## December 13, 2018

### Concept of Mathematical Laboratory

Mathematics has always been considered a dreadful subject and you might not have heard the word phobia associated with other subjects except Mathematics. The main reason is we fail to associate mathematics with the daily life situation. A concept can only be termed as permanent if it attracts the learner. Since the birth of an individual we start calculating weight, timing, diameter of head etc. that is nothing but a part of hidden mathematics but as we grow, we start hating mathematics despite the fact that it is the integral part of every individual.
Now to link mathematics with the daily life situation, to engage the students to have a feeling of exploring the knowledge by doing a concept of laboratory has been introduced in curriculum. The mathematics laboratory is a palace where anybody can experiment and explore patterns and ideas because mathematics is nothing but the pattern recognition, playing with numbers to enjoy the inner beauty of mathematics.
A world where student can discover what mathematics is all about and explore the previous knowledge to give a new breakthrough by means of games, puzzles, three dimensional objects, theorem in geometry etc. All such activities create interest among students who wants to explore and test some of their ideas, beliefs about mathematics. It provides and opportunity to discover through doing. The activities help students to visualize, manipulate and reason out conjectures and test them, and to generalize observed patterns. It is a place where anyone can generate problems and struggle to get an answer, design new mathematical activities.
Saying all the above, I am of the opinion that mathematics is omnipresent and it can’t be learnt in a confined area that is now termed as Mathematics laboratory. Mathematics learning can take place everywhere even in the garden, park, playground, the shapes, number around us.
1.       A symmetry pattern can be learnt in nature by observing the leaves symmetric.
2.       A Fibonacci pattern can be understood by cutting the cross section of oranges, counting the anti-clock wise and clockwise lines on pine apples, counting the pattern in sunflower etc.
3.       The concept of mathematics applied in Football shape having pentagon and hexagons can be understood in the field itself.
4.        Nature can teach us many mathematical phenomena and for that we don’t need any laboratory as such but for the practical purpose it is not feasible to do all such experiment in open.
Importance of Mathematics Lab: -
a)       Lab offers more scope for individual participation through different activities. It encourages independent learners and allows them to learn at their own pace.
b)      It widens the experimental base and lays groundwork for later learning applicable in new learning.
c)       It helps in developing meta cognitive abilities as a student attempts several times without any interference from time constraints.
Few mathematical lab instruments and its applications: -
1)      Tower of Hanoi: - It is easy to construct with the help of rubber sheet and rods but it has practical implication not only for the purpose of entertainment but it has its history associated with Vishwanath Temple Varanasi. If there are 5 discs chosen then minimum number of moves required to transfer the discs to one of the other rods using the third rod is 31. If the number of discs is more than number of moves will be 2^n – 1. This puzzle will help students to find the number of moves starting from 2 discs and increasing the discs one can understand the concept of COMBINATIONS. Moreover, students can also raise questions like – a) Why we need three rods not two rods? b) What would happen if we add one more rod?
2)      Circle: - In Primary level we have the different components of circle like – diameter, chord, sector, arc length etc. In addition to this in secondary level we study the different theorems based on segments, cyclic quadrilateral, tangents etc. All such concepts can be understood with the help of mathematical laboratory.  A) Finding the area of circle b) relation of sides and radius in circum-circle, incircle etc. c) tangential circle d) angle subtended by an arc of a circle e) equidistant equal chords etc.
3) Graph: - A student can’t be taught the graph on a normal blackboard. The concept of ordinate, abscissa and co-ordinate can be taught to a student in laboratory,
4)      Polygon and Polyhedron: - In mathematical laboratory a student visualize the three-dimensional pattern that he /she can’t feel in the book or when drawn on blackboard. The Euler’s relation between Edges, Vertices and Face can only be understood with the proper visualization of the object.

5)     Poster and Chart: - Mathematics can be made inspiring by letting the students know about the story and achievement of mathematician. Importance of mathematical days like – Pie day (14th March), Pi Approximation day (22nd July), National Mathematics Day (22 December). Besides that poster of mathematicians with some information and tidbits can be kept in laboratory.
6)      Surveying Instruments: - In geometry and especially in trigonometry measuring the height of an object from a distant point depends on the angle of elevation or depression and it can be understood if we use – 1) angle mirror 2) plane table 3) hypsometer and clinometers  4) astrolabe 5) sextant 6) ellipsograph 7) opisometer can come handy to understand the concept of height and distance.

