Teacher professional learning (TPL) is crucial for improving student outcomes. Existing research indicates a connection between principal instructional leadership and TPL, often mediated by teacher attitudinal variables like self-efficacy. However, studies have yielded inconsistent results, suggesting the relationship might be more complex than a simple linear one. The dynamic nature of schooling, embedded within an open system, necessitates considering potential nonlinear relationships. This study addresses this gap by investigating both linear and nonlinear relationships between principal instructional leadership, teacher self-efficacy, and TPL. Specifically, it aims to determine: 1. The existence of a significant direct (linear) relationship between instructional leadership and TPL and teacher self-efficacy. 2. Whether teacher self-efficacy mediates the relationship between instructional leadership and TPL. 3. The existence of significant nonlinear relationships between instructional leadership and TPL and teacher self-efficacy.

The literature extensively supports the positive influence of principal leadership and teacher attitudinal variables on TPL. Principal instructional leadership is particularly significant in the Asian context. Prior studies primarily focused on linear relationships and mediating effects of teacher self-efficacy and trust. However, inconsistencies in findings highlight the need to explore non-linear relationships. Studies have shown a moderate positive correlation between instructional leadership and TPL, but some found non-significant relationships. These mixed results suggest the relationship may not be simply linear. The lack of comprehensive investigations into both linear and nonlinear relationships in a single model is a critical gap in the current literature. Advanced analytic methods capable of addressing nonlinearity are needed to generate more insightful findings.

A quantitative cross-sectional survey design was employed, targeting primary and secondary school teachers in Penang, Malaysia. A clustered sampling approach selected 335 participants (83.75% response rate) from 40 schools. Data collection used an online questionnaire via Google Forms, with ethical approvals obtained. The instruments used were: * **Principal Instructional Management Rating Scale (PIMRS) Teacher Short Form:** Assessed principal instructional leadership across three dimensions (defining school mission, managing instructional programs, developing a positive school climate). * **Teacher Sense of Efficacy Scale:** Measured teacher self-efficacy across three dimensions (efficacy for instructional strategies, classroom management, student engagement). * **Li et al.'s (2016) scale:** Assessed teacher professional learning. Partial least squares structural equation modeling (PLS-SEM) using SmartPLS 3.2.9 was employed to analyze the data. A two-step approach was used, first evaluating the reflective measurement model (assessing indicator reliability, convergent, and discriminant validity) and then the structural model (assessing linear and nonlinear relationships, predictive power using PLSpredict). The non-normality of data was addressed by using bootstrapping. Collinearity tests were conducted to rule out single-source bias.

The measurement model demonstrated good reliability and validity. The structural model revealed: * **Significant positive linear relationships:** Instructional leadership positively predicted both teacher self-efficacy and TPL. Teacher self-efficacy positively predicted TPL. * **Significant mediating effect:** Teacher self-efficacy significantly mediated the relationship between instructional leadership and TPL. * **Significant nonlinear relationships:** A significant nonlinear relationship was found between instructional leadership and teacher self-efficacy (U-shaped curve), and between teacher self-efficacy and TPL (U-shaped curve). The nonlinear relationship between instructional leadership and TPL was non-significant. * **High predictive power:** The structural model showed high in-sample and out-of-sample predictive power, as assessed by PLSpredict.

The findings support the positive linear relationship between instructional leadership and TPL, consistent with previous research. The mediating role of teacher self-efficacy is also confirmed. However, the significant nonlinear relationships add nuance. The U-shaped relationship between instructional leadership and teacher self-efficacy suggests that the impact of instructional leadership may not be uniformly positive across all levels; there's a possible initial negative relationship before turning positive. Similarly, the U-shaped relationship between teacher self-efficacy and TPL indicates that extremely high self-efficacy may not always translate to increased engagement in professional learning. The non-significant nonlinear effect of instructional leadership on TPL suggests that the linear relationship is robust. These nonlinear findings highlight the importance of considering the complex interplay of factors and the potential for non-linear effects in educational settings.

This study advances our understanding of the complex relationships between instructional leadership, teacher self-efficacy, and TPL by revealing significant nonlinear relationships alongside previously established linear ones. The findings have implications for school leaders and policymakers, highlighting the need for nuanced strategies to promote TPL. Future research could explore the contextual factors influencing these relationships, investigate additional variables like transformational leadership or learning-centered leadership, and use longitudinal designs to examine the dynamic interplay of these factors over time.

The study's generalizability is limited by its focus on a specific region (Penang, Malaysia) and sample size. The exclusion of additional mediating variables and the cross-sectional design prevent a full understanding of the causal relationships. Future research should address these limitations by using larger samples, incorporating additional variables, and employing longitudinal designs.