The rapid advancement of science and technology in the 21st century requires education systems to design learning environments that cultivate deep conceptual understanding and advanced cognitive competencies. Higher education institutions are expected to prepare students who demonstrate analytical precision, structured reasoning, and creative problem-solving abilities aligned with global academic standards (Zhu & Meijden, 2023). Within mathematics education, the development of higher-order thinking skills is positioned as a central objective because mathematical understanding depends on reasoning, representation, and conceptual integration. Among the cognitive competencies emphasized in contemporary mathematics learning are critical thinking skills, mathematical connection ability, and spatial ability ((Mix et al., 2022); (Nathan & Walkington, 2022)). These three constructs function as interconnected dimensions that shape students’ capacity to analyze problems, integrate concepts across representations, and construct coherent mental models of mathematical structures ((Lowrie & Logan, 2023); (Mix, 2021)).

Spatial ability occupies a particularly strategic position in mathematics education because it underlies the interpretation and transformation of geometric representations (Battista, 2022). Spatial ability involves the capacity to visualize objects, mentally rotate figures, perceive spatial relationships, and interpret representations across dimensions ((Gilligan-Lee et al., 2023); (Harris et al., 2023); (Hawes et al., 2022); (Verdine et al., 2022)). (Ramful, 2022) classifies spatial ability into five principal components: spatial perception, spatial visualization, mental rotation, spatial relations, and spatial orientation. These dimensions represent structured cognitive processes that enable learners to associate visual representations with abstract mathematical reasoning ((Izzati & Farizi, 2025); (Lowrie & Logan, 2023); (Resnick & Newcombe, 2023)). Empirical findings demonstrate that spatial ability significantly predicts achievement in geometry, trigonometry, and calculus ((Poltz et al., 2025); (Xu et al., 2025); (Zhang & Lin, 2024)). Students with strong spatial competence exhibit accuracy in interpreting three-dimensional objects, transforming symbolic expressions into graphical representations, and analyzing geometric configurations ((Izzati et al., 2024); (Medina Herrera, 2024)). Further reports a positive correlation between spatial ability and students’ performance in geometric transformations, while (Sorby, 2022) confirms that spatial competence contributes to overall mathematics achievement. These findings establish spatial ability as a critical cognitive outcome in mathematics learning.

Despite its importance, National and International assessment data indicate that Indonesian students continue to experience difficulties in reasoning and representation-based mathematics tasks. The (Pisa, 2022) reports that Indonesian students obtained an average mathematics score of 366, considerably below the OECD average of 472. Similarly, the 2019 Trends in International Mathematics and Science Study (TIMSS) highlights persistent challenges in items requiring interpretation of geometric relationships and spatial configurations. These results reflect limitations in conceptual integration, reasoning accuracy, and representational fluency (Mulligan et al., 2021). Such patterns suggest that spatial reasoning remains an area requiring systematic investigation, particularly within mathematics education programs responsible for preparing future teachers (Yang et al., 2024).

Within higher education contexts, students enrolled in mathematics education programs are expected to master geometric visualization, representational transformation, and spatial explanation skills as part of their professional preparation (Izzati & Farizi, 2025). In the Mathematics Education Study Program at UIN Siber Syekh Nurjati Cirebon, geometry and advanced mathematics courses require students to interpret multidimensional representations and connect symbolic, graphical, and spatial forms. However, learning experiences frequently reveal challenges in mentally rotating objects, identifying spatial relationships, and integrating algebraic expressions with geometric structures. These instructional observations underscore the relevance of examining cognitive predictors that may influence students’ spatial ability.

Critical thinking skills represent one cognitive construct that potentially contributes to spatial reasoning development. (Ennis, 2018) defines critical thinking as disciplined judgment grounded in systematic reasoning, while (Facione, 2015) operationalizes it into six dimensions: interpretation, analysis, evaluation, inference, explanation, and self-regulation. In mathematical contexts, these processes guide learners in examining geometric structures, evaluating transformation procedures, and drawing logically consistent conclusions (Uttal et al., 2022). (Cheng & Mix, 2021) emphasize that analytical reasoning supports the identification of structural errors and improves strategic decision-making in problem solving. (Lowrie & Logan, 2023) further demonstrate that structured reasoning enhances the interpretation of visual-spatial information. Students with limited critical thinking often depend on algorithmic memorization without conceptual justification (Yanuari, 2023), which restricts their capacity to engage in complex spatial transformations.

