DEVELOPING STUDENTS' COGNITIVE ACTIVITY THROUGH THE CONSTRUCTION OF MATHEMATICAL MODELS OF PROBLEMS

SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 66 DEVELOPING STUDENTS’ COGNITIVE ACTIVITY THROUGH THE CONSTRUCTION OF MATHEMATICAL MODELS OF PROBLEMS N.B. Shamsiddinov1, B.S. Quvonov2 Mathematics Instructor, Academic Lyceum of the Tashkent State Technical University, PhD (Physics and Mathematics)1 Mathematics Instructor, Academic Lyceum of the Tashkent State Technical University, PhD (Technical Sciences)2 https://doi.org/10.5281/zenodo.17936352 Abstract. This article examines the development of students’ intellectual activity through the construction of mathematical models of problems. The concept of a mathematical model, its didactic significance, and its role in developing logical, analytical, and creative thinking skills are analyzed. The study highlights the effectiveness of mathematical modeling in the educational process for fostering independent thinking, problem formalization, and reasoned decision-making in solving mathematical problems. Keywords: mathematical model, mathematical modeling, intellectual activity, logical thinking, problem-based learning, analysis and synthesis. Introduction. It is known that the modern education system requires that students be developed not as recipients of ready-made knowledge, but as individuals capable of independently analyzing a problem, drawing conclusions, and applying them in practice. From this perspective, constructing a mathematical model of a problem in mathematics is one of the most effective means of developing students' mental activity. Mathematical modeling allows students to analyze and solve real-world problems, expressing them in mathematical language and evaluating the results. This process promotes the development of logical, abstract, and creative thinking in students. 1. The Concept of a Mathematical Model and Mathematical Modeling The mathematical model is an abstract image representing a real object, process, or phenomenon through mathematical concepts, symbols, and relationships. Mathematical modeling is the process of transforming a problem into mathematical form, solving it, and interpreting the results. The modeling process consists of the following stages: 1. Problem Analysis; 2. Identifying Key Variables; 3. Establishing Mathematical Relationships; 4. Finding a Solution Based on the Model; 5. Analysis and Verification of the Result. Let's analyze the modeling process by solving the following problem: Task. The artificial reservoir is rectangular with sides 1 km apart. Two people, standing at one end of this rectangle, set off for the opposite end. One person rows vertically in a boat, while the other walks along the shore. If both people are traveling at a speed of v km/h, and one arrives SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 67 at their destination 30 minutes before the other, find the dimensions of the reservoir (calculate the value for v = 4). Solution. Problem 1. Analysis: Let's draw a schematic representation of the problem—an image of the reservoir as a rectangle. Two passengers depart from corner A to corner C of the rectangle. The first passenger walks along diagonal AC, and the second along sides AB and BC. We obtain the following model of the problem (Figure 1). 2. Determine the key variables: Given: ABCD is a rectangle; AB =BC+1; The first passenger walked for t hours; the second for t + 0.5 hours; We need to find AB and BC. For solving the problem, we introduce the following notations: BC = 𝑥 and AB = 𝑥 + 1the distance traveled by the first passengerAC = √𝑥2+(𝑥 +1)2=vt. The path taken by the second passenger AB+BC = 2𝑥 + 1 = 𝑣(𝑡+0,5) 2𝑥 +1 = vt+𝑣 2. 3-Establish mathematical connections: We find the distance from the expression for the path traveled by both passengers and equate them: √2𝑥2+2𝑥 +1 = 2𝑥 +1 −𝑣 2. We solve the resulting equation to find the unknown𝑥 = 𝑓(𝑣) Finding a solution based on model 4: Square both sides of the equation to obtain the following quadratic equation: √2𝑥2+2𝑥 +1 = 2𝑥 +1 −𝑣 2 2𝑥2+(2−2𝑣)𝑥 +𝑣2 4−𝑣 = 0 . Quadratic equation 𝑥 = −(2−2𝑣)+√2𝑣2+4 4 to determine the solution. So, the sides of the water body BC =2𝑣−2+√2𝑣2+4 4 and AB =2𝑣−2+√2𝑣2+4 4+1 = 2𝑣+2+√2𝑣2+4 4 is expressed by the formula. Analysis and verification of result 5: Given that the passengers have speeds v=4, the sides of the rectangle are BC =2⋅4−2+√2⋅16+4 4=12 4= 3 and AB = 4 equal. All these stages activate students' mental activity. 2. The concept of mental activity and its components B A C D 1Figure x x+1 SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 68 Mental activity refers to the human intellectual processes: analysis, synthesis, comparison, generalization, imagination, and drawing conclusions. In the process of mathematical modeling, these types of activity manifest themselves as a single mechanism. For example: • Problem analysis – correct understanding of the conditions; ABCD – a rectangular rectangle; AB =BC+1 ; The first passenger traveled t hours; the second t + 0.5 hours. Find AB and BC. • Synthesis – combining elements into a single model; By finding and comparing the paths traveled by two passengers, construct the following mathematical model √2𝑥2+ 2𝑥 +1 = 2𝑥 +1 −𝑣 2. • generalization — applying the solution to other situations. Knowing that the solution obtained can be calculated for other values of speed v. Because the sides of a rectangle depend on speed v: BC =2𝑣−2+√2𝑣2+4 4 and 2 2 2 2 4 4 vv AB     . 