Young Students Exploring Cardinality by Constructing Infinite Processes
 Ken Kahn,
 Evgenia Sendova,
 Ana Isabel Sacristán,
 Richard Noss
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In this paper, we describe the design and implementation of computer programming activities aimed at introducing young students (9–13 years old) to the idea of infinity, and in particular, to the cardinality of infinite sets. This research was part of the WebLabs project where students from several European countries explored topics in mathematics and science by building computational models and programs, which they shared and discussed. We focus on a subset of the work in which students explored concepts of cardinality of infinite sets by interpreting and constructing computer programs in ToonTalk, a programming language and environment that is especially wellsuited for young students. Our hypothesis is that via carefully designed computational explorations within an appropriately constructed medium, infinity can be approached in a learnable way that does not sacrifice the rigour necessary for mathematical understanding of the concept, and at the same time contributes to introducing the real spirit of mathematics to the school classroom.
 Dauben, J. W. (1990) Georg cantor: His mathematics and philosophy of the infinite. Princeton University Press: Princeton, NJ.
 diSessa, A. (2000). Changing minds. Computers, learning and literacy. Cambridge, MA: MIT Press.
 Falk, R., & BenLavy, S. (1989) ‘How big is an infinite set? Exploration of children’s ideas’. In Proceedings of the Thirteenth Annual Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 252–259.
 Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3–40. CrossRef
 Hilton, P. (1991). The mathematical component of a good education. Miscellanea Mathematica. Berlin: Springer.
 Kahn, K. (2001) ‘Generalizing by Removing Detail’, Communications of the ACM, 43(3), March 2000. An extended version is in Henry Lieberman, editor, Your Wish Is My Command: Programming by Example, Morgan Kaufmann, 2001.
 Kaput, J., Noss, R., & Hoyles, C. (2002) Developing New Notations for a Learnable Mathematics in the Computational Era. In English, L. (Ed.), Handbook of international research in mathematics education, (pp. 51–75). London: Lawrence Erlbaum (Reprinted in Second Edition, 2008).
 Lang, S. (1985). Math!: encounters with high school students. New York Inc: Springer.
 Maor, E. (1987). To infinity and beyond: A cultural history of the infinite. Boston: Birkhäuser.
 Monaghan, J. (2001). Young peoples’ ideas of infinity. Educational Studies in Mathematics, 48, 239–257. CrossRef
 Mor, Y., & Sendova, E. (2003). ToonTalking about mathematics. In I. Derzhanski, H. Dimitrova, S. Grozdev, E. Sendova (Eds.), History and education in mathematics and informatics, attracting talent to science, (pp. 36–43) Proceedings of the International Congress MASSEE 2003, September 15–21. Borovets, Bulgaria.
 Noss, R. & Hoyles, C. (1996) Windows on mathematical meanings. Learning cultures and computers. Kluwer Academic Publishers: Dordrecht, Boston, London.
 Papert, S. (1972). Teaching children to be mathematicians versus teaching about mathematics. International Journal of Mathematics Education and Science and Technology, 3, 249–262. CrossRef
 Piaget, J., & Garcia, R. (1989). Psychogenesis and the history of science. Columbia University Press: New York.
 Sacristán, A. I. (1997). Windows on the infinite: Constructing Meanings in a Logobased Microworld. PhD. Dissertation, Institute of Education, University of London, UK.
 Sacristán, A. I., & Noss, R. (2008). Computational construction as a means to coordinate representations of infinity. International Journal of Computers for Mathematical Learning, 13(1), 47–70. CrossRef
 Simon, M. A., & Tzur, R. (2004). Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory. In Mathematical Thinking and Learning, 6(2), 91–104. CrossRef
 Tall, D. O. (2001). A child thinking about infinity. In Journal of Mathematical Behavior, 20, 7–19. CrossRef
 Tsamir, P. (2001). When ‘The Same’ is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48, 289–307. CrossRef
 Wilensky, U., & Papert, S. (2010). Restructurations: Reformulations of Knowledge Disciplines through new representational forms. In J. Clayson & I. Kallas (Eds.), Proceedings of the constructionism 2010 Conference. Paris, France.
 Title
 Young Students Exploring Cardinality by Constructing Infinite Processes
 Journal

Technology, Knowledge and Learning
Volume 16, Issue 1 , pp 334
 Cover Date
 20110401
 DOI
 10.1007/s1075801191750
 Print ISSN
 22111662
 Online ISSN
 15731766
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Infinity
 Cardinality
 ToonTalk
 Constructionism
 Programming
 Authors

 Ken Kahn ^{(1)}
 Evgenia Sendova ^{(2)}
 Ana Isabel Sacristán ^{(3)}
 Richard Noss ^{(1)}
 Author Affiliations

 1. London Knowledge Lab, Institute of Education, 2329 Emerald Street, London, WC1N 3QS, UK
 2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev bl. 8, 1113, Sofia, Bulgaria
 3. Department of Mathematics Education, Centre for Research and Advanced Studies (Cinvestav), Av. IPN 2508, 07360, Mexico, DF, Mexico