Langage et représentation dans l’apprentissage d’une démarche déductive”, Proceedings of the 13th PME International Conference 1 (1989), 228–235.

[35] Duval R., “Argumenter, demontrer, expliquer: continuité ou rupture cognitive?” Petit x 31 (1992-1993), 37–61.

[36] Duval R., “Cognitive functioning and the understanding of mathematical processes of proof”, En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 137–161), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[37] Fiallo J., Enseñanza de las razones trigonométricas en un ambiente cabri para el desarrollo de las habilidades de la demostración, Memoria de investigación, Universidad de Valencia, Valencia, España, 2006.

[38] Fiallo J., Estudio del proceso de demostración en el aprendizaje de las razones trigonométricas en un ambiente de geometría dinámica, Tesis doctoral, Universidad de Valencia, Valencia, España, 2010.

[39] Fiallo J., Gutiérrez Á. “Tipos de demostración de estudiantes del grado 10 o en Santander (Colombia)”. En M. Camacho, P. Flores, P. Bolea (Eds.), Investigación en Educación Matemática XI (2007), 355–368.

[40] Fischbein E., Intuition in science and mathematics, Dordrecht, Los Países Bajos, D. Reidel, 1987.

[41] Furinghetti F., Morselli F., “Every unsuccessful problem solver is unsuccessful in his or her own way: affective and cognitive factors in proving”, Educational Studies in Mathematics 70 (2008), 71–90.42] Garuti R., Boero P., Lemut E., “Cognitive unity of theorems and difficulty of proof”,Proceedings of the 22th PME International Conference 2 (1998), 345–352.

[43] Godino J., Recio A., “Significados institucionales de la demostración. Implicaciones para la educación matemática”, Enseñanza de las Ciencias 19 (2001), no. 3, 405–414.

[44] Grabiner J.V., “Why proof? A historian’s perspective”. En Hanna, G., de Villiers, M. (Eds.), Proof and proving in mathematics education. The 19th ICMI Study (pp. 147–167), Dordrecht, Los Países Bajos, Springer, 2012.

[45] Groman M., “Integrating Geometer’s Sketchpad into geometry course for secondary education mathematics major”, Proceedings of ASCUE (1996), 9–13.

[46] Gutiérrez Á., Fiallo J., “Analysis of conjectures and proofs produced when learning trigonometry”, Proceeding of the 5th Conference of European Society for Research in Mathematics Education (2007), 622–632.

[47] Hanna G., “Some pedagogical aspects of proof”, Interchange 21 (1990), no. 1, 6–13.

[48] Hanna G., “The ongoing value of proof”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 3–18), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[49] Hanna G., de Villiers M. (Eds.), “Proof and proving in mathematics education”. The 19th ICMI Study, Dordrecht, Los Países Bajos, Springer, 2012.

[50] Hanna G., Jahnke N., “Proof and proving”. En A. Bishop y otros (Eds.), International handbook of mathematics education (pp. 877–908), Dordrecht, Los Países Bajos, Kluwer, 1996.

[51] Harel G., “Two dual assertions: The first on learning and the second on teaching (or viceversa)”, The American Mathematical Monthly 105 (1998), 497–507.

[52] Harel G., “The development of mathematical induction as a proof scheme: a model for DNR-based instruction”, En S. Campbell, R. Zaskis (Eds.), Learning and teaching number theory, journal of mathematical behavior (pp. 185–212), New Jersey, EE.UU., Ablex, 2001.

[53] Harel G., “Students’ proof schemes revisited”, En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 65–78), Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[54] Harel G., Martin W.G., “Proof frames of preservice elementary teachers”, Journal for Research in Mathematics Education 20 (1989), no. 1, 41–51.

[55] Harel G., Sowder L., “Student’s proof schemes: results from exploratory studies”. En A. Schoenfeld y otros (Ed.), Research in collegiate mathematics education III (pp. 234–283), Providence, EE.UU., American Mahematical Society, 1998.

[56] Harel G., Sowder L., “Toward comprehensive perspectives on the learning and teaching of proof”. En F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 805–842), Reston, VA, EE.UU., National Council of Teachers of Mathematics, 2007.

[57] Healy L., Hoyles C., “Student’s performance in proving: competence or curriculum?” Proceedings of the 1st Conference of the European Society for Research in Mathematics Education (CERME1), (1998), 153–167.

