Education of Primary School Mathematics (한국수학교육학회지시리즈C:초등수학교육)

Korean Society of Mathematical Education (한국수학교육학회)
권호 표지
DOI QR코드

DOI QR Code

An Analysis on the Contents of Fractional Operations in CCSSM-CA and its Textbooks

CCSSM-CA와 미국 교과서에 제시된 분수의 연산 내용 분석

  • Received : 2019.03.11
  • Accepted : 2019.04.26
  • Published : 2019.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

Because of the various concepts and meanings of fractions and the difficulty of learning, studies to improve the teaching methods of fraction have been carried out. Particularly, because there are various methods of teaching depending on the type of fractions or the models or methods used for problem solving in fraction operations, many researches have been implemented. In this study, I analyzed the fractional operations of CCSSM-CA and its U.S. textbooks. It was CCSSM-CA revised and presented in California and the textbooks of Houghton Mifflin Harcourt Publishing Co., which reflect the content and direction of CCSSM-CA. As a result of the analysis, although the grades presented in CCSSM-CA and Korean textbooks were consistent in the addition and subtraction of fractions, there are the features of expressing fractions by the sum of fractions with the same denominator or unit fraction and the evaluation of the appropriateness of the answer. In the multiplication and division of fractions, there is a difference in the presentation according to the grades. There are the features of the comparison the results of products based on the number of factor, presenting the division including the unit fractions at first, and suggesting the solving of division problems using various ways.

본 연구에서는 CCSSM-CA와 그에 따른 미국 초등 교과서에 제시된 분수의 연산 내용을 분석하였다. 분석 결과, 분수를 단위분수나 분모가 같은 분수의 합으로 표현하게 하여 분수 개념과 연산을 연결 짓는 특징이 있었다. 또 분수의 곱셈에서는 곱하는 한 수의 크기에 기초하여 다른 수의 곱의 결과를 비교하도록 하거나, 나눗셈에서는 단위분수가 포함된 나눗셈을 먼저 다루고, 다양한 방법으로 계산을 하도록 제시하는 특징 등이 있었다.

Keywords

SHGHD@_2019_v22n2_129_f0001.png 이미지

[그림 1] 분수의 합(Dixon, Burger & Leinwand, 2015a,p. 286) [Fig. 1] Addition of fractions(Dixon, Burger &Leinwand, 2015a, p. 286)

SHGHD@_2019_v22n2_129_f0002.png 이미지

[그림 2] 대분수의 변환(Dixon, Burger & Leinwand, 2015a, p. 301) [Fig. 2] Conversion of mixed number(Dixon, Burger & Leinwand, 2015a, p. 301)

SHGHD@_2019_v22n2_129_f0003.png 이미지

[그림 3] 분모가 10, 100인 분수의 합(Dixon, Burger & Leinwand, 2015a, p. 381) [Fig. 3] Addition of fractional parts of 10,100(Dixon, Burger & Leinwand, 2015a, p. 381)

SHGHD@_2019_v22n2_129_f0004.png 이미지

[그림 4] 분수 띠를 이용한 분수의 합(Dixon, Burger & Leinwand, 2015b, p. 257) [Fig. 4] Addition of fraction using fraction strips(Dixon, Burger & Leinwand, 2015b, p. 257)

SHGHD@_2019_v22n2_129_f0005.png 이미지

[그림 5] 분수의 합(Dixon, Burger & Leinwand, 2015b, p. 273) [Fig. 5] Addition of fraction(Dixon, Burger & Leinwand, 2015b, p. 273)

SHGHD@_2019_v22n2_129_f0006.png 이미지

[그림 6] 동치분수에 대한 영역 모델(The California Department of Education, 2015, p. 252) [Fig. 6] Area model to show equivalence(The California Department of Education, 2015, p. 252)

SHGHD@_2019_v22n2_129_f0007.png 이미지

[그림 7] 분수의 합의 어림(Dixon, Burger & Leinwand, 2015b, p. 261) [Fig. 7] Estimation of fractional sum(Dixon, Burger & Leinwand, 2015b, p. 261)

SHGHD@_2019_v22n2_129_f0008.png 이미지

[그림 8] $3{\times}\frac{2}{5}$ 의 시각적 모델(The California Department of Education, 2015, p.216) [Fig. 8] The visual model of $3{\times}\frac{2}{5}$(The California Department of Education, 2015, p.216)

