3. Mathematics Education

Mathematics is the art and science of discovering patterns and explaining them. These patterns are all around us, in nature, in technology, and in the motion of the earth, sun, moon, and stars. There is Mathematics in everything that we do and see, from shopping and cooking, to throwing a ball and playing games, to solar eclipses and climate patterns. Mathematics thus gives us the foundational concepts and capacities required to think about the world around us and the world beyond us. But most of all, when taught well, Mathematics is truly enjoyable and can become a lifelong passion. The goal of Mathematics Education is to bring to life these aspects of Mathematics.

Mathematics, including Computational Thinking, has never been more important globally, for students and for society, with the growing challenges with respect to artificial intelligence, machine learning, data science, climate modelling, infrastructure development, and the numerous other related scientific issues faced by India and all nations today. Quality education in mathematics and mathematical thinking will thus be indispensable for India’s future, and indeed for ensuring India’s leadership role in these critically important and emerging fields.

Mathematics Education aims to develop capacities of logical thinking, finding patterns, explaining patterns, making, refuting, and proving conjectures, problem solving, computing fluently, and communicating clearly and precisely — through content areas such as arithmetic, algebra, geometry, probability, statistics, trigonometry, and calculus.

Section 3.1 - Aims

Mathematics helps students develop not only basic arithmetic skills, but also the crucial capacities of logical reasoning, creative problem solving, and clear and precise communication (both oral and written). Mathematical knowledge also plays a crucial role in understanding concepts in other school subjects, such as Science and Social Science, and even Art, Physical Education, and Vocational Education. Learning Mathematics can also contribute to the development of capacities for making informed choices and decisions. Understanding numbers and quantitative arguments is necessary for effective and meaningful democratic and economic participation.

Mathematics thus has an important role to play in achieving the overall Aims of School Education.

The specific aims of Mathematics Education in this NCF are as follows:

a. Basic Numeracy. Numbers and quantities along with words (language) are the two ways in which human beings understand and interpret the world. Numbers and quantities also play a very important role in day-to-day interactions within a complex society. Fluency in quantifying and performing calculating is essential for basic daily interactions, such as shopping and banking. Mathematics Education in schools should ensure that all students are fluent in basic numeracy. This would include not just fluency in numbers and number operations using Indian numerals, but also the capacities to handle situations that involve space and measurement.

b. Mathematical Thinking. Mathematical thinking involves systematic and logical ways to think about and interpret the world. The capacities for identifying patterns, explaining patterns, quantifying and measuring, using deductive reasoning, working with abstractions, and communicating clearly and precisely are some illustrations of mathematical thinking. Mathematics Education in schools should aim for developing such mathematical thinking in all students.

c. Problem Solving. The capacity to formulate well-defined problems that can be solved through mathematical thinking is an important aspect of learning Mathematics. Clear and precise formulation of problems and puzzles, knowing the appropriate mathematical concepts and techniques that can model the problems, and possessing the techniques and the creativity to solve the problems are core aspects of problem solving. Mathematics Education in schools should aim for developing such problem-solving capacities in all students. Problem solving also develops the capacities of perseverance, curiosity, confidence, and rigour.

d. Mathematical Intuition. Developing an intuition for what should or should not be true in Mathematics is often just as important as the more formal ‘paper - pencil’ doing of Mathematics. Focusing on the common themes and patterns of reasoning across mathematical areas, guessing correct answers (in terms of, e.g., ‘order of magnitude’) before working out precise answers, and engaging in informal argumentation before carrying out rigorous proofs are all effective ways of developing such mathematical intuition in students. Developing such mathematical intuition in all students should be one of the aims of Mathematics Education in schools.

e. Joy, curiosity, and wonder. Discovering, understanding, and appreciating patterns and other mathematical concepts, ideas, and models can require great creativity and often generates great wonder and joy. To see Mathematics as merely calculations and mechanical procedures is very limiting. Mathematics Education in schools should nurture this sense of joy, curiosity, aesthetics, creativity, and wonder in all students.

Section 3.2 - Nature of Knowledge

Unlike any other subject, the notion of truth in Mathematics is timeless and absolute. In other words, once assumptions (sometimes called axioms) are agreed upon, and a mathematical truth is established based on those assumptions through logical and rigorous reasoning (sometimes called proof), then that truth cannot be refuted or debated and is true for all time. On occasion, mathematicians may find completely new logical arguments or proofs to establish the same truth, and this too is considered a breakthrough; this is because Mathematics is not just a collection of truths, but is also a framework of methods, tools, and arguments used to arrive at these truths.

Over thousands of years, the mathematical truths that are known to humans have grown in number and scope. Quite often, new mathematical truths that are discovered and established build on previously known truths. For that reason, mathematical education, like mathematics knowledge, is cumulative — new concepts that are learned often build on those learned previously.

Mathematical knowledge is built through finding patterns, making conjectures (i.e., proposed truths), and then verifying/refuting those conjectures through logical and rigorous reasoning (i.e., through explanations/proofs or counterexamples). The process of finding patterns, making conjectures, and finding proofs or counterexamples often involves a tremendous amount of creativity, sense of aesthetics, and elegance. Often, there are many different ways to arrive at the same mathematical truth and many different ways of solving the same problem. It is for that reason that mathematicians often refer to their own subject as more of an art than a science.

Mathematics often uses a formal, stylised, and symbolic language for communication — in order to be abstract and provide rigorous explanations of claims. In reality, mathematical discovery is characterised by informal arguments based on the development of reliable intuition. It is for this reason that developing intuition is described as an important aspect of learning and doing Mathematics.

