Towards understanding the error analysis thinking of prospective mathematics teachers

Zoltán Paulovics; Csaba Csapodi

doi:10.29333/iejme/16678

Full Text (PDF)

Zoltán Paulovics ¹ ^* , Csaba Csapodi ¹ ²

More Detail

¹ Eötvös Loránd University, HUNGARY² Alfréd Rényi Institute of Mathematics, HUNGARY^* Corresponding Author

Abstract

High school teachers often encounter incorrect solutions from students, especially when teaching combinatorics. This study investigates the ability of prospective mathematics teachers to assess the correctness of solutions to combinatorial problems and to falsify incorrect ones. 39 second-year prospective teachers participated in the experiment, analysing both correct and incorrect solutions to combinatorial tasks. They were asked not only to judge the correctness but also to identify the cause of procedural or conceptual errors. The results indicate that for incorrect solutions, 44.16% of the error identifications were accurate, and only 36.36% of cases could be successfully falsified. Furthermore, when evaluating correct solutions, 57.1% of the correct solutions were mistakenly judged as incorrect. Overall, 31 out of the 39 participants were unable to accurately determine the correctness of the solutions, despite their advanced mathematical background. These findings highlight the need to foster error-detection skills and thorough reasoning in teacher training programs.

Keywords

error analysis
prospective mathematics teachers
mathematics teaching practices

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 20, Issue 4, November 2025, Article No: em0848

