Mathematical conceptual understanding in vocational education is conceptualized in this study as a multidimensional latent construct that reflects how students interpret, represent, and apply mathematical ideas in technical and real-world contexts. Drawing on contemporary research in mathematics education and psychometrics, conceptual understanding is not treated as a single skill but as an integrated system of cognitive and representational processes that enable meaningful problem solving in vocational settings. Consistent with prior measurement and construct validation studies, this framework assumes that complex educational constructs are best represented as interrelated latent dimensions rather than isolated abilities. ^{12, 13} Such an approach aligns with multidimensional models used in recent CFA and measurement invariance research, which emphasize the need to capture both commonality and diversity in how learners conceptualize academic constructs. ¹⁴

In this study, mathematical conceptual understanding is operationalized through four interrelated dimensions: conceptual reasoning, mathematical representation, problem modeling, and knowledge transfer. Conceptual reasoning refers to students’ ability to understand the underlying principles of mathematical procedures rather than merely applying formulas mechanically. This dimension is closely related to deeper cognitive engagement in learning and has been associated with higher academic motivation and persistence. ¹⁴

Mathematical representation captures students’ capacity to translate problems among verbal, graphical, and symbolic forms. This skill is critical in vocational contexts where mathematical ideas must be communicated through diagrams, technical drawings, or data visualizations. Prior research suggests that representational competence is a key component of mathematical proficiency and is influenced by both instructional quality and learner beliefs. ¹⁵ Problem modeling reflects students’ ability to frame real-world vocational challenges—such as machine calibration, energy efficiency, or production optimization—into mathematical formulations. This dimension resonates with findings that meaningful learning in technology-enhanced vocational education depends on students’ ability to connect abstract mathematics with practical applications. ¹⁶

Knowledge transfer refers to students’ capacity to apply mathematical concepts to new and unfamiliar technical situations. This ability is particularly relevant in rapidly changing industrial environments where workers must adapt their skills to emerging technologies and sustainability-oriented practices. ¹⁷ Together, these four dimensions form a coherent higher-order representation of mathematical conceptual understanding in vocational education. The framework assumes that while these dimensions are distinct, they are theoretically and empirically correlated, reflecting a broader latent construct of mathematical thinking.

Research in STEM education consistently shows that students’ learning processes and beliefs are shaped by contextual factors such as teaching practices, classroom climate, and socio-cultural background. ¹⁸ In vocational settings, these influences may be further shaped by school specialization, regional industrial characteristics, and access to technology-enhanced learning environments. Studies on mathematics self-beliefs and performance demonstrate that gender and contextual factors mediate how students perceive and engage with mathematics. ¹⁹ Similarly, cross-cultural and cross-group studies indicate that constructs such as motivation, attitudes, and teaching presence are not always interpreted equivalently across populations, necessitating rigorous measurement invariance testing. ¹³ Within vocational education, teacher–student interaction and technology-enhanced instruction have been shown to influence students’ engagement and learning outcomes, suggesting that mathematical understanding may vary across different educational environments. ¹⁶ Moreover, recent work on TVET and indigenous cultural integration in Indonesia highlights the importance of contextualized learning, which may further shape how students conceptualize mathematical ideas. ¹⁷

Given these theoretical and empirical considerations, it is insufficient to assume that vocational students across regions, school types, or gender groups conceptualize mathematics in the same way. Instead, measurement invariance analysis is required to determine whether the structure of mathematical conceptual understanding is comparable across groups.

Based on the conceptual framework and prior research, the following hypotheses are proposed:

Figure 1 illustrates the conceptual framework of the study, linking grouping variables (region, school specialization, and gender) to four latent dimensions of mathematical conceptual understanding—Conceptual Reasoning (CR), Mathematical Representation (MR), Problem Modeling (PM), and Knowledge Transfer (KT). The diagram represents three analytical stages: (1) validation of the four-factor structure (H1), (2) testing of measurement invariance across groups using multi-group CFA (H2–H4), and (3) comparison of latent means across groups after establishing partial scalar invariance (H5).

This study employed a cross-sectional quantitative research design based on structural equation modeling (SEM). ²⁶ The analysis was conducted in two sequential stages. First, Confirmatory Factor Analysis (CFA) was used to validate the measurement model of mathematical conceptual understanding among vocational students. ²⁷ Second, Multi-Group Measurement Invariance (MGI) analysis was conducted to examine whether this construct was interpreted equivalently across different groups. ²⁸ The overall analytic framework assumes that mathematical conceptual understanding is a latent construct that cannot be directly observed but can be inferred from multiple observed indicators. In matrix form, the CFA measurement model can be expressed as: x = Λξ + δ (1) where: •

X = represents the vector of observed items,

This formulation follows standard SEM conventions for latent variable modeling.

