STUDENTS’ THINKING PROCESSES IN INVERSE PROPORTION PROBLEM SOLVING
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May 31, 2026
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Abstract
This study describes students’ thinking processes in solving inverse proportion problems based on mathematical ability. The study employed a qualitative descriptive design involving three students categorized as having high, moderate, and low mathematical ability. Data were collected through a written inverse proportion task and task-based interviews organized according to four problem-solving stages: understanding the problem, devising a plan, carrying out the plan, and looking back. The task asked students to determine the completion time when 20 workers who could finish a job in 30 days were increased to 50 workers. The results showed that the high-ability student recognized the inverse relation, used the constant-product model 20 × 30 = 50 × x, obtained 12 days, and verified the answer. The moderate-ability student treated the situation as direct proportion by writing 20 : 50 = 30 : x, indicating relational misconception. The low-ability student recognized the inverse relation but had difficulty with multiplication and division and did not check the answer. The findings imply that inverse proportion instruction should emphasize contextual relations, strategy selection, computational fluency, and reflective checking.
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[1]
M. Basalama, “STUDENTS’ THINKING PROCESSES IN INVERSE PROPORTION PROBLEM SOLVING”, JURNAL EDUCATION AND DEVELOPMENT, vol. 14, no. 2, pp. 1055-1062, May 2026.
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Ayuningtyas, W., Mardiyana, and I. Pramudya. 2019. “Analysis of Students’ Mathematical Problem Solving Skill Viewed from Mathematical Ability.” Journal of Physics: Conference Series 1188:12089. doi: 10.1088/1742-6596/1188/1/012089.
Boonen, A. J. H., B. B. de Koning, J. Jolles, and M. van der Schoot. 2016. “Word Problem Solving in Contemporary Math Education: A Plea for Reading Comprehension Skills Training.” Frontiers in Psychology 7:191. doi: 10.3389/fpsyg.2016.00191.
Boudreaux, A., S. E. Kanim, and S. W. Brahmia. 2016. “Student Facility with Ratio and Proportion: Mapping the Reasoning Space in Introductory Physics.” Physical Review Physics Education Research 12(2):20102. doi: 10.1103/PhysRevPhysEducRes.12.020102.
Brahmia, S. W., A. Olsho, T. I. Smith, A. Boudreaux, P. Eaton, and C. Zimmerman. 2021. “The Physics Inventory of Quantitative Literacy: A Tool for Assessing Mathematical Reasoning in Introductory Physics.” Physical Review Physics Education Research 17(2):20129. doi: 10.1103/PhysRevPhysEducRes.17.020129.
Cypress, B. S. 2017. “Rigor or Reliability and Validity in Qualitative Research: Perspectives, Strategies, Reconceptualization, and Recommendations.” Dimensions of Critical Care Nursing 36(4):253–63. doi: 10.1097/DCC.0000000000000253.
Daroczy, G., M. Wolska, W. D. Meurers, and H. C. Nuerk. 2016. “Word Problems: A Review of Linguistic and Numerical Factors Contributing to Their Difficulty.” Frontiers in Psychology 7:348. doi: 10.3389/fpsyg.2016.00348.
Fuchs, L. S., D. Fuchs, A. S. Malone, A. Y. Wang, C. L. Hamlett, and S. Krowka. 2021. “Effects of Intervention to Improve At-Risk Fourth Graders’ Understanding, Calculations, and Word Problems with Fractions.” Exceptional Children 87(3):290–308. doi: 10.1177/0014402920985226.
Hidayat, R., H. Zulnaidi, and S. N. A. S. Zamri. 2018. “Roles of Metacognition and Achievement Goals in Mathematical Modeling Competency: A Structural Equation Modeling Analysis.” PLOS ONE 13(11):e0206211. doi: 10.1371/journal.pone.0206211.
Huang, X., S. D. Craig, J. Xie, A. Graesser, and X. Hu. 2016. “Intelligent Tutoring Systems Work as a Math Gap Reducer in 6th Grade After-School Program.” Learning and Individual Differences 47:258–65. doi: 10.1016/j.lindif.2016.01.012.
