Computational Construction as a Means to Coordinate Representations of Infinity

  • Ana Isabel SacristánEmail author
  • Richard Noss


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’.


Computational microworld Infinity Infinite sequences Logo Representational moderation Situated abstractions 


  1. Abramovich, S., Brantlinger, A., & Norton, A. (1999). Exploring quadratic-like sequences through a tool kit approach. In G. Goodell (Ed.), Proceedings of the 11th Annual International Conference on Technology in Collegiate Mathematics. Reading, MA: Addison Wesley Longman.Google Scholar
  2. Balacheff, N. (1994). La transposition informatique. Note sur un nouveau problème pour la didactique. In M. Artigue, et al. (Eds.), Vingt ans de didactique des mathématiques en France (pp. 364–370). Grenoble: La Pensée Sauvage.Google Scholar
  3. Cornu, B. (1986). Les Principaux Obstacles à l’Apprentissage de la Notion de Limite. Bulletin IREM-APMEP de Grenoble.Google Scholar
  4. Cornu, B. (1991). Limits. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Dordrecht: Kluwer Academic Publishers.Google Scholar
  5. Cuoco, A., & Goldenberg, P. (1992). Mathematical induction in a visual context. Interactive Learning Environments, 2(3–4), 181–203.CrossRefGoogle Scholar
  6. diSessa, A. (2000). Changing minds: Computers, learning, and literacy. Cambridge, MA.: MIT Press.Google Scholar
  7. Dreyfus, T., & Eisenberg, T. (1990). On difficulties with diagrams: Theoretical issues. In G. Booker, P. Cobb, & T. N. De Mendicuti (Eds.), Proceedings of the 14th International Conference on the Psychology of Mathematics Education (Vol. 1, pp. 27-31). Mexico.Google Scholar
  8. Dugdale, S. (1998). A spreadsheet investigation of sequences and series for middle grades through precalculus. Journal of Computers in Mathematics and Science Teaching, 17(2–3), 203–222.Google Scholar
  9. Duval, R. (1999). Representations, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings 21st Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 3–26). Columbus, Ohio: ERIC Clearinghouse for Science, Mathematics, & Environmental Education.Google Scholar
  10. Edwards, L. (1998). Embodying mathematics and science: Microworlds as representations. The Journal of Mathematical Behavior, 17, 53–78.CrossRefGoogle Scholar
  11. Espinoza, L., & Azcárate, C. (1995). A study on the secondary teaching system about the concept of limit. In Proceedings of the 19th International Conference for the Psychology of Mathematics Education (Vol. 2, pp. 11–17). Recife, Brazil.Google Scholar
  12. Ferrari, E., Laganà, A., Luzi, E., & Trovini, E. (1995). Il Concetto di Infinito nell’ Intuizione Matematica. L’Insegnamento della Matematica e delle Scienze Integrate, Paderno, Italy, 18B(3), 211–235.Google Scholar
  13. Fischbein, E. (1987). Intuition in science and mathematics. Dordrecht, Holland: Reidel.Google Scholar
  14. Fischbein, E., Tirosh, D., & Hess, P. (1979). The intuition of infinity. Educational Studies in Mathematics, 10, 3–40.CrossRefGoogle Scholar
  15. Galileo Galilei (1954). Dialogues concerning two new sciences (trans: Crew, H., & de Salvio, A.). New York: Dover Publications (Original work published 1638).Google Scholar
  16. Garbin, S., & Azcárate, C. (2002). Infinito Actual e Inconsistencias: Acerca de las incoherencias en los esquemas conceptuales de alumnos de 16–17 años. Enseñanza De Las Ciencias, 20(1), 87–113.Google Scholar
  17. Harel, I., & Papert, S. (Eds.) (1991). Constructionism. Norwood, NJ: Ablex Publishing Corporation.Google Scholar
  18. Harvey, B. (1997). Computer science logo style, Vols. 1–3. Cambridge, MA: MIT Press.Google Scholar
  19. Imaz, C. (1991). Infinitesimal models for real analysis. International Journal of Mathematical Education in Science and Technology, 22, 2.Google Scholar
  20. Jones, C. V. (1987). Las paradojas de Zenón y los primeros fundamentos de las matemáticas. Mathesis, 3(1), 3–14.Google Scholar
  21. Keisler, J. (1986). Elementary calculus: An approach using infinitesimals (2nd ed.). Boston, MA: Prindle, Weber & Schmidt. Retrieved 31 October, 2004, from
  22. Kissane, B. (2003). The calculator and the curriculum: The case of sequences and series. In W.-C. Yang, S.-C. Chu, T. de Alwis, & M.-G. Lee (Eds.), Proceedings of the 8th Asian Technology Conference in Mathematics: Technology Connecting Mathematics (pp. 357-366). Hsinchu, Taiwan: ATCM Inc.Google Scholar
