[논문]독일 고등학교 수학에서 행렬 교수·학습 내용 분석

안은경; 고호경

doi:10.7468/mathedu.2023.62.2.269

독일 고등학교 수학에서 행렬 교수·학습 내용 분석
Analysis of teaching and learning contents of matrix in German high school mathematics 원문보기

Journal of the Korean Society of Mathematical Education. Series A. The Mathematical Education, v.62 no.2, 2023년, pp.269 - 287

안은경 (아주대학교) , 고호경 (아주대학교)

초록
AI-Helper

행렬이론은 수학, 자연과학, 공학뿐 아니라 사회과학과 인공지능 분야에까지 다양하게 활용되고 있다. 중·고등학교 수학에서 행렬은 학습 부담 경감을 위해 2009 개정 수학과 교육과정에서 삭제되었다가 인공지능 시대를 맞이하여 2022 개정 교육과정에 재편성될 예정이다. 이에 다른 나라에서 다루고 있는 행렬 내용을 분석함으로써 행렬 지도를 위한 의미 있는 방향을 제시하고 교과서 구성을 위한 시사점을 도출할 필요성이 있다. 이를 위해 본고에서는 독일 수학과 표준교육과정과 독일 헤센주의 수학과 교육과정을 분석하고, 독일 수학 교과서의 행렬 단원의 내용 요소 및 전개 방식의 특징을 분석하였다. 분석 결과 독일 교과서는 선형연립방정식의 풀이를 위한 행렬, 일차변환을 설명하기 위한 행렬, 전환과정을 설명하기 위한 행렬로 나누어 행렬 단원을 다루고 있으며 모두 역행렬을 다루고 있고 수학적 추론 및 수학적 모델링에 중점을 두고 행렬을 학습하는 것으로 나타났다. 분석 결과로부터 학교 수학에 행렬을 재편성할 경우 깊이 있는 개념적 이해와 수학적 추론 및 수학적 모델링에 중점을 두어 교육내용을 구성할 것을 제안하는 바이다.

Abstract ▼ AI-Helper

Matrix theory is widely used not only in mathematics, natural sciences, and engineering, but also in social sciences and artificial intelligence. In the 2009 revised mathematics curriculum, matrices were removed from high school math education to reduce the burden on students, but in anticipation of the age of artificial intelligence, they will be reintegrated into the 2022 revised education curriculum. Therefore, there is a need to analyze the matrix content covered in other countries to suggest a meaningful direction for matrix education and to derive implications for textbook composition. In this study, we analyzed the German mathematics curriculum and standard education curriculum, as well as the matrix units in the German Hesse state mathematics curriculum and textbook, and identified the characteristics of their content elements and development methods. As a result of our analysis, it was found that the German textbooks cover matrices in three categories: matrices for solving linear equations, matrices for explaining linear transformations, and matrices for explaining transition processes. It was also found that the emphasis was on mathematical reasoning and modeling when learning matrices. Based on these findings, we suggest that if matrices are to be reintegrated into school mathematics, the curriculum should focus on deep conceptual understanding, mathematical reasoning, and mathematical modeling in textbook composition.

주제어

표/그림 (25)

