Developing critical thinking skills has become a central objective in modern educational systems, particularly within the context of 21st-century learning demands. Critical thinking is a fundamental skill in 21st-century education that enables individuals to analyze, evaluate, and synthesize information systematically before making reasoned judgments ((Facione, 2020); (Malcolm, 2020)). The rapid advancement of technology and the increasing complexity of real-world problems have further emphasized the need for critical thinking in various academic disciplines, including mathematics ((Ali, 2025); (Siregar et al., 2024)). In the context of mathematics education, critical thinking plays a crucial role in problem-solving, as students frequently encounter complex and contradictory information that requires logical reasoning and evidence-based decision-making ((Arifin, 2021); (Lestari et al., 2024); (Sachdeva & Eggen, 2021)). Therefore, fostering students' critical thinking abilities is essential for equipping them to navigate mathematical challenges effectively and make well-founded decisions in both academic and real-life contexts.

Understanding how students engage with mathematical problem-solving has become an essential concern in both educational research and classroom practice, particularly as curricula emphasize critical and analytical thinking. Mathematical problem-solving is not merely about obtaining correct answers but involves a comprehensive process of interpreting information, analyzing relationships, and verifying conclusions ((Nurwita et al., 2022); (Susanti & Wulandari, 2021)). Particularly in mathematical problems that present contradictory data or inconsistencies, students are challenged to utilize higher-order thinking skills to identify, question, and reconcile conflicting elements ((Darhim et al., 2020); (Eviota & Liangco, 2020); (Naim et al., 2025)). However, empirical studies have consistently revealed that many junior high school students exhibit difficulties in this regard, often defaulting to rote memorization and procedural responses rather than engaging in reflective and analytical reasoning ((Bonyah et al., 2023); (Maryani et al., 2021)). This recurring problem indicates a gap in the cultivation of critical thinking within mathematics instruction and highlights the urgent need for pedagogical approaches that can guide students in examining and resolving inconsistencies in mathematical contexts ((Ali, 2025);(Lestari et al., 2024); (Safitri et al., 2024)). Therefore, addressing students' challenges in dealing with contradictory information requires intentional instructional strategies that foster deeper cognitive engagement and support the development of critical thinking skills in mathematics.

The conceptualization of critical thinking has been a central focus in educational theory, with various scholars offering diverse frameworks to explain its underlying cognitive dimensions. Critical thinking is inherently multi-dimensional and involves cognitive processes such as interpretation, analysis, evaluation, inference, explanation, and self-regulation ((Facione, 2020); (Malcolm, 2020)). While existing frameworks like Bloom’s taxonomy and Polya’s problem-solving stages provide a foundation for understanding these processes, they often address the skill in general terms and may overlook the specific cognitive challenges posed by contradictory mathematical information ((Ali, 2025); Suharyat et al., 2022). Consequently, there is a growing need for models that not only encompass general critical thinking processes but also account for the unique reasoning demands encountered in complex mathematical problem-solving contexts.

The specific cognitive processes underlying students' responses to contradictions in mathematical problem-solving remains an underexplored area within critical thinking research. Although numerous studies have explored students' critical thinking in mathematical settings, there is a notable lack of research that specifically investigates how students cognitively navigate contradictions within mathematical problems ((Naim et al., 2025); (Sabiq et al., 2025)). Most existing research focuses on overall performance in problem-solving rather than dissecting the discrete cognitive stages students experience when confronted with anomalies or inconsistencies in data ((Eviota & Liangco, 2020); (Fitriyah et al., 2022)). The development of critical thinking is further influenced by several interrelated variables—cognitive ability, instructional design, and the learning environment ((Lestari et al., 2024); (Rivas et al., 2022)). Among these, the instructional approach has a pivotal role, with research confirming that methods such as inquiry-based learning and problem-based learning can significantly enhance students' engagement in reflective and logical reasoning ((Ali, 2025); (Lestari et al., 2024); (Safitri et al., 2024)). Therefore, it is essential to conduct in-depth investigations into students' cognitive processes when encountering contradictory mathematical information, particularly within pedagogical contexts that aim to foster higher-order thinking.

Adolescence, particularly during the junior high school years, constitutes a pivotal stage in cognitive development that warrants close educational attention. According to Piagetian theory, this phase represents a developmental transition from concrete operational to formal operational thinking, making it critical for the cultivation of higher-order thinking skills (Adolph & Hospodar, 2024). Students at this level often require structured scaffolding to effectively develop skills such as evaluating conflicting information and formulating coherent solutions ((Ali, 2025); (Safitri et al., 2024)). Research indicates that students who acquire the ability to think critically tend to show improvements in mathematical reasoning and efficiency in problem-solving ((Katende, 2023); (Katsamunska & Rosenbaum, 2020); (Siregar et al., 2024)). Nevertheless, little is known about the precise mental stages they pass through when encountering mathematical contradictions—an area that deserves focused scholarly attention (Malcolm, 2020)(Mohammed et al., 2024).

