The
proposed Russian Math Books for primary school mathematics course is part of a unified continuous
mathematics course for preschoolers, primary school, and grades 4-5 of
secondary school. It is designed to achieve new educational objectives,
including personal, cross-curricular, and subject-specific learning outcomes,
as well as readiness for self-development based on fostering students'
cognitive motivation, universal learning actions, and overall learning skills.
Mathematical knowledge in the course is not considered an end in itself but rather a means of developing specific personal and cross-curricular educational outcomes, mathematical activities, and the development of children's thinking, senses, emotions, creative abilities, and motivational factors. The stated objective is achieved through the use of the didactic system of the activity-based method by
L. Peterson. The technology of the activity-based method involves the following structure for introducing new knowledge in lessons:
1. Motivation (self-determination) for the activity.2. Activation of prior knowledge and identification of difficulties in a trial learning action.3. Identification of the place and cause of the difficulty.4. Construction of a project to overcome the difficulty.5. Implementation of the constructed project.6. Initial consolidation with verbalization in external speech.7. Independent work with self-assessment against a benchmark.8. Integration into the knowledge system and review.9. Reflection on the activity (lesson summary).Lessons of other types follow a similar structure: reflection (i.e., review and consolidation of knowledge, self-assessment, and correction of errors), as well as lessons focused on building knowledge systems and developmental assessment. Such lesson construction not only helps students develop a solid foundation of mathematical knowledge but also engages them in performing a comprehensive set of universal learning actions during each lesson.
The complex of pedagogical conditions that ensure the implementation of the activity-based method technology includes the following didactic principles: activity, continuity, holistic representation of the world, minimax, psychological comfort, variability, and creativity. These principles maintain their significance in the system of educational and management support for children's well-being. Thus,
L. Peterson's educational system enables the establishment of a unified educational, developmental, and health-preserving process based on an activity-based approach.
The educational system can be implemented at different levels: the basic level,
the technological level, and the system-technological level. The basic level of
the activity-based method (ABM) includes the following 7 steps:
When working at the basic level of ABM within the framework of didactic principles, the principle of activity is transformed into the principle of activity activation within the traditional education system. Special attention should be paid to the principles of minimax and psychological comfort, as their proper application allows each student to progress at their own pace, tackling challenges at their "maximum" but manageable level of difficulty. Ignoring these principles, on the other hand, can lead to student overload.
The described lesson structure systematizes the innovative experience of Russian schools, making it an accessible step for every teacher. It yields rapid results, including positive dynamics in students' knowledge acquisition, and development of their thinking, speech, and cognitive interest. The basic level of ABM can be easily mastered by any teacher during their initial acquaintance with the educational system by
L. Peterson, and it serves as a starting point for self-development when fully adopting the activity-based method.
The technological level of ABM implementation refers to the level of teacher's
work where the transitional structure (8 steps) and the system of didactic
principles of Peterson’s educational system are implemented. Concepts such as
benchmark, self-assessment benchmark, and detailed model are included in the
practice, and motivation for cognitive activities is organized at the levels of
"I want" and "I can".
The technological level of ABM implementation allows for the following:
1. Ensure all the outcomes of the basic level of ABM implementation.2. Create conditions for the development of general learning skills, including the ability to learn.The system-technological level of ABM implementation refers to the level of teacher's work where the comprehensive structure of educational activities (9 steps) and the system of didactic principles of Peterson's educational system are implemented. Concepts such as educational activities and their structure are included in the practice.
The system-technological level of ABM implementation allows for the following:
1. Ensure all the outcomes of the basic level of ABM implementation.2. Develop general learning skills specified in the educational standards.
Connection with practice, real-world problems
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 school curriculum is to reveal to students all three stages of mathematical knowledge formation.
These stages are:
1. The stage of mathematization involves constructing a mathematical model of a fragment of the real world.2. The stage of studying the mathematical model involves constructing a mathematical theory that describes the properties of the constructed model.3. The stage of applying the obtained results to the real world.For example, natural numbers are not initial abstractions, so their study is preceded by familiarizing students with finite collections of objects. Similarly, the study of addition and subtraction of natural numbers begins with the consideration of concrete operations of combining finite collections and removing a part of a collection. The study of formal operations of addition and subtraction of two-digit numbers is based on operations with symbolized representations of these numbers using dots and shapes (following the historical development of these operations).
The above-mentioned demonstrates how the first stage of mathematical modeling, the construction of mathematical models of the surrounding world, is reflected in the mathematics curriculum for first grade. The second stage, intra-model investigation, is associated with the study of addition and subtraction operations with single-digit numbers, constructing an addition table, and studying operations with two-digit numbers. Finally, the third stage is reflected in solving word problems where the learned operations with numbers are applied in practical contexts.
The curriculum achieves continuity between preschool education, primary school, and secondary school through the integration of technology, content, and methodologies, ensuring a seamless educational process across all stages of learning. The selection of content and the sequence of studying key mathematical concepts were based on a systemic approach.