Participants were vocational high school students enrolled in public vocational schools in Indonesia. A stratified sampling approach was used to ensure representation across different regions and school types. Students were grouped by: •

Region: Java vs. non-Java

A minimum total sample of approximately 400 students was targeted to ensure stable parameter estimates in CFA and multi-group analysis, with at least 200 students in each major comparison group, consistent with SEM recommendations. Participation was voluntary. Written informed consent was obtained from students and schools, and ethical approval was granted by the relevant institutional review board.

Mathematical conceptual understanding was operationalized as a multidimensional construct with four correlated latent factors ^29–31 : •

Conceptual Reasoning (CR)

Each factor was measured using four to six Likert-type items (1 = strongly disagree to 5 = strongly agree), ³² resulting in 18–24 total items. Content validity was ensured through expert review by three mathematics educators and two vocational teachers. A pilot study with 60 students confirmed clarity and acceptable reliability (Cronbach’s α ≥ .80 for all subscales). ³³

Data were collected using a paper-based questionnaire administered during regular class hours. ³⁴ Students completed the survey in approximately 20–25 minutes. Cases with more than 10% missing responses were removed. ³⁵ For remaining missing data, Full Information Maximum Likelihood (FIML) estimation was used. Item distributions were inspected for normality (|skewness| < 2; |kurtosis| < 7), and multivariate outliers were checked using Mahalanobis distance. ³⁶

A four-factor correlated model was tested. Model fit was evaluated using the following indices: •

CFI ≥ .95

Convergent validity was assessed using: AV E j = ∑ i = 1 k λ ij 2 ∑ i = 1 k λ ij 2 + ∑ i = 1 θ ij k (2) where λ _ij is the standardized factor loading and θ _ij is the error variance for item i on factor j. AVE ≥ .50 was considered acceptable.

Composite reliability was computed as: CR j = ( ∑ λ ij ) ( ∑ λ ij ) + ∑ θ ij (3)

Discriminant validity was evaluated using the Fornell–Larcker criterion: AV E j > r jk (4) where r _jk the correlation between factors j and k.

MGI testing followed a hierarchical procedure across region, specialization, and gender.

Step 1 — Configural invariance

The same factor structure was specified for all groups without constraining parameters. Configural invariance implies that students conceptualize mathematics using the same underlying model.

Step 2 — Metric invariance (weak invariance)

Factor loadings were constrained equal across groups: Λ g 1 = Λ g 2 (5)

Metric invariance was supported if: Δ CFI ≤ 0.01 (6)

Step 3 — Scalar invariance (strong invariance)

Item intercepts were constrained equal: T g 1 = T g 2 (7)

If full scalar invariance was not achieved, partial invariance was implemented by releasing constraints on non-invariant items.

Step 4 — Latent mean comparison

Once at least partial scalar invariance was established, latent mean differences were estimated as: Δ μ = μ g 1 − μ g 2 (8)

Effect sizes were reported using Cohen’s ddd.

Primary analyses were conducted in R (lavaan) and cross-validated in Mplus. The robust maximum likelihood estimator (MLR) was used to accommodate potential non-normality.

Three additional analyses were performed: a.

Known-groups validity: Comparison between high- and low-achieving students.

Data for this study were collected from 125 vocational high school students enrolled in public vocational schools in Indonesia. Schools were selected using a stratified cluster sampling approach to ensure representation from different geographic, institutional, and instructional contexts. Two main regional clusters were included: schools located in Java (n = 64) and non-Java regions (n = 61). Within each region, schools represented a range of specializations, including technical/engineering programs (e.g., mechanical, electrical, and manufacturing) and non-technical programs (e.g., business, administration, and services). Within each participating school, intact classes were invited to participate during regular mathematics lessons. The questionnaire was administered in a paper-based format by trained research assistants. Students were informed that participation was voluntary, that responses would be anonymous, and that their data would be used solely for research purposes. The administration took approximately 20–25 minutes, and no identifying personal information was collected. Before conducting any structural modeling, the dataset underwent rigorous screening. Cases with more than 10% missing responses were removed from analysis. For the remaining cases, missing values were handled using Full Information Maximum Likelihood (FIML) estimation, which is appropriate for SEM-based analyses. Item distributions were inspected for normality; skewness and kurtosis values fell within acceptable ranges (|skewness| < 2; |kurtosis| < 7). Multivariate outliers were assessed using Mahalanobis distance, and three extreme cases were excluded to prevent distortion of parameter estimates. These steps ensured that the final dataset was suitable for CFA and multi-group measurement invariance analysis, consistent with best practices in psychometric research.