To provide a more specific understanding of students’ critical thinking in mathematical contexts, this study refines general stage of critical thinking into a domain-specific framework known as stage mathematical critical thinking. In this study, the researcher develops general critical thinking stages into mathematical critical thinking stages, which consist of the following phases: (1) Initial Stage –The ability to sense an anomaly in a problem but without the ability to trace the specific components causing the anomaly; (2) Tracing Stage – The ability to detect an anomaly in a problem and identify the components that contribute to the inconsistency; (3) Global View Stage – The ability to explain the problem from a global perspective and interpret it from multiple viewpoints. These stages provide a structured pathway for understanding how students progressively engage with and resolve contradictory mathematical problems using critical thinking.

This formulation is grounded in Facione’s core critical thinking competencies and represents a novel contribution to the field, as such stages—especially in the context of contradictory mathematical problems—have not been explicitly examined in previous studies ((Facione, 2020); (Lestari et al., 2024)). The three stages of critical thinking in this study have not been previously examined, making them one of the novelties of this research. The relevance and urgency of this study stem from both theoretical and practical dimensions. Theoretically, it enriches the understanding of cognitive stages in mathematical critical thinking, a topic underexplored in current literature. Practically, it offers a diagnostic lens for educators to identify students’ cognitive positions and tailor instructional interventions accordingly. The study holds potential implications for curriculum development, teacher training, and assessment design in mathematics education.

Previous empirical studies reinforce the need for this exploration. For example, (Putri et al., 2024) conducted a meta-analysis showing that junior high school students' critical thinking in science particularly physics—remains underdeveloped. (Imayanti et al., 2021) observed students’ inability to meet critical thinking benchmarks when solving mathematical problems related to relations and functions. (Sa’diyah et al., 2024) emphasized the absence of learning strategies that fully engage students' cognitive potential, while (Harahap et al., 2024) highlighted a general tendency among students to accept information passively without critical interrogation.

These findings converge on a central issue: students are not sufficiently equipped to engage in deep analytical thinking, particularly in resolving contradictory information in mathematics. This problem is not merely cognitive but also pedagogical, rooted in instructional practices that do not adequately support the development of critical thinking.

Accordingly, this study aims to explore the stages of critical thinking among junior high school students when solving contradictory mathematical problems. By identifying the specific cognitive stages students experience, this research seeks to inform educators on how to design instructional strategies that nurture critical thinking more effectively. The outcomes of this study are expected to contribute meaningfully to both theoretical discourses and practical implementations in mathematics education, ultimately advancing students’ analytical capacity in solving complex, real-world problems ((Ali, 2025); Suharyat et al., 2022 ; (Siregar et al., 2024))

The findings of this study are presented by analyzing students’ responses according to the stages of critical thinking, beginning with the initial stage (interpretation), followed by the tracing stage (analysis), and culminating in the global view stage (evaluation and inference). The following section illustrates the responses of both subjects at each stage.

The comparative analysis of the two subjects highlights distinct trajectories in their critical thinking development across the stages of interpretation, analysis, evaluation, and inference. At the Initial Stage (Interpretation), both subjects demonstrated an ability to extract key information from the problem and expressed suspicion toward the plausibility of the distribution. Subject 1 exhibited stronger conceptual awareness by identifying that the fractional distribution appeared excessive before engaging in calculation, whereas Subject 2 relied primarily on procedural computation to validate suspicions. This contrast reflects differences in the depth of interpretation, where Subject 1 showed more advanced reflective judgment, while Subject 2 operated within a procedural orientation. Recent studies have emphasized that effective interpretation requires both recognition of surface features and anticipation of underlying inconsistencies ((Li & Schoenfeld, 2019); (Yulia & Salirawati, 2023)).

In the Tracing Stage (Analysis), both subjects accurately performed computations to demonstrate that the total distribution exceeded the available quantity (25/24 > 1). However, Subject 1 integrated numerical and verbal reasoning, demonstrating cognitive flexibility in representing the problem, while Subject 2 confined the analysis to algorithmic procedures. This distinction suggests that Subject 1 engaged in deeper structural analysis, whereas Subject 2 remained at the level of procedural verification. Prior research indicates that such differences are critical, as analytical reasoning grounded in multiple representations is associated with stronger transfer of critical thinking skills ((Unknown Author, 2025); (Savaş et al., 2024)). At the Global View Stage (Evaluation and Inference), both subjects recognized that the original distribution was flawed. Subject 1 evaluated the problem holistically and proposed an alternative solution (½, ¼, ⅙, 1/12) that satisfied all constraints, thereby demonstrating logical coherence, evaluative rigor, and sound inferential judgment. In contrast, Subject 2 adjusted the distribution (assigning ⅛ to both Teguh and Dedi), which corrected the numerical inconsistency but violated the problem condition requiring different shares. This indicates that Subject 2’s evaluation was only partially accurate and that inferential reasoning lacked attention to contextual constraints. Such findings resonate with recent scholarship showing that students often succeed in detecting contradictions but struggle to integrate all problem conditions into coherent solutions ((Chirove, 2023); (Lutfi Fauzi, 2025)).