The development of the thinking style necessary for the successful use of ICT (Information and Communication Technology) leads to a reassessment of the importance of many skills and abilities. For example, the ability to create and implement an action plan, strict adherence to given rules and algorithms, assess the plausibility of the obtained answer, explore various solution options, organize information search required to solve a given problem, and more, gain particular significance. Therefore, Peterson’s mathematics course effectively addresses all the objectives of the subject area "Mathematics and IT.«
The course has a differentiated approach, allowing children of different levels of preparation in schools and classes of all types to work with it based on the principles of minimax and psychological comfort. The selection of children to work with the course is not assumed based on their level of preparation, but rather on the teacher's level of preparation.
The instruction is conducted at a high level of difficulty (the "maximum" level), meaning it is within the "zone of proximal development" of the most advanced children. It takes into account each child's individual characteristics and abilities and fosters each child's belief in themselves and their abilities.
Practically, this means that BYOM presents tasks and the pace of learning that are of a sufficiently high level. From the very beginning, all children are placed in situations that require intellectual effort and productive actions. However, in instructional tasks and independent work, only the child's success and progress relative to themselves are evaluated. Mistakes are seen as working situation that requires correction, identification of the cause, and rectification. Ongoing and final assessments are conducted at a lower level than classroom work, which practically leads to the disappearance of low grades. Final grades are based on the number of "achievements" (which are evaluated with high grades only) and marks for tests. Low scores may appear very rarely, only when a child has shown negligence or failed to complete an assigned task that is clearly within their capabilities. It is preferable if the child themselves gives them a negative grade in accordance with the norms established in the class.
However, the high level of material presentation is not considered as a mandatory requirement, but rather as a suggestion and an opportunity for each child to achieve success, motivating them 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 accurate judgment, or simply an attempt to propose their own hypothesis. A student's incorrect answer should not evoke a negative reaction from the teacher, annoyance, or lecture. It is better if one of the students corrects the answer, saying, "What do you guys think?" In this situation, it is the teacher's role to provide moral support to the one who made the mistake: "Well done! You helped us understand!" or "Do you agree? Do you understand now? Well done!" and so on.
The principle of minimax is a self-regulating mechanism for multi-level learning. As mentioned above, it does not require any special selection of children to work with it. Furthermore, engagement in educational activities, internal activity, and the development of a habit of reflecting on each step is particularly important for children with developmental issues. However, working at a high level of difficulty must be accompanied by creating an atmosphere of trust, respect, and friendliness in the classroom, allowing each student to believe in their abilities and truly "unfold themselves." The teacher must believe in the student, saying, "You can do it!" The students must believe in themselves, thinking, "I can do it!" and all the other students in the class should believe in their classmates, saying, "They can do it!" Otherwise, education will lose its personal significance for the child, and the school will not be able to fulfill its main mission - to help the child achieve their individual maximum.
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The volume of tasks in the course sets the level of the individual educational trajectory for the most prepared children. Therefore, it is not expected that every child will complete all the tasks. Only 3-4 key tasks on each new topic and review tasks are mandatory for everyone, which reinforces the essential learning outcomes. The range of tasks can be expanded for more advanced children. However, overloading children, including with homework, should be avoided.
The consolidation and reinforcement of knowledge in the main content and methodological lines of the course (numerical line, text-based problem-solving) are conducted in parallel with the exploration of new mathematical ideas in additional lines (geometry, algebra, data analysis, etc.). Therefore, the practice exercises do not tire the children, especially since they are usually presented in a playful form (code cracking, riddles, etc.). Each child with a lower level of preparation has the opportunity to gradually master the necessary skills from the essential learning outcomes, while more advanced children are constantly provided with intellectual stimulation, making mathematics lessons appealing to all children, both strong and less prepared.
It is fundamentally important for each child to experience the joy of discovery and develop confidence in their abilities and curiosity in every lesson. Interest and success in learning are the key parameters that determine the comprehensive moral, intellectual, and physiological development of a child and, therefore, the quality of working with children.
Some of the topics included in the 4th-grade curriculum have traditionally been taught in the 5th grade. The decision to introduce them at an earlier stage is driven by several reasons. One of the most significant reasons is the need to take into account sensitive periods, which are the most favorable periods from a psychological perspective for acquiring certain knowledge. For example, in the 3rd grade, learning about multi-digit numbers within the range of 12 digits is easier than in the 5th grade, because children enjoy working with "long" numbers.
By the 4th grade, students accumulate fatigue from complex calculations, and they are not only logically but also emotionally ready for the next step - the introduction of fractions. The new numbers, their properties, and the algorithms that students construct on a tangible basis evoke surprise, joy, and sometimes even enthusiasm from their independent victories (which teachers in higher grades do not often observe when teaching fractions in the 5th or 6th grades). Through this emotional reinforcement, necessary training in operations with whole numbers is conducted simultaneously with the study of fractions, and the corresponding techniques for mental and written calculations are brought to the level of automated skill.
