Problem-based learning (PBL), a teaching strategy where students solve real-world problems in small groups with facilitator assistance, aims to enhance knowledge application and positive attitudes towards learning. In mathematics, PBL contextualizes knowledge, helping students understand when and how to apply it. Students identify their existing knowledge, determine missing information, search for it, and apply it to a new context. This approach should improve students' ability to apply mathematical knowledge and foster positive attitudes towards mathematics. While previous research supports PBL's effectiveness, there's a lack of understanding regarding the impact of teacher training on PBL implementation and student outcomes. This study addresses this gap by investigating how teacher training in PBL affects students' mathematical application and attitudes towards the subject.

Existing literature demonstrates PBL's effectiveness in improving knowledge application and fostering positive attitudes across various grade levels, particularly in fields like medicine. However, research in K-12 settings, especially primary schools, is limited. While teacher training in PBL is widely considered crucial for successful implementation, empirical evidence on its impact on student performance remains scarce. Studies highlight the importance of teachers' facilitation skills in guiding students' learning processes, suggesting that teacher training plays a vital role in maximizing PBL's benefits. There is a need for more research to explicitly investigate the effect of teacher training on student outcomes within a PBL framework, particularly in the context of mathematics.

This study employed a quasi-experimental design with a mixed-methods approach. 127 third-grade students from a randomly selected private school in Hail, Saudi Arabia, were divided into three groups: Group A (52 students) taught by a trained teacher using PBL, Group B (39 students) taught by an untrained teacher using traditional teaching methods (TTM), and Group C (36 students) taught by an untrained teacher using PBL. The topic was 'data display'. Data collection involved pre- and post-tests measuring knowledge application (adapted from TIMSS) and attitudes towards mathematics (also adapted from TIMSS). Qualitative data were gathered through classroom observations and semi-structured teacher interviews to assess PBL implementation. The trained teacher underwent a week-long PBL training program focusing on facilitating group learning and employing metacognitive strategies. The untrained teachers received PBL materials but no formal training. Quantitative data were analyzed using mixed-factor ANOVA, with Tukey's post hoc test used for pairwise comparisons. Qualitative data were analyzed deductively using a pre-defined categorization matrix based on existing literature, focusing on 'understanding the problem' and 'using metacognitive strategies'. Methodological triangulation was employed by comparing observational data with interview data.

Qualitative findings revealed that the trained teacher effectively implemented PBL, providing more time for students to understand problems and employing more metacognitive strategies compared to the untrained teachers. The trained teacher actively encouraged students to explain problems in their own words and used metacognitive questioning to guide their thinking. In contrast, the untrained teachers often proceeded with problem-solving after a simple affirmation of understanding, employing fewer metacognitive strategies. Quantitative analysis showed a significant improvement in knowledge application across all groups from pre-test to post-test (F(1, 121) = 76.795, p < 0.001, η²p = 0.388). Post-hoc analysis indicated a significantly greater improvement in knowledge application for Group A (trained teacher using PBL) compared to Group B (TTM group) (p = 0.009). Regarding attitudes towards mathematics, there was no significant overall improvement across groups. However, a significant interaction effect between time and group was found (F(3, 121) = 12.486, p < 0.001, η²p = 0.237). Post-hoc tests revealed that both PBL groups (Groups A and C) showed significantly more positive changes in attitudes towards mathematics than the TTM group (Group B) (p < 0.01 and p = 0.008, respectively). No significant difference was found between the attitudes of Group A and Group C.

The study's findings support the importance of teacher training in effectively implementing PBL to enhance student learning. The trained teacher's greater use of metacognitive strategies significantly improved students' ability to apply mathematical knowledge. This aligns with existing literature emphasizing the role of teacher facilitation in PBL's success. The significant improvement in attitudes towards mathematics in the PBL groups (regardless of teacher training) indicates that the real-life problem-solving context and active learning aspects of PBL are key factors in fostering positive student attitudes. This underscores that although teacher training is crucial for maximizing PBL's potential, PBL itself holds inherent benefits for enhancing student attitudes toward mathematics. The results strongly suggest that effective implementation of PBL necessitates teachers trained in facilitating students' learning processes and employing metacognitive strategies to support student independence.

This study demonstrates that PBL is an effective strategy for improving students' attitudes toward mathematics, even with untrained teachers. However, teacher training significantly enhances the effectiveness of PBL in improving students’ ability to apply mathematical knowledge. Future research should explore the long-term effects of teacher training on student outcomes, investigate the generalizability of these findings across different age groups and cultural contexts, and examine the optimal methods for training teachers in PBL.

The study's limitations include its non-randomized sample selection, the specific focus on third-grade students and mathematics, and the homogenous gender composition of the participants (all male). These factors limit the generalizability of the findings. The limited duration of the intervention and the specific context of the study also constrain the extent to which the results can be generalized.