참고문헌
- 교육부 (2015a). 수학 6-1. 서울: 천재교육. (Ministry of Education (2015a). Elementary mathematics 6-1. Seoul: Chunjae Education.)
- 교육부 (2015b). 수학 6-2. 서울: 천재교육 (Ministry of Education (2015b). Elementary mathematics 6-2. Seoul: Chunjae Education.)
- 김경선, 박영희 (2007). 초등학생의 비례적 추론 지도에 관한 연구. 학교수학, 9(4), 447-466. (Kim, K. S. & Park, Y. H. (2007). A study on the proportional reasoning instruction for elementary school children. School Mathematics, 9(4), 447-466.)
- 김숙진(2011). 초등학교 학생들의 비례 추론 능력에 시각적 표현이 미치는 영향 - 5, 6학년을 대상으로. 경인교육대학교 교육대학원 초등수학교육학전공 석사학위논문. (Kim, S. J. (2011). A study on the effects of visual representation on elementary school students' proportional reasoning ability - with a focus on the fifth and sixth graders. Major in Elementary Mathematics Education, Graduate School of Education, Gyeongin National University of Education.)
- 박정숙 (2009). 학생의 비례추론의 분석 모형과 특성 분석. 서울대학교 대학원 수학교육과 수학교육전공 박사학위논문. (Park, J. S. (2009). Analyzing the models and the characteristics of students' proportional reasoning. Major in Mathematics Education, The Graduate School, Seoul National University.)
- 서은미, 방정숙, 이지영 (2017). 시각적 모델을 활용한 비례 추론 수업 분석. 수학교육학연구, 27(4), 791-810. (Seo, E. M., Pang, J. S., & Lee, J. Y. (2017). An analysis of lessons to teach proportional reasoning with visual models - focused on ratio table, double number line, and double tape diagram. Journal of Educational Research in Mathematics, 27(4), 791-810.)
- 임재훈, 이형숙 (2015). 비례 추론을 돕는 시각적 모델에 대하여: 초등 수학 교과서의 비례식과 비례배분 실생활 문제를 대상으로. 수학교육학연구, 25(2), 189-206. (Yim, J. H., & Lee, H. S. (2015). Visual representations for improving proportional reasoning in solving word problems. Journal of Educational Research in Mathematics, 25(2), 189-206.)
- 장혜원, 박혜민, 김주숙, 임미인, 유미경, 이화영 (2017). 비례식과 비례배분에 대한 초등 수학교과서 비교 분석. 학교수학, 19(2), 229-248. (Chang, H. W., Park, H. M., Kim, J. S., Lim, M. I., Yu, M. G., & Lee, H. Y. (2017). A comparative analysis of proportional expression and proportional distribution in elementary mathematics textbooks. School Mathematics, 19(2), 229-248.)
- 장혜원, 임미인, 유미경, 박혜민, 김주숙, 이화영 (2017). 비와 비율에 대한 초등 수학 교과서 비교 분석. 한국초등수학학회, 21(1), 135-160. (Chang, H. W., Lim, M. I., Yu, M. G., Park, H. M., Kim, J. S., & Lee, H. Y. (2017). A comparative analysis of ratio and rate in elementary mathematics textbooks. Journal of Elementary Mathematics Education in Korea, 21(1), 135-160.)
- 정영옥 (2015). 초등학교에서 비례 추론 지도에 관한 논의. 수학교육학연구, 25(1), 21-58. (Chong, Y. O. (2015). Teaching proportional reasoning in elementary school mathematics. Journal of Educational Research in Mathematics, 25(1), 21-58.)
- 정은실(2010). 초등학교 수학 교과에서의 비례 추론에 대한 연구. 수학교육학연구, 23(4), 505-516. (Jeong, E. S. (2010). Study on proportional reasoning in elementary school mathematics. Journal of Educational Research in Mathematics, 23(4), 505-516.)
- 藤井齊亮外. (2015a). 新しい算數 5下. 東京: 東京書籍.
- 藤井齊亮外. (2015b). 新しい算數 6上. 東京: 東京書籍.
- 藤井齊亮外. (2015c). 新しい算數 6下. 東京: 東京書籍.
- 靑水靜海外. (2015). れくれく算數 6上. 啓林館
- 小山正孝外. (2015a). 小學算數 6年上. 日本文敎出板.
- 小山正孝外. (2015b). 小學算數 6年下. 日本文敎出板.
- Primary Mathematics Project Team (2015a). Primary Mathematics 5A Textbook. Singapore: Marshall Cavendish Education.
- Primary Mathematics Project Team (2015b). Primary Mathematics 6A. Textbook. Singapore: Marshall Cavendish Education.
- Beckmann, S., & Izsak, A. (2015). Two perspectives on proportional relationships: Extending complementary origins of multiplication in Terms of quantities. Journal for Research in Mathematics Education, 46(1), 17-38. https://doi.org/10.5951/jresematheduc.46.1.0017
- Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational numbers: Toward a semantic analysis-emphasis on the operator construct. Rational numbers: An integration of research, 13-47.
- Bell et al. (2015). Everyday Mathematics 4, Grade 6, Student Reference Book. McGraw-Hill Education.
