Analysis of Middle School Students’ Mathematical Critical Thinking Skills in Remote Areas on Integer Multiplication

Analysis of Middle School Students’ Mathematical Critical Thinking Skills in Remote Areas on Integer Multiplication | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Analysis of Middle School Students’ Mathematical Critical Thinking Skills in Remote Areas on Integer Multiplication Abu Moh. Rasyid Ridho, Al Jupri Al Jupri, Sumanang Muhtar Gozali This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9065392/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 12 You are reading this latest preprint version Abstract Mathematical critical thinking skills are essential competencies that must be developed in students, particularly when addressing learning challenges in remote areas. This study aims to analyze the mathematical critical thinking skills of junior high school students in remote areas on the topic of integer multiplication using the Facione framework. A qualitative descriptive approach was employed, involving twenty-four eighth-grade students from a public junior high school in Lembata Regency as research participants. The study was conducted during the odd semester of the 2025/2026 academic year. Data were collected through a critical thinking test consisting of four essay questions and semi-structured interviews. Data analysis was based on four Facione indicators: interpretation, analysis, evaluation, and inference. The results indicate that students’ mathematical critical thinking skills were in the low category, with an average score of 49.74%. Specifically, the interpretation aspect reached 55.2%, analysis 48.96%, evaluation 41%, and inference 54.2%. The findings revealed that students experienced difficulties in analyzing conceptual relationships, evaluating arguments, and drawing logical conclusions. These results have important implications for the development of learning strategies that are responsive to the characteristics and needs of students in remote areas. Facione critical thinking framework integer multiplication junior high school students mathematical critical thinking skills remote area education INTRODUCTION Mathematical critical thinking skills are widely recognized as essential 21st-century competencies that students must master to address the complexity of global challenges [ 1 – 4 ]. In mathematics learning, these skills are not only important for solving everyday problems but also encompass the ability to analyze information, evaluate arguments, and draw logically justified conclusions [ 5 – 7 ]. Therefore, enhancing mathematical critical thinking skills has become a crucial educational agenda, as mathematics functions not only as a formal discipline but also as a way of thinking that supports sound decision-making in real-life contexts [ 8 , 9 ]. The importance of critical thinking becomes even more evident when applied to conceptual and challenging mathematical content, such as integer multiplication. This topic is a fundamental component of the junior high school curriculum and serves as the basis for understanding more advanced concepts, including equations and functions. Nevertheless, numerous studies indicate that students continue to experience difficulties in understanding integer operations, particularly those involving negative numbers [ 10 , 11 ]. These difficulties occur not only at the procedural level but also at the conceptual level, such as interpreting signs, selecting appropriate operations, and connecting symbolic representations with problem contexts [ 12 , 13 ]. Consequently, integer multiplication is a relevant topic for examining how students employ mathematical critical thinking skills during problem-solving processes. These conceptual challenges are closely related to the diverse learning contexts experienced by students, especially with respect to geographical and social factors. This complexity is particularly evident in remote areas, where students often face limitations in educational infrastructure, access to learning resources, and the availability of qualified teachers [ 14 ]. Such conditions can negatively affect the quality of learning interactions and ultimately result in lower levels of mathematical critical thinking skills compared to students in urban areas [ 15 , 16 ]. However, rural contexts should not be viewed solely in terms of limitations. Remote areas also possess unique potential that can be leveraged to support context-based learning and generate meaningful educational impact [ 17 ]. To date, research on mathematical critical thinking skills has predominantly focused on students in urban settings or schools with relatively advanced educational facilities. Studies that specifically explore the mathematical critical thinking skills of students in remote areas, particularly in relation to foundational topics such as integer multiplication, remain limited. However, such investigations are essential for developing a more comprehensive understanding of how contextual factors influence students’ cognitive development in mathematics learning. Consistent with this urgency, several previous studies have identified gaps in critical thinking skills between urban and rural students. For instance, Gallardo-Estrada et al. [ 18 ] reported that although students in rural schools demonstrated improvements in critical thinking following collaborative learning interventions, they still lagged behind their urban counterparts in the consistent application of cognitive strategies. Similarly, Susanta et al. [ 19 ] found differences in mathematical comprehension based on geographical location, with urban students achieving higher scores. In contrast, Palinussa et al. [ 20 ] demonstrated that appropriate instructional approaches, such as Realistic Mathematics Education or higher-order thinking skills (HOTS)-oriented learning, can substantially enhance the critical thinking abilities of rural students, even bringing their performance closer to that of urban students. These findings suggest that geographical disparities are not fixed barriers but challenges that can be addressed through adaptive pedagogical strategies. Based on these considerations, this study aims to analyze the mathematical critical thinking skills of students in remote areas with respect to integer multiplication. The findings are expected to contribute theoretically to the literature on critical thinking across diverse geographical contexts and practically to the development of relevant and adaptive instructional strategies for learners in remote areas. Thus, this study not only enriches academic understanding but also offers actionable insights for improving the quality of mathematics education in regions that continue to face educational limitations. RESEARCH METHOD This study employed a qualitative descriptive approach with a research design focused on an in-depth analysis of students’ mathematical critical thinking skills in integer multiplication. A qualitative approach was chosen because it aims to describe and deeply understand the phenomena that occur, particularly the types of errors students make when applying mathematical critical thinking skills. The descriptive design was selected because it enables the presentation of a clear and comprehensive picture of students’ mathematical critical thinking skills within the context of this study. The investigation was conducted during the odd semester of the 2025/2026 academic year. The research subjects consisted of 24 eighth-grade students from a public junior high school located in Lembata Regency, East Nusa Tenggara Province, Indonesia. This school was selected because it represented learning conditions in remote areas, particularly island regions in East Nusa Tenggara with challenging geographical characteristics. The school was located approximately 45 km from the district center, with transportation access highly dependent on weather conditions. In addition, the school faced limited access to technology, digital learning resources, and teacher professional development, which are typical constraints that influence the mathematics learning process in remote areas. The test instruments were validated by two mathematics education lecturers and one experienced junior high school mathematics teacher. Semi-structured interviews were conducted with 24 selected students using purposive sampling based on variations in ability. These students became the main focus of the analysis to explore their profiles of mathematical critical thinking skills related to integer multiplication. Table 1 presents the written test instrument, while interview guidelines were used to confirm students’ mathematical critical thinking processes when completing the test. Table 1 Mathematical Critical Thinking Test Instrument 1 An employee is fined Rp2,000 for each time he is late. Over the last 14 days, he has been recorded as being late every day. a. Explain how you understand this situation in the form of a mathematical statement to determine the total fine that must be paid! b. Calculate the total fine that must be paid! 2 A tour bus departs with 48 passengers. At each terminal, 4 people get off. If the bus stops at 3 consecutive terminals: a. How many passengers get off in total? b. Explain the steps you took to find the answer! 3 Budi participated in a math competition. For every correct answer, he received 8 points, and for every incorrect answer, he lost 6 points. In one session, Budi answered 3 questions correctly and 7 questions incorrectly. The committee recorded that Budi’s total score was − 18 points. a. Check whether the recorded score is correct or incorrect! b. Explain the steps you took to find the answer! 4 A merchant sells oranges with the following data: Day 1: 25 kg; Day 2: 30 kg; Day 3: 35 kg. Sales increase by 5 kg each day. A friend concludes, “On the sixth day, the vendor will sell 50 kg of oranges.” a. Check whether your friend’s conclusion is correct! b. Explain how you drew your conclusion from the given facts! The test instrument consisted of four essay questions designed to encourage students to apply various aspects of mathematical critical thinking. Although the questions were presented in different contexts, they were closely related to the topic of integer multiplication, allowing for a comprehensive analysis of students’ abilities. The test was administered individually to ensure that students completed the tasks independently, without peer influence. The interviews conducted in this study were unstructured and aimed to confirm the results of the mathematical critical thinking test. This stage also sought to further explore students’ strategies, reasoning patterns, and approaches when solving the test problems. After the test data were collected, students’ responses were assessed using mathematical critical thinking skill assessment guidelines. The scores obtained for each indicator and the total score were then converted into percentages. These percentage scores were categorized according to the levels of mathematical critical thinking skills shown in Table 2 [ 21 , 22 ]. Table 2 Categories of Mathematical Critical Thinking Skill Percentages Interpretation (%) Category 80 < x ≤ 100 Very High 65 < x ≤ 80 High 55 < x ≤ 65 Moderate 40 < x ≤ 55 Low 0 < x ≤ 40 Very Low RESEARCH RESULTS AND DISCUSSION The analysis of students’ written responses for each indicator, supported by interview excerpts, is presented as follows. Interpretation Indicator The interpretation indicator appeared when students were able to read and restate the problem, but failed to fully understand the mathematical meaning of the situation described. This indicator is illustrated in Table 3, which presents the written test results for question number 1, and is further supported by evidence from the interview data. Table 3. Written Test Question and Results for Question Number 1 Question Number 1 Written Test Results for Question Number 1 An employee is fined Rp2,000 for each time he is late. Over the last 14 days, he has been recorded as being late every day. Explain how you understand this situation in the form of a mathematical statement to determine the total fine that must be paid! Calculate the total fine that must be paid! Based on the written test results, students generally identified the context of the problem but produced inaccurate mathematical representations. This finding is consistent with the interview results. When asked to explain what the problem was, the student stated, “The problem is about an employee who is fined Rp2,000 every time he is late for 14 days. What is being asked is the total fine that must be paid.” This response indicates that the student was able to comprehend the narrative context and the goal of the problem. Similarly, when identifying known information, the student correctly mentioned, “The fine is two thousand rupiah, and the lateness lasted for fourteen days.” These responses show that the student did not experience difficulty in reading the problem or extracting explicit information from it. However, difficulties became evident when the student was asked to express the situation in mathematical form. The student explained, “I wrote 2,000 + 14 because there are many fines.” This response revealed a misinterpretation of the relationship between the fine amount and the number of days. Instead of viewing the situation as repeated equal groups, the student treated the quantities as separate values to be added. When further asked why addition was chosen instead of multiplication, the student responded, “Because I thought the fines were added continuously, not multiplied.” This statement suggests that the student associated repetition with addition but lacked a clear conceptual understanding of multiplication as structured repeated addition. The interview also revealed partial conceptual awareness. When prompted about alternative strategies, the student admitted, “It can be multiplied, but I am not confident about how to write it.” This indicates that the student had some awareness of multiplication in similar contexts but lacked confidence in applying it independently. After guided questioning, the student was able to identify the correct operation, stating, “It should be multiplication, two thousand times fourteen days.” This progression demonstrates that the student’s difficulty was not due to an inability to perform multiplication, but rather to weaknesses in interpreting contextual information and selecting the appropriate mathematical operation without assistance. Overall, the interview quotations strengthen