• Iliada Elia
  • Athanasios Gagatsis
  • Areti PanaouraEmail author
  • Theodosis Zachariades
  • Fotini Zoulinaki


The present study explores students’ abilities in conversions between geometric and algebraic representations, in problem- solving situations involving the concept of “limit” and the interrelation of these abilities with students’ constructed understanding of this concept. An attempt is also made to examine the impact of the “didactic contract” on students’ performance through the processes they employ in tackling specific tasks on the concept of limit. Data were collected from 222 12th-grade high school students in Greece. The results indicated that students who had constructed a conceptual understanding of limit were the ones most probable to accomplish the conversions of limits from the algebraic to the geometric representations and the reverse. The findings revealed the compartmentalized way of students’ thinking in non-routine problems by means of their performance in simpler conversion tasks. Students who did not perform under the conditions of the didactic contract were found to be more consistent in their responses for various conversion tasks and complex problems on limits, compared to students who, as a consequence of the didactic contract, used only algorithmic processes.

Key words

“didactic contract” geometric and algebraic representations the concept of “limit” 


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  1. Alcock, L. & Simpson, A. (2004). Convergence of sequences and series: interactions between visual reasoning and the learners’ beliefs about their own role. Educational Studies in Mathematics, 57, 1–32.CrossRefGoogle Scholar
  2. Artigue, M. (1998). L’évolution des problématiques en didactique de l’analyse. Recherches en Didactique des Mathématiques, 18(2), 231–262.Google Scholar
  3. Bloch, I. (2003). Teaching functions in a graphic milieu: what forms of knowledge enable students to conjecture and prove? Educational Studies in Mathematics, 52, 3–28.CrossRefGoogle Scholar
  4. Bodin, A., Coutourier, R., & Gras, R. (2000). CHIC: Classification Hiérarchique Implicative et Cohésive-Version sous Windows - CHIC 1.2. Rennes: Association pour la Recherche en Didactique des Mathématiques. Google Scholar
  5. Brousseau, G. (1990). Le contract didactique: Le milieu. Recherches en Didactique de Mathématiques, 9, 308–336.Google Scholar
  6. Brousseau, G. (1997). Theory of didactical situations in mathematics (edited and translated by N. Balacheff, M. Cooper, R. Sutherlandd & V. Warfield). Dordrecht, Netherlands: Kluwer.Google Scholar
  7. Cornu, B. (1983). Quelques obstacles à l’apprentissage de la notion de limite. Recherches en Didactique des Mathematiques, 4, 236–268.Google Scholar
  8. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 153–166. )Dordrecht: Kluwer.Google Scholar
  9. Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process schema. Journal of Mathematical Behavior, 15, 167–192.CrossRefGoogle Scholar
  10. Davis, R. & Vinner, S. (1986). The notion of limit: some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281–303.Google Scholar
  11. DeWindt-King, A. M. & Goldin, G. A. (2003). Children’s visual imagery: Aspects of cognitive representation in solving problems with fractions. Mediterranean Journal for Research in Mathematics Education, 2(1), 1–42.Google Scholar
  12. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 95–126. )Dordrecht: Kluwer.Google Scholar
  13. Duval, R. (2002). The cognitive analysis of problems of comprehension in the learning of mathematics. Mediterranean Journal for Research in Mathematics Education, 1(2), 1–16.Google Scholar
  14. Elia, I., Gagatsis, A. & Gras, R. (2005). Can we “trace” the phenomenon of compartmentalization by using the I.S.A.? An application for the concept of function. In R. Gras, F. Spagnolo & J. David (Eds.), Proceedings of the Third International Conference I.S.A. Implicative Statistic Analysis (pp. 175–185. )Palermo, Italy: Universita degli Studi di Palermo.Google Scholar
  15. Elia, I., Panaoura, A., Eracleous, A. & Gagatsis, A. (2007). Relations between secondary pupils’ conceptions about functions and problem solving in different representations. International Journal of Science and Mathematics Education, 5(3), 533–556.CrossRefGoogle Scholar
