Students conceptual mode and analytical thinking: Its role during mathematical problem posing and solving
DOI:
https://doi.org/10.30862/jhm.v8i2.749Keywords:
Analytical Thinking, Conceptual Thinking, Guided Problem-Posing Method, Misconceptions, Problem-SolvingAbstract
In mathematics education, learners frequently rely on procedural imitation when solving problems, even in contexts that demand deep conceptual understanding. This tendency can obscure underlying structural reasoning, yet the extent to which surface-level cues constrain preservice teachers’ mathematical reasoning remains underexplored. Addressing this gap, the present study investigates how third-year secondary mathematics preservice teachers engage with problems requiring conceptual insight while highlighting potential limitations of procedural imitation. The study involved 15 preservice teachers at a state university during the 2023–2024 academic year. Data were collected using multiple standardized instruments, including a Weekly Log-Journal template, End-of-Week Summary sheets, an Instructor Field-Note protocol, and post-task semi-structured interviews, all validated by experts for clarity and content. Credibility was ensured through triangulation and double coding. Analysis employed directed content analysis with theory-informed a priori codes, refined inductively, alongside reflexive thematic analysis and descriptive cross-case synthesis. Findings revealed that routine problems were predominantly addressed through familiar procedures, with learners focusing on surface similarities in equations, leading to errors and the use of spurious methods. These results suggest that superficially correct solutions may mask inadequate structural understanding, underscoring the necessity of cultivating representational fluency, critical thinking, and deeper conceptual knowledge. To address this, problem-posing rubrics should explicitly define invariant conditions and learning objectives to differentiate isomorphic from non-isomorphic situations and reduce superficial copying. The study’s implications extend to instructional design, advocating interventions that promote structural thinking and computational reasoning. Future research may include quasi-experimental investigations, longitudinal tracking of preservice teachers’ practicum performance, and the integration of tools such as GeoGebra, generative AI software, and spreadsheet packages to enhance structural reasoning and procedural flexibility.
References
Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198. https://doi.org/10.1016/j.learninstruc.2006.04.001
Australian Association of Mathematics Teachers. (2002). Standards for excellence in teaching mathematics in Australian schools. AAMT.
Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77–101. https://doi.org/10.1191/1478088706qp063oa
Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Lawrence Erlbaum Associates.
Brown, S. I., & Walter, M. I. (2005). The art of problem posing (3rd ed.). Lawrence Erlbaum Associates. https://doi.org/10.4324/9781410611833
Cai, J. (2023). Impact of prompts on students’ mathematical problem posing. The Journal of Mathematical Behavior, 72, 101087. https://doi.org/10.1016/j.jmathb.2023.101087
Cai, J. (2024). Advances in research on mathematical problem posing: Focus on task variables. The Journal of Mathematical Behavior, 73, 101116. https://doi.org/10.1016/j.jmathb.2023.101116
Crespo, S., & Sinclair, N. (2008). What makes a problem mathematically interesting? Insights from teachers’ analyses of problem posing tasks. Journal of Mathematics Teacher Education, 11(5), 395–415. https://doi.org/10.1007/s10857-008-9081-0
Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. In F. Hitt & M. Santos (Eds.), Proceedings of the 21st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME-NA) (Cuernavaca, Mexico).
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1–2), 103–131. https://doi.org/10.1007/s10649-006-0400-z
English, L. D. (2020). Teaching and learning through mathematical problem posing: Commentary. International Journal of Educational Research, 102, 101451. https://doi.org/10.1016/j.ijer.2019.06.014
Ferretti, F., Gambini, A., & Spagnolo, C. (2024). Management of semiotic representations in mathematics: Quantifications and new characterizations. European Journal of Science and Mathematics Education, 12(1), 11–20. https://doi.org/10.30935/scimath/13827
Freudenthal, H. (1973). Mathematics as an educational task. D. Reidel. https://doi.org/10.1007/978-94-010-2903-2
Hsieh, H.-F., & Shannon, S. E. (2005). Three approaches to qualitative content analysis. Qualitative Health Research, 15(9), 1277–1288. https://doi.org/10.1177/1049732305276687
Leavy, A. (2024). Attending to task variables when engaging in group mathematical problem posing. The Journal of Mathematical Behavior, 72, 101108. https://doi.org/10.1016/j.jmathb.2024.101128
Leavy, A., & Hourigan, M. (2020). Posing mathematically worthwhile problems: Developing the problem-posing skills of prospective teachers. Journal of Mathematics Teacher Education, 23(4), 341–361. https://doi.org/10.1007/s10857-018-09425-w
Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. Sage.
Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Pearson.
Merriam, S. B., & Tisdell, E. J. (2016). Qualitative research: A guide to design and implementation (4th ed.). Jossey-Bass.
Ministry of Education of the People’s Republic of China. (2011). Compulsory education mathematics curriculum standards (2011 edition). Beijing Normal University Press.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Author.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Author.
Pólya, G. (1957). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton University Press.
Presmeg, N., Radford, L., Roth, W.-M., & Kadunz, G. (Eds.). (2016). Semiotics in mathematics education. Springer. https://doi.org/10.1007/978-3-319-31370-2
Saldaña, J. (2021). The coding manual for qualitative researchers (4th ed.). SAGE.
Schoenfeld, A. H. (1985). Mathematical problem solving. Academic Press.
Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27(5), 521–539. https://doi.org/10.2307/749846
Stake, R. E. (1995). The art of case study research. Sage.
Stoyanova, E. (1998). Problem posing in mathematics classroom. In A. McIntosh & N. Ellerton (Eds.), Research in mathematics education: A contemporary perspective (pp. 163–185). MASTEC, Edith Cowan University.
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in mathematics education: Proceedings of the 19th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 518–525). MERGA.
Vinner, S. (1997). The pseudo-conceptual and the pseudo-analytical thought processes in mathematics learning. Educational Studies in Mathematics, 34(2), 97–129. https://doi.org/10.1023/A:1002998529016
Yin, R. K. (2014). Case study research: Design and methods (5th ed.). Sage.
Zhang, L., Stylianides, G. J., & Stylianides, A. J. (2024). Enhancing mathematical problem-posing competence: A meta-analysis of intervention studies. International Journal of STEM Education, 11, 48. https://doi.org/10.1186/s40594-024-00507-1
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Adriano Patac Jr, Louida Patac
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
License and Copyright Agreement
In submitting the manuscript to the journal, the authors certify that:
- They are authorized by their co-authors to enter into these arrangements.
- The work described has not been formally published before, except in the form of an abstract or as part of a published lecture, review, thesis, or overlay journal. Please also carefully read Journal of Honai Math Posting Your Article Policy at http://journalfkipunipa.org/index.php/jhm/about
- That it is not under consideration for publication elsewhere,
- That its publication has been approved by all the author(s) and by the responsible authorities – tacitly or explicitly – of the institutes where the work has been carried out.
- They secure the right to reproduce any material that has already been published or copyrighted elsewhere.
- They agree to the following license and copyright agreement.
Copyright
Authors who publish with Journal of Honai Math agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY-NC-SA 4.0) that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