The transfer of this material from the 5th grade to the early years of the school became possible thanks to the new methods proposed in the curriculum, such as graphical modeling of text-based problems, associative techniques for solving equations, and others, which significantly reduced the time required for studying many topics in the curriculum.
In the 4th grade, there are several successful methodological innovations. Among them, we can mention the technique for solving fraction (percentage) problems, problems involving simultaneous motion, solving inequalities with natural numbers, exploring the properties of geometric shapes through constructions and measurements, reading and constructing motion graphs, and more. For example, the timely introduction of the percentage symbol (%) in the 4th grade as one of the notations for a hundredth part of a quantity helps eliminate the difficulties that students in higher grades face when studying percentages. It also improves the quality of solving simple fraction and percentage problems (for cases involving operations with numbers) in the future.
Thanks to the described transfer of material from higher grades to the early years of school, time is freed up in the 5th and 6th grades to address topics that are significant for 11-12-year-old children from both psychological and preparatory perspectives for their further study of the mathematics curriculum in grades 7-9. These topics include logic, modeling, the development of geometric concepts, variational and functional thinking, and more. Moreover, the quality of learning the middle school mathematics curriculum is significantly enhanced because the methodologies presented in the primary school course eliminate the need to translate the conclusions into the language used in higher grades.
Creative assignments in the textbook's work system
A highly effective means of allowing each child in the class to unfold and realize their potential is through creative work. Creative assignments, where children come up with ideas, create, and invent, should be regularly introduced, preferably 2-3 times a week. In these assignments, children can come up with examples for a learned computational technique, create a problem based on a given expression (e.g., 60 + 15 × 3 or a ÷ 5 − b ÷ 8), solve a problem of a specific type (e.g., related to motion, cost, work) or based on a given theme (e.g., about sports, animals, historical events), draw patterns or geometric shapes with specified properties (e.g., a right triangle, a pattern of circles), decode or encode the name of a city, book, or movie using computational examples, and so on.
Creative assignments are typically given as additional homework alongside the mandatory portion and are never graded poorly. The most successful creative works can be compiled into a "Problem Book" at the end of the year, with the students themselves being the authors of their respective works. Such assignments, where children act not as mere performers but as creators, have a profoundly positive impact on the development of children's personalities and contribute to a deeper and more solid assimilation of knowledge.
Volume and difficulty level of homework
It is recommended that students be given two-level homework assignments, consisting of a mandatory part and an optional (additional) part.
The mandatory part should be manageable for the child 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 task from #4-7 that you like."
The optional part, which is completed at the student's discretion, can include additional tasks, asterisk-marked assignments, and so on.
Types and forms of classroom work
It is necessary to diversify the types and forms of work in the classroom. The lesson should include collective, group, and individual forms of work, as well as oral work and work in grid notebooks. The practice of computational skills should be systematic and sufficiently intensive during lessons, but should not exceed 3-4 minutes. It is advisable to give computational exercises a developmental character by selecting answer numbers in such a way that the resulting series can be analyzed and classified, and patterns can be identified by the children. This will help not only to reinforce counting skills but also to prepare children’s thinking for activity-based methods.
When forming concepts using the methods adopted in the course, all types of memory are engaged in students – not only visual and auditory memory but also motor, imagery, tactile, and others.
Working in the workbook should not exceed, as a rule, 10-12 minutes. It mainly involves the independent completion of tasks by the students, which have been prepared beforehand during frontal work with similar but different tasks.
The time for independent task completion is usually limited (typically, from one to three to four minutes). Then, the task is checked using a Smartboard. Children compare their solution with the model answer or sample for self-checking and mark themselves with a "+" or "-." As a result, the child's ability for self-monitoring is purposefully developed.
Since the tasks are checked by the children themselves when completed independently, the teacher focuses primarily on the children's developed self-monitoring skills and record-keeping accuracy during their assessment.
The
system of knowledge assessment
The course includes a multi-level system of knowledge assessment: self-monitoring when introducing new material, peer monitoring during its practice, instructional control through independent assignments, ongoing assessment through periodic tests during the academic year, and a final assessment comprising two stages: a transfer test ("minimum") and a final test (to assess the level of program mastery).
The independent assignments are set at a high level of difficulty, and therefore only success is evaluated. Specifically, if the entire independent task is completed without errors (typically achieved by 3-5 children in the class), a grade of A is awarded. After each independent task, children who made mistakes work on correcting them.
If the error correction is successful and the teacher observes that the child has understood the material, a grade of B or even A can be given for that task. Grades of F or D are not given for independent assignments. "No grade" (indicating that nothing was earned) serves as a more significant signal for the child to be active and correct their own actions than receiving poor grades. The teacher's task is to motivate each child to understand their mistakes and correct them.
