Volume 11 Issue 2 (2022)

Assessing the Quality of Arguments in Students’ Mathematical Problem Solving

pp. 28-40  |  Published Online: December 2022  |  DOI: 10.22521/unibulletin.2022.112.2

Hendra Kartika, Mega Teguh Budiarto


Background/purpose – Argumentation plays an essential role in higher-order activities and in the communication of mathematical knowledge. Although the purposes of argumentation have piqued the interest of many researchers, few have simultaneously investigated the quality of argument in students’ mathematical problem solving.

Materials/methods – In this case study, 41 middle school students in Indonesia solved an argumentative task. The students’ responses were then analyzed for the quality of their arguments. In line with the study’s goal of assessing the quality of the students’ arguments in mathematical problem solving, their responses were coded according to the Polya method and the CER model.

Results – The results revealed that more than half of the students misunderstood the given mathematical problems, which were not appropriately evidenced or reasoned in their response.

Conclusion – These results indicate that further handling is needed to improve the quality of students' arguments and to develop the importance of activities that support students in explaining, justifying, and correcting their reasoning during mathematical argumentation.

Keywords: argumentation, argument, CER model, middle school, Polya method, problem solving


Adams, D. M., McLaren, B. M., Durkin, K., Mayer, R. E., Rittle-Johnson, B., Isotani, S., & van Velsen, M. (2014). Using erroneous examples to improve mathematics learning with a web-based tutoring system. Computers in Human Behavior, 36, 401-411. http://doi.org/10.1016/j.chb.2014.03.053 

Cardetti, F., & LeMay, S. (2018). Argumentation: Building students’ capacity for reasoning essential to learning mathematics and sciences. PRIMUS, 29(8), 775-798. http://doi.org/10.1080/10511970.2018.1482581

Carrascal, B. (2015). Proofs, mathematical practice and argumentation. Argumentation, 29(3), 305-324. http://doi.org/10.1007/s10503-014-9344-0

Chen, X., Mitrovic, A., & Mathews, M. (2019). Investigating the effect of agency on learning from worked examples, erroneous examples and problem-solving. International Journal of Artificial Intelligence in Education, 29, 394-424. http://doi.org/10.1007/s40593-019-00179-x

Dogruer, S. S., & Akyuz, D. (2020). Mathematical practices of eighth-graders about 3d shapes in an argumentation, technology, and design-based classroom environment. International Journal of Science and Mathematics Education, 18, 1485-1505 http://doi.org/10.1007/s10763-019-10028-x

Fielding-Wells, J. (2016). “Mathematics is just 1 + 1 = 2, what is there to argue about?”: Developing a framework for Argument-Based Mathematical Inquiry. In B. White (Eds.), Opening up mathematics education research (Proceedings of the 39th Annual Conference of the Mathematics Education Research Group of Australasia) (pp. 214-221). MERGA.

Foster, N. L., Rawson, K. A., & Dunlosky, J. (2018). Self-regulated learning of principle-based concepts: Do students prefer worked examples, faded examples, or problem solving? Learning and Instruction, 55, 124-138. http://doi.org/10.1016/j.learninstruc.2017.10.002

Freeman, J. B. (2011). Argument structure: Representation and theory. Springer.

Graham, M., & Lesseig, K. (2018). New teachers can immediately begin using these classroom-tested ways to incorporate mathematical argumentation in their classrooms on a daily basis. Mathematics Teachers, 112(3), 173-178.

Grobe, C. S. (2018). “Copying allowed-but be careful, errors included!”-effects of copying correct and erroneous solutions on learning outcomes. Learning and Instruction, 58, 173-181. http://doi.org/10.1016/j.learninstruc.2018.06.004

Heemsoth, T., & Heinze, A. (2016). Secondary school students learning from reflections on the rationale behind self-made errors: A Field Experiment. The Journal of Experimental Education, 84(1), 98-118. http://doi.org/10.1080/00220973.2014.963215

Isotani, S., Adams, D., Mayer, R. E., Durkin, K., Rittle-Johnson, B., & McLaren, B. M. (2011). Can erroneous examples help middle-school students learn decimals?. In C. D. Kloos, D. Gillet, R. M. Crespo García, F. Wild, & M. Wolpers (Eds.), Towards Ubiquitous Learning. EC-TEL 2011. Lecture Notes in Computer Science (Vol. 6964, pp. 181-195). Springer. http://doi.org/10.1007/978-3-642-23985-4_15

Jonassen, D. H. (2011). Learning to solve problems. Routledge.

Kartika, H., Budiarto, M. T., & Fuad, Y. (2021). Argumentation in K-12 mathematics and science education: A content analysis of articles. International Journal of Research in Education and Science (IJRES), 7(1), 51-64. http://doi.org/10.46328/ijres.1389

Klopp, E., & Stark, R. (2020). Learning to argue from others’ erroneous arguments-fostering argumentation skills through learning from advocatory errors. Frontiers in Education, 5(126). http://doi.org/10.3389/feduc.2020.00126

Knipping, C., & Reid, D. (2015). Reconstructing argumentation structures: A perspective on proving processes in secondary mathematics classroom interactions. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education, Advances in mathematics education (pp. 75-101). Springer. https://doi.org/10.1007/978-94-017-9181-6_4

Liua, Y., Tague, J., & Somayajulu, R. (2016). What do eighth-grade students look for when determining if a mathematical argument is convincing. International Electronic Journal of Mathematics Education, 11(7), 2373-2404. https://www.iejme.com/article/what-do-eighth-grade-students-look-for-when-determining-if-a-mathematical-argument-is-convincing

