Browse > Article

The Metaphorical Model of Archimedes' Idea on the Sum of Geometrical Series  

Lee, Seoung Woo (Seoul Science High School)
Publication Information
School Mathematics / v.18, no.1, 2016 , pp. 215-229 More about this Journal
Abstract
This study aims to identify Archimedes' idea used while proving proposition 23 in 'Quadrature of the Parabola' and to provide an alternative way for finding the sum of geometric series without applying the concept of limit by extending the idea though metaphor. This metaphorical model is characterized as static and thus can be complimentary to the dynamic aspect of limit concept adopted in Korean high school mathematics textbooks. In addition, middle school students can understand $0.999{\cdots}=1$ with this model in a structural way differently from the operative one suggested in Korean middle school mathematics textbooks. In this respect, I argue that the metaphorical model can be an useful educational tool for Korean secondary students to overcome epistemological obstacles inherent in the concepts of infinity and limit by making it possible to transfer from geometrical context to algebraic context.
Keywords
Archimedes' idea; Sum of Geometric Series; Metaphorical Model; Epistemological Obstacles;
Citations & Related Records
연도 인용수 순위
  • Reference
1 조한혁.최영기 (1999). 정적 동적 관점에서의 순환소수. 학교수학, 1(2), 605-615.
2 황선욱 외 8인 (2015). 중학교 수학2. 서울: 신사고
3 황선욱 외 10인 (2015). 미적분 I. 서울: 신사고
4 Apostol, T. M. (1981). Mathematical Analysis(2nd ed.). Addison-Wesley Publishing Company.
5 Borasi, R. (1994). Capitalizing on Errors as "Springboards for Inquiry": A Teaching Experiment. Journal for Research in Mathematics Education, 25(2), 166-208.   DOI
6 Boyer, C. B. (1959). The history of the calculus and its conceptual development: The concepts of the calculus. Mineola, NY: Dover Publications.
7 Cornu, B. (1991). Limits. In D. Tall(Ed.), Advanced Mathematical Thinking(pp. 153-166). Dordrecht, The Netherlands: Kluwer Academic Publishers.
8 Christianidis, J., & Demis, A. (2010). Archimedes' quadratures. In S. A. Paipetis, & M. Ceccarelli (Eds.), The Genius of Archimedes-23 Centuries of Influence on Mathematics, Science and Engineering (pp. 57-68). Dordrecht, The Netherlands: Springer.
9 DeSouza, C. E. (2012) The Greek method of exhaustion: Leading the way to modern integration. (Master degree paper, Ohio State University).
10 Edwards, C. J. (1979). The historical development of the calculus. NY: Springer-Verlag.
11 Judith V. Grabiner, J. V. (1983). Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus. The American Mathematical Monthly, 90(3), 185-194.   DOI
12 Heath, T. (2002). The Works of Archimedes. Mineola, NY: Dover Publications.
13 Heath, T. (2003). A Manual of Greek Mathematics. Mineola, NY: Dover Publications.
14 Katz, V. J. (1993). A history of mathematics: An introduction. NY: HarperCollins College Publishers.
15 White, M. J. (1992). The continuous and the discrete: Ancient physical theories from a contemporary perspective. Oxford: Oxford University Press.
16 Nelsen, R. B. (2000). Proofs Without Words II: More Exercises in Visual Thinking. Washington, DC: MAA
17 Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36.   DOI
18 Toeplitz, O. (1963). The calculus: a genetic approach. Chicago: University of Chicago Press.
19 Knopp, K. (1956). Infinite sequences and series. Mineola, NY: Dover Publications.
20 King, D. Albert. (1968). A Hisotyr of Infinite Series (Doctoral dissertation Paper, Peabody College for Teachers of Vanderbilt University).
21 Randolph, J. F. (1957). Limits. In NCTM (Ed.) Insights into Modern Mathematics (pp.200-240). Washington, DC: NCTM.
22 이승우 (2015b). 학교수학적 지식의 성장: 고등학교 영재 학생들의 위키(Wiki) 기반 협력 문제해결 활동을 중심으로. 수학교육학연구, 25(4), 717-754
23 이승우 (2015a). 학교수학이란 무엇인가? 수학교육학연구, 25(3), 381-405