  16. Even, R. (1998). Factors involved in linking representations of functions. The Journal of Mathematical Behavior, 17(1), 105–121.CrossRefGoogle Scholar
  17. Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38, 11–50.CrossRefGoogle Scholar
  18. Furinghetti, F. & Paola, D. (1991). The construction of a didactic itinerary of calculus starting from students’ concept images (ages 16–19). International Journal of Mathematical Education in Science and Technology, 22(5), 719–729.CrossRefGoogle Scholar
  19. Goldin, G. A. (1998). Representational systems, learning, and problem solving in mathematics. Journal of Mathematical Behavior, 17(2), 137–165.CrossRefGoogle Scholar
  20. Greeno, J. G. & Hall, R. P. (1997). Practicing representation: Learning with and about representational forms. Phi Delta Kappan, 78, 361–367.Google Scholar
  21. Juter, K. (2006). Limits of functions as they developed through time and as students learn them today. Mathematical Thinking and Learning, 8(4), 407–431.CrossRefGoogle Scholar
  22. Kaldrimidou, M. & Ikonomou, A. (1998). Factors involved in the learning of mathematics: The case of graphic representations of functions. In H. Steinbring, M.G. Bartolini Bussi & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 271–288. )Reston, Va: NCTM.Google Scholar
  23. Kaput, J. (1989). Linking representations in the symbolic systems of algebra. In S. Wagner & C. Kieran (Eds.), Research agenda for mathematics education: Research issues in the learning and teaching of algebra (pp. 167–194. )Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  24. Kaput, J. (1992). Technology and mathematics education. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556. )New York: Macmillan.Google Scholar
  25. Lerman, I. C. (1981). Classification et analyse ordinale des données. Paris: Dunod.Google Scholar
  26. Mamona-Downs, J. (2001). Letting the intuitive bear on the formal; A didactical approach for the understanding of the limit of sequence. Educational Studies in Mathematics, 48, 259–288.CrossRefGoogle Scholar
  27. Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20–24.Google Scholar
  28. Pedagogical Institute (2005). Instructions for the didactic content and the teaching of mathematics in the school year 2005–2006. Athens: Publishing Organization of Didactic Books (in Greek).Google Scholar
  29. Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge.Google Scholar
  30. Schwarz, B. & Dreyfus, T. (1995). New actions upon old objects: a new ontological perspective on functions. Educational Studies in Mathematics, 29, 259–291.CrossRefGoogle Scholar
  31. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.CrossRefGoogle Scholar
  32. Sierpinska, A. (1985). Obstacles épistémologiques relatifs à la notion de limite. Recherches en Didactique des Mathématiques, 6(1), 5–67.Google Scholar
  33. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397.CrossRefGoogle Scholar
  34. Tall, D. (1980). Mathematical intuition, with special reference to limiting processes. Proceedings of the Fourth International Congress on Mathematical Education (pp. 170–176). Berkeley.Google Scholar
  35. Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 495–511. )New York: Macmillan.Google Scholar
  36. Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.CrossRefGoogle Scholar
  37. Williams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22, 219–236.CrossRefGoogle Scholar
  38. Williams, S. (2001). Predications of the limit concept: An Application of Repertory Grids. Journal for Research in Mathematics Education, 32, 341–367.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2009

Authors and Affiliations

  • Iliada Elia
    • 1
  • Athanasios Gagatsis
    • 1
  • Areti Panaoura
    • 2
    Email author
  • Theodosis Zachariades
    • 3
  • Fotini Zoulinaki
    • 3
  1. 1.Department of EducationUniversity of CyprusNicosiaCyprus
  2. 2.Department of EducationFrederick UniversityNicosiaCyprus
  3. 3.Department of MathematicsUniversity of AthensNicosiaGreece

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