BYOM 2nd Grade Workbook

Overview

Main Teaching Features

Our Educational Focus

We develop the whole child, not just math skills.

This includes:

• Creative thinking
• Problem-solving abilities
• Interest in learning
• Emotional growth

Our Teaching Method

We use L. Peterson's activity method. Each lesson follows a structured approach.

Lesson structure:

• Motivation (self-determination) for activity
• Actualization of knowledge and fixing difficulties
• Identification of the difficulty and its cause
• Development of a project to overcome the difficulty
• Implementation of the constructed project
• Initial consolidation with verbal repetition
• Independent work with self-checking
• Integration into the system of knowledge
• Reflection on the activity (lesson summary)

Didactic Principles

The activity method requires specific teaching conditions. These principles guide our approach.

Key principles:

• Activity-based learning
• Continuity and progression
• Holistic worldview
• Minimax principle (each student at their own pace)
• Psychological comfort
• Variability and creativity

Ignoring these principles can lead to student overload. Each student progresses at their maximum but manageable level.

Connection to Real-World Practice

Math education must show children where concepts come from. Students need to understand math's role in the world.

Three stages of mathematical knowledge:

• Mathematization: Building a mathematical model from real life
• Model Study: Constructing a theory describing the model
• Application: Applying results back to the real world

For example, natural numbers aren't abstract starting points. Students first work with finite sets of objects.

Addition and subtraction begin with combining and removing parts of sets. Operations on two-digit numbers use dots and figures matching historical development.

Developing Information Technology Skills

The computerized world requires new thinking skills. Our program builds these abilities.

Important skills developed:

• Creating and executing action plans
• Following rules and algorithms
• Evaluating answer credibility
• Considering different solutions
• Finding necessary information

Multi-Level Program Design

Students with different preparation levels can succeed. The minimax principle makes this possible.

Training occurs at a high level of difficulty. This means the "zone of proximal development" for the most prepared students.

However, individual characteristics matter. Each child develops faith in their abilities.

Assessment focuses on success and progress. Errors are working situations requiring correction.

Tests are given at a lower level than classroom work. This nearly eliminates poor grades.

Final grades are based on achievements (high grades only). Low grades appear rarely - only for non-compliance with agreed tasks.

Supporting Every Student

High material presentation is an opportunity, not a requirement. Teachers notice and support even the smallest success.

Incorrect answers shouldn't trigger negative reactions. Other students often provide corrections.

The teacher's role is providing moral support: "Well done! You helped us understand! Do you agree now?"

Internal activity and reflection habits are especially important for students with difficulties. High-difficulty work must combine with trust, respect, and friendliness.

Three beliefs must exist:

• "You can do it!" - teacher believes in student
• "I can do it!" - student believes in themselves
• "They can do it!" - classmates believe in each other

Otherwise, learning loses personal meaning. The school cannot help students reach their individual maximum.

Task Volume and Individual Trajectories

The course task volume sets the level for the most prepared children. Not every child completes all tasks.

Only 3-4 key tasks on new topics are mandatory for everyone. Review tasks developing required outcomes are also required.

More prepared children may complete additional tasks. However, overloading children is not allowed.

Practice occurs parallel to exploring new ideas. Training exercises don't tire children - they're presented in game format.

Less prepared children can work slowly on necessary skills. Prepared children constantly receive "food for thought."

Math lessons become attractive for all students - both strong and less prepared.

Joy of Discovery

Every child must experience discovery at every lesson. This develops faith in their abilities and cognitive interests.

Interest and successful learning are the main quality parameters. These determine full development of the child.

Creative Work as Self-Expression

Creative work allows each child to express themselves. Creative assignments should be offered systematically.

Examples of creative tasks:

• Inventing examples of a studied technique
• Creating problems based on expressions
• Creating problems of a given type
• Drawing patterns or geometric shapes
• Deciphering or encoding names using examples

Creative assignments are additional homework. They're never graded poorly.

At year end, the best works can be collected in a "Problem Ledger" authored by students. Children act as creators, not just performers.

This positively affects personality development. It contributes to deeper, more solid knowledge acquisition.

Homework Volume and Difficulty

Two-level homework is recommended. It consists of mandatory and optional parts.

Mandatory part characteristics:

• Manageable for independent completion
• Takes approximately 15-20 minutes
• Selected by the teacher
• May involve student choice

Optional part completed at student's discretion. May include additional asterisk-marked assignments.

Lesson Work Variety

Diversify types and forms of work. Lessons should include collective, group, and individual activities.

Include both oral work and notebook work. Computational practice should be systematic and intensive but brief (3-4 minutes).

Give computational exercises a developmental character. Select answer numbers allowing analysis, classification, and pattern identification.

This consolidates counting skills and prepares thinking for activity-based methods.

Multi-Sensory Learning

Students activate all memory types. This includes visual, auditory, motor, figurative, and tactile memory.

