Abstract

General Background: Critical thinking is foundational in mathematics education, especially in probabilistic reasoning, which underpins decision-making under uncertainty. Specific Background: Despite its relevance, probability theory is often taught procedurally rather than conceptually, limiting students' reasoning skills. Knowledge Gap: Existing studies rarely integrate cognitive pedagogy with probability instruction, particularly in under-researched educational contexts like Uzbekistan. Aims: This study aims to evaluate the impact of a pedagogical model—incorporating inquiry, paradox analysis, and reflection—on developing tenth-grade students’ critical thinking in probability. Results: Using a quasi-experimental design, the intervention group significantly outperformed the control group in post-test scores (83.1 vs. 70.2), with a large effect size (Cohen’s d = 1.12). Improvements were particularly strong in logical inference, assumption evaluation, and problem analysis. Novelty: This research uniquely applies metacognitive strategies to probabilistic education within a context lacking prior integrative frameworks. Implications: Findings suggest that embedding critical thinking in probability instruction not only enhances cognitive skills but also supports broader curriculum reform and teacher development. The model holds promise for adaptation across diverse educational settings, contributing to a deeper, more transferable understanding of mathematics.
HIghlight : 

• Significant Skill Gains – Students in the experimental group improved critical thinking scores by over 20 points, nearly triple the control group’s gains.

• Active Learning Methods – Paradox analysis, reflection, and inquiry-based tasks fostered deeper reasoning and better problem-solving in probability.

• Educational Reform Implications – The study supports shifting from rote teaching to metacognitive, student-centered strategies in math instruction.

Keywords : Critical Thinking, Probability Theory, Inquiry Learning, Problem Solving, Metacognitive Strategy

Introduction

With uncertainty, data and risk all around us, it’s never been more important to think logically and make good choices. Using probability theory in mathematics, students learn to measure uncertain events and make smart judgments about them. Still, being good at probability means using logic, noticing patterns and reviewing data, all vital parts of critical thinking. Although the subject is important, research and practical experience reveal that students frequently do not use reasoning well in probabilistic situations but rely on their intuition or learned rules. According to Ennis, critical thinking which involves using reason to make important decisions, is an important aim in education. Teaching mathematics and probability to students means encouraging them to challenge ideas, break down situations, consider various responses and explain why they think certain things. This view is supported by cognitive science because experts such as Kahneman and Gigerenzer (2002) indicate that biases, heuristics and a large amount of information often influence how probabilistic reasoning is processed. This evidence seems to be ignored, as instruction in probability generally involves explaining procedures and guiding students instead of pushing them to use their minds [1], [2]. In recent years, Watson and Moritz, Garfield and Ben-Zvi and various others have started considering how inquiry, reflection and related situations can be used in probability education. Yet, literature on the subject has not yet given us a detailed framework for linking cognitive thinking with classroom practice to help students deal with probabilistic questions. Few studies have looked at learning systems in countries where theories introduced in the 20th century have not yet been fully formed. Without a better understanding, the field struggles to create interventions suitable for everyone and easy to scale [3], [4]. The main idea of this study is to meet that gap by outlining and testing a model using collaborative learning, looking at paradoxes and planned metacognitive activities. This study applies a quasi-experimental design and compares critical thinking enhanced instruction with the use of typical instruction in tenth-grade classrooms. Critical thinking skills such as inference, noticing assumptions and the evaluation of arguments were studied using a preschool test and a post-test. The depth and nature of how students think were studied by watching classes and talking with them in person [5], [6]. It is expected that students taught in this manner will make major improvements in probability skills and also in reasoning about situations that are uncertain. Initial research shows that when students think about paradoxes and face open-ended activities, they develop better reasoning and become more flexible with problem-solving. Should it be validated, this method might guide changes in both the math curriculum and professional development for teachers. The goal is to help expand the research base that suggests mathematics education needs to build critical thinking, not only teach basic calculations [7], [8], [9].

Methodology

A quasi-experimental design with mixed methods was used to assess whether a teaching model helped students develop critical thinking in solving problems about probability theory. During eight weeks, the research was done in two public high schools in Samarkand, with 60 of the students in the 10th grade aged 15 to 16. Everyone taking part was distributed between an experimental group of 30 students and a control group of 30 students. The members of the control group were taught probability traditions by following commonly used exercises from the national curriculum. In comparison, the experimental group experienced learning through inquiry-based activities, discussion groups, reflective writing, group problem-solving and probabilistic puzzles such as the Monty Hall question and the Birthday riddle. Instructors broke down concepts by having students explore more complex questions and to solve problems from the real world. Students’ progress was checked by giving both groups a pre-test and post-test using a rubric from Ennis’ framework which looks at inference, assumption evaluation and logical reasoning. Paired sample t-tests were used to check for significance differences among groups and effect size measures were used to understand the practical effect of the intervention. As well, information was obtained through interviews and from classroom observation which was arranged thematically to highlight changes in thought and behaviors linked to critical thinking. Using both kinds of data allowed us to see clearly how the instructional model developed students’ probabilistic reasoning and analysis skills.