the written test findings by showing that students’ errors originated from misinterpretation at the conceptual level. Students were able to read, recall, and verbally explain the situation, yet struggled to translate real-life contexts into correct mathematical representations. This highlights the need for instructional strategies that emphasize conceptual interpretation of word problems, particularly in recognizing relational cues that indicate multiplication rather than simple addition. Analysis Indicator The analysis indicator was identified when students experienced difficulty recognizing and explaining the relationships among statements, quantities, and concepts contained in the problem. This indicator is reflected in Table 4, which presents the written test results for question number 2, and is further supported by the interview findings. Table 4. Written Test Question and Results for Question Number 2 Question Number 2 Written Test Results for Question Number 2 A tour bus departs with 48 passengers. At each terminal, 4 people get off. If the bus stops at 3 consecutive terminals: How many passengers get off in total? Explain the steps you took to find the answer! Based on the written test results in Table 4, many students were unable to clearly articulate the relationship between “4 passengers getting off” and “3 terminals.” Although some students arrived at the correct numerical answer, they often failed to explain the reasoning process or the relationship among the given quantities in a structured manner. The interview results provide clearer insight into students’ analytical thinking. When asked what happened to the passengers at each terminal, the student stated, “At each terminal, four people get off.” When further asked about the number of terminals, the student responded, “At three terminals.” These responses indicate that the student was able to correctly identify individual pieces of information presented in the problem. Importantly, when asked about the relationship between the number of passengers and the number of terminals, the student explained, “Four people get off every time the bus stops.” This response shows an emerging ability to recognize a repeating pattern and to relate quantities conceptually rather than treating them as isolated values. This understanding was reflected in the student’s choice of operation. When asked which operation should be used, the student answered, “Multiplication, four times three.” When probed further about why simple addition was not chosen, the student explained, “Because the event happens repeatedly at each terminal.” These responses demonstrate that the student was able to analyze the structure of the problem and justify the use of multiplication based on repetition. Finally, when asked to determine the total number of passengers who got off, the student concluded, “Twelve people.” This indicates that once the relationship among the quantities was correctly identified, the student was able to apply the appropriate operation and reach the correct solution. Overall, the interview findings suggest that, unlike students who relied solely on surface-level reading, this student demonstrated the ability to analyze relationships among quantities and recognize repetition as a key indicator for multiplication. However, the written test results show that this level of analytical reasoning was not consistently demonstrated across all students. Many students still struggled to decompose the problem, explain relationships explicitly, and organize solution steps logically. These findings indicate that students’ analytical skills remain uneven and require instructional reinforcement, particularly in helping students articulate relationships between quantities and justify their choice of operations. Evaluation Indicator The evaluation indicator of mathematical critical thinking emerged when students were unable to verify the correctness of solution steps, assess the validity of their reasoning, and ensure that the chosen strategy and final result were consistent with the problem’s requirements. This indicator is presented in Table 5, which shows the written test results for question number 3, and is further strengthened by evidence from the interview data. Table 5. Written Test Question and Results for Question Number 3 Question Number 3 Written Test Results for Question Number 3 Budi participated in a math competition. For every correct answer, he received 8 points, and for every incorrect answer, he lost 6 points. In one session, Budi answered 3 questions correctly and 7 questions incorrectly. The committee recorded that Budi’s total score was −18 points. Check whether the recorded score is correct or incorrect! Explain the steps you took to find the answer! Based on the written test results in Table 5, many students directly accepted the recorded score of −18 points without critically examining whether the result was reasonable or whether the calculation process was correct. This tendency is clearly reflected in the interview responses. When asked about the points obtained from correct answers, the student stated, “Three times eight, so twenty-four.” Similarly, when asked about the points lost due to incorrect answers, the student answered, “Seven times six, so forty-two.” These responses indicate that the student was able to apply the appropriate operations to calculate gains and losses separately. However, when determining the total score, the student explained, “Twenty-four minus forty-two.” After calculating the result, the student stated, “Negative eighteen.” Although this computation is mathematically correct, the evaluation process did not extend beyond performing the calculation. When asked whether the result was correct, the student confidently responded, “Yes, because the calculation is already correct.” This response shows that the student equated computational accuracy with overall correctness, without evaluating whether the result aligned with the context or checking the reasoning process. More importantly, when asked whether another method was used to verify the answer, the student admitted, “No, I immediately trusted the result of the calculation.” This statement highlights a key weakness in the evaluation stage: the student did not attempt to recheck the answer using alternative strategies, such as rewriting the score calculation in a single expression (3 × 8 − 7 × 6), using a table of gains and losses, or reconsidering whether the negative result was reasonable in the context of the competition. Overall, the interview findings reinforce the written test results by showing that students tended to stop at obtaining an answer rather than evaluating its validity. Students were able to carry out calculations correctly, but did not engage in verification, justification, or reflection on their solutions. This indicates that students’ evaluation skills remain weak, particularly in verifying results and critically assessing whether their answers are logically and contextually appropriate. Therefore, instructional efforts should emphasize not only obtaining correct answers but also developing habits of checking, justifying, and evaluating solution processes and outcomes. Inference Indicator The inference indicator of mathematical critical thinking was observed when students experienced difficulty drawing logical conclusions from available information or failed to clearly connect identified patterns to justify their conclusions. This indicator is illustrated in Table 6, which presents the written test results for question number 4, and is further supported by evidence from the interview data. Table 6. Question and Written Test Results for Question Number 4 Question Number 4 Written Test Results for Question Number 4 A merchant sells oranges with the following data: Day 1: 25 kg; Day 2: 30 kg; Day 3: 35 kg. Sales increase by 5 kg each day. A friend concludes, “On the sixth day, the vendor will sell 50 kg of oranges.” Check whether your friend’s conclusion is correct! Explain how you drew your conclusion from the given facts! Based on the written test results in Table 6, many students were able to recognize that the sales data formed an increasing pattern, yet they struggled to explicitly articulate how the pattern led to the final conclusion. Although some students reached the correct numerical answer, their explanations often lacked clear inference processes or generalization of the pattern. The interview results provide insight into how students formed their conclusions. When asked about the daily sales trend, the student stated, “Every day it increases by five kilograms.” When asked whether the pattern from the first day to the third day remained consistent, the student confirmed, “Yes, it always increases by five kilograms.” These responses indicate that the student was able to identify and recognize a consistent pattern in the given data. However, when asked how to determine sales on the sixth day, the student explained, “From the first day, add five repeatedly until the sixth day.” While this explanation shows procedural reasoning, it relies on step-by-step continuation rather than an explicit inference based on a general rule. When asked whether the situation could be expressed using multiplication, the student responded, “It can be written as five times five, then added to twenty-five.” This statement indicates partial inferential reasoning, as the student attempted to generalize the repeated increase into a multiplicative form, although the explanation remained informal and incomplete. The student ultimately concluded, “Fifty kilograms.” When asked why they were confident in this conclusion, the student stated, “Because every day it always increases by five kilograms.” This justification reflects an intuitive inference based on pattern recognition but lacks a clear articulation of how the number of days and the rate of increase combine into a generalized expression (25 + 5 × 5). Overall, the interview findings reinforce the written test results by showing that students were often able to identify patterns but struggled to clearly express the inferential reasoning that connected the given facts to the final conclusion. Students tended to rely on repeated examples or intuitive reasoning rather than explicitly stating the rule or generalizing the pattern. These results indicate that students’ inference skills remain underdeveloped, particularly in forming and articulating logical conclusions based on mathematical patterns and relationships. Therefore, instructional strategies should emphasize helping students generalize patterns, justify conclusions, and clearly communicate inferential reasoning in mathematical problem solving. A summary of students’ mathematical critical thinking skills for each indicator is presented in Table 7. Table 7 . Students’ Mathematical Critical Thinking Skills Based on Indicators Indicator Percentage (%) Interpretation 55.2 Analysis 48.96 Evaluation 41 Inference 54.2 Average 49.74 The average percentage score of students’ mathematical critical thinking skills was 49.74%, which falls into the low category. The interpretation indicator achieved a percentage of 55.20%, which was also categorized as low. Similarly, the analysis (48.96%), evaluation (41.00%), and inference (54.20%) indicators were all classified as low. The findings of this study reinforce a growing body of literature indicating that students in remote or rural contexts tend to demonstrate lower mathematical critical thinking skills, particularly in the analysis, evaluation, and inference components of Facione’s framework. The overall mean score of 49.74% confirms that students’ engagement with integer multiplication problems remains largely procedural rather than reflective or metacognitive. Similar patterns have been reported in recent studies emphasizing that critical thinking weaknesses often emerge when students are required to justify reasoning, verify results, or generalize patterns rather than merely perform calculations. In terms of interpretation, students in this study showed relatively higher performance compared to other indicators, yet still struggled to translate contextual situations into correct mathematical representations. This finding is consistent with Amir et al. [12] and Vlassis and Demonty [11], who reported that students frequently misinterpret symbolic meanings in integer operations, especially when negative quantities are embedded in real-life contexts. Recent evidence from Rodríguez-Sánchez et al. [10] further suggests that such misinterpretations stem from limited exposure to multiple representations and insufficient emphasis on conceptual modeling during instruction. In remote learning contexts, this issue is often exacerbated by limited instructional scaffolding and reduced opportunities for guided discussion [14]. The analysis indicator, which yielded one of the lowest scores in this study, highlights students’ difficulty in identifying relationships among quantities and selecting appropriate operations. This aligns with findings by Dorimana et al. [2] and Samura and Darhim [3], who argue that students’ analytical weaknesses are closely linked to teacher-centered instructional practices that prioritize answer accuracy over reasoning processes. Moreover, Gallardo-Estrada et al. [18] demonstrated that rural students often fail to recognize multiplicative structures unless explicitly guided through collaborative or inquiry-based learning. These results suggest that analytical thinking in mathematics is not an innate ability but one that requires sustained exposure to structured problem decomposition and reasoning-oriented tasks. The evaluation indicator emerged as the weakest aspect of students’ critical thinking skills. Students rarely verified their answers or questioned the validity of given results, tending instead to accept conclusions at face value. This phenomenon has been widely reported in recent studies, particularly in contexts where assessment practices emphasize final answers rather than reflective justification [23, 24]. Ibrahim et al. [25] found that without explicit instruction on self-checking strategies, students seldom engage in evaluative thinking independently. In remote areas, limited feedback cycles and high student-teacher ratios may further restrict opportunities for students to practice evaluative reasoning, reinforcing passive learning habits. Regarding inference, although some students were able to reach correct numerical conclusions, many failed to articulate the logical basis for their answers or generalize observed