Educational Studies in Mathematics

, Volume 77, Issue 2–3, pp 247–265 | Cite as

Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects

  • Juan D. GodinoEmail author
  • Vicenç Font
  • Miguel R. Wilhelmi
  • Orlando Lurduy


The semiotic approach to mathematics education introduces the notion of “semiotic system” as a tool to describe mathematical activity. The semiotic system is formed by the set of signs, the production rules of signs and the underlying meaning structures. In this paper, we present the notions of system of practices and configuration of objects and processes that complement the notion of semiotic system and help to understand the complex nature of mathematical objects. We also show in what sense these notions facilitate the description and comprehension of building and communicating mathematical knowledge, by applying them to analyze semiotic systems involved in the teaching and learning of some elementary arithmetic concepts.


Onto-semiotic approach Object Meaning Mathematics Natural number Learning decimal numeration Semiotic system 



This research work has been carried out as part of the projects SEJ2007-60110/EDUC, MEC-FEDER, and EDU 2009-08120/EDUC.


  1. Arrieche, M. (2002). La teoría de conjuntos en la formación de maestros: Facetas y factores condicionantes del estudio de una teoría matemática. [The theory of sets in the training of primary teachers: Facets and factors conditioning the study of a mathematical theory]. Unpublished Doctoral Dissertation. Departamento de Didáctica de la Matemática. España: Universidad de Granada.Google Scholar
  2. Arzarello, F. (2006). Semiosis as a multimodal process. Revista Latinoamericana de Investigación en Matemática Educativa, 9(Especial), 267–299.Google Scholar
  3. Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13(1), 33–57.CrossRefGoogle Scholar
  4. Chevallard, Y. (1992). Concepts fondamentaux de la didactique: perspectives apportées par une approche anthropologique. [Fundamental concepts of didactic: Contributed perspectives by an anthropological approach]. Recherches en Didactique des Mathématiques, 12(1), 73–112.Google Scholar
  5. DeBlois, L. (1996). Une analyse conceptuelle de la numeration de position au primaire. [A conceptual analysis of positional numeration in primary school]. Recherches en Didactique des Mathématiques, 16(1), 71–128.Google Scholar
  6. Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensée. [Semiotic registers of representation and cognitive functioning of thinking]. Annales de Didactique et de Sciences Cognitives, 5, 37–65.Google Scholar
  7. Duval, R. (2006). Quelle sémiotique pour l’analyse de l’activité et des productions mathématiques? [What semiotics for the analysis of mathematical activity and results?]. Revista Latinoamericana de Investigacion en Matematica Educativa, 9(1), 45–82.Google Scholar
  8. Eco, U. (1978). A theory of semiotics. Bloomington: Indiana University Press.Google Scholar
  9. Elia, I., Gagatsis, A., & Gras, R. (2005). Can we “trace” the phenomenon of compartmentalization by using the implicative statistical method of analysis? An application for the concept of function. Third International Conference A.S.I.-Analyse Statistique Implicative, 175–185.Google Scholar
  10. Ernest, P. (1998). Social constructivism as a philosophy of mathematics. New York: State University of New York.Google Scholar
  11. Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number. Educational Study in Mathematics, 61, 67–101.CrossRefGoogle Scholar
  12. Font, V., & Contreras, A. (2008). The problem of the particular and its relation to the general in mathematics education. Educational Studies in Mathematics, 69, 33–52.CrossRefGoogle Scholar
  13. Font, V., Godino, J. D., & Contreras, A. (2008). From representation to onto-semiotic configurations in analysing mathematics teaching and learning processes. In L. Radford, G. Schubring, & F. Seeger (Eds.), Semiotics in mathematics education: Epistemology, history, classroom, and culture (pp. 157–173). Rotterdam: Sense Publishers.Google Scholar
  14. Giroux, J., & Lemoyne, G. (1998). Coordination of knowledge of numeration and arithmetic operations on first grade students. Educational Studies in Mathematics, 35, 283–301.CrossRefGoogle Scholar
  15. Godino, J. D., & Batanero, C. (1998). Clarifying the meaning of mathematical objects as a priority area of research in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity (pp. 177–195). Dordrecht: Kluwer, A. P.Google Scholar
  16. Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM. The International Journal on Mathematics Education, 39(1–2), 127–135.CrossRefGoogle Scholar
  17. Hjelmslev, L. (1943). Omkring sprogteoriens grundlæggelse. In Festkrift udg. af KøbenhavnsUniversitet, Københavns. English translation by Whitfield, F. J. (1963). Prolegomena to a Theory of Language (pp. 1–113). Madison, WI: The University of Wisconsin Press.Google Scholar
  18. Maddy, P. (1990). Realism in mathematics. Oxford: Clarendon.Google Scholar
  19. Radford, L. (2002). The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22(2), 14–23.Google Scholar
  20. Radford, L. (2006). The anthropology of meaning. Educational Studies in Mathematics, 61, 39–65.CrossRefGoogle Scholar
  21. Rotman, B. (1988). Toward a semiotics of mathematics. Semiotica, 72(1/2), 1–35.CrossRefGoogle Scholar
  22. Sáenz-Ludlow, A. (2004). Metaphor and numerical diagrams in the arithmetical activity of a fourth grade class. Journal for Research in Mathematics Education, 35(1), 34–56.CrossRefGoogle Scholar
  23. Sáenz-Ludlow, A. (2006). Classroom interpreting games with an illustration. Educational Studies in Mathematics, 61, 183–218.CrossRefGoogle Scholar
  24. Steffe, L. P., & von Glasersfeld, E. (1985). Helping children to conceive of number. Recherches en Didactique des Mathématiques, 6(2–3), 269–303.Google Scholar
  25. Wilhelmi, M. R., Godino, J. D., & Lacasta, E. (2007). Didactic effectiveness of mathematical definitions: The case of the absolute value. International Electronic Journal of Mathematics Education, 2(2), 72–90.Google Scholar
  26. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356–266.CrossRefGoogle Scholar
  27. Wittgenstein, L. (1953). Philosophische Untersuchungen/Philosophical investigations. New York: MacMillan.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Juan D. Godino
    • 1
    Email author
  • Vicenç Font
    • 2
  • Miguel R. Wilhelmi
    • 3
  • Orlando Lurduy
    • 4
  1. 1.Departamento de Didáctica de la Matemática, Facultad de EducaciónUniversidad de GranadaGranadaSpain
  2. 2.Departament de Didàctica de les Ciències Experimentals i la Matemàtica, Facultat de Formació del ProfessoratUniversitat de BarcelonaBarcelonaSpain
  3. 3.Departamento de MatemáticasUniversidad Pública de NavarraPamplonaSpain
  4. 4.UD–Licenciatura en Educación Básica con Énfasis en MatemáticasUniversidad Distrital “Francisco José de Caldas”BogotáColombia

Personalised recommendations