Mathematical connection ability also plays a strategic role in supporting spatial cognition. (Bower & Liben, 2021) explain that connection skills facilitate integration across mathematical topics and representations. The (National Council of Teachers of Mathematics, 2020) identifies connections as a core process standard that enables learners to recognize relationships among ideas, representations, and structures. (Sumarmo, 2010) outlines indicators of mathematical connection ability, including intra-topic connections, inter-domain connections, and application to real-life contexts. Empirical studies confirm that students who effectively connect algebraic, graphical, and geometric representations demonstrate stronger conceptual coherence and improved representational flexibility ((Hegarty, 2021); (Izzati, 2024); (Resnick & Newcombe, 2023)). Because spatial reasoning requires coordination between symbolic structures and visual representations, mathematical connection ability may function as a cognitive bridge linking abstract reasoning with spatial interpretation. Preliminary observations in geometry courses within the Mathematics Teaching Study Program indicated that some students encountered difficulties in visualizing three-dimensional representations and translating them into formal mathematical reasoning.

Previous research has predominantly examined the relationship between critical thinking skills and mathematics achievement or between mathematical connection ability and problem-solving performance ((Gilligan, 2022); (Schenck & Nathan, 2024)). Other investigations have analyzed spatial ability as an independent predictor of mathematical success (Uttal et al., 2022). However, empirical studies that simultaneously model the structural influence of critical thinking skills and mathematical connection ability on spatial ability remain limited ((Lowrie & Logan, 2023); (Uttal, 2021)). Existing investigations often focus on dual-variable correlations and do not examine how reasoning competencies interact collectively to shape spatial cognition within mathematics education students. This gap restricts comprehensive understanding of the integrated cognitive mechanisms underlying spatial reasoning.

Based on this theoretical and empirical foundation, the present study aims to examine the structural influence of mathematical critical thinking skills and mathematical connection ability on students’ spatial ability in the Mathematics Education Study Program at UIN Siber Syekh Nurjati Cirebon. Employing a quantitative correlational framework, this research analyzes the magnitude and direction of relationships among these cognitive constructs. The findings are expected to contribute to theoretical advancement in understanding the integration of reasoning and spatial cognition within mathematics education. Practically, the results are anticipated to inform instructional design that systematically strengthens analytical reasoning, conceptual linkage, and spatial competence in mathematics learning.

Based on the proposed conceptual framework, this study formulates three research hypotheses aligned with multiple regression analysis. First, mathematical critical thinking ability significantly predicts students’ mathematical spatial ability. Second, mathematical connection ability significantly predicts students’ mathematical spatial ability. Third, mathematical critical thinking ability and mathematical connection ability jointly and significantly predict students’ mathematical spatial ability, as reflected in the overall model significance (F-test) and the proportion of explained variance (R²).

This study employed a quantitative approach using an explanatory correlational design to examine the structural relationships between mathematical critical thinking ability, mathematical connection ability, and students’ mathematical spatial ability. The design was selected to analyze predictive relationships among variables without implementing instructional interventions or experimental treatments. Accordingly, the study focuses on modeling the magnitude and direction of relationships among constructs within a defined academic cohort.

The population of this study consisted of students enrolled in the Spatial Geometry course in the Mathematics Teaching Study Program at UIN Siber Syekh Nurjati Cirebon during the 2025/2026 academic year. The accessible population comprised one intact class consisting of 23 students. All students who met the inclusion criterion, namely having completed prerequisite geometry coursework, were included in the study. Therefore, the sampling procedure functioned as a census of the accessible cohort rather than purposive or random sampling. Because the total number of students in the defined population was limited and entirely involved, sample size estimation formulas such as Slovin were not applied. The findings are interpreted within the framework of analytical generalization, emphasizing explanation of regression-based relationships within the defined cohort rather than statistical generalization to a broader population. However, given the relatively small sample size, statistical power is inherently limited, and the results should be interpreted with caution. The findings are context-bound and may not be directly generalizable to

larger or different populations. In terms of statistical adequacy, multiple regression analysis involving two predictors can be conducted with small samples provided that the model remains parsimonious and assumptions are satisfied (Hair et al., 2019). Prior to hypothesis testing, diagnostic tests were conducted to ensure that all regression assumptions were fulfilled.

