The main methodological features of this book:
1) The focus is on developing personal and meta-subject educational outcomes, developing the child's spiritual potential, creative abilities, and interest in the subject. Mathematical knowledge in the course is considered not as an end in itself but as a means of developing certain personal and meta-subject educational outcomes, mathematical activity, children's thinking, feelings and emotions, creative abilities, and motives for activity. The set goal is achieved through the use of the didactic system of the activity method by L. Peterson. The activity method technology involves the following structure of lessons for introducing new knowledge:
2) The complex pedagogical conditions that ensure the implementation of the activity method technology include the following didactic principles: activity, continuity, a holistic representation of the world, minimax, psychological comfort (which, when used correctly, allow each student to move forward at their own pace, at their "maximum," but manageable level of difficulty, and, conversely, ignoring these principles can lead to student overload), variability, and creativity.
3) Connection to practice, real problems of the surrounding world. Full-fledged mathematics education is impossible without children understanding the origin and significance of mathematical concepts and the role of mathematics in the system of sciences. Therefore, one of the main tasks of the course is to reveal to students all three stages of the formation of mathematical knowledge. These are:
4) Formation of the thinking style necessary for the successful use of information and communication technologies. The computerization of the world around us leads to a reassessment of the importance of many skills and abilities. For example, special significance is gained in skills such as the ability to create and execute an action plan, the ability to strictly follow given rules and algorithms, the ability to evaluate the credibility of the obtained answer, the ability to consider different solutions, organize a search for information necessary to solve a problem, and more.
5) The multi-level nature of the program.
The course materials provide the opportunity for children with different levels of preparation to work on it based on the principles of minimax and psychological comfort. There is no selection of children to work with the course, as the level of preparation of the teacher is more important than the level of preparation of the children.
The training is conducted at a high level of difficulty (at the "maximum" level), i.e., in the "zone of proximal development" of the most prepared children, but with mandatory consideration of their individual characteristics and abilities, and the formation of faith in themselves and their own abilities for each child. In practice, this means that a sufficiently high level of tasks and the pace of their study are proposed. From the very first lesson, all children are placed in a situation that requires intellectual effort and productive actions. However, in training tasks and independent work, only the child's success and progress relative to themselves are evaluated. Errors, on the other hand, are considered as a working situation requiring correction, identification of their causes, and correction.
The current and final test is carried out at a lower level than the work in the class, which leads to the almost complete disappearance of poor grades. Final grades are given based on the number of "achievements" (which are evaluated only with high grades) and grades for control work. Low grades may appear very rarely - only when the child shows non-compliance and did not perform the agreed-upon task, which is clearly within their abilities. It is better if the child assigns themselves a negative grade in accordance with the norms accepted in the class.
6) At the same time, the high level of material presentation is not considered a mandatory requirement, but rather a proposal, an opportunity for each child to achieve success and be encouraged to take action. Therefore, the teacher should notice and support any, even the smallest, success of the child - their activity, involvement in the process of finding a solution, their correct judgment, or just an attempt to put forward their own hypothesis. An incorrect answer from a student should not provoke a negative reaction from the teacher, irritation, or lecturing. It is better if one of the students corrects the answer:
"What do you guys think?" It is the teacher's job in this situation to provide moral support to the one who made the mistake:
"Well done! You helped us understand! Do you agree? Do you understand now? Well done!" and so on.
Involvement in learning activities, internal activity, and the development of a habit of reflecting on each step is especially important for children with developmental difficulties. However, working at a high level of difficulty must be combined with creating an atmosphere of trust, respect, and friendliness in the classroom, allowing each student to truly "unfold" and believe in their abilities.
"You can do it!" - the teacher must believe in the student, "I can do it!" - they must believe in themselves, "They can do it!” – all the other students in the class must believe. Otherwise, learning will lose personal meaning for the child, and the school will not be able to fulfill its main mission - to help them reach their individual maximum.
The amount of tasks in the course sets the level of the individual educational trajectory for the most prepared children.