Mayer, R. E. (2013). Problem Solving. In D. Reisberg (Ed.), The Oxford handbook of cognitive psychology (pp. 769-778). Oxford University Press. http://doi.org/10.1093/oxfordhb/9780195376746.013.0048

McLaren, B. M., Adams, D., Durkin, K., Goguadze, G. Mayer, R. E., Rittle-Johnson, B., Sosnovsky, S., Isotani, S., & Van Velsen, M. (2012). To err is human, to explain and correct is divine: A study of interactive erroneous examples with middle school math students. In A. Ravenscroft, S. Lindstaedt, C. D. Kloos, & D. Hernández-Leo (Eds.), Proceedings of ECTEL 2012: Seventh European Conference on Technology Enhanced Learning, LNCS 7563 (pp. 222-235). Springer. https://doi.org/10.1007/978-3-642-33263-0_18

Ministry of Education and Culture of the Republic of Indonesia. (2018). Core competencies and competencies basic lessons in the 2013 curriculum in elementary and junior secondary education. Kemendikbud.

Mohaghegh, M., & Grobler, A. (2020). The dynamics of operational problem-solving: A dual-process approach. Systemic Practice and Action Research, 33, 27-54. http://doi.org/10.1007/s11213-019-09513-9

National Governors Association Center for Best Practices. (2010). Common core state standards for mathematics.

Nordin, A. K., & Boistrup, L. B. (2018). A framework for identifying mathematical arguments as supported claims created in day-to-day classroom interactions. The Journal of Mathematical Behavior, 51, 15-27. http://doi.org/10.1016/j.jmathb.2018.06.005

Nussbaum, E. M. (2011). Argumentation, dialogue theory, and probability modeling: Alternative frameworks for argumentation research in education. Educational Psychologist, 46(2), 84-106. http://doi.org/10.1080/00461520.2011.558816

Organisation for Economic Co-operation and Development. (2003). The PISA 2003 Assessment framework. mathematics, reading, science, and problem solving knowledge and skills. OECD Publishing.

Osborne, J. F., Henderson, J. B., Macpherson, A., Szu, E., Wild, A., & Yao, S. Y. (2016). The development and validation of a learning progression for argumentation in science. Journal of Research in Science Teaching, 53(6), 821-846. https://doi.org/10.1002/tea.21316

Polya, G. (1973). How to solve it: A new aspect of mathematical method. Princeton University Press.

Rapanta, C. (2019). Argumentation as critically oriented pedagogical dialogue. Informal Logic, 39(1), 1-31. http://doi.org/10.22329/il.v39i1.5116

Rushton, S. J. (2018). Teaching and learning mathematics through error analysis. Fields Mathematics Education Journal, 3, Article 4. http://doi.org/10.1186/s40928-018-0009-y

Schwarz, B. B., Hershkowitz, R., & Prusak, N. (2010). Argumentation and mathematics. In C. Howe & K. Littleton (Eds.), Educational dialogues: Understanding and promoting productive interaction (pp. 115-141). Routledge.

Shi, Y. (2020). Talk about evidence during argumentation. Discourse Processes, 57(9), 770-792. http://doi.org/10.1080/0163853x.2020.1777498

Sitzman, D. M., Rhodes, M. G., Tauber, S. K., & Liceralde, V. R. T. (2015). The role of prior knowledge in error correction for younger and older adults. Aging, Neuropsychology, and Cognition, 22(4), 502-516. http://doi.org/10.1080/13825585.2014.993302

Staples, M., & Newton, J. (2016). Teachers’ contextualization of argumentation in the mathematics classroom. Theory Into Practice, 55(4), 294-301. https://doi.org/10.1080/00405841.2016.1208070

Stylianides, A. J. (2019). Secondary students’ proof constructions in mathematics: The role of written versus oral mode of argument representation. Review of Education, 7(1), 156-182. https://doi.org/10.1002/rev3.3157

Toulmin, S. (2003). The uses of argument. Cambridge University Press.

Tsovaltzi, D., Melis, E., McLaren, B. M., Meyer, A. K., Dietrich, M., & Goguadze, G. (2010). Learning from erroneous examples: When and how do students benefit from them? In M. Wolpers, P. A. Kirschner, M. Scheffel, S. Lindstaedt, & V. Dimitrova (Eds.), Sustaining TEL: From Innovation to Learning and Practice, 5th European Conference on Technology Enhanced Learning (pp. 357-373). Springer.

Yopp, D. A. (2018). When an argument is a content: Rational number comprehension through conversions across registers. Journal of Mathematical Behaviour, 50, 42-56. http://doi.org/10.1016/j.jmathb.2018.01.001

Youkap, P. T., Ngansop, J. N., Tieudjo, D., & Ntam, L. N. (2019). Influence of drawing and figures on secondary school students’ argumentation and proof: An investigation on parallelogram. Acta Didactica Napocensia, 12(2), 133-144. http://doi.org/10.24193/adn.11.2.10

Zambak, V. S., & Magiera, M. T. (2020). Supporting grades 1-8 preservice teachers’ argumentation skills: Constructing mathematical arguments in situations that facilitate analyzing cases. International Journal of Mathematical Education in Science and Technology, 51(8), 1196-1223. http://doi.org/10.1080/0020739X.2020.1762938



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