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Educational Studies in Mathematics

, Volume 67, Issue 3, pp 217–235 | Cite as

The completeness property of the set of real numbers in the transition from calculus to analysis

  • Analía BergéEmail author
Article

Abstract

This paper focuses on teaching and learning the set of real numbers R and its completeness property at the university level. It studies, in particular, the opportunities for understanding this property that students are offered in four undergraduate correlative courses in Calculus and Analysis. The theoretical framework used in the study draws on concepts developed in the Anthropological Theory of Didactics, especially the notions of praxeology and mathematical organization. The paper shows different expectations concerning the same notion (R and its completeness) through different school levels, and intends to bring up some reflections about the transition from Calculus to Analysis.

Keywords

Real numbers Completeness University level Institutional rapport Mathematical organization Praxeology 

Notes

References

  1. Artigue, M. (1990). Epistémologie et didactique. Recherches en Didactique des Mathématiques, 10(2–3), 241–286.Google Scholar
  2. Artigue, M. (1998). L’évolution des problématiques en didactique de l’Analyse. Recherches en Didactique des Mathématiques, 18(2), 231–262.Google Scholar
  3. Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: The case of limits of functions in Spanish high schools. In C. Laborde, M.-J. Perrin-Glorian, & A. Sierpinska (Eds.), Beyond the apparent banality of the mathematics classroom (pp. 235–268), Springer: New York.CrossRefGoogle Scholar
  4. Bergé, A. (2004). Un estudio de la evolución del pensamiento matemático: el ejemplo de la conceptualización del conjunto de los números reales y de la noción de completitud en la enseñanza universitaria. Doctoral thesis, University of Buenos Aires, Argentine. Available at Biblioteca Central Luis Federico Leloir, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina.Google Scholar
  5. Bergé, A., & Sessa, C. (2003). Completitud y continuidad revisadas a través de 23 siglos. Aportes a una investigación didáctica. Revista Latinoamericana en Matemática Educativa, 6(3), 163–197.Google Scholar
  6. Cantor, G. (1871). Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Mathematische Annalen V, 123–132Google Scholar
  7. Chevallard, Y. (1997). La transposicion didáctica. Aique Grupo Editor S.A., Argentina.Google Scholar
  8. Chevallard, Y. (1998). Analyse des pratiques enseignantes et didactique des mathématiques : l’approche antropologique. Actes de l’École d’été de la Rochelle, 91–118Google Scholar
  9. Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.Google Scholar
  10. Chevallard, Y. (2002a). Organiser l'étude 1. Structures et fonctions. In J.-L. Dorier et al. (Eds.), Actes de la 11e école d'été de didactique des mathématiques – Corps 21–30 Août, 2001 (pp. 3–22). Grenoble: La Pensée Sauvage Éditions.Google Scholar
  11. Chevallard, Y. (2002b). Organiser l'étude 3. Écologie et régulation. In J.-L. Dorier et al. (Eds.), Actes de la 11e école d'été de didactique des mathématiques – Corps 21–30 Août, 2001 (pp. 41–56). Grenoble: La Pensée Sauvage Éditions.Google Scholar
  12. Dedekind, J. W. R. (1963). Essays on the theory of numbers: I. Continuity and irrational numbers. II. The nature and meaning of numbers. New York: Dover Publications.Google Scholar
  13. Dorier, J.-L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29(2), 175–197.CrossRefGoogle Scholar
  14. Hilbert, D. (1899). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung Math Verein, 8, 180–184.Google Scholar
  15. Hilbert, D. (1971). Foundations of geometry (2nd ed.). La Salle, Illinois: The Open Court Publishing Company.Google Scholar
  16. Laborde, C., & Perrin-Glorian, M.-J. (2005). Introduction. Teaching situations as object of research: Empirical studies within theoretical perspectives. In C. Laborde, M.-J. Perrin-Glorian, & A. Sierpinska (Eds.), Beyond the apparent banality of the mathematics classroom (pp. 1–12), Springer: New York.CrossRefGoogle Scholar
  17. Maschietto, M. (2002). L’enseignement de l’analyse au lycée: les débuts du jeu global/local dans l’environment de calculatrices. Thèse doctorale, Université Paris VII.Google Scholar
  18. Robert, A., & Robinet, J. (1996). Prise en compte du méta en didactique des mathématiques. Recherches en Didactique des Mathématiques, 16(2), 145–176.Google Scholar
  19. Sierpinska, A. (2005). Beyond the apparent banality of the mathematics classroom. Preface. In C. Laborde, M.-J. Perrin-Glorian, & A. Sierpinska (Eds.), Beyond the apparent banality of the mathematics classroom (pp. v–viii). Springer: New York.Google Scholar
  20. Zariski, O. (1926). Essenza e significato dei numeri. Continuitá e numeri irrazionali, translation to Italian with notes. Rome: Casa Editrice Alberto Stock.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Cégep de Rimouski, Département de MathématiquesRimouskiCanada

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