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A Historical and Mathematical Analysis on the Radian  

Yoo, Jaegeun (Graduate School, Seoul National University)
Lee, Kyeong-Hwa (Seoul National University)
Publication Information
Journal of Educational Research in Mathematics / v.27, no.4, 2017 , pp. 833-855 More about this Journal
Abstract
This study aims to reinvestigate the reason for introducing radian as a new unit to express the size of angles, what is the meaning of radian measures to use arc lengths as angle measures, and why is the domain of trigonometric functions expanded to real numbers for expressing general angles. For this purpose, it was conducted historical, mathematical and applied mathematical analyzes in order to research at multidisciplinary analysis of the radian concept. As a result, the following were revealed. First, radian measure is intrinsic essence in angle measure. The radian is itself, and theoretical absolute unit. The radian makes trigonometric functions as real functions. Second, radians should be aware of invariance through covariance of ratios and proportions in concentric circles. The orthogonality between cosine and sine gives a crucial inevitability to the radian. It should be aware that radian is the simplest standards for measuring the length of arcs by the length of radius. It can find the connection with sexadecimal method using the division strategy. Third, I revealed the necessity by distinction between angle and angle measure. It needs justification for omission of radians and multiplication relationship strategy between arc and radius. The didactical suggestions derived by these can reveal the usefulness and value of the radian concept and can contribute to the substantive teaching of radian measure.
Keywords
radian; mathematical analysis; arc measure; angle measure; ratios and proportions;
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1 강미광(2011). 호도법에 관한 교수학적 고찰. 한국수학교육학회지 시리즈A <수학교육>, 50(3), pp. 355-365.
2 강향임, 최은아(2015). 예비교사의 라디안에 대한 이해. 학교수학, 17(2), pp. 309-329.
3 김완재(2009). 라디안의 속성에 관한 연구: 1rad은 각인가 실수인가?. 수학교육학연구, 19(3), pp. 443-459.
4 남진영, 임재훈(2008). 라디안에 대한 교수학적 분석. 수학교육학연구, 18(2), pp. 263-281.
5 송은영(2008). 삼각함수 개념의 지도에 관한 연구. 서울대학교 대학원 석사학위논문.
6 우정호(2010). 수학 학습-지도 원리와 방법 제2개 정판. 서울: 서울대학교 출판문화원.
7 우정호, 이광연, 박세원, 신범영, 이계세, 김정화, 박문환, 윤정호, 박상의, 서원호, 전제동, 이동흔(2014). 미적분II. 서울: 두산동아(주).
8 유재근(2014). 삼각함수 개념의 역사적 분석. 수학교육학연구, 24(4), pp. 599-614.
9 이준열, 최부림, 이동재, 한대희, 전용주, 장희숙, 조석연, 조성철, 황선미, 박성훈(2014). 미적분II. 서울: (주)천재교육.
10 최영기(1999). 중학교 수학에서 평행공리의 의미. 학교수학, 1(1), pp. 7-17.
11 최영기(2016). 교사를 위한 기하학의 공리론적 접근. 2016 수학 핵심교원 특별연수 자료집, 20-21주차(한국과학창의재단 비출판물), pp. 1-10.
12 최은아, 강향임(2015). 호의 측도로 도(Degree)와 라디안 이해하기. 학교수학, 17(3), pp. 447-467.
13 Clayton, D. G. (2010). A trigonometrical ratio to replace the dimensionless angle in radians. International Journal of Mechanical Engineering Education, 38(2), pp. 132-134.   DOI
14 Akkoc, H. (2008). Pre-service mathematics teachers' concept image of radian. International Jouranl of Mathematical Education in Science and Technology, 39(7), pp. 857-878.   DOI
15 Aubrecht, G. J., French, A. P., Iona, M., Welch, D. W. & The AAPT Metric Education and SI Practices Committee. (1993). The radian- That troublesome unit. The Physics Teacher, 31, pp. 84-87.   DOI