Therefore, it is not assumed that every child will complete all tasks. Only 3-4 key tasks on the new topic and review tasks, which develop the required learning outcomes, are mandatory for everyone. For more prepared children, the range of tasks may be expanded. However, overloading children, including with homework, cannot be allowed.
Practice and consolidation of knowledge of the basic content and methodological lines of the course (numerical, lines of textual problems) are carried out in parallel with the exploration of new mathematical ideas in additional lines (geometric, algebraic, data analysis, etc.). Therefore, the training exercises do not tire the children, especially since they are usually presented in a game format (coding and decoding, riddles, etc.). Each child with a low level of preparation has the opportunity to "slowly" work on the necessary skill from the mandatory learning outcomes, and more prepared children constantly receive "food for thought," which makes math lessons attractive for all children - both the "strong" and less prepared ones.
It is crucially important that every child experiences the joy of discovery at every lesson and develops faith in their own abilities and cognitive interests. Interest and successful learning are the main parameters that determine the full moral, intellectual, and physiological development of the child and, therefore, the quality of work with children.
7) A powerful tool that allows each child in the class to express themselves and realize their potential is creative work. Creative assignments, where children invent, create, and compose, should be offered systematically. They can come up with examples of a studied computational technique, create a problem based on a given expression (e.g. 85 ÷ 5 · 9 or x · 5 + y · 8), a problem of a given type (such as multiple comparisons, by sum and difference, etc.) or based on a given plot (about sports, animals, a fairy tale problem, etc.), draw patterns or geometric shapes of a specified property (e.g. ray KM intersecting line AB and not intersecting segment CD), decipher or encode the name of a city, book, or movie using computational examples, and so on.
Creative assignments are usually offered as additional homework to the mandatory part and are never graded poorly. At the end of the year, the most successful creative works can be collected in a "Problem Ledger," which will be authored by the students themselves. Such assignments, where children act not as performers but as creators, have the most positive effect on the development of children's personalities and contribute to a deeper and more solid acquisition of knowledge by them.
8) The volume and level of difficulty of homework. It is recommended to offer students two-level homework assignments, consisting of a mandatory and an optional (additional) part. The mandatory part should be manageable for the student to complete independently and should not exceed 15-20 minutes of their independent work time. It is also recommended to give assignments based on the children's own choice, for example: "Choose and complete one assignment from #4-7 that you like." The optional part, which is completed at the student's discretion, may include additional assignments marked with an asterisk, etc.
9) Types and forms of work in the lesson. It is necessary to diversify the types and forms of work in the lesson. The lesson should include collective, group, and individual forms of work, oral work and work in notebooks in a cell. Practice of computational skills should be systematic and intensive, but not take more than 3-4 minutes. It is advisable to give computational exercises a developmental character, selecting answer numbers so that children can analyze, classify, and identify patterns in the obtained series. This will not only help to consolidate counting skills but also prepare children's thinking for activity-based work methods.
When forming concepts using the methods adopted in the course, students activate all types of memory—not only visual and auditory but also motor, figurative, tactile, etc. Special attention is paid to rhythmic games, which in Grade 1 already help children master counting by 2, 3, 4, etc. up to 9, thus preparing a solid foundation for their further study of multiplication tables in Grade 2.
| Private Lessons | | | Group Lessons |
When conducting rhythmic games, attention should be paid to the composition of movements, starting with counting by 5, by the children themselves - in this case, the movements are easier and faster to remember, and as a result, the memorization of multiples of one-digit numbers is faster.
The work in the notebook should not exceed, as a rule, 10-12 minutes. It mainly involves the independent completion of tasks by students, which have been prepared beforehand during frontal work with similar but different tasks. The time for the independent completion of a task is usually limited (from one to 4 minutes). Then the task is checked. The children compare their solution with the standard for self-checking or with a sample and give themselves a "+" or a "-". As a result, the child’s ability to self-monitor is deliberately developed. Since the tasks completed independently are checked by the children themselves, the teacher, when checking them, pays attention primarily to the formation of self-monitoring skills and the accuracy of record keeping.
