거리함수와 속력함수의 관계에서 거리함수의 상수항에 대한 학생들의 인식과 표현

• Received : 2017.09.21
• Accepted : 2017.11.27
• Published : 2017.11.30

Abstract

The purpose of this study is to investigate the change of students 'perception and expression about the motion of object following distance function $={x \atop 3}$ and distance function $y=\frac{x^3}{3}+3$ according to the necessity of research on students' perception and expression about integral constant. In this paper, we present the recognition and the expression of the difference of the constant in the relationship between the distance function and the speed function of the students, while examining the process of constructing the speed function and the inverse process of the distance function. This provides implications for the relationship between the derivative and the indefinite integral corresponding to the inverse process. In particular, in a teaching experiment, a constructive activity was performed to analyze the motion of two distance functions, where the student had a difference of the constant term. At this time, the students used the expression 'starting point' for the constants in the distance function, and the motion was interpreted by using the meaning. This can be seen as a unique 'students' mathematics' in the process of analyzing the motion of objects. These scenes, in introducing the notion of the relation between differential and indefinite integral, it is beyond the comprehension of the integral constant as a computational procedure, so that the learner can understand the meaning of the integral constant in relation to the motion of the object. It is expected that it will be a meaningful basic research on the relationship between differential and integral.

References

1. 권오남, 박재희, 조경희, 박정숙, 박지현 (2015). 학습자 중심의 미적분 교육과정과 교실 문화, 학습자중심교과교육연구 15(6), 617-642. (Kwon, O.N., Park, J.H., Joe, K.H., Park, J.S., & Park, J.H. (2015). Learner-centered calculus curriculum and classroom culture, Journal of Learner-Centered Curriculum and Instruction 15(6), 617-642.)
2. 민숙, 최성원 (2016). 자기주도적 팀 활동을 적용한 대학 미적분수업 사례, 학습자중심교과교육연구 16(10), 1159-1180. (Min, S. & Choi, S.W. (2016). A Case Study of College Calculus Courses with Self-Directed Team Learning: Qualitative Analyses with Social Learning Theory, Journal of Learner-Centered Curriculum and Instruction 16(10). 1159-1180.)
3. 신보미 (2009). 고등학생들의 정적분 개념 이해, 학교수학 11(1), 93-110. (Shin, B.M. (2009). High School Students' Understanding of Definite Integral. School Mathematics 11(1), 93-110.)
4. 이동근 (2017). 고등학교 1학년 학생들의 시간, 속력, 거리의 관계에서 평균속력에 대한 인식과 평균속력 함수 구성에 대한 연구. 박사학위 논문, 한국교원대학교. (Lee, D.G. (2017). A Study on 1st Year High School Students' Construction of Average Speed Concept and Average Speed Functions in Relation to Time, Speed, and Distance. Unpublished doctoral dissertation. Korea National University of Education.)
5. 이동근, 안상진, 김숙희, 신재홍 (2016). 거리함수와 속력함수에서, 거리와 속력의 관계에 대한 학생들의 인식과 표현의 변화과정에 대한 연구, 학교수학 18(4), 333-354. (Lee, D.G., Kim, S.H., Ahn, S.J., & Shin, J.H. (2016). A Study on the Change Process of Students" Perception and Expression About Distance and Speed in Distance Function and Speed Function. School Mathematics 18(4), 333-354.)
6. 이동근, 신재홍 (2017). 구간에서의 변화율에 대한 인식과 표현에 대한 연구, 수학교육학연구 27(1), 1-22. (Lee, D.G. & Shin, J.H. (2017). Students' Recognition and Representation of the Rate of Change in the Given Range of Intervals. The Journal of Educational Research in Mathematics 27(1), 1-22.)