7)    Conic Section: - In geometry the concept of parabola, hyperbola, ellipse can’t be easily explained on blackboard. The concept of focus, latus rectum, eccentricity can be explained in lab with proper explanation and experiment to students.

8)  Mathematical models: - Various models on plane geometry, three-dimensional geometry can be stored in a lab to understand the abstract mathematical proofs, principles or statements in mathematics. Moreover, the concept of symmetry, rotational symmetry can be explained in laboratory.

9) Algebraic/ Geometrical proof: - Mathematics is all about knowing the proofs of identities. In algebra there are many identities that can be explained with the help of some puzzles. Even theorems can be explained in lab.

There are 100 of mathematical activities that can be explained in laboratory. Even working models can be explained or students can be encouraged to make mathematical models.

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

## December 4, 2018

### Maths Phobia - Causes and Remedies -- By K Bhanumoorthy

What is so important about Mathematics Phobia?

- K Bhanumoorthy
Rtd. Principal , KVS

Let me say what Mathematics is? Maths is a language of pattern. It is a vehicle to train the mind. People think Maths is all about numbers only and equate with “Arithmetic” in minds, of course numbers and Arithmetic are only part of maths. The subject improves logical thinking, reasoning, and a way of rigorous thinking.  Its applications   are applied in our daily life.  One  can  never  think  of  a situation where  mathematics  is not applied,  applied  in  all  branches  of  science, Commerce, Engineering , Sports, Music, Dance , History, geography, literature, religion and so on. It is a pre-requisite for our work.

I have been teaching Mathematics since long.  Students dislike maths, perhaps may not know what mathematics is all about.  It is not, people do not like mathematics; I would say people do not understand Maths.  With due apology, I would like to say ,some if not many humanity faculty  people ( inclusive of  some parents) who proudly  say/talk about how they can’t do mathematics  and  they are quite proud that they don’t understand anything  about  Mathematics.  This could be one of the reasons for the irrational fear in learning mathematics.

Some  kind  of  remedies:   First and foremost to create interest in the subject and make it exciting. Fun with numbers,  interesting  puzzles to solve, quizzes, by narrating some interesting incidences / stories in life of Indian mathematicians especially Shri. SRINIVASA  RAMANUJAN, who was born at KUMBAKONAM, Tamilnadu, contributions relevant to subject maths. For certain questions, given by him, solutions are yet to be found out.  Contributions from Great mathematicians from other countries, History of maths right from ancient period. Greek has contributed quite a lot. Easy way  of  calculations referring from Vedic maths, how number system developed, to find  the  easy way to solve the question. There are interesting numbers connected with famous mathematicians.  Ex:  Srinivasa Ramanujan Number is 1729. It is the only number which can be written as sum of cubes of two numbers ie 1729 =10^3 +9^3 , 12^3 + 1^3. Similarly Caprikar  number, harshad number, Triangular number, so on . These kinds of numbers are formed by simple operations of mathematics. So one has to have curiosity to find out, most of the matters are available on the ancient books/ internet. People have to find some time to know about them out of their own interest, not under compulsion. Teacher has to take some effort to find out what additional information can be given, though it may not be asked in the examination but yet to know/create curiosity or interest.

A simple statement:   To concentrate on class room teaching.  One thing I would like to share, we do give more importance to procedural learning .There are some constraints. Time  limit on completion of syllabus.  Teachers are not having sufficient time to provide for more in depth of the subject, to orient students on open ended approach, allowing investigation, explore, etc. Just like science one has to explore and investigate why it is so. On some occasions teachers are deputed on duty to escort students, other  govt  directed and assigned works (for election, census work, etc)  one  cannot deny. Quotation  from  Einstein “ If you have not made any mistake , you have not  tried anything new “.The emphasis is on explore to meet the challenges later on, spoon feeding can’t help to solve a problem at unknown situation where difficult and challenging questions might come to solve.

