SECONDARY STUDENTS’ COVARIATIONAL REASONING IN SOLVING THE FILLING BOTTLE PROBLEM
DOI:
https://doi.org/10.24127/ajpm.v14i3.12981Keywords:
Covariational reasoning, filling bottle problem, functional graphAbstract
Covariational reasoning is closely related to the concept of functions, as both involve relationships between two quantities. Although it is useful for understanding change and interdependence between quantities, the notion of covariation is rarely introduced directly to students since it is not explicitly included in the school curriculum. This study aims to explore students’ covariational reasoning in constructing function graphs in the context of a filling bottle problem. Three aspects of covariational reasoning were analyzed: identifying variables, ways of coordinating variables, and quantifying the rate of change. This qualitative research employed a case study design and involved three 11th-grade students selected through purposive sampling. Data were obtained from students’ responses to the filling bottle task and interviews, and analyzed through data condensation, data display, and conclusion drawing. The findings indicate that (1) students had difficulty identifying the independent and dependent variables in a functional problem, especially graph, (2) students tended to rely on secondary variables when coordinating the independent and dependent variables, and (3) students’ ability to quantify the rate of change depended on their ability to identify and coordinate the two variables.References
Bagossi, S. (2024). Second-order covariation: enlarging the theoretical framework of covariational reasoning. Mathematical Thinking and Learning, 1–22. https://doi.org/10.1080/10986065.2024.2393889
Bagossi, S., Ferretti, F., & Arzarello, F. (2022). Assessing covariation as a form of conceptual understanding through comparative judgement. Educational Studies in Mathematics, 111(3), 469–492. https://doi.org/10.1007/s10649-022-10178-w
Belin, M., Ä°ÅŸler-baykal, I., Akar, G. K., Belin, M., Ä°ÅŸler-baykal, I., Karagöz, G., & An, A. (2023). An investigation of two mathematics teachers’ covariational reasoning. Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13), 3711–3718. https://www.researchgate.net/publication/383603322_An_investigation_of_two_mathematics_teachers’_covariational_reasoning
Büchter, T., Eichler, A., Böcherer-Linder, K., Vogel, M., Binder, K., Krauss, S., & Steib, N. (2024). Covariational reasoning in Bayesian situations. Educational Studies in Mathematics, 115(3), 481–505. https://doi.org/10.1007/s10649-023-10274-5
Dinkelman, M. O., & Cavey, L. O. (2015). Learning about Functions through Learner-Generated Examples. The Mathematics Teacher, 109(2), 104–110. https://doi.org/10.5951/mathteacher.109.2.0104
Donovan, C., Johnson, H. L., Knurek, R., Whitmore, K. A., & Bechtold, L. (2024). Validating a measure of graph selection and graph reasoning for dynamic situations. Journal of Mathematical Behavior, 73, 1–22. https://doi.org/10.1016/j.jmathb.2024.101137
Johnson, H. L., McClintock, E. D., & Gardner, A. (2020). Opportunities for Reasoning: Digital Task Design to Promote Students’ Conceptions of Graphs as Representing Relationships between Quantities. Digital Experiences in Mathematics Education, 6(3), 340–366. https://doi.org/10.1007/s40751-020-00061-9
Johnson, H. L., McClintock, E., & Hornbein, P. (2017). Ferris wheels and filling bottles: a case of a student’s transfer of covariational reasoning across tasks with different backgrounds and features. ZDM - Mathematics Education, 49(6), 851–864. https://doi.org/10.1007/s11858-017-0866-4
Kalpokaite, N., & Radivojevic, I. (2019). Demystifying qualitative data analysis for novice qualitative researchers. Qualitative Report, 24(13), 44–57. https://doi.org/10.46743/2160-3715/2019.4120
Kertil, M. (2020). Covariational reasoning of prospective mathematics teachers: How do dynamic animations affect? Turkish Journal of Computer and Mathematics Education, 11(2), 312–342. https://doi.org/10.16949/turkbilmat.652481
Kertil, M., Erbas, A. K., & Cetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21(3), 207–233. https://doi.org/10.1080/10986065.2019.1576001
Moleko, M. M. (2021). Teachers’ Perspectives on Addressing Linguistic Factors Affecting Visualisation of Mathematics Word Problems. Eurasia Journal of Mathematics, Science and Technology Education, 17(11), 1–18. https://doi.org/10.29333/ejmste/11248
Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students’ graphing activity. RUME, 782–789. https://bit.ly/Shapethinking
Paoletti, T., Gantt, A. L., & Vishnubhotla, M. (2022). Constructing a system of covariational relationships: two contrasting cases. Educational Studies in Mathematics, 110(3), 413–433. https://doi.org/10.1007/s10649-021-10134-0
Patterson, C. L., & McGraw, R. (2018). When time is an implicit variable: An investigation of students’ ways of understanding graphing tasks. Mathematical Thinking and Learning, 20(4), 295–323. https://doi.org/10.1080/10986065.2018.1509421
Stalvey, H. E., & Vidakovic, D. (2015). Students’ reasoning about relationships between variables in a real-world problem. Journal of Mathematical Behavior, 40, 192–210. https://doi.org/10.1016/j.jmathb.2015.08.002
Sterner, H. (2024). Using Graphical Representations to Develop Students’ Correspondence Relationships and Covariational Thinking in Pattern Generalizations in Primary School. International Journal of Science and Mathematics Education, 0123456789. https://doi.org/10.1007/s10763-024-10520-z
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of mathematical thinking. Compendium for Research in Mathematics Education, January, 421–456. https://bit.ly/variationcovariation
Wilkie, K. J. (2020). Investigating Students’ Attention to Covariation Features of their Constructed Graphs in a Figural Pattern Generalisation Context. International Journal of Science and Mathematics Education, 18(2), 315–336. https://doi.org/10.1007/s10763-019-09955-6
Yemen-Karpuzcu, S., Ulusoy, F., & Işıksal-Bostan, M. (2017). Prospective Middle School Mathematics Teachers’ Covariational Reasoning for Interpreting Dynamic Events During Peer Interactions. International Journal of Science and Mathematics Education, 15(1), 89–108. https://doi.org/10.1007/s10763-015-9668-8
Downloads
Published
Issue
Section
License
Authors who publish with AKSIOMA: Jurnal Program Studi Pendidikan Matematika agree to the following terms:
Creative Commons License
AKSIOMA:Â Jurnal Program Studi Pendidikan Matematika is licensed under a Creative Commons Attribution 4.0 International License.
This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. AKSIOMA offers all authors of journal articles allows their research openly available, free access and time restrictions.
All articles published Open Access will be immediately and permanently free for everyone to read and download. Under the CC-BY license, authors retain ownership of the copyright for their article, but authors grant others permission to use the content of publications in AKSIOMA in whole or in part provided that the original work is properly cited. Users (redistributors) of AKSIOMA are required to cite the original source, including the author's names, AKSIOMA as the initial source of publication, year of publication, volume number and DOI (if available).
