Educational Studies in Mathematics

, Volume 91, Issue 1, pp 107–122 | Cite as

Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA

  • Vicenç Font Moll
  • María Trigueros
  • Edelmira Badillo
  • Norma Rubio


This paper presents a networking of two theories, the APOS Theory and the ontosemiotic approach (OSA), to compare and contrast how they conceptualize the notion of a mathematical object. As context of reflection, we designed an APOS genetic decomposition for the derivative and analyzed it from the point of view of OSA. Results of this study show some commonalities and some links between these theories and signal the complementary nature of their constructs.


Networking of theories mathematical objects Onto-semiotic approach APOS theory Encapsulation Thematization Derivative 



This research has been carried out as part of projects EDU2012-32644 and EDU2012-31464, and by Asociación Mexicana de Cultura A. C, and ITAM.


  1. Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., et al. (2014). APOS Theory: A framework for research and curriculum development in mathematics education. New York: Springer.CrossRefGoogle Scholar
  2. Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. (1997). The development of students’ graphical understanding of the derivative. Journal of Mathematical Behaviour, 16(4), 399–431.CrossRefGoogle Scholar
  3. Badillo, E. (2003). La derivada como objeto matemático y como objeto de enseñanza y aprendizaje en profesores de matemática de Colombia [The derivative as a mathematical object and purpose of teaching and learning in mathematics teachers in Colombia] (Unpublished doctoral dissertation). University Autònoma of Barcelona, Spain.Google Scholar
  4. Badillo, E., Azcárate, C., & Font, V. (2011). Análisis de los niveles de comprensión de los objetos f’(a) y f’(x) de profesores de matemáticas [Analysis of mathematics teachers’ level of understanding of the objects f’(a) and f’(x)]. Enseñanza de las Ciencias, 29(2), 191–206.Google Scholar
  5. Bikner-Ahsbahs, A., & Prediger, S. (2010). Networking of theories—an approach for exploiting the diversity of theoretical approaches. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers. Advances in mathematics education series (Vol. 1, pp. 483–506). New York: Springer.Google Scholar
  6. Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: A framework and an example. Journal for Research in Mathematics Education, 38(4), 370–392.Google Scholar
  7. Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses: A reflective analysis of student learning with computer algebra from instrumental and onto-semiotic perspectives. Educational Studies in Mathematics, 82, 23–49.CrossRefGoogle Scholar
  8. Dubinsky, E. (1997). A reaction to a critique of the selection of mathematical objects as a central metaphor for advanced mathematical thinking by Confrey and Costa. International Journal of Computers for Mathematical Learning, 2, 67–91.CrossRefGoogle Scholar
  9. Ernest, P. (1994). Varieties of constructivism: Their metaphors, epistemologies and pedagogical implications. Hiroshima Journal of Mathematics Education, 2, 1–14.Google Scholar
  10. Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69(1), 33–52.CrossRefGoogle Scholar
  11. Font, V., Godino, J. D., & Gallardo, J. (2013). The emergence of objects from mathematical practices. Educational Studies in Mathematics, 82, 97–124.CrossRefGoogle Scholar
  12. Haspekian, M., Bikner-Ahsbahs, A., & Artigue, M. (2013). When the fiction of learning is kept: A case of networking two theoretical views. In A. Lindmeier, & A. Heinze (Eds.), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 9–16). Kiel, Germany: PME.Google Scholar
  13. Radford, L. (2008). Connecting theories in mathematics education: Challenges and possibilities. ZDM. The International Journal on Mathematics Education, 40(2), 317–327.CrossRefGoogle Scholar
  14. Rondero, C., y Font, V. (2015). Articulación de la complejidad matemática de la media aritmética [Articulation of the mathematical complexity of the arithmetic mean]. Enseñanza de las Ciencias, 33(2), 29--49.Google Scholar
  15. Sánchez-Matamoros, G., Fernández, C., & Llinares, S. (2014). Developing pre-service teachers’ noticing of students’ understanding of the derivative concept. International Journal of Science and Mathematics Education. doi: 10.1007/s10763-014-9544-y Google Scholar
  16. Trigueros, M., & Martínez-Planell, R. (2010). Geometrical representations in the learning of two-variable functions. Educational Studies in Mathematics, 73(1), 3–19.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Vicenç Font Moll
    • 1
  • María Trigueros
    • 2
  • Edelmira Badillo
    • 3
  • Norma Rubio
    • 4
  1. 1.Departament de Didàctica de les CCEE i la Matemàtica, Facultat de Formació del ProfessoratUniversitat de BarcelonaBarcelonaSpain
  2. 2.Departamento de MatemáticasInstituto Tecnológico Autónomo de MéxicoSan AngelMéxico
  3. 3.Departament de Didàctica de la Matemàtica i de les Ciències Experimentals, Facultat d’EducacióUniversitat Autònoma de BarcelonaBarcelonaSpain
  4. 4.Departamento Académico de CienciasPontificia Universidad Católica del Perú Av. UniversitariaSan MiguelPerú

Personalised recommendations