The level of tests should be lower than that of independent assignments (but higher than administrative control), and all children should be evaluated. It is recommended that test tasks be designed so that approximately three-quarters of the class can handle them and achieve grades of A or B.
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Intensive exercises to practice computational skills should be included in almost every lesson. It is advisable to give these exercises a developmental character by selecting answer numbers that allow children to analyze, classify, and identify patterns in the resulting series. This approach will help not only consolidate counting skills but also prepare children for active problem-solving methods.
CONTENT OF THE 4th GRADE MATH COURSE
Numbers and arithmetic operations on them
Estimating and approximating sums, differences, products, and quotients.
Division by two-digit and three-digit numbers. Division of whole numbers (with remainder). Division of multi-digit numbers in general. Checking the accuracy of calculations (algorithm, inverse operation, estimation of results, assessing reliability, using a calculator for computation).
Measurements and fractions. Insufficiency of natural numbers for practical measurements. Practical measurements as a source of expanding the concept of numbers.
Fractions. Visual representation of fractions using geometric shapes and number lines. Comparing fractions with the same denominators and fractions with the same numerators. Division and fractions.
Finding a part of a number, a number based on its part, and the part that one number represents of another. Finding a percentage of a number and a number based on its percentage.
Adding and subtracting fractions with the same denominators.
Proper and improper fractions. Mixed numbers. Extracting the whole part from an improper fraction. Representing a mixed number as an improper fraction. Adding and subtracting mixed numbers (with the same denominators of the fractional part).
Constructing and using algorithms for the studied cases of operations with fractions and mixed numbers.
Working with word problems
Independent analysis of the problem, building models, planning, and implementing the solution. Exploring different solution methods. Relating the obtained result to the problem statement, assessing its plausibility. Checking the problem.
Composite problems involving 2-5 steps with natural numbers, involving all arithmetic operations, and multiple comparisons. Problems involving addition, subtraction, and comparison of fractions and mixed numbers.
Problems involving unit conversion (fourth proportion).
Problems involving finding a fraction of a whole and finding the whole based on its fraction.
Three types of fraction problems: finding a part of a number, finding a number based on its part, and finding a fraction that one number represents of another. Problems involving finding a percentage of a number and finding a number based on its percentage.
Problems involving the simultaneous uniform motion of two objects (approaching each other, moving in opposite directions, chasing, lagging): determining the distance between them at a given moment, the time until they meet, and rates of convergence (divergence).
Problems involving calculating the area of a right triangle and the areas of shapes.
Geometric shapes and quantities
Right triangle, its angles, sides (legs and hypotenuse), area, and relationship with a rectangle. Reflex angle. Adjacent and vertical angles. Central angle and inscribed angle in a circle. Angle measurement. Protractor. Constructing angles using a protractor.
Units of area: square millimeter, square centimeter, square decimeter, square meter, are hectare, and their relationships. Estimation of area. Approximate calculation of areas using a grid. Investigation of properties of geometric shapes through measurements. Transformation, comparison, addition, and subtraction of homogeneous geometric quantities. Multiplication and division of geometric quantities by a natural number.
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Quantities and dependencies between them
Dependencies between components and results of arithmetic operations. The formula for the area of a right triangle: A = (a · b) / 2. Scales. Number line. Coordinate line. Distance between points on the coordinate line. The uniform motion of points on the coordinate line is a model of the uniform motion of real objects. Approaching speed and receding speed of two objects during simultaneous uniform motion. Formulas for approaching speed and receding speed. Formulas for distance d between two objects moving uniformly at time t for motion towards each other (d0 = s - (v1 + v2) · t), in opposite directions (d0 = s + (v1 + v2) · t), chasing each other (d0 = s - (v1 - v2) · t), and with one object lagging behind (d0 = s + (v1 - v2) · t). The formula for simultaneous motion s = v(approach) · t(meet).
Coordinate angle. Motion graph. Observing dependencies between quantities and documenting them using formulas, tables, and graphs (motion). Constructing motion graphs using formulas and tables. Transformation, comparison, addition, and subtraction of homogeneous quantities, as well as their multiplication and division by a natural number.
Algebraic concepts
Inequality. The solution is set to inequality. Strict and non-strict inequality. Symbols ≥,≤≥,≤. Double inequality. Solving simple inequalities on the set of non-negative integers using a number line. Using symbolic notation to generalize and systematize knowledge.
Mathematical language and elements of logic
Introduction to symbolic notation for fractions, decimals, percentages, inequality notation, coordinate notation on a line and in a plane, the language of diagrams and graphs. Definition of the truth value of statements. Constructing statements using logical connectives and words such as "true/false that...", "not", "if... then...", "each", "all", "there exists", "always", "sometimes", "and/or".
Working with information and data analysis
Circular, bar, and line diagrams, motion graphs: reading, interpreting data, construction.
