Educational Studies in Mathematics

, Volume 93, Issue 3, pp 315–332

# Three concepts or one? Students’ understanding of basic limit concepts

Article

## Abstract

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of limit connect them. This paper examines the nature of connections made by students exposed to this fragmented curriculum. The study adopted a phenomenographic approach and used card sorting and comparison tasks to expose students to symbols representing these different types of limit. The findings suggest that, while some students treat limit cases as separate, some can draw connections, but often do so in ways which are at odds with the formal mathematics. In particular, while there are occasional, implicit uses of neighbourhood notions, no student in the study appeared to possess a unifying organisational framework for all three basic uses of limit.

### Keywords

Limits Advanced mathematical thinking Definitions Card-sorting Phenomenography

### References

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): 1–32.
2. Alcock, L., Simpson A. (2005) Convergence of sequences and series 2: Interactions between nonvisual reasoning and the learner’s beliefs about their own role. Educational Studies in Mathematics 58 (1): 77–100.
3. Borovik, A., Katz M. G. (2012) Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Foundations of Science 17 (3): 245–276.
4. Bryant, V. (1990) Yet another introduction to analysis. Cambridge University Press, Cambridge.
5. Bussolon, S., Russi B., Missier F. D. (2006) Online card sorting: As good as the paper version. Proceedings of the 13th European conference on Cognitive ergonomics: trust and control in complex socio-technical systems, 113–114.Google Scholar
6. Chi, M. T., Feltovich P. J., Glaser R. (1981) Categorization and representation of physics problems by experts and novices. Cognitive Science 5 (2): 121–152.
7. Cottrill, J., Dubinsky E., Nichols D., Schwingendorf K., Thomas K., Vidakovic D. (1996) Understanding the limit concept: Beginning with a coordinated process scheme. The Journal of Mathematical Behavior 15 (2): 167–192.
8. Dubinsky, E., Elterman F., Gong C. (1988) The student’s construction of quantification. For the learning of mathematics 8 (2): 44–51.Google Scholar
9. Elia, I., Gagatsis A., Panaoura A., Zachariades T., Zoulinaki F. (2009) Geometric and algebraic approaches in the concept of “limit” and the impact of the “didactic contract”. International Journal of Science and Mathematics Education 7 (4): 765–790.
10. Ely, R. (2010) Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education 41 (2): 117–146.Google Scholar
11. Fincher, S., Tenenberg J. (2005) Making sense of card sorting data. Expert Systems 223: 89–93.
12. Font, V., Bolite J., Acevedo J. (2010) Metaphors in mathematics classrooms: Analyzing the dynamic process of teaching and learning of graph functions. Educational Studies in Mathematics 75 (2): 131–152.
13. Güçler, B. (2013) Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics 82 (3): 439–453.
14. Jones, S. R. (2015) Calculus limits involving infinity: The role of students’ informal dynamic reasoning. International Journal of Mathematical Education in Science and Technology 46 (1): 105–126.
15. Keisler, H. J. (1986) Elementary calculus: An infinitesimal approach. Prindle Weber & Schimidt, Boston.Google Scholar
16. Kidron, I. (2011) Constructing knowledge about the notion of limit in the definition of the horizontal asymptote. International Journal of Science and Mathematics Education 9 (6): 1261–1279.
17. Lakoff, G., Núñez R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic books, New York.Google Scholar
18. 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 (2–3): 259–288.
19. Marton, F. (1986) Phenomenography—a research approach to investigating different understandings of reality. Journal of Thought: 28–49.Google Scholar
20. McDonald, M. A., Mathews D. M., Strobel K. H. (2000) Understanding sequences: A tale of two objects. Research in Collegiate Mathematics Education IV: 77–102.Google Scholar
21. Monaghan, J. (1991) Problems with the language of limits. For the learning of mathematics 11 (3): 20–24.Google Scholar
22. Nosofsky, R. M. (1986) Attention, similarity, and the identification–categorization relationship. Journal of Experimental Psychology: General 115 (1): 39.
23. Oehrtman, M. (2008) Layers of abstraction: Theory and design for the instruction of limit concepts. In: Carlson M. Rasmussen C. (eds) Making the connection: research and teaching in undergraduate mathematics, 65–80.. Mathematical Association of America Washington, Washington.
24. Oehrtman, M. (2009) Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education 40 (4): 396–426.Google Scholar
25. Przenioslo, M. (2004) Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics 55 (1–3): 103–132.
26. Raman, M. (2004) Epistemological messages conveyed by three high-school and college mathematics textbooks. The Journal of Mathematical Behavior 23 (4): 389–404.
27. Roh, K. H. (2008) Students’ images and their understanding of definitions of the limit of a sequence. Educational Studies in Mathematics 69 (3): 217–233.
28. Sierpínska, A. (1987) Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics 18 (4): 371–397.
29. Spivak, M. (2006) Calculus corrected third edition. Cambridge University Press, Cambridge.Google Scholar
30. Swinyard, C. (2011) Reinventing the formal definition of limit: The case of Amy and Mike. The Journal of Mathematical Behavior 30 (2): 93–114.
31. Szydlik, J. E. (2000) Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education 31 (3): 258–276.
32. Tall, D., Thomas M., Davis G., Gray E., Simpson A. (1999) What is the object of the encapsulation of a process? The Journal of Mathematical Behavior 18 (2): 223–241.
33. 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.
34. Trigueros, M., Ursini S. (2003) First-year undergraduates’ difficulties in working with different uses of variable. CBMS Issues in Mathematics Education 8: 1–26.
35. Weber, K. (2005) Problem-solving, proving, and learning: The relationship between problem-solving processes and learning opportunities in the activity of proof construction. The Journal of Mathematical Behavior 24 (3): 351–360.
36. Williams, S. R. (1991) Models of limit held by college calculus students. Journal for Research in Mathematics Education 22 (3): 219–236.