The Effect of a Multiple-Solution Approach to Solving Open-Ended Mathematical Modeling Problems on Students’ Independence and Self-Efficacy

Abstract

Purpose: This study aimed to determine the effect of a multiple-solution approach to open-ended mathematical modeling problems on the independence and self-efficacy of ninth-grade female students.

Methodology: An explanatory mixed-methods design was employed, comprising a one-group quasi-experimental pretest–posttest component with repeated assessments and a qualitative case study. The participants were 76 ninth-grade female students selected through convenience sampling from a public school in District 3 of Tehran. The intervention consisted of seven 45-minute sessions involving explicit instruction in the mathematical modeling cycle, open-ended problems containing sufficient, missing, or irrelevant data, and the generation and comparison of multiple solutions in heterogeneous groups of three or four students. Mathematical independence was assessed using a six-item, five-point Likert questionnaire. Cronbach’s alpha coefficients across the assessment sessions ranged from 0.842 to 0.936. Semi-structured interviews were also conducted with nine purposively selected students representing low, moderate, and high levels of mathematical performance. Quantitative data were analyzed using the Friedman test.

Findings: The Friedman test demonstrated statistically significant session-related changes in five of the six mathematical independence items. The significance levels for these items ranged from 0.008 to 0.049, and their mean ranks generally increased across the intervention sessions. One questionnaire item did not show a statistically significant difference across sessions (p=0.069). Friedman statistics for the significant items ranged from 9.514 to 13.931, indicating significant improvements in dimensions associated with organizing learning activities, choosing participation and problem-solving strategies, making independent decisions, and reviewing and correcting solutions without direct dependence on the teacher.

Conclusion: Integrating explicit mathematical modeling instruction with open-ended tasks, multiple-solution strategies, and small-group collaborative learning can strengthen students’ learning independence and perceived self-efficacy. By allowing students to select strategies, articulate personal ideas, evaluate alternative solutions, and assume responsibility for correcting their work, this instructional approach can facilitate a transition from teacher-dependent learning toward active, self-directed mathematical learning.