This study examined three variables, namely mathematical critical thinking ability (X₁) and mathematical connection ability (X₂) as independent variables, and mathematical spatial ability (Y) as the dependent variable. The structural relationship investigated in this study positions mathematical critical thinking and mathematical connection abilities as simultaneous predictors of mathematical spatial ability within a multiple linear regression framework. The conceptual model of this relationship is presented in Figure 1.

The relationship among variables is mathematically expressed as:

(1)

where represents mathematical spatial ability, represents mathematical critical thinking ability, represents mathematical connection ability, denotes the intercept, and represent regression coefficients, and represents the error term.

The instruments used to measure these variables were developed systematically beginning with the identification of theoretical dimensions drawn from established literature, followed by the preparation of a table of specifications to ensure alignment between indicators and test items. The mathematical critical thinking instrument was constructed based on indicators including interpretation, analysis, evaluation, inference, explanation, and self-regulation. Initially, 18 items were developed, and after item analysis, 13 items were retained. The mathematical connection instrument was developed based on indicators covering intra-mathematical connections, connections between mathematical concepts and other disciplines, and the application of mathematics in real-life contexts. From the initial 9 items constructed, 5 items met the validity criteria and were retained. The mathematical spatial ability instrument was developed based on indicators including spatial perception, spatial visualization, mental rotation, spatial relations, and spatial orientation. Of the 9 items initially developed, 6 items were retained following item analysis.

Content validity was established through a structured expert judgment process involving two mathematics education experts who evaluated each item in terms of conceptual relevance, clarity of formulation, and alignment with the intended theoretical constructs. The review also considered cognitive demand and the suitability of spatial and representational prompts to ensure accurate construct representation. Revisions were implemented based on expert feedback, including refinement of wording, adjustment of problem contexts, and clarification of task instructions. The instruments were subsequently pilot-tested with students possessing characteristics comparable to the target cohort to examine empirical item quality. Item validity was analyzed using item–total correlation coefficients to assess each item’s contribution to the overall construct, while discrimination indices were calculated to determine the ability of items to differentiate between higher- and lower-performing students. The difficulty index was examined to ensure an appropriate distribution of item complexity. Only items meeting acceptable psychometric criteria were retained in the final instruments. Reliability analysis was then conducted to evaluate internal consistency, and all three instruments demonstrated acceptable reliability levels for explanatory correlational research. Prior to hypothesis testing, the overall quality of the instruments was confirmed through

combined analyses of validity, reliability, difficulty level, and discrimination power, with the results for mathematical critical thinking ability presented in Table 1

As shown in Table 1 all items of the mathematical critical thinking instrument meet the criteria for empirical validity, as indicated by item–total correlation coefficients exceeding the minimum threshold. The reliability coefficient also demonstrates that the instrument possesses satisfactory internal consistency. Furthermore, the distribution of difficulty indices ranges from moderate to acceptable levels, indicating that the items are neither excessively easy nor overly difficult. The discrimination indices reveal that the items are capable of differentiating effectively between high- and low-ability students. Collectively, these results confirm that the mathematical critical thinking

instrument is psychometrically sound and suitable for use in subsequent regression analyses. Following the analysis of the critical thinking instrument, the same validation procedures were applied to the mathematical connection ability instrument.

Table 2 indicates that all items of the mathematical connection instrument meet the established empirical validity criteria based on item total correlation analysis, confirming that each item contributes meaningfully to the measurement of the intended construct. The reliability coefficient reflects satisfactory internal consistency, indicating that the items function cohesively in assessing students’ ability to relate mathematical concepts across representations and domains. The difficulty indices are distributed within an acceptable range, ensuring that the instrument captures varying levels of relational complexity without clustering at extreme levels. Furthermore, discrimination index analysis demonstrates that each item possesses adequate power to differentiate between students with higher and lower levels of mathematical connection ability. Collectively, these psychometric indicators confirm the adequacy and measurement precision of the instrument for assessing relational understanding in mathematics. Subsequently, the instrument designed to measure mathematical spatial ability underwent the same systematic validation and reliability procedures to ensure comparable psychometric rigor across constructs.

