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Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis

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An Erratum to this article was published on 01 July 2016

Abstract

Mathematical working space (MWS) is a model that is used in research in mathematics education, particularly in the field of geometry. Some MWS elements are independent of the field while other elements must be adapted to the field in question. In this paper, we develop the MWS model for the field of analysis with an identification of paradigms. We show the advantages of this MWS model, which takes into account the epistemological and cognitive aspects of mathematical work, and more specifically the semiotic, instrumental and discursive geneses, by making them function as one system. By using examples and data from three countries, we illustrate how this model can be used to perform a priori analyses and analyses of class situations and individual student work.

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Notes

  1. This paper focuses only on one-dimensional real analysis, which is taught at the upper secondary levels and in the first years of university (including teacher training programs).

  2. We do not use the term calculus, which is subject to different interpretations.

  3. The letter ε is representative of a type of work or thought that is used even in non-formal mathematicians’ language to mention a negligible quantity, but work in RA is not restricted to this kind of work with ε (see Sect. 3). RA paradigm brings also a topological idea of analysis in the case of R equipped with the usual topology.

  4. a√3/2 is the height CD and a 2√3/8 is half the total area of triangle ABC.

  5. Local reasoning, which could lead to a switch to the RA paradigm, is then possible but not necessary. Through semiotic genesis, the curve through R may be shown to be a function z→r(z). At a/2, in RA r can be said to be a function of z locally, while in CA it is probable that only the relevant formula r(z) dominating the calculations may be identified (i.e., the one that takes the value 0 at a/2).

  6. There is no a priori separation of the CA and RA paradigms. Both paradigms can coexist in the mathematical activity.

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Acknowledgments

This study was supported by the ECOS-Sud C13H03 project.

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Authors and Affiliations

  1. IMA, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile

    Elizabeth Montoya Delgadillo

  2. LDAR, Université Paris Diderot, Paris, France

    Laurent Vivier

  3. IMAG, Université de Montpellier, Montpellier, France

    Laurent Vivier

Authors
  1. Elizabeth Montoya Delgadillo
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  2. Laurent Vivier
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Correspondence to Laurent Vivier.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11858-016-0793-9.

Appendix 1: Curves

Appendix 1: Curves

Instructions: For each of the following curves, draw a tangent line at the point indicated when such a tangent exists, or provide an explanation if it does not.

figure a

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Delgadillo, E.M., Vivier, L. Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis. ZDM Mathematics Education 48, 739–754 (2016). https://doi.org/10.1007/s11858-016-0777-9

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