The content of the 4th-grade mathematics course is aimed at achieving the following personal, meta-subject, and subject-specific results:
Personal results
The student will develop:
• Motivational foundation for academic activities:
1. Understanding the purpose of learning and adopting the model of a "good student."2. Positive attitude towards school.3. Belief in one's abilities.• Holistic perception of the surrounding world, understanding the history of mathematical knowledge development, and the role of mathematics in the system of knowledge.
• Self-control ability based on standards, focusing on understanding the reasons for success/failure and correcting mistakes.
• Reflective self-assessment based on criteria of academic success, readiness to understand, and consider suggestions and evaluations from teachers, peers, parents, and others.
• Independence and personal responsibility for one's results in both executive and creative activities.
• Acceptance of values: knowledge, creativity, development, friendship, cooperation, health, responsible attitude towards one's health, ability to apply rules of health preservation, and support in academic activities.
• Academic and cognitive interest in studying mathematics and methods of mathematical activities.
• Respectful and positive attitude towards oneself and others, understanding oneself as an individual and a member of the class collective, a citizen of one's country, awareness and demonstration of responsibility for the common well-being and success.
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• Knowledge of the basic moral norms necessary for success in learning and orientation towards their application in academic activities.
• Formation of ethical feelings (shame, guilt, conscience) and empathy (understanding, tolerance towards the individual characteristics of others, compassion) as regulators of moral behavior through learning activities.
• Development of aesthetic feelings through the perception of harmony in mathematical knowledge, internal unity of mathematical objects, and the universality of the mathematical language.
• Acquisition of initial skills for adaptation in a dynamically changing world based on the method of reflexive self-organization.
• Experience in independent successful mathematical activities according to the 4th-grade curriculum.
The student will have the opportunity to develop:
• Internal position as a student and a positive attitude towards school and learning, expressed in a predominance of academic and cognitive motives.
• Stable academic and cognitive motivation and interest in new general problem-solving methods.
• Positive attitude towards the results of one’s own and classmates’ academic activities created by the students themselves.
• Adequate understanding of the reasons for success/failure in academic activities.
• Manifestation of civic identity in actions and activities.
• Ability to solve moral problems based on moral norms, considering the positions of partners and ethical requirements.
• Ethical feelings and empathy, expressed in understanding the feelings of others, empathy, and assistance.
• Ability to perceive the aesthetic value of mathematics, its beauty, and harmony.
• Adequate self-assessment of one's own actions based on the criteria of a "good student" role, creating an individual diagram of one's qualities as a student and being focused on self-development.
Metasubject Results
Regulatory
The student will learn:
• to accept and maintain a learning task;
• to apply self-motivation techniques to learning activities;
• to plan, including internally, their learning activities in accordance with the clarified structure (15 steps);
• to consider the guidelines provided by the teacher for new learning material in collaboration with the teacher;
• to apply learned methods and algorithms for performing key steps in learning activities:
— trial learning action,
— identifying individual difficulties,
— identifying the location and cause of difficulties,
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— creating a plan to overcome difficulties (setting goals, selecting methods for implementation, developing an action plan, choosing resources, determining timeframes),
— implementing the devised plan and documenting new knowledge as a standard,
— assimilating new knowledge,
— self-monitoring learning outcomes,
— self-evaluating learning activities based on success criteria;
• to differentiate between knowledge, skills, projects, goals, plans, methods, means, and outcomes of learning activities;
• to perform learning actions in tangible, media-based, oral, and mental forms;
• to apply learned methods and algorithms for performing key steps in corrective activities:
— independent work,
— self-assessment (based on examples, detailed samples, and standards),
— identifying errors,
— determining the cause of errors,
— correcting errors based on a general error correction algorithm,
— self-monitoring the outcome of corrective activities,
— self-evaluating corrective activities based on success criteria;
• to use mathematical terminology learned in the 4th grade to describe the results of their learning activities;
• to perceive and take into account suggestions and evaluations from teachers, peers, parents, and other individuals;
• to make necessary adjustments to their actions after completion based on evaluation and considering the nature of the mistakes made, using suggestions and evaluations to create new, improved results;
• to apply an algorithm for reflecting on their learning activities.
The student will have the opportunity to learn:
• to transform practical tasks into cognitive ones;
• to independently consider the guidelines provided by the teacher for new learning material;
• to document the steps of the clarified structure of learning activities (15 steps) and independently implement it in its entirety;
• to conduct self-evaluation of the ability to apply learned techniques of positive self-motivation to learning activities,
— self-evaluation of the ability to apply learned methods and algorithms for performing key steps in learning activities,
— self-evaluation of the ability to demonstrate responsibility in learning activities,
— self-evaluation of the ability to apply an algorithm for reflecting on their learning activities;
• to document the steps of the clarified structure of corrective activities (15 steps) and independently implement it in its entirety;
• to set new learning tasks in collaboration with the teacher;
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• to determine the types of projects based on the learning objectives and independently engage in project-based activities.
