The Mathematics Curriculum in Primary and Lower Secondary Grades
For all three stages of general education (Grades 1 to 11), the mathematics national curriculum stipulates the goals of mathematics education, the requirements for student achievement (what students should know and be able to do at the end of each stage of general school), and the instructional content that must be presented to ensure achievement of these requirements.
The goal of mathematics education in primary school is the mathematical and general intellectual development of students, with special attention given to the formation of the ability to solve various problems using mathematics.
Students at the fourth grade level who participated in TIMSS 2015 were taught in accordance with the Federal State Educational Standards for Primary General Education (issued in 2009 and introduced from Grade 1 in 2011)3 and the Model Basic Educational Program for Educational Institutions—Primary School, issued in 2011.4 The second document includes a model program for mathematics in Grades 1 to 4, which prescribes the basic content of mathematics instruction and the expected student outcomes (i.e., requirements for the attainment of primary school graduates). While the model program defines the basic mathematics content and skill objectives, more detailed curricula are written for specific textbooks, which are developed by authors and used in schools.
The objectives for student achievement are presented in two categories: objectives all primary school students are required to master, and objectives all primary school students have the opportunity to learn but are not required to master.
The program includes new mathematical content (e.g., fractions of the type 1n , spatial geometric shapes, and creation and verification of the truth of statements), as well as the following new content areas:
- Numbers and Magnitude—This content area focuses on formation of the understanding of concepts
- Working with Word Problems—This content area emphasizes formation of the ability to solve problems
- Working with Information—This content area contains elements of descriptive statistics related to working with tables and charts, material of great practical significance for orientation in everyday life
Exhibit 1 presents the recommended mathematics content for primary school students.
Exhibit 1: Mathematics Content for Primary School, Grades 1–4
| Topic Area | Recommended Content |
| Numbers and Values | Reading and writing numbers from 0 to 1 million; classes and categories; units of measurement of weight (grams, kilograms, hundredweight, and tons), capacity (liters), and time (seconds, minutes, and hours); comparing and ordering homogeneous quantities; and fractions (12 , 13 ,14 , 110 , 1100 ,11000) |
| Arithmetic Operations | Naming the components of arithmetic operations; addition and multiplication tables; the relationship among the arithmetic operations; finding unknown components of arithmetic operations; division with a remainder; numeric expressions (determining the order of operations and solving); using the properties of arithmetic operations in calculations (permutations and grouping terms in a sum; factors in a product; multiplying a sum and difference by a given number); algorithms of written operations (addition, subtraction, multiplication, and division of multi-digit numbers); methods of verifying computations (e.g., algorithms, reverse calculation, reliability assessment, estimation, and using a calculator) |
| Solving Word Problems | Solving word problems by arithmetic methods; solving problems involving the relationships “more (or less) on” and “more (or less) in”; dependencies between quantities describing movement, work, purchase and sale, etc.; speed, time, and distance; measuring work, time, and labor productivity; quantity of goods, including their price and value; planning solutions to a problem; representing information using charts, tables, and other models; and finding parts of a whole and a whole based on its parts |
| Spatial Relations and Geometric Figures | Corresponding arrangements of objects in space and on a plane; recognizing and depicting geometric figures including points, line segments (curved and straight), broken lines, angles, polygons, triangles, rectangles, squares, circumference, and circles; using drawing tools to perform constructions; geometric shapes in the world; recognizing and naming cubes, spheres, parallelepipeds, pyramids, cylinders, and cones |
| Geometric Quantities | Geometric quantities and their measurement; measuring length of a segment; units of length (mm, cm, dm, m, km); perimeter; calculating perimeter of a polygon; area of geometric figures; units of area (cm2, dm2, m2); exact and approximate measurement of area of geometric figures; calculating area of a rectangle |
| Working with Information | Collecting and representing information related to counting and measurement; recording and analyzing information; drafting simple expressions using the logical connectives and words “and,” “no,” “if … then,” “true or false,” “every,” “all,” and “some;” verifying the truth of statements; drafting finite sequences of objects, numbers, geometric figures, etc., according to a rule; drawing up, writing, and executing simple algorithms, planning for information retrieval; reading, completing, and interpreting data in a table; reading bar graphs; creating simple information models (e.g., diagrams, tables, and chains) |
By the end of primary education, student achievement in mathematics must meet the requirements presented in Exhibit 2.
