The historical roots of the limit notion: Cognitive development and the development of representation registers

Canadian Journal of Science, Mathematics and Technology Education

Volumn 5, Issue 4, 2005, Pages 453-468

  Bagni, Giorgio T ^a  

^a UNIVERSITY OF UDINE   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85032069143     PISSN: 14926156     EISSN: 19424051     Source Type: Journal    
DOI: 10.1080/14926150509556675     Document Type: Article

References (94)

1. AIbree, J. (1994). An alternative to the conventional wisdom of a term paper in history of mathematics survey courses. MAA Noies, 24, 175-182.
2. Anglin, W.S. (1992). Mathematics and history. Mathematical Intelligencer, 14(4), 6-12.
3. Anglin, W.S. (1994). Mathematics: A concise history and philosophy. Berlin: Springer.
4. Arrigo, G., & D’Amore, B. (1992). Infiniti. Milano: Angeli.
5. Artigue, M. (1998). L’évolution des problématiques en didactique de l’analyse. Recherches en didactique des mathématiques, 18(2), 231-262.
6. Artigue, M., Defouad, B., Dupérier, M., Juge, G., & Lagrange, J.B. (1997). L'intégration de calculatrices complexes dans l'enseignement des mathématiques au lycée (Rapport de recherche. Équipe DID1REM). Paris: IREM Paris 7.
7. Bagni, G.T. (1998). Dimostrare e convincere. Bollettino dei docenti di matematica del Canton Ticino, 36, 53-60.
8. Bos, H.J.M. (1975). Differentials, high-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1-90.
9. Bos, H.J.M. (1976). History of mathematics in the mathematics curriculum at Utrecht University. Historia Math, 3, 473-476.
10. Bostock, D. (1972). Aristotle, Zeno and the potential infinite. Proceedings of the Aristotelian Society, 73, 37-51.
11. Bottazzini, U. (1990). Il flauto di Hilbert. Storia della matematica moderna e contemporanea. Torino, Italy: UTET.
12. Bourbaki, N. (1963). Elementi di storia della matematica. Milano: Feltrinelli.
13. Boyer, C.B. (1982). Storia della matematica (A. Carugo, Trans.). Milano: Mondadori. (Original work published in 1968).
14. Carruccio, E. (1972). Matematiche elementari da un punto di vista superiore. Bologna, Italy: Pitagora.
15. Castelnuovo, G. (1938). Le origini del calcolo infinitesimale. Bologna, Italy: Zanichelli.
16. Cauchy, A.L. (1884-1897). Oeuvres complètes. Paris: Gauthier-Villars.
17. Chevallard, Y. (1985). La transposition didactique, du savoir savant au savoir enseigné. Grenoble, France: La Penseé Sauvage.
18. Confrey, J. (1992). Function probe. Santa Barbara, CA: Intellimation Library for Macintosh.
19. Cornu, B. (1980). Interference des modèles spontanés dans l’apprentissage de la notion de limite. Cahier du séminaire de didactique des mathématiques et informatique, 8, 57-83.
20. Cornu, B. (1981). Grandes Iignes de revolution historique de la notion de limite. Cahier du seminaire de didactique des mathematiques et de Vinformatique, 26, 305-326.
21. Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Dordrecht, The Netherlands: Kluwer Academic.
22. Davis, R., & Vinner, S. (1986). The notion oflimit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.
23. Dieudonné, J. (1989). L'arte dei numeri. Milano: Mondadori.
24. Dimarakis, I., & Gagatsis, A. (1997). Alcune difficoltà nella comprensione del concetto dilimite. La matematica e la sua didattica, 2, 132-149.
25. Dittrich, A.B. (1973). An experiment in teaching the history of mathematics. Mathematics Teacher, 66, 35-38.
26. Dubinsky, E. (1995). ISETL: A programming language for learning mathematics. Communications in Pure and Applied Mathematics, 48, 1-25.
27. Dunham, W. (1993). A mathematics seminar from the National Endowment for the Humanities. MAA Notes, 32, 177-182.
28. Edwards, C.H., Jr. (1994). The historical development of the calculus. Berlin: Springer.
29. Enriques, F. (1938). Le matematiche nella storia e nella cultura. Bologna, Italy: Zanichelli.
30. Euclid E. (1970). Elementi (A. Frajese & L. Maccioni, Eds.). Torino, Italy: UTET.
31. Euclid E. (1952). The thirteen books of Euclid’s Elements (Sir T.L. Heath, Trans.). In R.M. Hutchins (Ed.), The Great hooks of the western world (pp. 1-402). Chicago: Encyclopaedia Britannica.
32. Euler, L. (1755-1787). Institutiones calculi differentialis cum eius usu in analyst finitorum ac doctrina serierum (Vols. 1-2). Pavia, Italy: Galeati.