Cognitive
The student will learn:
• to understand and apply mathematical terminology to solve learning tasks according to the 4th-grade curriculum, using symbolic notation, including models and diagrams, for problem-solving;
• to perform logical operations based on learned algorithms — analyzing objects by identifying essential features, synthesis, comparison, and classification based on given criteria, generalization, and analogy, and concept formation;
• to establish cause-and-effect relationships within the scope of studied phenomena;
• to apply learned algorithms of cognitive methods — observation, modeling, investigation — in learning activities;
• to engage in project-based activities using various project structures depending on the learning objectives;
• to apply rules for working with text, extracting essential information from different types of messages (primarily texts);
• to apply basic methods of integrating new knowledge into their existing knowledge system;
• to search for the necessary information to complete learning tasks using textbooks, encyclopedias, and reference materials (including electronic and digital resources), in open information space, including controlled Internet environments;
• to record selective information about the surrounding world and oneself, including using ICT tools, and systematize it;
• to be open to diverse problem-solving approaches;
• to construct oral and written statements and arguments about an object, its structure, properties, and relationships;
• to possess a set of general problem-solving techniques;
• to understand and apply basic interdisciplinary concepts according to the 4th-grade curriculum (evaluation, estimation, diagrams: circular, bar, linear, graphs, etc.);
• to create and solve their own problems, examples, and equations according to the 4th-grade curriculum;
• to understand and use the signs and symbols used in the 4th-grade textbook and workbook for organizing learning activities.
The student will have the opportunity to learn:
• to conduct self-evaluation of the ability to apply the algorithm of analogical reasoning;
• to conduct self-evaluation of the ability to apply observation and investigation methods to solve learning tasks;
• to conduct self-evaluation of the ability to create and modify models and diagrams for problem-solving;
• to conduct self-evaluation of the ability to use reading comprehension strategies;
• to establish and apply basic rules for information retrieval.
• Presenting projects based on the learning objective;
• Conducting extensive information searches using library resources and the Internet;
• Presenting information and documenting it in various ways for communication purposes;
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• Understanding that new knowledge helps solve new problems and is an element of the knowledge system;
• Constructing coherent and deliberate oral and written messages;
• Choosing the most effective problem-solving methods based on specific conditions;
• Building logical reasoning that includes establishing cause-and-effect relationships;
• Voluntarily and consciously applying learned general problem-solving techniques;
• Applying knowledge from the 4th-grade curriculum in modified conditions;
• Solving creative and exploratory problems in accordance with the 4th-grade curriculum.
Communicative
The student will learn:
• Distinguishing essential differences between discussions and arguments, applying rules of conducting discussions, and formulating their own position;
• Acknowledging the possibility of different perspectives, respecting others' opinions, and demonstrating tolerance towards the characteristics of the interlocutor;
• Striving for consensus among different positions in collaborative activities, negotiating and reaching a common decision based on communicative interaction (including situations involving conflicting interests);
• Allocating roles in communicative interactions, formulating the functions of "author," "listener," "critic," "organizer," and "arbiter," and applying rules while performing these roles (constructing coherent expressions understood by the partner, asking questions for comprehension, using agreed-upon standards to justify one's own point of view, etc.);
• Appropriately utilizing linguistic means to solve communicative tasks, constructing monologues, and possessing dialogic speech form;
• Understanding the significance of teamwork in achieving positive results in collaborative activities and applying rules of teamwork;
• Understanding the importance of cooperation in teamwork and applying rules of cooperation;
• Understanding and applying recommendations for student adaptation in a new collective.
The student will have the opportunity to learn:
• To conduct self-evaluation of the ability to apply rules of conducting discussions based on a standard;
• To conduct self-evaluation of the ability to perform the roles of "arbiter" and "organizer" in communicative interactions;
• To conduct self-evaluation of the ability to justify one's own position;
• To conduct self-evaluation of the ability to consider others' positions in communicative interactions;
• To conduct self-evaluation of the ability to participate in teamwork and contribute to the team's successful outcomes;
• To conduct self-evaluation of the ability to demonstrate respect and tolerance towards others in collaboration;
• To engage in mutual control and provide necessary assistance in collaboration.
SUBJECT OUTCOMES
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Numbers and arithmetic operations on them
The student will learn:
• to estimate and approximate sums, differences, products, and quotients;
• to perform division of multi-digit numbers by two-digit and three-digit numbers;
• to verify the accuracy of calculations using the algorithm, inverse operations, estimation, approximation of the result, and calculation using a calculator;
• to perform mental calculations with multi-digit numbers, involving operations within the range of 100;
• to calculate the values of numerical expressions with known natural numbers within the range of 1,000,000,000, involving 4-6 operations (with and without parentheses), based on the rules of the order of operations;
• to identify fractions, visually represent them using geometric shapes and a number line, compare fractions, find a fraction of a number and a number based on a fraction;
• to read and write fractions, represent them visually using geometric shapes and a number line, compare fractions with the same denominators and fractions with the same numerators;
• to find a part of a number, a number based on its part, and the part that one number represents of another;
• to add and subtract fractions with the same denominators;
• to read and write mixed numbers, represent them visually using geometric shapes and a number line, separate the whole part from an improper fraction, represent a mixed number as an improper fraction, add and subtract mixed numbers (with the same denominators of the fractional part);
• to apply the properties of arithmetic operations to a set of fractions.
