Linear and nonlinear relationships between instructional leadership and teacher professional learning through teacher self-efficacy as a mediator: a partial least squares analysis

The study addresses inconsistencies in prior research regarding the relationship between principals’ instructional leadership (IL) and teacher professional learning (TPL), suggesting potential nonlinearity in these associations. Grounded in the view that schooling is an open, complex system, the study examines both linear and nonlinear relationships among IL, teacher self-efficacy (TSE), and TPL, with TSE as a mediator. Research questions: (1) Are there significant direct (linear) relationships between IL and TPL, and between IL and TSE? (2) Does TSE mediate the relationship between IL and TPL? (3) Are there significant nonlinear relationships between IL and TPL, and between IL and TSE? The context is Malaysia’s centralized, hierarchical education system, where instructional leadership is prioritized but principals face constraints. The purpose is to provide more robust empirical insight by modeling both linear and nonlinear effects to better capture the dynamics influencing teacher outcomes.

Theoretical foundations and hypotheses: - Teacher professional learning (TPL) is an ongoing, dynamic process encompassing formal and informal, job-embedded activities that enhance teacher knowledge and practice to support student learning. It is linked to school effectiveness and student outcomes. - Instructional leadership (IL), widely studied since the 1980s, is framed by Hallinger and Murphy (1985) in three dimensions: defining the school mission, managing the instructional program, and developing a positive school learning climate. IL facilitates structures and environments that support TPL. - Empirical studies generally find positive linear links between IL and TPL across contexts (e.g., Turkey, China, Malaysia), though some report nonsignificant results, hinting at complexity and possible nonlinearities. - IL and Teacher Self-Efficacy (TSE): Drawing on social cognitive theory, IL can enhance TSE via vicarious experiences and verbal persuasion. Meta-analytic and empirical studies show positive associations between IL and TSE. - TSE and TPL: TSE relates positively to engagement in professional learning; TSE often mediates the IL→TPL relationship in diverse contexts. - Complexity theory motivates examining nonlinear relationships in educational systems, where variable interactions may not be linear. Hypotheses: H1: IL has a linear, positive relationship with TPL. H2: IL has a linear, positive relationship with TSE. H3: TSE has a linear, positive relationship with TPL. H4: TSE mediates the IL→TPL relationship. H5: IL has a nonlinear relationship with TPL. H6: IL has a nonlinear relationship with TSE. H7: TSE has a nonlinear relationship with TPL. Research context: Malaysia’s centralized system prioritizes IL but limits principal autonomy; professional development is mandated via in-house training, with potential information loss in cascading delivery. Prior Malaysian studies have not simultaneously tested linear and nonlinear relations among IL, TSE, and TPL.

Design and sample: Quantitative cross-sectional survey of primary and secondary school teachers in Penang, Malaysia. Clustered sampling across four districts (North, Central, Northeast, Southwest): 10 primary and 10 secondary schools per district (target 40 schools); 10 teachers randomly selected per school (target n=400). Final sample n=335 (response rate 83.75%), collected May–August 2020. Ethics approvals: USM/JEPeM/20020077 and Ministry of Education Malaysia (KPM.600-3/2/3-eras (7462)). Participants were informed and consented; responses confidential and voluntary. T-tests indicated no significant differences between primary and secondary teachers on IL, TSE, and TPL. Measures: - Instructional Leadership (IL): Principal Instructional Management Rating Scale (PIMRS) Teacher Short Form (Hallinger & Wang, 2015), 22 items, 5-point Likert; dimensions: Defining school mission (DE), Managing instructional programme (MA), Positive school climate (PC). Reliability and validity acceptable (Table 2). - Teacher Self-Efficacy (TSE): Teacher Sense of Efficacy Scale—short form (Tschannen-Moran & Woolfolk Hoy, 2001), 12 items, 9-point Likert; dimensions: Instructional strategies (IS), Classroom management (CM), Student engagement (SG). Reliability acceptable. - Teacher Professional Learning (TPL): Li et al. (2016) unidimensional scale, 8 items, 6-point Likert; one negatively worded item recoded (TP3_R). Cronbach’s alpha 0.883. Data screening and bias checks: Data non-normal by Mardia’s tests (skewness β=8.078, p<0.01; kurtosis β=81.398, p<0.01). Single-source bias assessed via full collinearity VIFs: IL=1.288, TSE=1.247, TPL=1.365 (<3.3), indicating no serious common method bias. Analysis approach: Partial least squares structural equation modeling (PLS-SEM) using SmartPLS 3.2.9. Second-order modeling for IL and TSE (higher-order constructs with first-order dimensions). Two-step evaluation: (1) Reflective measurement models: indicator reliability (loadings ≥0.70), convergent validity (CR>0.80; AVE>0.50), discriminant validity via HTMT<0.90. (2) Structural model: 10,000-subsample bootstrap, path significance, effect sizes (f²), and predictive power via PLSpredict (Q²predict, RMSE vs linear model). Nonlinearity tested by including quadratic terms (IL×IL, TSE×TSE) with percentile bootstrapping (95% CI, two-tailed, 10,000 subsamples).

