You are here

Teaching the Fundamental Theorem of Calculus: A Historical Reflection - References

Author(s): 
Omar A. Hernandez Rodriguez (University of Puerto Rico) and Jorge M. Lopez Fernandez (University of Puerto Rico)

[1] M. Artigue, “Analysis.” In D. Tall (Ed.), Advanced Mathematical Thinking, Kluwer Academic Publishers, Dordrecht, 1992. pp. 167-198.

[2] M. E. Baron, The Origins of Infinitesimal Calculus, Dover Publications, New York, 1969.

[3] N. Bourbaki, Elements of Mathematics, Functions of a Real Variable, Elementary Theory, Springer, Heidelberg, 2003.

[4] R. E. Bradley and C. Sandifer, Cauchy’s Cours d’Analyse, An Annotated Translation, Springer Verlag, Doordrecht, 2009.

[5] D. M. Bressoud, “Historical reflections on teaching the fundamental theorem of integral calculus,” American Mathematical Monthly 118 (2011), pp. 99–115.

[6] F. Cajori, “The history of notations of the calculus,” Annals of Mathematics, Second Series 25 (1923), pp. 1–46.

[7] F. Cajori, A History of Mathematics, The Macmillan Company, New York, 1931.

[8] A. L. Cauchy, Résumé des Leçons Données a L’Ecole Royale Polytechnique, Oevres Complétes, vol. IV of 11, Gauthier-Villard, Paris, 1899.

[9] J. M. T. Child, The Geometrical Lectures of Isaac Barrow, Translated with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus., no. 3 in Series of Classics of Science and Philosophy, The Open Court Publishing Company, Chicago, 1916.

[10] B. Cornu, “Limits.In D. Tall (Ed.), The transition to advanced mathematical thinking: Functions, limits, infinity and proofs, Kluwer Academic Publishers, Dordrecht, 1992. pp. 153-166.

[11] J. Cottrill, D. Nichols, K. Schwingendorf, K. Thomas, and D. Vidakovic, “Understanding the limit concept: Beginning with a coordinated process schema,” Journal of Mathematical Behavior 15 (1996), pp. 167–192.

[12] R. B. Davis and S. Vinner, “The notion of limit: Some seemingly unavoidable misconception stages,” Journal of Mathematical Behavior (1986), pp. 281–303.

[13] G. Ervynck, “Conceptual difficulties for first year university students in acquisition for the notion of limit of a function.” In L. P. Mendoza and E. R. Williams (Eds.), Canadian Mathematics Education Study Group: Proceedings of the Annual Meeting, Kingston, Ontario: Memorial University of Newfoundland, 1988. pp. 330-333.

[14] L. Euler, Introductio in analysin infinitorum, vol. 1, Real Sociedad Matemática Española, Sevilla, edición facsimilar, “thales” ed., 1748.

[15] L. Euler, Introductio in analysin infinitorum, Vol. 2; Introduction to the Analysis of the Infinite, Book II (translation of vol. 2), Springer Verlag, New York, 1748.

[16] L. Euler, Institutiones calculi differentialis, 1755; Foundations of Differential Calculus (translation), Springer Verlag, New York, 2000.

[17] H. Freudenthal, Mathematics as an Educational Task, D. Reidel Publishing Co., Doordrecht, 1973.

[18] H. Freudenthal, Didactical Phenomenology of Mathematical Structures, D. Reidel Publishing Co., Doordrecht, 1999.

[19] M. N. Fried, “Some reflections on Hernández and López’s reflections on the chain rule,” The Montana Mathematics Enthusiast 7 (2010), pp. 333–338.

[20] L. Gillman, “An axiomatic approach to the integral,” American Mathematical Monthly 100 (1993), pp. 16–25. Available at http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2913

[21] L. Gillman and R. H. MacDowell, Calculus, W. W. Norton, New York, 1973.

[22] L. Gillman and R. H. MacDowell, Calculus, W. W. Norton, New York, second ed., 1978.

[23] N. Guicciardini, Isaac Newton on Mathematical Certainty and Method, The MIT Press, Cambridge, Massachusetts, London, 2009.

