The Korea Society Of Educational Studies In Mathematics

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Journal of Educational Research in Mathematics - Vol. 31 , No. 1

[ Article ]
Journal of Educational Research in MathematicsVol. 30, No. 4, pp.649-673
Abbreviation: JERM
ISSN: 2288-7733 (Print) 2288-8357 (Online)
Print publication date 30 Nov 2020
Received 01 Oct 2020 Revised 08 Nov 2020 Accepted 17 Nov 2020

Pre-service Teachers’ Conceptions Revealed from Interpreting and Using Rate of Change
Lee, Jihyun* ; Yi, Gyuhee**,
*Professor, Incheon National University, South Korea (
**Teacher, Namsung Middle School, South Korea (

Correspondence to : Teacher, Namsung Middle School, South Korea,


The function concept arose from analyzing the relationships between co-varying quantities to model change. From the covariation perspective of function, we analyzed the responses of 13 pre-service teachers to the tasks to interpret rates of change or reason the corresponding change of one quantity from the changes of the other using the given rate of change; by using the rate of change, a change of one quantity can be directly deduced from the given change of the other quantity. Most of the pre-service teachers reasoned the change correctly. However, a portion of pre-service teachers reflexively calculated the formula of the function and the difference between the two values of the function. In addition, some teachers revealed their insufficient understanding of the slope of the linear function. The conceptions of pre-service teachers help elucidate what and how students might learn and might not learn about functions from the school mathematics curriculum.

Keywords: function, rate of change, change in quantity, covariation

1. Arons, A. B. (1997). Teaching introductory physics. New York: Wiley.
2. Ayalon, M., Watson, A., & Lerman, S. (2015). Progression Towards Functions: Students’ Performance on Three Tasks About Variables from Grades 7 to 12. International Journal of Science and Mathematics Education, 14(6), 1153-1173. doi:10.1007/s10763-014-9611-4
3. Ayalon, M., Watson, A., & Lerman, S. (2017). Students’ conceptualisations of function revealed through definitions and examples. Research in Mathematics Education, 19(1), 1-19. doi:10.1080/14794802.2016.1249397
4. Byerley, C., Hatfield, N., & Thompson, P. W. (2012). Calculus students’ understandings of division and rate. In Research in Undergraduate Mathematics Education.
5. Byerley, C., & Thompson, P. W. (2014). Secondary teachers’ relative size schemes. North American Chapter of the International Group for the Psychology of Mathematics Education.
6. Byerley, C., & Thompson, P. W. (2017). Secondary mathematics teachers’ meanings for measure, slope, and rate of change. The Journal of Mathematical Behavior, 48, 168-193. doi:10.1016/j.jmathb.2017.09.003
7. Byun, H. H. & Ju, M. K. (2012). Korean middle school students' conception of function. Journal of Educational Research in Mathematics, 22(3), 353-370.
8. Callahan, J. J. (2010). Advanced calculus: a geometric view: Springer Science & Business Media.
9. 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, 352-378.
10. Carlson, M., & Moore, K. (2015). The role of covariational reasoning in understanding and using the function concept. In E. A. Silver & P. A. Kenney (Eds.), Lessons learned from research: Useful and useable research related to core mathematical practices (Vol. 1, pp. 279-291): National Council of Teachers of Mathematics.
11. Chimhande, T., Naidoo, A., & Stols, G. (2017). An analysis of Grade 11 learners’ levels of understanding of functions in terms of APOS theory. Africa Education Review, 14(3-4), 1-19. doi:10.1080/18146627.2016.1224562
12. Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. In P. Cobb (Ed.), Learning mathematics constructivist and interactionist theories of mathematical development (pp. 31-60). Dordrecht: Springer.
13. 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.
14. Ellis, A. B. (2011). Algebra in the middle school: Developing functional relationships through quantitative reasoning. In Early algebraization A global dialogue from multiple perspectives (pp. 215-238). Berlin, Heidelberg: Springer.
15. Even, R. (1990). Subject Matter Knowledge for Teaching and the Case of Functions. Educational Studies in Mathematics, 21(6), 521-544.
16. Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 94-116.
17. Hatisaru, V., & Erbas, A. K. (2015). Mathematical Knowledge for Teaching the Function Concept and Student Learning Outcomes. International Journal of Science and Mathematics Education, 15(4), 703-722. doi:10.1007/s10763-015-9707-5
18. Hong, D. S., & Choi, K. M. (2018). A comparative analysis of linear functions in Korean and American standards-based secondary textbooks. International Journal of Mathematical Education in Science and Technology, 1-27.
19. Hughes-Hallett, D., Gleason, A., McCallum, W., Lomen, D., Lovelock, D., & Tecosky-Feldman, J. e. a. (2009). Calculus single variable (5 ed.). U.S.A: John Wiley & Sons.
20. Kang, Y. S., & Jun, S. A. (2006). An Inquiry on the Building Process of Pedagogical Content Knowledge of Prospective Mathematics Teachers-centered at function concepts. The Mathematical Education, 45(2), 217-230.
21. Kim, H. K. et al. (2019). Middle school mathematics 2. Seoul: Sinsago publisher.
22. Kwon, S. A. (2009). 9th grade students' perception of induced-unit appeared in middle school science textbooks. Unpublished master’s thesis. Korea National University of Education.
23. Lee, S. Y. (2004). Middle school students' perception about science units & factors affecting their understanding of units appeared from problem solving activities. Unpublished master’s thesis. Seoul National University.
24. Monk, S. (1992). Students' understanding of a function given by a physical model. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175-194), MAA Notes, 25. Washington, DC: Mathematical Association of America.
25. Nyikahadzoyi, M. R. (2015). Teachers' knowledge of the concept of a function: A theoretical framework. International Journal of Science and Mathematics Education, 13(2), 261-283.
26. Panorkou, N., Maloney, A. P., & Confrey, J. (2014). Expressing covariation and correspondence relationships in elementary schooling.
27. Sajka, M. (2003). A secondary school student's understanding of the concept of function-A case study. Educational Studies in Mathematics, 53(3), 229-254.
28. Sin, J. & Park, S.(2014) Analysis of units of velocity in elementary science textbooks. Paper presented at the Conference of Korean Society of Elementary Science Education.
29. Simon, M. A. (1993). Prospective Elementary Teachers’ Knowledge of Division. Journal for Research in Mathematics Education, 24(3), 233-254.
30. Smith, E., & Confrey, J. (1994). Multiplicative structures and the development of logarithms: What was lost by the invention of function. In The development of multiplicative reasoning in the learning of mathematics (pp. 331-360).
31. Son. J. H. (2013). A Study in teaching units of middle school : Comparison in terms of calculations and units of the formula calculation. Unpublished master’s thesis. Ewha University.
32. Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope. Mathematics Education Research Journal, 11(2), 124-144.
33. Stump, S. L. (2001). High school precalculus students’ understanding of slope as measure. School science and mathematics, 101(2), 81-89.
34. Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. Research in collegiate mathematics education, 1, 21-44.
35. Thompson, P. W., & Milner, F. (2019). Teachers’ Meanings for Function and Function Notation in South Korea and the United States. In The Legacy of Felix Klein (pp. 55-66).
36. Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34(2), 97-129.
37. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 356-366.