33. Euler, L. (1796). Introduction a I'analyse infinitesimale (Vols. 1-2). Paris: Barrois.
34. Fauvel, J., & van Maanen, J. (2000). History in the mathematics education: The ICMI study. Dordrecht, The Netherlands: Kluwer Academic.
35. Furinghetti, F. & Radford, L. (2002). Historical conceptual developments and the teaching of mathematics: From philogenesis and ontogenesis theory to classroom practice. In L. English (Ed.), Handbook of international research in mathematics education (pp. 631-654). Hillsdale, NJ: Lawrence Erlbaum.
36. Furinghetti, F., & Somaglia, A. (1997). Storia della matematica in classe. L’educazione matematica, Anno 18, Serie 5, 2, 1.
37. Geymonat, L. (1970). Storia del pensiero filosofico e scientifico. Milano: Garzanti.
38. Giusti, E. (1983). Analisi matematica (Vol. 1). Torino, Italy: Boringhieri.
39. Goldstine, H.H. (1995). A history of the calculus of variations from the 17th through the 19th century (Rev. ed.). Berlin: Springer.
40. Gray, E.M., & Tall, D.O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal of Research in Mathematics Education, 25(2), 115-141.
41. Grugnetti, L. (1992). L’histoire des mathématiques: une experience interdisciplinaire fondée sur l’histoire des mathématiques. Plot, 60, 17-21.
42. Grugnetti, L., & Rogers, L. (2000). Philosophical, multicultural and interdisciplinary issues. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education: The ICMI study (pp. 39-62). Dordrecht, The Netherlands: Kluwer Academic.
43. Jahnke, H.N. (1995). Historische Reflexion im Unterricht. Mathematica Didáctica, 18(2), 30-58.
44. Katz, VJ. (1986). Using history in teaching mathematics. For the Learning of Mathematics, 6, 13-19.
45. Kleiner, I. (1986). Famous problems in mathematics: An outline of a course. For the Learning of Mathematics, 6(1), 31-38.
46. Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.
47. Lakoff, G., & Nunez, R. (2000). Where mathematics come from? How the embodied mind brings mathematics into being. New York: Basic.
48. Laubenbacher, R.C., & Pengelley, D.J. (1992). Great problems of mathematics: A course based on original sources. American Mathematical Monthly, 99, 313-317.
49. Leibniz, G.W. (1849-1863). Mathematische Schriften (Vols. 1-7) (C.I. Gerhardt, Ed.). BerlinHalle: Ascher-Schmidt.
50. Loria, G. (1982). Storia delle matematiche dall’alba delle civiltà al tramonto del secolo XIX. Milano: Cisalpino-Goliardica. (Originai work published in 1929-1933)
51. Mamona, J. (1987). Students interpretations of some concepts of mathematical analysis. Unpublished doctoral dissertation, University of Southampton.
52. Mamona-Downs, J. (1990). Pupils’ interpretations of the limit concept: A comparison study between Greek and English. PME-14, 69-76.
53. Marchisotto, E.A. (1993). Teaching mathematics humanistically: A new look at an old friend. MAA Notes, 32, 183-190.
54. McKinzie, M., & Tuckey, C.D. (2001). High trigonometry, hyperreal numbers and Euler’s Analysis of infinities. Mathematics Magazine, 74(5), 339-368.
55. Menghini, M. (1982). Cavalieri e Leibniz: dagli indivisibili al differenziale. In O. Montaldo & L. Grugnetti (Eds.), La storia delle matematiche in Italia (pp. 385-394). Cagliari, Italy: Università Cagliari.
56. Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20-24.
57. Nemirovsky, R., & Noble, T. (1997). On mathematical visualization and the place where we live. Educational Studies in Mathematics, 33, 99-131.
58. Nobre, S. (Ed.). (1994). Meeting of the International Study Group on Relations between History and Pedagogy of Mathematics. Blumenau, Brasil: UNESP.
59. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235-250.
60. Piaget, J., & Garcia, R. (1983). Psychogenèse et histoire des sciences. Paris: Flammarion.
61. Rabardel, P. (1995). Les hommes et les technologies: Approche cognitive des instruments contemporains. Paris: A. Colin.
62. Robinson, A. (1974). Non-standard analysis. London: North-Holland.
63. Rufini, E. (1961). Il « Metodo » di Archimede e le origini del calcolo infinitesimale nell’antichità. Milano: Feltrinelli. (Original work published 1926)
64. Saccheri, G. (1904). Euclides ab omni naevo vindicatus (G. Boccardini, Ed.). Milano: Hoepli.