[24] H. Hahn and A. Rosenthal, Set Functions, The University of New Mexico Press, 1948.

[25] O. Hernández Rodríguez and J. López Fernández, “A semiotic reflexion on the didactics of the chain rule,” The Montana Mathematics Enthusiast 7 (2010), pp. 321–332.

[26] V. J. Katz, A History of Mathematics, An Introduction, Addison Wesley, Pearson, New York, third edition, 2009.

[27] J. L. Lagrange, Theorie des fonctions analytiques, L’Imprimerie de la republique, Paris, 1797.

[28] S. Lang, A First Course in Calculus, Addison Wesley, third printing (1974), 1968.

[29] R. Laubenbacher and D. Pengelley, Mathematical Expeditions, Chronicles by the Explorers, Undergraduate Texts in Mathematics, Springer Verlag, 1999.

[30] L. Le and D. Tall, “Constructing different concept images of sequences and limits by programming,” Proceedings of the Seventeenth Conference for the Psychology of Mathematics Education, Tsukuba, Japan (1993), pp. 41–48.

[31] A. Leahy, “An Introduction to James Gregory’s Geometriae Pars Universalis,” Proceedings of the Eighth Midwest History of Mathematics Conference (2000).

[32] A. Leahy, “A Euclidean Approach to the FTC,” Loci: Convergence (2004). DOI: 10.4169/loci002156 http://mathdl.maa.org/mathDL/46/?pa=content&sa=viewDocument&nodeId=2156

[33] H. Levi, Polynomials, Power Series and Calculus, Van Nostrand Company, Inc., Princeton, 1968.

[34] G. F. A. L’Hospital, Analyse des infiniment petits pour l’ intelligence des lignes courbes, L’Imprimerie Royale, Paris, 1696.

[35] G. F. A. L’Hospital, The method of fluxions both direct and inverse (translated by Edmund Stone), printed for William Innys, London, 1730.

[36] J. Mamona-Downs, “Pupils’ interpretations of limit concept: A comparison between Greeks and English,” Proceedings of the Fourteenth Conference for the Psychology of Mathematics Education, Mexico City, Mexico (1990), pp. 69–76.

[37] J. J. O'Connor and E. F. Robertson, "Stanislaw Saks," MacTutor History of Mathematics Archive, 2000, http://www.gap-system.org/~history/Biographies/Saks.html

[38] J. J. O'Connor and E. F. Robertson, "Hans Hahn," MacTutor History of Mathematics Archive, 2006, http://www.gap-system.org/~history/Biographies/Hahn.html

[39] A. Robert, “L’acquisition de la notion de convergence des suites numeriques dans l’enseignement superieur,” Reserches en Didactique des Mathematiques 3 (1982), pp. 307–341.

[40] A. Sierpinska, “Humanities students and epistemological obstacles related to limits,” Educational Studies in Mathematics (1987), pp. 371–397.

[41] D. J. Struik, A Source Book in Mathematics, 1200-1800, Dover Publications Inc., New York, 1969.

[42] D. O. Tall, “The transition to advanced mathematical thinking: Functions, limits, infinity and proof.” In D. A. Grows (Ed.), The Handbook of research on mathematics teaching and learning, Macmillan, New York, 1992, pp. 495-511.

[43] D. O. Tall and S. Vinner, “Concept image and concept definition in mathematics, with particular reference to limits and continuity,” Educational Studies in Mathematics (1981), pp. 151–169.

[44] D. T. Whiteside, “Patterns of mathematical thought in the latter seventeenth century,” Archive for History of Exact Sciences 1 (1961), pp. 179–388.

[45] S. R. Williams, “Model of limit held by college calculus students,” Journal for Research in Mathematics Education (1991), pp. 219–236.

Omar A. Hernandez Rodriguez (University of Puerto Rico) and Jorge M. Lopez Fernandez (University of Puerto Rico), "Teaching the Fundamental Theorem of Calculus: A Historical Reflection - References," Loci (January 2012), DOI:10.4169/loci003803

Dummy View - NOT TO BE DELETED