[58] Healy L., Hoyles C., “A study of proof conceptions in algebra”, Journal for Research in Mathematics Education 31 (2000), no. 4, 396–428.

[59] Healy L., Hoyles C., “Software tools for geometrical problem solving: potentials and pitfalls”, International Journal of Computers for Mathematical Learning 6 (2001), 235–256.

[60] Hemmi K., Approaching proof in a community of mathematical practice, Tesis doctoral, Department of Mathematics, Stockholm University, 2006.

[61] Herbst P.G., “Acerca de la demostración y la lógica de la práctica en la enseñanza de la geometría: Observaciones sobre la forma de prueba a dos columnas”, Proof Newsletter, (1999), Enero-Febrero. Revista electrónica accesible en
http://www.lettredelapreuve.it/OldPreuve/Newsletter/990102.html.

[62] Hollebrands K., Conner A., Smith R., “The nature of arguments provided by college geometry students whith access to technology while solving problems”, Journal for Research in Mathematics Education 41 (2010), no. 4, 324–350.

[63] Hoyles C., “The curricular shaping of students’ approaches”, For the Learning of Mathematics 17 (1997), no. 1, 7–16.

[64] Hoyles C., Küchemann D., “Student’s understandings of logical implication”, Educational Studies in Mathematics 51 (2002), 193–223.

[65] Ibañes M., Ortega T., “Reconocimiento de procesos matemáticos en alumnos de primer curso de bachillerato”, Enseñanza de las Ciencias 21 (2003), no. 1, 49–63.

[66] Ibañes M., Ortega T., “Un análisis del tratamiento de la demostración matemática en los libros de texto de bachillerato”, Números 57 (2004), 19–32.

[67] Jones K., “Providing a foundation for deductive reasoning: students’ interpretation when using dynamic geometry software and their evolving mathematical explanations”, Educational Studies in Mathematics 44 (2000), 55–85.

[68] Knipping C., “A method for revealing structures of argumentations in classroom proving processes”, ZDM The International Journal on Mathematics Education 40 (2008), 427– 441.

[69] Knuth E.J., “Teachers’ conceptions of proof in the context of secondary school mathematics”, Journal of Mathematics Teacher Education 5 (2002), 61–88.

[70] Küchemann D., Hoyles C., “Investigating factors that influence students’ mathematical reasoning”, Proceedings of the 25th PME International Conference 3 (2001), 257–264.

[71] Laborde C., “Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving”, Educational Studies in Mathematics 44 (2000), 151–161.

[72] Laborde C., Kynigos C., Hollebrands K., Sträesser R., “Teaching and learning geometry with technology”. En Á. Gutiérrez, P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (pp. 275–304), Rotterdam, Los Países Bajos, Sense Publishers, 2006.

[73] Lampert M., “When the problem is not the question and the solution is not the answer: mathematical knowing and teaching”, American Educational Research Journal, 27 (1990), no. 1, 29–63.

[74] Mariotti M.A., “Justifying and proving: figural and conceptual aspects”. En M. Hejny, J. Novotna (Eds.), Proceedings of the European Research Conference on Mathematical Education, 1997.

[75] Mariotti M.A., “Introduction to proof: the mediation of a dynamic software environment”, Educational Studies in Mathematics 44 (2000), 25–53.

[76] Mariotti M.A., “Proof and Proving in Mathematics Education”. En Á. Gutiérrez, P. Boero (Eds.), Handbook of research on the psychology of mathematics education. Past, present and future (pp. 173–204), Rotterdam, Los Países Bajos, Sense Publishers, 2006.

[77] Mariotti M. A., Bartolini M., Boero P., Ferri F., Garuti R., “Approaching geometry theorems in contexts: from history and epistemology to cognition”, Proceedings of the 21th PME International Conference 1 (1997), 180–195.

[78] Marrades R., Gutiérrez A., “Proofs produced by secondary school students learning geometry in a dynamic computer environment”, Educational Studies in Mathematics 44 (2000), 87–125.

[79] Martin T.S., Soucy McCrone S.M., Wallace M.L., Dindyal J., “The interplay of teacher and student actions in the teaching and learning of geometric proof”, Educational Studies in Mathematics 60 (2005), 95–124.

[80] Nardi E., Amongst mathematicians. Teaching and learning mathematics at university level, New York, USA, Springer, 2008.

[81] NCTM, Curriculum and evaluation standards for school mathematics, Reston, VA,
EE.UU., 1989.

