USE OF STUDENT'S BASIC KNOWLEDGE IN TEACHING MATHEMATICAL ANALYSIS IN HIGHER SCHOOLS

ИСПОЛЬЗОВАНИЕ БАЗОВЫХ ЗНАНИЙ СТУДЕНТОВ ПРИ ПРЕПОДАВАНИИ МАТЕМАТИЧЕСКОГО АНАЛИЗА В ВЫСШИХ УЧЕБНЫХ ЗАВЕДЕНИЯХ

Aliyeva Z.

08.02.2025 61

2(128)

02. Теория и методика обучения и воспитания

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Aliyeva Z. USE OF STUDENT'S BASIC KNOWLEDGE IN TEACHING MATHEMATICAL ANALYSIS IN HIGHER SCHOOLS // Universum: психология и образование : электрон. научн. журн. 2025. 2(128). URL: https://7universum.com/ru/psy/archive/item/19294 (дата обращения: 27.03.2025).

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DOI - 10.32743/UniPsy.2025.128.2.19294

ABSTRACT

This article focuses on ensuring continuity and interrelation between the Mathematical Analysis course in undergraduate mathematics teacher training and the general education school mathematics curriculum. The study compares the topics covered in the Mathematical Analysis course at the higher education level with the knowledge and skills that students acquire from the school mathematics curriculum. The primary objective of the research is to identify opportunities for maintaining continuity and connection while teaching Mathematical Analysis at the university level. The research methodology includes an analysis of existing curricula, textbooks, and lecture materials, as well as an experiment conducted through the application of the proposed methodological system. The research hypothesis suggests that if the interrelation between the university-level Mathematical Analysis course and the general school mathematics curriculum is properly structured and the principle of continuity is maintained, students will achieve a higher level of comprehension, the allocated instructional time will be used more efficiently, and future professionals will develop a deeper understanding of mathematical analysis concepts in their careers.

АННОТАЦИЯ

Статья посвящена обеспечению связи и преемственности между курсом «Математический анализ» в программе подготовки учителей математики бакалавриата и общеобразовательной школьной программой по математике. В исследовании сравнивались темы, преподаваемые на курсе математического анализа в университете со знаниями и навыками приобретенными учащимися на уроках математики в средней школе. Целью исследования является определение возможностей наследования и связи при преподавании математического анализа в высшей школе. Методология исследования включает анализ существующих программ, учебников и лекционных материалов, а также эксперимент, проведенный с использованием предложенной методической системы. Гипотеза исследования состоит в том, что при правильном построении преподавания курса математического анализа в высшей школе во взаимодействии с общеобразовательным курсом математики и соблюдении принципа преемственности повышается качество освоения студентами данного предмета, эффективно используется время, отведенное на преподавание, и сегодняшние студенты могут лучше усваивать понятия математического анализа в будущей профессиональной деятельности.

Keywords: mathematical analysis, Continuity

Ключевые слова: математический анализ, наследование.

Introduction

One of the key responsibilities of higher education institutions is to define directions that contribute to shaping a younger generation capable of understanding and addressing both current and future societal challenges. This involves designing and implementing a methodological system aligned with these directions. The teacher plays a central role in fulfilling these objectives. To achieve effective results, educators must, among other criteria, have a comprehensive understanding of the content, scope, and placement of their subject at various levels and stages of education.

A university lecturer teaching Mathematical Analysis cannot determine effective teaching methods and approaches without a thorough understanding of the knowledge, skills, and competencies students have acquired from the general education school curriculum.

The Law on Education of the Republic of Azerbaijan [1] states: "Educational activities at different levels and stages are regulated by relevant legislative acts. The achievements obtained by a learner in the previous stage must be considered in the continuation of education at the next stage". This principle highlights the necessity of ensuring continuity and interrelation in the teaching of Mathematical Analysis at the university level.

Our research has revealed that many professors and lecturers teaching at the undergraduate level in institutions that train mathematics teachers are either unaware of or do not give sufficient importance to the Mathematics Curriculum of general education schools [8]. Instead, they primarily focus on teaching the rigorous theoretical foundations of the subject, assuming that addressing these gaps falls under the domain of the methodology of teaching mathematics. However, this approach not only leads to wasted classroom time but also negatively impacts the professional competence of future mathematics teachers.

