The Korea Society Of Educational Studies In Mathematics

Current Issue

Journal of Educational Research in Mathematics - Vol. 30

[ Special ]
Journal of Educational Research in Mathematics - Vol. 30, No. SP1, pp.213-227
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 Aug 2020
DOI: https://doi.org/10.29275/jerm.2020.08.sp.1.213

An Analysis of Fifth and Sixth Graders’ Algebraic Thinking about Reverse Fraction Problems
JeongSuk Pang* ; SeonMi Cho**, ; Minsung Kwon***
*Professor, Korea National University of Education, South Korea
**Teacher, Seoul Sungsan Elementary School, South Korea
***Professor, California State University at Northridge, USA

Correspondence to : Email: jeongsuk@knue.ac.kr, halifax01@naver.com, minsung.kwon@csun.edu

Please cite this article as: Pang, J., Cho, S., & Kwon, M. An analysis of fifth and sixth graders’ algebraic thinking about reverse fraction problems.


Abstract

In light of a growing emphasis on the relationship between fractional knowledge and algebraic thinking, the purpose of this study was to compare algebraic thinking about reverse fraction problems between fifth graders who had not been taught fraction division and sixth graders who had been taught fraction division. For this purpose, we conducted cognitive interviews with 30 students (15 fifth graders and 15 sixth graders) who used different solution methods in a paper-and-pencil test dealing with reverse fraction problems. We analyzed their algebraic thinking in terms of problem structure reasoning, generalization-representation, and justification. The results showed that fifth graders could generalize their solution methods but sixth graders who learned only the algorithm of fraction division had difficulties with generalization and justification. However, sixth graders who reasoned about the problem structure demonstrated proficiency in algebraic thinking. Although some students used the same solution methods in the paper-and-pencil test, they had different capacities in generalizing, representing, and justifying them during the cognitive interviews. The results of this study provide implications for teaching fraction operations in a way that develops algebraic thinking by attending to the mathematical structure and relation.


Keywords: Reverse fraction, Algebraic thinking, Reasoning the problem structure, Generalization, Representation, Justification

References
1. Barnett-Clarke, C., Fisher, W., Marks, R., & Ross, S. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
2. 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.
3. Blanton, M. et al. (2018). Implementing a framework for early algebra. In C. Kieran (Ed.), Teaching and learning algebraic thinking with 5- to 12-year-olds (pp. 27-49). Switzerland: Springer.
4. 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.
5. 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.
6. Cooper, T., & Warren, E. (2011). Years 2 to 6 students' ability to generalise: Models, representations and theory for teaching and learning. In J. Cai, & E. Kunth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 187-214). New York: Springer.
7. 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.
8. 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.
9. Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York: Lawrence Erlbaum.
10. Kieran, C., Pang, J., Schifter, D., & Ng, S. F. (2016). Early algebra: Research into its nature, its learning, its teaching. New York: Springer Open eBooks.
11. 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.
12. 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.
13. 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 Journal of Mathematics Education, 14(1), 109-122.
14. Lobato, J., Ellis, A. B., Charles, B. I., & Zbiek, R. M. (2010). Developing essential understanding of ratio, proportions, and proportional reasoning for teaching mathematics in grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
15. Moss, J., & McNab, S. L. (2011). An approach to geometric and numeric patterning that fosters second grade students’reasoning and generalizing about functions and co-variation. In J. Cai, & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 277-301). New York: Springer.
16. Ministry of Education (2018). Elementary mathematics 6-1 (in Korean). Seoul: Chunjae Education.
17. Ministry of Education (2019). Teacher manual for Elementary Mathematics 6-2 (in Korean). Seoul: Chunjae Education.
18. Ng, S. F. (2015). A theoretical framework for understanding the different attention resource demands of letter-symbolic verse model method. In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education (pp. 2129-2137). Cham: Springer.
19. 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.
20. Pang, J. S., & Kim, S. M. (2018). An analysis of the elementary mathematics textbooks on the generalization of properties of numbers and operations. School Mathematics, 20(1), 251-267.
21. Pang, J. S., & Cho, S. M. (2019a). An analysis of solution methods by fifth grade students about ‘reverse fraction problems’. The Mathematical Education, 58(1), 1-20.
22. 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.
23. Pearn, C., Pierce, R., & Stephens, M. (2017). In B. Kaur, W. K. Ho, T. L. Toh, & B. H. Choy (Eds.), Reverse fractional tasks reveal algebraic thinking. Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, (pp. 1-8). Singapore: PME.
24. 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.
25. Radford, L. (2011). Embodiment, perception and symbols in the development of early algebraic thinking. In B. Ubuz (Ed.). Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, (pp. 17-24). Ankara, Turkey: PME.
26. Radford, L., & Roth, W. M. (2011). Intercorporeality and ethical commitment: An activity perspective on classroom interaction. Educational Studies in Mathematics, 77, 227-245.
27. Russell, S. J., Shifter, D., & BasTable, V. (2011). Connecting arithmetic to Algebra. Portsmouth, NH: Heinemann.
28. Siegler, R. et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(10), 691-697.