Intuitive and reflective mathematical reasoning in solving quadratic function problems: A dual-process perspective
DOI:
https://doi.org/10.58524/jasme.v6i2.1237Abstract
Background: Understanding students’ mathematical reasoning in solving quadratic function problems is important because it reveals how they construct and regulate their thinking during problem-solving activities. According to Dual-Process Theory, reasoning involves intuitive processes (System 1) and reflective processes (System 2).
Aim: This study aimed to investigate how students engage System 1 and System 2 when solving quadratic function problems.
Method: A qualitative descriptive approach was employed involving two tenth-grade students representing lower and higher mathematical ability levels. Data were collected through written tasks, video recordings, and semi-structured interviews. Data credibility was established through method triangulation.
Results: The findings showed that both System 1 and System 2 were involved in solving quadratic function problems. Lower-ability students relied more frequently on System 1, resulting in automatic and less reflective reasoning. In contrast, higher-ability students demonstrated a more coordinated use of System 1 and System 2, leading to more systematic and accurate solutions.
Conclusion: Quadratic function problem solving requires the integration of intuitive and reflective reasoning. System 2 plays a critical role in evaluating and validating initial intuitive responses, thereby supporting more effective mathematical reasoning.
References
Ahmed, Z., Mohamed, K., Zeeshan, S., & Dong, X. (2020). Artificial intelligence with multi-functional machine learning platform development for better healthcare and precision medicine. Database, 2020, Article baaa010. https://doi.org/10.1093/database/baaa010
Alsina, A., Maurandi, A., Ferre, E., & Coronata, C. (2021). Validating an instrument to evaluate the teaching of mathematics through processes. International Journal of Science and Mathematics Education, 19(3), 559–577. https://doi.org/10.1007/s10763-020-10064-y
Augusto, R. (2024). Two kinds of process or two kinds of processing? Disambiguating dual-process theories. Review of Philosophy and Psychology, 15(1), 277–298. https://doi.org/10.1007/s13164-023-00673-1
Borodin, A. (2016). The need for an application of dual-process theory to mathematics education. CORERJ: Cambridge Open-Review Educational Research e-Journal, 3, 2056–7804.
Darmawan, P. (2019). Aplikasi dual-process theory: Karakteristik proses mental siswa dalam memecahkan masalah segi banyak. Prosiding: Konferensi Nasional Matematika dan IPA Universitas PGRI Banyuwangi, 1(1), 204–214.
Darmawan, P., Murtafiah, W., & Osman, S. (2024). Exploring default-interventionist interaction of thinking activity types on probability problem-solving. Journal on Mathematics Education, 15(1), 295–316. https://doi.org/10.22342/jme.v15i1.pp295-316
Díaz, V., Aravena, M., & Flores, G. (2020). Solving problem types contextualized to the quadratic function and error analysis: A case study. Eurasia Journal of Mathematics, Science and Technology Education, 16(11), Article em1896. https://doi.org/10.29333/ejmste/8547
Evans, J. S. B. T. (2019). Hypothetical thinking: Dual processes in reasoning and judgement. Routledge. https://doi.org/10.4324/9780367823832
Evans, V. (2007). A glossary of cognitive linguistics. Edinburgh University Press. https://doi.org/10.1515/9780748629862