patterns into formal rules. This partial inference aligns with findings by Palinussa et al. [20], who noted that rural students often rely on empirical pattern recognition without progressing toward abstraction. Recent research by Popova et al. [8] emphasizes that inferential reasoning in mathematics develops optimally when students are encouraged to move from concrete examples to symbolic generalization through guided questioning and reflective discussion, some elements that are often underrepresented in remote-area classrooms. From a broader perspective, these findings support the argument that geographical location alone is not the primary determinant of students’ critical thinking skills. Meta-analytic evidence by Suparman et al. [24] indicates that instructional design, learning strategies, and cognitive engagement play a more significant role than school location. However, the persistent low performance observed in this study suggests that students in remote areas remain structurally disadvantaged due to limited access to adaptive pedagogies, professional teacher development, and contextualized learning materials [16, 17]. Hence, this study contributes empirical support to recent theoretical positions asserting that mathematical critical thinking, particularly at the levels of evaluation and inference, requires intentional instructional intervention. Without explicit emphasis on reasoning verification, argument evaluation, and generalization, students are unlikely to develop these higher-order skills organically. Therefore, the low critical thinking performance observed among students in Lembata Regency should not be viewed as a deficit inherent to rural learners, but rather as an indicator of pedagogical gaps that can be addressed through context-sensitive, reasoning-focused instructional models. This study has several limitations that should be considered when interpreting the findings. The primary limitation concerns the small sample size, which involved only twenty-four students (n = 24). This limited sample reduces the statistical power of the study and significantly constrains the generalizability of the results. Consequently, the findings cannot be directly applied to broader student populations across different geographical, social, or academic contexts. Another limitation is the absence of moderator or mediator variables, such as students’ and families’ socioeconomic status, prior mathematical ability, teacher quality, and parental support. These factors play an important role in shaping the development of mathematical critical thinking skills and should be included to provide a more comprehensive explanatory model. In addition, limitations were identified in the test instrument used. The items did not equally represent all mathematical operations, such as addition, subtraction, and division. This imbalance may have influenced the accuracy of the overall assessment of students’ numeracy skills, as certain operational competencies may have been underrepresented, thereby reducing the completeness of the analysis. Given these limitations, the findings of this study should be interpreted with caution. Future research should involve larger and more diverse samples, include schools from various regions, and employ more balanced instruments that comprehensively cover all mathematical operations to obtain a deeper understanding of students’ mathematical critical thinking skills. The use of mixed-method designs incorporating control groups from both urban and rural settings is also recommended to allow direct comparison of the influence of geographical context. Furthermore, research instruments should be further developed to ensure greater rigor, including a minimum of ten items, construct validity testing through factor analysis, and adequate reliability measures. Inter-rater reliability in qualitative assessments is also essential to enhance objectivity and consistency. Future studies should also incorporate relevant moderator or mediator variables, such as socioeconomic status, initial mathematical ability, teacher quality, and parental support, to better model the factors influencing mathematical critical thinking. Longitudinal designs spanning at least one academic semester are also needed to monitor students’ critical thinking development over time and to examine the sustained effects of instructional interventions. Finally, experimental studies employing local context-based learning strategies, such as Problem-Based Learning (PBL), Realistic Mathematics Education (RME), or the GASING method, integrated with local wisdom, may provide valuable insights into their effects on each critical thinking indicator proposed by Facione. The use of simple technologies, including mobile applications or interactive digital media, may also be explored to support students’ evaluation and generalization processes, particularly in areas with limited internet access. CONCLUSION The results of this study indicate that the mathematical critical thinking skills of eighth-grade students at a public junior high school in Lembata Regency remained relatively low. This was reflected in the overall average score of 49.74%, which fell into the low category. Among the four aspects assessed, interpretation showed the highest achievement at 55.20%, while evaluation was the lowest at 41.00%. With regard to interpretation, the students were generally able to identify known and unknown information in mathematical problems. However, they frequently misinterpreted contextual information or numerical signs, leading to inaccurate mathematical representations. The average achievement for this indicator was 55.20%, indicating that students still require reinforcement in translating contextual problems into appropriate mathematical symbols and models. Students’ analytical skills were also found to be relatively low, with an average score of 48.96%. Many students experienced difficulty explaining relationships among problem components, particularly in distinguishing between repeated addition and integer multiplication. This finding suggests the need for more intensive instruction focused on analyzing problem structures, decomposing problems into simpler elements, and selecting solution strategies that align with the characteristics of integer operations. The evaluation indicator emerged as the weakest aspect, with an average achievement of only 41.00%. Students rarely reviewed their solution steps or verified their final answers, and were not accustomed to assessing the validity of mathematical arguments. This situation highlights the importance of fostering reflective habits and strengthening logical reasoning skills, enabling students to evaluate and refine their own thinking processes. For the inference indicator, students demonstrated the ability to draw conclusions from specific examples but showed limitations in generalizing patterns into formal rules or justifying hypotheses. The average score for this indicator was 54.20%, which remains in the low category. This finding underscores the need for instructional strategies that emphasize pattern recognition, generalization, and the use of logical reasoning to establish cause-and-effect relationships, thereby enhancing students’ ability to formulate valid and systematic mathematical conclusions. Overall, this study provides insight into the current condition of junior high school students’ mathematical critical thinking skills in remote areas. These findings may serve as a reference for teachers and researchers in selecting, designing, and developing learning approaches that promote mathematical critical thinking. Contextual learning strategies that emphasize accurate context translation, structured problem-solving processes, systematic evaluation practices, and generalization activities are recommended to improve the quality of students’ critical thinking skills. The findings of this study have important implications for mathematics instruction, teacher practice, and educational policy in remote-area contexts. The consistently low performance across all four Facione indicators indicates that mathematics learning in remote schools remains largely procedural, with limited emphasis on reasoning quality. This suggests the need to shift instructional practices from answer-focused approaches toward reasoning-oriented learning that explicitly develops interpretation, analysis, evaluation, and inference skills. Teachers should place greater emphasis on explaining why strategies are used, how relationships among quantities are constructed, and whether conclusions are logically justified. Although students showed relatively better performance in interpretation, they still experienced difficulties translating contextual situations into appropriate mathematical representations. This implies the importance of instructional modeling that highlights how real-life problems are represented mathematically, particularly in distinguishing additive and multiplicative structures in integer operations. The use of multiple representations, such as number lines, tables, verbal reasoning, and symbolic expressions, may help students connect contextual understanding with formal mathematical reasoning. The weakest performance in the evaluation indicator reveals that students are not accustomed to verifying results, questioning given answers, or reflecting on solution validity. Therefore, assessment practices should move beyond rewarding correct answers to valuing justification, verification, and reflective reasoning. Targeted professional development may also be needed to support teachers in integrating evaluative questioning strategies into daily instruction. Similarly, low inference scores indicate that while students can recognize patterns, they struggle to generalize these patterns into formal rules. Instructional strategies should therefore guide students from concrete examples toward abstraction through structured generalization and reasoning tasks. At a broader level, these findings suggest that low critical thinking performance in remote areas reflects pedagogical and structural constraints rather than students’ abilities. Consequently, context-sensitive learning resources, teacher training focused on higher-order thinking, and reasoning-based assessment frameworks should be prioritized. This study also provides a diagnostic basis for future intervention research aimed at strengthening mathematical critical thinking in remote-area schools. Declarations FUNDING The authors would like to express their sincere gratitude to Universitas Pendidikan Indonesia and Yayasan Surya Institut. ETHICS APPROVAL AND CONSENT TO PARTICIPATE This study was conducted during the odd semester of the 2025/2026 academic year and involved 24 eighth-grade students at SMP Negeri 3 Wulandoni Satu Atap Labala, Wulandoni District, Lembata Regency, East Nusa Tenggara, Indonesia. Permission to conduct the research was granted by the principal of SMP Negeri 3 Wulandoni Satu Atap Labala prior to data collection, as documented in the official approval letter No. 01.19/KS/1,24,14/050/X/2025, dated 23 October 2025. All procedures involving human participants were carried out in accordance with relevant institutional procedures and with the ethical principles for research involving human participants, in line with the Declaration of Helsinki and its later amendments or comparable ethical standards. Informed consent to participate in the study was obtained from all participants’ parents or legal guardians prior to their involvement in the research. The participants and their guardians were informed about the purpose of the study, procedures, potential risks and benefits, and their right to withdraw at any time without consequences. Participation was voluntary, and all collected data were anonymized to ensure confidentiality and protect participant privacy. AUTHOR CONTRIBUTIONS Conceptualization: Abu Moh. Rasyid Ridho. Data Curation: Abu Moh. Rasyid Ridho, Al Jupri, and Sumanang Muhtar Gozali. Formal Analysis: Abu Moh. Rasyid Ridho. Investigation: Abu Moh. Rasyid Ridho. Methodology: Al Jupri and Sumanang Muhtar Gozali. Project Administration: Abu Moh. Rasyid Ridho. Supervision: Al Jupri and Sumanang Muhtar Gozali. Validation: Al Jupri and Sumanang Muhtar Gozali. Visualization: N Abu Moh. Rasyid Ridho. Writing – Original Draft: Abu Moh. Rasyid Ridho. Writing – Review & Editing: Abu Moh. Rasyid Ridho, Al Jupri, and Sumanang Muhtar Gozali. CONFLICT OF INTEREST The authors declare that there are no potential conflicts of interest related to the research, authorship, or publication of this work. DATA AVAILABILITY The data supporting the findings of this study are not publicly available due to ethical and confidentiality considerations involving student participants. However, the data may be made available upon reasonable request by contacting the corresponding author ( [email protected] ). CONSENT TO PUBLISH DECLARATION Not applicable. CONSENT TO PARTICIPATE DECLARATION Not applicable. References Arista E, Mahmudi A. Mathematical creative thinking ability in solving open-ended problems of PISA type based on school level. PYTHAGORAS: Jurnal Matematika dan Pendidikan Matematika. 2020;15(1):87–99. https://doi.org/10.21831/pg.v15i1.34606 . Dorimana A, Uworwabayeho A, Nizeyimana G. Enhancing upper secondary learners’ problem-solving abilities using problem-based learning in mathematics. Int J Learn Teach Educational Res. 2022;21(8):235–52. https://doi.org/10.26803/ijlter.21.8.14 . Samura AO, Darhim. Improving mathematics critical thinking skills of junior high school students using blended learning model (BLM) in GeoGebra Assisted Mathematics Learning. Int J Interact Mob Technol. 2023;17(2):101–17. https://doi.org/10.3991/ijim.v17i02.36097 . Stephens M, Buteau C. Introduction to the special issue on Computational thinking and mathematics teaching and learning. J Pedagogical Res. 2023. https://doi.org/10.33902/jpr.202313362 . Ennis RH. The nature of critical thinking: An outline of critical thinking dispositions and abilities. Univ Ill. 2011;2(4):1–8. Facione PA. (2020). Critical Thinking: What It Is and Why It Counts . Insight Assessment. https://www.insightassessment.com/wp-content/uploads/ia/pdf/whatwhy.pdf Stohlmann M, Yang Y. Growth mindset in high school mathematics: A review of the literature since 2007. J Pedagogical Res. 2024. https://doi.org/10.33902/jpr.202424437 . Popova Y, Abdualiyeva M, Torebek Y, Yelshibekov N, Omashova G. Improving the effectiveness of senior graders’ education based on the development of mathematical intuition and logic. Front Educ. 2022;7:986093. https://doi.org/10.3389/feduc.2022.986093 . Syafril S, Aini NR, Netriwati, Pahrudin A, Yaumas NE, Engkizar. (2020). Spirit of mathematics critical thinking skills (CTS). Journal of Physics: Conference Series , 1467 (1), 012069. https://doi.org/10.1088/1742-6596/1467/1/012069 Rodríguez-Sánchez MM, Sánchez-García AB, López-Fernández R. Subtraction: More than an algorithm? Sustainability. 