Young Students Exploring Cardinality by Constructing Infinite Processes
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Young Students Exploring Cardinality by Constructing Infinite Processes Ken Kahn • Evgenia Sendova • Ana Isabel Sacrista´n • Richard Noss
Published online: 19 April 2011 Springer Science+Business Media B.V. 2011
Abstract 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 well-suited 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. Keywords
Infinity Cardinality ToonTalk Constructionism Programming
K. Kahn (&) R. Noss London Knowledge Lab, Institute of Education, 23-29 Emerald Street, London WC1N 3QS, UK e-mail: [email protected] R. Noss e-mail: [email protected] E. Sendova Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev bl. 8, 1113 Sofia, Bulgaria e-mail: [email protected] A. I. Sacrista´n Department of Mathematics Education, Centre for Research and Advanced Studies (Cinvestav), Av. IPN 2508, 07360 Mexico, DF, Mexico e-mail: [email protected]
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1 Introduction The infinite! No other question has ever moved so profoundly the spirit of men! These words of David Hilbert (quoted in Maor 1987) have been a motivating challenge for a number of mathematics educators in their attempt to convey the real spirit of mathematics, by discussing deep ideas with students at school age. In this paper we present our own attempt at introducing young students to this topic, through computational constructions that lead to the exploration of the cardinality of infinite sets. We were mindful that the infinite is particularly complex due to its contradictory nature. Tall (2001) has noted that the ideas that adolescents have of infinite concepts are influenced by their prior experiences; and since they tend to meet the notion of ‘‘limit’’ in some dynamic sense before the notion of one to one correspondences between infinite sets, this potential view of infinity clashes with the notion of infinite cardinals. He thus attempted to introduce a young 7-year old student to the concept of cardinal infinity by comparing infinite sets to see what this did to his intuitions; the results were mixed, since the child developed conflicting interpr
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