Skip to main content
SpringerLink
Log in

Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory

  • Published:
Educational Studies in Mathematics Aims and scope Submit manuscript

Abstract

The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory – the concepts of inclusion and belonging. To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of mathematical objects draws its meaning from different semiotic systems framed by their own cultural context. Our results suggest that the successful accomplishment of knowledge attainment seems to be linked to the students' ability to suitably distinguish and coordinate the meanings and symbols of the various semiotic systems (e.g. verbal, diagrammatic and symbolic) that encompass their mathematical experience.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Artigue, M.: 1998, ‘L'évolution des problématiques en didactique de l'analyse’, Recherches en Didactique des Mathématiques 18(2), 231–262.

    Google Scholar 

  • Arzarello, F. and Bartolini Bussi, M.: 1998, ‘Italian trends of research in mathematics education: A national case study in the international perspective’, in J. Kilpatrick and A. Sierpinska (eds.), Mathematics Education as a Research Domain: The Search for Identity, Vol. 2, Kluwer, Dordrecht, pp. 243–262.

    Google Scholar 

  • Arzarello, F., Bartolini Bussi, M.G. and Robutti, O.: 2002, ‘Time(s) in didactics of mathematics: A methodological challenge’, in L.D. English, M. Bartolini Bussi, G.A. Jones, R.A. Lesh and D. Tirosh (eds.), Handbook of International Research in Mathematics Education, Erlbaum, Mahwah, NJ, pp. 525–552.

    Google Scholar 

  • Bagni, G.T.: in press, ‘Historical roots of limit notion. Development of its representative registers and cognitive development’, Canadian Journal of Science, Mathematics and Technology Education.

  • Balacheff, N.: 1991, ‘The benefits and limits of social interactions: The case of mathematical proof’, in A.J. Bishop, A.J. Bishop, S. Mellin-Olsen and J. van Dormolen (eds.), Mathematical Knowledge: Its Growth Through Teaching, Kluwer, Dordrecht, pp. 175–192.

    Google Scholar 

  • Boero, P. and Szendrei, J.: 1998, ‘Research and results in mathematics education: Some contradictory aspects’, in J. Kilpatrick and A. Sierpinska (eds.), Mathematics Education as a Research Domain: The Search for Identity, Vol. 1, Kluwer, Dordrecht, pp. 197–212.

    Google Scholar 

  • Bourguignon, J.-P.: 2001, ‘A basis for a new relationship between mathematics and society’, in B. Engquist and W. Schmid (eds.), Mathematics Unlimited-2001 and Beyond, Vol. I, Springer-Verlag, Berlin, pp. 171–188.

    Google Scholar 

  • Boyer, C.B.: 1985, A History of Mathematics, Princeton University Press, Princeton, NJ (1st ed., Wiley, New York, 1968).

    Google Scholar 

  • Casari, E.: 1964, Questioni di filosofia della matematica, Feltrinelli, Milano.

    Google Scholar 

  • Dodero, N., Baroncini, P. and Manfredi, R.: 1999, Lineamenti di matematica 1, Ghisetti e Corvi, Milano.

    Google Scholar 

  • Duval, R.: 1993, ‘Registres de représentations sémiotiques et fonctionnement cognitif de la pensée’, Annales de Didactique et de Sciences Cognitives IREM Strasbourg, 5, 37–65.

    Google Scholar 

  • Duval, R.: 1995, Sémiosis et pensée humaine, Lang, Paris.

    Google Scholar 

  • Duval, R.: 2000, ‘Basic issues for research in Mathematics Education’, in T. Nakahara and M. Koyama (eds.), Proceedings of the 24th PME, Vol. 1, Nishiki Print, Hiroshima, pp. 55–69.

    Google Scholar 

  • Ferrari, P.L.: 2004, ‘Mathematical language and advanced mathematics learning’, in M. Johnsen Høines and A. Berit Fuglestad (eds.), Proceedings of the 28th PME, Vol. 2, Bergen, Norway, pp. 383–390.

    Google Scholar 

  • Ferro, R.: 1993, ‘La teoria degli insiemi p. II.’, L’insegnamento della matematica e delle scienze integrate 16(11/12), 1077–1099.

    Google Scholar 

  • Fischbein, E.: 1993, ‘The theory of figural concepts’, Educational Studies in Mathematics 24, 139–162.

    Article  Google Scholar 

  • Fischbein, E. and Baltsan, M.: 1999, ‘The mathematical concept of set and the “collection” model’, Educational Studies in Mathematics 37, 1–22.

    Article  Google Scholar 

  • Freudenthal, H.: 1983, Didactical Phenomenology of Mathematical Structures, Reidel, Dordrecht.

    Google Scholar 

  • Halliday M.A.K.: 1985, An Introduction to Functional Grammar, Arnold, London.

    Google Scholar 

  • Johnson, M.: 1987, The Body in the Mind. The Bodily Basis of Meaning, Imagination, and Reason, University of Chicago Press, Chicago.

    Google Scholar 

  • Kaput, J.J.: 1993, ‘The representational roles of technology in connecting mathematics with authentic experience’, in R. Bieler, R.W. Scholz, R. Strässer and B. Winkelman (eds.), Didactics of Mathematics as a Scientific Discipline, Kluwer, Dordrecht, pp. 379–397.

    Google Scholar 

  • Kline, M.: 1972, Mathematical Thought from Ancient to Modern Times, Oxford University Press, New York.