As presented in Table 3 the mathematical spatial ability instrument demonstrates acceptable empirical validity across all items. The reliability analysis indicates adequate internal consistency. The difficulty indices suggest a proportional distribution of item complexity, while the discrimination indices confirm that the items function effectively in differentiating students’ spatial ability levels. Therefore, the spatial ability instrument fulfills the required psychometric standards for inferential statistical analysis.

Taken together, the overall results indicate that the instruments demonstrate sufficient psychometric robustness to support explanatory multiple regression analysis within the defined cohort. As the study aims to examine predictive structural relationships rather than to inform high-stakes assessment decisions, the observed reliability coefficients were deemed adequate for inferential modeling purposes. Following the comprehensive evaluation of content validity, empirical item validity, reliability, difficulty level, and discrimination power, a systematic item refinement process was undertaken to ensure optimal measurement precision. Items that did not meet the established psychometric thresholds were excluded to enhance construct coherence and statistical reliability. Consequently, only items demonstrating acceptable validity and discrimination indices with appropriate levels of difficulty were retained for subsequent analysis. Table 4 provides a detailed summary of this refinement process, specifying the item-level decisions and identifying which items were retained and which were removed based on the combined psychometric evaluation criteria.

Table 2 presents the distribution of retained and discarded items for each instrument following the validity and item quality analysis. For the mathematical critical thinking instrument, 13 out of 18 items were retained, while 5 items were excluded due to not meeting the established validity criteria or demonstrating insufficient discrimination. In the mathematical connection instrument, 5 items were retained and 4 items were discarded. For the mathematical spatial ability instrument, 6 items were retained, and 3 items were excluded from the final version of the test.

The exclusion of certain items was based on statistical considerations, including item-total correlation values below the required threshold and weak discrimination indices. The retained items collectively represent the theoretical dimensions specified in the instrument blueprint, ensuring that the construct coverage remained intact despite item reduction. This refinement process strengthens the internal consistency of the instruments and enhances the accuracy of measurement used in subsequent regression analysis.

All instruments were administered within the same instructional topic, namely Spatial Geometry, to ensure construct consistency across variables. However, the three tests were conducted across three separate class meetings to minimize fatigue effects and reduce potential testing bias. Each test session lasted approximately 90 minutes and was administered under standardized classroom conditions. Conducting the tests within the same topic ensured that all measured variables reflected mastery within the same content domain, thereby strengthening the internal validity of the regression analysis and minimizing the risk that differences in subtopic exposure would affect the interpretation of relationships among variables.

Data were analyzed using multiple linear regression at a significance level of 0.05 to examine both partial and simultaneous effects of mathematical critical thinking ability and mathematical connection ability on students’ mathematical spatial ability. Prior to conducting hypothesis testing, regression assumptions were evaluated, including normality using the Kolmogorov–Smirnov test and P–P plot, multicollinearity using tolerance and variance inflation factor values, and homoscedasticity using scatterplot and Glejser test procedures. The diagnostic results indicated that all assumptions were satisfied, confirming that the regression model was appropriate for examining the proposed relationships.

The findings indicate that mathematical critical thinking ability and mathematical connection ability are significant predictors of students’ mathematical spatial ability. These results suggest that spatial ability is not merely a perceptual skill, but is cognitively grounded in higher-order reasoning processes and relational understanding within mathematics. Spatial performance appears to emerge from structured analytical thinking and the integration of conceptual relationships rather than from isolated visualization skills.

The significant predictive role of mathematical critical thinking ability supports the view that analytical reasoning, evaluation of arguments, and logical inference contribute to the construction of accurate mental representations. Students who demonstrate stronger critical thinking skills are more likely to analyze spatial configurations systematically, evaluate solution strategies, and reflect on the validity of their reasoning (Bruce et al., 2023). This interpretation aligns with theoretical perspectives on critical thinking proposed by (Ennis, 2018) and (Facione, 2015), which conceptualize critical thinking as a core component of higher-level cognitive processing. Empirically, the present finding is consistent with recent regression-based studies indicating that reasoning abilities significantly predict performance in spatial and geometric tasks (Lowrie & Logan, 2023). The result therefore confirms that spatial ability is cognitively intertwined with structured reasoning processes.