2020;12(21):9148. https://doi.org/10.3390/su12219148 . Vlassis J, Demonty I. The role of algebraic thinking in dealing with negative numbers. ZDM – Math Educ. 2022;54(6):1243–55. https://doi.org/10.1007/s11858-022-01402-1 . Amir MF, Wardana MDK, Zannah M, Rudyanto HE, Nawafilah NQ. Capturing strategies and difficulties in solving negative integers: A case study of instrumental understanding. Acta Sci. 2022;24(2):64–87. https://doi.org/10.17648/acta.scientiae.6432 . Brahmia SW, Olsho A, Smith TI, Boudreaux A. Framework for the natures of negativity in introductory physics. Phys Rev Phys Educ Res. 2020;16(1):010120. https://doi.org/10.1103/PhysRevPhysEducRes.16.010120 . Kurniawan H, Purwoko RY, Setiana DS. Integrating cultural artifacts from remote regions in developing mathematics lesson plans to enhance mathematical literacy. J Pedagogical Res. 2023;7(1):1–15. https://doi.org/10.33902/JPR.202423016 . Darmaji D, Kurniawan DA, Astalini A, Perdana R, Kuswanto K, Ikhlas M. Do science process skills affect critical thinking? Differences in urban and rural contexts. Int J Evaluation Res Educ. 2020;9(4):874–84. https://doi.org/10.11591/ijere.v9i4.20687 . Tanti T, Kurniawan DA, Kuswanto K, Utami W, Wardhana I. Science process skills and critical thinking in science: Urban and rural disparity. Jurnal Pendidikan IPA Indonesia. 2020;9(4):489–98. https://doi.org/10.15294/jpii.v9i4.24139 . Wulandari DU, Mariana N, Wiryanto W, Amien MS. Integration of ethnomathematics teaching materials in mathematics learning in elementary school. IJORER: Int J Recent Educational Res. 2024;5(1):204–18. https://doi.org/10.46245/ijorer.v5i1.542 . Gallardo-Estrada C, Nussbaum M, Pinto M, Alvares D, Alario-Hoyos C. Enhancing grit and critical thinking in rural primary students: Impact of a targeted educational intervention. Educ Sci. 2024;14(9):1009. https://doi.org/10.3390/educsci14091009 . Susanta A, Susanto E, Rusnilawati R, Sumardi H, Ali SRB. Literacy skills through the use of digital STEAM-inquiry learning modules: A comparative study of urban and rural elementary schools in Indonesia. Eurasia J Math Sci Technol Educ. 2025;21(4):em2615. https://doi.org/10.29333/ejmste/16170 . Palinussa AL, Molle JS, Gaspersz M. Realistic mathematics education: Mathematical reasoning and communication skills in rural contexts. Int J Evaluation Res Educ. 2021;10(2):522–31. https://doi.org/10.11591/ijere.v10i2.20640 . Hendriana H, Rohaeti EE, Sumarmo U. Hard Skill dan Soft Skill Matematik Siswa. PT Refika Aditama; 2017. Susilo BE, Darhim D, Prabawanto S. (2020). Critical thinking skills in integral calculus lecture based on mathematical dispositions. Journal of Physics: Conference Series , 1521 (3), 032045. https://doi.org/10.1088/1742-6596/1521/3/032045 Uddin MR, Shimizu K, Sharmin H, Widiyatmoko A. (2023). Comparing critical thinking skills between rural and urban students at secondary level education. AIP Conference Proceedings, 2601 , 020035. https://doi.org/10.1063/5.0127674 Suparman S, Juandi D, Martadiputra BAP. Students’ heterogeneous mathematical critical thinking skills in problem-based learning: A meta-analysis investigating the involvement of school geographical location. Al-Jabar: Jurnal Pendidikan Matematika. 2023;14(1):37–53. https://doi.org/10.24042/ajpm.v14i1.16200 . Ibrahim NN, Ayub AFM, Yunus ASM. Impact of higher order thinking skills (HOTS) module based on the cognitive apprenticeship model on students’ performance. Int J Learn Teach Educational Res. 2020;19(7):246–62. https://doi.org/10.26803/ijlter.19.7.14 . Suparman S, Juandi D, Tamur M. (2021). Does problem-based learning enhance students’ higher order thinking skills in mathematics learning? A systematic review and meta-analysis. 2021 4th International Conference on Big Data and Education , 44–51. https://doi.org/10.1145/3451400.3451408 Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 05 May, 2026 Reviews received at journal 04 May, 2026 Reviews received at journal 03 May, 2026 Reviews received at journal 30 Apr, 2026 Reviewers agreed at journal 26 Apr, 2026 Reviewers agreed at journal 24 Apr, 2026 Reviewers agreed at journal 24 Apr, 2026 Reviewers invited by journal 25 Mar, 2026 Editor invited by journal 24 Mar, 2026 Editor assigned by journal 16 Mar, 2026 Submission checks completed at journal 14 Mar, 2026 First submitted to journal 14 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9065392","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":612015659,"identity":"d297f757-ab82-4b51-8f11-1ef042e7816b","order_by":0,"name":"Abu Moh. Rasyid Ridho","email":"","orcid":"","institution":"Universitas Pendidikan Indonesia","correspondingAuthor":false,"prefix":"","firstName":"Abu","middleName":"Moh. Rasyid","lastName":"Ridho","suffix":""},{"id":612015660,"identity":"0373ed9a-84e4-49e1-9b33-e7d74ba79a25","order_by":1,"name":"Al Jupri Al Jupri","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVRIiWNgGAWjYDACdoYEICkhx8DA2MDAwEaMFmaIFmOStIBBYgOYIkYLfzPDM+nCPRbpG243NzB8KDtMWIvEYYY06RnPJHI33DnYwDjjHBFaGEBaeA4AtdxIbGDmbSNCizxUS7oBSMtfYrQYQLUkgLUwEqPF8DBDsvWMAxKGM4F+OdhzLp2wFrnjPYm3Cw7UyfPdbn/44EeZNWEtDAw8CZCokWBgOECMeiBgPwDXMgpGwSgYBaMAKwAAD/U6vdA7w5UAAAAASUVORK5CYII=","orcid":"","institution":"Universitas Pendidikan Indonesia","correspondingAuthor":true,"prefix":"","firstName":"Al","middleName":"Jupri Al","lastName":"Jupri","suffix":""},{"id":612015664,"identity":"dd17e68a-7371-4c43-958d-994184736ad3","order_by":2,"name":"Sumanang Muhtar Gozali","email":"","orcid":"","institution":"Universitas Pendidikan Indonesia","correspondingAuthor":false,"prefix":"","firstName":"Sumanang","middleName":"Muhtar","lastName":"Gozali","suffix":""}],"badges":[],"createdAt":"2026-03-08 15:55:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9065392/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9065392/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105728873,"identity":"a13b5f79-c7bc-42b7-9196-42d04a9892f5","added_by":"auto","created_at":"2026-03-30 11:12:56","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2658815,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9065392/v1/ea37a88a-a452-449c-b571-61986f967bcd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eAnalysis of Middle School Students’ Mathematical Critical Thinking Skills in Remote Areas on Integer Multiplication\u003c/p\u003e","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eMathematical critical thinking skills are widely recognized as essential 21st-century competencies that students must master to address the complexity of global challenges [\u003cspan additionalcitationids=\"CR2 CR3\" citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. In mathematics learning, these skills are not only important for solving everyday problems but also encompass the ability to analyze information, evaluate arguments, and draw logically justified conclusions [\u003cspan additionalcitationids=\"CR6\" citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e\u0026ndash;\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. Therefore, enhancing mathematical critical thinking skills has become a crucial educational agenda, as mathematics functions not only as a formal discipline but also as a way of thinking that supports sound decision-making in real-life contexts [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eThe importance of critical thinking becomes even more evident when applied to conceptual and challenging mathematical content, such as integer multiplication. This topic is a fundamental component of the junior high school curriculum and serves as the basis for understanding more advanced concepts, including equations and functions. Nevertheless, numerous studies indicate that students continue to experience difficulties in understanding integer operations, particularly those involving negative numbers [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e]. These difficulties occur not only at the procedural level but also at the conceptual level, such as interpreting signs, selecting appropriate operations, and connecting symbolic representations with problem contexts [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e, \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. Consequently, integer multiplication is a relevant topic for examining how students employ mathematical critical thinking skills during problem-solving processes.\u003c/p\u003e \u003cp\u003eThese conceptual challenges are closely related to the diverse learning contexts experienced by students, especially with respect to geographical and social factors. This complexity is particularly evident in remote areas, where students often face limitations in educational infrastructure, access to learning resources, and the availability of qualified teachers [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e]. Such conditions can negatively affect the quality of learning interactions and ultimately result in lower levels of mathematical critical thinking skills compared to students in urban areas [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHowever, rural contexts should not be viewed solely in terms of limitations. Remote areas also possess unique potential that can be leveraged to support context-based learning and generate meaningful educational impact [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. To date, research on mathematical critical thinking skills has predominantly focused on students in urban settings or schools with relatively advanced educational facilities. Studies that specifically explore the mathematical critical thinking skills of students in remote areas, particularly in relation to foundational topics such as integer multiplication, remain limited. However, such investigations are essential for developing a more comprehensive understanding of how contextual factors influence students\u0026rsquo; cognitive development in mathematics learning.\u003c/p\u003e \u003cp\u003eConsistent with this urgency, several previous studies have identified gaps in critical thinking skills between urban and rural students. For instance, Gallardo-Estrada et al. [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] reported that although students in rural schools demonstrated improvements in critical thinking following collaborative learning interventions, they still lagged behind their urban counterparts in the consistent application of cognitive strategies. Similarly, Susanta et al. [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] found differences in mathematical comprehension based on geographical location, with urban students achieving higher scores. In contrast, Palinussa et al. [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] demonstrated that appropriate instructional approaches, such as Realistic Mathematics Education or higher-order thinking skills (HOTS)-oriented learning, can substantially enhance the critical thinking abilities of rural students, even bringing their performance closer to that of urban students. These findings suggest that geographical disparities are not fixed barriers but challenges that can be addressed through adaptive pedagogical strategies.\u003c/p\u003e \u003cp\u003eBased on these considerations, this study aims to analyze the mathematical critical thinking skills of students in remote areas with respect to integer multiplication. The findings are expected to contribute theoretically to the literature on critical thinking across diverse geographical contexts and practically to the development of relevant and adaptive instructional strategies for learners in remote areas. Thus, this study not only enriches academic understanding but also offers actionable insights for improving the quality of mathematics education in regions that continue to face educational limitations.\u003c/p\u003e"},{"header":"RESEARCH METHOD","content":"\u003cp\u003eThis study employed a qualitative descriptive approach with a research design focused on an in-depth analysis of students’ mathematical critical thinking skills in integer multiplication. A qualitative approach was chosen because it aims to describe and deeply understand the phenomena that occur, particularly the types of errors students make when applying mathematical critical thinking skills. The descriptive design was selected because it enables the presentation of a clear and comprehensive picture of students’ mathematical critical thinking skills within the context of this study.\u003c/p\u003e \u003cp\u003eThe investigation was conducted during the odd semester of the 2025/2026 academic year. The research subjects consisted of 24 eighth-grade students from a public junior high school located in Lembata Regency, East Nusa Tenggara Province, Indonesia. This school was selected because it represented learning conditions in remote areas, particularly island regions in East Nusa Tenggara with challenging geographical characteristics. The school was located approximately 45 km from the district center, with transportation access highly dependent on weather conditions. In addition, the school faced limited access to technology, digital learning resources, and teacher professional development, which are typical constraints that influence the mathematics learning process in remote areas.\u003c/p\u003e \u003cp\u003eThe test instruments were validated by two mathematics education lecturers and one experienced junior high school mathematics teacher. Semi-structured interviews were conducted with 24 selected students using purposive sampling based on variations in ability. These students became the main focus of the analysis to explore their profiles of mathematical critical thinking skills related to integer multiplication. Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e presents the written test instrument, while interview guidelines were used to confirm students’ mathematical critical thinking processes when completing the test.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\"\u003e\u003c/div\u003e\u003ctable id=\"Tab1\" border=\"1\"\u003e \u003ccaption\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eMathematical Critical Thinking Test Instrument\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003c/colgroup\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\"\u003e \u003cp\u003eAn employee is fined Rp2,000 for each time he is late. Over the last 14 days, he has been recorded as being late every day.