Section 3.3 - Current Challenges

Our current education system has faced multiple challenges with respect to Mathematics learning.

a. Currently, a large proportion of students in the early grades are not achieving Foundational Literacy and Numeracy. This makes it difficult for students to achieve any further higher learning in Mathematics and excludes them from effective economic and democratic participation in later years as described in earlier sections. Attaining Foundational Literacy and Numeracy for all students must therefore become an immediate national mission and a central goal of the Foundational Stage curriculum.

b. Mathematics learning has traditionally been more ‘robotic’ and ‘procedural’ rather than creative and aesthetic. This is a misrepresentation of the nature of Mathematics and must be addressed in the school curriculum.

c. Very often, the content presented in textbooks to illustrate mathematical concepts is far removed from the contextual realities of the learners. Young students find some mathematical concepts easier to absorb when they are directly connected to their experiences. Textbooks, classroom activities, and examples should aim to be motivated by and related to students’ lives whenever possible.

d. There has also been a mistaken and exclusive emphasis on symbolic language and formalism in Mathematics teaching and learning, rather than on the informal argumentation and development of mathematical intuition that is so important for mathematical discovery.

e. Methods of assessment, too, have encouraged rote learning and meaningless drills and exercises, which in turn have promoted the perception of Mathematics as highly mechanical and computational. Assessment must move towards encouraging genuine understanding of core mathematical capacities, competencies, and creativity rather than mechanical procedures and rote learning.

f. Ultimately, many students in the current system have unfortunately developed a real fear of Mathematics. Tackling this will require some changes in how society perceives and talks about Mathematics but can also be addressed through the use of teaching-learning methods that encourage students to find meaning and joy in Mathematics and assessment methods that do not kill this joy. Interactive teaching-learning methods involving play, exploration, discovery, discussion, games, and puzzles may also help counter this fear.

Fear of Mathematics There are two major aspects that cause fear of Mathematics: (1) the nature of the subject and how it is taught and assessed; and (2) how it is perceived in society. 1. The nature of Mathematics and how it is taught: a. Concepts in Mathematics are often cumulative in nature. If students struggle with place value, then certainly they will struggle with all four basic operations and decimal numbers, and hence in word problems. In early grades, the Teacher must provide differentiated learning experiences to ensure that each student has mastered the foundational concepts in Mathematics.
b. When symbols — part of the ‘language’ of Mathematics — are manipulated without understanding, after a point, boredom and bewilderment dominate many students, and dissociation develops. Hence, it is important for the Teacher to start teaching the concept by making connections to real life using the local language (especially up to Preparatory Stage), providing exposure to explore using concrete objects or examples, and gradually shift to more algebraic language.
c. Most of the assessment techniques and questions focus on facts, procedures, and memorisation of formulas. However, assessment should focus on understanding, reasoning, and when and how a mathematical technique is to be used in different contexts. 2. Societal perceptions and expectations: a. A large number of parents expect their children to choose a career in the Science stream, regardless of their children’s individual passions and interests — this inhibits them from the enjoying the process of mathematical discovery.
b. Similarly, mathematical ability is seen as central to ‘cracking’ competitive entrance exams for professional courses, such as those in engineering. Due to immense competition in these exams, parents sometimes end up burdening their children with immense pressure to go to coaching classes and get a high score in Mathematics, instead of allowing them to proceed at their own pace and appreciate its joy and wonder.
Hence, we must rethink the approach of teaching to one where students see Mathematics as a part of their life, and enjoy it with a greater focus on reasoning and creative problem solving. As with Language learning, students should not be allowed to fall behind in Mathematics and should be immediately supported to catch up if they do fall behind. NEP 2020 already has suggested delinking competitive entrance exams and the ‘coaching culture’ from the scheme of studies in schools. These measures should help redress this situation.

Section 3.4 - Learning Standards

In the Foundational Stage, attaining foundational numeracy represents the key focus of Mathematics. Foundational numeracy includes understanding Indian numerals, adding and subtracting with Indian numerals, developing a sense of basic shapes and measurement using non-standard tools, and early mathematical thinking through play.

In the Preparatory Stage, while the focus is on to work on building conceptual understanding of numbers, operations (all four basic operations), shapes and spatial sense, measurement (standard tools and units) and data handling, the objective is to develop capacities in procedural fluency, and mathematical and computational thinking to solve problems from daily life.

In the Middle Stage, the emphasis moves towards abstracting some of the concepts learned in the Preparatory Stage to make them more widely applicable. Algebra, in particular, is introduced at this stage via which students are able to, for example, form rules to understand, extend, and generalise patterns. More abstract geometric ideas are also introduced at this Stage and relations with algebra are explored to solve problems and puzzles.

Finally, the Secondary Stage focusses on further developing the ability to justify claims and arguments through logical reasoning. Students become comfortable in working with abstractions and other core techniques of Mathematics and Computational Thinking, such as the mathematical modelling of phenomena and the development of algorithms to solve problems.

Across the Stages, students develop mathematical skills such as problem solving, visualisation, optimisation, representation, and communication, and thereby develop the capacities of Mathematics and Computational Thinking. Through creating and solving puzzles, pictorials, word problems, and optimisation problems, various values and dispositions such as perseverance, curiosity, confidence, rigour, and honesty would be developed across grades.

Finally, Mathematics has an extremely rich history in India spanning thousands of years. India is where the place value number system (including zero) — that we all use today to write numbers — was first developed and used and is where many of the key foundations of algebra, geometry, trigonometry, and calculus were laid. By learning about the development of Mathematics in India as well as throughout the world, the rootedness in India can be enhanced, along with a more general appreciation of the history of Mathematics, and of the remarkable evolution and development of mathematical concepts through time (and India’s critical roles in these developments).

