Investigating the performance, representation application, and metacognitive knowledge for solving fraction problems among pupils

Understanding fractions poses a significant challenge for many pupils, primarily due to the complexity of representing and conceptualizing fractional relationships. The importance of representation in learning fractions cannot be overlooked, as it serves as the foundation for pupils' comprehension and problem-solving abilities. Metacognitive knowledge plays a crucial role in this process. It enables pupils to reflect on their understanding, choose appropriate strategies, and adjust their approaches based on the representation used. As pupils encounter various forms of representation, including enactive, iconic, and symbolic, their metacognitive knowledge guide them in navigating these challenges and deepening their conceptual grasp of fractions. While the focus of mathematics learning and teaching is on the mathematical process, it is necessary to investigate how pupils adapt this process. However, there is still a scarcity of studies that explicitly examine the intersection of representation and metacognitive knowledge in learning fraction. There is still a lack of studies on representation developed from metacognitive knowledge, which shows there is a research gap. Hence, this study aims to investigate the pupils’ performance in fractions, their application of representation, metacognitive knowledge, and the relationship between the performance in fractions and metacognitive knowledge. Particularly, this study investigates the support of metacognitive knowledge on the application of representations in fractions learning among primary school pupils in Perak. This research employed an explanatory sequential design to collect data using fraction test, metacognitive knowledge questionnaire and clinical interviews. Samples for the quantitative data were chosen using cluster random sampling for quantitative data and purposive sampling for qualitative data. The fraction test was used to collect data on fractions understanding and representations used. The metacognitive knowledge questionnaire was used to collect data on metacognitive knowledge. The clinical interviews were applied to collect data in depth, both on representations and metacognitive knowledge. The findings revealed that pupils have a moderate level of understanding of fractions with different types of representations were used. But they relied much on symbolic representations. The relationship showed the significant roles of metacognitive knowledge in fraction learning in terms of choosing suitable representations. The qualitative data revealed that mathematical processes of representations was achieved in solving fractions if pupils are prepared with metacognitive knowledge. The interview also revealed those pupils with limited metacognitive knowledge experience developmental growth within their abilities through various pathways which the proficiency in representation consistently contributes to the advancement of metacognitive knowledge, encompassing both tasks and strategies. Pupils who actively engaged in metacognitive knowledge, 57.14% of them primarily rely on symbolic representation before switching to other representations based on their suitability. Representations and metacognitive knowledge are not associated, but the interchange of the use of representation as well as the abilities (task, strategy, person) of using representation depends on the metacognitive knowledge engagement. The chi-square test value of p = 0.102 showed that there is no statistically significant association between representation and metacognitive knowledge. The research suggests that teachers should encourage pupils to use multiple representations when solving fraction problems to enhance their understanding of the concepts. Besides, both representation and metacognitive knowledge are important in learning fractions. The strategies of flexibility in using representations and frequent feedback through questioning in developing self-monitoring for metacognition development are expected to happen in the learning of mathematics. In addition, teachers can start the teaching using symbolic before switching to other types of representations since pupils easily applying symbolic representation.