3. Developing logical thinking through constructing a mathematical model While constructing a model, students identify cause-and-effect relationships. This leads to the development of logical thinking. For example, translating a word problem into an algebraic equation or system requires logical thinking. In non-standard problems, students are forced to compare several model options and select the most appropriate one. This process develops critical and analytical thinking. 4. Developing creative thinking through mathematical modeling A mathematical model is not always unique. A single problem can be represented by several different models. This situation develops students' creativity. The student: independently selects a model; tests their hypothesis; applies existing knowledge to a new situation. This develops a high level of mental activity. Task 2. A circle and a rectangle made of wire are arranged so that the circle passes through two ends of the rectangle and points in a direction not passing through these ends. If the diameter of a circle is 2R, and the perimeter of a rectangle is three times the diameter, find the sides of the rectangle. Solution: Let's represent the problem schematically: a circle and a rectangle, as specified. Let's assume that vertices A and B of the rectangle lie on the circle, and side CD touches the circle. O is the center of the circle, and AO = R. We obtain the following model of the problem (Fig. 2). Given: R is a circle with radius ; ABCD is a rectangle; AO = 𝑅; 𝑃 = 3⋅2𝑅; We need to find AB and AD. SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 69 For solving the problem, we draw a diameter EF perpendicular to the side AB. Since the perpendicular to the diameter is bisected at the point of intersection, AN=NB. As a result of these considerations, it is determined from the given drawing that AON is a right-angled triangle, and we apply the Pythagorean theorem to this triangle: AO2=AN2+ON2. (1) Here AO = 𝑅; AN =AB 2 and ON =AD−AO =AD−𝑅 put the expressions in 𝑅2= (AB 2)2+(AD−𝑅)2. From this AB2 4+AD2−2⋅AD⋅𝑅 = 0 (2) is formed. Since the perimeter of a rectangle is 3 times greater than the circumference of a circle, according to the problem statement: 𝑃 = 2(AB+AD)= 6𝑅. From here, we find AB = 3𝑅−AD and substitute into (2): (3𝑅−AD)2 4+AD2−2⋅AD⋅𝑅 = 0 9𝑅2−6⋅𝑅⋅AD−AD2 4+AD2−2⋅AD⋅𝑅 = 0 9𝑅2−14⋅ 𝑅⋅AD+5AD2= 0 we form a quadratic equation. From using our existing knowledge of algebra, we solve the equation in this situation. Since the equation is a single equation in two variables, we solve the equation with respect to the parameterized unknown: AD = 𝑅 and AD =9 5𝑅 get the result. It follows AB = 2𝑅 and AB =6 5𝑅 5. The Practical Importance of Mathematical Modeling In problem-based education, knowledge is not provided in a ready-made form. Mathematical modeling is one of the key tools of problem-based education, encouraging students to actively explore. During the lesson, posing a problem based on a real-life situation, constructing a mathematical model based on it, and searching for a solution encourages students to think actively. Mathematical modeling prepares students to solve everyday problems, including economic issues, technical processes, and environmental and social problems. This allows students to connect mathematics with real life and enhance their interest in the subject. Conclusion A B C D O N 2pic E F SCIENCE AND INNOVATION INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 4 ISSUE 12 DECEMBER 2025 ISSN: 2181-3337 | SCIENTISTS.UZ 70 Finally, it is supposed that constructing a mathematical model of a problem is an important pedagogical tool for developing students' mental activity. With the help of mathematical modeling, students develop logical, analytical, and creative thinking, independent decision-making skills, and the practical application of knowledge. Therefore, it is advisable to devote special attention to mathematical modeling in mathematics education. REFERENCES 1. Fields G. How to solve it. – Princeton University Press, 2004. 2. Abdukodirov A. A. Modern Pedagogical Technologies. – Tashkent, 2019. 3. Zokirov B. Methods of Teaching Mathematics. – Tashkent, 2017. 4. Lesch R., Doerr H. Beyond Constructivism: Models and Prospects of Modeling. – Springer, 2003. 5. Khasanov B. Fundamentals of Mathematical Modeling. – Tashkent, 2021. 6. Khusanov D. Kh., Shamsiddinov N. B. "The Role of Geometric Problems in the Development of Independent Creative Thinking of Young People". Scientific and Methodological Journal "Public Education". 2020. Issue 5. 7. Khusanov D. Kh., Shamsiddinov N. B. 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