    Google Scholar 

  • Lakoff, G. and Núñez, R.: 2000, Where Mathematics Come From? How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.

    Google Scholar 

  • Leckie-Tarry, H.: 1995, Language and Context –A Functional Linguistic Theory of Register, Pinter, London.

    Google Scholar 

  • Linchevski, L. and Livneh, D.: 2002, ‘The competition between numbers and structure: Why expressions with identical algebraic structures trigger different interpretations’, Focus on Learning Problems in Mathematics 24(2), 38–20.

    Google Scholar 

  • Linell, P.: 1998, ‘Approaching dialogue: Talk and interactions in dialogical perspective’, John Benjamins, Philadelphia.

    Google Scholar 

  • Mevarech, Z.R. and Kramarsky, B.: 1997, ‘From verbal description to graphic representations: Stability and change in students' alternative conceptions’, Educational Studies in Mathematics 32, 229–263.

    Article  Google Scholar 

  • Otte, M.: 2001, ‘Mathematical epistemology from a semiotic point of view’, Paper presented to the Discussion Group on Semiotics at the 25th PME.

  • Rabardel, P.: 1995, Les hommes et les technologies: Approche cognitive des instruments contemporains, Colin, Paris.

    Google Scholar 

  • Radford, L.: 2000, ‘Signs and meanings in the students' emergent algebraic thinking’, Educational Studies in Mathematics 42, 237–268.

    Article  Google Scholar 

  • Radford, L.: 2002a, ‘The object of representations: Between wisdom and certainty’, in F. Hitt (ed.), Representations and Mathematics Visualization, Cinvestav-IPN, Mexico, pp. 219–240.

    Google Scholar 

  • Radford, L.: 2002b, ‘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 

  • Radford, L.: 2003a, ‘Gestures, speech and the sprouting of signs’, Mathematical Thinking and Learning 5(1), 37–70.

    Article  Google Scholar 

  • Radford, L.: 2003b, ‘On culture and mind. A post-Vygotskian semiotic perspective, with an example from Greek mathematical thought’, in M. Anderson, A. Sáenz-Ludlow, S. Zellweger and V.V. Cifarelli (eds.), Educational Perspectives on Mathematics as Semiosis: From Thinking to Interpreting to Knowing, Legas, Ottawa, pp. 49–79.

    Google Scholar 

  • Radford, L.: 2003c, ‘On the epistemological limits of language: Mathematical knowledge and social practice during the Renaissance’, Educational Studies in Mathematics 52, 123–150.

    Article  Google Scholar 

  • Radford, L.: 2005, ‘The semiotics of the schema. Kant, Piaget, and the Calculator’, in M.H.G. Hoffmann, J. Lenhard and F. Seeger (eds.), Activity and Sign. Grounding Mathematics Education, Springer, New York, pp. 137–152.

    Google Scholar 

  • Ryve, A.: 2004, ‘Can collaborative concept mapping create mathematical productive discourses?’, Educational Studies in Mathematics 26, 157–177.

    Article  Google Scholar 

  • Schoenfeld, A.H.: 1986, ‘On having and using geometric knowledge’, in J. Hiebert (ed.), Conceptual and Procedural Knowledge: The Case of Mathematics, Erlbaum, Hillsdale, NJ, pp. 225–263.

    Google Scholar 

  • Steinbring, H.: 2002, ‘What makes a sign a mathematical sign? An epistemological perspective on mathematical interaction’, Paper presented to the Discussion Group on Semiotics at the 26th PME.

  • Tall, D. and Vinner, S.: 1981, ‘Concept image and concept definition in mathematics with special reference to limits and continuity’, Educational Studies in Mathematics 12, 151–169.

    Article  Google Scholar 

  • Tsamir, P. and Tirosh, D.: 1994, ‘Comparing infinite sets: Intuition and representation’, Proceedings of the XVIII PME. Lisboa, 2, 345–352.

    Google Scholar 

  • Tsamir, P.: 1999, ‘The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers’, Educational Studies in Mathematics 38, 209–234.

    Article  Google Scholar 

  • Tsamir, P. and Tirosh, D.: 1999, ‘Consistency and representations: The case of actual infinity’, Journal for Research in Mathematics Education 30, 213–219.

    Article  Google Scholar 

  • Tsamir, P. and Dreyfus, T.: 2002, ‘Comparing infinite sets – a process of abstraction. The case of Ben’, Journal of Mathematical Behavior 221(1), 1–23.

    Article  Google Scholar 

  • van Heijenort, J.: 1967, From Frege to Gödel, Harvard University Press, Cambridge, UK.

    Google Scholar 

  • Vinner, S. and Dreyfus, T.: 1989, ‘Images and definitions for the concept of a function’, Journal for Research in Mathematics Education 20, 356–366.

    Article  Google Scholar 

  • Vygotsky, L.S.: 1962, Thought and Language. MIT Press, Cambridge, MA (Mylenie i re, Gosudarstvennoe Social'no-Ekonomieskoe Izdatel'stvo, Moskva-Leningrad 1934).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Udine, Via delle Scienze 206, I-33100, Udine, Italy

    Giorgio T. Bagni

Authors
  1. Giorgio T. Bagni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giorgio T. Bagni.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bagni, G.T. Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory. Educ Stud Math 62, 259–280 (2006). https://doi.org/10.1007/s10649-006-8545-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-006-8545-3

Key Words

Search

Navigation