3.4.1 Curricular Goals & Competencies

3.4.1.1 Preparatory Stage

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 recognises patterns in number sequences	C-1.1: Represents numbers using the place value structure of the Indian number system, compares 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 ½, ¼) as parts of unit wholes, as locations on number lines and as divisions of whole numbers. C-1.3: Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10×10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems. C-1.4: 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: Describes location and movement using both common language and mathematical vocabulary; understands the notion of a map (najri naksha). C-2.3: Recognises and creates symmetry (reflection, rotation) in familiar 2D and 3D shapes. C-2.4: Discovers, recognises, 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, weight, area, volume, and time using non-standard and standard units	C-3.1: Measures in non-standard and standard units and evaluates the need for standard units. C-3.2: Uses an appropriate unit and tool for the attribute (like length, perimeter, time, weight, volume) being measured. C-3.3: Carries out simple unit conversions, such as from centimetres to metres, within a system of measurement. C-3.4: Understands the definition and formula for the area of a square or rectangle as length × breadth. C-3.5: Devises strategies for estimating the distance, length, time, perimeter (for regular and irregular shapes), area (for regular and irregular shapes), weight, and volume and verifies the same using standard units. C-3.6: Deduces that shapes having equal areas can have different perimeters and shapes having equal perimeters can have different areas. C-3.7: Evaluates the conservation of attributes like length and volume, and solves daily-life problems related to them.
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 (including word puzzles and puzzles from ‘recreational’ areas, such as the construction of magic squares). C-4.2: Learns to systematically count and list all possible permutations or combinations given a constraint, in simple situations (e.g., how to make a committee of two people from a group of five people). C-4.3: Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation, or paper-pencil calculation, in accordance with the context.
CG-5: Knows and appreciates the development in India of the decimal place value system that is used around the world today	C-5.1: Understands the development of zero in India and the Indian place value system for writing numerals, the history of its transmission to the world, and its modern impact on our lives and in all technology.

***3.4.1.2 Middle Stage***

CG-1: Understands numbers and sets of numbers (whole numbers, fractions, integers, rational numbers, and real numbers), looks for patterns, and appreciates relationships between numbers	C-1.1: Develops a sense for and an ability to manipulate large whole numbers of up to 20 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. C-1.3: Learns about the inclusion of zero and negative quantities as numbers, and arithmetic operations on them. C-1.4: Explores and understands sets of numbers and visualises them on the number line. C-1.5: Explores the idea of percentage and applies it to solve problems. C-1.6: Explores and applies fractions in daily-life situations.
CG-2: Understands the concepts of variable, constant, coefficient, expression, and equation, and uses these concepts to solve meaningful daily-life problems	C-2.1: Understands equality between numerical expressions and checks arithmetical equations. C-2.2: Represents numbers using variables and algebraic expressions. C-2.3: Forms and manipulates algebraic expressions. C-2.4: Solves linear equations, including puzzles and word problems. C-2.5: Develops own methods to solve 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 geometric shapes. C-3.2: Outlines properties of lines, angles, triangles, quadrilaterals, and polygons. C-3.3: Works hands-on to construct 3D shapes and visualizes them using 2D representations. C-3.4: Draws and constructs geometric shapes using a compass and straightedge. C-3.5: Understands congruence and similarity in geometric shapes.
CG-4: Develops understanding of perimeter and area for 2D shapes and applies them to solve real-life problems	C-4.1: Understands and uses formulae to determine the area of various 2D shapes. C-4.2: Learns and proves the Baudhayana-Pythagoras theorem using geometric methods. C-4.3: Creates designs using tiling and appreciates their artistic applications. C-4.4: Understands the concept of fractals and identifies their occurrences in nature and art.
CG-5: Collects, organises, represents, and interprets data from daily-life experiences	C-5.1: Uses measures like average, mode, and median to interpret data. C-5.2: Creates and uses various graphical representations to analyze data.
CG-6: Develops mathematical thinking and the ability to communicate mathematical ideas logically	C-6.1: Uses inductive and deductive logic to develop definitions, conjectures, and proofs in algebra, number theory, and geometry.
CG-7: Engages with puzzles and mathematical problems and develops creative problem-solving methods	C-7.1: Finds unique solutions to puzzles and appreciates different approaches. C-7.2: Engages in and enjoys puzzle-making and problem-solving.
CG-8: Develops basic skills in computational thinking, including pattern recognition, data representation, and algorithmic problem-solving	C-8.1: Uses programmatic thinking techniques such as iteration, symbolic representation, and logical operations. C-8.2: Learns systematic counting, iterative patterns, and algorithmic reasoning.
CG-9: Appreciates the development of mathematical ideas and contributions of mathematicians	C-9.1: Recognises the evolution of mathematical concepts over time. C-9.2: Knows and appreciates contributions of Indian mathematicians like Baudhayana, Aryabhata, Brahmagupta, and Ramanujan.
CG-10: Recognises the interaction of Mathematics with other school subjects	C-10.1: Understands how Mathematics connects with Science, Social Science, Arts, Music, Vocational Education, and Sports.