Overall, both subjects demonstrated engagement in critical thinking, yet their characteristics diverged significantly: Subject 1 exhibited a coherent progression across interpretation, analysis, evaluation, and inference, reflecting both conceptual and procedural fluency. Subject 2, while procedurally competent, revealed limitations in evaluative judgment and inferential precision. These findings align with recent research emphasizing that critical thinking development requires not only computational fluency but also metacognitive reflection to ensure logical consistency and constraint integration ((Anggo et al., 2021); (Nobutoshi, 2023)).

The strengths and weaknesses of the two subjects are further distinguished and made more evident in the following Table 3. The comparison presented in Table 3 highlights the complementary yet contrasting profiles of the two subjects. Subject 1 demonstrated stronger conceptual awareness and evaluative rigor, while Subject 2 showed procedural fluency but struggled to integrate contextual constraints into coherent solutions. These findings resonate with recent studies emphasizing that students often exhibit imbalances between procedural and conceptual dimensions of critical thinking ((Li & Schoenfeld, 2019); (Lutfi Fauzi, 2025)). Moreover, the strengths of Subject 1 in integrating multiple

representations align with research suggesting that cognitive flexibility enhances the transferability of critical thinking across problem contexts ((Unknown Author, 2025); (Savaş et al., 2024)). Conversely, the weaknesses observed in Subject 2 reflect patterns reported in more recent scholarship, where students tend to focus narrowly on algorithmic accuracy while overlooking logical coherence and problem-specific conditions (Khairunnisa et al., 2022; (Putri et al., 2024)). These findings underscore the importance of fostering both computational fluency and metacognitive reflection, as supported by (Anggo et al., 2021), to ensure that critical thinking development is balanced and sustainable.

This study explored the critical thinking stages of two junior high school students in solving contradictory mathematical problems, focusing on interpretation, analysis, evaluation, and inference. The findings revealed both similarities and differences in their approaches. At the interpretation stage, both subjects successfully identified key information and expressed suspicion regarding the plausibility of the distribution, yet Subject 1 demonstrated stronger conceptual awareness by recognizing inconsistencies prior to calculation, while Subject 2 relied more on procedural computation. During the analysis stage, both subjects detected over-distribution; however, Subject 1 integrated multiple representations and verbal reasoning, whereas Subject 2 confined the analysis to algorithmic verification. At the evaluation and inference stages, both identified flaws in the original distribution, but Subject 1 proposed a logically coherent alternative that satisfied all constraints, while Subject 2 suggested a numerically correct yet logically inconsistent correction. Overall, Subject 1 displayed a coherent progression across all stages, reflecting both conceptual and procedural fluency, whereas Subject 2 showed procedural accuracy but limited evaluative judgment and inferential precision.

The practical implication of these findings is that contradictory problems can serve as effective instructional tools to balance procedural and conceptual dimensions of critical thinking. Teachers can employ such problems not only to test computational accuracy but also to encourage students to reflect on the coherence of their reasoning, question underlying assumptions, and propose alternative solutions. By integrating contradictory problems into regular instruction, educators can foster metacognitive reflection and enhance students’ ability to reconcile surface-level computations with deeper conceptual understanding.

For future research, several directions are recommended. First, expanding the sample size and including more diverse participants would provide broader insights into the variability of students’ critical thinking. Second, longitudinal studies are needed to trace how exposure to contradictory problems contributes to the sustained development of critical thinking over time. Third, experimental research should investigate the effectiveness of instructional interventions, such as scaffolding strategies, collaborative problem-solving, and digital learning environments, in supporting students’ ability to integrate conceptual and procedural reasoning. Fourth, cross-cultural and cross-curricular comparisons could illuminate contextual factors shaping students’ responses to contradictory problems. Finally, examining the relationship between cognitive processes and affective dispositions, such as curiosity, persistence, and openness to ambiguity, could enrich understanding of how critical thinking skills are fostered in mathematical learning.

Collectively, these findings and implications highlight the pedagogical value of contradictory mathematical problems in fostering balanced and sustainable critical thinking, while also pointing toward fertile avenues for further research to strengthen both theory and practice in mathematics education.

The author would like to express sincere gratitude to the students and teachers for their valuable participation in this paper. The author also extends special thanks to the supervisors, Mr. Purwanto and Mr. Rustanto Rahardi, for their assistance in the preparation of this paper.