The sub-dimensions of metacognition and their influence on modeling competency

The study addresses the need to better understand how metacognition influences mathematical modeling competence, given widespread student difficulties with modeling and the growing curricular emphasis on authentic problem solving (e.g., PISA). It focuses on two sub-constructs of modeling competence—horizontal mathematization (translating real-world contexts into mathematical representations) and vertical mathematization (working within mathematics to solve and interpret)—and examines whether metacognitive sub-dimensions (awareness, cognitive strategy, planning, self-checking) predict these competencies. Research questions: (a) Do awareness, cognitive strategy, planning, and self-checking predict horizontal mathematization? (b) Do these sub-constructs predict vertical mathematization? The work aims to inform instruction by identifying which metacognitive processes are most relevant to different phases of modeling.

The theoretical framing draws on the Models and Modeling Perspective (MMP), which views models as conceptual tools emerging from real-world contexts and emphasizes iterative modeling cycles. Mathematical modeling is conceptualized as a process of mathematization with complementary forms: horizontal mathematization (simplifying, formulating, translating real-world problems into mathematical terms) and vertical mathematization (operating within mathematics, interpreting and validating results). Metacognition comprises knowledge of and regulation over one’s cognitive processes, often operationalized via awareness, cognitive strategies, planning, and self-checking. Prior work suggests metacognition supports problem solving and modeling, but evidence is mixed regarding which sub-dimensions matter most, and transfer to modeling contexts can be constrained by mathematical knowledge and task complexity. Hypotheses posited significant relationships between each metacognitive sub-dimension and both horizontal and vertical mathematization.

Design: Correlational study using structural equation modeling (SEM) to examine relations between metacognitive sub-dimensions and modeling competence. Participants: N=538 university students in mathematics education programs (Riau Province, Indonesia); first-year 24.7% (n=133), second-year 41.4% (n=223), third-year 33.8% (n=182); 89.8% female (n=483), 10.2% male (n=55). Cluster random sampling across universities with similar characteristics. Ethical approval and informed consent obtained. Testing time: 60 minutes for metacognition inventory and modeling test. Measures:

• Modeling competence: Adapted from Haines and Crouch (2001). Two sub-constructs: horizontal mathematization (18 items) and vertical mathematization (4 items). Multiple-choice items scored 0 (wrong), 1 (partially correct), 2 (correct); total 22 items (max score 44). Reliability: horizontal α=0.861; vertical α=0.740. Composite reliability (CR)=0.775–0.925; AVE=0.500–0.501; discriminant validity supported.
• Metacognition: O’Neil and Abedi (1996) inventory with four 5-item subscales: awareness, cognitive strategy, planning, self-checking. Example items tailored to modeling tasks. Reliability: awareness α=0.825; cognitive strategy α=0.853; planning α=0.842; self-checking α=0.828. CR=0.775–0.925; AVE=0.500–0.526; discriminant validity supported. Analysis Strategy: Descriptive statistics, missing data checks, outlier detection, normality (skewness ±2.0; kurtosis ±8.0). Pearson correlations among latent variables (multicollinearity threshold r<0.900). Confirmatory factor analyses (CFAs) for metacognition (20 indicators) and modeling competence (22 indicators), followed by SEM to test hypothesized paths from metacognitive sub-dimensions to horizontal and vertical mathematization. Fit indices and thresholds: χ²/df<5, RMSEA<0.08, SRMR<0.08, CFI>0.95, TLI>0.95, GFI>0.90. Software: AMOS 18.0.

Descriptive statistics: Metacognition subscales were moderate (means: awareness 3.940, cognitive strategy 3.737, planning 3.951, self-checking 3.910). Modeling competence was moderate (horizontal M=0.914, vertical M=0.848). Normality criteria met for skewness and kurtosis. Highest correlation observed between awareness and cognitive strategy (r=0.677); correlation between horizontal and vertical mathematization was r=0.342. Measurement models: Metacognition CFA fit: χ²=325.454, χ²/df=1.984, RMSEA=0.043, SRMR=0.036, CFI=0.965, GFI=0.955, TLI=0.959. Modeling competence CFA fit: χ²=261.077, χ²/df=1.305, RMSEA=0.024, SRMR=0.041, CFI=0.975, GFI=0.958, TLI=0.971. All factor loadings >0.50 and significant (P<0.001). Structural model: Excellent fit: χ²=1163.570, df=797, χ²/df=1.460, RMSEA=0.029, SRMR=0.043, CFI=0.950, GFI=0.908, TLI=0.950. Path coefficients:

• Horizontal mathematization: cognitive strategy β=0.26, t=2.535, P=0.011 (significant); planning β=0.23, t=2.369, P=0.018 (significant); self-checking β=0.23, t=2.470, P=0.014 (significant); awareness β=0.17, t=1.685, P=0.092 (ns).
• Vertical mathematization: self-checking β=0.27, t=2.138, P=0.033 (significant); awareness β=0.08, t=0.635, P=0.526 (ns); cognitive strategy β=0.24, t=1.763, P=0.078 (ns); planning β=0.15, t=1.180, P=0.238 (ns). Interpretation: Cognitive strategy, planning, and self-checking significantly predicted horizontal mathematization, while only self-checking significantly predicted vertical mathematization. Awareness did not significantly predict either form. The authors note these standardized effects as approximately 26%, 23%, and 23% for horizontal and 27% for vertical, respectively.

Findings address the research questions by showing that metacognitive regulation components—cognitive strategy use, planning, and especially self-checking—are linked to students’ capacity to translate real-world situations into mathematical representations (horizontal mathematization). In contrast, only self-checking relates significantly to operating within the mathematical domain and interpreting results (vertical mathematization). The non-significant role of awareness suggests that simply being conscious of one’s thinking may be insufficient without active regulation strategies. The differential pattern implies that vertical mathematization may demand more advanced mathematical knowledge and abstraction, making general cognitive strategies and planning less directly impactful unless coupled with robust monitoring. Pedagogically, embedding explicit instruction and scaffolds for self-monitoring, structured planning, and strategic processing during modeling tasks can strengthen modeling proficiency, particularly in the transition from contextual formulation to within-mathematics reasoning and validation.

The study clarifies how specific metacognitive sub-dimensions relate to distinct aspects of modeling competence. Cognitive strategy, planning, and self-checking bolster horizontal mathematization, while self-checking uniquely supports vertical mathematization; awareness alone shows no direct effect. These results underscore the importance of fostering metacognitive regulation—especially monitoring/self-checking—in modeling instruction. The work contributes by distinguishing metacognitive influences across modeling phases and suggests instructional designs emphasizing planning and strategic organization for formulation tasks and sustained self-monitoring for within-mathematics processing and interpretation. Future research should identify which concrete strategies, planning routines, and monitoring practices most effectively promote horizontal and vertical mathematization and examine how mathematical content knowledge moderates these effects.

• Some hypothesized relationships were unsupported, indicating other factors (e.g., prior mathematical knowledge, motivation, instructional quality) may be influential.
• Correlational design precludes causal inference; experimental or intervention studies are needed to establish causality.
• Reliance on self-reported metacognition may introduce bias; incorporating objective measures, observations, or multiple data sources would strengthen validity.