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].

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.

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.