Exhibit 2: Mathematics Requirements for Primary School, Grades 1–4
| Topic Area | Minimum Learning Requirements | Additional Learning Opportunities |
| Numbers and Values | Read, write, and compare numbers up to 1 million; establish rules for number sequences and write sequences according to a given or a self-determined rule; group numbers according to given or self-determined criteria; classify numbers according to one or more criteria and explain; and read, write, and compare quantities (e.g., mass, time, length, area, and speed) | Select units of measurement for given quantities (e.g., length, mass, area, and time) and explain |
| Arithmetic Operations | Carry out written calculations (i.e., addition, subtraction, multiplication, and division of numbers up to 10,000 into one-digit and two-digit numbers) using addition and multiplication tables and algorithms of written arithmetic operations (including division with a remainder); perform oral addition, subtraction, multiplication, and division on one-digit, two-digit, and three-digit numbers within 100 (including 0 and the number 1); solve arithmetic operations for unknown components; calculate the value of a numeric expression (containing 2 or 3 arithmetic operations, with and without brackets) | Perform operations with known quantities; use the properties of arithmetic operations to perform calculations; and verify calculations using reverse action, estimation, assessment of results, etc. |
| Working with Word Problems | Use mathematics to represent the relationship between given quantities in a problem, develop a plan for solving the problem, and explain the chosen strategy; apply arithmetic methods to solving educational problems (1 to 2 steps) and problems in everyday life; solve problems involving fractions (12 , 13 ,14 , 15 ,110); verify solutions step by step and evaluate how realistic they are | Solve problems in 3 to 4 steps; find multiple ways of solving a problem |
| Spatial Relations and Geometric Figures | Describe the relative position of objects in space and in the coordinate plane; recognize, name, and represent geometric figures (e.g., points, lines, broken lines, right angles, polygons, triangles, rectangles, squares, circles, and circumference); construct geometric shapes with specified dimensions (e.g., lines, squares, and rectangles) with a ruler and a set square; use the properties of rectangles and squares to solve problems; recognize and name geometrical solids (e.g., cubes and spheres); and relate real world objects to models of geometric shapes | Recognize, distinguish, and name geometric solids (e.g., parallelepipeds, pyramids, cylinders, and cones) |
| Geometric Quantities | Measure the length of a segment; calculate the perimeter of a triangle, a rectangle, and a square, and the area of a rectangle and a square; and approximate (by eye) the size of geometric objects and distances | Calculate the perimeter of a polygon, and the area of a figure made up of rectangles |
| Working with Information | Read and fill in simple tables; read simple bar graphs; and understand simple expressions containing logical connectives and words (e.g., “and,” “if … then,” “true or false,” “every,” “all,” “some,” and “no”) | Read simple pie charts; complete simple bar graphs; draw up, write, and execute instructions (e.g., easy algorithms); plan a search for information; recognize the same information presented in different forms (e.g., tables and charts); plan simple research, and collect and present the information using tables and charts; interpret the information obtained through simple research (e.g., explain, compare, and summarize data, draw conclusions, and make forecasts) |
The eighth grade students who participated in TIMSS 2015 studied in accordance with the Federal State Education Standards for Basic Education (Grades 5 to 9) issued in 2004.5 The standards include a practical orientation for instruction (including statistics and probability in all textbooks), which emphasizes mathematical modeling, the universality of mathematical language, and the application of mathematics to solving practical problems related to everyday life.
In Grades 5 to 9, basic general mathematics education is directed toward achieving the following goals:
- Mastering the system of mathematics knowledge and the skills necessary for its application in practical activities, related subjects, and continuing education
- Developing intellectually, forming the characteristics necessary for a productive life in modern society, such as clarity and accuracy of thinking, critical thinking, intuition, logical thinking, elements of algorithmic culture, spatial sense, and the ability to overcome difficulties
- Developing understanding of the concepts and methods of mathematics as the universal language of science and technology and a means of simulating phenomena and processes
- Developing one’s attitude toward mathematics as part of a universal culture and understanding the importance of mathematics to scientific and technological progress
Exhibit 3 presents the compulsory minimum mathematics content for basic general education (Grades 5 to 9) and includes four topic areas: Arithmetic; Algebra; Geometry; and Elements of Logic, Combinatorics, and Statistics and Probability.
Exhibit 3: Compulsory Minimum Mathematics Content for Educational Programs, Grades 5–9
| Topic Area | Minimum Content |
| Arithmetic | Natural numbers, fractions, rational numbers, real numbers, word problems, measurement, approximation, and estimation |
| Algebra | Algebraic expressions, properties of exponents, equations, and inequalities, solving word problems algebraically, number sequences, functions, and coordinates |
| Geometry | Basic geometry concepts and theorems, angles, lines, circumference and circles, visualizing spatial figures, triangles, trigonometry, quadrangles, polygons, geometric measurement, area of plane figures, volume of solids (e.g., cubes, spheres, cylinders, cones), vectors, geometric transformations, and geometric constructions using a ruler and a compass |
| Elements of Logic, Combinatorics, and Statistics and Probability | Proofs, sets and combinatorics, and statistical data and probability |
Exhibit 4 presents the mathematics achievement requirements for students in basic education (Grades 5 to 9).