Similarly, the significant predictive role of mathematical connection ability indicates that students’ capacity to relate concepts across mathematical domains and representations contributes meaningfully to spatial understanding. The ability to coordinate algebraic, geometric, symbolic, and visual representations appears to facilitate flexible cognitive transitions that support spatial reasoning processes. This finding is consistent with broader empirical evidence demonstrating a positive association between spatial skills and mathematical performance across domains, including regression-based analyses highlighting the predictive role of spatial reasoning in mathematics achievement (Xu et al., 2025). Moreover, meta-analytic syntheses have reported robust positive relationships between spatial abilities and mathematical performance, reinforcing the interpretation that representational and relational processes contribute meaningfully to mathematical cognition (Atit et al., 2022). While many prior studies emphasize spatial reasoning as a predictor of mathematics outcomes, the present study extends this line of research by demonstrating that mathematical connection ability, as a relational construct internal to mathematics, retains a unique predictive contribution to spatial ability within a simultaneous regression framework.

The joint significance of mathematical critical thinking and mathematical connection abilities further suggests that spatial ability reflects coordinated higher-order cognitive functioning rather than isolated visualization skills. Multivariate research in mathematics education increasingly supports the view that multiple cognitive constructs collectively explain substantial variance in complex mathematical performance. The present findings align with this perspective, as meta-analytic evidence indicates that spatial and mathematical abilities are systematically interrelated across diverse samples (Atit et al., 2022). However, by modeling mathematical critical thinking and connection abilities simultaneously within a single regression framework, this study extends prior regression-based research by demonstrating how analytical reasoning and conceptual linkage jointly account for substantial variation in spatial ability among pre-service mathematics teachers. The relatively high proportion of explained variance indicates that, within this bounded higher-education context, spatial competence is strongly associated with coordinated reasoning and representational integration processes rather than isolated visualization skills (Oliveira et al., 2023). This integrative relationship is conceptually represented in Figure 4, which illustrates the structural model derived from the regression analysis. The model emphasizes the complementary and mutually reinforcing roles of critical reasoning and mathematical connections in shaping students’ spatial performance.

From a theoretical perspective, these findings contribute to the understanding of spatial ability as a higher-order mathematical construct shaped by reasoning and representational integration. Spatial cognition in mathematics can therefore be conceptualized as an outcome of coordinated analytical and relational processes. Practically, the results suggest that the development of spatial ability may benefit from instructional emphases that cultivate critical reasoning and encourage meaningful connections among mathematical representations. Importantly, these implications are derived from relational findings rather than from experimental intervention, and therefore should be interpreted within the correlational scope of the present study.

This study demonstrates that mathematical critical thinking ability and mathematical connection ability are significant predictors of students’ mathematical spatial ability. Both variables individually contribute to variations in spatial performance, and their simultaneous effect indicates that spatial ability develops through the integration of analytical reasoning and conceptual linkage processes. These findings position spatial ability not merely as a perceptual skill, but as a higher-order cognitive construct grounded in structured reasoning and relational understanding.

The results imply that efforts to strengthen students’ spatial ability should emphasize the cultivation of critical analysis, logical evaluation, and meaningful connections among mathematical representations. Enhancing these higher-order cognitive processes may indirectly support the development of spatial competence within mathematics learning contexts. The findings therefore contribute to a more integrated understanding of how reasoning and representational coherence shape spatial cognition.

This study is limited by its cross-sectional design and relatively small sample size, which restrict causal interpretation and generalizability. Future research may employ longitudinal designs to examine how the interaction between critical thinking, mathematical connections, and spatial ability develops over time across different academic levels. Such investigations could clarify whether improvements in reasoning and relational understanding precede and predict growth in spatial competence, thereby providing a more comprehensive explanation of the developmental trajectory of spatial cognition in mathematics.

The author would like to thank the campus and the Laboratory Unit of Mathematics and Natural Sciences UIN Siber Syekh Nurjati Cirebon.

Determine which cuboid has the greater volume and justify your answer mathematically.

Explain mathematically the difference in both volume and surface area between the two cuboids, even though they share some identical dimensions.

both with a volume of 1000 cm³. Determine the dimensions of each package so that they have the same volume. Then compare their surface areas and explain which design is more efficient in terms of material usage.

Provide mathematical justification for your conclusions.