Faradiba, S. S., Alifiani, & Nasir, N. A. M. (2024). The resolution of quadratic inequality problems in mathematics: Discrepancies between thought and action. Infinity Journal, 13(1), 61–82. https://doi.org/10.22460/infinity.v13i1.p61-82
Faturohman, I., & Afriansyah, E. A. (2020). Peningkatan kemampuan berpikir kreatif matematis siswa melalui creative problem solving. Mosharafa: Jurnal Pendidikan Matematika, 9(1), 107–118. https://doi.org/10.31980/mosharafa.v9i1.562
Frankish, K. (n.d.). Playing double: Implicit bias, dual levels, and self-control
Gawronski, B., Luke, D. M., & Creighton, L. A. (2013). Dual-process theories. In The Oxford handbook of social cognition (pp. 282–312). Oxford University Press. https://doi.org/10.1093/oxfordhb/9780199730018.013.0014
Hadi, S., & Zaidah, A. (2020). Analysis of student quotient adversity in problem solving HOTS (High Order Thinking Skill) mathematics problems. Path of Science, 6(12). https://doi.org/10.22178/pos.65-4
Hjelte, A., Schindler, M., & Nilsson, P. (2020). Kinds of mathematical reasoning addressed in empirical research in mathematics education: A systematic review. Education Sciences, 10(10). https://doi.org/10.3390/educsci10100289
Hu, Q., Son, J.-W., & Hodge, L. (2022). Algebra teachers’ interpretation and responses to student errors in solving quadratic equations. International Journal of Science and Mathematics Education, 20(3), 637–657. https://doi.org/10.1007/s10763-021-10166-1
Istiandaru, A. (2017). Krulik and Rudnick model heuristic strategy in mathematics problem solving. International Journal on Emerging Mathematics Education, 1(2), 205–210. https://doi.org/10.12928/ijeme.v1i2.5708
Kahneman, D. (2020). Thinking, fast and slow (Z. Anshor, Trans.).
Kholifah, F. N., & Darmawan, P. (2025). Berpikir siswa SMP dalam memecahkan masalah luas permukaan dan volume gabungan bangun ruang berdasarkan teori dual process. [Nama jurnal tidak tersedia], 6(2).
Kollosche, D. (2021). Styles of reasoning for mathematics education. Educational Studies in Mathematics, 107(3), 471–486. https://doi.org/10.1007/s10649-021-10046-z
Kurniasari, Y., Sugandi, A. I., & Sariningsih, R. (2021). Analisis kesalahan siswa kelas X dalam [judul tidak lengkap]. JPMI: Jurnal Pembelajaran Matematika Inovatif, 4(6), 1561–1568. https://doi.org/10.22460/jpmi.v4i6.1561-1568
Leron, U., & Hazzan, O. (2006). The rationality debate: Application of cognitive psychology to mathematics education. Educational Studies in Mathematics, 62(2), 105–126. https://doi.org/10.1007/s10649-006-4833-1
Manik, M., Candrama, T., Darmawan, P., & Basri, H. (2023). High school student’s thinking in solving curved sides space numeracy problem on AKM based on dual-process theory. Jurnal Riset Pendidikan dan Inovasi Pembelajaran Matematika, 7(1), 1–25. https://doi.org/10.26740/jrpipm.v7n1.p1-25
Milner, M., & Wolfer, T. A. (2023). Using the reflective judgment model to understand and appraise effective reasoning by social work students. Journal of Social Work Education, 59(1), 78–90. https://doi.org/10.1080/10437797.2021.1970064
Mohamad, S. K., & Tasir, Z. (2023). Exploring how feedback through questioning may influence reflective thinking skills based on association rules mining technique. Thinking Skills and Creativity, 47, Article 101231. https://doi.org/10.1016/j.tsc.2023.101231
Mohammed, M. A., Abdulkareem, K. H., Al-Waisy, A. S., Mostafa, S. A., Al-Fahdawi, S., Dinar, A. M., Alhakami, W., BAZ, A., Al-Mhiqani, M. N., Alhakami, H., Arbaiy, N., Maashi, M. S., Mutlag, A. A., García-Zapirain, B., & De La Torre Díez, I. D. L. T. (2020). Benchmarking methodology for selection of optimal COVID-19 diagnostic model based on entropy and TOPSIS methods. IEEE Access, 8, 99115–99131. https://doi.org/10.1109/ACCESS.2020.2995597