Results and Discussion

This study found a major difference in how much students’ critical thinking skills developed after using the cognitive-pedagogical model. After analysis, we observed that the experimental group gained from 62.5 to 83.1, but the control group’s progress was smaller at 63.0 to 70.2. The p-value from the paired t-test was less than 0.01 which means that the noticed difference was not random [10], [11]. In addition, a very strong impact on students’ ability to analyze problem situations was shown by the very large effect size (Cohen’s d = 1.12). People could see that students improved the most in weighting assumptions, justifying how they solved problems and using logic with uncertain questions. Unlike the other group, the control group grew mainly by mastering the steps of the activity and did not develop analytical skills. The difference in critical thinking scores between pretest and posttest for probability problems is shown in Table 1, separated for control and experimental groups.

Here is the table displaying the pre-test and post-test mean scores along with the score gain for both the control and experimental groups. The accompanying bar chart visually illustrates the difference in score improvement between the two groups, highlighting the significantly higher gain achieved by the experimental group.

Figure 1.

The conclusions were supported by observations from the groups. The codes for students’ interviews and observations in the classroom pointed to an increase in engagement, a move to more rational reasoning and the practice of metacognitive learning. Learners in the experimental group became more curious, confidently debated with their peers during work sessions and clearly described various probabilities. Furthermore, students who at first only relied on easy interpretations of probability problems started to find connections between conditions and address confusing cases by using clear techniques. The findings demonstrate that adding critical thinking aspects to probability teaching helps students develop a deeper grasp of problems and become mentally more flexible in solving unknown or unusual problems [12], 13]. The research supports the importance of learning theories such as constructivist and metacognitive, that promote focusing on students when helping them become more thoughtful. It follows the work of Piaget and Vygotsky, who considered both mental disagreements and interaction with others important for gaining knowledge. Practically, the results point out that teaching only the standard facts in mathematics may not be enough, particularly in probability, since this area tends to promote wrong and erroneous ways of thinking. It is clear from paradox-based learning and reflective journaling that teacher training should have sections on sparking classroom talks and supporting noteworthy struggle among students [14], [15]. Yet, it has become clear that this research points to areas that still need research. The first point is that this study demonstrates quick advances in critical thinking, yet more research over time is important to find out if these gains last. In addition, research should analyze how students’ styles of thinking and awareness of their mental processes change as they handle different kinds of probabilistic tasks. Additionally, considering various learning settings and students of different ages could increase the usefulness of the findings to everyone. Even more work on underlying theories is required to fine-tune the taxonomy for critical thinking in mathematical subjects, because existing frameworks might not cover all the important features of probabilistic cognition. In conclusion, this study provides compelling evidence that a well-designed instructional model centered on critical thinking significantly enhances students’ problem-solving skills in probability theory. By bridging theoretical insights with classroom practice, the research opens new avenues for pedagogical reform and lays the groundwork for more rigorous studies in the cognitive development of mathematical reasoning.

Conclusion

According to the study, teaching probability theory with a critical thinking focus greatly boosts students’ ability to analyze and solve problems. Students in the experimental group, involved in inquiry-based tasks, focused on paradoxes and encouraged to reflect, did much better and showed more convincing ways of reasoning, more understanding of their own thinking and more interest in probabilities when compared to students in the control group. Based on these research findings, it is important to switch teaching methods from simple steps to richer learning settings that urge students to make smart and honest judgements. Both constructivist and metacognitive learning are shown to be relevant theory, according to research, especially in math fields where uncertainty and abstract concepts are common. At the same time, the study reports on unanswered questions, including if critical thinking progress is kept over time, how diverse tasks may affect reasoning and how a finer version of a taxonomy for critical thinking in mathematics can be created. Researchers should go on to test how these pedagogical strategies work on a bigger scale and in various educational environments, as well as how they might be enforced with younger students, in different sociocultural environments and across various subjects. In general, the findings emphasize that using critical thinking in math classes is important for having smart problem solvers and thoughtful thinkers prepared for a complex and ever-changing world.

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