Recognizing the significance of ensuring continuity and interrelation between school mathematics and university-level Mathematical Analysis, we have developed a correspondence table to bridge the gap between these two levels of education.

Materials and Methods

The Mathematical Analysis course is a mandatory subject in the undergraduate mathematics teacher training program. This requirement is due to the broad application of mathematical analysis concepts in various fields, including geometry, physics, chemistry, biology, mechanics, computer science, statistics, and economics. Similarly, mathematical analysis concepts play a significant role in the general education school mathematics curriculum.

When designing the learning outcomes of the Mathematical Analysis course in higher education, instructors must first accurately determine the place, scope, and content of these concepts at the preceding educational level. A university lecturer teaching Mathematical Analysis must assess the students' foundational knowledge in this subject, evaluate their prior understanding, and structure the course accordingly. This approach not only prevents time loss but also enhances the quality of learning outcomes through an effective interrelation between educational levels.

During the research process, we analyzed the "Mathematics Curriculum for General Education Schools of the Republic of Azerbaijan" [8], the teaching programs of relevant undergraduate specialties ([2], [3], [4]), as well as university-level textbooks and methodological resources on Mathematical Analysis ([5]-[15]). We conducted surveys and interviews with several university instructors teaching Mathematical Analysis. Additionally, we carried out a questionnaire-based assessment to evaluate the initial mathematical analysis proficiency of first-year undergraduate students enrolled in the Mathematics Teaching program.

Based on these analyses, we formulated the following hypotheses:

If the principle of continuity is maintained between the general education school mathematics curriculum and the Mathematical Analysis course in undergraduate teacher training, the allocated teaching time will be used more efficiently.
Ensuring this continuity will facilitate better mastery of mathematical analysis concepts, enabling future teachers to more effectively convey these concepts to students in general education schools.

Throughout the study, we examined various aspects of the general education mathematics curriculum, including its content framework, learning outcomes, and algebra and functions components. We also analyzed content standards related to mathematical analysis concepts, the expected learning outcomes for students at different grade levels, and how these concepts are introduced in school textbooks through definitions, theorems, and explanatory notes. Based on these findings, we developed a methodological guide titled "Ensuring Continuity in the Teaching of the Mathematical Analysis Course" ([5]).

The proposed methodological system was discussed with university instructors teaching Mathematical Analysis, and explanatory sessions were conducted to facilitate its implementation. The system was then tested on first-year Mathematics Teaching students at Nakhchivan Teachers’ Institute and Nakhchivan State University. The results of the experiment were analyzed, and conclusions were drawn accordingly.

Table 1.

The topics covered in the first-year university Mathematical Analysis course were aligned with the knowledge and skills acquired by students from the general education school mathematics curriculum to ensure continuity and a smooth transition between educational levels