The student will have the opportunity to learn:
• to independently develop and use algorithms for oral and written operations with multi-digit numbers, fractions, and mixed numbers;
• to perform division of rounded numbers (with remainder);
• to find a percentage of a number and a number based on its percentage using general problem-solving methods;
• to create and present their own project on the history of fractions and operations with them;
• to solve problems involving order of operations with numerical expressions;
• to create and solve their own problems based on the studied cases of numerical operations.
Working with word problems
The student will learn:
• to independently analyze problems, build models, plan and implement solutions, explain the solution process, search for different solution methods, compare the obtained result with the problem statement, assess its plausibility, and solve problems with questions;
• to solve composite problems involving 2-5 operations with natural numbers using the concepts of arithmetic operations, comparison by difference and multiples, and uniform processes (of the form a = bc);
• to solve problems involving reducing to unity (fourth proportional);
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• to solve simple and composite problems involving 2-5 operations of addition, subtraction, and comparison by difference with fractions and mixed numbers;
• to solve problems involving finding a fraction of a number and a number based on its fraction;
• to solve three types of problems involving fractions: finding a part of a number, a number based on its part, and a fraction representing one number as a fraction of another;
• to solve problems involving the simultaneous uniform motion of two objects approaching each other, moving in opposite directions, following each other, with one object lagging behind: determining the rate of approach and the rate of separation, the distance between moving objects at a given moment, and the time until they meet;
• to solve problems of all studied types with algebraic data and create word problems based on given algebraic expressions;
• to independently create their own problems based on mathematical models such as numerical and algebraic expressions, diagrams, and tables;
• to perform all arithmetic operations with the studied quantities when solving problems.
The student will have the opportunity to:
• independently develop and use algorithms for solving text problems;
• analyze, model, and solve text problems involving 6-8 operations with all the studied arithmetic operations;
• solve problems involving finding a percentage of a number and finding a number based on its percentage, as a particular case of part problems;
• solve problems involving the calculation of the area of a right-angled triangle and the areas of figures composed of rectangles, squares, and right-angled triangles;
• solve non-standard problems related to the topics studied and use motion graphs to solve text problems.
Geometric shapes and quantities
The student will learn:
• to recognize a right-angled triangle, its angles, and sides (legs and hypotenuse), and find its area based on its relationship with a rectangle;
• to find the areas of shapes composed of squares, rectangles, and right-angled triangles;
• to directly compare angles using the overlay method;
• to measure angles using various measuring tools;
• to measure angles using a protractor and express them in degrees;
• to find the sum and difference of angles;
• to construct an angle of a given magnitude using a protractor;
• to recognize a straight angle, adjacent and vertical angles, a central angle, and an angle inscribed in a circle, and investigate their basic properties through measurements.
The student will have the opportunity to:
• independently establish methods for comparing angles, measuring them, and constructing them using a protractor;
• when exploring the properties of geometric shapes through practical measurements and physical models, formulate their own hypotheses (properties of adjacent and vertical angles, the sum of angles in a triangle, quadrilateral, pentagon; properties of central and inscribed angles, etc.);
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• draw conclusions that the properties observed in specific shapes cannot be generalized to all geometric shapes of the same type, as it is impossible to measure each one of them.
Quantities and their relationships
The student will learn:
• to use the relationships between the studied units of length, area, volume, mass, and time in calculations;
• to convert, compare, add, and subtract homogeneous quantities, multiply and divide quantities by natural numbers;
• to use new units of area in the sequence of studied units - 1 mm², 1 cm², 1 dm², 1 m², 1 a, 1 ha, 1 km²; to convert them, compare them, and perform arithmetic operations on them;
• to estimate the area and approximate the calculation of areas using a pallet;
• to establish the relationship between the sides and the area of a right-angled triangle and express it using the formula A = (a · b) ÷ 2;
• to find the scale division, use the scale to determine the value of a quantity;
• to recognize the number line, identify its essential features, determine the position of a number on the number line, and add and subtract numbers using the number line;
• to identify the essential features of the coordinate line, determine the coordinates of points on the coordinate line with non-negative integer coordinates, construct and use the distance formula between its points to solve problems;
• to construct models of simultaneous uniform motion of objects on the coordinate line;
• to observe, using the coordinate line and tables, the dependencies between quantities describing the simultaneous uniform motion of objects, construct formulas for closing and separating velocities for all cases of simultaneous uniform motion, and use the constructed formulas to solve problems;
• to recognize the coordinate angle, name its essential features, determine the coordinates of points on the coordinate angle, and construct points based on their coordinates;
• to read and, in simple cases, construct circular, linear, and bar graphs;
• to read and construct motion graphs, determine from them: the time of departure and arrival of an object; the direction of its motion; the location and time of meeting with other objects; the time, location, duration, and number of stops;
• to create stories based on motion graphs, reflecting events that could be represented by the given motion graphs;
• to use the dependencies between the components and results of arithmetic operations to estimate the sum, difference, product, and quotient.
