The relation between mathematical object/mathematical name: Conceptual changes between designation, description, denomination and definition

Giorgio Bolondi 1, Federica Ferretti 1 * , Alessandro Gambini 2
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1 Department of Mathematics, Alma Mater Studiorum University of Bologna, Bologna, Italy
2 Department of Mathematics, University of Ferrara. Ferrara, Italy
* Corresponding Author
EUROPEAN J SCI MATH ED, Volume 2, Issue 2A, pp. 169-176.
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The actions of designation, description, denotation, denomination and definition are crucial in the didactic activity in the classroom (D'Amore and Fandiño Pinilla, 2012) since they embody different interplays between objects, representations, properties, names (in the sense of Duval (2008)). Switching from one action to the other may be the result of a conceptual change (diSessa, 2006). We present the result of a teaching experiment in classes of grades from 2 to 4 where the relation object/name is investigated in the case of the circle. The experiment makes use of a particular artefact, the Lénárt Spheres (Lénárt, 1996). Comparative geometry activities allow to deal with geometrical objects in a learning environment where the relations between objects, representations and properties are different from the usual ones, hence implying a restructuration of the interplays between them. As a result of the teaching experiment, as can be seen, in particular, from the comparison of initial and conclusive questionnaires, children started a change of their way of associating a name to an object. We argue that this is due also to a conceptual change and not only to “learning what was taught”.


Bolondi, G., Ferretti, F., & Gambini, A. (2014). The relation between mathematical object/mathematical name: Conceptual changes between designation, description, denomination and definition. European Journal of Science and Mathematics Education, 2(2A), 169-176.


  • Antonini, S. and Maracci, M. (2013). Straight on the Sphere: Meanings and Artefacts. In A.M. Lindmeier and A. Heinze (Eds), Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol.2 (pp. 36-40). Kiel, Germany: PME.
  • Bolondi, G., Ferretti, F. and Maffia, A. (2014). Monomials and Polynomials: the long march towards a definition (to appear).
  • Crowley, Mary L. (1987). "The van Hiele Model of the Development of Geometric Thought". In Learning and Teaching Geometry, K-12, 1987 Yearbook of the National Council of Teachers of Mathematics, edited by Mary Montgomery Lindquist (pp.1-16). Reston, Va.: National Council of Teachers of Mathematics.
  • D'Amore, B., and Fandiño Pinilla, M.I. (2012). Su alcune D in didattica della matematica: designazione, denotazione, denominazione, descrizione, definizione, dimostrazione. Riflessioni matematiche e didattiche che possono portare lontano. Bollettino dei docenti di matematica, n.64, 33-46.
  • Deliyianni, E., Elia, I., Gagatsis, A., Monoyiou, A., and Panaoura, A. (2006). A theoretical model of students’geometrical figure understanding. In V. Durand-Guerrier, S. Soury-Lavergne and F. Arzarello (eds.), Proceedings of CERME 6. Lyon, France: Cerme.
  • diSessa, A. A. (2006). A history of conceptual change research: Threads and fault lines. In K. Sawye (Ed.), Cambridge Handbook of the learning sciences (pp. 265-281). Cambridge, UK: Cambridge University Press.
  • Duval, R. (1995). Geometrical Pictures: Kinds of Representation and Specific Processes. In R. Sutherland and J. Mason (eds.), Exploiting mental imagery with computers in mathematical education (pp.142-157). Berlin, DE: Springer.
  • Duval, R. (1999). Representation, Vision and Visualization: Cognitive Functions in Mathematical Thinking. Basic Issues for learnin., Retrieved from ERIC ED 466 379.
  • Duval, R. (2008). Eight problems for a Semiotic approach in Mathematics Education. In L. Radford, G. Schubring and F. Seeger (Eds.), Semiotic in Mathematics Education (pp.39.61). Rotterdam, NL: Sense Publishers.
  • Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24 (2), 139-162.
  • Fischbein, E., and Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education 20(10), 1193- 1211.
  • Lénárt, I. (1993). Non-Euclidean Adventures in the Lénàrt Sphere. Emeryville, Ca: Key Curriculum Press.
  • Lénárt, I. (1996). Alternative models on the drawing ball. Educational Studies in Mathematics, 24, 277-312.
  • Radford, L. (2003). Gestures, speech, and the sprouting of signs: a semiotic-cultural approach to students’ types of generalizatio. Mathematical Thinking and Learning, 5(1), 37–70.
  • van Hiele, P.M. (1986). Structure and insight. A theory of Mathematics Education, Orlando, Fl: Academic Press.
  • Vygotsky, L. S. (1978). Mind in Society. The Development of Higher Psychological Processes, Harvard, Ms: Harvard University Press.