AN HISTORICAL – EPISTEMOLOGICAL ANALYSIS OF FUNCTION
Main Article Content
Abstract
This article presents a synthesis of historical epistemological analysis and adds some new results on the formation and development of functions; identifying concepts influencing the development process and epistemological characteristics of functions. The research was conducted using the method of historical epistemological analysis of documents on the history of Calculus and real functions. The results show that the function developed in six periods: the Ancient period, from the Middle Ages to the end of the 15th century, the Renaissance, the 18th century, the 19th century, and from the 20th century to the present. The conceptions of geometry, algebra, calculus, metric, topology, and arithmetization have strongly influenced the formation and development of functions. In addition, an obstacle to the historical epistemology of functions is the distinction between the definition and representation of functions. The research results contribute to the epistemological analysis of the history of mathematics and further research on the obstacles faced by pupils and students when learning the concept of functions.
Keywords
analytic expression, function, function’s definition, historical epistemological analysis, obtacle
Article Details
References
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https://www.researchgate.net/publication/242490242_Operational_origins_of_mathematical_objects_and_the_quandary_of_reification-The_case_of_function
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Stallings, L. (2000). A Brief History of Algebraic Notation. School Science and Mathematics, 100(5), 230-235. http://doi.org/10.1111/j.1949-8594.2000.tb17262.x
Youschkevitch, A. P. (1981). Le concept de fonction jusqu’au milieu du XIXe siècle [The concept of function until the mid-19th century]. In Fragments of the History of Mathematics, Brochure A.P.M.E.P (Vol. 1, pp. 7-68). https://www.apmep.fr/IMG/pdf/0041_fragments_d_histoire_des_mathematiques_i_apmep_1981.pdf
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Burnett-Bradshaw, C. S. (2007). From functions as process to functions as object: a review of reification and encapsulation. [Thesis for Doctor of Philosophy in Mathematics Education
Tufts University]. https://dl.tufts.edu/concern/pdfs/j6731f78v
Chauvat, G. (1999). Courbes et fonctions au collège [Curves and functions in college]. Petit x, 51,
23-44. https://publimath.univ-irem.fr/biblio/IGR99012.htm
Chevallard, Y. (1985/1991). La transposition didactique, du savoir savant au savoir enseigné [Didactic transposition, from scholarly knowledge to taught knowledge]. la pensée sauvage.
Dhombres, J. (1987). Sur un texte d’Euler relatif à une équation fonctionnelle. Archaïsme, pédagogie, et syle d’écriture [On an Euler text relating to a functional equation. Archaism, pedagogy and writing style]. Historia Scientiarum: International journal of the Science Society of Japan, (33), 77-99. https://dl.ndl.go.jp/pid/11023744/1/52
Douady, R. (1984). French (Jeux de cadres et dialectiques outil-objet dans l'enseignement des Mathématiques. Une réalisation dans tout le cursus primaire... Histoire et perspectives sur les mathématiques [math.HO]. [Games with frame games and tool-object dialectics in the teaching of Mathematics. An achievement throughout the primary curriculum... History and perspectives on mathematics [math.HO]]. University of Paris VII. https://theses.hal.science/tel-01250665
Douady, R. (1986). Games with frames and the dialectic tool-object . Recherches en didactique des mathématiques, 7(2), 5-31. https://revue-rdm.com/1986/jeux-de-cadres-et-dialectique/
Ferraro, G. (2000). Jeux de cadres et dialectique outil-objet [Functions, Functional Relations, and the Laws of Continuity in Euler]. Historia Mathematica, 27(2), 107-132. http://doi.org/10.1006/hmat.2000.2278
Grattan-Guinness, I. (1970). The Development of the Foundations of Mathematical Analysis from Euler to Riemann. M.I.T.
Janvier, C. (1998). The Notion of Chronicle as an Epistemological Obstacle to the Concept of Function. Journal of Mathematical Behavior, 17(1), 79-103.
https://doi.org/10.1016/S0732-3123(99)80062-5
Klein, F., Menghini, M., & Schubring, G. (2016). Elementary mathematics from a higher standpoint.
Springer.
Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282-300. http://doi.org/10.2307/2686848
Malik, M. A. (1980). Historical and pedagogical aspects of the definition of function. International Journal of Mathematics Education in Science and Technology, 11(4), 489-492. http://doi.org/10.1080/0020739800110404
Marsden, J. E., & Hoffman, M. J. (1993). Elementary classical analysis (2nd ed.). W. H. Freeman and Company.