Teaching is an art, differs from individual to individual. So care is to be taken by the teaching   fraternity.  The teaching   has to be student centered.  Basics, I mean fundamentals at primary level to be taught thoroughly, linking with daily life situation, then learning becomes easier.  Conceptual teaching learning process shall definitely help students for retention.  Learning outcomes are specified for every class, with specific Minimum level of competencies if not all abilities are acquired/ equipped, so to say minimum level of learning (MLL). It is the duty of teacher concerned to see that the minimum specified abilities are acquired. Now a day’s text books are well written by subject experts and students are to be guided by subject teachers, to read and go through the text book, just like other subject books. This would help students to a large extent for self learning and ask to clarify doubts. A teacher has to prepare, modify according to the level of students and see that how much learning has taken place. Sometimes the same method may not help, if a teacher teaches more than one section, there requires skills to modify/ that suits the students in teaching learning process, so to say need based. There is a saying if you want to teach mathematics to a student, say “KIM”, it is not enough the teacher knows mathematics, he must know “KIM” also.  Quite sometimes back, I have heard from my friend, who is an academician.  I am reproducing what I heard.  He narrated like this: A student asked him, “Sir, I am not able to understand the way by which you teach me, could you please teach the way by which I can understand.”  Case study of a student is very much required.

Teacher has to adopt certain techniques so that learning becomes easier. There are various activities could be carried out either in the classroom or outside the classroom by group activities. One would have become a teacher either by choice or by chance, but the accountability in the profession is a must. If all  students say “Sir, today class has been so interesting and enjoyable”. Then it is a big reward /award for any teacher. Teacher has to use simple language, make the students understand the concepts clearly. We can also use synonyms for easy understanding.
Teachers should adopt collaborative approach to promote positive attitude among students towards math subject. Also, teachers can reduce fear of math by being friendly, encouraging self confidence among students, providing support and guidelines, aiming at, student entered learning, accommodating students to adopt activity based learning and learning by doing. Students could be motivated to learn subject by simplifying concepts through math lab activities, mathematical games, hand outs, worksheets, math quizzes, puzzles etc.
“The Only way to learn Mathematics is to do Mathematics.”
“ Believe You can and You’ re  halfway there.”----- Theodore Roosevelt

## June 23, 2017

### 23 and Mathematics

Today is 23rd day of the month. Watch this video to understand the number 23 mathematically.

## February 20, 2017

### How to find Rational Number between two numbers

Number between two Rational Number:-
There are infinitely many Rational number between two rational numbers. If you are asked to find 5 Natural numbers between 5 and 15 then you would have begun with the smallest number 5 and added 1 to get 6, then consecutively and continuously adding 1 to the previous number you would have obtained the number between two Natural numbers.

Question: - Can you find 5 natural number or whole number between 3 and 4?
Answer:- Absolutely not, there is no natural or whole number lying between 3 and 4.
Now come to rational number and here you can find the infinite many rational numbers between two numbers.

First Method: - If m and n be two rational numbers such that m < n then 1/2 (m + n) is a rational number between m and n. Question:- Find 3 rational Number between 3 and 4? Answer: - 1st number between 3 and 4 is ( 3 + 4 )/2 = 7/2 2nd Number between 3 and 7/2 is (3 + 3.5) / 2 = 3.25 3rd Number between 3 and 3.25 is ( 3 + 3.25) / 2 = 3.125 Second Method:- Multiply the numerator and denominator of both the number by 10, 100 …

Third Method:- To find the rational number between two number p and q there is a beautiful formula that will help you to find the rational number easily.