10) System of knowledge control. The course provides a multi-level system of knowledge control: self-control when introducing new material, peer control during the practice of the material, educational control in the system of independent work, ongoing control during quizzes throughout the academic year, and final control. Independent work is conducted at a high level of difficulty, so only success is evaluated. Specifically, if the entire independent work is done without mistakes, the highest grade is awarded. After each independent work, students who made mistakes work on correcting them. If the work on mistakes is successfully completed and the teacher sees that the child has understood the material being studied, then a high grade can be awarded for this work. Poor grades are not given for independent study assignments: "no grade" (nothing to grade, "not earned") is a much more significant signal for the child to be active and correct their own activities than poor grades. The teacher's task is to encourage each child to understand their mistakes and correct them.
In the course for Grade 2, universal learning activities (personal, regulatory, cognitive, and communicative) are formed during the study of the following basic questions: numeration, comparison, addition, and subtraction of numbers within 1000, multiplication and division tables, and multiplication and division of numbers within 100. Techniques for mental calculations within 100 are reinforced and brought to the level of automatic skill.
Tasks with letter data, problems with unit conversion, and composite problems involving all four arithmetic operations are introduced. Special attention is paid to the development of the ability to independently analyze word problems, the formation of geometric concepts in children, the ability to identify patterns, and the development of cognitive processes, logical and variational thinking.
Almost every lesson should include intensive exercises to develop computational skills. Computational exercises should be given a developmental character, selecting answer numbers so that the resulting series can be analyzed, classified, and patterns identified. This will not only help consolidate counting skills but also prepare children for activity-based learning.
Results of studying the mathematics course for Grade 2:
The content of the mathematics course for Grade 2 is aimed at achieving the following personal, meta-subject, and subject results:
Personal results:
The student will learn to:
The student will also have the opportunity to learn:
• to conduct a self-assessment of their ability to apply the algorithm for analyzing objects and comparing two objects, and to list the means they used to discover new knowledge based on a standard;
• to explore non-standard situations;
• to apply knowledge according to the 2nd-grade program in altered conditions;
• to solve problems of a creative and exploratory nature in accordance with the 2nd-grade program.
Subject outcomes "Numbers and arithmetic operations on them":
The student will learn:
• To apply mental addition and subtraction techniques with two-digit numbers;
• To perform column addition and subtraction with two-digit numbers;
• To add and subtract two-digit and three-digit numbers (all cases);
• To read, write, order, and compare three-digit numbers, represent them as a sum of hundreds, tens, and ones (decimal composition);
• To perform calculations according to a program given in parentheses;
• To determine the order of operations in expressions containing addition, subtraction, multiplication, and division (with or without parentheses);
• To use the associative property of addition, subtraction of a sum from a number, subtraction of a number from a sum to rationalize calculations;
• To understand the meaning of multiplication and division operations, justify the choice of these operations when solving problems;
• To perform multiplication and division of natural numbers, apply multiplication and division signs ( ·, : ), name the components and results of multiplication and division, and establish the relationship between them;
| Private Lessons | | | Group Lessons |
• Calculate specific cases of multiplication and division with 0 and 1;
• Conduct multiple comparisons of numbers (greater... smaller...), name divisors and multiples;
• Use specific cases of multiplication and division with 0 and 1;
• Use the commutative property of multiplication;
• Find the results of table multiplication and division using a multiplication table;
• Use the associative property of multiplication, multiply and divide by 10 and 100, multiply and divide round numbers;
• Calculate the values of arithmetic expressions with the learned natural numbers containing 3-4 operations (with and without parentheses) based on knowledge of the order of operations;
• Use arithmetic properties to rationalize calculations;
• Perform division with the remainder using models, find the components of division with the remainder, and their relationship, perform the algorithm for division with the remainder, and check the division with the remainder;
• Mentally add, subtract, multiply, and divide numbers up to 1000 in cases that can be reduced to operations within 100;
• Write addition and subtraction problems for numbers up to 1000.
