Challenges in teaching students to plot equations: Another impact of graphing procedures

Authors

DOI:

https://doi.org/10.29408/jel.v10i3.25278

Keywords:

covariational reasoning, didactical obstacles, graphs, mathematical understanding

Abstract

Graphs are essential representations in mathematics, but many students only focus on procedural rules when sketching graphs of equations. This study aims to describe the errors and obstacles experienced by students in drawing graphics of equations. The research method uses a qualitative approach by collecting data through tests and interviews. The research results show that students' mistakes in drawing graphs can be caused by dependence on procedural knowledge in plotting points without conceptual understanding. Students who make mistakes do not understand points as multiplicative objects, lack covariational reasoning, need help understanding the concept of complex numbers, and show reflective thinking in completing assignments. This research implies that teachers must consider the students' difficulties and mistakes to avoid becoming didactic learning obstacles. In studying plotting point procedures, students' knowledge must include understanding concepts related to graphs, such as the meaning of points and graph concavity, graph shifts, and the relationship between the discriminant of a quadratic equation and its graph.

Author Biographies

Ulumul Umah, Universitas Pesantren Tinggi Darul Ulum

Mathematics Education Department

Ana Rahmawati, Universitas Pesantren Tinggi Darul Ulum

Mathematics Education Department

References

Andalia, N., AG, B., & Zulfajri, M. (2020). The student's ability in graph understanding for mastering natural science concepts through the process skills approach. International Journal of Instruction, 13(4), 145–160.

Barbieri, C. A., Miller-Cotto, D., & Booth, J. L. (2019). Lessening the load of misconceptions: Design-based principles for algebra learning. Journal of the Learning Sciences, 28(3), 381–417.

Barmby, P., Harries, T., Higgins, S., & Suggate, J. (2007). How can we assess mathematical understanding. Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, 41–48.

Birgin, O. (2012). Investigation of eighth-grade students' understanding of the slope of the linear function. Bolema - Mathematics Education Bulletin, 26(42 A), 139–162. https://doi.org/10.1590/S0103-636X2012000100008

Bossé, M. J., Adu-Gyamfi, K., & Cheetham, M. R. (2011). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6(3), 113–133. https://doi.org/10.29333/iejme/264

Bragdon, D., Pandiscio, E., & Speer, N. (2019). University students' graph interpretation and comprehension abilities. Investigations in Mathematics Learning, 11(4), 275–290.

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23(3), 247–285. https://doi.org/10.1007/BF02309532

Brousseau, G. (2002). Epistemological obstacles, problems, and didactical engineering. In B. Nicolas, C. Martin, S. Rosamund, & W. Virginia (Ed.), Theory of Didactical Situations in Mathematics (Nomor 19, hal. 79–117). https://doi.org/10.1007/0-306-47211-2_6

Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. Research in Collegiate Mathematics Education, 1(7), 115–162. https://doi.org/10.1090/cbmath/007/04

Carlson, M. P., 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

Carlson, M. P., Madison, B., & West, R. D. (2015). A study of students' readiness to learn calculus. International Journal of Research in Undergraduate Mathematics Education, 1(2), 209–233. https://doi.org/10.1007/s40753-015-0013-y

Castillo-Garsow, C., Johnson, H. L., & Moore, K. C. (2013). Chunky and smooth images of change. For the Learning of Mathematics, 33(3), 31–37.

Cavey, L. O., Totorica, T., Libberton, J., Carney, M., Souders, K., & Lowenthal, P. (2019). A framework for analyzing secondary students' covariational reasoning. Annual research conference of the National Council of Teachers of Mathematics.

Chin, C., Brown, D. E., & Bruce, B. C. (2002). Student-generated questions: A meaningful aspect of learning in science. International Journal of Science Education, 24(5), 521–549. https://doi.org/10.1080/09500690110095249

Cho, P., & Nagle, C. (2017). Procedural and conceptual difficulties with slope: an analysis of students' mistakes on routine tasks. International Journal of Research in Education and Science (IJRES), 3(1), 135–150. https://ijres.net/index.php/ijres/article/view/142

Conru, B. (2002). Limits. In D. Tall (Ed.), Advanced mathematical thinking (Vol. 11, hal. 153–165). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_10

Cortina, J. L., Visnovska, J., & Zúñiga, C. (2003). Equipartition as a didactical obstacle in fraction instruction. Acta Didactica Universitatis Comenianae Mathematics, May 2016, 1–17.

Ellis, A., Ely, R., Singleton, B., & Tasova, H. (2020). Scaling-continuous variation : supporting students' algebraic reasoning.

Ennis, R. H. (1993). Critical thinking assessment. Theory Into Practice, 32(3), 179–186. https://doi.org/10.1080/00405849309543594

Frank, K. . (2016). Plotting points: implications of "over and up" on students' covariational reasoning. Proceedings of the 38th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, November, 573–580.

Frank, K. M. (2017). Examining the development of students' covariational reasoning in the context of graphing. In Dissertation. Arizona State University.

Gilmore, C., Keeble, S., Richardson, S., & Cragg, L. (2017). The interaction of procedural skill, conceptual understanding and working memory in early mathematics achievement. Journal of Numerical Cognition, 3(2).