Send your comments to
rkthakur1974@gmail.com

## October 22, 2016

### Uses of Quadrilateral in our life

Uses of Quadrilateral in our life:-
a) In architecture quadrilateral are the most common shape used in architecture. Triangles and quadrilaterals can both make amazing shapes. Here is an architectural design of a house and as you can see that in involves only the use of different shapes of quadrilaterals.

b) The vast majority of properties are bounded by quadrilaterals. Nearly all papers and magazines are quadrilateral, as the footprints of most boxes, the shapes of many rooms, the walls of all houses and the floor in most of the cases are in in shape of quadrilaterals. A general quadrilateral with all sides of different lengths and no parallel sides may not be suitable for such tiling when it is repeated. We tend to use/choose those shapes that are suitable for packing and tiling. The use of golden triangle is evident in the construction of the pyramid. The value of Golden number is 1.618033989 and an angle based on this will have size arc sec (1.618033989) = 510 50’ and the sides of Pyramid rises at an angle of 510 52’. The second interesting part of pyramid is about its perimeter that is 365.24 the number of days in the year. The famous Parathenon temple in Athenes have also been based on the theory of Golden rectangle.

c) Diagonal of a rectangle divides it into two congruent triangles and the idea of congruency especially in triangles had been used by Egyptians to build The Great Pyramids of Giza
The idea of congruency of triangles initially from diagonal of quadrilaterals also helped Leonardo Da Vinci to paint the world famous 'Monalisa'!!!!! The Monaslisha painting is of dimension 73 cm x 53 cms which is obviously a rectangular shape. Some mathematicians believe that the Leonardo da Vinci used the principle of Golden rectangle while painting Monalisa. A group of researcher from University of California, San Diego and the University of Toronto discovered that the distance between a wonan’s eyes and the distance between her eyes and her mouth are the key factors in determining how attractive she is to others and this ratio is nothing but the Golden ratio discussed above.

d)

e)

So enjoy reading

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

### Quadrilateral during Vedic Time

Quadrilateral during Vedic Time:-
Vedas are the rich source of our knowledge. During this period the rituals to please gods by means of Yagna were high. The rituals was an extremely important part of the ancient Hindu religion. Fire altars were constructed to perform yagna and they were made with precision in different shapes and size. Geometrical rules found in the Sulvasutras, therefore, refers to the construction of squares and rectangles, the relation of the diagonal to the sides, equivalent rectangles and squares, equivalent circles and squares, conversion, of oblongs into squares and vice versa, and the construction of squares equal to the sum or difference of two squares. In such relations a prior knowledge of the Pythagorean Theorem, that the square of the hypotenuse of a right-angled triangle is equal to the sum of squares of the other two sides, is disclosed.
Sulba means pieces of Chord or string and Sutra means formula. The Sulba sutras are the mathematical discoveries by famous Indian rishis turned mathematicians at about 1000 BC to 200 BC, using a piece of chord for constructions of various fire sacrifice altars.

Fire altar

The purpose of the rituals was to build an immortal body that would transcend suffering and death, both hallmarks of mortal existence. According to Plofker:-
Many of the altar shapes involved simple symmetrical figures such as squares and rectangles, triangles, trapezia, rhomboids, and circles. Frequently, one such shape was required to be transformed into a different one of the same size. Hence the Åšulba-sÅ«tra rules often involve what we would call area–preserving transformations of plane figures, and thus include the earliest known Indian versions of certain geometric formulas and constants. . In an article published in Indian Journal of Histroy of Science “Ritual Geometry in India and its Parallelism in other Cultural areas” – by Mr. A. K. Bag where he writes that the construction of altars having drawn on a base of different figures such as square, circle, semi- circle, isosceles trapezium, triangle, rhombus, falcon or tortoise shape and other led to the development of various geometrical figures, their transformations and calculation of areas involving many Pythagorean relations with rational and irrational numbers leading to its general statement, approximation of the value of √2 and others. Here is the shape of Mahavedi that is in shape of Isosceles Trapezoid trapezium of area 972 sq. units with 24 and 30 units of length for parallel sides and 36 units of length for altitude with the shape of Square and Rectangle within.

Dr Rajesh Kumar Thakur
rkthakur1974@gmail.com

### Quadrilateral -1

Can you guess which one is the largest highway project in India? Don’t worry, I am giving you a clue. This is the project started in 2001 by the PM Atal Bihari Vajpayee. See the map below and concentrate on a mathematical term emerging out from this map. One more thing about this project is that in this project four major metropolitan cities – Delhi, Mumbai, Kolkatta and Chennai will be linked with each other.