7. 이현주, 류중현, 조완영 (2015). 통합적 이해의 관점에서 본 고등학교 학생들의 미분계수 개념 이해 분석, 수학교육논문집 29(1), 131-155. (Lee, H.J., Ryu, J.H., & Joe, W.Y. (2015). An Analysis on the Understanding of High School Students about the Concept of a Differential Coefficient Based on Integrated Understanding, Communications of Mathematical Education 29(1), 131-155.)
8. 정연준, 이경화 (2009a). 미적분의 기본정리에 대한 고찰 - 속도 그래프 아래의 넓이와 거리의 관계를 중심으로, 수학교육학연구 19(1), 123-142. (Joung, Y.J. & Lee, K.H. (2009a). A Study on the Fundamental Theorem of Calculus : Focused on the Relation between the Area Under Time-velocity Graph and Distance, The Journal of Educational Research in Mathematics 19(1), 123-142.)
9. 정연준, 이경화 (2009b). 부정적분과 정적분의 관계에 관한 고찰, 학교수학 11(2), 301-316. (Joung, Y.J. & Lee, K.H. (2009b). A study on the Relationship between Indefinite Integral and Definite, School Mathematics 11(2), 301-316.)
10. 최영주, 홍진곤 (2014). 도함수의 성질에 관련한 학생들의 사고에 대하여, 수학교육 53(1), 25-40. (Choi, Y.J. & Hong, J.G. (2014). On the students' thinking of the properties of derivatives, The Mathematical Education 53(1), 25-40.)
11. 한대희 (1999). 미적분학의 기본정리에 대한 역사-발생적 고찰, 수학교육학연구 9(1), 217-228. (Han, D.H. (1999). A study on a genetic history of the fundamental theorem of calculus, The Journal of Educational Research in Mathematics 9(1), 217-228.)
12. 황선욱, 강병개, 김영록, 윤갑진, 김수영, 송미현, 이성원, 도종훈, 이문호, 박효정, 박진호 (2014). 미적분 I, 서울: 신사고. (Hwang, S.W., Kang, B.G., Kim, Y.R., Youn, G.J., Kim, S.Y, Song, M.H., Lee, S.W., Do, J.H., Lee, M.H., Park, H.J., & Park, J.H. (2014). Calculus I, Seoul: ShinSaGo.)
13. Boyer, C. (1959). 미분적분학사-그 개념의 발달 (김경화 역), 서울: 교우사.
14. Courant, R. (1970). Differential and Integral Calculus (E. J. Mcshane, Trans.). New York: John Wiley & Sons.
15. Courant, R. & Robbins, H. (1996). What is Mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press.
16. Glasersfeld, E. (1995). 급진적 구성주의 (김판수, 박수자, 심성보, 유병길, 이형철, 임채성, 허승희 역), 서울: 원미사.
17. Gravemeijer, K. & Doorman, M.(1999). Context Problems in Realistic Mathematics Education : a Calculus Course as an Example, Educational Studies in Mathematics, 39(1), 111-30. https://doi.org/10.1023/A:1003749919816
18. Hackenberg, A. J. (2005). A model of mathematical learning and caring relations, For the Learning of Mathematics 25(1), 45-51.
19. Merriam, S. B. (1997). 질적 사례연구법 (허미화 역), 서울: 양서원.
20. Stake, R. E.(2006). Multiple case study analysis, NY: The Guilford Press.
21. Steffe, L. P. & Gale, J. E. (1995). 구성주의와 교육 (조연주, 조미헌, 권형규 역), 서울: 학지사.
22. Steffe, L. P. & Tzur, R. (1994). Interaction and children's mathematics. In P. Ernest (Ed.), Constructing mathematical knowledge: Epistemology and mathematics education (Studies in Mathematics Education Vol. 4, pp. 8-32). London: Falmer.
23. Steffe, L. P. & Thompson, P. W. (2000). Interaction or intersubjectivity? A reply to Lerman, Journal for Research in Mathematics Education, 31(2), 191-209. https://doi.org/10.2307/749751
24. Steffe, L. P. & Wiegel, H. G. (1996). On the nature of a model of mathematical learning. In L.P. Steffe, P. Nesher, P. Cobb, G. A. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 477-498), Mahwah, NJ: Erlbaum.
25. Toeplitz, O. (1963). The Calculus -a Genetic Approach. Chicago: The Press of Chicago University.
26. Zandieh, M. J. (1998). The Evolution of Students Understanding of the Concept of Derivative. Unpublished doctoral dissertation, Oregon State University.