한국, 일본, 미국의 초등학교 수학교과서에서 범자연수 곱셈의 연산 성질을 지도하는 방안에 대한 비교·분석

• Accepted : 2019.07.25
• Published : 2019.07.31

Abstract

Even though the properties of operations in multiplication serve a fundamental basis of conceptual understanding the multiplication with whole numbers for elementary students, there has been lack of research in this field. Given this, the purpose of this study was to analyze instructional methods related to the properties of operations in multiplication (i.e., commutative property of multiplication, associative property of multiplication, distributive property of multiplication over addition) in a series of mathematics textbooks of Korea, Japan, and the US. The overall analysis was conducted in the following two aspects: (a) when and how to deal with the properties of multiplication in three instructional context (i.e., introduction, application, generalization), and (b) what models use to represent the properties of multiplication. The results of this showed that overall similarities in introducing the properties of multiplication .in (one digit) ${\times}$ (one digit) as well as emphasizing the divers representation. However, subtle but meaningful differences were analyzed in applying and generalizing the properties of multiplication. Based on these results, this paper closes with some implications on how to teach the properties of operations in multiplication properties in elementary mathematics.

범자연수의 곱셈을 개념적으로 의미 있게 이해하기 위해서는 곱셈의 연산 성질에 대한 이해가 뒷받침되어야 한다. 이러한 필요성에 따라, 본 논문은 한국, 일본, 미국의 초등학교 수학교과서에서 범자연수 곱셈의 연산 성질을 어떻게 지도하는지 비교 분석하였다. 구체적으로 곱셈의 교환법칙, 결합법칙, 분배법칙을 처음 도입하는 맥락, 연산 성질을 활용하는 맥락, 연산 성질을 일반화하는 맥락으로 나누어 분석하였으며, 각각의 지도 맥락에서 어떠한 시각적 모델을 사용하는지도 함께 분석하였다. 분석 결과, 세나라는 (한 자리 수)${\times}$(한 자리 수)의 지도 맥락에서 곱셈의 연산 성질을 처음 도입한다는 점, 곱셈의 연산 성질을 지도할 때 세 나라가 모두 유사한 시각적 모델을 사용한다는 점 등에서 공통적인 경향성을 확인하였다. 그러나 두 자리 수 이상의 곱셈에서 곱셈의 연산 성질을 활용하거나 일반화하는 맥락에서는 나라별로 지도 방안의 측면에서 미묘한 차이가 있었다. 연구 결과를 토대로 국내의 초등학교 수학 교육에서 범자연수 곱셈의 연산 성질을 지도하는 방안에 관한 시사점을 논의하였다.

Keywords [그림 1] 교환법칙 지도 시 사용할 수 있는 손 동작(Bell et al., 2016a, p. 281) [Fig. 1] The gesture that can be used to teach commutative property (Bell et al., 2016a, p. 281) [그림 2] 결합법칙을 활용하는 예(藤井斉亮외, 2015b, p. 106) [Fig. 2] An example using associative property (藤井斉亮et al., 2015b, p. 106) [그림 3] 분배법칙을 활용하는 예(교육부, 2018c, p. 64) [Fig. 3] An example using distributive property (Ministry of Education, 2018c, p. 64) [그림 4] (두 자리 수)×(한 자리 수)를 지도하는 활동의 예(藤井斉亮외, 2015b, pp. 18-19) [Fig. 4] An example teaching the (two digits) x (one digit) (藤井斉亮et al., 2015b, pp. 18-19)

[표 1] 한국, 일본, 미국의 교육과정 비교 분석 [Table 1] Comparative analysis of curriculum in Korea, Japan, and US [표 2] 분석 대상 [Table 2] Textbooks to be analyzed [표 3] 한국, 일본, 미국의 수학교과서에서 범자연수 곱셈을 지도하는 단원 정리 [Table 3] The units on multiplication for whole numbers in mathematics textbooks of Korea, Japan, and US. [표 4] 분석 기준 및 내용 [Table 4] Criteria of analysis [표 5] 교환법칙의 도입 [Table 5] Introduction for commutative property of multiplication [표 6] 교환법칙의 활용 [Table 6] Application for commutative property of multiplication [표 7] 교환법칙의 일반화 [Table 7] Generalization for commutative property of multiplication [표 8] 결합법칙의 도입 [Table 8] Introduction for associative property of multiplication [표 9] 결합법칙의 활용 [Table 9] Application for associative property of multiplication [표 10] 결합법칙의 일반화 [Table 10] Generalization for associative property of multiplication [표 11] 분배법칙의 도입 [Table 11] Introduction for distributive property of multiplication [표 12] 분배법칙의 활용 [Table 12] Application for distributive property of multiplication [표 13] 분배법칙의 일반화 [Table 13] Generalization for distributive property of multiplication [표 14] 한국, 일본, 미국의 주요 시사점 비교 [Table 14] Comparative analysis of main implications in Korea, Japan, and USA References