Glazer, N. (2011). Challenges with graph interpretation: A review of the literature. Studies in Science Education, 47(2), 183–210. https://doi.org/10.1080/03057267.2011.605307

Hurrell, D. (2021). Conceptual knowledge or procedural knowledge or conceptual knowledge and procedural knowledge: Why the conjunction is important to teachers. Australian Journal of Teacher Education, 46(2), 57–71. https://doi.org/10.14221/ajte.2021v46n2.4

Knuth, E. J. (2020). Understanding connections between equations and graphs. The Mathematics Teacher, 93(1), 48–53. https://doi.org/10.5951/mt.93.1.0048

Manrique, H. M., Read, D. W., & Walker, M. J. (2024). On some statistical and cerebral aspects of the limits of working memory capacity in anthropoid primates, with particular reference to pan and homo, and their significance for human evolution. Neuroscience & Biobehavioral Reviews, 105543.

Metcalfe, J. (2017). Learning from errors. Annual review of psychology, 68, 465–489.

Mhlolo, M. K., Schafer, M., & Venkat, H. (2012). The nature and quality of the mathematical connections teachers make. pythagoras, 33(1), 1–9.

Murniasih, T. R., Suwanti, V., Syaharuddin, S., Rahaju, R., & Farida, N. (2022). Prospective teachers' perceptions of didactic obstacles in the online mathematics learning. Jurnal Elemen, 8(2), 619–630. https://doi.org/10.29408/jel.v8i2.5740

Nurrahmawati, Sa’dijah, C., Sudirman, & Muksar, M. (2021). Assessing students' errors in mathematical translation: From symbolic to verbal and graphic representations. International Journal of Evaluation and Research in Education, 10(1), 115–125.

Oehrtman, M., Carlson, M. P., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students' function understanding. In M. P. Carlson & C. Rasmussen (Ed.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education (hal. 27–42). Mathematical Association of America.

Retnawati, H., Apino, E., & Santoso, A. (2020). High school students' difficulties in making mathematical connections when solving problems. International Journal of Learning, Teaching and Educational Research, 19(8), 255–277.

Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346.

Rolfes, T., Roth, J., & Schnotz, W. (2021). Mono ‑ and multi-representational learning of the covariational aspect of functional thinking. Journal for STEM Education Research, 1–27. https://doi.org/10.1007/s41979-021-00060-4

Ryan, J., & Williams, J. (2007). Children's mathematics 4-15: Learning from errors and misconceptions: learning from errors and misconceptions. McGraw-Hill Education (UK).

Sandie, Purwanto, Subanji, & Hidayanto, E. (2019). Student difficulties in solving covariational problems. International Journal of Humanities and Innovation (IJHI), 2(2), 42–47. https://doi.org/10.33750/ijhi.v2i2.38

Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. https://doi.org/10.5951/at.26.3.0009

Stevens, I. E., Paoletti, T., Moore, K. C., Liang, B., & Hardison, H. (2017). Principles for designing tasks that promote covariational reasoning. Proceeding of the 20th Annual Conference on Research in Undergraduate Mathematics Education, 928–936.

Sunariah, L., & Mulyana, E. (2020). The didactical and epistemological obstacles on the topic of geometry transformation. Journal of Physics: Conference Series, 1521(3). https://doi.org/10.1088/1742-6596/1521/3/032089

Susilo, B. E., Darhim, D., & Prabawanto, S. (2021). Students' learning difficulties in integral calculus based on critical thinking skills. Journal of Physics: Conference Series, 1918(4), 042058. https://doi.org/10.1088/1742-6596/1918/4/042058

Sutini, Aaidati, I. F., & Kusaeri. (2020). Identifying the structure of students' argumentation in covariational reasoning of constructing graphs. Beta: Jurnal Tadris Matematika, 13(117), 61–80. https://doi.org/10.20414/betajtm.v13i1.374

Syarifuddin, S., Nusantara, T., Qohar, A., & Muksar, M. (2020). Students' thinking processes connecting quantities in solving covariation mathematical problems in high school students of indonesia. Participatory Educational Research, 7(3), 59–78. https://doi.org/10.17275/per.20.35.7.3

Tasova, H. I., Liang, B., & Moore, K. C. (2020). The role of lines and points in the construction of emergent shape thinking.

Tasova, H. I., & Moore, K. C. (2021). Framework for representing a multiplicative object in the context of graphing. December, 236–245. https://doi.org/10.51272/pmena.42.2020-24

Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Ed.), New perspectives and directions for collaborative research in mathematics education. WISDOMe Monographs (Nomor 1, hal. 33–57).

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 (hal. 421–456). Compendium for research in mathematics education.

Veloo, A., Krishnasamy, H. N., & Wan Abdullah, W. S. (2015). Types of student errors in mathematical symbols, graphs and problem-solving. Asian Social Science, 11(15), 324–334. https://doi.org/10.5539/ass.v11n15p324

Whitmire, B. J. (2014). Undergraduate students' development of covariational reasoning. Arizona State University.

Zeytun, A. S., Cetinkaya, B., & Erbas, A. K. (2010). Mathematics teachers' covariational reasoning levels and predictions about students' covariational reasoning abilities. Educational Sciences: Theory and Practice, 10(3), 1601–1612.

Zohar, A., & Zohar, A. (2004). Teachers' knowledge about the treatment of students' wrong answers. Higher Order Thinking in Science Classrooms: Students' Learning and Teachers' Professional Development, 139–156.

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Published

14-09-2024

How to Cite

Umah, U., & Rahmawati, A. (2024). Challenges in teaching students to plot equations: Another impact of graphing procedures. Jurnal Elemen, 10(3), 501–515. https://doi.org/10.29408/jel.v10i3.25278

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