\u003c/p\u003e \u003cp\u003ea. Explain how you understand this situation in the form of a mathematical statement to determine the total fine that must be paid!\u003c/p\u003e \u003cp\u003eb. Calculate the total fine that must be paid!\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eA tour bus departs with 48 passengers. At each terminal, 4 people get off. If the bus stops at 3 consecutive terminals:\u003c/p\u003e \u003cp\u003ea. How many passengers get off in total?\u003c/p\u003e \u003cp\u003eb. Explain the steps you took to find the answer!\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eBudi participated in a math competition. For every correct answer, he received 8 points, and for every incorrect answer, he lost 6 points. In one session, Budi answered 3 questions correctly and 7 questions incorrectly. The committee recorded that Budi’s total score was − 18 points.\u003c/p\u003e \u003cp\u003ea. Check whether the recorded score is correct or incorrect!\u003c/p\u003e \u003cp\u003eb. Explain the steps you took to find the answer!\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eA merchant sells oranges with the following data:\u003c/p\u003e \u003cp\u003eDay 1: 25 kg; Day 2: 30 kg; Day 3: 35 kg.\u003c/p\u003e \u003cp\u003eSales increase by 5 kg each day. A friend concludes, \u003cem\u003e“On the sixth day, the vendor will sell 50 kg of oranges.”\u003c/em\u003e\u003c/p\u003e \u003cp\u003ea. Check whether your friend’s conclusion is correct!\u003c/p\u003e \u003cp\u003eb. Explain how you drew your conclusion from the given facts!\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/table\u003e\u003c/div\u003e \u003cp\u003e\u003c/p\u003e \u003cp\u003eThe test instrument consisted of four essay questions designed to encourage students to apply various aspects of mathematical critical thinking. Although the questions were presented in different contexts, they were closely related to the topic of integer multiplication, allowing for a comprehensive analysis of students’ abilities. The test was administered individually to ensure that students completed the tasks independently, without peer influence. The interviews conducted in this study were unstructured and aimed to confirm the results of the mathematical critical thinking test. This stage also sought to further explore students’ strategies, reasoning patterns, and approaches when solving the test problems.\u003c/p\u003e \u003cp\u003e After the test data were collected, students’ responses were assessed using mathematical critical thinking skill assessment guidelines. The scores obtained for each indicator and the total score were then converted into percentages. These percentage scores were categorized according to the levels of mathematical critical thinking skills shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e\u003cdiv class=\"gridtable\"\u003e\u003cdiv align=\"left\" class=\"colspec\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\"\u003e\u003c/div\u003e\u003ctable id=\"Tab2\" border=\"1\"\u003e \u003ccaption\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eCategories of Mathematical Critical Thinking Skill Percentages\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003c/colgroup\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\"\u003e \u003cp\u003eInterpretation (%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\"\u003e \u003cp\u003eCategory\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e80 \u0026lt; x ≤ 100\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eVery High\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e65 \u0026lt; x ≤ 80\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eHigh\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e55 \u0026lt; x ≤ 65\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eModerate\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e40 \u0026lt; x ≤ 55\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eLow\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\"\u003e \u003cp\u003e0 \u0026lt; x ≤ 40\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\"\u003e \u003cp\u003eVery Low\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/table\u003e\u003c/div\u003e "},{"header":"RESEARCH RESULTS AND DISCUSSION","content":"\u003cp\u003eThe analysis of students\u0026rsquo; written responses for each indicator, supported by interview excerpts, is presented as follows.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInterpretation Indicator\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe interpretation indicator appeared when students were able to read and restate the problem, but failed to fully understand the mathematical meaning of the situation described. This indicator is illustrated in Table 3, which presents the written test results for question number 1, and is further supported by evidence from the interview data.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 3.\u0026nbsp;\u003c/strong\u003eWritten Test Question and Results for Question Number 1\u003c/p\u003e\n\u003ctable style=\"width: 4.8e+2pt;\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eQuestion Number 1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eWritten Test Results for Question Number 1\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eAn employee is fined Rp2,000 for each time he is late. Over the last 14 days, he has been recorded as being late every day.\u003c/p\u003e\n \u003col\u003e\n \u003cli\u003eExplain how you understand this situation in the form of a mathematical statement to determine the total fine that must be paid!\u003c/li\u003e\n \u003cli\u003eCalculate the total fine that must be paid!\u003c/li\u003e\n \u003c/ol\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\n \u003cv:shapetype id=\"_x0000_t75\" coordsize=\"21600,21600\" o:spt=\"75\" o:preferrelative=\"t\" path=\"m@4@5l@4@11@9@11@9@5xe\" filled=\"f\" stroked=\"f\"\u003e\u0026nbsp;\u003cv:stroke joinstyle=\"miter\"\u003e\u0026nbsp;\u003cv:formulas\u003e\u0026nbsp;\u003cv:f eqn=\"if lineDrawn pixelLineWidth 0\"\u003e\u0026nbsp;\u003cv:f eqn=\"sum @0 1 0\"\u003e\u0026nbsp;\u003cv:f eqn=\"sum 0 0 @1\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @2 1 2\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @3 21600 pixelWidth\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @3 21600 pixelHeight\"\u003e\u0026nbsp;\u003cv:f eqn=\"sum @0 0 1\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @6 1 2\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @7 21600 pixelWidth\"\u003e\u0026nbsp;\u003cv:f eqn=\"sum @8 21600 0\"\u003e\u0026nbsp;\u003cv:f eqn=\"prod @7 21600 pixelHeight\"\u003e\u0026nbsp;\u003cv:f eqn=\"sum @10 21600 0\"\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:f\u003e\u0026nbsp;\u003c/v:formulas\u003e\n \u003cv:path o:extrusionok=\"f\" gradientshapeok=\"t\" o:connecttype=\"rect\"\u003e\u0026nbsp;\u003c/v:path\u003e \u0026nbsp;\u0026nbsp;\u0026nbsp;\u0026nbsp;\n \u003c/v:stroke\u003e\u0026nbsp;\u003c/v:shapetype\u003e\u003cimg src=\"https://myfiles.space/user_files/127393_c7e80a1c9bb65875/127393_custom_files/img1774638550.png\" alt=\"image\" style=\"width: 284px;\"\u003e\n \u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;Based on the written test results, students generally identified the context of the problem but produced inaccurate mathematical representations. This finding is consistent with the interview results. When asked to explain what the problem was, the student stated, \u003cem\u003e\u0026ldquo;The problem is about an employee who is fined Rp2,000 every time he is late for 14 days. What is being asked is the total fine that must be paid.\u0026rdquo;\u003c/em\u003e This response indicates that the student was able to comprehend the narrative context and the goal of the problem. Similarly, when identifying known information, the student correctly mentioned, \u003cem\u003e\u0026ldquo;The fine is two thousand rupiah, and the lateness lasted for fourteen days.\u0026rdquo;\u003c/em\u003e These responses show that the student did not experience difficulty in reading the problem or extracting explicit information from it. However, difficulties became evident when the student was asked to express the situation in mathematical form. The student explained, \u003cem\u003e\u0026ldquo;I wrote 2,000 + 14 because there are many fines.\u0026rdquo;\u003c/em\u003e This response revealed a misinterpretation of the relationship between the fine amount and the number of days. Instead of viewing the situation as repeated equal groups, the student treated the quantities as separate values to be added.\u003c/p\u003e\n\u003cp\u003eWhen further asked why addition was chosen instead of multiplication, the student responded, \u003cem\u003e\u0026ldquo;Because I thought the fines were added continuously, not multiplied.\u0026rdquo;\u003c/em\u003e This statement suggests that the student associated repetition with addition but lacked a clear conceptual understanding of multiplication as structured repeated addition. The interview also revealed partial conceptual awareness. When prompted about alternative strategies, the student admitted, \u003cem\u003e\u0026ldquo;It can be multiplied, but I am not confident about how to write it.\u0026rdquo;\u003c/em\u003e This indicates that the student had some awareness of multiplication in similar contexts but lacked confidence in applying it independently. After guided questioning, the student was able to identify the correct operation, stating, \u003cem\u003e\u0026ldquo;It should be multiplication, two thousand times fourteen days.\u0026rdquo;\u003c/em\u003e This progression demonstrates that the student\u0026rsquo;s difficulty was not due to an inability to perform multiplication, but rather to weaknesses in interpreting contextual information and selecting the appropriate mathematical operation without assistance.\u003c/p\u003e\n\u003cp\u003eOverall, the interview quotations strengthen the written test findings by showing that students\u0026rsquo; errors originated from misinterpretation at the conceptual level. Students were able to read, recall, and verbally explain the situation, yet struggled to translate real-life contexts into correct mathematical representations. This highlights the need for instructional strategies that emphasize conceptual interpretation of word problems, particularly in recognizing relational cues that indicate multiplication rather than simple addition.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAnalysis Indicator\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe analysis indicator was identified when students experienced difficulty recognizing and explaining the relationships among statements, quantities, and concepts contained in the problem. This indicator is reflected in Table 4, which presents the written test results for question number 2, and is further supported by the interview findings.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 4.\u0026nbsp;\u003c/strong\u003eWritten Test Question and Results for Question Number 2\u003c/p\u003e\n\u003ctable style=\"width: 4.9e+2pt;\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eQuestion Number 2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eWritten Test Results for Question Number 2\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eA tour bus departs with 48 passengers. At each terminal, 4 people get off. If the bus stops at 3 consecutive terminals:\u003c/p\u003e\n \u003col\u003e\n \u003cli\u003eHow many passengers get off in total?\u003c/li\u003e\n \u003cli\u003eExplain the steps you took to find the answer!\u003c/li\u003e\n \u003c/ol\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/127393_c7e80a1c9bb65875/127393_custom_files/img1774638552.png\" alt=\"image\" style=\"width: 310px;\"\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;Based on the written test results in Table 4, many students were unable to clearly articulate the relationship between \u0026ldquo;4 passengers getting off\u0026rdquo; and \u0026ldquo;3 terminals.\u0026rdquo; Although some students arrived at the correct numerical answer, they often failed to explain the reasoning process or the relationship among the given quantities in a structured manner. The interview results provide clearer insight into students\u0026rsquo; analytical thinking. When asked what happened to the passengers at each terminal, the student stated, \u003cem\u003e\u0026ldquo;At each terminal, four people get off.\u0026rdquo;\u003c/em\u003e When further asked about the number of terminals, the student responded, \u003cem\u003e\u0026ldquo;At three terminals.\u0026rdquo;\u003c/em\u003e These responses indicate that the student was able to correctly identify individual pieces of information presented in the problem. Importantly, when asked about the relationship between the number of passengers and the number of terminals, the student explained, \u003cem\u003e\u0026ldquo;Four people get off every time the bus stops.\u0026rdquo;\u003c/em\u003e This response shows an emerging ability to recognize a repeating pattern and to relate quantities conceptually rather than treating them as isolated values.\u003c/p\u003e\n\u003cp\u003eThis understanding was reflected in the student\u0026rsquo;s choice of operation. When asked which operation should be used, the student answered, \u003cem\u003e\u0026ldquo;Multiplication, four times three.\u0026rdquo;\u003c/em\u003e When probed further about why simple addition was not chosen, the student explained, \u003cem\u003e\u0026ldquo;Because the event happens repeatedly at each terminal.\u0026rdquo;\u003c/em\u003e These responses demonstrate that the student was able to analyze the structure of the problem and justify the use of multiplication based on repetition. Finally, when asked to determine the total number of passengers who got off, the student concluded, \u003cem\u003e\u0026ldquo;Twelve people.\u0026rdquo;\u003c/em\u003e This indicates that once the relationship among the quantities was correctly identified, the student was able to apply the appropriate operation and reach the correct solution.