References
- Blanton, M., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understanding of algebraic thinking for teaching mathematics in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
- Brizuela, B. M., Blanton, M., Sawrey, K., Newman-Owens, A., & Gardiner, A. M. (2015). Children's use of variables and variable notation to represent their algebraic ideas. Mathematical Thinking and Learning, 17(1), 34-63. https://doi.org/10.1080/10986065.2015.981939
- Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 669-705). Reston, VA: National Council of Teachers of Mathematics.
- Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. 김수환, 박영희, 백선수, 이경화, 한대희 공역(2006). 어떻게 수학을 배우지? 서울: 경문사.
- Chang, H. W., Lim, M. I., Yu, M. K., 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.
- Empson, S. B., Levi, L., & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277-301). New York: Springer.
- Hackenberg, A. J., & Lee, M. Y. (2015). Relationships between students' fractional knowledge and equation writing. Journal for Research in Mathematics Education, 46(2), 196-243. https://doi.org/10.5951/jresematheduc.46.2.0196
- Kang, H. K. (2009). An alternative program for the teaching of multiplication concept based on times idea. School Mathematics, 11(1), 17-37.
- Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York: Lawrence Erlbaum.
- Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. New York: Springer.
- Kieran, C. (2018). Seeking, using, and expressing structure in numbers and numerical operation: a fundamental path to developing early algebraic thinking. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 79-105). Switzerland: Springer.
- Kim, J. H. (2020). Analysis of the Transition Process of Fraction Division Teaching Method. Journal of Educational Research in Mathematics, 30(1), 67-88. https://doi.org/10.29275/jerm.2020.02.30.1.67
- Kim, S. H., Shin, J. H., & Lee, S. J. (2019). Algebraic representations of middle school students with different fraction knowledge. Journal of Educational Research in Mathematics, 29(4), 625-654. https://doi.org/10.29275/jerm.2019.11.29.4.625
- Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed). New York: Routledge.
- Lee, M. Y., & Hackenberg, A. J. (2014). Relationships between fractional knowledge and algebraic reasoning: The case of Willa. International Journal of Science and Mathematics Education, 12(4), 975-1000. https://doi.org/10.1007/s10763-013-9442-8
- Lee, M. Y. (2019). A case study examining links between fractional knowledge and linear equation writing of seventh-grade students and whether to introduce linear equations in an earlier grade. International Electronic J ournal of Mathematics Education, 14(1), 109-122.
- Lee, J. Y. (2015). Development of fraction division learning trajectory based on quantitative reasoning with unit of elementary school students. Korea National University thesis of doctor.
- Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. New York: Routledge.
- National Mathematics Advisory Panel (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
- Pang, J. S., & Cho, S. M. (2019a). An analysis of solution methods by fifth grade students about 'reverse fraction problems'. The Mathematics Education, 58(1), 1-20.
- Pang, J. S., & Cho, S. M. (2019b). An analysis of solution methods by sixth grade students about reverse fraction problems. Journal of Educational Research in Mathematics, 29(1), 71-91. https://doi.org/10.29275/jerm.2019.2.29.1.71
- Pang, J. S., Cho, S. M, & Kim, J. W. (2017). An analysis of variable in the elementary mathematics textbooks and workbooks. The Mathematics Education, 56(1), 81-100. https://doi.org/10.7468/mathedu.2017.56.1.81
- Pang, J. S., Cho, S. M., & Kwon, M. S. (2020). An analysis of fifth and sixth graders' algebraic thinking about reverse fraction problems. Journal of Educational Research in Mathematics, Special Issue, 213-227.
- Pang, J. S., & Kim, J. W. (2018). Characteristics of Korean students' early algebraic thinking: A generalized arithmetic perspective. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 141-165). Switzerland: Springer.
- Pang, J. S., & Lee, J. Y. (2009). An analysis of the multiplication and division of fractions in elemetary mathematics instructional materials. School Mathematics, 11(4), 723-743.
- Park, K. S., Song, S. H., & Yim, J. H. (2004). A study on understanding of the elementary teachers in pre-service with respect to fractional division. School Mathematics, 6(3), 235-249.
- Pearn, C., & Stephens, M. (2018). Generalizing fractional structures: A critical precursor to algebraic thinking. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 237-260). Switzerland: Springer.
- Russell, S. J., Schifter, D., & Bastable, V. (2011). Developing algebraic thinking in the context of arithmetic. In J. Cai & E. Knuth (Eds.), Early algebraization (pp. 43-69). New York: Springer.
- Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (Vol. 2, pp. 41-52). Reston, VA: Erlbaum.
- Shin, J. S. (2013). A proposal to the construction of textbook contents of fraction division connected to problem context. The Mathematics Education, 52(2), 217-230. https://doi.org/10.7468/mathedu.2013.52.2.217
- Siebert, I. (2002). Connecting informal thinking and algorithms: The case of division of fraction. In B. Litwiller, & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 247-256). Reston, VA: NCTM.
- Siegler, R., Duncan, G., Davis-Kean, P., Duckworth, K., Claessens, A., Engel, M. et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(10), 691-697. https://doi.org/10.1177/0956797612440101
- Sinicrope, R., Mick, H. W., & Kolb, J. R. (2002). Interpretations of fraction division. In B. Litwiller, & G. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 153-161). Reston, VA: NCTM.
- Yim, J. H. (2007). Division of fractions in the contexts of the inverse of a cartesian product. School Mathematics, 9(1), 13-28.
- Yim, J. H., Kim, S. M., & Park, K. S. (2005). Different approaches of introducing the division algorithm of fractions: comparison of mathematics textbooks of North Korea, South Korea, China, and Japan. School Mathematics, 7(2), 103-121.
- Wu, H. H. (2001). How to prepare students for algebra. American Educator, 25, 1-7.