Source: - National Highway Authority in India

Don’t worry, Let me tell you the name of this project so that you can guess the topic we will discuss now in details. The name of this project is Golden Quadrilateral.
The above examples are enough to show that quadrilateral is a geometrical shape that consists of four edges (Sides) and four vertices (corner).Quadrilaterals, like triangles are the second most common shape used in geometry. It is a geometric shape that consists of four vertices sequentially joined by straight line segments.
The word ‘quadrilateral is derived from the two Latin word – quadri and latus. The first word quadri means four whereas the word latus refers for side. In other words, quadrilateral is a figure having four sides.
According to the Merriam- Webster Dictionary a quadrilateral is a geometrical shape having four sides. The Collins Dictionary defines quadrilateral as a polygon having four sides and a complete quadrilateral consists of four lines and their six points of intersections. The other words sometimes used for quadrilateral are quadrangle and tetragon.
A quadrilateral is generally of three types – Convex quadrilateral, Concave quadrilateral and Simple quadrilateral.
A quadrilateral that has an angle more than 1800 is called Concave Quadrilateral. A quadrilateral is called simple if the four sides of a quadrilateral meet at vertices. Here we shall only focus on convex quadrilateral where each angle is less than 180.

Dr Rajesh Kr Thakur
rkthakur1974@gmail.com

## October 16, 2016

### Fundamental of Mathematics for Secondary and Senior Secondary level

Secondary and Senior Secondary level

Secondary level is the stage where mathematics comes to the students as an academic discipline. It is a stage where students begin to perceive the structure of mathematics. Mathematical terminology gets sophisticated and the concept of proof becomes the central to curriculum. The new branch of mathematics Trigonometry and Co-ordinate geometry get introduced. The working of secondary level is somehow based on the learning the concept of primary and upper primary level. Mathematics at this stage is totally based on concept and reasoning. The theoretical part of mathematics begins and students find it irritating. A bigger question now arises: - What does mathematics really consist of? Mathematics in secondary and senior secondary level consists of Axioms, Theorems, Definitions, Theories, Formulas and Methods. Mathematics could surely not exist without these ingredients; they are all essential. Now the question is what a secondary student should do to enhance his mathematical skill. Polya says – To a mathematician, who is active in research, mathematics may appear sometimes as a guessing game; you have to guess a mathematical theorem before you prove it, you have to guess the idea of the proof before you carry through all the details. Mathematical facts are first guessed and then proven. If the learning of mathematics has anything to do with the discovery to do problems in which he first guesses and then proves some mathematical facts on an appropriate level.

• The student needs to integrate the techniques and basics learnt at the previous level.

• Students at this stage should have practical knowledge of why this and why not this. Reasoning helps to retain the learning for long and you not only understand the principle of mathematics but also understand its practical application.

• Students should have the knowledge of finding square, square root, cube and cube root of any number. It is also expected from the student in secondary level to have memorized the square up to 50 and cube up to 30 so that calculation can be done with ease without error. If possible learn the fourth power and fifth power up to 5 of first 10 numbers.

• They should learn some technique that can help them to calculate speedy. There are ample books available in market which will prove beneficial in order to calculate faster. Vedic Mathematics is now a popular book used by students for fast calculation. The best part of learning some trick to do arithmetical calculation faster is that you save ample time in your examination and the chance of your doing right calculation increase. Besides that student should also learn method to counter check calculation so that the chance of error is avoided. There is one simple and sophisticated technique to counter check the calculation in seconds. The method CASTING OUT NINES method will be beneficial to students in checking the calculation.

• In geometry, first learn about different shapes and its properties. Since the geometry of secondary level is highly based on some theorems so it is advised to students to learn the important statement of theorems so that they can write the statement or name of the theorem in proving a geometrical proof.

• Word problem is another area that needs to be addressed. Understanding mathematical language is equally important to do word problem. So before going through the exercise learn the most frequently used mathematical terms.

