Exploring students’ understanding of linear and quadratic relationships in a projectile motion context
DOI:
https://doi.org/10.24071/icre.v1i1.2Keywords:
covariational reasoning, linear relationship, projectile motion, quadratic relationshipAbstract
Previous research has shown that students often struggle to develop an understanding of linear and quadratic relationships. Covariational reasoning has been identified as a way to support this development. This study aims to investigate how covariational reasoning supports students in developing understandings of linear and quadratic relationships within a projectile motion context. A teaching experiment was conducted with two middle school students who engaged in a digital task exploring the relationship between height and time. The analysis characterizes how the students’ covariational reasoning evolved as they interpreted the changing quantities in the task. The findings suggest that prompts encouraging students to compare linear and quadratic relationships can foster more sophisticated forms of covariational reasoning. The discussion highlights how specific features of the task design, including the affordances of technology, the emphasis on conceiving graphs as representations of covarying quantities, and the use of non-canonical graphing tasks, can support covariational reasoning.
Downloads
References
Abrams, W. (2023). Baseball as a quantitative reasoning course. PRIMUS, 33(1), 84–96. https://doi.org/10.1080/10511970.2022.2032507
Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2–3), 131–152. https://doi.org/10.1016/S0360-1315(99)00029-9
Altindis, N. (2025). Networking theories of quantitative reasoning and mathematical reasoning to explore students’ understanding of functions. The Journal of Mathematical Behavior, 80, Article 101276. https://doi.org/10.1016/j.jmathb.2025.101276
Clement, J. (1989). The concept of variation and misconceptions in Cartesian graphing. Focus on Learning Problems in Mathematics, 11(1–2), 77–87. https://people.umass.edu/~clement/pdf/concept_of_variation.pdf
Czocher, J. A., Hardison, H. L., & Kularajan, S. S. (2022). A bridging study analyzing mathematical model construction through a quantities-oriented lens. Educational Studies in Mathematics, 111(2), 299–321. https://doi.org/10.1007/s10649-022-10163-3
Hadjidemetriou, C., & Williams, J. (2002). Children’s graphical conceptions. Research in Mathematics Education, 4(1), 69–87. https://doi.org/10.1080/14794800008520103
Harel, G. (2013). Intellectual need. In K. R. Leatham (Ed.), Vital directions for mathematics education research (pp. 119–151). Springer New York. https://doi.org/10.1007/978-1-4614-6977-3_6
Hohensee, C., Gartland, S., Melville, M., & Willoughby, L. (2024). Comparing contrasting instructional approaches: A way for research to develop insights about backward transfer. Research in Mathematics Education, 1–21. https://doi.org/10.1080/14794802.2024.2388067
Jiménez, F. R. S., & Sierra, G. M. (2025). Mathematical modelling and covariational reasoning: A multiple case study. International Journal of Mathematical Education in Science and Technology, 56(10), 1964–1993. https://doi.org/10.1080/0020739X.2024.2379485
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
Kendon, A. (2000). Language and gesture: Unity or duality? In D. McNeill (Ed.), Language and Gesture (1st ed., pp. 47–63). Cambridge University Press. https://doi.org/10.1017/CBO9780511620850.004
Kong, A. P.-H., Law, S.-P., & Chak, G. W.-C. (2017). A comparison of coverbal gesture use in oral discourse among speakers with fluent and nonfluent aphasia. Journal of Speech, Language, and Hearing Research, 60(7), 2031–2046. https://doi.org/10.1044/2017_JSLHR-L-16-0093
Kristanto, Y. D., & Lavicza, Z. (2025). Down-to-earth mathematics: Transforming curriculum and teaching on function transformations. In Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14): Vol. TWG22: Curricular Resources and Task Design in Mathematics Education. Free University of Bozen-Bolzano; ERME. https://hal.science/hal-05294249
Kristanto, Y. D., Lavicza, Z., & Martinelli, A. (2025). It’s time to build skyscrapers: A digital task sequence on graphs as representations of covarying quantities. In D. Koperová & M. Rusek (Eds.), Project-Based Education and Other Student-Activating Strategies and Issues in STE(A)M Education XXII. (pp. 23–33). Charles University.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64. https://doi.org/10.3102/00346543060001001
Martínez-Sierra, G., & Villadiego Franco, K. J. (2025). Exploring students’ covariational reasoning in sine and cosine functions: A comparison of expected and manifested learning trajectories with dynamic tasks. The Journal of Mathematical Behavior, 80, 101260. https://doi.org/10.1016/j.jmathb.2025.101260
Moore, K. C. (2021). Graphical shape thinking and transfer. In C. Hohensee & J. Lobato (Eds.), Transfer of learning (pp. 145–171). Springer International Publishing. https://doi.org/10.1007/978-3-030-65632-4_7
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 Annual Conference on Research in Undergraduate Mathematics Education (pp. 782–789). West Virginia University.
