Educational Studies in Mathematics

, Volume 69, Issue 3, pp 217–233 | Cite as

Students’ images and their understanding of definitions of the limit of a sequence

  • Kyeong Hah RohEmail author


There are many studies on the role of images in understanding the concept of limit. However, relatively few studies have been conducted on how students’ understanding of the rigorous definition of limit is influenced by the images of limit that the students have constructed through their previous learning. This study explored how calculus students’ images of the limit of a sequence influence their understanding of definitions of the limit of a sequence. In a series of task-based interviews, students evaluated the propriety of statements describing the convergence of sequences through a specially designed hands-on activity, called the ɛ–strip activity. This paper illustrates how these students’ understanding of definitions of the limit of a sequence was influenced by their images of limits as asymptotes, cluster points, or true limit points. The implications of this study for teaching and learning the concept of limit, as well as on research in mathematics education, are also discussed.


Asymptotes Cluster points Images Limits Rigorous definitions Sequences 



The author thanks Professor Sigrid Wagner, Professor Alfinio Flores, and three anonymous reviewers for their helpful comments on earlier versions of the article.


  1. Alcock, L., & Simpson, A. (2004). Convergence of sequences and series: Interactions between visual reasoning and the learner’s beliefs about their own role. Educational Studies in Mathematics, 57, 1–32.CrossRefGoogle Scholar
  2. Alcock, L., & Simpson, A. (2005). Convergence of sequences and series 2: Interactions between visual reasoning and the learner’s beliefs about their own role. Educational Studies in Mathematics, 58, 77–100.CrossRefGoogle Scholar
  3. Apostol, T. (1974). Mathematical analysis. Reading, MA: Addison-Wesley.Google Scholar
  4. Bezuidenhout, J. (2001). Limits and continuity: Some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32, 487–500.CrossRefGoogle Scholar
  5. Brackett, J. D. (1991). The association of mathematical context with students’ responses to tasks involving infinity. Dissertation, University of Georgia.Google Scholar
  6. Brousseau, G. (1997). Theory of didactical situations in mathematics (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Trans.) Dordrecht, The Netherlands: Kluwer Academic.Google Scholar
  7. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153–166). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar
  8. 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
  9. Davis, R. B., & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281–303.Google Scholar
  10. Ferrini-Mundy, J., & Lauten, D. (1993). Teaching and learning calculus. In P. S. Wilson (Ed.), Research ideas for the classroom: High school mathematics (pp. 155–176). New York: Macmillan.Google Scholar
  11. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: D. Reidel.Google Scholar
  12. Hitt, F., & Lara-Chavez, H. (1999). Limits, continuity and discontinuity of functions from two points of view: That of the teacher and that of the student. In L. Bills (Ed.), Proceeding of the British Society for Research into Learning Mathematics, 19, 49–54.Google Scholar
  13. Kawski, M. (1997). How CAS and visualization lead to a complete rethinking of an introduction to vector calculus. In W. Fraunholz (Ed.), Proceeding of the Third International Conference on Technology in Mathematics Teaching. Koblenz, Germany.Google Scholar
  14. Kidron, I., & Zehavi, N. (2002). The role of animation in teaching the limit concept. The International Journal of Computer Algebra in Mathematics Education, 9, 205–227.Google Scholar
  15. Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limit of a sequence. Educational Studies in Mathematics, 48, 259–288.CrossRefGoogle Scholar
  16. Marsden, J. E., & Hoffman, M. J. (2000). Elementary classical analysis (2nd ed.). New York: Freeman.Google Scholar
  17. Navarro, M., & Carreras, P. (2006). Constructing a concept image of convergence of sequences in the van Hiele framework. Research in Collegiate Mathematics Education, VI, 61–98.Google Scholar
  18. Neuhauser, C. (2004). Calculus for biology and medicine (2nd ed.). Upper Saddle River, New Jersey: Pearson Education.Google Scholar
  19. Oehrtman, M. (2002). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts: An instrumentalist investigation into calculus students’ spontaneous reasoning. Dissertation, The University of Texas.Google Scholar
  20. Piaget, J., & Inhelder, B. (1967). The child’s conception of space (F. J. Langdon & J. L. Lunzer., Trans.). New York: The Norton Library.Google Scholar
  21. Pinto, M., & Tall, D. (2002). Building formal mathematics on visual imagery: A case study and a theory. For the Learning of Mathematics, 22, 2–10.Google Scholar
  22. Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55, 103–132.CrossRefGoogle Scholar
  23. Roh, K. (2005). College students’ intuitive understanding of the limit of a sequence and their levels of reverse thinking. Dissertation, The Ohio State University.Google Scholar
  24. Sierpińska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397.CrossRefGoogle Scholar
  25. Stewart, J. (2003). Calculus: Early transcendental single variable (5th ed.). Belmont, CA: Brooks/Cole-Thomas Learning.Google Scholar
  26. Stroyan, K. D. (1998). Mathematical background: Foundations of infinitesimal calculus. Retrieved from (June 8, 2007)
  27. Szydlik, J. E. (2000). Mathematical beliefs and conceptual understanding of the limit of function. Journal for Research in Mathematics Education, 31, 256–276.CrossRefGoogle Scholar
  28. Taback, S. (1975). The child’s concept of limit. In M. F. Rosskopf (Ed.), Children’s mathematical concepts (pp. 111–144). New York: Teachers College Press.Google Scholar
  29. 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
  30. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics, with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar
  31. Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht, The Netherlands: Kluwer Academic.Google Scholar
  32. Williams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22, 219–236.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsArizona State UniversityTempeUSA

Personalised recommendations