• Solving algebraic problem is considered a tough task but I have noticed students performing well in algebra once they understand the language of the word problem. Confusion creates when you don’t remember the formula to be applied at the appropriate problem and look for another way of solving the problem. Algebraic problem may be turned into a joyful game once you understand how to proceed with the problem. So before doing the algebra, learn algebraic identities, get mastery over understanding mathematical language of the problem.

• Trigonometry is introduced in Class 10th in India and students feel uneasy when they solve the problem or prove the identities. You must understand the fact that Trigonometry is the backbone of mathematics in senior secondary level. If you are not good at Trigonometry in secondary level, you will have baskets of problem as Differential and Integral Calculus taught in class 12 can’t be done without clear understanding of Trigonometry. In trigonometry, you must learn the formulas based on trigonometric ratios and understand the principle on which the finding of different angles such as 0, 30, 45, 60 and 90 are based so that when you go in class 11, you can extend these principle in finding the value of different angles such as-120, 135,150,180,---360 and more.

• Last but not the least, at the secondary stage; a special emphasis should be done on experimentation and exploration. Make model and visit mathematics laboratory whenever possible. Mathematics can’t learn alone by merely learning formulas until you master mathematics by practicing. This practical aspect of learning can be achieved only if you make mathematical model, play mathematical brain twister and visit mathematical laboratory to understand the mathematical concept with the help of diagram and mathematical shapes kept in laboratory. Activities in practical mathematics help students immensely in visualization.

Dr Rajesh Thakur
rkthakur1974@gmail.com

### Fundamental of Mathematics for Primary and Middle School Students

Fundamentals of Mathematics

Mathematics is like a pyramid. Every new skill requires an understanding of pre requisites to do well. Mathematics is also an analytical subject. It opens the closed gate of our mind. If you are good at math your ways of critical thinking will certainly be far better than those who are not so well in math. Now the BIG question is--- Why someone is good at math and others are not?

The first one is a Kuchha House made by mud and the second one is a Pucca House constructed with bricks, iron rod, cement etc with proper design. Please answer me a simple question--- Which house is stronger and you would love to live in?
Of course, the second one, because it is strong and well constructed. Its foundation is strong as good quality of bricks, cement and iron rods are used while constructing the house. A well planned building last long, so as if the initial preparation is done for mathematics, it will make the mathematical foundation of a student strong enough to cope up with the syllabus of higher class. Now the big question arises --- What one should know in order to make his/ her mathematics good?
I have divided the fundamental principles into two parts:-

a) For Primary and Upper Primary Level
b) For Secondary and Senior Secondary Level

For Primary and Upper Primary Level

The Annual Status of Education Report 2011 has shown an alarming decline in mathematics skill in 6-14 age group. It states that less than a third of class students in rural Indian school can solve simple two digit subtraction problem.

I see many students saying—I hate math. It is not that they hate math, it was that they hated the fact that they didn’t understand math and that they didn’t understand math because they were missing basic building block. Mathematics has a distinction of being most unpopular subject because it requires the learner to think correctly. Most people love to speak about any issue but hate to accept that they are wrong. Mathematics tells right is right and wrong is wrong. The best art to learn mathematics is that you should have the pre-requisite knowledge of the subject. You can’t do addition until you have the understanding of numbers.

The ASER report indicates the same. Two years ago I read an article describing that more than 50% of students passing out class 5 don’t have the adequate knowledge of summing up two numbers. It shows that the fundamental of such students are weak enough to pursue math as carrier. Now the bigger problem is – if students are not aware of eight fundamentals till upper primary level, you can’t think of their doing well in secondary level math.

