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A Historical Study on the Continuity of Function - Focusing on Aristotle's Concept of Continuity and the Arithmetization of Analysis -  

Baek, Seung Ju (Gajaeul High School)
Choi, Younggi (Seoul National University)
Publication Information
Journal of Educational Research in Mathematics / v.27, no.4, 2017 , pp. 727-745 More about this Journal
Abstract
This study investigated the Aristotle's continuity and the historical development of continuity of function to explore the differences between the concepts of mathematics and students' thinking about continuity of functions. Aristotle, who sought the essence of continuity, characterized continuity as an 'indivisible unit as a whole.' Before the nineteenth century, mathematicians considered the continuity of functions based on space, and after the arithmetization of nineteenth century modern ${\epsilon}-{\delta}$ definition appeared. Some scholars thought the process was revolutionary. Students tended to think of the continuity of functions similar to that of Aristotle and mathematicians before the arithmetization, and it is inappropriate to regard students' conceptions simply as errors. This study on the continuity of functions examined that some conceptions which have been perceived as misconceptions of students could be viewed as paradigmatic thoughts rather than as errors.
Keywords
continuity of function; Aristotle; indivisible unit as a whole; arithmetization; paradigmatic thought;
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1 강현영.이동환(2007). 수학교육에서 상보성. 수학교육학연구, 17(4), 437-452.
2 국립국어연구원(1999). 표준국어대사전. 서울 : 두산동아.
3 김선희(2004). 수학적 지식 점유에 관한 기호학적 고찰. 이화여자대학교 대학원 박사학위논문.
4 백승주(2015). 함수의 연속 개념의 역사적 고찰. 서울대학교 대학원 석사학위 논문.
5 유재민(2014). 아리스토텔레스의 연속 이론 연구. 서울대학교 대학원 박사학위 논문.
6 이경화.신보미(2005). 상위 집단 학생들의 함수의 연속 개념 이해. 수학교육학연구, 15(1), 39-569.
7 이지현(2011). 학교수학에서 대학수학으로 공리적 방법의 이행에 관한 연구. 서울대학교 대학원 박사학위 논문.
8 이진영(2011). 교수학적 변환의 관점에서 한 점에서 함수의 연속.불연속, 연속함수 정의의 검토. 이화여자대학교 대학원 석사학위논문.
9 정동명.조승제(1992). 실해석학개론. 서울: 경문사.
10 아리스토텔레스(1963). Aristotle's Categories and De Interpretatione, (Translated with note by Ackrill, J. L.). Oxford University Press.
11 아리스토텔레스(2008a). 범주들.명제에 관하여. (김진성 역). 서울: 이제이북스.
12 아리스토텔레스(2008b). Themistius: On Aristotle Physics 5-8. (Translated by Robert B. Todd). London: Gerald Duckworth & Co,Ltd.
13 Boyer, C. B. (1949). The history of the calculus and its conceptual development. New York: Dover Publications.
14 Bridgers, L. C. (2007). Conceptions of continuity: An investigation of high school calculus teachers and their students. Unpublished doctoral dissertation, Syracuse University, New York.
15 Cauchy, A. L. (2009). Cauchy's Cours d'analyse. (Translated by Bradley, Robert E. & Sandifer, C. Edward). New York: Springer. (The original edition was published in 1821)
16 Dimitric, R. M. (2003). Is the function continuous at 0?. arXiv:1210.2930 [math.GM].
17 Euler, L. (2000). Foundations of Differential Calculus. (Translated by John D. Blanton). New York: Springer. (The original edition was published in 1755)
18 Ely, R. (2007). Student obstacles and historical obstacles to foundational concepts of calculus. Unpublished doctoral dissertation, university of Wisconsin-Madison, Madison, Wisconsin.
19 Euler, L. (1988). Introduction to Analysis of the Infinite: Book I. (Translated by John D. Blanton). New York: Springer-Verlag. (The original edition was published in 1748)
20 정연준.김재홍(2013). 함수의 연속성 개념의 역사적 발달과정 분석-직관적 지도의 보완을 중심으로- 수학교육학연구, 23(4), 567-584.
21 Ferraro, G. (2001). Analytical Symbols and Geometrical Figures in Eighteenth-Century Calculus. Studies in History and Philosophy of Science, 32(3), 535-555.   DOI
22 Kleiner, I. (1989). Evolution of the function concept: a brief survey. College mathematics journal, 20, 282-300.   DOI
23 Grabiner, J. V. (1983). Who gave you the epsilon? Cauchy and the origins of rigorous calculus. American Mathematical Monthly, 90(3), 185-194   DOI
24 Jourdain, E. B. (1913). The Origin of Cauchy's Conceptions of a Definite Integral and of the Continuity of a Function. Isis, 1(4), 1-703.   DOI
25 Klein, F. (1896). The Arithmetizing of Mathematics. Translated by Isabel Maddison, Bryn Mawr College. Bulletin of The American Mathematical Society, 2(8), 241-249.   DOI
26 Lakoff, G., & Nunez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
27 Koetsier, T. (1991). Lakatos' Philosophy of Mathematics: A Historical Approach. Amsterdam: Elsevier Science Publishers B.V.
28 Lakatos. I. (1996). 수학, 과학 그리고 인식론. (이영애 역). 서울: 민음사. (영어원작은 1978년 출판)
29 Lakatos. I. (2001). 수학적 발견의 논리. (우정호 역). 서울: 아르케. (영어원작은 1976년 출판)
30 Nunez, R. E., & Lakoff, G. (1998). What Did Weierstrass Really Define? The Cognitive Structure of Natural and continuity. Mathematical Cognition, 4(2), 85-101.   DOI
31 Pierpont, J. (1899). On the Arithmetization of Mathematics. Bulletin of the American Mathematical Society, 5, 394-406.   DOI
32 Russell, B. (2002). 수리철학의 기초. (임정대 역). 서울: 경문사. (영어원작은 1919년 출판)
33 Schubring, G. (2005). Conflicts between generalization, rigor, and intuition. New York: Springer.
34 Shipman, B. A. (2012). A comparative study of definitions on limit and continuity of functions. Primus, 22(8), 609-633.   DOI
35 Youschkevitch, A. P. (1976). The concept of function up to the middle of the 19th century. Archive for History of Exact Sciences, 16(1), 37-85.   DOI
36 Tall, D. (2008). The Transition to Formal Thinking in mathematics. Mathematics Education Research Journal, 20(2), 5-24.   DOI
37 Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy's conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86, 97-124.   DOI
38 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.   DOI
39 White, M. J. (1988). On Continuity: Aristotle versus Topology? History and philosophy of logic, 9(1), 1-12.   DOI
40 Wilder, R. L. (1967). The Role of the Axiomatic Method. The American Mathematical Monthly, 74(2), 115-127.   DOI
41 박달원.홍순상.신민영(2012). 연속함수에 대한 고등학교 교과서의 정의와 고등학생들의 이해. 한국학교수학회논문집, 15(3), 453-456.