Special attention goes to rhythmic games. In Grade 1, these help master counting by 2, 3, 4, up to 9.

This prepares a solid foundation for multiplication tables in Grade 2.

Starting with counting by 5, children compose movements themselves. Movements are easier and faster to remember.

Memorization of multiples of one-digit numbers happens faster.

Notebook Work Guidelines

Notebook work should not exceed 10-12 minutes. It mainly involves independent task completion.

Tasks are prepared during frontal work. Students complete similar but different tasks independently.

Time is usually limited (one to four minutes). Then tasks are checked.

Children compare solutions with standards or samples. They give themselves a "+" or "-".

This deliberately develops self-monitoring ability. Teachers check primarily for self-monitoring skills and record-keeping accuracy.

Knowledge Control System

The course provides multi-level knowledge control.

Control types:

• Self-control when introducing new material
• Peer control during practice
• Educational control in independent work
• Ongoing control during quizzes
• Final control

Independent work is conducted at high difficulty. Only success is evaluated.

If work is done without mistakes, the highest grade is awarded. After each work, students who made mistakes work on corrections.

If correction work succeeds, a high grade can be awarded. Poor grades are not given.

"No grade" is a more significant signal than poor grades. The teacher encourages understanding and correcting mistakes.

Grade 2 Course Content

Universal learning activities are formed during study of core questions.

Core topics:

• Numeration, comparison, addition, and subtraction within 1000
• Multiplication and division tables
• Multiplication and division within 100
• Mental calculations within 100 (reinforced to automatic skill)

Tasks with letter data are introduced. Unit conversion and composite problems involving all four operations are studied.

Special attention goes to independent word problem analysis. Geometric concept formation and pattern identification abilities develop.

Logical and variational thinking are emphasized.

Almost every lesson includes intensive computational skill exercises. Computational exercises have developmental character.

Select answer numbers allowing analysis and classification. This consolidates skills and prepares for activity-based learning.

Results of Studying the Mathematics Course for Grade 2

The content aims at achieving personal, meta-subject, and subject results.

Personal Results

• Understanding of student academic activity and "learner" social role
• Initial understanding of corrective activity
• Understanding value of knowledge for personal development
• Basic understanding of generalized nature of mathematical knowledge
• Motivation to work for results
• Experience of independence and personal responsibility
• Experience of self-control according to samples and standards
• Calm attitude towards mistakes as working situations
• Experience applying health preservation rules
• Ability to work in pairs and groups
• Knowledge of basic communication rules
• Experience coordinating actions and results in groups
• Demonstrating activity, kindness, honesty, and patience
• Showing respect towards teachers, family, peers, and country
• Understanding self and classmates as people with good qualities
• Knowledge of methods for identifying positive qualities
• Knowledge of emotional management techniques
• Understanding goal-orientation and independence as means to achieve results
• Experience resolving disputes through agreed norms
• Experience independently completing mathematical tasks

The Student Will Learn To:

• Understand and apply mathematical terminology
• Apply algorithms for analyzing and comparing objects
• Make generalizations and specify general concepts
• List means used to discover new knowledge
• Read and construct graphic models and diagrams
• Relate real objects to geometric shape models
• Comment on progress of completing tasks
• Use standards to justify correctness
• Identify extra or missing data in problems
• Create and solve own problems and equations
• Understand basic interdisciplinary concepts
• Understand signs and symbols in textbook and workbook

The Student Will Also Have the Opportunity to Learn:

• To conduct self-assessment of their abilities
• To explore non-standard situations
• To apply knowledge in altered conditions
• To solve problems of a creative and exploratory nature

Subject Outcomes: Numbers and Arithmetic Operations

The Student Will Learn:

• To apply mental addition and subtraction 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
• To represent numbers as sum of hundreds, tens, and ones
• To perform calculations according to programs in parentheses
• To determine operation order in expressions
• To use associative property of addition
• To understand meaning of multiplication and division
• To perform multiplication and division of natural numbers
• To calculate specific cases with 0 and 1
• To conduct multiple comparisons of numbers
• To name divisors and multiples
• To use commutative property of multiplication
• To find results using multiplication table
• To multiply and divide by 10 and 100
• To calculate values of arithmetic expressions (3-4 operations)
• To use arithmetic properties to rationalize calculations
• To perform division with remainder
• To mentally calculate with numbers up to 1000
• To 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 operations
• Express numbers in different units of measurement
• Independently derive multiplication and division techniques
• Graphically interpret multiplication, division, and comparisons
• See analogy in relationships between operation components

Working with Word Problems

The Student Will Learn:

• To solve simple problems involving multiplication and division
• To solve simple problems involving multiple comparisons
• To create simple expressions and solve inverse problems
• To analyze simple and composite problems involving all operations
• To create graphical models and tables
• To plan and implement solutions
• To perform arithmetic operations with quantities
• To solve problems calculating perimeter and area

The Student Will Have the Opportunity to Learn:

