[ Article ]
Journal of Educational Research in Mathematics - Vol. 30, No. 2, pp.281-305
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 31 May 2020
Received 06 Apr 2020 Revised 13 May 2020 Accepted 14 May 2020

# Relationship Between Proportional Reasoning and Covariational Reasoning of 7th Grade Students

Park, JongHee* ; Lee, Soo Jin**,
*Teacher, Cheonan Yonggok Middle School, South Korea cyber-iris@hanmail.net
**Professor, Korea National University of Education, South Korea sjlee@knue.ac.kr

Correspondence to: Professor, Korea National University of Education, South Korea, sjlee@knue.ac.kr

## Abstract

To understand the relationship between students’ proportional reasoning and covariational reasoning, clinical interview was conducted with four 7th grade students. Students participated in four interviews each of which took about three-four hours. The analysis of the data suggests the following results: First, all four students conceived ratio as ‘internalized ratio’ but how they reasoned with the ratio concept were somewhat different, which were categorized into three different types of proportional reasoning. Secondly, there were differences in how the four students conceived two quantities in graph construction and interpretation tasks. Specifically, two of the four students showed ‘coordination of values’ level, one showed ‘chunky continuous covariational reasoning’, and the other showed ‘smooth continuous covariational reasoning’. Thirdly, there was little relationship as to how students conceived ratio and how they reasoned with quantities in function task. All four students perceived the continuity of function in various ways. We have finalized our study by suggesting students' ways of partitioning and iterating of two quantities in proportion and function tasks as potential theoretical constructs to distinguish how students reason.

## Keywords:

proportional reasoning, covariational reasoning, proportion, function, constant comparative method

## Acknowledgments

본 연구는 제1저자의 2019년 박사학위 논문의 내용을 토대로 재구성, 수정 및 보완하여 작성하였음.

## References

• Beckmann, S., & Izsák, 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]
• Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers' meanings for measure, slope, and rate of change. Journal of Mathematics Behavior, 48, 168-193. [https://doi.org/10.1016/j.jmathb.2017.09.003]
• Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352-378. [https://doi.org/10.2307/4149958]
• Castillo-Garsow, C. W. (2012). Continuous quantitative reasoning. In Mayes, R., Bonillia, R., Hatfield, L. L., and Belbase, S. (Eds.), Quantitative reasoning and Mathematical modeling: A driver for STEM Integrated Education and Teaching in Context. WISDOMe Monographs, Volume 2 (pp. 55-73). Laramie, WY: University of Wyoming Press.
• Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In Lesh, R. and Kelly, A., Handbook of research methodologies for science and mathematics education (pp. 341-385). Hillsdale, NJ: Lawrence Erlbaum.
• Ellis, A. (2013). Teaching ratio and proportion in the middle grades: Ratio and proportion. Research Brief. Reston, VA: National Council of Teachers of Mathematics.
• Glaser, B., & Strauss, A. (1999). Discovery of Grounded Theory: Strategies for Qualitative Research. London & New York: Routledge.
• Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. Los Angeles: Sage.
• Karplus Robert, Steven Pulos, & Elizabeth Stage. (1983). Early Adolescents' Proportional Reasoning on Rate Problems. Educational Studies in Mathematics. 14. 219-233. [https://doi.org/10.1007/BF00410539]
• Kim, C. Y., & Shin, J. (2018). A Case Study of Students` Constructions and Interpretations of Informal Graphs. Journal of Korea Society Educational Studies in Mathematics School Mathematics, 20(1), 107-130. [https://doi.org/10.29275/sm.2018.03.20.1.107]
• Kwon, O. N., Park, J. S., & Park, J. H. (2007). An analysis on mathematical concepts for proportional reasoning in the middle school mathematics curriculum. The Mathematical Education, 46(3), 315-319.
• National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards for school mathematics. Reston, VA:NCTM.
• Leinhardt, G., Zaslavsky, O., & Stein, M. M. (1990). Functions, graphs, and graphing: Tasks, learning and teaching. Review of Educational Research, 60, 1-64. [https://doi.org/10.3102/00346543060001001]
• 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.
• Ma, M. Y., & Shin, J. (2017). Two Middle School Students' Proportional Reasoning Emerging through the Process of Expressing and Interpreting the Function Graphs. Journal of Korea Society Educational Studies in Mathematics School Mathematics, 19(2), 345-367.
• Merriam, S. B. (1998). Qualitative Research and Case Study Applications in Education. Revised and Expanded from "Case Study Research in Education.". San Francisco: Jossey-Bass Publishers.
• Ministry of Education. (2015a). Mathematics curriculum. Notification of the Ministry of Education No. 2015-74. [Vol. 8]
• Ministry of Education. (2015b). Elementary mathematics teacher's guide book 6-1. Seoul: Chunjae Education.
• Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students' graphing activity. In T. Fukawa-Connelly, N. E. Infante, K. Keene & M. Zandieh (Eds.), Proceedings of the 18th Meeting of the MAA Special Interest Group on Research in Undergraduate Mathematics Education, pp. 782-789. Pittsburgh, PA: RUME.
• Park, J. H., Shin, J., Lee, S. J., & Ma, M. Y. (2017). Analyzing Students’ Works with Quantitative and Qualitative Graphs Using Two Frameworks of Covariational Reasoning. Journal of Educational Research in Mathematics. 28(1), 1-26. [https://doi.org/10.29275/jerm.2018.02.28.1.1]
• Park, J. H., & Lee, S. J. (2018). A Comparative Study of the Mathematics Textbooks of Korea and the United States based on a Learning Trajectory for the Concept of Slope. Journal of Educational Research in Mathematics, 28(1), 1-26. [https://doi.org/10.29275/jerm.2018.02.28.1.1]
• Post, T, Behr, M., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings. In Algebraic concepts in the curriculum K-12 (1988 Yearbook). Reston, VA: National Council of Teachers of Mathematics. 33 Schwartz, J. L. (1983). The semantic calculator users manual. New York: Sunburst Communication.
• Ryu, K., Jung, J. W., Kim, Y. S., & Kim, H. B. (2018). Understanding Qualitative Research Methods. Seoul; Park Young Sa.
• Shin, J., & Lee, S. J. (2019). Conceptual Analysis for Solving a Missing Value Problem Using a Proportional Relationship. Journal of Educational Research in Mathematics, 29(2), 227-250. [https://doi.org/10.29275/jerm.2019.5.29.2.227]
• Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.
• Thompson, P. W. (1994). The Development of the Concept of Speed and its Relationship to Concepts of Rate. In G. Harel & J. Confrey(Eds.), The Development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.
• Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education, WISDOMe Monographs (Vol. 1, pp. 33-57). Laramie, WY: University of Wyoming.
• Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics.