Maths Learning Objectives and Outcomes 

Curricular Goals(Learning Objectives) & Competencies

Curricular Goals, Competencies and 

Illustrative Learning Outcomes (LOs) Issued by CBSE & NCF

LEARNING OBJECTIVES

Introduction:

Learning objectives in mathematics play a crucial role in guiding the teaching and learning process. They serve as specific, measurable goals that help educators design effective lessons and assessments while providing students with a clear understanding of what is expected of them.

In summary, learning objectives in mathematics are essential for creating a structured, focused, and effective learning experience. They benefit both educators and students by providing a roadmap for instruction, assessment, and continuous improvement.

Here are some key reasons why learning objectives in mathematics are important:

• Learning objectives provide clarity on what students are expected to learn and achieve in a particular lesson, unit, or course.

• Learning objectives are often aligned with educational standards, ensuring that the curriculum meets established benchmarks and requirements

• Teachers use learning objectives to plan their lessons effectively, selecting appropriate instructional methods, resources, and assessments that align with the desired outcomes.

• Learning objectives serve as a basis for creating assessments that measure student understanding and mastery of specific mathematical concepts and skills
• Clearly stated objectives can motivate students by providing a sense of purpose and direction in their learning.

LEARNING OBJECTIVES

PREPARATORY STAGE (Nursery to 5nd)
CG-1 Understands numbers (counting numbers and fractions), represents whole numbers using the Indian place value system, understands and carries out the four basic operations with whole numbers, and discovers and recognizes patterns in number sequences.	C-1.1 Represents numbers using the place-value structure of the Indian number system, appreciates the key role of zero in this system, compares the sizes of whole numbers, and knows and can read the names of very large numbers. C-1.2 Represents and compares commonly used fractions in daily life (such as 1 ⁄ 2, 1 ⁄ 4, etc.) as parts of unit wholes, as locations on number lines, and as divisions of whole numbers. C-1.3 Identifies relationships amongst operations and applies the four basic operations on whole numbers to solve daily life problems. C-1.4 Discovers, recognises, describes, and extends simple number patterns such as odd numbers, even numbers, square numbers, cubes, powers of 2, powers of 10, and Virahanka--Fibonacci numbers.
CG-2 Analyses the characteristics and properties of two- and three-dimensional geometric shapes, specifies locations and describes spatial relationships, and recognises and creates shapes that have symmetry.	C-2.1 Identifies, compares, and analyses attributes of two- and three-dimensional shapes and develops vocabulary to describe their attributes/properties. C-2.2 Identifies and builds a three-dimensional object from two-dimensional representations of that object. C-2.3 Describes location and movement using both common language and mathematical vocabulary; understands the notion of map (najri naksha). C-2.4 Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes. C-2.5 Discovers, recognizes, describes, and extends patterns in 2D and 3D shapes.
CG-3 Understands measurable attributes of objects and the units, systems, and processes of such measurement, including those related to distance, length, mass, weight, area, volume, and time, using non-standard and standard units.	C-3.1 Measures using non-standard and standard units and recognises and appreciates the need for standard units. C-3.2 Uses an appropriate unit and tool for the attribute being measured. C-3.3 Carries out simple unit conversions, such as from centimetres to metres, within a system of measurement, and solves daily life problems. C-3.4 Devises strategies for estimating the distance, length, time, perimeter (for regular and irregular shapes), area (for regular and irregular shapes), weight and volume. C-3.5 Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas. C-3.6 Measures distance, length, perimeter, time, weight, area, and volume and to solve daily life problems.
CG-4 Develops problem solving skills with procedural fluency, to solve mathematical puzzles as well as daily life problems, and as a step towards developing computational thinking.	C-4.1 Solves puzzles and daily life problems involving one or more operations on whole numbers. C-4.2 Selects appropriate methods and tools for computing with whole numbers such as mental computation, estimation, or paper and pencil calculation, in accordance with the context.
CG-5 Knows and appreciates the development of numeration through human history including the major contributions of India.	C-5.1 Understands the development of the representation of numbers through human history, from tallying (e.g., on the Lebombo bones), to Roman numerals, to the Mayan and Babylonian systems, leading up to the development of zero in India and the modern Indian system of writing numerals (from Yajurveda, story of Buddha, Bakshali Manuscript, Vasavadatta, Aryabhatiya, Brahmasphutasiddanta, Gwalior inscription, etc.) and its transmission to the world (due to Al-Kharizmi, Al-Kindi, Fibonacci, etc.).
MIDDLE STAGE (6th to 8th)
CG-1 Understands numbers and sets of numbers (Whole numbers, Fractions, Integers, and Rational numbers) looks for patterns, and appreciates relationships between numbers.	C-1.1 Develops a sense for and an ability to manipulate (e.g., read, write, form, compare, estimate, and apply operations) large whole numbers of up to 10 digits and expresses them in scientific notation using exponents and powers. C-1.2 Discovers, identifies, and explores patterns in numbers and describes rules for their formation (e.g., prime numbers, powers of 3, etc.) and explain relations between different patterns. C-1.3 Explores and understands sets of numbers such as whole numbers, fractions, integers, and rational numbers, and their properties. C-1.4 Represents rational numbers in decimal form as an extension of the Indian system of numeration `past the decimal point’. C-1.5 Explores the idea of percentage and apply it in solving problems. C-1.6 Explores and applies fractions (both as ratios and in decimal form) in daily life situations.
CG-2 Understands the concepts of variable, constant, coefficient, expression, and (one-variable) equation, and uses these concepts to solve meaningful daily life problems with procedural fluency.	C-2.1 Extends the abstract representation of a number in the form of a variable or an algebraic expression using a variable. C-2.2 Forms algebraic expressions using variables, coefficients, and constants, and manipulates them through addition, subtraction, and multiplication. C-2.3 Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems. C-2.4 Develops own methods to solve puzzles and problems using algebraic thinking.
CG-3 Understands, formulates, and applies properties and theorems regarding simple geometric shapes (2D and 3D).	C-3.1 Describes, classifies, and understands relationships among different types of two and three-dimensional shapes using their defining properties/attributes. C-3.2 Knows properties of lines, angles, triangles, quadrilaterals, and polygons, and applies them to solve related problems. C-3.3 Identifies attributes of three-dimensional shapes (cubes, parallelepipeds, cylinders, cones, etc.) and uses two-dimensional representations of three-dimensional objects to visualise and solve problems. C-3.4 Draws and constructs geometric shapes such as lines, parallel lines, angles, and simple triangles, with specified properties, using compass and straightedge.
CG-4: Develops understanding of perimeter and area for 2D shapes and uses them to solve day-to-day life problems.	C-4.1 Identifies, selects, and uses units of appropriate size and type to measure and examine the relationship between perimeter and area for 2D shapes (both regular and irregular shapes). C-4.2 Discovers, understands, and uses formulas to determine the circumference of a circle and the area of a triangle, parallelogram, and trapezium, and develops strategies to find the areas of more complex 2D shapes. C-4.3 Explores and uses Baudhayana’s Theorem on right triangles and other fundamental geometric theorems to solve puzzles and everyday problems. C-4.4 Discovers and constructs tilings of the plane using 2D shapes and identifies and appreciates their appearances in art in India and around the world. C-4.5 Develops the notion of fractal and identifies and appreciates the appearances of fractals in nature and art in India and around the world.
CG-5 Collects, organises, represents (graphically and in tables), and interprets data/ information from daily life experiences.	C-5.1 Collects, organises data, and applies measures of central tendencies such as average/mean, mode, and median. C-5.2 Selects, creates, and uses appropriate graphical representations of data, including pictographs, bar graphs, histograms, line graphs, and pie charts.
CG-6 Develops mathematical thinking and the ability to logically and precisely communicate mathematical ideas.	C-6.1 Applies both inductive and deductive logic to formulate definitions and conjectures, evaluates and produces convincing arguments/proofs to turn these definitions and conjectures into theorems or correct statements, particularly in the areas of algebra, elementary number theory, and geometry.
CG-7 Engages with puzzles and mathematical problems and develops own creative methods and strategies to solve them.	C-7.1 Applies creativity to develop one’s own solutions to puzzles and other problems and appreciates the work of others to develop their own solutions. C-7.2 Engages in and appreciates the artistry and aesthetics of puzzle-making, puzzle-posing, and puzzle-solving.
CG-8   Knows and appreciates the development of mathematical ideas over human history, and the contributions of past and modern mathematicians from India and across the world.	C-8.1 Recognises important mathematical contributions of India (e.g., zero, Indian numerals, ideas around infinity, concepts of algebra, etc.) as well as the contributions of specific Indian mathematicians (such as Baudhayana, Panini, Pingala, Aryabhata, Brahmagupta, Virahanka, Bhaskara, Madhava, and Ramanujan). C-8.2 Recognizes and appreciates how concepts (like the notion of number, from counting numbers, to 0, to negative numbers, to rational evolved over a period of time across different civilizations.
CG-9 Develops basic skills and capacities of computational thinking, namely, decomposition, pattern recognition, data representation, generalization, abstraction, and algorithms, in order to solve problems where such techniques of computational thinking are effective.	C-9.1 Approaches problems using programmatic thinking techniques such as iteration, symbolic representation, and logical operations and reformulates problems into series of ordered steps (algorithmic thinking). C-9.2 Identifies, analyses, and implements possible solutions to problems, with the goal of achieving the most efficient and effective combination of steps and resources and generalizes this process to a wide variety of problems.
SECONDARY STAGE (9th & 10th)
CG-1 Understands numbers, ways of representing numbers, relationships among numbers, and number sets.	C-1.1 Develops a deeper understanding of numbers, including the set of real numbers and its properties. C-1.2 Uses deductive logic to prove theorems such as ‘√2 is an irrational number’ and `there are infinitely many prime numbers’. C-1.3 Uses inductive logic to prove theorems such as the recursion relation for Virahanka numbers, `the sum of consecutive odd numbers starting with 1 is a square number’, `the sum of consecutive cubes starting with 1 is the square of a triangular number’, etc. C-1.4 Explores that every counting number has a unique factorisation into prime numbers (fundamental theorem of arithmetic). C-1.5 Recognises and appropriately uses powers and exponents. C-1.6 Computes powers and roots and applies them to solve problems. C-1.7 Computes simple and compound interest and solve real-life problems.
CG-2 Discovers and proves algebraic identities and uses such identities to solve equations.	C-2.1 Learns the art of factoring polynomials. C-2.2 Applies the division algorithm to both integers and polynomials in order to solve problems such as those involving GCDs and LCMs. C-2.3 Models and solves contextualised problems using equations (e.g., simultaneous linear equations in two variables or single polynomial equations) and draws conclusions about a situation being modelled.
CG-3 Analyses characteristics and properties of two-dimensional geometric shapes and develops mathematical arguments to explain geometric relationships.	C-3.1 Describes relationships including congruence of two-dimensional geometric shapes (such as lines, angles, triangles) to make and test conjectures and solve problems. C-3.2 Proves theorems using Euclid’s axioms and postulates – for triangles, quadrilaterals, and circles and applies them to solve geometric problems. C-3.3 Specifies locations and describes spatial relationships using coordinate geometry, e.g., plotting a pair of linear equations and graphically finding solution, or finding the area of triangle with given coordinates as vertices.
CG-4 Derives and uses formulas to calculate areas of plane figures, and surface areas and volumes of solid objects.	C-4.1 Visualises, represents, and calculates the area of a triangle using Heron’s formula. C-4.2 Visualises and uses mathematical thinking to discover formulas to calculate surface areas and volumes of solid objects (cubes, cuboids, spheres, hemispheres, right circular cylinders/cones, and their combinations).
CG-5 Analyses and interprets data using statistical concepts (such as measures of central tendency, standard deviations) and probability.	C-5.1 Applies measures of central tendencies such as mean, median, and mode. C-5.2 Applies concepts from probability to solve problems on the likelihood of everyday events.
CG-6 Begins to perceive and appreciate the axiomatic and deductive structure of mathematics. Uses stated assumptions, axioms, postulates, definitions, and mathematics vocabulary to prove mathematical statements and carry out geometric constructions.	C-6.1 Uses deductive and inductive logic to prove theorems about numbers, measurements such as areas and shapes. C-6.2 Visualises and appreciates geometric proofs for algebraic identities and other `proofs without words’. C-6.3 Proves theorems using Euclid’s axioms and postulates– for angles, triangles, quadrilaterals, circles, area-related theorems for triangles and parallelograms. C-6.4 Constructs different geometrical shapes like bisectors of line segments, angles and their bisectors, triangles, and other polygons, satisfying given constraints.
CG-7 Appreciates important contributions of mathematicians from India and around the world.	C-7.1 Recognises the important contributions made by Indian mathematicians in the field of mathematics. C-7.2 Recognizes how concepts (like evolution of numbers, geometry, etc.) evolved over a period of time across different civilizations.
CG-8 Sharpens skills such as visualisation, optimisation, representation, and mathematical modelling, and their application in daily life.	C-8.1 Models daily life phenomena and uses representations such as graphs, tables, and equations to draw conclusions. C-8.2 Uses two-dimensional representations of three-dimensional objects to visualise and solve problems such as those involving surface area and volume. C-8.3 Employs optimisation strategies to maximise desired quantities (such as area, volume, or other output) under given constraints.
CG-9  Develops computational thinking, i.e., deals with complex problems and is able to break them down into a series of simple problems that can then be solved by suitable procedures/algorithms.	C-9.1 Decomposes a problem into sub problems. C-9.2 Describes and analyses a sequence of instructions being followed. C-9.3 Analyses similarities and differences among problems to make one solution or procedure work for multiple problems. C-9.4 Engages in algorithmic problem solving to design such solutions.
CG-10  Explores connections of mathematics with other subjects.	C-10.1 Applies mathematical knowledge and tools to analyse problems/situations in multiple subjects across science, social science, visual arts, music, and sports.

 

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GENERAL LEARNING OUTCOMES CLASSWISE

Introduction:

The learning outcomes of mathematics play a crucial role in education and have broad implications for individuals and society. Here are some key reasons highlighting the importance of learning outcomes in mathematics.

learning outcomes of mathematics are essential for individual development, societal progress, and global competitiveness. They guide the teaching and assessment of mathematical concepts, ensuring that students acquire the skills and knowledge necessary for success in various aspects of life.

Here are some key reasons why learning outcomes in mathematics are important:

• Mathematics develops logical reasoning and critical thinking skills. Learning outcomes in mathematics ensure that students acquire the ability to analyse problems, think logically, and make informed decisions.
• Mathematics teaches problem-solving techniques that are applicable in various real-life situations. The learning outcomes emphasize the application of mathematical concepts to solve practical problems, fostering a problem-solving mindset.
• Many careers and professions require a solid foundation in mathematics. Learning outcomes help students acquire the mathematical skills and knowledge necessary for success in fields such as science, technology, engineering, economics, finance, and more.
• Understanding mathematical concepts is essential for scientific literacy. Learning outcomes in mathematics ensure that students are equipped with the skills needed to comprehend and contribute to scientific advancements.

LEARNING OUTCOMES FOR 
PREPARATORY STAGE 

Curricular Goal

Understands numbers (counting numbers and fractions), represents whole numbers using the Indian place value system, understands and carries out the four basic operations with whole numbers, and discovers and recognizes patterns in number sequences.

Competency

Represents numbers using the place-value structure of the Indian number system, appreciates the key role of zero in this system, compares the sizes of whole numbers, and knows and can read the names of very large numbers.

Represents numbers using the place-value structure of the Indian number system, appreciates the key role of zero in this system, compares the sizes of whole numbers, and knows and can read the names of very large numbers.

LEARNING OUTCOMES FOR 
3rd to 5th (AGES 9 TO 11)

GRADE 3 (LOs) 

Recognises, reads, and writes number names and numerals up to 999 using place value concept.

Compares and forms the greatest and smallest three-digit number (with and without repetition of given digits) using the place value concept.

GRADE 4 (LOs)

Recognises, reads, and writes number names and numerals up to 9999 using place value concept.

Compares and forms the greatest and smallest four-digit number (with and without repetition of given digits) using the place value concept.

GRADE 5 (LOs)

Reads, writes, and compares numbers bigger than 9999 (being used in her/his surroundings) using Indian numeration system.

LEARNING OUTCOMES
MIDDLE STAGE 6th to 8th (Ages 12 to 14)

Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems.

Curricular Goal

Understands the concepts of variable, constant, coefficient, expression, and (one-variable) equation, and uses these concepts to solve meaningful daily life problems with procedural fluency.

Competency

Poses and solves linear equations to find the value of an unknown, including to solve puzzles and word problems.

GRADE 6 (LOs)

Uses variable(s) to write down formulas and equation.

GRADE 7 (LOs)

Uses number and variable with different operations and expresses a real-life situation in the form of a simple linear equation and vice versa. Uses trial and error method and determines the solution of a simple equation.

GRADE 8 (LOs)

Reads, writes, and compares numbers bigger than 9999 (being used in her/his surroundings) using Indian numeration system. Writes simple contextual problems as linear equations in one variable, finds its solution, and verifies.

 LEARNING OUTCOMES FOR SECONDARY STAGE  

9th & 10th (AGES 15 TO 16)

Curricular Goal

Begins to perceive and appreciate the axiomatic and deductive structure of mathematics. Uses stated assumptions, axioms, postulates, definitions, and mathematics vocabulary to prove mathematical statements and carry out geometric constructions.

Competency 

Uses deductive and inductive logic to prove theorems about numbers, measurements (such as areas), and shapes.

Uses deductive and inductive logic to prove theorems about numbers, measurements (such as areas), and shapes.

GRADE 9 (LOs)

Applies deductive logic to prove theorems related to parallel lines. Applies deductive logic to proves theorems related to triangles, congruence of triangles.

GRADE 10 (LOs)

Applies deductive logic to prove statements like - √2 is an irrational number, sum of two odds is even etc. Applies deductive logic to prove theorems related to properties of quadrilaterals, areas of parallelograms and triangles, mid-point theorem and theorems related to circles.

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