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Computational Construction as a Means to Coordinate Representations of Infinity

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Abstract

In this paper, we describe a design experiment aimed at helping students to explore and develop concepts of infinite processes and objects. Our approach is based on the design and development of a computational microworld, which afforded students the means to construct a range of representational models (symbolic, visual and numeric) of infinity-related objects (infinite sequences, in particular). We present episodes based on four students’ activities, seeking to illustrate how the available tools mediated students’ understandings of the infinite in rich ways, allowing them to discriminate subtle process-oriented features of infinite processes. We claim that the microworld supported students in the coordination of hitherto unconnected or conflicting intuitions concerning infinity, based on a constructive articulation of different representational forms we name as ‘representational moderation’.

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Notes

  1. The best-known theory is Robinson’s semantic model, which extends the set of real numbers to include infinitely large and infinitesimal numbers in the set of hyperreal numbers (*R), where an infinitesimal is defined as a number smaller than every positive real number and bigger than every negative real number (Robinson 1974/1996). Other approaches include Nelson (1977)’s axiomatic Internal Set Theory.

  2. This is independent of the convergence or divergence of the series, although when there is convergence it is easier to think of the series as a “complete” object, e.g. when \( {\sum\limits_{n = 1}^\infty {\frac{1} {{2^{n} }}} } = 1 \).

  3. The study took place in Mexico, so all data was translated into English from Spanish.

  4. A Logo programming course was advertised in several schools across Mexico City. The course was free if students agreed to participate in the study that followed the course. Thus, the researcher never met beforehand any of the students, and did not chose them: these participants signed up for the Logo course and volunteered to participate in the study.

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Authors and Affiliations

  1. Department of Mathematics Education, Centre for Research and Advanced Studies (Cinvestav), Av. Instituto Politécnico Nacional 2508, Col. San Pedro Zacatenco, Mexico, DF, 07360, Mexico

    Ana Isabel Sacristán

  2. London Knowledge Lab, Institute of Education, University of London, 23-29 Emerald Street, London, WC1N 3QS, England, UK

    Richard Noss

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  1. Ana Isabel Sacristán
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Correspondence to Ana Isabel Sacristán.

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Sacristán, A.I., Noss, R. Computational Construction as a Means to Coordinate Representations of Infinity. Int J Comput Math Learning 13, 47–70 (2008). https://doi.org/10.1007/s10758-008-9127-5

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  • DOI: https://doi.org/10.1007/s10758-008-9127-5

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