표 Table 1. Textbook unit contents of 'matrix' and 'linear transformation' according to curriculum changes
표 Table 2. Content areas of Standard Curriculum in Mathematics for Primary and Secondary Education
표 Table 3. Description of the content area 'Algorithms and Numbers' (KMK, 2012, p.18)
표 Table 4. Achievement criteria for content area of 'Algorithm and Number' in grades 11~12 of gymnasium secondary level Ⅱ (KMK, 2012, p.18)
표 Table 5. An overview of matrices in the Hessian state mathematics core curriculum (KCGO, 2016, p.35)
표 Table 6. Achievement criteria for matrices in Hessian state mathematics core curriculum (KCGO, 2016, p.37)
표 Table 7. Unit titles of the selected textbook
그림 Figure 1. Linear equations to a matrix (Lambacher Schweizer, Analytic Geometry, 2022, p.14)
그림 Figure 2. Solving system of linear equations using Gauss-elimination method (Lambacher Schweizer, Analytic Geometry, 2022, p.14)
그림 Figure 3. 'V. Matrices and mapping' cover page (Lambacher Schweizer, Analytic Geometry, 2022, p.134~135)
표 Table 8. 'Ⅴ. Matrices and Transformations' Unit's objectives and contents
그림 Figure 4. 'Ⅴ-1. 'Geometric mapping' introductory task (Lambacher Schweizer, Analytic Geometry, 2022, p.138)
그림 Figure 5. 'V-5. Parallel Projection of an Object on a Plane' introductory task (Lambacher Schweizer, Analytic Geometry, 2022, p.150)
그림 Figure 6. 'V-6. Composite mapping-Matrix Multiplication' introductory task (Lambacher Schweizer Analytic Geometry, 2022, p.153)
그림 Figure 7. 'V-7. Inverse Matrix-Inverse mapping' introductory task (Lambacher Schweizer Analytic Geometry, 2022, p. 156)
그림 Figure 8. Defnition of determinant and inverse matrix in German textbooks (Lambacher Schweizer Analytic Geometry, 2022, p.156)
그림 Figure 9. 'VI. Matrices and Processes' cover page (Lambacher Schweizer Analytic Geometry, 2022, p.168-169)
표 Table 9. 'VI. Matrices and Processes' unit learning objectives and instructional contents
그림 Figure 10. 'VI-1. Description of a single process through a matrix' introduction task (Lambacher Schweizer Analytic Geometry, 2022, p.172)
그림 Figure 11. 'VI-2. Matrix Operation' introductory task and concept explanation (Lambacher Schweizer Analytic Geometry, 2022, p.175)
그림 Figure 12. 'VI-3. Two-step Process-Matrix Multiplication' concept explanation (Lambacher Schweizer Analytic Geometry, 2022, p.178)
그림 Figure 13. 'VI-4. Inverse Matrix' introducing and explaining the concept of inverse matrix (Lambacher Schweizer Analytic Geometry, 2022, p.181)
그림 Figure 14. Inverse of a cubic square matrix (Lambacher Schweizer Analytic Geometry, 2022, p.182)
그림 Figure 15. 'VI-4. Inverse Matrix' sub-unit exercises (Lambacher Schweizer Analytic Geometry, 2022, p.184)
그림 Figure 16. 'VI-5. Stochastic Process' practice problem(Lambacher Schweizer Analytic Geometry, 2022, p.189)

참고문헌 (42)

Andreas C. M. & Sarah G. (2016). Introduction to Machine Learning with Python. O'Reilly Media.
Baum, M. et al. (2012). Lambacher Schweizer mathematik analytische geometrie und lineare algebra GK/LK - Bundesausgabe ab 2012. Klett Ernst.
Bishop, C. M. (2006). Pattern recognition and machine learning. Springer
Choi, E. & Kwon, O. N. (2020). Comparison of trigonometry in mathematics textbooks in Korea, Australia, and Finland. Communications of？Mathematical Education, 34(3), 393-419. https://doi.org/10.7468/jksmee.2020.34.3.393

원문보기 상세보기
Choi, H. M. (2010). A comparative analysis of matrix unit in national textbooks by country [Unpublished Master's thesis, Korea University].
Chong, Y. O. (2004). Reflections on mathematics curriculum of germany -Focusing on Nordrhein-Westfalen-, School Mathematics, 6(2), 181-211.
Edge, J., & Wharton, S.(1998). Autonomy and development: Living in the materials world. In B. Tomlinson (Ed.), Materials development in？language teaching (pp. 295-310). Cambridge University Press.
Foxman, D. (1999). Mathematics textbooks across the world: Some evidence from the third iInternational mathematics and science study (TIMSS).？NFER
Goodfellow, I.., Bengio, Y.., & Courville, A. (2016). Deep Learning. MIT Press.
Hessisches Kultusministerium (2016). Kerncurriculum gymnasiale Oberstufe Mathematik. Retrieved from https://kultusministerium.hessen.de/sites/kultusministerium.hessen.de/files/2021-07/kcgo-m.pdf
Heo, N. G. (2020). Analysis of secondary mathematics knowledge for AI learning through the AI related R&E program. Journal of Learner-Centered Curriculum and Instruction, 20(16), 673-689. http://doi.org/10.22251/jlcci.2020.20.16.673

상세보기
Huh, E. S. (2005). Study on the teaching of the concepts of linear algebra in the high school mathematics : Focused on the mathematical connections？[Unpublished Master's thesis, Graduate School of Seoul National University].
Huh, N., Ahn, E. K., & Ko, H. K. (2011). Comparative analysis of the contents of functions in the middle school mathematics textbooks in Korea？and Germany. School Mathematics, 13(2), 323-343.
Kang, H. Y. (2022). A study on digital and AI literacy education to prepare for Future. Journal of Digital Contents Society, 23(6), 1067-1075. http://doi.org/10.9728/dcs.2022.23.6.1067

상세보기
Kim, M. (2011). How are textbooks made? Ewha Womans University Press.
Kim, M. H, &. Kim, G. Y. (2013). The analysis of mathematical tasks in the high school mathematics. School Mathematics, 15(1), 37-59.
Kim, M. J. (2011). A study on Teaching of Matrices in the High School Math [Master's thesis, Graduate School of Education of Pusan National？University].
Kim, S. K., & Lee, M. Y. (2021). Analysis of the German mathematics standard curriculum: Focusing on the comparison with the 2015 revised？mathematics curriculum of Korea. School Mathematics, 23(1), 147-168. https://doi.org/10.29275/sm.2021.03.23.1.147

