An Analysis of Third Graders’ Understanding of the Principle of Computation Between Two-Digit Number and Two-Digit Number: Focused on the Array Model
Abstract
Along with the significance of distributive property playing a fundamental role in providing explanations for multi-digit multiplication of whole numbers, the array model has been widely used to teach distributive property effectively. Given this, the purpose of this study was to investigate how third graders could understand multiplication in (two digits) × (two digits) and how their understanding could be expanded and applied to algebraic ideas after being taught reconstructed units of multiplication using the array model. The results of this study showed that most students represented the principle of computation between two-digit number and two-digit number using the array model as well as reasoned which picture represented the product of two numbers. Also, the participating students successfully applied the principle to the computation between three-digit number and two-digit number, and some students represented the product of two numbers by using variables (e.g., □, △). This paper concludes with some implications on how to teach multiplication of whole numbers in elementary mathematics.
Keywords:
computation between two-digit and two-digit number, distributive property, array model, generalizationAcknowledgments
본 연구 결과의 일부는 2019 수학교육관련학회 연합국제학술대회(2019.12.13.~12.15.)에서 발표하였음.
References
- Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2009). The array representation and primary children’s understanding and reasoning in multiplication. Educational Studies in Mathematics, 70(3), 217-241. [https://doi.org/10.1007/s10649-008-9145-1]
- 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, 34-63. [https://doi.org/10.1080/10986065.2015.981939]
- Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann.
- Clivaz, S. (2017). Teaching multidigit multiplication: Combining multiple frameworks to analyse a class episode. Educational Studies in Mathematics, 96(3), 305-325. [https://doi.org/10.1007/s10649-017-9770-7]
- Jeon, P. G., & Park, S. S. (2013). Research methods in mathematics education: Theory and Practice. Seoul: KyoWooSa.
- Kang, H. G. & Shim, S. Y. (2010). A 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.
- Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
- Kim, N. G. & Kim, J. E. (2009). A Study of the development of children's multiplication strategies and the computational resources. School Mathematics, 11(4), 745-771.
- 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.
- Larson, M. R. & Kanold, T. D. (2016). Balancing the equation: A guide the school mathematics for educators & parents. Bloomington, IN: Solution Tree Press.
- Ministry of Education (2018a). Elementary mathematics 3-1. Seoul: Chunjae Education.
- Ministry of Education (2018b). Elementary mathematics 3-2. Seoul: Chunjae Education.
- Ministry of Education (2018c). Elementary mathematics 4-1. Seoul: Chunjae Education.
- National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: The Author.
- 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.
- Reys, R. E., Lindquist, M. M., Lambdin, D. V., & Smith, N. L. (2015). Helping children learn mathematics (11th Ed.). New York: John Wiley & Sons.
- Sunwoo, J. (2019). A comparative analysis of instructional methods on the properties of multiplication in elementary mathematics textbooks of Korea, Japan, and the US. Education of Primary School Mathematic, 22(3), 181-203.
- Sunwoo, J. & Pang, J. S. (2019). An analysis of third graders’ understanding of the properties of multiplication by elementary mathematics instruction. Journal of Elementary Mathematics Education in Korea, 23(1), 143-168.
- Van De Walle, J. A., Karp, K., Bay-Williams, J. M., Wray, J. A., & Rigelman, N. R. (2014). Elementary and middle school mathematics: Teaching developmentally (8th ed.). Harlow: Pearson.
- 藤井斉亮외 41명(2015a). 新しい算數2下. 東京: 東京書籍.
- 藤井斉亮외 41명(2015b). 新しい算數3上. 東京: 東京書籍.