The student will have the opportunity to learn:
• create graphic models of three-digit numbers and arithmetic operations with them, express them in different units of measurement, and, based on this, see the analogy between the decimal notation system for numbers and the decimal system of measures;
• independently derive techniques and methods for multiplication and division of numbers;
• graphically interpret multiplication, division, and multiple comparisons of numbers, as well as the properties of multiplication and division;
• see an analogy in the relationships between the components and results of addition and subtraction operations and the operations of multiplication and division.
Working with word problems
The student will learn:
• to solve simple problems involving the concept of multiplication and division (equal parts and content), and represent them using tables;
• to solve simple problems involving multiple comparisons (using the relations "more (less) than by...") ;
• to create simple expressions and solve mutually inverse problems involving multiplication, division, and multiple comparisons;
• to analyze simple and composite problems involving all arithmetic operations within 1000, create graphical models and tables, plan and implement the solution;
• to perform arithmetic operations with the quantities studied while solving problems;
• to solve problems involving the calculation of the length of a broken line, the perimeter of a triangle and quadrilateral, and the area and perimeter of a rectangle and square.
The student will have the opportunity to learn:
• to solve simple word problems involving letters as data;
• to create letter expressions based on word problems and graphical models, and vice versa, to create word problems for given letter expressions;
• to solve problems of studied types with incorrect formulations (extra or incomplete data, unrealistic conditions);
• to model and solve word problems involving all arithmetic operations within 1000, involving 4-5 steps;
| Private Lessons | | | Group Lessons |
• to independently find and justify methods for solving problems involving multiplication, division, and multiple comparisons;
• to find and justify various methods of solving a problem;
• to establish an analogy between solutions to problems with different storylines;
• to compare the obtained result with the problem condition, and assess its plausibility;
• to solve problems involving finding the "hidden number," involving 3-4 steps.
Geometric shapes and quantities
The student will learn:
• to recognize, label, and draw a straight line, ray, and line segment using a ruler;
• to measure the length of a line segment using a ruler, find the length of a polyline, and find the perimeter of a polygon;
• to identify a rectangle and a square among other shapes using a drafting triangle;
• to construct a rectangle and a square on graph paper given their side lengths, compute their perimeter and area;
• to recognize a rectangular prism and a cube, their vertices, faces, and edges;
• to construct a circle using a compass, differentiate between a circle and a disk, label and name their center, radius, and diameter;
• to express lengths in different units of measurement - millimeter, centimeter, decimeter, meter, kilometer;
• to determine the area of a geometric shape using a given scale from a prepared drawing, compare shapes directly, and using measurements;
• to express the area of shapes in different units of measurement - square centimeter, square decimeter, square meter;
• to convert, compare, add, and subtract homogeneous geometric quantities.
The student will have the opportunity to learn:
• to independently identify the properties of geometric shapes;
• to recognize and name acute, obtuse, and right angles;
• to determine intersecting, parallel, and perpendicular lines;
• to draw patterns from circles using a compass;
• to compose shapes from parts and break figures into parts, find the intersection of geometric shapes;
• to calculate the areas of shapes composed of rectangles and squares;
• to find the volume of a rectangular prism and a cube using units of volume (cubic centimeter, cubic decimeter, cubic meter) and their relationships.
Quantities and their relationships
The student will learn:
• to differentiate between the concepts of quantity and units of measurement;
| Private Lessons | | | Group Lessons |
• to recognize, compare (directly), and order quantities of length, area, and volume;
• to measure area and volume using a prepared drawing with arbitrary units, use new units of length measurement;
• to convert learned units of length, area, and volume based on relationships between homogeneous units of measurement, compare and add or subtract quantities using different units of measurement;
• to observe the dependency of measurement results of length, area, and volume on the choice of units, express the observed dependencies in speech, and use formulas (A = a · b; V = (a · b) · c).The student will be able to:
• Make an independent choice of convenient units of measurement for length, area, and volume for a specific situation.
• Observe, in simple cases, the dependencies between variables using tables.
• Establish a relationship between the components and results of multiplication and division, express them in speech, and use them to simplify problem-solving and examples.
Algebraic concepts
The student will learn:
• To read and write numerical and algebraic expressions containing addition, subtraction, multiplication, and division (with and without parentheses).