\u003c/p\u003e\n\u003cp\u003eOverall, the interview findings suggest that, unlike students who relied solely on surface-level reading, this student demonstrated the ability to analyze relationships among quantities and recognize repetition as a key indicator for multiplication. However, the written test results show that this level of analytical reasoning was not consistently demonstrated across all students. Many students still struggled to decompose the problem, explain relationships explicitly, and organize solution steps logically. These findings indicate that students\u0026rsquo; analytical skills remain uneven and require instructional reinforcement, particularly in helping students articulate relationships between quantities and justify their choice of operations.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEvaluation Indicator\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe evaluation indicator of mathematical critical thinking emerged when students were unable to verify the correctness of solution steps, assess the validity of their reasoning, and ensure that the chosen strategy and final result were consistent with the problem\u0026rsquo;s requirements. This indicator is presented in Table 5, which shows the written test results for question number 3, and is further strengthened by evidence from the interview data.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 5.\u0026nbsp;\u003c/strong\u003eWritten Test Question and Results for Question Number 3\u003c/p\u003e\n\u003ctable style=\"width: 4.8e+2pt;\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eQuestion Number 3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eWritten Test Results for Question Number 3\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eBudi participated in a math competition. For every correct answer, he received 8 points, and for every incorrect answer, he lost 6 points. In one session, Budi answered 3 questions correctly and 7 questions incorrectly. The committee recorded that Budi\u0026rsquo;s total score was \u0026minus;18 points.\u003c/p\u003e\n \u003col\u003e\n \u003cli\u003eCheck whether the recorded score is correct or incorrect!\u003c/li\u003e\n \u003cli\u003eExplain the steps you took to find the answer!\u003c/li\u003e\n \u003c/ol\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/127393_c7e80a1c9bb65875/127393_custom_files/img1774638551.png\" alt=\"image\" style=\"width: 276px;\"\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;Based on the written test results in Table 5, many students directly accepted the recorded score of \u0026minus;18 points without critically examining whether the result was reasonable or whether the calculation process was correct. This tendency is clearly reflected in the interview responses. When asked about the points obtained from correct answers, the student stated, \u003cem\u003e\u0026ldquo;Three times eight, so twenty-four.\u0026rdquo;\u003c/em\u003e Similarly, when asked about the points lost due to incorrect answers, the student answered, \u003cem\u003e\u0026ldquo;Seven times six, so forty-two.\u0026rdquo;\u003c/em\u003e These responses indicate that the student was able to apply the appropriate operations to calculate gains and losses separately. However, when determining the total score, the student explained, \u003cem\u003e\u0026ldquo;Twenty-four minus forty-two.\u0026rdquo;\u003c/em\u003e After calculating the result, the student stated, \u003cem\u003e\u0026ldquo;Negative eighteen.\u0026rdquo;\u003c/em\u003e Although this computation is mathematically correct, the evaluation process did not extend beyond performing the calculation.\u003c/p\u003e\n\u003cp\u003eWhen asked whether the result was correct, the student confidently responded, \u003cem\u003e\u0026ldquo;Yes, because the calculation is already correct.\u0026rdquo;\u003c/em\u003e This response shows that the student equated computational accuracy with overall correctness, without evaluating whether the result aligned with the context or checking the reasoning process. More importantly, when asked whether another method was used to verify the answer, the student admitted, \u003cem\u003e\u0026ldquo;No, I immediately trusted the result of the calculation.\u0026rdquo;\u0026nbsp;\u003c/em\u003eThis statement highlights a key weakness in the evaluation stage: the student did not attempt to recheck the answer using alternative strategies, such as rewriting the score calculation in a single expression (3 \u0026times; 8 \u0026minus; 7 \u0026times; 6), using a table of gains and losses, or reconsidering whether the negative result was reasonable in the context of the competition.\u003c/p\u003e\n\u003cp\u003eOverall, the interview findings reinforce the written test results by showing that students tended to stop at obtaining an answer rather than evaluating its validity. Students were able to carry out calculations correctly, but did not engage in verification, justification, or reflection on their solutions. This indicates that students\u0026rsquo; evaluation skills remain weak, particularly in verifying results and critically assessing whether their answers are logically and contextually appropriate. Therefore, instructional efforts should emphasize not only obtaining correct answers but also developing habits of checking, justifying, and evaluating solution processes and outcomes.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eInference Indicator\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe inference indicator of mathematical critical thinking was observed when students experienced difficulty drawing logical conclusions from available information or failed to clearly connect identified patterns to justify their conclusions. This indicator is illustrated in Table 6, which presents the written test results for question number 4, and is further supported by evidence from the interview data.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 6.\u0026nbsp;\u003c/strong\u003eQuestion and Written Test Results for Question Number 4\u003c/p\u003e\n\u003ctable style=\"width: 4.7e+2pt;\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eQuestion Number 4\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eWritten Test Results for Question Number 4\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eA merchant sells oranges with the following data: Day 1: 25 kg; Day 2: 30 kg; Day 3: 35 kg.\u003c/p\u003e\n \u003cp\u003eSales increase by 5 kg each day. A friend concludes, \u003cem\u003e\u0026ldquo;On the sixth day, the vendor will sell 50 kg of oranges.\u0026rdquo;\u003c/em\u003e\u003c/p\u003e\n \u003col\u003e\n \u003cli\u003eCheck whether your friend\u0026rsquo;s conclusion is correct!\u003c/li\u003e\n \u003cli\u003eExplain how you drew your conclusion from the given facts!\u003c/li\u003e\n \u003c/ol\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cimg src=\"https://myfiles.space/user_files/127393_c7e80a1c9bb65875/127393_custom_files/img1774638700.png\" style=\"width: 190px;\"\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;Based on the written test results in Table 6, many students were able to recognize that the sales data formed an increasing pattern, yet they struggled to explicitly articulate how the pattern led to the final conclusion. Although some students reached the correct numerical answer, their explanations often lacked clear inference processes or generalization of the pattern. The interview results provide insight into how students formed their conclusions. When asked about the daily sales trend, the student stated, \u003cem\u003e\u0026ldquo;Every day it increases by five kilograms.\u0026rdquo;\u003c/em\u003e When asked whether the pattern from the first day to the third day remained consistent, the student confirmed, \u003cem\u003e\u0026ldquo;Yes, it always increases by five kilograms.\u0026rdquo;\u003c/em\u003e These responses indicate that the student was able to identify and recognize a consistent pattern in the given data. However, when asked how to determine sales on the sixth day, the student explained, \u003cem\u003e\u0026ldquo;From the first day, add five repeatedly until the sixth day.\u0026rdquo;\u003c/em\u003e While this explanation shows procedural reasoning, it relies on step-by-step continuation rather than an explicit inference based on a general rule.\u003c/p\u003e\n\u003cp\u003eWhen asked whether the situation could be expressed using multiplication, the student responded, \u003cem\u003e\u0026ldquo;It can be written as five times five, then added to twenty-five.\u0026rdquo;\u003c/em\u003e This statement indicates partial inferential reasoning, as the student attempted to generalize the repeated increase into a multiplicative form, although the explanation remained informal and incomplete. The student ultimately concluded, \u003cem\u003e\u0026ldquo;Fifty kilograms.\u0026rdquo;\u003c/em\u003e When asked why they were confident in this conclusion, the student stated, \u003cem\u003e\u0026ldquo;Because every day it always increases by five kilograms.\u0026rdquo;\u003c/em\u003e This justification reflects an intuitive inference based on pattern recognition but lacks a clear articulation of how the number of days and the rate of increase combine into a generalized expression (25 + 5 \u0026times; 5).\u003c/p\u003e\n\u003cp\u003eOverall, the interview findings reinforce the written test results by showing that students were often able to identify patterns but struggled to clearly express the inferential reasoning that connected the given facts to the final conclusion. Students tended to rely on repeated examples or intuitive reasoning rather than explicitly stating the rule or generalizing the pattern. These results indicate that students\u0026rsquo; inference skills remain underdeveloped, particularly in forming and articulating logical conclusions based on mathematical patterns and relationships. Therefore, instructional strategies should emphasize helping students generalize patterns, justify conclusions, and clearly communicate inferential reasoning in mathematical problem solving.\u003c/p\u003e\n\u003cp\u003eA summary of students\u0026rsquo; mathematical critical thinking skills for each indicator is presented in Table 7.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003e7\u003c/strong\u003e\u003cstrong\u003e.\u0026nbsp;\u003c/strong\u003eStudents\u0026rsquo; Mathematical Critical Thinking Skills Based on Indicators\u003c/p\u003e\n\u003ctable\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eIndicator\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003ePercentage (%)\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eInterpretation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e55.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eAnalysis\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e48.96\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eEvaluation\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e41\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003eInference\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e54.2\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003eAverage\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd\u003e\n \u003cp\u003e\u003cstrong\u003e49.74\u003c/strong\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u0026nbsp;The average percentage score of students\u0026rsquo; mathematical critical thinking skills was 49.74%, which falls into the low category. The interpretation indicator achieved a percentage of 55.20%, which was also categorized as low. Similarly, the analysis (48.96%), evaluation (41.00%), and inference (54.20%) indicators were all classified as low.\u003c/p\u003e\n\u003cp\u003eThe findings of this study reinforce a growing body of literature indicating that students in remote or rural contexts tend to demonstrate lower mathematical critical thinking skills, particularly in the analysis, evaluation, and inference components of Facione\u0026rsquo;s framework. The overall mean score of 49.74% confirms that students\u0026rsquo; engagement with integer multiplication problems remains largely procedural rather than reflective or metacognitive. Similar patterns have been reported in recent studies emphasizing that critical thinking weaknesses often emerge when students are required to justify reasoning, verify results, or generalize patterns rather than merely perform calculations. In terms of interpretation, students in this study showed relatively higher performance compared to other indicators, yet still struggled to translate contextual situations into correct mathematical representations. This finding is consistent with Amir et al. [12] and Vlassis and Demonty [11], who reported that students frequently misinterpret symbolic meanings in integer operations, especially when negative quantities are embedded in real-life contexts. Recent evidence from Rodr\u0026iacute;guez-S\u0026aacute;nchez et al. [10] further suggests that such misinterpretations stem from limited exposure to multiple representations and insufficient emphasis on conceptual modeling during instruction. In remote learning contexts, this issue is often exacerbated by limited instructional scaffolding and reduced opportunities for guided discussion [14].\u003c/p\u003e\n\u003cp\u003eThe analysis indicator, which yielded one of the lowest scores in this study, highlights students\u0026rsquo; difficulty in identifying relationships among quantities and selecting appropriate operations. This aligns with findings by Dorimana et al. [2] and Samura and Darhim [3], who argue that students\u0026rsquo; analytical weaknesses are closely linked to teacher-centered instructional practices that prioritize answer accuracy over reasoning processes. Moreover, Gallardo-Estrada et al. [18] demonstrated that rural students often fail to recognize multiplicative structures unless explicitly guided through collaborative or inquiry-based learning. These results suggest that analytical thinking in mathematics is not an innate ability but one that requires sustained exposure to structured problem decomposition and reasoning-oriented tasks. The evaluation indicator emerged as the weakest aspect of students\u0026rsquo; critical thinking skills. Students rarely verified their answers or questioned the validity of given results, tending instead to accept conclusions at face value. This phenomenon has been widely reported in recent studies, particularly in contexts where assessment practices emphasize final answers rather than reflective justification [23, 24]. Ibrahim et al. [25] found that without explicit instruction on self-checking strategies, students seldom engage in evaluative thinking independently. In remote areas, limited feedback cycles and high student-teacher ratios may further restrict opportunities for students to practice evaluative reasoning, reinforcing passive learning habits. Regarding inference, although some students were able to reach correct numerical conclusions, many failed to articulate the logical basis for their answers or generalize observed patterns into formal rules. This partial inference aligns with findings by Palinussa et al. [20], who noted that rural students often rely on empirical pattern recognition without progressing toward abstraction. Recent research by Popova et al. [8] emphasizes that inferential reasoning in mathematics develops optimally when students are encouraged to move from concrete examples to symbolic generalization through guided questioning and reflective discussion, some elements that are often underrepresented in remote-area classrooms.