Exhibit 4: Mathematics Requirements for Basic Education, Grades 5–9
| General Facts and Methods | |
| Know or Understand | Mathematical proofs and the concept of algorithms; how to use mathematical formulas, equations, and inequalities for solving mathematical and practical problems; the probabilistic character of many laws of the natural world; examples of statistical regularities and conclusions; how geometry has arisen from practical geodetic problems; examples of geometric objects and statements that are relevant to educational and real life problems |
| Arithmetic | |
| Be Able to | Carry out arithmetic operations orally; convert numbers from one form to another; compare and carry out arithmetic operations with rational numbers; solve simple expressions with an integer exponent and roots; solve numerical expressions; find approximations with integers and decimals; use basic units of length, weight, time, speed, area, and volume; and solve word problems, including problems involving ratio and proportion, fractions, and percentages |
| Use Acquired Knowledge and Skills in Practical Activities and Daily Life to | Solve simple practical problems using reference materials, a calculator, and a computer; estimate calculations orally; examine calculation results using various methods; and interpret results considering restrictions connected with the problems and solution processes |
| Algebra | |
| Be Able to | Write, transform, and solve algebraic expressions and formulas; carry out operations with exponents, polynomials, and algebraic fractions; transform and solve numeric expressions containing square roots; solve linear, quadratic, and rational equations and systems of equations and inequalities; solve word problems using algebraic methods; determine coordinates of a point and locate points on the Cartesian plane; recognize arithmetic and geometric progressions and solve problems using formulas for the general term and the sum of the first several terms of a progression; find values of functions represented in various forms and define properties of functions using graphs; use graphical methods for solving equations, systems, and inequalities; and graph certain functions |
| Use Acquired Knowledge and Skills in Practical Activities and Daily Life to | Calculate using formulas, and write formulas expressing the relationship among real data; find necessary formulas in reference materials; model practical situations and research constructed models using algebraic methods; describe dependencies between physical variables in simple practical situations using corresponding formulas; and interpret graphs representing real dependencies between variables |
| Geometry | |
| Be Able to | Use geometric language for describing objects; recognize geometric figures; draw geometric figures; carry out drawings according to the context of a problem; carry out transformations of figures; recognize basic spatial figures in drawings, models, and the environment and draw them; draw parts of simple spatial figures; carry out operations using vectors and calculate length, the coordinates of a vector, and the angle between vectors; calculate length, angle, area, and volume, and define values of trigonometric functions for angles from 0° to 180°; find values of trigonometric functions using the value of a given angle; calculate the sides, angles, and area of triangles, length of broken lines, arcs of circumference, and area of basic geometric figures and composite figures; solve geometric problems using properties of figures and relationships among them and apply additional constructions, algebraic and trigonometric methods, and ideas of symmetry; reason when solving problems using known theorems and find opportunities for their application; and solve elementary planimetric problems in space |
| Use Acquired Knowledge and Skills in Practical Activities and Daily Life to | Describe real situations in the language of geometry; calculate with elementary trigonometric formulas; solve geometric problems using trigonometry; solve practical problems connected with geometric measurement (using reference books and instruments if necessary); and make constructions using geometric tools (e.g., rulers, set squares, compasses, and protractors) |
| Elements of Logic, Combinatorics, and Statistics and Probability | |
| Be Able to | Perform simple proofs; draw elementary conclusions from statements; evaluate reasoning logic, using examples for illustration and counterexamples for refutation of statements; retrieve information from tables, diagrams, and graphs; create tables, diagrams, and graphs; solve combinatorial problems by systematically sorting possible outcomes and using the rule of multiplication; calculate average measurement; calculate the frequency of events using personal observations and given statistical data; and calculate the probability of simple random events |
| Use Acquired Knowledge and Skills in Practical Activities and Daily Life to | Defend a proof orally; recognize logically incorrect reasoning; record mathematical statements and proofs; analyze real numerical data presented in diagrams, graphs, and tables; solve practical problems in daily and professional life using operations with numbers, percentages, length, area, volume, time, and speed; solve educational and practical problems that require the systematic sorting of possible outcomes; compare the chance of occurrence of random events, estimate the probability of random events in practical situations, and match models to real situations; and understand statistical statements |
In 2015, the new Federal State Education Standards for General Basic Education6 and the Model Basic Educational Program of Basic General Education,7 issued in 2011, were introduced in Grades 5 to 9. The role of practical activities in mathematics, including making discoveries and participating in individual and group projects, was increased to stimulate students’ cognitive activity.