16 Clairaut, A. C. (2005). 클레로의 기하학 원론. (장혜원, 역). 서울: 경문사. (불어 원작은 1741년 출판).
17 Clayton, D. G. (1998). Making the Radian Less Special. International Journal of Mechanical Engineering Education, 26(3), pp. 253-257. Available from: http://journals.sagepub.com/doi/pdf/10.1177/030641909802600310.   DOI
18 Euclid & Heath, T. (1998) 기하학원론 마. 이무현 역, 서울: 교우사.
19 Fredenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel Publishing Company.
20 Giambattista, A., Richardson, B. M. & Richardson, R. C. (2008). 대학물리학 I. (김용은, 역). 서울: 북스힐.
21 Kendal, M. & Stacey, K. (1997). Teaching trigonometry. Australian Mathematics Teacher, 54(1), pp. 34-39.
22 Klein, R. J. & Hamilton, I (1997). Using technology to introduce radian measure. The Mathematics Teacher, 90(2), pp. 168-172.
23 Kreyszig, E. (2012). Kreyszig 공업수학 개정 10판. (서진헌, 심형보, 이상구, 유일, 배현덕, 양영균, 김희택, 이성철, 허건수, 한광희, 함운철, 최항석, 박제남, 역). 서울: 범한서적. (영어 개정판 원작은 2012년 출판).
24 Moore, K. C. (2010). The role of quantitative reasoning in precalclus students learning central concepts of trigonometry. Ph.D. dissertation, Arizona State University, USA.
25 Moore, K. C. (2013). Making sense by measuring arcs: a teaching experiment in angle measure. Educational Studies in Mathematics, 83(2), pp. 225-245.   DOI
26 Moore, K. C. & LaForest, K. R. (2014). Approach to Circle Trigonometry. Mathematics teacher, 107(8), pp. 617-623.
27 Newton Highlight 84. (2016). 삼각함수의 세계. 서울: (주)아이뉴턴.
28 Shibuya, M. (2006). 만화로 쉽게 배우는 푸리에 해석. (홍희정, 역). 서울: 성안당. (일본어 원작은 2005년 출판).
29 Stewart, I. (2016). 교양인을 위한 수학사 강의. (노태복, 역). 서울: 반니. (영어 원작은 2008년 출판).
30 Toeplitz, O. (2006). 퇴플리츠의 미분적분학. (우정호, 임재훈, 박경미, 이경화, 역). 서울: 경문사. (영어 원작은 1963년 출판).
31 Thompson, P., Carlson, M. and Silverman, J. (2007). The design of tasks in support of teachers' development of coherent mathematical meanings. Journal of Mathematics Teacher Education. 10: pp. 415-432.   DOI
32 Topcu, T., Kertil, M., Akkoc, H., Kamil, Y. & Onder, O. (2006). Pre-service and in-service mathematics teachers' concept images of radian. In J. Novotna, H. Moraova, M. Kratka, & N. Stehlikova (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 5, pp. 281-288). Prague: PME.
33 Watson, A. (2009b). Working group on trigonometry: meeting 4. In Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 29(3), pp. 121-123.
34 Senk, S. L., Thompson, D. R., Viktora, S. S., Rubenstein, R. Halvorson, J. Flanders, J., Jakucyn, N., Pillsbury, G., & Usiskin, Z. (1993). UCSMP Advanced Algebra. Illinois: Scott Foresman.
35 Watson, A. (2008). Working group on trigonometry: meeting 1. In Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 28(3), pp. 148-150.
36 Watson, A. (2009a). Working group on trigonometry: meeting 2-3. In Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 29(2), pp. 94-97.
37 Watson, A. (2010). Working group on trigonometry: meeting 5. In Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics, 30(2), pp. 68-69.
38 Whitehead, A. N. (2009). 화이트헤드의 수학이란 무엇인가. (오채환, 역). 서울: 궁리. (영어 원작은 1948년 출판).
39 Janke, H. N. & Otte, M. (1982). Complementarity of Theoretical Terms-Ratio and Proportion as an Example. Conference on Function, (pp. 97-113). SLO Foundation for Curriculum Development.