1. Kang, H. K. & Sim, S. Y. (2010). Design of multiplication unit of elementary mathematics textbook by making the best use of diversity of algorithm. Journal of Elementary Mathematics Education in Korea, 14(2), 287-314.
2. Ministry of Education (2015). Mathematics Curriculum, Ministry of Education Notification No, 2015-74.
3. Ministry of Education (2017). Elementary mathematics 2-2. Seoul: Chunjae Education.
4. Ministry of Education (2018a). Elementary mathematics 3-1. Seoul: Chunjae Education.
5. Ministry of Education (2018b). Elementary mathematics 3-2. Seoul: Chunjae Education.
6. Ministry of Education (2018c). Elementary mathematics 4-1. Seoul: Chunjae Education.
7. Kim, M. H.; Lee, S. E. & Kim, S. M. (2017). The Analysis of Korean Elementary mathematics textbooks and workbooks with respect to distributive principles. Journal of Educational Research in Mathematics, 27(3), 451-467.
8. Kim, S. M. (2012). The transition of error patterns and error rates in elementary students' arithmetic performance by going up grades and its instructional implication. Journal of Elementary Mathematics Education in Korea, 16(1), 125-143.
9. Kim, J. W. & Pang, J. S. (2017). Exploring the principle of computation between two-digit number and one-digit number: A case study of using cuisenaire rods and array models. Journal of Educational Research in Mathematics, 27(2), 249-267.
10. Pang, J. S. & Choi, J. Y. (2011). An analysis of the whole numbers and their operations in mathematics textbooks: Focused on algebra as generalized arithmetic. The Mathematical Education, 50(1), 41-59. https://doi.org/10.7468/mathedu.2011.50.1.041
11. Pack, D. H. (2017). A note on the use of properties of operations and the equal sign in elementary school mathematics. Journal of Elementary Mathematics Education in Korea, 21(4), 643-662.
12. Byun, H. H. (2011). A comparative analysis on the distributive property in Korean and Japanese elementary textbooks. Journal of Elementary Mathematics Education in Korea, 15(1), 39-56.
13. Seo, K. H. (2003). The Korean national curriculum, everyday mathematics, and investigations in number, data, and space curriculum: A comparative analysis of patterns and functions in elementary school mathematics textbooks. Journal of educational studies, 34(1), 163-180.
14. Seo, E. M.; Cho, S. M. & Pang, J. S. (2018). An analysis of double scale models in the Japanese elementary mathematics textbooks. Education of Primary School Mathematics. 22(1), 29-48. https://doi.org/10.7468/JKSMEC.2019.22.1.29
15. Lee, D. H. (2018). A Comparative analysis on the fraction contents of Korean, Japanese, Singaporean, American, and Finnish mathematics textbooks. Education of Primary School Mathematics, 21(2), 111-130. https://doi.org/10.7468/JKSMEC.2018.21.2.111
16. Chan, H. W. (2017). Research on teaching method for the properties of arithmetic based on analysis of elementary school mathematics textbooks. Journal of Elementary Mathematics Education in Korea, 21(1), 1-22.
17. Joung, Y. J. & Cho, Y. M. (2012). Comparative research on teaching and learning of algorithm of natural number multiplication: Focused on the elementary textbooks of South Korea, USA, Singapore, and Japan. Journal of Educational Research in Mathematics, 22(2), 293-309.
18. Bell, J.; Bell, M.; Bretzlauf, J.; Dairyko, M. E.; Dillard, A.; Hartfield, R., et al. (2016a). Everyday mathematics 4, grade 3-1. Columbus, OH: McGraw-Hill Education.
19. Bell, J.; Bell, M.; Bretzlauf, J.; Dairyko, M. E.; Dillard, A.; Hartfield, R. et al. (2016a). Everyday mathematics 4, grade 3-2. Columbus, OH: McGraw-Hill Education.
20. Bell, M.; Bretzlauf, J.; Dillard, A.; Hartfield, R.; Isaacs, A.; Maxcy, R. W. et al. (2016c). Everyday mathematics 4, grade 4-1. Columbus, OH: McGraw-Hill Education.
21. Blanton, M.; Levi, L.; Crites, T. & Dougherty, B. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5. In B. J. Dougherty, & R. M. Zbiek (Eds.), Essential understandings series. Reston, VA: National Council of Teachers of Mathematics. 방정숙, 최지영, 이지영, 김정원 공역 (2017). 대수적 사고의 필수 이해. 서울: 교우사.
22. Common Core State Standards Initiative (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/Math/.
23. Larson, M. R. & Kanold, T. D. (2016). Balancing the equation: A guide the school mathematics for educators & parents. Bloomington, IN: Solution Tree Press.
24. Otto, A. D.; Caldwell, J. H.; Lubinski, C. A. & Hancock, S. W. (2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3-5. In E. C. Rathmell & R. M. Zbiek (Eds.), Essential understandings series. Reston, VA: National Council of Teachers of Mathematics. 백석윤, 류현아, 이종영, 도주원 공역 (2016). 곱셈과 나눗셈의 필수 이해. 서울: 교우사.
25. Reys, R. E.; Lindquist, M. M.; Lambdin, D. V. & Smith, N. L. (2015). Helping children learn mathematics (11th Ed.). New York: John Wiley & Sons. 박성선, 김민경, 방정숙, 권점례 공역(2017). 초등교사를 위한 수학과 교수법. 서울: 교우사.
26. 藤井斉亮외 41명 (2015a). 新しい算數2下. 東京: 東京書籍.
27. 藤井斉亮외 41명 (2015b). 新しい算數3上. 東京: 東京書籍.
28. 藤井斉亮외 41명 (2015c). 新しい算數3下. 東京: 東京書籍.
29. 藤井斉亮외 41명 (2015d). 新しい算數4上. 東京: 東京書籍.
30. 藤井斉亮외 41명 (2015e). 新しい算數4下. 東京: 東京書籍.
31. 文部科學省 (2017). 小學校學習指導要領解説. 文部科學省.