P.№	Mathematical Analysis Course Topics for the 1st Year of the Mathematics Teaching Specialty	The Learning Outcomes Acquired by the Student from the General Education School Mathematics Curriculum
1	2	3
1.	The Concept of Sets and Operations on Them Plan: Fundamental logical symbols, mathematical language. Mathematical induction method. The concept of sets and operations on them. De Morgan's laws. Cartesian product of sets. The concept of relations.	*Learning Outcomes Achieved by the Student in Grade VI [11]:* 1.Performs operations on sets. Finds the difference, union, and intersection of two finite sets. 2.Determines whether there is direct proportionality between the coordinates of given pairs in the set of integers. 3.Expresses direct and inverse proportional dependencies in the form of a function *Learning Outcomes Achieved by the Student in Grade VII [12]:* 1.Applies the properties of operations on sets. 2.Determines whether there is linear dependence between the coordinates of given pairs in the set of rational numbers. 3.Expresses the dependencies between quantities encountered in daily life using functions. 4Expresses the relationship between distance and time in uniform rectilinear motion, as well as the relationship between temperature measured in Celsius and Fahrenheit, as a linear function. *Learning Outcomes Achieved by the Student in Grade VIII [13]:* 1.Determines whether there is quadratic dependence between the coordinates of given pairs in the set of real numbers. 2.Expresses the dependencies between quantities encountered in daily life using functions. 3.Expresses the relationship between distance and time in uniform rectilinear motion, as well as the relationship between temperature measured in Celsius and Fahrenheit, as a linear function. *Learning outcomes achieved by the student in Grade IX [14]:* 1.Expresses the dependencies between quantities encountered in daily life using functions. The student can handle quadratic functions and their graphs, present quadratic functions in various forms, y = a(x- m)² + n the intersection points of the parabola with the x-axis, y = ax²+ bx + c the graph of the function, y = ax²+ bx + c the analysis of the function, solve problems using quadratic functions, and construct and graph y = \|x\| functions and y= x³ functions. *Learning outcomes achieved by the student in Grade X [15]:* 1.Knows the concept of a function, builds mathematical models for real-life problems, and solves these problems with the help of the properties of functions. 2.Knows the definition and methods of representing numerical functions, understands the concepts of their domain and range. 3.Knows the concept of the graph of a function, investigates its periodicity, oddness, evenness, and monotonicity, and can transform graphs. 4.Knows the concepts of composite functions and inverse functions, and finds the inverse functions of some functions. 5.Recognizes the basic trigonometric functions and their inverse functions, and constructs their graphs. 6.Knows the definition and properties of power functions, and constructs their graphs. 7.Knows the definition and properties of exponential functions, and constructs their graphs. 8.Knows the definition and properties of logarithmic functions, and constructs their graphs. Learning outcomes achieved by the student in Grade XI [16]: 1.Applies mathematical operations and procedures, and determines the relationships between them. 2.Knows the definition of a numerical sequence and its limit, and applies the properties of convergent sequences. 3.Knows the concept of a function's limit, its properties, and notable limits, and uses them to calculate the limits of functions. 4.Knows the concept of continuity of functions and applies the basic properties of continuous functions. 5.Uses the derivative of a function to find its stationary points and checks whether these points are extremum points. 6.Applies differential calculus to analyze functions and construct their graphs. 7.Knows the concept of an indefinite integral, and uses the integral table for elementary functions and integration rules to compute the integrals of functions. 8.Knows the definition of a definite integral and applies the Newton-Leibniz formula. 9.Uses definite integrals to calculate the area of a curvilinear trapezoid. 10.Uses definite integrals to calculate the volume of solids of revolution. 11.Uses the properties of evenness, oddness, and periodicity of functions in the efficient calculation of definite integrals. 12.Uses definite integrals to find the area of a curvilinear trapezoid and other plane figures. Top of Form Bottom of Form
2.	Reflection, Simple Classification of Reflections Plan: Definition of reflection, domain of definition, range of values. The image and preimage of a set. Surjective, injective, and bijective reflections. Composition of reflections.
3.	The Set of Real Numbers Plan: The set of rational and irrational numbers. The set of real numbers, its ordering, and continuity. The concept of a cut. The boundary of a numerical set.
4.	Single-variable Functions Plan: Definition of a function. Composite functions. Operations on functions. Numerical sequences. The modulus of a real number. Methods of defining a function. Graph of a function. Bounded and unbounded functions. Monotonic functions. Odd and even functions. Periodic functions. Inverse functions. Inverse trigonometric functions. Functions given in parametric form.
5.	Limit of a Function Plan: Concept of the neighborhood of a real number (point). Limit point of a set. Interior, exterior, and boundary points of a set. The limit of a function at a point and its geometric interpretation.
6.	Limit over a Set. Limit of a Composite Function. Finite and Infinite Limits. Plan: Infinite limits and their geometric meanings. Limit of a numerical sequence. Some important limits. One-sided limits. Limit of a composite function.
7.	Properties of Functions with Limits Related to Inequalities. Plan: Properties of limits related to inequalities based on the concept of a neighborhood. Theorem on the limit of an intermediate variable.