[82] NCTM, Principles and standards for school mathematics, Reston, VA, EE.UU., 2000.

[83] NCTM, Principios y estándares para la educación matemática, Sociedad Andaluza de Educación Matemática Thales, Sevilla, España, 2003.

[84] Otte M., “Proof and explanation from a semiotical point of view”, Relime, número especial (2006), 23–43.

[85] Parenti L., Barberis M., Pastorino M., Viglienzone P., “From dynamic exploration to “theory” and “theorems” (from 6th to 8th grades)”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 265–284), Rotterdam, Los Países Bajos, Sense Publishers, (2007).

[86] Pedemonte B., Etude didactique et cognitive des rapports de l’argumentation et de la démonstration dans le apprentisage des mathématiques, Tesis doctoral, Université Joseph Fourier-Grenoble I, Grenoble, Francia, 2002.

[87] Pedemonte B., “Quelques outils pour l‘analyse cognitive du rapport entre argumentation et démonstration”, Recherches en Didactique des Mathématiques 25 (2005), no. 3, 313–348.

[88] Pedemonte B., “How can the relationship between argumentation and proof be analysed?”, Educational Studies in Mathematics 66 (2007), 23–41.

[89] Pedemonte B., “Argumentation and algebraic proof”, ZDM the International Journal on Mathematics Education 40 (2008), 385–400.

[90] Perks P., Prestage S., “Why don’t they prove?” Mathematics in School 24 (1985), no. 3, 43–45.

[91] Radford L., “La enseñanza de la demostración: aspectos teóricos y prácticos”, Educación Matemática 6 (1994), no. 3, 21–36.

[92] Reid D.A., Knipping C., Proof in mathematics education, Rotterdam, Los Países Bajos, Sense Publishers, 2010.

[93] Sackur C., Drouhard J.P., Maurel M., “Experiencing the necessity of a mathematical statement”, Proceedings of the 24th PME International Conference 4 (2000), 105–112.

[94] Sinclair N., Jones K., “Geometrical reasoning in the primary school, the case of parallel lines”, Proceedings of the British Society for Research into Learning Mathematics 29 (2009), no. 2, 88–93.

[95] Stacey K., Vincent J., “Modes of reasoning in explanations in year 8 textbooks”, Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australasia, (2008), 475–481.

[96] Stylianides A.J., “Toward a comprehensive knowledge package for teaching proof: A focus on the misconception that empirical arguments are proofs”, Pythagoras 32 (2011), no. 1, 10 p.

[97] Stylianides A.J., Stylianides G.J., “Proof constructions and evaluations”, Educational Studies
in Mathematics 72 (2009), no. 2, 237–253.

[98] Stylianides G.J., Stylianides A.J., “Facilitating the transition from empirical arguments to proof”, Journal for Research in Mathematics Education 40 (2009), no. 3, 314–352.

[99] Szendrei-Radnai J., Török J., “The tradition and role of proof in mathematics education in Hungary”. En P. Boero (Ed.), Theorems in school: From history, epistemology and cognition to classroom practice (pp. 117–134). Rotterdam, Los Países Bajos, Sense Publishers, 2007.

[100] Tall D., “The cognitive development of proof: is mathematical proof for all or for some”, Paper presented at the UCSMP Conference, Chicago University, Chicago, EE.UU., 1998.

[101] Toulmin S.E., The use of argument, Cambridge, Gran Bretaña, Cambridge University Press, 1958.

[102] Tsamir P., Tirosh D., Dreyfus T., Barkai R., Tabach M., “Should proof be minimal? Ms. T’s evaluation of secondary school students’ proofs”, The Journal of Mathematical Behavior 28 (2009), 58–67.

[103] Weber K., Maher C., Powell A., “Learning opportunities from group discussions: warrants become the objects of debate”, Educational Studies in Mathematics 68 (2008), 247–261.

[104] Weiss M., Herbst P., Chen C., “Teachers’ perspectives on “authentic mathematics” and the two-column proof form”, Educational Studies in Mathematics 70 (2009), 275–293.

[105] Yackel E., “Explanation justification and argumentation in mathematics classrooms”, Proceedings of the 25th PME International Conference 1 (2001), 9–24.

[106] Yackel E., Cobb P., “Sociomathematical norms, argumentation, and autonomy in mathematics”, Journal for Research in Mathematics Education 27 (1996), no. 4, 458–477.