The student will have the opportunity to learn:
• to independently construct a scale with a given division, a coordinate line, and formulas for the distance between points on the coordinate line, formulas for the dependency of the coordinate of a moving point on the time of motion, etc.;
• to observe, using tables and the number line, the dependencies between variables, and express them in simple cases using formulas;
• to determine, from formulas of the form x = a + bt, x = a - bt, expressing the dependency of the coordinate x of a moving point on the time of motion t;
• to construct and use formulas for the distance d between two uniformly moving objects at a given time t for motion towards each other (d = s0 - (v1 + v2) · t), in opposite directions (d = s0 + (v1 + v2) · t), chasing (d = s0 - (v1 - v2) · t), and falling behind (d = s0 + (v1 - v2) · t); • to encode figures of a coordinate angle using the coordinates of points, independently create figures using broken lines, transmit encoded images "at a distance," and decode the codes;
• to determine the velocities of objects based on the motion graph;
• to independently create motion graphs and come up with stories based on them.
Algebraic concepts
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The student will learn:
• to read and write expressions involving 2-3 arithmetic operations, starting with the name of the last operation;
• to write in symbolic form the commutative, associative, and distributive properties of addition and multiplication, rules for subtracting a number from a sum and a sum from a number, dividing a sum by a number, special cases of operations with 0 and 1, and use all these properties to simplify calculations;
• to extend the properties of arithmetic operations to the set of fractions;
• to solve simple equations involving all arithmetic operations of the form a + x = b, a - x = b, x - a = b, a · x = b, a ÷ x = b, x ÷ a = b mentally, at the level of automated skill, and be able to justify their choice of operation based on a graphical model, explain the steps of the solution, naming the components of the actions;
• to solve compound equations that can be reduced to a sequence of simple steps (3-4 steps) and explain the steps of the solution in terms of components of the actions;
• to read and write strict, non-strict, and double inequalities using the symbols >,
• to solve simple inequalities on the set of non-negative integers using the number line and mental reasoning, record the sets of their solutions using set notation.
The student will have the opportunity to learn:
• based on the general properties of arithmetic operations in simple cases:
• to determine the set of roots of non-standard equations;• to simplify algebraic expressions; • to use symbolic notation to generalize and systematize knowledge.Mathematical language and elements of logic
Students will learn:
• to recognize, read, and apply new symbols of mathematical language, including fractional notation, fractions, percentages (symbol %), strict, non-strict, and double inequalities using the symbols >, ≤≤, ≥≥, the approximate equality symbol ≈, coordinate notation on a line and a plane, circular, bar, and line diagrams, and motion graphs;
• to determine the truth and falsehood of statements in simple cases, construct simple statements using logical connectors and words like "true/false," "not," "if... then...," "each," "all," "there exists," "always," "sometimes," "and/or";
• to justify their judgments using the rules and properties learned in the 4th grade, and make logical conclusions;
• to conduct simple logical reasoning under adult guidance, using logical operations and connectors.
Students will have the opportunity to learn:
• to justify statements of a general nature and statements of existence in simple cases, based on common sense;
• to solve logical problems using graphical models, tables, graphs, and Euler-Venn diagrams;
• to construct (under adult guidance and independently) and master problem-solving techniques of a logical nature according to the 4th-grade curriculum.
Working with information and data analysis
Students will learn:
• to use tables, circular, linear, and bar diagrams, and motion graphs to analyze, represent, and systematize data; compare values using these tools, interpret data from tables, diagrams, and graphs;
• to work with text: identify parts of the educational text - introduction, main idea, important notes, examples illustrating the main idea and important notes, check comprehension of the text;
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• to carry out project work on topics such as "From the History of Fractions," "Sociological Survey" (on a given or self-selected topic), develop a plan for information search, select information sources (reference books, encyclopedias, controlled internet space, etc.), choose methods of information presentation;
• to complete creative tasks on topics such as "Transmission of Information Using Coordinates," "Motion Graphs";
• to work in the material and informational environment of primary general education (including with educational models) in accordance with the content of the subject "Mathematics, Grade 4."
Students will have the opportunity to learn:
• to summarize educational texts;
• to carry out extracurricular project work (under adult guidance and independently), gather information from reference books, encyclopedias, and controlled internet sources, and present information using available technical means;
• using the information found in various sources, create their own problems based on the 4th-grade curriculum and become co-authors of the "Problem Book for Grade 4," which includes the best problems created by students; compile a portfolio of a 4th-grade student.
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