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

Mathematical modeling is a crucial skill for solving real-world problems, yet many students struggle with it. This study addresses the gap in research on the relationship between metacognition and mathematical modeling competency. The current curriculum emphasizes problem-solving skills beyond rote memorization and formula application. Mathematical modeling, a key component in PISA assessments, fosters deeper understanding and versatility in problem-solving by connecting mathematics to real-world scenarios. However, traditional assessment methods for modeling proficiency are insufficient. This study aims to understand the role of metacognition—the awareness and control of one's thinking processes—in predicting students' modeling abilities. The study explores whether specific metacognitive sub-dimensions (awareness, cognitive strategy, planning, and self-checking) predict success in horizontal (translating real-world problems into mathematical representations) and vertical (working within mathematics to solve the problem) mathematization.

The Models and Modeling Perspective (MMP) framework is introduced, emphasizing the cyclical nature of modeling and the integration of real-world context into problem-solving. The study distinguishes between horizontal and vertical mathematization, with horizontal involving translating real-world problems into mathematical terms, and vertical focusing on applying mathematical tools to solve the translated problem. The literature supports the role of metacognition in problem-solving, especially in complex tasks. Previous research suggests that metacognitive strategies such as planning, monitoring, and reflection improve problem-solving skills. However, there's limited research on the specific metacognitive strategies most effective for improving mathematical modeling, particularly the different effects on horizontal and vertical mathematization. This research gap motivated the present study to explore the influence of metacognition's sub-dimensions on these two aspects of mathematical modeling competency.

This study employed a correlational research design using structural equation modeling (SEM) with AMOS version 18.0. The participants were 538 Indonesian university students studying mathematics education, selected using cluster random sampling from universities with similar characteristics. Data collection involved a 60-minute assessment comprising a metacognitive inventory (measuring awareness, cognitive strategy, planning, and self-checking) and a modeling test (measuring horizontal and vertical mathematization). The modeling test used a three-level scoring system (0=wrong, 1=partially correct, 2=correct). Descriptive statistics, Pearson correlations, and SEM were used to analyze the data. The measurement model assessed the fit of the instruments to the data using CFA, ensuring the reliability and validity of the measures before testing the hypothetical model. Goodness-of-fit indices (χ²/df, RMSEA, SRMR, CFI, GFI, TLI) were used to evaluate the model fit. Prior to SEM, the data was checked for missing data, outliers, normality (using skewness and kurtosis), and multicollinearity among latent variables. The hypothesized model tested the relationships between the metacognitive sub-dimensions and the two types of mathematization.

The measurement model showed a satisfactory fit for both metacognition and modeling competency. Descriptive statistics revealed moderate levels of metacognitive skills and modeling competency among the participants. SEM analysis revealed the following significant relationships: * Cognitive strategy, planning, and self-checking positively predicted horizontal mathematization (β = 0.26, 0.23, and 0.23 respectively, all p < 0.05). * Only self-checking positively predicted vertical mathematization (β = 0.27, p < 0.05). * Awareness showed no significant relationship with either horizontal or vertical mathematization. Cognitive strategy, planning, and self-checking together accounted for 26%, 23%, and 23% of the variance in horizontal mathematization, while self-checking alone accounted for 27% of the variance in vertical mathematization.

The findings partially support the hypotheses. The significant positive relationships between cognitive strategy, planning, self-checking, and horizontal mathematization suggest that metacognitive control processes are crucial for translating real-world problems into mathematical models. The significant relationship between self-checking and vertical mathematization emphasizes the role of monitoring and error correction in the mathematical manipulation phase of modeling. The lack of a significant relationship between awareness and mathematization suggests that simply being aware of one's thinking processes is not sufficient for successful modeling; rather, active control and regulation of those processes are key. The differing effects of metacognitive sub-dimensions on horizontal and vertical mathematization indicate the complexity of the modeling process and the need for nuanced teaching strategies.

This study provides valuable insights into the role of metacognition in mathematical modeling competency. While awareness alone doesn't significantly influence success, the active use of cognitive strategies, planning, and self-checking are vital for improving both horizontal and vertical mathematization. Future research should explore specific cognitive strategies, planning techniques, and self-checking methods that enhance mathematical modeling. Further investigation into other factors like prior mathematical knowledge, motivation, and teaching quality is also warranted. The findings inform the development of teaching approaches that specifically target metacognitive skills to enhance students’ mathematical modeling abilities.

The study's correlational design does not establish causality. The reliance on self-reported measures of metacognition might introduce bias. Future research could benefit from experimental designs and objective measures to address these limitations. The sample consisted of university students in Indonesia, limiting the generalizability of the findings to other populations and contexts. Exploring different cultural contexts and age groups could provide broader insights into the role of metacognition in mathematical modeling.