What is the minimum competency a primary student should possess?

a) Learn Table:- Student should learn table up to 20.

b) Command over Four Fundamental operations:- In primary level students are expected to be well off in four fundamentals of mathematics (Add, Subtract, Multiply, Divide). Not only this student should have a clear understanding of the facts- when to add, subtract, multiply or divide.

c) Number and their properties:- They should be aware of numbers and their properties. Till primary level one should know about Natural Numbers, Whole Numbers, Prime and Composite Numbers, Even and Odd Numbers. On the other hand a student passing class 8 should be aware of Rational Number, Irrational Numbers, Real Numbers, and Perfect Numbers etc. Besides that, student should know the properties of these numbers. Moreover, it is expected from a bright student to be familiar with Triangular numbers, Square Numbers, Cube Numbers, Pythagorean Numbers, and Ramanujan Numbers etc.

d) Place Value System:- The discovery of ZERO made our life easy and we are now in a position to write big- big numbers. In primary level (up to class 5), students should learn to write numbers up to 100 and they should also have sound knowledge of Place Value System. Students should also learn how to write numbers. I have seen many students of class 6 in government school in Delhi fail to differentiate between 40023 and 423. I would advise students that they should develop the ability to recognize the pattern of writing numbers with the help of place value system. They should have enough practice of writing big numbers at home, at school so that they never get confused in writing big numbers either in words or figures. This can only be achieved if students are taught the place value system in an effective manner.

Before decimal After Decimal
Lac Ten-thousand thousand hundred tens one Tenth Hundredth Thousandths
100000 10000 1000 100 10 1 1/10 1/100 1/1000

Clear understanding of the diagram will not only help them understand how the number is written but also it will help student understand the facts why 22-48 is read as Twenty two decimal four eight not forty eight.

e) Conversion of Units:- Students till primary level should also be taught the addition, subtraction, multiplication etc of Decimals, Hours –Minute- Seconds, Rs –Paisa, etc.. They should also know the conversion of one unit into another. 1 inch = 2.54 cms I foot = 12 inch 1 Kg = 1000 gram 1 meter = 100 Centimeter 1 Hour = 3600 seconds 1 Hectare =10000 Sq metre etc.

f) Divisibility Test of Numbers:- It makes me sad when I see students of upper primary level (Class 6 – 8) struggling in converting a fraction into simplest form. This is because they do not remember the divisibility test taught at primary level. Divisibility test pays important role in solving the numerical problem whether it is from simple arithmetic or mensuration. Every student should learn the divisibility of 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. Divisibility rule plays a vital role when a fractional number is simplified. It also helps us in deciding the factor of any particular number. In upper primary level or secondary level you will have problem on Factorization of a number, finding square root, cube root by factor method and there you will find the divisibility rule handy.

g) Mathematical Laws: - When I teach the secondary school students and find them not knowing about Commutative law, Associative law, Distributive law, Identity law, Inverse law, I feel very sad about it. The first three rules are taught in Primary level and the last two rules are taught in upper primary level. If a student is not familiar with these laws then they won’t be able to solve the problem of Binary operations in secondary level. Moreover, in secondary level when they are taught about characteristics of numbers, matrix they will hardly understand about such laws. Hence, it becomes the duty of teachers and parents to teach these rules to students.
A + B = B + A A X B = B X A --------- Commutative law for + and x
A + (B + C) = (A+B) +C A x (B x C )=( A x B) x C – Associative law for + and x
A x (B + C) = A x B + A x C Distributive law
A +0 = A Additive identity (0)
A x 1 = A Multiplicative identity (1)
A + ( - A ) = 0 Additive inverse
A x 1/A = 1 Multiplicative inverse

h) Knowledge about Mathematical Shapes and their Properties:- In Primary level, students should be given the detailed knowledge about all geometrical shapes ( Types of Triangle and Quadrilaterals). The properties of geometrical shapes such as Triangle (Right angle, Equilateral, Isosceles, Scalene, Acute angle, Obtuse angle), Quadrilateral (Parallelogram, Trapezium, Rhombus, Square, and Rectangle) should be understood by students. Mere recognizing the mathematical shapes do not work unless you also know about the properties of the geometrical figures. Moreover, students should be shown the 2-Dshapes like Circle, Triangles, Quadrilaterals etc and 3-D shapes like Cone, Cylinder, Sphere, Hemisphere, Cube and Cuboids so as to use it explicitly when the application of such mathematical figure come. In primary level wherever possible students should be made acquaintance with such 2-D and 3-D shapes with mathematical models, power-point presentation etc so as to understand the concept of areas and volumes clearly. In villages, women draw Rangoli where they use curves, mathematical 2-D shapes, properties of symmetries. