Three concepts or one? Students’ understanding of basic limit concepts Article First Online: 19 July 2016 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 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.

CrossRef Google Scholar 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.

CrossRef Google Scholar 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.

CrossRef Google Scholar Bryant, V. (1990) Yet another introduction to analysis. Cambridge University Press, Cambridge.

CrossRef Google Scholar 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 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.

CrossRef Google Scholar 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.

CrossRef Google Scholar Dubinsky, E., Elterman F., Gong C. (1988) The student’s construction of quantification. For the learning of mathematics 8 (2): 44–51.

Google Scholar 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.

CrossRef Google Scholar Ely, R. (2010) Nonstandard student conceptions about infinitesimals. Journal for Research in Mathematics Education 41 (2): 117–146.

Google Scholar Fincher, S., Tenenberg J. (2005) Making sense of card sorting data. Expert Systems 223: 89–93.

CrossRef Google Scholar 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.

CrossRef Google Scholar Güçler, B. (2013) Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics 82 (3): 439–453.

CrossRef Google Scholar 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.

CrossRef Google Scholar Keisler, H. J. (1986) Elementary calculus: An infinitesimal approach. Prindle Weber & Schimidt, Boston.

Google Scholar 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.

CrossRef Google Scholar Lakoff, G., Núñez R. E. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. Basic books, New York.

Google Scholar 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.

CrossRef Google Scholar Marton, F. (1986) Phenomenography—a research approach to investigating different understandings of reality. Journal of Thought: 28–49.

Google Scholar 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 Monaghan, J. (1991) Problems with the language of limits. For the learning of mathematics 11 (3): 20–24.

Google Scholar Nosofsky, R. M. (1986) Attention, similarity, and the identification–categorization relationship. Journal of Experimental Psychology: General 115 (1): 39.

CrossRef Google Scholar 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.

CrossRef Google Scholar 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 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.

CrossRef Google Scholar Raman, M. (2004) Epistemological messages conveyed by three high-school and college mathematics textbooks. The Journal of Mathematical Behavior 23 (4): 389–404.

CrossRef Google Scholar 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.

CrossRef Google Scholar Sierpínska, A. (1987) Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics 18 (4): 371–397.

CrossRef Google Scholar Spivak, M. (2006) Calculus corrected third edition. Cambridge University Press, Cambridge.

Google Scholar Swinyard, C. (2011) Reinventing the formal definition of limit: The case of Amy and Mike. The Journal of Mathematical Behavior 30 (2): 93–114.

CrossRef Google Scholar 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.

CrossRef Google Scholar 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.

CrossRef Google Scholar 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.

CrossRef Google Scholar Trigueros, M., Ursini S. (2003) First-year undergraduates’ difficulties in working with different uses of variable. CBMS Issues in Mathematics Education 8: 1–26.

CrossRef Google Scholar 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.

CrossRef Google Scholar Williams, S. R. (1991) Models of limit held by college calculus students. Journal for Research in Mathematics Education 22 (3): 219–236.

CrossRef Google Scholar © Springer Science+Business Media Dordrecht 2016

Authors and Affiliations 1. Department of Didactics of Mathematics University of Granada Granada Spain 2. School of Education Durham University Durham UK