Molerov, D., Zlatkin-Troitschanskaia, O., Nagel, M.-T., Brückner, S., Schmidt, S., & Shavelson, R. J. (2020). Assessing university students’ critical online reasoning ability: A conceptual and assessment framework with preliminary evidence. Frontiers in Education, 5. https://doi.org/10.3389/feduc.2020.577843
Musriroh, R. Z., Hidayanto, E., & Rahardi, R. (2021). Penalaran spasial matematis dimensi persepsi dan visualisasi kelas VIII dalam pemecahan masalah geometri. Jurnal Pendidikan: Teori, Penelitian, dan Pengembangan, 6(11), 1774–1781. https://doi.org/10.17977/jptpp.v6i11.15144
Odutayo, A. O., & Fonseca, K. (2024). Making quadratic functions interesting: Students teams-achievement division instructional strategy. Eurasia Journal of Mathematics, Science and Technology Education, 20(1), Article em2386. https://doi.org/10.29333/ejmste/14092
Parta, I. N. (2021). Teacher interventions to induce students’ awareness in controlling their intuition. Bolema: Boletim de Educação Matemática, 35(70), 745–765. https://doi.org/10.1590/1980-4415v35n70a10
Reid O’Connor, B., & Norton, S. (2024). Exploring the challenges of learning quadratic equations and reflecting upon curriculum structure and implementation. Mathematics Education Research Journal, 36(1), 151–176. https://doi.org/10.1007/s13394-022-00434-w
Richards, J. B., Hayes, M. M., & Schwartzstein, R. M. (2020). Teaching clinical reasoning and critical thinking: From cognitive theory to practical application. Chest, 158(4), 1617–1628. https://doi.org/10.1016/j.chest.2020.05.525
Riset, J., & Pendidikan, H. (2023). Kognitif. Kognitif, 3, 21–34. https://doi.org/10.51574/kognitif.v3i1.657
Ruhma, S. Z., Ratnaningsih, N., & Rahayu, D. V. (2023). Analisis kesulitan siswa dalam memecahkan masalah pada materi persamaan kuadrat berdasarkan prosedur Polya. Primatika: Jurnal Pendidikan Matematika, 12(2), 139–152. https://doi.org/10.30872/primatika.v12i2.2699
Safitri, A. D., & Darmawan, P. (2022). Karakteristik proses mental siswa dalam memecahkan masalah peluang berdasarkan teori dual-process. Prosiding: Konferensi Nasional Matematika dan IPA Universitas PGRI Banyuwangi, 2(1), 54–61.
Säfström, A. I., Lithner, J., Palm, T., Palmberg, B., Sidenvall, J., Andersson, C., Boström, E., & Granberg, C. (2024). Developing a diagnostic framework for primary and secondary students’ reasoning difficulties during mathematical problem solving. Educational Studies in Mathematics, 115(2), 125–149. https://doi.org/10.1007/s10649-023-10278-1
Sigalingging, A., & Siregar, N. (2025). Kajian literatur tentang konsep, sifat, dan aplikasi fungsi kuadrat dalam pembelajaran matematika menengah. Harmoni Pendidikan: Jurnal Ilmu Pendidikan, 2(4), 178–184. https://doi.org/10.62383/hardik.v2i4.2406
Sukma, A. (2011). Profil berfikir intuitif matematik.
Wahyuni, S., Darmawan, P., & Syarifudin, S. (2023). Berpikir mahasiswa dalam menyelesaikan masalah kurva tertutup sederhana ditinjau dari gaya belajar berdasarkan dual process theory. Jurnal Pendidikan MIPA, 13(3), 670–680. https://doi.org/10.37630/jpm.v13i3.1138
Wijaya, M. A. W., & Khabibah, S. (2025). MATHE dunesa. Jurnal Ilmiah Pendidikan Matematika, 14(6), 39–58. https://doi.org/10.26740/mathedunesa.v14n1.p245-260
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Evi Triani Puspitasari, Subanji, Santi Irawati, Lathiful Anwar, Puguh Darmawan
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