\u003c/p\u003e\n\u003cp\u003eFrom a broader perspective, these findings support the argument that geographical location alone is not the primary determinant of students\u0026rsquo; critical thinking skills. Meta-analytic evidence by Suparman et al. [24] indicates that instructional design, learning strategies, and cognitive engagement play a more significant role than school location. However, the persistent low performance observed in this study suggests that students in remote areas remain structurally disadvantaged due to limited access to adaptive pedagogies, professional teacher development, and contextualized learning materials [16, 17]. Hence, this study contributes empirical support to recent theoretical positions asserting that mathematical critical thinking, particularly at the levels of evaluation and inference, requires intentional instructional intervention. Without explicit emphasis on reasoning verification, argument evaluation, and generalization, students are unlikely to develop these higher-order skills organically. Therefore, the low critical thinking performance observed among students in Lembata Regency should not be viewed as a deficit inherent to rural learners, but rather as an indicator of pedagogical gaps that can be addressed through context-sensitive, reasoning-focused instructional models.\u003c/p\u003e\n\u003cp\u003eThis study has several limitations that should be considered when interpreting the findings. The primary limitation concerns the small sample size, which involved only twenty-four students (n = 24). This limited sample reduces the statistical power of the study and significantly constrains the generalizability of the results. Consequently, the findings cannot be directly applied to broader student populations across different geographical, social, or academic contexts. Another limitation is the absence of moderator or mediator variables, such as students\u0026rsquo; and families\u0026rsquo; socioeconomic status, prior mathematical ability, teacher quality, and parental support. These factors play an important role in shaping the development of mathematical critical thinking skills and should be included to provide a more comprehensive explanatory model. In addition, limitations were identified in the test instrument used. The items did not equally represent all mathematical operations, such as addition, subtraction, and division. This imbalance may have influenced the accuracy of the overall assessment of students\u0026rsquo; numeracy skills, as certain operational competencies may have been underrepresented, thereby reducing the completeness of the analysis.\u003c/p\u003e\n\u003cp\u003eGiven these limitations, the findings of this study should be interpreted with caution. Future research should involve larger and more diverse samples, include schools from various regions, and employ more balanced instruments that comprehensively cover all mathematical operations to obtain a deeper understanding of students\u0026rsquo; mathematical critical thinking skills. The use of mixed-method designs incorporating control groups from both urban and rural settings is also recommended to allow direct comparison of the influence of geographical context. Furthermore, research instruments should be further developed to ensure greater rigor, including a minimum of ten items, construct validity testing through factor analysis, and adequate reliability measures. Inter-rater reliability in qualitative assessments is also essential to enhance objectivity and consistency. Future studies should also incorporate relevant moderator or mediator variables, such as socioeconomic status, initial mathematical ability, teacher quality, and parental support, to better model the factors influencing mathematical critical thinking. Longitudinal designs spanning at least one academic semester are also needed to monitor students\u0026rsquo; critical thinking development over time and to examine the sustained effects of instructional interventions.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFinally, experimental studies employing local context-based learning strategies, such as Problem-Based Learning (PBL), Realistic Mathematics Education (RME), or the GASING method, integrated with local wisdom, may provide valuable insights into their effects on each critical thinking indicator proposed by Facione. The use of simple technologies, including mobile applications or interactive digital media, may also be explored to support students\u0026rsquo; evaluation and generalization processes, particularly in areas with limited internet access.\u003c/p\u003e"},{"header":"CONCLUSION","content":"\u003cp\u003eThe results of this study indicate that the mathematical critical thinking skills of eighth-grade students at a public junior high school in Lembata Regency remained relatively low. This was reflected in the overall average score of 49.74%, which fell into the low category. Among the four aspects assessed, interpretation showed the highest achievement at 55.20%, while evaluation was the lowest at 41.00%. With regard to interpretation, the students were generally able to identify known and unknown information in mathematical problems. However, they frequently misinterpreted contextual information or numerical signs, leading to inaccurate mathematical representations. The average achievement for this indicator was 55.20%, indicating that students still require reinforcement in translating contextual problems into appropriate mathematical symbols and models. Students\u0026rsquo; analytical skills were also found to be relatively low, with an average score of 48.96%. Many students experienced difficulty explaining relationships among problem components, particularly in distinguishing between repeated addition and integer multiplication. This finding suggests the need for more intensive instruction focused on analyzing problem structures, decomposing problems into simpler elements, and selecting solution strategies that align with the characteristics of integer operations.\u003c/p\u003e \u003cp\u003eThe evaluation indicator emerged as the weakest aspect, with an average achievement of only 41.00%. Students rarely reviewed their solution steps or verified their final answers, and were not accustomed to assessing the validity of mathematical arguments. This situation highlights the importance of fostering reflective habits and strengthening logical reasoning skills, enabling students to evaluate and refine their own thinking processes. For the inference indicator, students demonstrated the ability to draw conclusions from specific examples but showed limitations in generalizing patterns into formal rules or justifying hypotheses. The average score for this indicator was 54.20%, which remains in the low category. This finding underscores the need for instructional strategies that emphasize pattern recognition, generalization, and the use of logical reasoning to establish cause-and-effect relationships, thereby enhancing students\u0026rsquo; ability to formulate valid and systematic mathematical conclusions.\u003c/p\u003e \u003cp\u003eOverall, this study provides insight into the current condition of junior high school students\u0026rsquo; mathematical critical thinking skills in remote areas. These findings may serve as a reference for teachers and researchers in selecting, designing, and developing learning approaches that promote mathematical critical thinking. Contextual learning strategies that emphasize accurate context translation, structured problem-solving processes, systematic evaluation practices, and generalization activities are recommended to improve the quality of students\u0026rsquo; critical thinking skills.\u003c/p\u003e \u003cp\u003eThe findings of this study have important implications for mathematics instruction, teacher practice, and educational policy in remote-area contexts. The consistently low performance across all four Facione indicators indicates that mathematics learning in remote schools remains largely procedural, with limited emphasis on reasoning quality. This suggests the need to shift instructional practices from answer-focused approaches toward reasoning-oriented learning that explicitly develops interpretation, analysis, evaluation, and inference skills. Teachers should place greater emphasis on explaining why strategies are used, how relationships among quantities are constructed, and whether conclusions are logically justified. Although students showed relatively better performance in interpretation, they still experienced difficulties translating contextual situations into appropriate mathematical representations. This implies the importance of instructional modeling that highlights how real-life problems are represented mathematically, particularly in distinguishing additive and multiplicative structures in integer operations. The use of multiple representations, such as number lines, tables, verbal reasoning, and symbolic expressions, may help students connect contextual understanding with formal mathematical reasoning. The weakest performance in the evaluation indicator reveals that students are not accustomed to verifying results, questioning given answers, or reflecting on solution validity. Therefore, assessment practices should move beyond rewarding correct answers to valuing justification, verification, and reflective reasoning. Targeted professional development may also be needed to support teachers in integrating evaluative questioning strategies into daily instruction. Similarly, low inference scores indicate that while students can recognize patterns, they struggle to generalize these patterns into formal rules. Instructional strategies should therefore guide students from concrete examples toward abstraction through structured generalization and reasoning tasks. At a broader level, these findings suggest that low critical thinking performance in remote areas reflects pedagogical and structural constraints rather than students\u0026rsquo; abilities. Consequently, context-sensitive learning resources, teacher training focused on higher-order thinking, and reasoning-based assessment frameworks should be prioritized. This study also provides a diagnostic basis for future intervention research aimed at strengthening mathematical critical thinking in remote-area schools.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eFUNDING\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors would like to express their sincere gratitude to Universitas Pendidikan Indonesia and Yayasan Surya Institut.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eETHICS APPROVAL AND CONSENT TO PARTICIPATE\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study was conducted during the odd semester of the 2025/2026 academic year and involved 24 eighth-grade students at SMP Negeri 3 Wulandoni Satu Atap Labala, Wulandoni District, Lembata Regency, East Nusa Tenggara, Indonesia. Permission to conduct the research was granted by the principal of SMP Negeri 3 Wulandoni Satu Atap Labala prior to data collection, as documented in the official approval letter No. 01.19/KS/1,24,14/050/X/2025, dated 23 October 2025. All procedures involving human participants were carried out in accordance with relevant institutional procedures and with the ethical principles for research involving human participants, in line with the Declaration of Helsinki and its later amendments or comparable ethical standards. Informed consent to participate in the study was obtained from all participants\u0026rsquo; parents or legal guardians prior to their involvement in the research. The participants and their guardians were informed about the purpose of the study, procedures, potential risks and benefits, and their right to withdraw at any time without consequences. Participation was voluntary, and all collected data were anonymized to ensure confidentiality and protect participant privacy.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAUTHOR CONTRIBUTIONS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConceptualization:\u003c/strong\u003e Abu Moh. Rasyid Ridho. \u003cstrong\u003eData Curation:\u003c/strong\u003e Abu Moh. Rasyid Ridho, Al Jupri, and Sumanang Muhtar Gozali. \u003cstrong\u003eFormal Analysis:\u003c/strong\u003e Abu Moh. Rasyid Ridho. \u003cstrong\u003eInvestigation:\u0026nbsp;\u003c/strong\u003eAbu Moh. Rasyid Ridho. \u003cstrong\u003eMethodology:\u003c/strong\u003e Al Jupri and Sumanang Muhtar Gozali. \u003cstrong\u003eProject Administration:\u003c/strong\u003e Abu Moh. Rasyid Ridho. \u003cstrong\u003eSupervision:\u0026nbsp;\u003c/strong\u003eAl Jupri and Sumanang Muhtar Gozali. \u003cstrong\u003eValidation:\u003c/strong\u003e Al Jupri and Sumanang Muhtar Gozali. \u003cstrong\u003eVisualization:\u003c/strong\u003e N Abu Moh. Rasyid Ridho. \u003cstrong\u003eWriting \u0026ndash; Original Draft:\u003c/strong\u003e Abu Moh. Rasyid Ridho. \u003cstrong\u003eWriting \u0026ndash; Review \u0026amp; Editing:\u003c/strong\u003e Abu Moh. Rasyid Ridho, Al Jupri, and Sumanang Muhtar Gozali.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCONFLICT OF INTEREST\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that there are no potential conflicts of interest related to the research, authorship, or publication of this work.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDATA AVAILABILITY\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data supporting the findings of this study are not publicly available due to ethical and confidentiality considerations involving student participants. However, the data may be made available upon reasonable request by contacting the corresponding author ([email protected]).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCONSENT TO PUBLISH DECLARATION\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCONSENT TO PARTICIPATE DECLARATION\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eArista E, Mahmudi A. Mathematical creative thinking ability in solving open-ended problems of PISA type based on school level. PYTHAGORAS: Jurnal Matematika dan Pendidikan Matematika. 