Measurement model: All first- and second-order constructs demonstrated acceptable reliability (CR>0.80) and convergent validity (AVE>0.50); discriminant validity supported (HTMT<0.90). No indicators were dropped despite a few loadings <0.70, as CR and AVE thresholds were met. Structural model—Linear effects: - H1: IL → TPL: β=0.364, t=4.275, p<0.001, PCI [0.175, 0.525], f²=0.112 (supported). - H2: IL → TSE: β=0.511, t=9.094, p<0.001, PCI [0.399, 0.623], f²=0.268 (supported). - H3: TSE → TPL: β=0.314, t=5.780, p<0.001, PCI [0.211, 0.420], f²=0.087 (supported). - H4 (mediation): IL → TSE → TPL: indirect β=0.160, t=4.994, p<0.001, PCI [0.102, 0.225], f²=0.160 (supported). Nonlinear effects: - H5: IL×IL → TPL: β=0.033, t=1.001, p=0.371, PCI [−0.034, 0.086], f²=0.004 (not supported). - H6: IL×IL → TSE: β=0.181, t=5.670, p<0.001, PCI [0.110, 0.240], f²=0.137 (supported; substantial effect). Relationship exhibits a U-shape. - H7: TSE×TSE → TPL: β=0.048, t=2.074, PCI [0.003, 0.095], f²=0.009 (supported; small effect). Relationship exhibits a U-shape. Predictive power: R² ≈ 0.23 for TSE (explained by IL) and R² ≈ 0.28 for TPL (explained by IL and TSE), indicating satisfactory in-sample predictive power. PLSpredict showed positive Q²predict values and lower RMSE for PLS vs linear model, indicating high out-of-sample predictive capability.

Findings confirm robust positive linear relationships among IL, TSE, and TPL, and a significant mediating role of TSE in the IL→TPL pathway. This addresses RQ1–RQ2 and aligns with prior literature, reinforcing the importance of principals organizing professional learning opportunities, providing feedback, and cultivating collaborative cultures. Nonlinear analyses (RQ3) reveal additional complexity: IL shows a U-shaped relationship with TSE, suggesting that at lower levels initial IL efforts may not immediately raise TSE (potential lag due to recognition and relationship-building), but beyond a threshold, stronger IL is associated with higher TSE. TSE also shows a U-shaped relationship with TPL, indicating that very low or very high TSE may be associated with lower TPL initially, with increased engagement in professional learning emerging as TSE grows further and teachers recognize areas for improvement. No significant nonlinearity was found for IL→TPL, suggesting the linear direct effect is robust. Collectively, results underscore the value of modeling both linear and nonlinear dynamics to better inform leadership practices and policy in complex educational systems.

The study advances educational leadership research by jointly modeling linear and nonlinear relationships among instructional leadership, teacher self-efficacy, and teacher professional learning in a Malaysian context. It confirms positive linear effects and TSE’s mediating role, and uncovers significant nonlinear (U-shaped) relationships for IL→TSE and TSE→TPL. Methodologically, it demonstrates the utility of PLS-SEM for higher-order constructs and robustness checks with nonlinearity and PLSpredict. Practically, it suggests principals should strengthen instructional leadership practices and invest time in sustained interactions to build TSE and foster professional learning. Future research should expand models with additional leadership and teacher variables, examine contextual moderators, and employ longitudinal and qualitative designs to capture temporal dynamics.

Generalizability is limited to 335 teachers from Penang, Malaysia. The study did not analyze differences by school type (primary vs secondary) or location (urban vs rural); contextual factors may influence results. The centralized Malaysian context may limit applicability to other cultures. The model explains about 28% of variance in TPL, indicating omitted variables (e.g., transformational or learning-centered leadership, teacher trust, agency, mastery goals). Future research should include additional predictors/mediators, use larger and more representative samples, and apply longitudinal methods (e.g., latent growth curve modeling) and qualitative approaches to deepen understanding of linear and nonlinear dynamics.