SHGHD@_2019_v22n2_129_f0009.png 이미지

[그림 9] 분수 곱셈 원리와 일반화(Dixon, Burger & Leinwand, 2015a, pp. 334-340) [Fig. 9] The principle and generalization of multiplication of fraction(Dixon, Burger & Leinwand, 2015a, pp. 334-340)

SHGHD@_2019_v22n2_129_f0010.png 이미지

[그림 10] 분수 곱셈(Dixon, Burger & Leinwand, 2015b, p. 316) [Fig. 10] The multiplication of fraction(Dixon, Burger & Leinwand, 2015b, p. 316)

SHGHD@_2019_v22n2_129_f0011.png 이미지

[그림 11] $\frac{2}{3}{\times}\frac{4}{5}$의 시각적 모델 [Fig. 11] The visual model of $\frac{2}{3}{\times}\frac{4}{5}$

SHGHD@_2019_v22n2_129_f0012.png 이미지

[그림 12] 분수 곱셈(Dixon, Burger & Leinwand, 2015b, p. 337) [Fig. 12] The multiplication of faction(Dixon, Burger & Leinwand, 2015b, p. 337)

SHGHD@_2019_v22n2_129_f0013.png 이미지

[그림 13] 단위분수 나눗셈(Dixon, Burger & Leinwand, 2015b, p. 371) [Fig. 13] The division of unit fraction(Dixon, Burger & Leinwand, 2015b, p. 371)

SHGHD@_2019_v22n2_129_f0014.png 이미지

[그림 14] 곱셈과 나눗셈 관계(Dixon, Burger & Leinwand, 2015b, p. 372) [Fig. 14] The relation between multiplication and division (Dixon, Burger & Leinwand, 2015b, p. 372)

SHGHD@_2019_v22n2_129_f0015.png 이미지

[그림 15] 분수 나눗셈(Dixon, Burger & Leinwand, 2015c, p. 77) [Fig. 15] The division of fraction(Dixon, Burger & Leinwand, 2015c, p. 77)

SHGHD@_2019_v22n2_129_f0016.png 이미지

[그림 16] $\frac{9}{10}{\div}\frac{3}{5}$ 의 시각적 모델(The California Department of Education, 2015, p. 292) [Fig. 16] The visual model of $\frac{9}{10}{\div}\frac{3}{5}$ (The California Department of Education, 2015, p. 292)

SHGHD@_2019_v22n2_129_f0017.png 이미지

[그림 17] 분수 나눗셈의 어림(Dixon, Burger & Leinwand, 2015c, p. 73) [Fig. 17] The of estimation dvision of fraction(Dixon, Burger & Leinwand, 2015c, p. 73)

SHGHD@_2019_v22n2_129_f0018.png 이미지

[그림 18] 실천 규준 예시(Dixon, Burger & Leinwand, 2015b p. 264) [Fig. 18] The example of mathematics practice(Dixon, Burger & Leinwand, 2015b p. 264)

SHGHD@_2019_v22n2_129_f0019.png 이미지

[그림 19] 분수의 덧셈(교육부, 2018c, p. 97) [Fig. 19] Addition of fraction(The ministry of Education, 2018c, p. 97)

[표 1] 분석 대상 교과서 [Table 1] Textbooks for an analysis

SHGHD@_2019_v22n2_129_t0001.png 이미지

[표 2] 분석의 주안점 [Table 2] Focus of analysis

SHGHD@_2019_v22n2_129_t0002.png 이미지

[표 3] 4학년 분수 연산의 내용(CCSSM-CA) [Table 3] Contents of Fractional Operation by Grade 4(CCSSM-CA)

SHGHD@_2019_v22n2_129_t0003.png 이미지

[표 4] 5학년 분수 연산의 내용(CCSSM-CA) [Table 4] Contents of Fractional Operation by Grade 5(CCSSM-CA)

SHGHD@_2019_v22n2_129_t0004.png 이미지

[표 5] 6학년 분수 연산의 내용(CCSSM-CA) [Table 5] Contents of Fractional Operation by Grade 6(CCSSM-CA)

SHGHD@_2019_v22n2_129_t0005.png 이미지

[표 6] 학년별 분수 연산의 내용(한국) [Table 6] Contents of Fractional Operation by Grades(Korea)