2020;15(1):87\u0026ndash;99. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.21831/pg.v15i1.34606\u003c/span\u003e\u003cspan address=\"10.21831/pg.v15i1.34606\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDorimana A, Uworwabayeho A, Nizeyimana G. Enhancing upper secondary learners\u0026rsquo; problem-solving abilities using problem-based learning in mathematics. Int J Learn Teach Educational Res. 2022;21(8):235\u0026ndash;52. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.26803/ijlter.21.8.14\u003c/span\u003e\u003cspan address=\"10.26803/ijlter.21.8.14\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSamura AO, Darhim. Improving mathematics critical thinking skills of junior high school students using blended learning model (BLM) in GeoGebra Assisted Mathematics Learning. Int J Interact Mob Technol. 2023;17(2):101\u0026ndash;17. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3991/ijim.v17i02.36097\u003c/span\u003e\u003cspan address=\"10.3991/ijim.v17i02.36097\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eStephens M, Buteau C. Introduction to the special issue on Computational thinking and mathematics teaching and learning. J Pedagogical Res. 2023. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.33902/jpr.202313362\u003c/span\u003e\u003cspan address=\"10.33902/jpr.202313362\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEnnis RH. The nature of critical thinking: An outline of critical thinking dispositions and abilities. Univ Ill. 2011;2(4):1\u0026ndash;8.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eFacione PA. (2020). \u003cem\u003eCritical Thinking: What It Is and Why It Counts\u003c/em\u003e. Insight Assessment. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://www.insightassessment.com/wp-content/uploads/ia/pdf/whatwhy.pdf\u003c/span\u003e\u003cspan address=\"https://www.insightassessment.com/wp-content/uploads/ia/pdf/whatwhy.pdf\" targettype=\"URL\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eStohlmann M, Yang Y. Growth mindset in high school mathematics: A review of the literature since 2007. J Pedagogical Res. 2024. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.33902/jpr.202424437\u003c/span\u003e\u003cspan address=\"10.33902/jpr.202424437\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePopova Y, Abdualiyeva M, Torebek Y, Yelshibekov N, Omashova G. Improving the effectiveness of senior graders\u0026rsquo; education based on the development of mathematical intuition and logic. Front Educ. 2022;7:986093. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3389/feduc.2022.986093\u003c/span\u003e\u003cspan address=\"10.3389/feduc.2022.986093\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSyafril S, Aini NR, Netriwati, Pahrudin A, Yaumas NE, Engkizar. (2020). Spirit of mathematics critical thinking skills (CTS). \u003cem\u003eJournal of Physics: Conference Series\u003c/em\u003e, \u003cem\u003e1467\u003c/em\u003e(1), 012069. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1088/1742-6596/1467/1/012069\u003c/span\u003e\u003cspan address=\"10.1088/1742-6596/1467/1/012069\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRodr\u0026iacute;guez-S\u0026aacute;nchez MM, S\u0026aacute;nchez-Garc\u0026iacute;a AB, L\u0026oacute;pez-Fern\u0026aacute;ndez R. Subtraction: More than an algorithm? Sustainability. 2020;12(21):9148. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3390/su12219148\u003c/span\u003e\u003cspan address=\"10.3390/su12219148\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eVlassis J, Demonty I. The role of algebraic thinking in dealing with negative numbers. ZDM \u0026ndash; Math Educ. 2022;54(6):1243\u0026ndash;55. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1007/s11858-022-01402-1\u003c/span\u003e\u003cspan address=\"10.1007/s11858-022-01402-1\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eAmir MF, Wardana MDK, Zannah M, Rudyanto HE, Nawafilah NQ. Capturing strategies and difficulties in solving negative integers: A case study of instrumental understanding. Acta Sci. 2022;24(2):64\u0026ndash;87. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.17648/acta.scientiae.6432\u003c/span\u003e\u003cspan address=\"10.17648/acta.scientiae.6432\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBrahmia SW, Olsho A, Smith TI, Boudreaux A. Framework for the natures of negativity in introductory physics. Phys Rev Phys Educ Res. 2020;16(1):010120. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1103/PhysRevPhysEducRes.16.010120\u003c/span\u003e\u003cspan address=\"10.1103/PhysRevPhysEducRes.16.010120\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKurniawan H, Purwoko RY, Setiana DS. Integrating cultural artifacts from remote regions in developing mathematics lesson plans to enhance mathematical literacy. J Pedagogical Res. 2023;7(1):1\u0026ndash;15. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.33902/JPR.202423016\u003c/span\u003e\u003cspan address=\"10.33902/JPR.202423016\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDarmaji D, Kurniawan DA, Astalini A, Perdana R, Kuswanto K, Ikhlas M. Do science process skills affect critical thinking? Differences in urban and rural contexts. Int J Evaluation Res Educ. 2020;9(4):874\u0026ndash;84. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.11591/ijere.v9i4.20687\u003c/span\u003e\u003cspan address=\"10.11591/ijere.v9i4.20687\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eTanti T, Kurniawan DA, Kuswanto K, Utami W, Wardhana I. Science process skills and critical thinking in science: Urban and rural disparity. Jurnal Pendidikan IPA Indonesia. 2020;9(4):489\u0026ndash;98. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.15294/jpii.v9i4.24139\u003c/span\u003e\u003cspan address=\"10.15294/jpii.v9i4.24139\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eWulandari DU, Mariana N, Wiryanto W, Amien MS. Integration of ethnomathematics teaching materials in mathematics learning in elementary school. IJORER: Int J Recent Educational Res. 2024;5(1):204\u0026ndash;18. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.46245/ijorer.v5i1.542\u003c/span\u003e\u003cspan address=\"10.46245/ijorer.v5i1.542\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGallardo-Estrada C, Nussbaum M, Pinto M, Alvares D, Alario-Hoyos C. Enhancing grit and critical thinking in rural primary students: Impact of a targeted educational intervention. Educ Sci. 2024;14(9):1009. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.3390/educsci14091009\u003c/span\u003e\u003cspan address=\"10.3390/educsci14091009\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSusanta A, Susanto E, Rusnilawati R, Sumardi H, Ali SRB. Literacy skills through the use of digital STEAM-inquiry learning modules: A comparative study of urban and rural elementary schools in Indonesia. Eurasia J Math Sci Technol Educ. 2025;21(4):em2615. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.29333/ejmste/16170\u003c/span\u003e\u003cspan address=\"10.29333/ejmste/16170\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003ePalinussa AL, Molle JS, Gaspersz M. Realistic mathematics education: Mathematical reasoning and communication skills in rural contexts. Int J Evaluation Res Educ. 2021;10(2):522\u0026ndash;31. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.11591/ijere.v10i2.20640\u003c/span\u003e\u003cspan address=\"10.11591/ijere.v10i2.20640\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHendriana H, Rohaeti EE, Sumarmo U. Hard Skill dan Soft Skill Matematik Siswa. PT Refika Aditama; 2017.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSusilo BE, Darhim D, Prabawanto S. (2020). Critical thinking skills in integral calculus lecture based on mathematical dispositions. \u003cem\u003eJournal of Physics: Conference Series\u003c/em\u003e, \u003cem\u003e1521\u003c/em\u003e(3), 032045. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1088/1742-6596/1521/3/032045\u003c/span\u003e\u003cspan address=\"10.1088/1742-6596/1521/3/032045\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eUddin MR, Shimizu K, Sharmin H, Widiyatmoko A. (2023). Comparing critical thinking skills between rural and urban students at secondary level education. \u003cem\u003eAIP Conference Proceedings, 2601\u003c/em\u003e, 020035. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1063/5.0127674\u003c/span\u003e\u003cspan address=\"10.1063/5.0127674\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSuparman S, Juandi D, Martadiputra BAP. Students\u0026rsquo; heterogeneous mathematical critical thinking skills in problem-based learning: A meta-analysis investigating the involvement of school geographical location. Al-Jabar: Jurnal Pendidikan Matematika. 2023;14(1):37\u0026ndash;53. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.24042/ajpm.v14i1.16200\u003c/span\u003e\u003cspan address=\"10.24042/ajpm.v14i1.16200\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eIbrahim NN, Ayub AFM, Yunus ASM. Impact of higher order thinking skills (HOTS) module based on the cognitive apprenticeship model on students\u0026rsquo; performance. Int J Learn Teach Educational Res. 2020;19(7):246\u0026ndash;62. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.26803/ijlter.19.7.14\u003c/span\u003e\u003cspan address=\"10.26803/ijlter.19.7.14\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e.\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSuparman S, Juandi D, Tamur M. (2021). Does problem-based learning enhance students\u0026rsquo; higher order thinking skills in mathematics learning? A systematic review and meta-analysis. \u003cem\u003e2021 4th International Conference on Big Data and Education\u003c/em\u003e, 44\u0026ndash;51. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1145/3451400.3451408\u003c/span\u003e\u003cspan address=\"10.1145/3451400.3451408\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"discover-education","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"diedu","sideBox":"Learn more about [Discover Education](https://www.springer.com/journal/44217)","snPcode":"44217","submissionUrl":"https://submission.nature.com/new-submission/44217/3","title":"Discover Education","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Facione critical thinking framework, integer multiplication, junior high school students, mathematical critical thinking skills, remote area education","lastPublishedDoi":"10.21203/rs.3.rs-9065392/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9065392/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eMathematical critical thinking skills are essential competencies that must be developed in students, particularly when addressing learning challenges in remote areas. This study aims to analyze the mathematical critical thinking skills of junior high school students in remote areas on the topic of integer multiplication using the Facione framework. A qualitative descriptive approach was employed, involving twenty-four eighth-grade students from a public junior high school in Lembata Regency as research participants. The study was conducted during the odd semester of the 2025/2026 academic year. Data were collected through a critical thinking test consisting of four essay questions and semi-structured interviews. Data analysis was based on four Facione indicators: interpretation, analysis, evaluation, and inference. The results indicate that students\u0026rsquo; mathematical critical thinking skills were in the low category, with an average score of 49.74%. Specifically, the interpretation aspect reached 55.2%, analysis 48.96%, evaluation 41%, and inference 54.2%. The findings revealed that students experienced difficulties in analyzing conceptual relationships, evaluating arguments, and drawing logical conclusions. These results have important implications for the development of learning strategies that are responsive to the characteristics and needs of students in remote areas.\u003c/p\u003e","manuscriptTitle":"Analysis of Middle School Students’ Mathematical Critical Thinking Skills in Remote Areas on Integer Multiplication","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-27 19:14:16","doi":"10.21203/rs.3.rs-9065392/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-05-05T11:48:59+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-04T06:47:20+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-03T13:06:27+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-01T02:04:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"184200203984499911830183417973214801516","date":"2026-04-26T06:23:39+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"281185653693203490580224678337672546566","date":"2026-04-25T01:34:56+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"257770286315437829405278260055978586009","date":"2026-04-24T23:53:31+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-03-25T12:22:48+00:00","index":"","fulltext":""},{"type":"editorInvited","content":"","date":"2026-03-24T09:59:19+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-16T10:12:50+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-14T12:57:05+00:00","index":"","fulltext":""},{"type":"submitted","content":"Discover Education","date":"2026-03-14T12:50:45+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"discover-education","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"diedu","sideBox":"Learn more about [Discover Education](https://www.springer.com/journal/44217)","snPcode":"44217","submissionUrl":"https://submission.nature.com/new-submission/44217/3","title":"Discover Education","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Discover Series","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"454ad64a-38f0-4996-ab8a-d4461920875e","owner":[],"postedDate":"March 27th, 2026","published":true,"recentEditorialEvents":[{"type":"decision","content":"Revision requested","date":"2026-05-05T11:48:59+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-04T06:47:20+00:00","index":50,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-03T13:06:27+00:00","index":49,"fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-05-01T02:04:39+00:00","index":48,"fulltext":""}],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-18T13:23:15+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-27 19:14:16","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9065392","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9065392","identity":"rs-9065392","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}