상세보기
Kim, Y. K. (2002). A study of practical life problems and solutions using the matrix [Unpublished Master's thesis, Suncheon National University].
KMK (2012). Bildungsstandards im Fach Mathematik fur die Allgemeine Hochschulreife. Retrieved from https://www.kmk.org/fileadmin/Dateien/veroeffentlichungen_beschluesse/2012/2012_10_18-Bildungsstandards-Mathe-Abi.pdf (검색일: 2022.10.25.)
Ko, H. K. (2020). A study on development of school mathematics contents for artificial intelligence (AI) capability. Journal of the Korean School？Mathematics, 23(2), 223-237. http://doi.org/10.30807/ksms.2020.23.2.003

원문보기 상세보기
Kuem, J. H., Ko, H. K., Kwon, O. N., Kim, M. H., Kim, Y. W., Nam, H. S., Song, S. H., Lee, K. W., Lee, S. H., Lee, Y. H., Jang, J. W. & Hwang, J.？M. (2018). An impact analysis of content reduction in high school mathematics curriculum on preparing next generation STEM majors. KAST？research report 125.
Lee, K. H., Kim, H. L., & Lee, S. H. (2022). Comparison of matrix in the mathematics textbooks in Korea, Singapore, and Australia. School？Mathematics, 24(1), 57-88. http://doi.org/10.29275/sm.2022.3.24.1.57
Lee, S. G., & Ham, Y. M. (2006). J. J. Sylvester, F. Klein and American mathematics in 19th century, Journal for history of mathematics, 19(2). 77-88.
Lee, S. G., Lee, J. H., & Ham, Y. M. (2012). Early history of linear algebra. Communications of mathematical education, 26(4), 351-362.
Lee, S. M. (2012). A case study on the competency-based curriculum development in Hessen. The Korean Society for Curriculum Studies, 30(1),？151-174.
Lee, T. H., & Kim, J. (2017). Understanding textbooks. Hakjisa.
Ministry of Education (2015). Mathematics and Curriculum. Ministry of Education. Announcement No. 2015-74 [Appendix 8].
Ministry of Education (2022). Mathematics and Curriculum. Ministry of Education. Announcement No. 2022-33 [Appendix 8].
Nam, G. Y., Maeng, U. K., & Ko, H. K. (2012). Korean language, English, and mathematics teachers' perceptions on selecting textbooks. Journal of？Research in Curriculum & Instruction, 16(3), 603-627. http://doi.org/10.24231/rici.2012.16.3.603

상세보기
Oh, E. J. (2013). A Study of the Efficient Ways to Teach Matrices in High School [Master's thesis, The Graduate School of Education of In-ha？University].
Pang, J. S., & Oh, M. Y. (2022). Research trends of mathematics textbooks: An analysis of the journal articles published from 1963 to 2021. The？Mathematical Education, 61(3), 457-476. http://doi.org/10.7468/mathedu.2022.61.3.457

원문보기 상세보기
Park, K. M. et al. (2015). Study on development of mathematics curriculum according to 2015 revised curriculum II. Korea Foundation for the？Advancement of Science and Creativity.
Park, S. Y., & Han, S. Y. (2018). Reconstruction and application of reforming textbook problems for mathematical modeling process. The？Mathematical Education, 57(3), 289-309. http://doi.org/10.7468/mathedu.2018.57.3.289

원문보기 상세보기
Rubdy, R.(2003). Selection of materials. In B. Tomlinson (Ed.), Developing materials for language teaching (pp. 37-57). Continuum.
Schmidt, W., & Tatto, M. T. (2017). The preparation gap: Teacher education for middle school mathematics in six countries. The Journal of？Mathematical Behavior, 47, 23-37.
Sheldon, L. E.(1988). Evaluating ELT textbooks and materials. ELT Journal, 42(4). 237-246. https://doi.org/10.1093/elt/42.4.237

상세보기
Shin, Y. S. et al. (2011). A study on mathematics and curriculum based on the 2009 revised national curriculum. Korea Foundation for the？Advancement of Science and Creativity.
Stein, M. K.., Remillard, J. T., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester Jr. (Ed.), Second handbook of？research on mathematics teaching and learning. (pp. 319-369). Information Age.
Strang, G. (2019). Linear agebra and its applications (4th edition). Cengage Learning.
Tatto, M. T., Schwille, J., Senk, S. L., Ingvarson, L., Rowley, G., Peck, R., & Rodriguez, M. (2008). Teacher education and development study in？mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics: conceptual framework. IEA.
Yoo, B. S. (2008). Comparative study on mathematics textbook of Korea and U.S.A.: Focused on matrix [Unpublished Master's thesis, Hanyang？University].

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