SHGHD@_2019_v22n2_129_t0006.png 이미지

References

  1. Kang, H. J. (2016). Study on california common core states standards for mathematics-focused on the geometry domain of elementary school-. Journal of Elementary Mathematics Education in Korea, 20(2) 239-257.
  2. Kang, H. K. (2005). A comparative study on didactical aspects of fraction concept and algorithm appeared in the textbook of McLellan, MiC, Korea. The Journal of Educational Research in Mathematics, 15(4), 375-399.
  3. Ministry of Education(2015). Mathematics Curriculum. The ministry of Education Notification No, 2015-74호[Separate Volume 8]
  4. Ministry of Education (2018a). Mathematics 3-2. Seoul: (Inc.) Cheonjaegyoyook.
  5. Ministry of Education (2018b). Mathematics 4-2. Seoul: (Inc.) Cheonjaegyoyook.
  6. Ministry of Education (2018c). Mathematics 5-1. Seoul: (Inc.) Cheonjaegyoyook.
  7. Ministry of Education (2018d). Mathematics 5-2. Seoul: (Inc.) Cheonjaegyoyook.
  8. Ministry of Education (2018e). Mathematics 6-1. Seoul: (Inc.) Cheonjaegyoyook.
  9. Ministry of Education (2018f). Mathematics 6-2. Seoul: (Inc.) Cheonjaegyoyook.
  10. Kim, Y. O., Choi, S. W., & Lee, S. M. (2010). The introduction of Common Core State Standards for Mathematics of U. S. The Proceedings of Mathematical Education, 38, 27-36.
  11. Park. E. H. (2007). Comparative Analysis of Mathematics Textbooks of Elementary School in Korea and Singaporefocused on Fractions and Decimals -. Master's thesis in KNUE.
  12. Seo, D. Y. (2005). Historico-Genetic Instruction on Fraction. The Journal of Educational Research in Mathematics. 15(3), 233-249.
  13. Ahn, J. Y., Jeon, Y. J., Youn, M. B., & Lee, J. H. (2014). A comparative analysis of the 2009 revised curriculum for mathematics in Korea and the common core state standard for mathematics (CCSSM) in the U.S.-Focus on the number and operation strand in elementary school. Journal of Elementary Mathematics Education in Korea 14 (4), 437-464.
  14. Won, S. H. (2016). A Comparative Analysis of Elementary Mathematics Textbooks Focused on Fractions. Master's thesis in SNUE.
  15. Lee, D. H. (2017). An analysis on the contents of fraction in CA-CCSSM and its textbook. Journal of Elementary Mathematics Education in Korea 21(4) 547-574.
  16. Lee, D. H., Park C. W., Bang, S. O., & Seong, C. G. (2017). Mathematics in Elementary Mathematics. Seoul: Kyowoosa.
  17. Lee, J. Y., Pang, J. S., Seo, E. M., & Kim, K. H. (2017). A Comparative Analysis of Graphical Representations Related to Addition of Fractions in Elementary Mathematics Textbooks of Korea and Singapore. Journal of Educational Research in Mathematics, 27(3), 537-555.
  18. Jang, H. W. (2012). Study on the Standards for Mathematical Practice of Common Core State Standards for Mathematics. Journal of Educational Research in Mathematics. 22(4), 557-580.
  19. Choi, K. B. (2015). A comparative study of elementary school mathematics textbooks of Korea(2007 Curriculums) and America(Harcourt Math). Journal of Elementary Mathematics Education in Korea, 19(1), 17-37.
  20. Baroody, A. J., & Coslick, R. T. (1998). Fostering Children's Mathematics Power: An Investigative Approach to K-8 Mathematics Instruction. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers.
  21. Dixon, J. K., Burger, E. B., & Leinwand S. J. (2015a). California Go Math!-4 Common Core. Orlando: Houghton Mifflin Harcourt Publishing Company.
  22. Dixon, J. K., Burger, E. B., & Leinwand S. J. (2015b). California Go Math!-5 Common Core. Orlando: Houghton Mifflin Harcourt Publishing Company.
  23. Dixon, J. K., Burger, E. B., & Leinwand S. J. (2015c). California Go Math!-6 Common Core. Orlando: Houghton Mifflin Harcourt Publishing Company.
  24. Empson, S. B., & Levi, L. (2011). Extending Children's Mathematics-Fractions and Decimals-. Portsmouth, NH: Heinemann.
  25. Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United State. New Jersey: Lawrence Erlbaum Associates, Publishers.
  26. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
  27. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
  28. Reys, R. E., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2009) Helping Children Learn Mathematics, 9th Edition. John Wiley & Sions.
  29. The California Department of Education (2013). California Common Core State Standards-Mathematics. The California Department of Education.
  30. The California Department of Education (2015). Mathematics Framework for California Public Schools-Kindergarten through Grade Twelve-. The California Department of Education.
  31. The Mathematics in Context Development Team (1997). Measure for Measure. Naon.