, Volume 48, Issue 6, pp 809–826 | Cite as

Facilitating the genesis of functional working spaces in guided explorations

  • Vicente Carrión Miranda
  • François PluvinageEmail author
  • Robert Adjiage
Original Article


Approximating given real-valued functions by affine functions is among the most basic activities with functions. In this study we examine two contexts in which two such approximations are performed. The first involves a microscopic representation of functions for the study of tangents; the second a macroscopic representation of functions for the study of asymptotes. In the proposed research, we conducted three sessions to observe how small groups of college freshmen worked in a setting of multiple dynamical representations including algebraic, graphic and CAS (computer algebra system) views. This enabled the observation of individual Mathematical Working Spaces (iMWS). The analysis of students’ answers leads us to propose an enrichment of the MWS model. Specifically, this analysis suggests that educational resources could foster the geneses described in the MWS model: observation for visualizing, drawing for constructing and justification for proving.


Mathematical Working Spaces Calculus Zoom in Zoom out Tangent Asymptote 


  1. Artigue, M. (1988). Ingénierie didactique. Recherche en Didactique des Mathématiques, 9(3), 281–308.Google Scholar
  2. Biza, I., Christou, C., & Zachariades, T. (2008). Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis. Research in Mathematics Education, 10(1), 53–70.CrossRefGoogle Scholar
  3. Carrión Miranda, V., & Pluvinage, F. (2014). Registros y estratos en ETM al servicio del pensamiento funcional. Revista Latinoamericana de Investigación en Matemática Educativa, 17(4-II), 267–286.Google Scholar
  4. Douady, R. (1986). Jeux de cadres et dialectique outil-objet. Recherche en Didactique des Mathématiques, 7(2), 5–31.Google Scholar
  5. Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée. Annales de Didactique et de Sciences Cognitives, 5, 37–65.Google Scholar
  6. Duval, R. (1995). Sémiosis et pensée humaine. Bern: Peter Lang.Google Scholar
  7. Hitt, F. (2007). Utilisation de calculatrices symboliques dans le cadre d’une méthode d’apprentissage collaboratif, de débat scientifique et d’autoréflexion. In M. Baron, D. Guin et L. Trouche (Éds.), Environnements informatisés pour l’éducation et la formation scientifique et technique: modèles, dispositifs et pratiques (pp. 65–88). Paris: Hermes.Google Scholar
  8. Hitt, F. (2011). Construction of mathematical knowledge using graphic calculators (CAS) in the mathematics classroom. International Journal of Mathematical Education in Science and Technology, 42(6), 723–735.CrossRefGoogle Scholar
  9. Hitt, F., & González-Martín, A. (2015). Covariation between variables in a modeling process: The ACODESA (Collaborative learning, Scientific debate and Self-reflection) method. Educational Studies in Mathematics, 88(2), 201–219.CrossRefGoogle Scholar
  10. Kuhn T. S. (1970). The structure of scientific revolutions (2nd ed. enlarged). The University of Chicago.Google Scholar
  11. Kuzniak, A. & Richard, P. R. (2014). Spaces for Mathematical Work: Viewpoints and perspectives. Revista Latinoamericana de Investigación en Matemática Educativa, 17 (4-I), 17-27.Google Scholar
  12. Minh, T. K. & Lagrange, J. B. (2016). Connected functional working spaces: a framework for the teaching and learning of functions at upper secondary level. ZDM Mathematics Education, Published online: 09 March 2016.Google Scholar
  13. Montoya Delgadillo, E., & Vivier, L. (2014). Les changements de domaine dans le cadre des espaces de travail mathématique. Annales de Didactique et de Sciences Cognitives, 19, 73–101.Google Scholar
  14. Morgan, C., & Kynigos, C. (2014). Digital artefacts as representations: forging connections between a constructionist and a social semiotic perspective. Educational Studies in Mathematics, 85(3), 357–379.CrossRefGoogle Scholar
  15. Radford, L. (2014). On the role of representations and artefacts in knowing and learning. Educational Studies in Mathematics, 85(3), 405–422.CrossRefGoogle Scholar
  16. Retrieved on 09/28/2015 from
  17. Roth, W. M., & McGinn, M. K. (1998). Inscriptions: Toward a Theory of Representing as Social Practice. Review of Educational Research, 68(1), 35–59.CrossRefGoogle Scholar
  18. Tanguay, D. & Geeraerts, L. (2014). Conjectures, postulats et vérifications expérimentales dans le paradigme du géomètre-physicien: comment intégrer le travail avec les LGD ? Revista Latinoamericana de Investigación en Matemática Educativa, 17, (4-II), 287–302.Google Scholar
  19. Vivier, L. (2011). La noción de tangente en la educación media superior. El Cálculo y su Enseñanza. Vol. II, año 2010–2011. México: Cinvestav-IPN.Google Scholar
  20. Wagenschein, M. (1977). Verstehen lernen, genetisch-sokratisch-exemplarisch, Beltz Verlag: Basel; Weinheim.Google Scholar

Copyright information

© FIZ Karlsruhe 2016

Authors and Affiliations

  • Vicente Carrión Miranda
    • 1
  • François Pluvinage
    • 1
    Email author
  • Robert Adjiage
    • 2
  1. 1.Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico NacionalCiudad de MéxicoMexico
  2. 2.IREM StrasbourgStrasbourgFrance

Personalised recommendations