***3.4.1.3 Secondary Stage***

CG-1: Understands numbers (natural, whole, integer, rational, irrational, and real), ways of representing numbers, relationships amongst numbers, and number sets	C-1.1: Develops understanding of numbers, including the set of real numbers and its properties.
CG-2: Builds deductive and inductive logic to prove theorems related to numbers and their relationships (such as ‘√2 is an irrational number’, recursion relation for Virahanka numbers, formula for the sum of the first n square numbers)	C-2.1: Extends the understanding of powers (radical powers) and exponents.
CG-3: Discovers and proves algebraic identities and models real-life situations in the form of equations to solve them	C-3.1: States and motivates/proves remainder theorem, factor theorem, and division algorithm. C-3.2: 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. C-3.3: Learns Brahmagupta’s quadratic formula (in both symbolic and poetic form) and its derivation, and uses it to solve some of the poetic puzzles of Bhaskara as well as modern-day problems.
CG-4: Analyses characteristics and properties of two-dimensional geometric shapes and develops mathematical arguments to explain geometric relationships	C-4.1: Describes relationships including congruence of two-dimensional geometric shapes (such as lines, angles, triangles) to make and test conjectures and solve problems. C-4.2: Proves theorems using Euclid’s axioms and postulates for triangles and quadrilaterals, and applies them to solve geometric problems. C-4.3: Proves theorems about the geometry of a circle, including its chords, subtended angles, inscribed polygons, and area in terms of π. C-4.4: Understands the irrationality of π, the best approximations to π discovered over human history, and the first exact formula (infinite series) for π given by Madhava. C-4.5: Specifies locations and describes spatial relationships using coordinate geometry, e.g., plotting a pair of linear equations and graphically finding the solution, or finding the area of a triangle with given coordinates as vertices. C-4.6: Understands the definitions of the basic trigonometric functions, their history and motivation (including the introduction of the sin and cos functions by Aryabhata using chords), and their utility across the sciences.
CG-5: Derives and uses formulae to calculate areas of plane figures, and surface areas and volumes of solid objects	C-5.1: Visualises, represents, and calculates the area of a triangle using Heron’s formula and its generalisation to cyclic quadrilaterals given by Brahmagupta’s formula. C-5.2: Visualises and uses mathematical thinking to discover formulae to calculate surface areas and volumes of solid objects (cubes, cuboids, spheres, hemispheres, right circular cylinders/cones, and their combinations).
CG-6: Analyses and interprets data using statistical concepts (such as measures of central tendency, standard deviations) and probability	C-6.1: Applies measures of central tendencies such as mean, median, and mode. C-6.2: Applies concepts from probability to solve problems on the likelihood of everyday events.
CG-7: Begins to perceive and appreciate the axiomatic and deductive structure of Mathematics	C-7.1: Proves mathematical statements and carries out geometric constructions using stated assumptions, axioms, postulates, definitions, and mathematics vocabulary. C-7.2: Visualises and appreciates geometric proofs for algebraic identities and other ‘proofs without words’. C-7.3: Proves theorems using Euclid’s axioms and postulates – for angles, triangles, quadrilaterals, circles, area-related theorems for triangles and parallelograms. C-7.4: Constructs different geometrical shapes like bisectors of line segments, angles and their bisectors, triangles, and other polygons, satisfying given constraints.
CG-8: Builds skills such as visualisation, optimisation, representation, and mathematical modelling along with 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: Knows and appreciates important contributions of mathematicians from India and around the world	C-10.1: Recognises the important contributions made by mathematicians (Indian and others) in the field of Mathematics (such as the evolution of numbers, geometry, algebra). C-10.2: Recognises modern contributions to Mathematics made in both India and abroad, and understands the next frontiers and next major open questions in the field of Mathematics.
CG-11: Explores connections of Mathematics with other subjects	C-11.1: Applies mathematical knowledge and tools to analyse problems/situations in multiple subjects across Science, Social Science, Visual Arts, Music, Vocational Education, and Sports.

**3.4.2 Rationale for Selection of Concepts** The Learning Standards — the Curricular Goals and Competencies — defined here make choices for the concepts that will be taught and learnt in each of the Stages. The key principles that underlie these choices are described here. **a. Principle of Essentiality** This principle involves three key questions: What Mathematics is essential to learn so that one can solve one’s day-to-day problems, live a normal life, and ably participate in the democratic processes of the country? What Mathematics is essential to be able to adequately understand other essential school subjects, such as Science and Social Science? And, finally, what mathematical ideas are essential for developing interest in students to further pursue the discipline if one desires to do so? **b. Principle of Coherence** Concepts that are selected for each Stage must be in coherence with each other and with the overall Stage-specific Curricular Goals, Competencies, and Learning Outcomes. The goal must not be to bombard the student with all mathematical concepts at the expense of coherence. **c. Principle of Practicality and Balance** Due to a rush for completing the syllabus, the focus on building conceptual understanding often gets compromised and rote memorisation of formulae and direct use of algorithms becomes a central part of the teaching process. NEP 2020 strongly recommends reducing content to give time for discussion, analytical thinking, and fully appreciating concepts. At each Stage, while choosing the concepts for Mathematics, emphasis has been given to the idea of balancing content load with discussion, analytical thinking, and true conceptual understanding. The selection of concepts in each Stage must aim to increase the space for balancing conceptual and procedural understanding of the concepts. This will create space for Teachers to focus more on building conceptual understanding and meaningful practice. With this rationale, Learning Standards have been configured to give emphasis to understanding Mathematics as a discipline by the end of Grade 10, so that students can appreciate its intrinsic beauty and value and, thereby, also pursue higher education in Mathematics if they so wish. Areas and concepts that are considered useful for all students to interact with the world over their lives, or study other subjects, are covered by Grade 10, so that if they decide to drop Mathematics after Grade 10, they are still equipped with essential skills, concepts, and Competencies in Mathematics. At each Stage, all concepts are included that may be needed as prerequisites for concepts in later Stages. ## Section 3.5 - Content The approach, principles, and methods of selecting content has commonalities across subjects — those have been discussed in Part A, Chapter 3, §3.2 of this document. This section focusses only on what is most critical to Mathematics Education in schools. Hence, it will be useful to read this section along with above-mentioned the section. **3.5.1 Principles for Content Selection ** The following principles will be followed while choosing topics of study for Mathematics classrooms. Stage-wise principles are laid down; for each Stage, principles for the previous Stage have also been considered, wherever applicable. **3.5.1.1 Preparatory Stage ** a. Plenty of space should be given to students’ local context and surroundings for developing concepts in Mathematics. Case studies, stories, situations from daily life, and vocabulary and phrasing in the home language should be brought in to help introduce and unfold a concept and its sub-concepts. b. The development of a culture of learning outside the classroom should be encouraged. More play-way methods (activities) should be included wherever possible. c. Avenues for mathematical reasoning should be created in all activities, projects, assignments, and exercises. The content should encourage students to articulate the reasons behind their observations and guesses/conjectures and ask why a pattern extends in a certain way and what the rule behind it is. d. The language of the content should be simple so that students can also express their thoughts using similar language; this should gradually enlarge their vocabulary and enable them to become more skilled over time in using precise mathematical vocabulary, symbols, and notation. e. Content should encourage learning processes (meaningful practice leading to building memory and procedural fluency) and cognitive skills (reasoning, comparing, contrasting, and classifying), as well as the acquisition of specific mathematical capacities. f. There should be consistency and coherence across the content, and the progression of the concepts should be spiral rather than linear g. For content selection, the focus should be on activities that are engaging and built around the daily-life experiences of students. It should cater to more than one Learning Objective/ Competency simultaneously and take into account one or more learning areas at the same time. h. More formal definitions should naturally evolve at the end of a more informal discussion, as students gradually develop a clear understanding of a concept. **3.5.1.2 Middle Stage** a. Content should allow students to explore several strategies for solving a problem or puzzle. b. Content should involve situations and problems that offer multiple correct answers. For this, open-ended questions should be given more space in the exercises. c. Content should provide opportunities for students to ‘talk’ Mathematics. Semi-formal language used by students in discussions should be accepted and encouraged. d. Problem posing is an important part of doing Mathematics. Exercises that require students to formulate and create a variety of problems and puzzles for their peers and others should be encouraged. e. Content should allow students to explore, create, appreciate, and understand instead of just memorising concepts and algorithms without understanding the rationale behind how they work. f. Content should offer meaningful practice (through worksheets, games, puzzles) that leads to working memory (smriti) and ultimately builds procedural/computational fluency. g. Mathematics should emerge as a subject of exploration, discovery, and creativity rather than a set of mechanical procedures. h. Content should give opportunities to naturally motivate the usefulness of abstraction > *** Finding π *** *Ever wondered if we can find Pi (π) on our own without any computers, calculators, or any textbooks? The answer is yes! And that too, just by using a thread, paper-pencil, and a geometry box. In my class, I do this exercise when we learn the properties of circles. The activity is quite simple. We take a piece of paper and draw multiple circles using a compass of varying diameters. Then we put the thread on the outline of the circle and measure the length of that thread. This length will be the circumference of the circle. Repeat this for all the circles and note down in table like below:

#	Circumference or Length of the Thread (C)	Diameter (D)	Ratio of Circumference to Diameter
1
2
3
4
5

At the end of exercise, I ask students to notice the ratio value…it is usually around 3. This is the famous constant, popularly known as the Pi, denoted by a Greek symbol ‘π’.
Once we do this, we solve many more related questions, like ‘If I know the radius of the circle, can we guess its circumference?’ and more. Using teaching aids with such activities keeps my students more alert and interested throughout the class.* **3.5.1.3 Secondary Stage** a. Content should be chosen and designed in a way that enables students to understand notions of abstraction, the axiomatic system, and deductive logic. b. More project-based work should be designed and given space in the content so that students have opportunities to weave together several concepts simultaneously. This will help students appreciate the unity and inter-relatedness of mathematical concepts. c. Interdisciplinary approaches should be kept in consideration while designing the content. Project-based work could be designed based on themes to ensure the integration of other subjects, e.g., linear variation and equation solving in the Science and Social Science. d. Content at this stage should allow students to develop and consolidate the mathematical knowledge and skills acquired during the Middle Stage. e. Students should develop necessary skills to work with tools, modern technological devices, and mathematical software useful in mathematical discovery and learning. f. Content should highlight the history of Mathematics and how mathematical concepts have developed over time, and, in particular, the contributions of Indian and other mathematicians. **3.5.2 Materials and Resources** Materials and resources form a critical part of content. Principles for selecting content for teaching and learning Mathematics include: a. **Concrete materials:** TLMs can be useful resources that make learning experiences more interesting and enjoyable. Such material can be used in understanding concepts, as well as in practice and assessment. These resources enable students to comprehend concepts more effectively, as they connect verbal instruction with real experience, concretise abstract concepts, and develop curiosity and interest in learning. Schools can establish Mathematics laboratories or corners with equipment for experimentation, exploration, demonstration, and verification of mathematical ideas. Some examples include electronic calculators, graph machines, mathematical games, puzzles, ganit malas, bundle sticks, geoboards, algebra tiles, Dienes Blocks, or flat long cards, dominoes, pentominoes, Mathematics-related videos, and inclinometers. >  **b. Textbooks:** Textbooks should provide factually correct information in an accessible manner. There should be broad narratives and motivations (including those specific to a State or region) that hold the content together, so it does not read like just a collection of techniques. The development of ideas should be coherent and sequential/spiral, with concrete examples leading to abstract concepts, and new concepts growing from old ones. The language should be simple and comprehensible and should give space for students to build their own definitions, and only gradually start using more formal mathematical terms. The content chosen should be in alignment with the pedagogical instructional practices specific to Mathematics (stated in Section 3.6). Content should include first concrete representations and visual representations, and only then abstract representations. There should be a balance between content and exercises/puzzles to ensure ‘learning by discovering/doing’. **c. Workbooks:** Workbooks are a useful tool in the teaching and learning of Mathematics. Workbooks can be designed to fulfil three purposes: (a) introducing new concepts, (b) practice for consolidating the understanding of concepts, and attaining procedural and computational fluency, and (c) self-assessment tools for students to track their own understanding, and to provide the same information to teachers also. Teacher handbooks accompanying student workbooks can also be very useful. **d. Technology:** Information and communication technologies provide additional opportunities for students to see and interact with mathematical concepts, making the teaching of mathematics more interactive and engaging. Making use of graphing calculators/software, computer algebra systems, and other digital tools allow students to experiment with and visualise mathematical objects and operations, and to explore and make discoveries through games and simulations. ## Section 3.6 - Pedagogy and Assessment The approach, principles, and methods of pedagogy and assessment has commonalities across subjects — those have been discussed in Part A, Chapter 3, §3.3 and §3.4 of this document. This section focusses only on what is most critical to Mathematics in schools. Hence, it will be useful to read this section along with the above-mentioned section. **3.6.1 Pedagogy for Mathematics** Traditional approaches to teaching mathematics directly jump into abstract symbolic manipulation. This is not very effective in making mathematics accessible to learners. There are several steps before the learner is ready for symbolic manipulations. The first step is to have concrete experiences that embody the mathematical concept involved. Once the learners have immersed themselves in this experience, discussing this experience using language is the next level of abstraction. This language use can then be represented as pictures or diagrams. Finally, these pictures can be converted into the symbols that are used in Mathematics to represent that particular concept or idea. Effective Mathematics pedagogy should take into consideration this sequence for developing a conceptual understanding of mathematics. For students, problem-solving and problem-posing are critical steps in learning Mathematics. Practice and independent problem solving help students process and remember difficult concepts, and this should be encouraged in the classroom as frequently as possible. Students should also be encouraged to solve problems and puzzles in groups, so that they can see different approaches towards solving a problem and have conversations about mathematical concepts, thereby making them more graspable. They should also be encouraged to pose questions and come up with new problems. Many students from the Preparatory Stage onwards enjoy learning via scientific experiments performed in laboratories. Students get to experience the following stages of scientific discovery: observing a phenomenon in nature, setting up an experiment in a lab, performing the experiment and noting down observations, trying to find a pattern, and then finally trying to explain the phenomenon. Unfortunately, current practices in Mathematics teaching does not expose students to such a journey. Mathematics is too often presented as a finished product which is purely demonstrative and formal. Guessing using increasingly developed intuition, a skill needed to discover new theorems and their proofs, is discouraged in Mathematics classes. It is possible to show this ‘experimental’ nature of Mathematics using an inductive method of teaching at the Preparatory, Middle, and Secondary Stages. The idea is to develop teaching material consisting of appropriate Mathematics experiments. Students can work in groups and guessing can be encouraged. For example, an experiment could students to write down the first few even numbers as sums of two smaller numbers and lead them to observe that they can always be written as sum of 2 primes. Guidance from the Teacher should lead them to conjecture that ‘any even number greater than 2 is a sum of 2 primes’. They should collect as much evidence as possible. Though this particular problem is unsolved, towards the end, each student could be given an opportunity to talk about their experiment (instead of proving/ disproving the statement). Many such experiments in number theory, geometry, and combinatorics could be prepared. Such activities have the potential to greatly enhance intuition and problem-solving ability. Mathematics also naturally provides many opportunities for critical thinking, in the form of interrogating definitions, formulating, or choosing alternative proofs, conjectures, explanations, representations, or generalisations. Curriculum and pedagogy need to provide room and educational opportunity for such thinking. For example, working on match-stick geometry helps interrogate geometric assertions. Students should be encouraged to define their own geometric objects and classes of numbers with specific properties to encourage experimentation, creativity, discovery, and critical thinking. **3.6.1.1 Instructional Practices** a. Students should be exposed to multiple ways of seeing the same mathematical concept. This could be through pictures, symbols, different motivations or applications, and different descriptions in spoken language. Each of these provides a distinct perspective, and they together help the student form their own understanding of the mathematical concept. b. The Teacher should encourage students to express their understanding in their own words using mathematical vocabulary and terms (including in their own home language when different from the medium of instruction). Opportunities should be created for mathematical conversations between students. c. The Teacher should provide opportunities for students to engage in meaningful discussions involving questions that require explanations (such as ‘How could you explain your thinking to someone just beginning to learn this?’ or ‘How do you know?’). Habits of verification should be inculcated from an early age. When a student has distributed 24 toffees among 8 students, it is important to not only ask ‘Has everyone gotten an equal number of toffees?’, but also to follow up with ‘How do you know?’ d. The Teacher can incorporate problem-solving tasks in the classroom that serve multiple purposes. Problems can be chosen to review concepts previously studied by students and link them to new concepts. The task can be designed so that students have to reason through questions, and then justify their thinking orally or in writing. e. Teachers can use physical models, diagrams, graphing calculators, simulations, computer algebra systems, games, and other tools to help students model situations, visualise concepts, think through a problem, and devise strategies for solutions. f. Small-group work can be an effective way of learning Mathematics. Discussions and problem solving in groups give students the opportunity to talk about Mathematics, ask questions they may be hesitant to ask the teacher, and work on harder problems by pooling together their understanding. However, it should be of short duration to manage the groups effectively. g. Meaningful practice, through worksheets, puzzles, games, mental and oral Mathematics, group work, and homework involving paper and pencil, should be an integral part of the Mathematics classroom. Practice problems should be designed so that students revisit concepts and techniques and see different situations where a certain technique can be used. When choosing problems and guiding students to solve them, teachers should ensure that students are actively learning and not just memorising techniques. h. During the Middle and Secondary stages, opportunities should be provided for reading simple mathematical text and writing mathematical content. Mathematical communication should be explicitly signalled as an item of teaching and learning. Books and online resource material should be provided by schools for this purpose. > ***Discovering the Magic of Mathematics!** In geometry, there is a lot of learning through axioms, corollaries, and theorems which often leave students wondering ‘Where did this come from!? Does it happen every time!?’ So, what I do is I usually introduce such things through activities. For example, to learn that ‘sum of all internal angles in a quadrilateral is equal to 360⁰,’ we do a simple activity. I ask students to draw any random quadrilateral and then label their internal angles as ∟1, ∟2, ∟3 and ∟4 as shown in the picture below. Next, I ask the students to measure the 4 angles to discover the sum is around 360⁰. Alternatively, when one cuts these angles and join them to meet their all four vertices at a point without leaving any gap (as seen below) to form a complete angle — one discovers that the sum is 360⁰.
 Here, my emphasis is always on designing activities that help my students learn mathematical concepts instead of just memorising them as facts and formulae. This is an example of inductive method of teaching theorems, one could also use deductive methods to teach these theorems. * **3.6.1.2 Some Suggested Methods of Teaching** **a. Play-way (activity-based) method:** In play-way or activity-based learning, students use toys, games, and puzzles to explore mathematics. This may involve physical games, or games/activities involving aids such as dice, puzzles, dominoes, and building blocks Incorporating play can help make some mathematical notions seem more natural and relevant to students, and also enables them to be creative, pose questions, collaborate with others, and use multiple senses to learn. For students who feel alienated by Mathematics, this may help them enjoy the subject and feel more confident in their ability to understand it. **b. Inquiry-based method:** This method allows students to explore mathematical content by posing, investigating, and answering questions and sharing their findings with their peers for them to critique. Through this method, the student learns to reason and collaborate with their peers to discover mathematical patterns and truths. **c. Problem-solving method:** Word and logic puzzles (including puzzles that use elimination grids to solve logic problems) are a fun way to teach deductive reasoning. Simple puzzles can help develop students’ logical and creative thinking skills in an enjoyable manner. **d. Inductive method:** Inductive method is based on the principle of induction. Induction is the process to establish a generalised truth by showing that if it is true for a reasonably adequate number of cases, then it is true for all such cases. Thus, the inductive method of teaching leads us from known to unknown, from a particular case to a general rule and from concrete to abstract. When a number of concrete cases have been understood, the student is able to attempt a generalisation. Students are presented with a series of individual concrete cases, and they are expected to come up with a generalised and abstract mathematical representation of these cases. This method can help students discover patterns in numbers or geometry, which they may later encounter as theorems or formulae. Such discoveries reveal the beauty that drives many people to study mathematics. **e. Deductive method:** Deduction is the process by which a particular fact is derived from some generally known truths. Thus, in the deductive method of teaching, the student proceeds from general to particular, from abstract to concrete, or from formula to examples. Here, a pre-established rule or formula is given to the student, and they are asked to solve the related problems by using that formula or to prove theorems using definitions, axioms, and postulates. Each of the methods above has its own advantages and limitations. It is also true that one method does not work for all students, and so the Teacher must draw on their understanding of their class and choose a combination of methods to ensure the learning of every student. The matrix below suggests methods in rows and Stages in three columns.

Suggestive Methods	Preparatory	Middle	Secondary
Play-way	✔✔	✔✔✔	✔
Discovery/Inquiry	✔✔	✔✔✔	✔✔
Problem Solving	✔✔	✔✔✔	✔✔✔
Inductive	✔✔	✔✔	✔
Deductive	✔	✔✔	✔✔

Recommendation on Use: ✔✔✔ — More Often, ✔✔ — Often & ✔ — Less Often

**3.6.1.3 Integrating Mathematics with Other Curricular Areas ** An interdisciplinary approach enables students to expand their horizons by allowing them to consider and tackle problems that do not fit exactly into one subject. It also changes how students learn by enabling them to synthesise multiple perspectives, instead of driving their thoughts unidirectionally based on the understanding of one discipline. It allows students to explore and involve multiple perspectives and dimensions from different curricular areas to deal with daily life problems. Hence, integrating Mathematics with other Curricular Areas can help students develop interest in the subject and build a holistic view of different disciplines. Mathematics learning can thus be made more meaningful and interesting by integrating it with other curricular areas. Some possibilities for doing this are described below: **a. Integrating Mathematics and Art:** Art and Mathematics are closely linked, with both disciplines playing an important role in understanding patterns, as well as enhancing spatial abilities and visualisation. Many activities that are a part of students’ lives, such as music, dance, needlework, and rangoli naturally lead them to see patterns, which can be described and further understood using mathematical language. Integrating the Arts with Mathematics can include art and craft activities that engage students in creating visual patterns, tessellations, and geometric objects, and can include exposure to examples of artworks that contain interesting patterns. Some ideas for integrating Art in the Mathematics classroom are: i. Creating and analysing different rangoli/kolam patterns. ii. Creating origami, and using it to understand angles, symmetries, and how a 2D object can be transformed into a 3D one. iii. Recognising geometries and symmetries in art and architecture. iv. Symmetry can also be explored through dance and movement by assigning mirroring exercises for students. This concept can also be explored through visual games, self designed board games, simple print-making activities based on traditional art forms like Rogan printing, and by viewing examples of architecture, painting, and sculpture. v. Pattern activities could also include art forms, like weaving, embroidery, and bead work, where patterning is heavily reliant on mathematical precision, grids, and matrices. vi. Ratio and proportion are fundamental to the arts. The technique of drawing the human body requires an understanding of proportion (e.g., the length of an arm is about thrice the length of the head). The study of ratios and proportions can also be related to different cultures and their canons of beauty being defined by specific ratios and proportions. vii. Music is filled with patterns. The joy of making music lies in creating innumerable permutations and combinations of patterns by grouping notes, sounds, and beats. Tempo determines how notes can be combined and fitted into specific rhythm cycles in multiple variations. Music is an extremely useful way to understand fractions since it uses full notes, half notes, quarter notes, and one-eighth notes which also relate to tempo in terms of ek gun, dugun, trigun, chaugun. Improvisation in the classical forms of music require an immense alertness and ability to do mental math. For example, creating note patterns in multiples of 3, 5, or 7 in a 4-beat rhythm can be, both challenging and aesthetically pleasing. The way frequencies are chosen in music also involves understanding simple fractions, due to what sounds good and most resonant to the ear. For example, the ratio of frequencies of the top and bottom Sa in a saptak is 2:1, and the ratio of frequencies of Pa and Sa is 3:2. There are reasons from Physics (namely, the notion of resonance) as to why particular combinations of notes sound good to the ear, and the notes (shrutis) that are used in Indian classical music (and also in music around the world), as explained in Bharata’s Natyashastra, is based on simple whole number ratios of frequencies. **b. Integrating Mathematics and Sports:** Integrating mathematics and Sports can benefit students who enjoy sports and see the relevance of measurement, unit conversion, probability and statistics, scoring systems, and trajectories of thrown objects in the context of sports. Student projects can explore mathematical connections such as in the Fosbury Flop in high jump or the Duckworth Lewis Scoring System in cricket. **c. Integrating Mathematics and Science.** The appearance of the Virahanka numbers and the golden angle in nature (e.g., in pinecones, sunflowers, daisies, kaner and tulsi plants) make for an excellent interdisciplinary journey of discovery. Similarly, other Curricular Areas can also be integrated with Mathematics to understand and see more meaning of Mathematics in daily life. **3.6.2 Assessment in Mathematics** Few key principles for assessment in Mathematics are: a. Students must be assessed for understanding of concepts and mathematical skills and capacities, such as procedural fluency, computational thinking, problem solving, visualisation, optimisation, representation, and communication. b. Students must be assessed through a variety of ways, e.g., solving a variety of problems testing procedural knowledge and conceptual understanding in key mathematical concepts, geometric reasoning, algebraic thinking, word problems, and working in groups to solve mathematical problems. c. Open book assessments can go a long way towards reducing anxiety in students. Examinations could provide ‘fact sheets’ consisting of information, such as formulae, and definitions, so that students need not memorise them but use them in actual problem solving. A few Teacher Voices illustrate assessment in Mathematics below. > ***Multiplication as Repeated Addition** I teach Grade 4. I wanted to assess my students’ understanding of multiplication as repeated addition and its application in daily life. The question below is directly related to their conceptual understanding of multiplication as repeated addition and the ability to apply that understanding in a new situation. I preferred this to just asking them to solve questions from the textbook using vertical and horizontal multiplication.  Given below are groups of flowers. Which of the following shows the fastest way to find out the total number of flowers in the given picture?* A. 4 + 5 B. 4 + 4 + 4+ 4 + 4 C. 3 + 2 D. 5 x 4 E. 4 x 5 I used the following marking scheme:

