# The Boundary Between Finite and Infinite States Through the Concept of Limits of Sequences

## Abstract

In this article, attempts were made to examine students’ thinking about the concepts of infinity and their ideas about transiting from finite to infinite states through the concept of limits of sequences. The participants included 78 senior high-school students ranging in age between 17 and 19 years old. The data were collected through a questionnaire and an interview with all of the subjects. The findings showed that the students’ understanding of infinity is related to finite situations and many students consider infinite processes as a generalized form of finite processes. In the present study, the most common mistakes committed by students were related to consideration of infinity as a number and application of known finite results to infinite states.

## Keywords

Boundary Finite states Generalization Infinite states Infinity Limits of sequences## References

- Cornu, B. (1991).
*Limits. Advanced mathematical thinking*. Dordrecht, The Netherlands: Kluwer Academic.Google Scholar - Dubinsky, E., Weller, K., McDonald, A. & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An APOS-based analysis: Part 2.
*Educational Studies in Mathematics, 60*, 253–266.CrossRefGoogle Scholar - Edwards, B. (1997). An undergraduate student’s understanding and use of mathematical definitions in real analysis. In J. Dossey, J. Swafford, M. Parmantie & A. Dossey (Eds.),
*Proceedings of the nineteenth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 17–22). Columbus, OH.Google Scholar - Fischbein, E., Tirosh, D. & Melamed, U. (1981). Is it possible to measure the intuitive acceptance of a mathematical statement?
*Educational Studies in Mathematics, 12*, 491–512.CrossRefGoogle Scholar - Gimenez, J. (1990). About intuitional knowledge of density in elementary school.
*Process PME, 14*(3), 19–26.Google Scholar - Hardy, G. H. (1955).
*A course of pure mathematics*. New York, NY: Cambridge University Press.Google Scholar - Kattou, M., Kontoyianni, K. & Christou, C. (2009).
*Teachers’ perception about infinity: A process or an object?*Paper presented at the sixth conference of the European Society for Research in Mathematics Education, France, Lyon.Google Scholar - Lakoff, G. & Nunez, R. (2000).
*Where mathematics come from: How the embodied mind brings mathematics into being*. New York, NY: Basic Books.Google Scholar - Mamolo, A. & Zazkis, R. (2008). Paradoxes as a window to infinity.
*Research in Mathematics Education, 10*(2), 167–182.CrossRefGoogle Scholar - Mamona-Downs, J. (2001). Letting the intuitive bear on the formal: A didactical approach for the understanding of the limits of a sequence.
*Educational Studies in Mathematics, 48*, 259–288.CrossRefGoogle Scholar - Monaghan, J. (2001). Young peoples’ ideas of infinity.
*Educational Studies in Mathematics, 48*, 239–257.CrossRefGoogle Scholar - Mundy, F. J. & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives and integrals.
*MAA Notes, 33*, 31–45.Google Scholar - Renyi, A. (1967).
*Dialogues on mathematics*(A. Renyi Trans.). San Francisco, CA: Holden Day.Google Scholar - Sacristan, A. I. & Noss, R. (2008). Computational construction as a means to coordinate representations of infinity.
*International Journal of Computers for Mathematical Learning, 13*, 47–70.CrossRefGoogle Scholar - Schwarzenberger, R. & Tall, D. (1978). Conflicts in the learning of real numbers and limits.
*Mathematics Teaching, 82*, 44–49.Google Scholar - Tall, D. (1980). The notion of infinite measuring numbers and its relevance in the intuition of infinity.
*Educational Studies in Mathematics, 11*, 271–284.CrossRefGoogle Scholar - Tall, D. (1990). Inconsistencies in the learning of calculus and analysis.
*Focus on Learning Problems in Mathematics, 12*(3&4), 49–64.Google Scholar - Tall, D. (1992).
*The transition to advanced mathematical thinking: Function, limits, infinity and proof*. In: D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495–511). New York, NY: Macmillan.Google Scholar - Tirosh, D. (1999). Finite and infinite sets: Definitions and intuitions.
*International Journal of Mathematical Education in Science & Technology, 30*(3), 341–349.CrossRefGoogle Scholar - Wheeler, M. M. (1987). Children’s understanding of zero and infinity.
*Arithmetic Teacher, 35*(3), 42–44.Google Scholar