Russian Federation: Description of the Advanced Mathematics Programs and Curriculum
High school programs for mathematics (Grades 10-11) are distinguished by the amount of the material being studied and the amount of instructional time. The Basic level program is designed for those students who plan to learn a profession that is not related to mathematics or plan to use mathematics as an auxiliary “tool” in their professional lives. The Profile level program provides sufficient depth of mathematics study to make it possible for students to enter a profession where mathematics is actively used. It includes a large amount of content and has higher requirements for its mastery. The mastery of this content makes it possible for students to continue to university-level studies in mathematical disciplines. Within the Profile level there is a subset of students in an even more intensive program taking six hours or more of mathematics lessons per week. The sample of students participating in the TIMSS Advanced 2015 Advanced Mathematics assessment included both Profile-level students and Intensive-level students. The results for students in the Intensive level were also reported separately as Russian Federation 6hr+.
The Profile level curriculum includes an explanation of the main goals of the program and provide for the organization and planning of mathematics courses, including:
- General characteristics of the profile course
- Teaching goals
- The number of lessons per week and per year
- General learning skills and activities
- Compulsory content and learning outcomes
The content of the Profile course is divided into two sections: Algebra and Calculus, and Geometry. The topics included in each section are listed below.
| Content Areas in Algebra and Calculus | |
|---|---|
| Grade 10 | |
| Polynomials | Transformation of polynomials, factorization; division of polynomials; Horner’s method; roots of polynomials; Bezout’s theorem; converting irrational expressions |
| Graphs of Functions | Complex functions; conversion of graphs; graphs of linear-fractional functions, asymptotes; graphs of functions which include a sign of a module (e.g., y=2x-6|3-x| or y=sin|x|); reciprocal functions and their graphs |
| Introduction to Calculus | Numerical sequences, limits of sequences, limits of functions, theorems on limits of functions; properties and continuity of elementary functions |
| Derivatives and their Applications | Geometric and physical meaning of the derivative, continuity and differentiability of functions, derivatives of sums, products, quotients, composites and exponential functions; second derivatives and higher order derivatives; application of derivatives to study functions; Lagrange’s theorem and its consequences; drawing graphs of functions |
| Trigonometric Functions | Trigonometric functions of numeric argument (sine, cosine, tangent and cotangent); trigonometric identities and their consequences; reduction formulas; identical transformation of trigonometric expressions; periodicity of trigonometric functions; properties, graphs, and derivatives of trigonometric functions |
| Grade 11 | |
| Integral and Differential Equations | Indefinite integrals; definite integrals and their properties, numerical approximation of definite integrals, approximate computation; Newton-Leibniz formula; application of integrals for calculating areas, volumes, and lengths of arcs in physical problems; solutions of simple differential equations |
| Exponential and Logarithmic Functions | Properties and graphs of exponential functions; logarithms, definitions, and properties; identical transformations of exponential and logarithmic expressions; exponential and logarithmic equations, inequalities and systems of inequalities, types and methods of solution; derivatives of exponential functions; natural logarithms, radioactive decay |
| Complex Numbers | Algebraic form, arithmetic operations, conjugating complex numbers; solutions of quadratic equations with complex coefficients; the complex plane; trigonometric form of complex numbers, multiplication, division, and raising to power; De Moivre’s formula; complex roots of polynomials; the Fundamental Theorem of Algebra |
| Elements of Combinatorics | Methods of mathematical induction; proofs of identities; factorials; the basic formulae of combinatorics; combinations and permutations; Binomial Theorem, Dirichlet’s Principle |
| Elements of the Theory of Probability and Mathematical Statistics | Classic definition of probability, calculating probabilities using combinatorics; conditional probability, the rules of addition and multiplication of probabilities, independent events, Bernoulli distribution; mathematical expectation and variance; the concept of the law of large numbers and a normal distribution law; parent population and sample, levels of significance and reliability; evaluation of probability using frequency; the concept of statistical hypothesis testing |
| Equations, Inequalities, Systems | General methods and approaches for solving equations; irrational equations; generalized method of intervals for solving inequalities; systems of equations and inequalities, basic methods for solving systems of equations; application of graphs to solve equations, inequalities and systems; approximate methods for solving equations; equations, inequalities, and systems with parameters |
| Content Areas in Geometry | |
|---|---|
| Grade 10 | |
| Axioms of Solid Geometry | |
| Parallel Lines and Planes | Mutual arrangement of lines and planes in space; theorems of parallelism of lines and planes |
| Perpendicularity of Lines and Planes | Theorems of dependences between parallelism and perpendicularity of lines and planes, the Theorem of the Three Perpendiculars; angles between straight lines and a plane |
| Coordinates and Vectors in a Space | Rectangular coordinate systems on a plane, the formula for distance between points, equations of straight lines and circumference; Cartesian coordinate system in a space, equations of straight lines and a plane; movements in a space and their properties (central symmetry, parallel translation, rotation), similarity in a space |
| Vectors in a Space | Decomposition of vectors into three non-coplanar vectors; scalar products; applications of coordinates and vectors to solve problems |
| Grade 11 | |
| Polyhedrons | Concepts of polyhedrons, prisms, rectangular parallelepipeds, and pyramids; areas of faces and surfaces; sections; regular polyhedrons; dihedral angles |
| Solids of Revolution | Bodies and surfaces of revolution, cylinders, cones, axial sections of cylinders and cones; spheres and solid spheres, sections of solid spheres, equation of a sphere; inscribed and circumscribed cylinder, cone, sphere |
| Volumes of Bodies | Volumes of polyhedrons (prisms, pyramids) and solids of revolution (cylinder, cone, sphere, part of the sphere) |
| The Surface Areas of Solids of Revolution | Areas of spheres, surface areas of cylinders and cones |
Learning outcomes are described in terms of what students should know and be able to do in each of these areas. Teachers have some discretion as to the introduction of optional topics.