A four-factor correlated model of mathematical conceptual understanding was tested using CFA on the full sample (N = 125). The model specified four latent dimensions: Conceptual Reasoning (CR), Mathematical Representation (MR), Problem Modeling (PM), and Knowledge Transfer (KT). To demonstrate that this structure was theoretically and empirically superior, two alternative models—a three-factor model and a two-factor model—were also estimated for comparison.

Table 1 Interpretation of the four-factor model exhibited excellent fit to the data (CFI = .961; TLI = .954; RMSEA = .046; SRMR = .041), outperforming both alternative models across all fit indices, including AIC and BIC. This confirms Hypothesis 1 (H1): mathematical conceptual understanding among vocational students is best represented as a multidimensional construct rather than a single or simplified factor.

Figure 2 shown all items loaded strongly on their respective latent factors (λ ≥ .62, p < .001), indicating that each indicator reliably represented its intended construct. The explained variance (R ²) ranged from .38 to .96, suggesting that a substantial portion of item variance was attributable to the underlying latent factors rather than measurement error.

Table 2 Shown that all four factors demonstrated high reliability (CR ≥ .82; AVE ≥ .55). The square root of AVE exceeded inter-factor correlations, confirming discriminant validity. Collectively, Tables 1– 3 provide strong evidence of construct validity, justifying progression to multi-group invariance testing.

MGI analysis was conducted to examine whether the measurement model functioned equivalently across three grouping variables: region (Java vs. non-Java), school specialization (technical vs. non-technical), and gender (male vs. female). The analysis followed a hierarchical procedure: configural, metric, and scalar invariance.

Table 3 is the interpretation from configural invariance was supported across all groups, confirming H2: vocational students from different regions, specializations, and genders share the same conceptual structure of mathematical understanding. Metric invariance was also supported (ΔCFI ≤ .01), indicating that items contributed similarly to latent constructs across groups, supporting H3. Full scalar invariance was not initially achieved; therefore, partial scalar invariance was implemented by freeing specific non-invariant items. This is common in cross-context research and still allows valid latent mean comparisons, supporting H4 (partial).

Table 4 shown that the items related to practical modeling (PM) were more sensitive to contextual differences, particularly between urban–rural regions and technical–non-technical programs. This suggests that students’ real-world exposure to mathematical applications influenced how they interpreted these items.

After establishing partial scalar invariance, latent means were compared across groups.

Key patterns emerging from Table 5: a.

Regional differences: Students from Java showed significantly higher scores in mathematical representation (MR) and problem modeling (PM). This likely reflects greater access to technology-enhanced learning, industry exposure, and data-driven instruction in urbanized regions.

These findings support H5, which predicted meaningful but context-dependent differences across groups.

Table 6 shown that all four dimensions were strongly correlated (r = .46–.55), supporting a higher-order interpretation of mathematical conceptual understanding in vocational education.

Figure 3 provides an integrated summary of the hypothesis testing process by linking the statistical evidence from Confirmatory Factor Analysis (CFA) and Multi-Group Measurement Invariance (MGI) to the final decisions regarding each hypothesis. The table serves as a synthesis of methodological rigor, empirical findings, and theoretical interpretation, ensuring transparency in how conclusions were derived from the data.

The CFA results confirmed that mathematical conceptual understanding among vocational students is best represented as a four-dimensional construct comprising conceptual reasoning, mathematical representation, problem modeling, and knowledge transfer. This finding supports H1 and aligns with contemporary views that mathematical understanding is not a unitary skill but a system of interrelated cognitive processes. Conceptual reasoning captures students’ ability to understand underlying principles rather than merely applying procedures. Mathematical representation reflects their capacity to move flexibly between verbal, graphical, and symbolic forms. Problem modeling indicates how students frame real-world vocational tasks mathematically, while knowledge transfer reflects their ability to apply mathematics in new technical situations. Together, these dimensions reflect the competencies required in modern, technology-driven workplaces.