Option	Points
Option A	0
Option B	1
Option C	0
Option D	2
Option E	0.5

> ***Percentages** I teach Grade 7. I wanted to assess my students’ ability to use the understanding of percentages in solving real-life contextual word problems. Students can solve routine problems from their textbooks, but tend to stumble when they need to comprehend a word problem by themselves and use the appropriate procedure to solve it. This is what I tried. Q: A cricket team played 20 matches in a tournament. If they lost 25% of the matches and won all the remaining matches, how many matches did the team lose? a. 2 matches b. 4 matches c. 5 matches d. 10 matches Once the class solved this problem, I also asked my students to frame a problem on percentages where the answer would be 10 matches. Each student came up with a different word problem! This was a real test of their mathematical capacities.* > ***Multiple Methods** I teach Grade 9. I find that my students usually solve mathematical problems using only one method. They rarely use a combination of methods to arrive at a solution. This is an important capacity in Mathematics as it helps in discerning when to use which method. So, I gave my students a problem and asked them to use at least 3 different ways of solving it. This was the problem: There are 48 students in Grade 8 in a school. If the number of girls is three times the number of boys, then how many girls and boys are there in that class? Solve this question using at least 3 different methods: Algebraic method, Ratio method, Section method, using patterns, or any other method. Since students may attempt this in very many ways, I also gave them these solved answers to self-assess their work after they worked out their methods*

Correct Answer 1: Algebraic Method (1)	Step 1: Number of boys = x Step 2: Number of girls = 3x Step 3: 3x + x = 48 Step 4: 4x = 48 Step 5: X = 12 Step 6: Number of boys = 12 Step 7: Number of girls = 36
Correct Answer 2: Algebraic Method (2)	Step 1: Number of girls = x Step 2: Number of boys = x/3 Step 3: X + (x/3) = 48 Step 4: (4x/3) = 48 Step 5: X = 36 Step 6: Y = 12 Step 7: Number of girls = 36 Step 8: Number of boys = 12
Correct Answer 3: Using Ratios	Step 1: The ratio of the number of girls to the number of boys is 3:1. Step 2: 3x + x = 48 Step 3: 4x = 48 Step 4: X = 12 Step 5: Number of boys = 12 Step 6: Number of girls = 36
Correct Answer 4: Using Section Method	Step 1: Girl’s section = 3/4 Step 2: Number of girls = 3/4 (48) Step 3: (3 x 48)/4 Step 4: 3 x 12 = 36 Step 5: Number of boys = 1/4 (48) = 12 or number of boys 48 - 36 = 12 Step 6: Number of girls = 36
Correct Answer 5: Solving using the pattern	Boys and Girls: 1 → 3 2 → 6 4 → 12 8 → 24 12 → 36 Number of girls = 36 Number of boys = 12 Pattern Approach: 36 + 12 = 48 Write 36 = 3 x 12 So, the number of girls = 36 and the number of boys = 12.
Correct Answer 6: Using Equations	Step 1: Number of girls = x Step 2: Number of boys = y Step 3: X + y = 48 Step 4: X = 3y Step 5: 3y + y = 48 Step 6: 4y = 48 Step 7: Y = 12 Step 8: X = 36 Step 9: Number of boys = 12 Step 10: Number of girls = 36