위상에 대한 교수학적 접근-수렴성과 연속성을 중심으로-
Didactical Approach on Topology -Centered on convergence and continuity-
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The purpose of this study is to show that the topology is closely related to some subjects learned in school mathematics and then to give motivations for learning of the topology. To do this, it is showed that the topology is an abstracted device that deal with structure of limit and continuity introduced in school mathematics. This study took a literature study. The results of this study are as follows. First, the formal definition of general topology to structure open sets was examined. The nearness relation together with the closure operation was introduced and used to characterize for construction of general topology. Second, as definitions for continuity of function, we considered the intuitive definition, definition, structured definitions using open intervals and definition using open sets and then we investigated their roles. We also examined equivalent definition using the nearness relation which is helpful to understand continuity of function. Third, the sequence and its limit are treated in terms of continuous functions having the set of natural numbers and its extended set as domains. From these, it can be concluded that the convergence of sequence and the continuity of function are identified as functions that preserve the nearness relation and that the topology is a specialized tool for dealing with convergence and continuity.
Abstract
The purpose of this study is to show that the topology is closely related to some subjects learned in school mathematics and then to give motivations for learning of the topology. To do this, it is showed that the topology is an abstracted device that deal with structure of limit and continuity introduced in school mathematics. This study took a literature study. The results of this study are as follows. First, the formal definition of general topology to structure open sets was examined. The nearness relation together with the closure operation was introduced and used to characterize for construction of general topology. Second, as definitions for continuity of function, we considered the intuitive definition, definition, structured definitions using open intervals and definition using open sets and then we investigated their roles. We also examined equivalent definition using the nearness relation which is helpful to understand continuity of function. Third, the sequence and its limit are treated in terms of continuous functions having the set of natural numbers and its extended set as domains. From these, it can be concluded that the convergence of sequence and the continuity of function are identified as functions that preserve the nearness relation and that the topology is a specialized tool for dealing with convergence and continuity.
참고문헌 (22)
- "2011", ~
- 교육부, "수학과 교육과정(교육부 고시 제 2015-74호)", 교육부, ~ (2015)
- 김남희, "예비교사와 현직 교사를 위한 수학교육과정과 교재연구" (2017), 경문사
- 김진환, "학교수학과 학문수학에서의 연속성 개념 정의의 분석", 수학교육학연구, 27(3), pp.375~389 (2017), 대한수학교육학회
- 박선용, "극한과 무한집합의 상호작용과 그 교육적 시사점에 대한 역사적 연구", 한국수학사학회지, 31(2), pp.73~91 (2018), 한국수학사학회[DOI]
- 우정호, "수학교육학 연구방법론" (2006), 경문사
- 우정호, "학교수학의 교육적 기초" (2008), 서울대학교출판부
- 우정호, "수학 학습-지도 원리와 방법" (2010), 서울대학교출판문화원
- Atanasov, A., "Topology and Continuity", Columbia Science Review, 6(2), pp.33~35 (2010)
- Bartle, R. G., "Introduction to real analysis" (2011), John Wiley & Sons
- Cameron, P., "Nearness-a better approach to continuity and limits", Amer. Math. Monthly, 81, pp.739~745 (1974)
- Foerster, P. A., "Calculus: concepts and applications" (2010), Key Curriculum Press
- Gauld, D. B., "Nearness-a better approach to topology", Math. Chronicle, 7, pp.84~90 (1977)
- Harcharras, A, "Calculus connections: Mathematics for middle school teachers" (2007), Pearson Prentice Hall
- Hocking, J. G., "Topology" (1961), Addison-Wesley Publishing Company, Inc
- Janvier, C., "Problem of representation in teaching and learning mathematics", pp.27~32 (1987), Lawrence Erlbaum Associates
- Kaput, J. J., "Problems of representation in mathematics learning and problem solving", pp.159~196 (1987), Lawrence Erlbaum Associates
- Lipschutz, S, "Schaum’s outlines General Topology" (2012), McGraw Hill
- Moore, G. H., "The emrgence of open sets, closed sets, and limit points in analysis and topology", Historia Mathematica, 35, pp.220~241 (2008)
- Munkres, J. R., "Topology-a first course" (1975), Prentice-Hall
- "History topic: A history of topology, MacTutor History of Mathematics" (1996)[URL]
- "Calculus Early Transcendentals(8E)" (2016)[URL]