8.	Theorem on Finite Limits. Plan: The boundedness of finite limits. The limit of a constant. The limit of a sum, product, and quotient. The limit of a function that decreases without bound. Inverse theorems.
9.	Theorem on Infinite Limits. Plan: The unboundedness of infinite limits. Finite and infinite functions and their limits.
10.	Limit of a Monotonic Bounded Function. The number "e". Plan: The limit of a monotonic function. Finite and infinite limits of numerical sequences. The number "e". The principle of nested intervals.
11.	Continuous Function at a Point. Points of Discontinuity. Plan: Argument and function growth. Continuous function at a point. Discontinuous functions. Points of discontinuity and their types. Operations on continuous functions.
12.	Local Properties of Continuous Functions at a Point Plan: Boundedness of a continuous function. Continuity over a set. One-sided continuity.
13.	Global Properties of Continuous Functions on an Interval Plan: Cauchy’s Theorem. Weierstrass Theorem. Uniformly continuous functions. Cantor’s Theorem. Bolzano-Cauchy Theorem.
14.	Existence and Properties of the Inverse Function. Parametrically Defined Function Plan: Necessary and sufficient condition for the existence of an inverse function. The exact lower and upper bounds of the function’s range. Parametrically defined function.
15.	Existence and Properties of Power, Exponential, Logarithmic, Trigonometric, and Hyperbolic Functions Plan: Existence and properties of power functions. Existence, continuity, and properties of exponential and logarithmic functions.Hyperbolic functions. Inverse hyperbolic functions.
16.	Derivative Plan: Physical and geometric problems related to the concept of a derivative, definition of the derivative. Derivatives of elementary functions. Derivatives of inverse, inverse trigonometric, and hyperbolic functions. Operations on differentiable functions.
17.	Higher-Order Derivatives Plan: Continuity of functions with finite derivatives. Derivative of composite functions. Formula for the increase of a function. Derivative of exponential functions. Derivative of power functions. One-sided and infinite derivatives. Derivative of a function given in parametric form. Approximation methods. Higher-order derivatives and some general formulas for them. Leibniz's rule.
18.	Differential and Higher-Order Differentials Plan: Differentiable functions. Operations on differentials. Differential table. Relationship between derivatives and differentials. Invariance property of differential forms. Higher-order differentials.
19.	Fundamental Theorems of Differential Calculus Plan: Fermat's Theorem. Rolle's Theorem. Lagrange's Theorem. Cauchy's Theorem.
20.	Taylor's Theorem Plan: Taylor's Theorem for Multivariable Functions. Maclaurin's Theorem. Taylor's Theorem for an Arbitrary Function. Taylor's Theorems for Some Specific Functions.
21.	Study of Funcations Using Derivatives Plan: Stability and monotonicity criteria of the function. Extrema. Necessary and sufficient conditions. Concavity and convexity. Necessary and sufficient conditions. The maximum and minimum values of a function on a segment. Asymptotes.
22.	L'Hôpital's Rules Plan: Resolution of indeterminate forms using differential calculus. L'Hôpital's rules.
23.	Indefinite Integral Plan: Primitive function, integrable function, indefinite integral and its basic properties. Integral table. Basic methods for calculating indefinite integrals. Substitution method and integration by parts.
24.	Integration of Simple Rational Fractions Plan: Functions that can be integrated in a finite form. Recurrence formula.
25.	Integration of Rational and Irrational Functions Plan: Integration of proper fractions. Integration of rational functions using the method of undetermined coefficients. Ostrogradsky formula. Integration of some functions containing radicals. Euler substitutions. Integration of trigonometric functions.
26.	Definition of the Definite Integral Plan: Geometric problem leading to the definite integral. Definition of the definite integral. Darboux sums. Properties of the definite integral. Existence conditions of the definite integral.
27.	Main Properties of Integrable Functions Plan: Certain classes of integrable functions. Properties of the definite integral related to equality. Property of the definite integral related to inequality. Mean value theorem.
28.	Fundamental Theorem of Integral Calculus Plan: Integral with a variable upper limit. Continuity of the integrable function. Newton-Leibniz formula. Substitution method in definite integrals, integration by parts, and Simpson’s formulas.
29.	Improper Integrals. Plan: Type I improper integral and its properties. Type II improper integral and its properties. Convergence and divergence properties of improper integrals. Poisson integral. Absolutely and conditionally convergent improper integrals. Singular integral.
30.	Application of Definite Integrals Plan: Squared figures and their properties. Calculation of integrals of functions given in polar and parametric form. Calculation of the length of irregular curves. Calculation of areas and volumes using integrals.

Conclusion

If a teacher who teaches the Mathematical Analysis course at the undergraduate level analyzes the content and scope of Mathematical Analysis concepts in the general education school mathematics curriculum, they will be able to organize their activities more efficiently.
If a teacher teaching the Mathematical Analysis course at the undergraduate level takes into account the knowledge and skills that the student (former pupil) has acquired in Mathematical Analysis from the general education school mathematics curriculum, they will be able to use the allocated teaching time more effectively.
If the conditions of the principle of continuity are followed in the teaching of the Mathematical Analysis course at the undergraduate level, students will be able to achieve higher results in their education.

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