  23. Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  24. Mamona-Downs, J. (1987). Students interpretations of some concepts of mathematical analysis. Doctoral Dissertation, Faculty of Mathematical Studies, University of Southampton.Google Scholar
  25. Monaghan, J. (2001). Young people’s ideas of infinity. Educational Studies in Mathematics, 48, 239–257.CrossRefGoogle Scholar
  26. Monaghan, J., Sun, S., & Tall, D. O. (1994). Construction of the limit concept with a computer algebra system. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th International Conference on the Psychology of Mathematics Education (pp. 279–286). Lisboa.Google Scholar
  27. Moreno, L., & Waldegg, G. (1991). The conceptual evolution of actual mathematical infinity. Educational Studies in Mathematics, 22(5), 211–231.Google Scholar
  28. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.Google Scholar
  29. Noss, R., & Hoyles, C. (1996). Windows on mathematical meanings. Learning cultures and computers. Dordrecht: Kluwer Academic Press.Google Scholar
  30. Noss R., & Hoyles, C. (2006). Exploring mathematics through construction and collaboration. In K. R. Sawyer (Ed.), Cambridge handbook of the learning sciences (pp. 389–405). Cambridge: Cambridge University Press.Google Scholar
  31. Núñez E., R. (1993). En Deçà du Transfini. Aspects psychocognitifs sous-jacents au concept d’infini en mathématiques (Vol. 4. Fribourg, Suisse: Éditions Universitaires).Google Scholar
  32. Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York: Basic Books.Google Scholar
  33. Robinson, A. (1974/1996). Non-standard analysis. Revised edition. Princeton, NJ: Princeton University Press. (Original work published 1974).Google Scholar
  34. Sacristán, A. I. (1991). Los Obstáculos de la intuición en el aprendizaje de procesos infinitos. Educación Matemática, 3(1), 5–18.Google Scholar
  35. Sacristán, A. I. (1997). Windows on the infinite: Creating meanings in a logo-based microworld. Doctoral Dissertation, Institute of Education, University of London, UK. Retrieved 8 December, 2007, from∼asacristan/PhDSacristan97.pdf.
  36. Schwarzenberger, R. L. E., & Tall, D. O. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49.Google Scholar
  37. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.CrossRefGoogle Scholar
  38. Sierpinska, A., & Viwegier, M. (1989). How and when attitudes towards mathematics and infinity become constituted into obstacles in students? In Proceedings of the 13th International Conference on the Psychology of Mathematics Education (Vol. 3, pp. 166–173). Paris, France.Google Scholar
  39. Struik, D. J. (1967). A concise history of mathematics. New York: Dover Publications.Google Scholar
  40. Tall, D. O. (1980). The notion of infinite measuring number and its relevance in the intuition of infinity. Educational Studies in Mathematics, 11, 271–284.CrossRefGoogle Scholar
  41. Tall, D. O. (1986). Building and testing a cognitive approach to the calculus using interactive computer graphics. Doctoral dissertation. University of Warwick, UK.Google Scholar
  42. Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199-238.CrossRefGoogle Scholar
  43. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151–169.CrossRefGoogle Scholar
  44. Tirosh, D., & Tsamir, P. (1996). The role of representations in students’ intuitive thinking about infinity. International Journal of Mathematics Education in Science and Technology, 27(1), 33–40.CrossRefGoogle Scholar
  45. Tsamir, P. (1999). The transition from the comparison of finite to the comparison of infinite sets: Teaching prospective teachers. Educational Studies in Mathematics, 38(1–3), 209–234.CrossRefGoogle Scholar
  46. Tsamir, P. (2001). When ‘The Same’ is not perceived as such: The case of infinite sets. Educational Studies in Mathematics, 48, 289–307.CrossRefGoogle Scholar
  47. Waldegg, G. (1987). Esquemas de Respuesta ante el Infinito Matemático. Transferencia de la Operatividad de lo Finito a lo Infinito. Doctoral dissertation. Centre for Research and Advanced Studies, Mexico.Google Scholar
  48. Waldegg, G. (1993). La comparaison des ensembles infinis: un cas de résistance à l’instruction. Annales de Didactique et de Sciences cognitives, 5, 19–36.Google Scholar
  49. Wilensky, U. (1991). Abstract meditations on the concrete, and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood, NJ: Ablex Publishing Corporation.Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematics EducationCentre for Research and Advanced Studies (Cinvestav)MexicoMexico
  2. 2.London Knowledge Lab, Institute of EducationUniversity of LondonLondonEngland, UK

Personalised recommendations