Moore, K. C., Silverman, J., Paoletti, T., & LaForest, K. (2014). Breaking conventions to support quantitative reasoning. Mathematics Teacher Educator, 2(2), 141–157. https://doi.org/10.5951/mathteaceduc.2.2.0141
Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019). Conventions, habits, and U.S. teachers’ meanings for graphs. The Journal of Mathematical Behavior, 53, 179–195. https://doi.org/10.1016/j.jmathb.2018.08.002
Paneth, L., Jeitziner, L. T., Rack, O., Opwis, K., & Zahn, C. (2024). Zooming in: The role of nonverbal behavior in sensing the quality of collaborative group engagement. International Journal of Computer-Supported Collaborative Learning, 19(2), 187–229. https://doi.org/10.1007/s11412-024-09422-7
Paoletti, T., & Vishnubhotla, M. (2022). Constructing covariational relationships and distinguishing nonlinear and linear relationships. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative reasoning in mathematics and science education (Vol. 21, pp. 133–167). Springer International Publishing. https://doi.org/10.1007/978-3-031-14553-7_6
Paoletti, T., Gantt, A. L., & Corven, J. (2023). A local instruction theory for emergent graphical shape thinking: A middle school case study. Journal for Research in Mathematics Education, 54(3), 202–224. https://doi.org/10.5951/jresematheduc-2021-0066
Paoletti, T., Hardison, H. L., & Lee, H. Y. (2022a). Students’ static and emergent graphical shape thinking in spatial and quantitative coordinate systems. For the Learning of Mathematics, 42(2), 48–50.
Paoletti, T., Lee, H. Y., Rahman, Z., Vishnubhotla, M., & Basu, D. (2022b). Comparing graphical representations in mathematics, science, and engineering textbooks and practitioner journals. International Journal of Mathematical Education in Science and Technology, 53(7), 1815–1834. https://doi.org/10.1080/0020739X.2020.1847336
Paoletti, T., Moore, K. C., & Vishnubhotla, M. (2024a). Intellectual need, covariational reasoning, and function: Freeing the horse from the cart. The Mathematics Educator, 32(1), 39–72.
Paoletti, T., Stevens, I. E., Acharya, S., Margolis, C., Olshefke-Clark, A., & Gantt, A. L. (2024b). Exploring and promoting a student’s covariational reasoning and developing graphing meanings. The Journal of Mathematical Behavior, 74, 101156. https://doi.org/10.1016/j.jmathb.2024.101156
Pittalis, M., Sproesser, U., & Demosthenous, E. (2025). Graphically representing covariational functional situations in an interactive embodied digital learning environment. International Journal of Mathematical Education in Science and Technology, 56(6), 1083–1113. https://doi.org/10.1080/0020739X.2024.2327552
Saldanha, L. A., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.), Proceedings of the 20th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 298–304). North Carolina State University.
Sokolowski, A. (2021a). Enabling covariational reasoning in Einstein’s formula for photoelectric effect. Physics Education, 56(3), 035029. https://doi.org/10.1088/1361-6552/abed3a
Sokolowski, A. (2021b). Parametrization of projectile motion. In A. Sokolowski, Understanding physics using mathematical reasoning (pp. 101–126). Springer International Publishing. https://doi.org/10.1007/978-3-030-80205-9_8
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–307). Erlbaum. https://www.taylorfrancis.com/chapters/mono/10.4324/9781410602725-18/teaching-experiment-methodology-underlying-principles-essential-elements-anthony-edward-kelly-richard-lesh
Stewart, J., Redlin, L., & Watson, S. (2016). Algebra and trigonometry (Fourth edition/Student edition). Cengage Learning.
Tallman, M. A., Weaver, J., & Johnson, T. (2024). Developing (Pedagogical) content knowledge of constant rate of change: The case of Samantha. The Journal of Mathematical Behavior, 76, 101179. https://doi.org/10.1016/j.jmathb.2024.101179
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). National Council of Teachers of Mathematics. https://www.nctm.org/Store/Products/Compendium-for-Research-in-Mathematics-Education/
Van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2008). The linear imperative: An inventory and conceptual analysis of students’ overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311–342.
Wilkie, K. J. (2019). Investigating secondary students’ generalization, graphing, and construction of figural patterns for making sense of quadratic functions. The Journal of Mathematical Behavior, 54, 100689. https://doi.org/10.1016/j.jmathb.2019.01.005
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
Wilkie, K. J., & Ayalon, M. (2018). Investigating years 7 to 12 students’ knowledge of linear relationships through different contexts and representations. Mathematics Education Research Journal, 30(4), 499–523. https://doi.org/10.1007/s13394-018-0236-8
Yu, F. (2024). Extending the covariation framework: Connecting covariational reasoning to students’ interpretation of rate of change. The Journal of Mathematical Behavior, 73, 101122. https://doi.org/10.1016/j.jmathb.2023.101122
Yu, F. (2025). A Quantitative and Covariational Reasoning Approach to Introducing Instantaneous Rate of Change: A Lesson Analysis Manuscript. PRIMUS, 1–17. https://doi.org/10.1080/10511970.2025.2518528
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Yosep Dwi Kristanto, Teo Paoletti, Russasmita Sri Padmi, Serli Evidiasari, Zsolt Lavicza, Tony Houghton, Houssam Kasti (Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
