Problem posing to enhance flexibility and fraction understanding: Teachers’ predictions and students’ performance

Abstract

[Objective] The purpose of this study is to investigate the effect of problem posing on students’ conceptual comprehension of fractions and their capacity for flexibility in mathematics and the limited empirical evidence that supports the connection between the teachers’ predictions and students’ flexible ability to solve problems of fractions. [Methodology] This was a multiple-case qualitative study of 100 fifth-grade students and five mathematics teachers from five elementary schools in Malang, Indonesia (2024/2025). The pupils engaged in problem-posing on fractions posed at three levels of challenge. Teachers’ predictions were received by interviews and questionnaires. Data were analyzed thematically with reference to Silver’s model of mathematical flexibility and triangulated across students’ tasks, teacher interviews, and classroom observations to measure flexibility levels and prediction accuracy. [Results] Findings revealed that there were three levels of flexibility: 53 students displayed limited flexibility due to applying the same denominator for routine procedural problems; 39 demonstrated moderate flexibility by involving varied operations and multi-step reasoning; and 8 had high flexibility by creating complex, open-ended problems with multiple valid answers. Teacher predictions were mostly consistent except that some discrepancies occurred in the medium group, indicating difficulty in predicting students’ partially developed conceptual understanding. [Conclusion] Problem posing served to support the students' conceptual understanding and mathematical flexibility within a constructivist learning context, thus extending previous research into the extent to which teachers predict correctly in mathematics education. The consistency of teachers’ predictions with students’ demonstrated flexibility emphasizes the necessity of adaptive instructional practices that combine problem posing to facilitate flexible reasoning and meaningful mathematics learning.