This multidimensional structure resonates with prior research emphasizing that meaningful mathematics learning requires integration of reasoning, representation, and application rather than rote computation. In vocational contexts, this integration is even more critical because students must translate abstract mathematics into concrete industrial practices such as machine calibration, energy efficiency calculations, and production optimization. The strong correlations among the four factors suggest the presence of a higher-order construct of mathematical thinking in TVET, where students’ abilities in reasoning, representation, modeling, and transfer reinforce one another. This supports a systemic rather than fragmented view of mathematical competence in vocational education.

The MGI results revealed that the four-factor model was configurally and metrically invariant across region, school specialization, and gender, supporting H2 and H3. This indicates that vocational students from different backgrounds share a common conceptual framework of mathematical understanding and that items function similarly across groups. This is a crucial methodological contribution. Without configural and metric invariance, comparisons of student performance would be biased and potentially misleading. The present findings demonstrate that the instrument measures the same latent constructs across diverse contexts in Indonesia, making it a valid tool for national-level evaluation of mathematical competencies in TVET.

However, full scalar invariance was not achieved, necessitating partial invariance. This pattern is common in cross-context studies and does not invalidate comparisons, but it highlights contextual sensitivity in certain items—particularly those related to problem modeling and representation. Items that showed non-invariance were primarily those involving practical workshop tasks or context-specific applications. This suggests that students’ interpretations of mathematical problems are shaped by their exposure to real-world industrial settings, which varies across regions and school types. In other words, differences in teaching practices, equipment availability, and industry partnerships influence how students perceive and respond to mathematical tasks. Rather than viewing this as a measurement flaw, this result can be interpreted as evidence that mathematical understanding is socially and contextually situated—a perspective widely supported in contemporary educational theory.

Latent mean comparisons revealed that students from Java outperformed non-Java students in mathematical representation and problem modeling, with moderate effect sizes. This finding supports H5 and suggests that regional disparities in educational resources shape mathematical learning. Java-based schools are more likely to have access to digital tools, simulation software, industry partnerships, and project-based learning environments. Such exposure may enhance students’ ability to translate real problems into mathematical representations and models. In contrast, non-Java schools—often located in more rural or remote areas—may rely more on traditional instruction, limiting students’ opportunities for applied mathematical reasoning. This finding has important policy implications. If national TVET standards aim to produce a uniformly skilled workforce, targeted investment is needed in non-Java regions—particularly in digital infrastructure, teacher professional development, and industry collaboration—to reduce inequities in mathematical learning.

Students in technical specializations demonstrated significantly higher performance in problem modeling and knowledge transfer compared to those in non-technical programs. This aligns with expectations, as technical curricula typically involve hands-on projects, real data analysis, and applied problem-solving. These results suggest that mathematical learning is deeply embedded in disciplinary practices. In technical programs, mathematics is not taught as an abstract subject but as a tool for solving authentic engineering and industrial problems. This contextualization likely strengthens students’ ability to apply mathematical concepts beyond the classroom. From a curriculum perspective, this finding highlights the value of integrating mathematics with vocational projects across all specializations—not only technical ones. Embedding real-world applications into business, administration, and service-related programs could enhance students’ mathematical reasoning and transfer skills.

Gender differences were present but relatively small. Female students showed slightly higher conceptual reasoning, while male students scored marginally higher in problem modeling. These small effect sizes suggest that gender gaps in mathematical understanding are not driven by innate ability but likely reflect differences in classroom experiences, self-beliefs, and task exposure. This finding aligns with prior research indicating that gender differences in mathematics are often mediated by confidence, teaching practices, and learning environments rather than cognitive capacity. It also suggests that equitable instructional strategies—such as collaborative problem-solving and inclusive classroom discourse—can further reduce gender disparities in vocational mathematics learning.

The study advances theory in three key ways: a.

Integration of mathematics education and TVET theory. The findings demonstrate that mathematical conceptual understanding in vocational settings is both cognitively structured and contextually shaped, bridging cognitive and sociocultural perspectives.

The findings have several implications for policy and practice: a.

Curriculum alignment.

National TVET curricula should emphasize not only procedural mathematics but also conceptual reasoning, representation, and real-world modeling.

Despite its contributions, the study has limitations. The sample size (N = 125) limits generalizability, and future research should include larger, nationally representative samples. Longitudinal studies are also needed to examine how mathematical understanding evolves over time in vocational education. Future research could also explore the role of teacher practices, classroom climate, and digital learning environments as mediators of mathematical understanding. Additionally, cross-national comparisons would help determine whether the observed patterns are unique to Indonesia or generalizable to other TVET systems.