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

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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.

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Notes

  1. The word “completeness” refers here to the property of R that can be stated as follows: every non-empty and upper-bounded set of real numbers has a least upper bound that belongs to R. There are other equivalent characterizations, which are listed in the Appendix. The word “continuity” refers to the analogous property of the straight line.

  2. By “pre-constructed notions” Chevallard (1997) meant those whose existence is taken for granted, with a representation that does not allow one to operate or make proofs.

  3. For the expert or for an advanced student this would not pose a problem. In fact, there are situations in mathematics where the experts or advanced students just compute something or are convinced by a graph without worrying about proofs. When they have doubts about something or they have to communicate to others they stop and think about the assumptions and how well they are founded. Such situations happen naturally side by side and this does not shake mathematicians’ understanding of the relevance of proofs in mathematics, because they have control over what they do. For a novice, there is the risk of perceiving validation as something external, that depends arbitrarily on the formulation of the tasks, instead of perceiving it as an internal need, a characteristic of the mathematical work at a certain level.

  4. Remark: an ordered field K that satisfies one of the statements 1, 4 or 5 is necessarily Archimedean.

  5. Proofs available in Bergé (2004) (go to Anexos al capitulo 3.doc).

References

  • Artigue, M. (1990). Epistémologie et didactique. Recherches en Didactique des Mathématiques, 10(2–3), 241–286.

    Google Scholar 

  • Artigue, M. (1998). L’évolution des problématiques en didactique de l’Analyse. Recherches en Didactique des Mathématiques, 18(2), 231–262.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

  • 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 

  • Cantor, G. (1871). Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Mathematische Annalen V, 123–132

    Google Scholar 

  • Chevallard, Y. (1997). La transposicion didáctica. Aique Grupo Editor S.A., Argentina.

  • Chevallard, Y. (1998). Analyse des pratiques enseignantes et didactique des mathématiques : l’approche antropologique. Actes de l’École d’été de la Rochelle, 91–118

  • Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en Didactique des Mathématiques, 19(2), 221–266.

    Google Scholar 

  • 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 

  • 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 

  • 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 

  • Dorier, J.-L. (1995). Meta level in the teaching of unifying and generalizing concepts in mathematics. Educational Studies in Mathematics, 29(2), 175–197.

    Article  Google Scholar 

  • Hilbert, D. (1899). Über den Zahlbegriff. Jahresbericht der Deutschen Mathematiker-Vereinigung Math Verein, 8, 180–184.

    Google Scholar 

  • Hilbert, D. (1971). Foundations of geometry (2nd ed.). La Salle, Illinois: The Open Court Publishing Company.

    Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

  • 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 

  • 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 

  • Zariski, O. (1926). Essenza e significato dei numeri. Continuitá e numeri irrazionali, translation to Italian with notes. Rome: Casa Editrice Alberto Stock.

    Google Scholar 

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Authors and Affiliations

  1. Cégep de Rimouski, Département de Mathématiques, Rimouski, Canada

    Analía Bergé

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  1. Analía Bergé
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Correspondence to Analía Bergé.

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Materials of Courses

Course I: Análisis Exactas-Ingeniería. Fundación Enseñar Ciencia (2000) Editorial Centro de Copiado La Copia.

Course II: Retrieved on November 22, 2007 from University of Buenos Aires, Faculty of Exact Sciences, Department of Mathematics site : http://www.dm.uba.ar/materias/analisis_1_M/2003/1/programa.pdf (Syllabus); http://www.dm.uba.ar/materias/analisis_1_M/2002/2/practica1.pdf and http://www.dm.uba.ar/materias/analisis_1_M/2002/2/practica2.pdf (Tasks)

Course III: Retrieved on November 22, 2007 from University of Buenos Aires, Faculty of Exact Sciences, Department of Mathematics site : http://www.dm.uba.ar/materias/analisis_2_CpM/2002/2/programa.htm (Syllabus); http://www.dm.uba.ar/materias/analisis_2_CpM/2002/2/practica1.pdf and http://www.dm.uba.ar/materias/analisis_2_CpM/2002/2/practica2.pdf (Tasks)

Course IV: Retrieved on November 22, 2007 from University of Buenos Aires, Faculty of Exact Sciences, Department of Mathematics site : http://www.dm.uba.ar/materias/calculo_avanzado/2002/2/programa.html (Syllabus) http://www.dm.uba.ar/materias/calculo_avanzado/2002/2/practica0.pdf (Tasks)

Some of the analyses included in this paper have also been presented in my article, ‘Análisis institucional a propósito de la noción de completitud del conjunto de los números reales’, RELIME, volume 9(1), 31–64. They are presented here with the permission of the publisher.

Appendix

Appendix

1.1 Different ways of defining completeness

There are several ways of thinking about and defining the property of completeness of R. We present a – non-exhaustive – list of different equivalent ways of characterizing it5:

  1. 1.

    Every bounded sequence of elements of R has a subsequence convergent in R.

  2. 2.

    Every Cauchy sequence is convergent in R.

  3. 3.

    If (I n ) n=1 is a nested sequence of closed intervals of R whose lengths tend to zero, then there exists an unique element x in R, x∈∩∞ n=1 I n .

  4. 4.

    Every bounded infinite subset of R has an accumulation point in R.

  5. 5.

    Every non-empty and upper-bounded subset of R has a least upper bound that belongs to R.

  6. 6.

    Every monotonic and bounded sequence of elements of R is convergent in R.

  7. 7.

    R is connected.

  8. 8.

    Every continuous function defined in R that takes values of different sign in a closed interval takes the zero value at an element of this interval.

  9. 9.

    Every cut of elements of R has a unique element of separation that belongs to R.

  10. 10.

    Every covering by open sets of a closed and bounded subset of R has a finite sub-covering.

  11. 11.

    Every decimal expansion is a number that belongs to R.

Each of these statements shows different aspects of the completeness of R. If they are thought of as hypotheses for proving theorems, each one involves different images, tools and ways of operating, that make us think of them as different conceptions (Artigue 1990) of completeness. Beyond the study of R, these properties reflect more general principles that go through all of mathematics: the construction of objects by approximation of others of a particular type; the completion of metric spaces, the individualization of an element by means of a nested sequence of closed sets, the attainment of extremes, etc.

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Bergé, A. The completeness property of the set of real numbers in the transition from calculus to analysis. Educ Stud Math 67, 217–235 (2008). https://doi.org/10.1007/s10649-007-9101-5

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