The Korea Society Of Educational Studies In Mathematics

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Journal of Educational Research in Mathematics - Vol. 30 , No. 4

[ Article ]
Journal of Educational Research in Mathematics - Vol. 30, No. 4, pp.705-732
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 30 Nov 2020
Received 08 Oct 2020 Revised 02 Nov 2020 Accepted 06 Nov 2020

Secondary Students’ Reasoning about Covarying Quantities in Quadratic Function Problems
Lee, Jin Ah* ; Lee, Soo Jin**,
*Teacher, Hyowon High School, South Korea (
**Professor, Korea National University of Education, South Korea (

Correspondence to : Professor, Korea National University of Education, South Korea,


The purpose of this study is to investigate how two 7th grade students with different concepts of ratio reason about change in covarying quantities in quadratic function problems and how these differences affect the process of expressing algebraic equations. To this end, we selected two 7th grade students who had no formal instruction in quadratic functions and four clinical interviews (each of which lasted about 90 minutes) were conducted every week from December 2019 to January 2020 with each of them. Our analysis revealed that one student whose conception of ratio was at the level of additive comparison attended to differences in amount of change in quadratic problem situations and interpreted the problems additively by repetitively adding the differences, whereas another student who operated at the interiorized ratio level could perceive the problem situations as the differences in amount of change per unit time, which resulted in their distinctive ways of expressing quadratic functions as equations. We have finalized our study by suggesting an explanatory hypothesis about students’ ratio concepts that seem highly pertinent to their ability to interpret quantitatively and to express equations of quadratic functions.

Keywords: ratio, internalized ratio, interiorized ratio, quantification, quantitative reasoning, rate of change, quadratic function

1. Ahn, S. H., & Pang, J. S. (2008). A survey on the proportional reasoning ability of fifth, sixth, and seventh graders. Journal of Educational Research in Mathematics, 18(1), 103-121.
2. Clement, J. (2000). Analysis of clinical interview: foundations and model viability. In Lesh, R. and Kelly, A., Handbook of research methodologies of science and mathematics education (pp. 341-385). Hillsdale, NJ: Lawrence Erlbaum.
3. 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.
4. Ellis, A. B. (2007). The influence of reasoning with emergent quantities on students’ generalizations. Cognition and Instruction, 25(4), 439-478.
5. Ellis, A. B. (2011). Algebra in the middle school: developing funtional relationship through quantitative reasoning. In J. Cai, & E. Kunth (Eds), Early algebraization (pp. 215-238). Springer-Verlag Berlin Heidelberg.
6. Fonger, N. C., Dogan, M. F., & Ellis, A. (2017). Students’ clusters of concepts of quadratic functions. In Kaur, B., Ho, W.K., Toh, T.L., &, B.H. (Eds.). Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education, (Vol. 2, pp. 329-336). Singapore: PME.
7. Jeon, H. O., Lee, K. H., & Pang, J. S. (2009). Case study of the sixth grade students' quantitative reasoning. Journal of Educational Research in Mathematics, 19(1), 81-98.
8. Jeong, E. S. (2013). Study on proportional reasoning in elementary school mathematics. Journal of Educational Research in Mathematics, 23(4), 505-516.
9. Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities. Mathematical Thinking and Learning, 17(1), 64-90.
10. Kang, H., & Choi, E. A. (2015). Teacher knowledeg necessary to address student errors and difficuities about ratio and rate. School Mathematics, 17(4), 613-632.
11. Kim, J. W. & Pang, J. S. (2013). An analysis on third graders’ multiplicative thinking and proportional reasoning ability. Journal for Research in Mathematics Education, 23(1), 1-16.
12. Ko, E. S., & Lee, K. H. (2007). Analysis on elementary students’ proportional thinking : a case study with two 6-graders. Journal for Research in Mathematics Education, 17(4), 359-380.
13. Kwon, O. N., Park, J. S., & Park, J. H. (2007). An analysis on mathematical concepts for proportional reasoning in the middle school mathematics curriculum. School Mathematics, 46(3), 315-329.
14. Lee, D. G., Kim, S. H., Ahn, S. J., & Shin, J. H. (2016). Analysis on high school students’ recognitions and expressions of changes in concentration as a rate of change. Journal of Educational Research in Mathematics, 26(3), 333-354.
15. Lobato, J., & Ellis, A. (2010). Developing essential understanding of ratios, proportions, and proportional reasoning for teaching mathematics, Grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
16. Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The Case of Quadratic Functions. Mathematical Thinking and Learning, 14, 85-119.
17. Ma, M., & Im, D. (2019). Two middle school students’ understanding of ‘constant rate of change’ and ‘constant rate of change of rate of change’. School Mathematics, 21(3), 607-624.
18. Ministry of Education (2015). Mathematics and curriculum. Notification of Ministry of Education No. 2015-74 [Vol. 8].
19. Park, J., & 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.
20. Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals. Mathematical Thinking and Learning, 8(4), 359-371.
21. Simon, M. A., & Placa, N. (2012). Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding. For the Learning of Mathematics, 32(2), 35-41.
22. Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Erlbaum.
23. Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebra. Center for Research in Mathematics & Science Education: San Diego State University.
24. 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.
25. 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.