Jitendra, A. K., D. N. Dupuis, J. R. Star, and M. C. Rodriguez. 2016. “The Effects of Schema-Based Instruction on the Proportional Thinking of Students with Mathematics Difficulties.” Journal of Learning Disabilities 49(4):354–67. doi: 10.1177/0022219414554228.
Johnson, J. L., D. Adkins, and S. Chauvin. 2020. “A Review of the Quality Indicators of Rigor in Qualitative Research.” American Journal of Pharmaceutical Education 84(1):7120. doi: 10.5688/ajpe7120.
Krawitz, J., S. Schukajlow, and W. Van Dooren. 2018. “Unrealistic Responses to Realistic Problems with Missing Information: What Are Important Barriers?” Educational Psychology 38(10):1221–38. doi: 10.1080/01443410.2018.1502413.
Levitt, H. M., M. Bamberg, J. W. Creswell, D. M. Frost, R. Josselson, and C. Suárez-Orozco. 2018. “Journal Article Reporting Standards for Qualitative Primary, Qualitative Meta-Analytic, and Mixed Methods Research in Psychology: The APA Publications and Communications Board Task Force Report.” American Psychologist 73(1):26–46. doi: 10.1037/amp0000151.
Liljedahl, P., M. Santos-Trigo, U. Malaspina, and R. Bruder. 2016. Problem Solving in Mathematics Education. Springer.
Mason, A., and C. Singh. 2016. “Using Reflection with Peers to Help Students Learn Effective Problem Solving Strategies.” Physical Review Physics Education Research 12(1):10103. doi: 10.1103/PhysRevPhysEducRes.12.010103.
Novotná, J., P. Eisenmann, J. Přibyl, J. Ondrušová, and J. Břehovský. 2019. “Problem Solving in School Mathematics Based on Heuristic Strategies.” Journal on Efficiency and Responsibility in Education and Science 12(1):1–6. doi: 10.7160/eriesj.2019.120101.
Phillips, J. A., J. E. McCallum, K. W. Clemmer, and T. M. Zachariah. 2016. “A Problem-Solving Framework to Assist Students and Teachers in STEM Courses.” Journal of College Science Teaching 46(2):33–39. doi: 10.2505/4/jcst16_046_02_33.
Pongsakdi, N., T. H. Laine, K. Veermans, M. M. Hannula-Sormunen, and E. Lehtinen. 2020. “Improving Word Problem Performance in Elementary School Students by Enriching Word Problems Used in Mathematics Teaching.” LUMAT: International Journal on Math, Science and Technology Education 8(1):23–44. doi: 10.31129/LUMAT.8.1.1318.
Rittle-Johnson, B. 2017. “Developing Mathematics Knowledge.” Child Development Perspectives 11(3):184–90. doi: 10.1111/cdep.12229.
Rott, B., B. Specht, and C. Knipping. 2021. “A Descriptive Phase Model of Problem-Solving Processes.” ZDM Mathematics Education 53:737–52. doi: 10.1007/s11858-021-01244-2.
Schukajlow, S., G. Kaiser, and G. Stillman. 2018. “Empirical Research on Teaching and Learning of Mathematical Modelling: A Survey on the Current State-of-the-Art.” ZDM Mathematics Education 50:5–18. doi: 10.1007/s11858-018-0933-5.
Smith, T. I., S. W. Brahmia, A. Olsho, A. Boudreaux, P. Eaton, P. J. Kelly, K. J. Louis, M. A. Nussenbaum, and L. J. Remy. 2020. “Developing a Reasoning Inventory for Measuring Physics Quantitative Literacy.” Physical Review Physics Education Research 16(2):20123. doi: 10.1103/PhysRevPhysEducRes.16.020123.
Verschaffel, L., S. Schukajlow, J. Star, and W. Van Dooren. 2020. “Word Problems in Mathematics Education: A Survey.” ZDM Mathematics Education 52:1–16. doi: 10.1007/s11858-020-01130-4.