https://doi.org/10.29333/iejme/16678

Publication date: 01 Oct 2025

Online publication date: 29 Jul 2025

Article Views: 64

Article Downloads: 30

Open Access References How to cite this article

References

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. https://doi.org/10.1016/j.chb.2014.03.053
As'ari, A. R., Mahmudi, A., & Nuerlaelah, E. (2017). Our prospective mathematic teachers are not critical thinkers yet. Journal on Mathematics Education, 8(2), 145-156.
Austin, J. (2015). Prospective teachers’ personal mathematics teacher efficacy beliefs and mathematical knowledge for teaching. International Electronic Journal of Mathematics Education, 10(1), 17-36. https://doi.org/10.29333/iejme/289
Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
Batanero, C., Navarro-Pelayo, V., & Godino, J. (1997). Effect of the implicit combinatorial model on combinatorial reasoning in secondary school pupils. Educational Studies in Mathematics, 32, 181-199. https://doi.org/10.1023/A:1002954428327
Borasi, R. (1994). Capitalizing on errors as “springboards for inquiry”: A teaching experiment. Journal for Research in Mathematics Education, 25(2), 166-208. https://doi.org/10.5951/jresematheduc.25.2.0166
Bransford, J., Sherwood, R., Vye, N., & Rieser, J. (1986). Teaching thinking and problem solving: Research foundations. American Psychologist, 41(10), 1078-1089. https://doi.org/10.1037/0003-066X.41.10.1078
Chick, H. L. (2011). God-like educators in a fallen world. AARE 2011 Conference Proceedings, Article 00667.
Dubois, J. G. (1984). Une systématique des configurations combinatoires simples [A systematic study of simple combinatorial configurations]. Educational Studies in Mathematics, 15, 37-57. https://doi.org/10.1007/BF00380438
Erdoğan, F. (2020). The relationship between prospective middle school mathematics teachers’ critical thinking skills and reflective thinking skills. Participatory Educational Research, 7(1), 220-241. https://doi.org/10.17275/per.20.13.7.1
Fischbein, E., & Gazit, A. (1988). The combinatorial solving capacity in children and adolescents. Zentralblatt für Didaktik der Mathematik, 5, 193-198.
Fredua-Kwarteng, E. (2015). How prospective teachers conceptualized mathematics: Implications for teaching. International Electronic Journal of Mathematics Education, 10(2), 77-95. https://doi.org/10.29333/iejme/293
Gordon Győri, J., Fried, K., Köves, G., Oláh, V., & Pálfalvi, J. (2020). The traditions and contemporary characteristics of mathematics education in Hungary in the post-socialist era. In Eastern European Mathematics Education in the Decades of Change (pp. 75-129). Springer. https://doi.org/10.1007/978-3-030-38744-0_3
Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Learning and Instruction, 17(6), 612-634. https://doi.org/10.1016/j.learninstruc.2007.09.008
Hiebert, J., & Lefevre, P. (2013). Conceptual and procedural knowledge in mathematics: An introductory analysis. In Conceptual and procedural knowledge (pp. 1-27). Routledge. https://doi.org/10.4324/9780203063538
Khasawneh, A. A., Al-Barakat, A. A., & Almahmoud, S. A. (2023). The impact of mathematics learning environment supported by error-analysis activities on classroom interaction. Eurasia Journal of Mathematics, Science and Technology Education, 19(2), Article em2227. https://doi.org/10.29333/ejmste/12951
Lampert, M. (2001). Teaching problems and the problems of teaching. Yale University Press.
Lockwood, E. (2013). A model of students’ combinatorial thinking. The Journal of Mathematical Behavior, 32(2), 251-265. https://doi.org/10.1016/j.jmathb.2013.02.008
Lockwood, E., Reed, Z., & Erickson, S. (2021). Undergraduate students’ combinatorial proof of binomial identities. Journal for Research in Mathematics Education, 52(5), 539-580. https://doi.org/10.5951/jresematheduc-2021-0112
Maharani, I. P., & Subanji, S. (2018). Scaffolding based on cognitive conflict in correcting the students' algebra errors. International Electronic Journal of Mathematics Education, 13(2), 67-74. https://doi.org/10.12973/iejme/2697
Mamona-Downs, J., & Downs, M. (2005). The identity of problem solving. The Journal of Mathematical Behavior, 24(3-4), 385-401. https://doi.org/10.1016/j.jmathb.2005.09.011
Mashiach Eizenberg, M., & Zaslavsky, O. (2004). Students' verification strategies for combinatorial problems. Mathematical Thinking and Learning, 6(1), 15-36. https://doi.org/10.1207/s15327833mtl0601_2
Mason, J., & Davis, B. (2013). The importance of teachers’ mathematical awareness for in-the-moment pedagogy. Canadian Journal of Science, Mathematics and Technology Education, 13, 182-197. https://doi.org/10.1080/14926156.2013.784830
Matteucci, M. C., Corazza, M., & Santagata, R. (2015). Learning from errors, or not. An analysis of teachers’ beliefs about errors and error-handling strategies through questionnaire and video. Progress in Education, 37, 33-54.
Mellone, M., Jakobsen, A., & Ribeiro, C. M. (2015). Mathematics educator transformation (s) by reflecting on students' non-standard reasoning. In CERME 9-Ninth Congress of the European Society for Research in Mathematics Education (pp. 2874-2880).
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. NCTM.
Paulovics, Z., Csapodi, C., & Nagy, Z. L. (2023). The impact of incorrect solutions on the attitude and problem-solving abilities of prospective mathematics teachers towards combinatorics. In Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13) (No. 12). Alfréd Rényi Institute of Mathematics; ERME.
Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press. https://doi.org/10.1515/9781400828678
Pólya, G. (1962). Mathematical discovery: On understanding, learning and teaching problem solving. Wiley.
Radatz, H. (1980). Students' errors in the mathematical learning process: A survey. For the Learning of Mathematics, 1(1), 16-20.
Rasiman, R. (2015). Leveling of critical thinking abilities of students of mathematics education in mathematical problem solving. Journal on Mathematics Education, 6(1), 40-52. https://doi.org/10.22342/jme.6.1.1941.40-52
Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189. https://doi.org/10.1037/0022-0663.91.1.175
Rushton, S. J. (2018). Teaching and learning mathematics through error analysis. Fields Mathematics Education Journal, 3(1), 1-12. https://doi.org/10.1186/s40928-018-0009-y
Salavatinejad, N., Alamolhodaei, H., & Radmehr, F. (2021). Toward a model for students’ combinatorial thinking. The Journal of Mathematical Behavior, 61, Article 100823. https://doi.org/10.1016/j.jmathb.2020.100823
Santagata, R. (2005). Practices and beliefs in mistake-handling activities: A video study of Italian and US mathematics lessons. Teaching and Teacher Education, 21(5), 491-508. https://doi.org/10.1016/j.tate.2005.03.004
Schoenfeld, A. H. (2014). Mathematical problem solving. Elsevier.
Schoenfeld, A. H. (2016). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (Reprint). Journal of Education, 196(2), 1-38. https://doi.org/10.1177/002205741619600202
Shaughnessy, M., DeFino, R., Pfaff, E., & Blunk, M. (2021). I think I made a mistake: How do prospective teachers elicit the thinking of a student who has made a mistake? Journal of Mathematics Teacher Education, 24, 335-359. https://doi.org/10.1007/s10857-020-09461-5
Siegler, R. S. (2002). Microgenetic studies of self-explanation. In N. Granott & J. Parziale (Eds.), Microdevelopment (pp. 31–58). Cambridge University Press. https://doi.org/10.1017/cbo9780511489709.002
Strauss, A., & Corbin, J. (1998). Basics of qualitative research techniques.
VanLehn, K. (1999). Rule-learning events in the acquisition of a complex skill: An evaluation of cascade. Journal of the Learning Sciences, 8(1), 71-125. https://doi.org/10.1207/s15327809jls0801_3
Watson, A., & Barton, B. (2011). Teaching mathematics as the contextual application of modes of mathematical enquiry. In T. Rowland, & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp. 65-82). Springer. https://doi.org/10.1007/978-90-481-9766-8_5

How to cite this article

APA

Paulovics, Z., & Csapodi, C. (2025). Towards understanding the error analysis thinking of prospective mathematics teachers. International Electronic Journal of Mathematics Education, 20(4), em0848. https://doi.org/10.29333/iejme/16678

Vancouver

Paulovics Z, Csapodi C. Towards understanding the error analysis thinking of prospective mathematics teachers. INT ELECT J MATH ED. 2025;20(4):em0848. https://doi.org/10.29333/iejme/16678

AMA

Paulovics Z, Csapodi C. Towards understanding the error analysis thinking of prospective mathematics teachers. INT ELECT J MATH ED. 2025;20(4), em0848. https://doi.org/10.29333/iejme/16678

Chicago

Paulovics, Zoltán, and Csaba Csapodi. "Towards understanding the error analysis thinking of prospective mathematics teachers". International Electronic Journal of Mathematics Education 2025 20 no. 4 (2025): em0848. https://doi.org/10.29333/iejme/16678

Harvard

Paulovics, Z., and Csapodi, C. (2025). Towards understanding the error analysis thinking of prospective mathematics teachers. International Electronic Journal of Mathematics Education, 20(4), em0848. https://doi.org/10.29333/iejme/16678

MLA

Paulovics, Zoltán et al. "Towards understanding the error analysis thinking of prospective mathematics teachers". International Electronic Journal of Mathematics Education, vol. 20, no. 4, 2025, em0848. https://doi.org/10.29333/iejme/16678