- Broekman, H., Van der Valk, T., & Wijers, M. (2000). Teacher knowledge needed to teach ratio and proportion in secondary school mathematics : on using the ratio table. http://www.fisme.science.uu.nl/bps/artikelen/ratiotable.htm
- Collars, C., Koay, P. L., Lee, N. H., Ong, B. L., & Tan, C. S. (2015a). Shaping Maths Coursebook 5A. Singapore: Marshall Cavendish Education.
- Collars, C., Koay, P. L., Lee, N. H., Ong, B. L., & Tan, C. S. (2015b). Shaping Maths Coursebook 6A. Singapore: Marshall Cavendish Education.
- Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft): 6-7, Ratio and proportional relationships. https://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf
- Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2-3), 135-164. https://doi.org/10.1007/BF01273661
- Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66-86. https://doi.org/10.2307/749228
- Gravemeijer, K.P.E. (1994). Developing Realistic Mathematics Education, Freudenthal Institute, Utrecht, The Netherlands.
- Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. New York, VA: State University of New York Press.
- Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
- Kuchemann, D., Hodgen, J., & Brown, M. (2011). Using the double number line to model multiplication. Paper presented at Seventh Annual Congress of the European Society for Research in Mathematics Education, Rzeszow, Poland.
- Kuchemann, D., Hodgen, J., & Brown, M. (2014). The use of alternative double number lines as models of ratio tasks and as models for ratio relations and scaling. Proceedings of the 8th British Congress of Mathematics Education (BCME8) (pp. 231-238). BSRLM: University of Nottingham.
- Chong, L. C., Soon, D, S., & Lian, T. K., (2015a) Discover Maths 5A Textbook. Singapore: Pan Pacific.
- Chong, L. C., Soon, D, S., & Lian, T. K., (2015b) Discover Maths 6A Textbook. Singapore: Pan Pacific.
- Lamon, S. J. (1993). Ratio and proportion: Connecting content and children's thinking. Journal for Research in Mathematics Education, 24(1), 41-61. https://doi.org/10.2307/749385
- Lamon (2012). Teaching Fractions and Ratios for Understanding - Essential Content Knowledge and Instructional Strategies for Teachers. New York, VA: Routledge.
- Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. Number concepts and operations in the middle grades (pp. 93-118). Reston, VA: National Council of Teachers of Mathematics.
- Lobato, J., & Ellis, A. B. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics: Grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
- National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.
- National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. http://www2.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf
- Shield, M., & Dole, S. (2013) Assessing the potential of mathematics textbooks to promote deep learning. Educational studies in mathematics, 82(2), 183-199. https://doi.org/10.1007/s10649-012-9415-9
- Stacey, K. (2003). Do mathematics textbooks cultivate shallow teaching? Applying the TIMMS video study criteria to Australian Eight-grade mathematics textbooks. Mathematics Education Research Journal, 20(1), 82-107. https://doi.org/10.1007/BF03217470
- Streefland, L. (1985). Search for the roots of ratio: Some thoughts on the long term learning process (towards... a theory). Educational Studies in Mathematics, 16(1), 75-94. https://doi.org/10.1007/BF00354884
- Sumarto, S. N. (2013). Design Research on Mathematics Education : Ratio Table in Developing The Students' Proportional Reasoning. Palembang: Sriwijaya University.
- Treffers, A. & Goffree, F. (1985). Rational analysis of realistic mathematics education -the Wiskobas program. Proceedings of the Ninth International Conference for the Psychology of Mathematics Education, 2, 97-121.
- Treffers, A.: 1987, Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction - The Wiskobas Project, Reidel Publishing Company, Dordrecht, The Netherlands.
- Van den Heuvel-Panhuizen, M. (1995). A representional model in a long term learning process - the didactical use of models in Realistic Mathematics Education. paper presented at the AERA conference, San Francisco, CA.
- Van den Heuvel-Panhuizen, M. (2002). Realistic Mathematics Education as work in progress. in F.L. Lin (ed.), Common Sense in Mathematics Education - Proceedings of 2001 The Netherlands and Taiwan Conference on Mathematics Education, Taipei, Taiwan, National Taiwan Normal University, Taipei, Taiwan, 1-42.
- Van den Heuvel-Panhuizen, M. (2003). The didactical use of models in realistic mathematics education: An example from a longitudinal trajectory on percentage. Educational Studies in Mathematics, 54, 9-35. https://doi.org/10.1023/B:EDUC.0000005212.03219.dc
- Van de Walle, J. A. (2008). 수학을 어떻게 가르칠 것인가. (남승인, 서찬숙, 최진화, 강영란, 홍우주, 배혜진 외 역) 서울: 경문사. (영어 원작은 2004 년 출판).
- Vergnaud, G. (1988). Multiplicative Structures. Number concepts and operations in middle grades (pp. 141-161). Reston, VA: National Council of Teachers of Mathematics.
- Webb, D. C., Boswinkel, N., & Dekker, T. (2008). Beneath the tip of the iceberg: Using representations to support student understanding. Mathematics Teaching in the Middle School, 14(2), 110-113.