Mawfik, N., Boukhssimi, D., & Lamrabet, D. (2001). Genèse historique de la notion de continuité d’une fonction. [Historical genesis of the notion of continuity of a function]. Bulletin Mathematical Association of Quebec, XLI(1), 31-44. https://www.amq.math.ca/ancien/archives/2001/1/2001-1-part9.pdf
Mendes, I. A. (2021). Historical Creativities for the Teaching of Functions and Infinitesimal Calculus. International Electronic Journal of Mathematics Education, 16(2), em0629. https://doi.org/10.29333/iejme/10876
Murillo Lopez, S. (2008). Étude d'une pratique ordinaire face à un obstacle didactique: La correction en classe de mathématiques dans le cas de la fonction réciproque [Study of an ordinary practice faced with a didactic obstacle: Correction in mathematics class in the case of the reciprocal function]. [Doctoral thesis, University of Toulouse].
National Council of Teachers of Mathematics. (NCTM) (1989). Curriculum and Evaluation Standards for School Mathematics. NCTM.
O’Connor, J. J. & Robertson, E. F. (2005). History topic: The function concept. In MacTutuor History of Mathematics. http://www-history.mcs.st-andrews.ac.uk/HistTopics/Functions.html
Panza, M. (2007). Euler’s introductio in analysin infinitorum and the program of algebraic analysis: Quantities, functions and numerical partitions [Euler's Introductio in analysin infinitorum and the program of algebraic analysis: quantities, functions and numerical partitions]. In R. Backer (Ed.), (pp.119-166). The Kendrick Press. https://hal.science/halshs-00162383/
Parmentier, M. (1989). La naissance du calcul différentiel de Leibniz [Leibniz’s The birth of differential calculus]. Vrin.
Petri B., & Schappacher N. (2007). On Arithmetization. In Goldstein C., Schappacher N., Schwermer J. (eds) The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae. Springer. https://doi.org/10.1007/978-3-540-34720-0_12
Ponte, J. P.(1992). The history of the concept of function and some educational implications. The Mathmeatics Educator, 3, 3-8. https://www.researchgate.net/publication/251211596_The_history_of_the_concept_of_function_and_some_educational_implications
Rogers, L, & Pope, S. (2016) Working Group report: A brief history of functions for mathematics educators. Adams. G. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 36(2), 76-81. https://bsrlm.org.uk/wp-content/uploads/2016/11/BSRLM-CP-36-2-14.pdf
Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification – the case of function. In E. Dubinsky & G. Harel (Eds.). The Concept of Function: Aspects of Epistemology and Pedagogy (pp. 59-84). Mathematical Association of America.
https://www.researchgate.net/publication/242490242_Operational_origins_of_mathematical_objects_and_the_quandary_of_reification-The_case_of_function
Sierpinska, A. (1987). On the relativity of Errors [Sur la relativité des Erreurs. Dans]. In Proceedings of the 39th CIEAEM meeting (pp. 70-87). Editions of the University of Sherbrooke.
Sierpinska, A. (1992). On Understanding the Notion of Function. Dans Harel, G. et Dubinsky, E. (éd.) The Concept of Function. Aspects of Epistemology and Pedagogy (pp. 25-58). Mathematical Association of America. https://www.researchgate.net/publication/238287243_On_understanding_the_notion_of_function
Stallings, L. (2000). A Brief History of Algebraic Notation. School Science and Mathematics, 100(5), 230-235. http://doi.org/10.1111/j.1949-8594.2000.tb17262.x
Youschkevitch, A. P. (1981). Le concept de fonction jusqu’au milieu du XIXe siècle [The concept of function until the mid-19th century]. In Fragments of the History of Mathematics, Brochure A.P.M.E.P (Vol. 1, pp. 7-68). https://www.apmep.fr/IMG/pdf/0041_fragments_d_histoire_des_mathematiques_i_apmep_1981.pdf
Gill, N. S. (2017). Babylonian Table of Squares. ThoughtCo. https://www.thoughtco.com/babylonian-table-of-squares-116682
Vnexpress. (2017). Bang luong giac 3700 nam cua nguoi Babylon [3700-year-old Babylonian trigonometry table]. https://vnexpress.net/bang-luong-giac-3700-nam-cua-nguoi-babylon-3633006.html