• To solve word problems involving letters as data
• To create letter expressions based on word problems
• To solve problems with incorrect formulations
• To model and solve problems involving 4-5 steps
• To find and justify solution methods
• To find various methods of solving a problem
• To establish analogy between different problem solutions
• To compare obtained results with problem conditions
• To solve problems involving "hidden numbers"

Geometric Shapes and Quantities

The Student Will Learn:

• To recognize, label, and draw straight lines, rays, and segments
• To measure line segment length using a ruler
• To find length of polylines and perimeter of polygons
• To identify rectangles and squares using drafting triangle
• To construct rectangles and squares on graph paper
• To compute perimeter and area of rectangles and squares
• To recognize rectangular prisms and cubes
• To construct circles using a compass
• To differentiate between circles and disks
• To label center, radius, and diameter
• To express lengths in different units
• To determine area using a given scale
• To express area in different units
• To convert, compare, add, and subtract geometric quantities

The Student Will Have the Opportunity to Learn:

• To independently identify properties of geometric shapes
• To recognize acute, obtuse, and right angles
• To determine intersecting, parallel, and perpendicular lines
• To draw patterns from circles using compass
• To compose shapes from parts and break figures into parts
• To calculate areas of shapes composed of rectangles
• To find volume of prisms and cubes

Quantities and Their Relationships

The Student Will Learn:

• To differentiate between quantity concepts and units
• To recognize, compare, and order quantities
• To measure area and volume using prepared drawings
• To use new units of length measurement
• To convert learned units based on relationships
• To compare and add or subtract quantities
• To observe dependency of measurement results on unit choice
• To express observed dependencies in speech
• To use formulas (A = a · b; V = (a · b) · c)

The Student Will Be Able To:

• Make independent choice of convenient units
• Observe dependencies between variables using tables
• Establish relationship between multiplication and division components

Algebraic Concepts

The Student Will Learn:

• To read and write numerical and algebraic expressions
• To find values of simple algebraic expressions
• To express relationships using algebraic equations
• To write properties of arithmetic operations in algebraic forms

Examples of algebraic properties:

1. a + b = b + a (commutative property of addition)
2. (a + b) + c = a + (b + c) (associative property of addition)
3. a · b = b · a (commutative property of multiplication)
4. (a · b) · c = a · (b · c) (associative property of multiplication)
5. (a + b) · c = a · c + b · c (distributive property)
6. (a + b) - c = (a - c) + b (subtraction of number from sum)
7. a - (b + c) = a - b - c (subtraction of sum from number)
8. (a + b) ÷ c = a ÷ c + b ÷ c (division of sum by number)

Students solve equations of the form a · x = b, x · a = b, a ÷ x = b, x ÷ a = b. They use the associative method based on rectangle area relationships.

The Student Will Be Able To:

• Independently identify and express properties in algebraic form
• Explain solutions to simple equations of all studied types

Mathematical Language and Elements of Logic

The Student Will Learn:

• To recognize, read, and apply new mathematical symbols
• To build simple statements (true/false, not, if...then...)
• To determine truth and falsehood of statements
• To establish patterns in simple cases

The Student Will Have the Opportunity To:

• Justify judgments using learned rules and properties
• Independently build and master techniques for solving logical problems

Working with Information and Data Analysis

The Student Will Learn To:

• Read and fill in tables according to given rules
• Create sequences of objects, numbers, and figures
• Determine operation, object, and result
• Perform direct and inverse operations
• Find unknowns in operations
• Execute algorithms (linear, branched, and cyclic)
• Perform ordered enumeration using tables
• Find information on given topics in different sources
• Work in material and informational environment

The Student Will Have the Opportunity to Learn To:

• Independently create algorithms and record them
• Collect and present information from references and encyclopedias
• Create own tasks based on obtained data
• Become co-author of "Grade 2 Problem Book"
• Create a portfolio of Grade 2 student

Program Structure and Progression

The first part begins with prolonged review. This occurs parallel to clarifying known concepts.

Students learn written and oral methods for two-digit numbers. Then they master three-digit numbers.

All computational methods are "discovered" by children themselves. They use particular models adopted in the program.

Working with particular models ("triangles and dots") is mandatory at this stage.

Introducing Three-Digit Numbers

When three-digit numbers are introduced, the meter is introduced as a new unit. The interrelation between meter, decimeter, and centimeter is revealed.

This is based on analogy between decimal number notation and decimal measurement system. This is demonstrated using "triangles and dots."

Equations and Text Problems

Throughout the first part, simple addition and subtraction equations are presented. Compound text problems contain new calculation cases.

When solving text problems, special attention goes to independent analysis. Children build graphic models.

Geometric Development

Geometric representation development continues. Children learn to mark points and draw straight lines.

They find points of intersection. To develop variational thinking, problems allow different answers.

Problems involving permutations of three elements are included. Development of analysis, comparison, and generalization continues.

Classification, pattern identification, and expressing patterns in speech are emphasized. Latin letters designate geometric objects.