Educational Studies in Mathematics

, Volume 61, Issue 3, pp 395–402 | Cite as

Convergence of Numerical Sequences – a Commentary on “the Vice: Some Historically Inspired and Proof Generated Steps to Limits of Sequences” By R.P. Burn

  • Analía Bergé


Burn (2005) proposes a genetic approach to teaching limits of numerical sequences. The article includes an explanation of the Method of Exhaustion,1 a generalization of this method, and a description of how this method was used for obtaining areas and lengths in the seventeenth century. The author uses these historical and mathematical analyses as a basis for proposing an alternative definition of the limit of a sequence. The paper focuses on the fine mathematical and historical detail of the notion of limit. Reading it made me reflect on the explicitly or implicitly involved didactical aspects, which I would like to share with ESM readers.


Mathematical Analysis Genetic Approach Seventeenth Century Historical Detail Generate Step 
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  1. Bergé, A. and Sessa, C.: 2003, ‘Completitud y continuidad revisadas a través de 23 siglos. Aportes a una investigación didáctica’, Revista Latinoamericana de Investigación en Matemática educativa 6(3), 163–197.Google Scholar
  2. Bkouche, R.: 1997, ‘Épistemologie, histoire et enseignement de mathématiques’, For the Learning of Mathematics 17(1), 34–42.Google Scholar
  3. Brousseau, G.: 1997, Theory of Didactical Situations in Mathematics, Kluwer Academic Publishers, Dordrecht.Google Scholar
  4. Chevallard, Y.: 1998, ‘Analyse des pratiques enseignantes et Didactique des Mathématiques: l’approche antropologique’, En Actes de l’École d’été de la Rochelle, 91–118.Google Scholar
  5. Cornu, B.: 1991, ‘Limits’, in D. Tall (ed.), Advanced Mathematical Thinking, chapter 10, Kluwer Academic Press, Dordrecht.Google Scholar
  6. Davis, R.B. and Vinner, S.: 1986, ‘The notion of limit: Some seemingly unavoidable misconception stages’, Journal of Mathematical Behavior 5, 281–303.Google Scholar
  7. Dubinsky, E., Weller, K., McDonald, M. and Brown, A.: 2005, ‘Some historical issues and paradoxes regarding the concept of infinity: An apos-based analysis: Part 1’, Educational Studies in Mathematics 58(3), 335–359.CrossRefGoogle Scholar
  8. Gobierno de la Ciudad de Buenos Aires, Secretaría de Educación, Dirección General de Planeamiento, Dirección de Curricula, Prediseño Curricular para la Educación General Básica: 1999, Marco General.Google Scholar
  9. Mamona-Downs, J.: 2001, ‘Letting the intuitive bear on the formal; a didactical approach for the understanding of the limit of a sequence’, Educational Studies in Mathematics 48(2/3), 259–288.Google Scholar
  10. Radford, L.: 1997, ‘On psychology, historical epistemology, and the teaching of mathematics: Towards a socio-cultural history of mathematics’, For the Learning of Mathematics 17(1), 26–32.Google Scholar
  11. Sierpinska, A.: 1990, ‘Some remarks on understanding in mathematics’, For the Learning of Mathematics 10(3), 24–41.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Concordia University Department of Mathematics and StatisticsMontrealCanada

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