65. Schwarzenberger, R. (1980). Why calculus cannot be made easy? Mathematical Gazette, 64, 158-166.
66. Scriba, C.J. (1993). Die Rolle der Geschichte der Mathematik in der Ausbildung von Schülern und Lehrern. Jahresberichte der Deutschen Mathematiker-Vereinigung, 85, 113-128.
67. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflection on processes and objects as different sides of the same coins. Educational Studies in Mathematics, 22, 1-36.
68. Shenitzer, A. (1995). A topics course in mathematics. In Swetz et al. (Eds.), Learn from the masters (pp. 283-295). Washington, DC: Mathematical Association of America.
69. Sierpinska, A. (1985). Obstacle épistémologiques relatifs à la notion de limite. Recherches en didactique des mathématiques, 5(1), 5-68.
70. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.
71. Siu, M.K. (1995). Mathematical thinking and history of mathematics. In Swetz et al. (Eds.), Learn from the masters (pp. 279-282). Washington, DC: Mathematical Association of America.
72. Smith, D.E. (1959). A source book in mathematics. New York: Dover. (Original work published in 1929)
73. Stillwell, J. (1997). Mathematics and its history. Berlin: Springer.
74. Swetz, F.J. (1982). The use of projects in teaching the history of mathematics. Historia Mathematica, 9, 201-205.
75. Swetz, F.J. (1989). Using problems from the history of mathematics in classroom instruction. Mathematics Teacher, 82, 370-377.
76. Swetz, F.J. (1992). Some not so random thoughts about the history of mathematics, its teaching, learning and textbooks. Primus, 5(2), 97-107.
77. Swetz, F.J. (1995). To know and to teach: Mathematical pedagogy from a historical context. Educational Studies in Mathematics, 29, 73-88.
78. 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.
79. Tall, D. (1982). Elementary axioms and pictures for infinitesimal calculus. Bulletin of the IMA, 14, 43-48.
80. Tall, D. (1985). Understanding the calculus. Mathematical Teaching, 110, 49-53.
81. Tall, D. (1986). Graphic Calculus (Versions 1, 2, & 3). [BBC compatible software]. London: Glentop Press.
82. Tall, D. (1990a). A versatile approach to calculus and numerical methods. Teaching Mathematics and Its Applications, 9(3), 124-130.
83. Tall, D. (1990b). Inconsistencies in the learning of calculus and analysis. Focus on Learning Problems in Mathematics, 12, 49-64.
84. Tall, D. (1994, July). A versatile theory of visualization and symbolisation in mathematics. Paper presented at the 46th Conference of CIEAEM, Toulouse, France.
85. Tall, D. (1996). Function and calculus. In A.J. Bishop et al. (Eds.), International Handbook of Mathematics Education (pp. 289-325). Dordrecht, The Netherlands: Kluwer Academic.
86. Tall, D., McGowen, M., & DeMarois, P. (2000). The function machine as a cognitive root for building a rich concept image of the function concept. In N. Fernandez (Ed.), Proceedings of the twenty-second annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 247-254). Columbus, OH: ERIC Clearinghouse for Science, Mathematics and Environmental Education.
87. Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48(2/3), 199-238.
88. Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., et al. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and Technology Education, I, 81-104.
89. Todorov, T.D. (2001). Back to classics: Teaching limits through infinitesimals. International Journal of Mathematical Education in Science and Technology, 32(1), 1-20.
90. Trouche, L. (1996). Etude des rapports entre processus de conceptualisation et processus d'instrumentation. Thèse de doctorat, Université de Montpellier 2.
91. Tsamir, P., & Tirosh, D. (1992). Students’ awareness of inconsistent ideas about actual infinity. PME-16, 90-97.
92. Vitali, G. (1979). Limiti, serie, frazioni continue, prodotti infiniti. In L. Berzolari, G. Vivanti, & D. Gigli (Eds.), Enciclopedia delle matematiche elementari (Vol. 1, p. 2). Milano: Hoepli.
93. Williamson, B. (1979). History of mathematics: A course outline, Historia Mathematica, 6, 318-320.
94. Yerushalmy, M. (1997). Reaching the unreachable: technology and semantics of asymptotes. International Journal of Computers for Mathematical Learning, 2, 1-25.