• To find the values of simple algebraic expressions for given values of variables.
• To express the relationships between multiplication and division using algebraic equations of the form: a · b = c, b · a = c, c ÷ a = b, c ÷ b = a.
• To write the properties of arithmetic operations in algebraic forms, such as:
• To solve and explain the steps taken to solve equations of the form a · x = b, x · a = b, a ÷ x = b, x ÷ a = b by using an associative method (based on the relationship between the sides and the area of a rectangle).
The student will be able to:
• Independently identify and express in algebraic form the properties of numbers and operations.
• Explain the solutions to simple equations of all types studied, identifying the components of the operations involved.
Mathematical language and elements of logic
The student will learn:
• to recognize, read, and apply new symbols of mathematical language, such as multiplication and division signs, brackets, and geometric shapes (such as points, lines, rays, line segments, angles, polygons, triangles, quadrilaterals, etc.);
• to construct simple statements such as "true/false that...", "not", "if..., then...";
• to determine the truth and falsehood of statements about studied numbers and quantities and their properties;
• to establish patterns in simple cases (such as the rule by which a sequence is composed, a table is filled in, continuing a sequence, filling in missing elements, completing empty cells in a table, etc.).
The student will have the opportunity to:
• justify their judgments using the rules and properties learned in the 2nd Grade, making logical conclusions;
• independently construct and master techniques for solving logical problems in accordance with the 2nd-grade curriculum.
Working with Information and Data Analysis
| Private Lessons | | | Group Lessons |
The student will learn to:
• Read and fill in tables according to a given rule, analyze data in tables;
• Create sequences (chains) of objects, numbers, figures, etc. according to a given rule;
• Determine the operation, object, and result of the operation;
• Perform direct and inverse operations on objects, figures, and numbers;
• Find unknowns: the object of the operation, the operation being performed, and the result of the operation;
• Execute algorithms of various types (linear, branched, and cyclic) recorded in the form of action plans in different ways (flowcharts, action plans, etc.);
• Perform an ordered enumeration of options using tables and a tree of possibilities;
• Find information on a given topic in different sources (textbook, reference book, encyclopedia, etc.);
• Work in the material and informational environment of primary education (including educational models) in accordance with the content of the subject "Mathematics, Grade 2."
The student will have the opportunity to learn to:
• Independently create algorithms and record them in the form of flowcharts and action plans;
• Collect and present information in reference books, encyclopedias, and controlled space of the Internet about the lifespan of different animals and plants, their sizes, and create their own tasks based on the obtained data involving all four arithmetic operations;
• Become a co-author of the "Grade 2 Problem Book" composed of the best problems created by the students themselves;
• Create a portfolio of a Grade 2 student.
The first part of the "Mathematics, Grade 2" program begins with a prolonged review stage, which is carried out in parallel with clarifying the content of concepts known to children (chain, point, line). Students learn written and oral methods of adding and subtracting two-digit numbers and then move on to mastering the numeration, comparison, addition, and subtraction of three-digit numbers. All computational methods are "discovered" by the children themselves using the particular models of numbers adopted in the program. Working with particular models (so-called "triangles and dots") is mandatory at this stage for each child.
When three-digit numbers are introduced, a new unit of length measurement is introduced —the meter. The interrelation between the meter, decimeter, and centimeter is revealed based on the analogy between the decimal notation of numbers and the decimal system of measures, which is also demonstrated using "triangles and dots."
Throughout the first part of the Grade 2 program, simple addition and subtraction equations and compound text problems containing new calculation cases are presented. When solving text problems, special attention is paid to teaching children Independent analysis and the construction of graphic models.
The development of geometric representations continues: children learn to mark points, draw straight lines using a ruler, and find points of intersection. To develop their variational thinking, children are offered problems that allow for different answers, such as problems involving permutations of three elements, etc. The development of abilities for analysis, comparison, generalization, classification, identification of patterns, and expressing them in speech also continues. The designation of geometric objects using the most common Latin letters is introduced.
| Private Lessons | | | Group Lessons |
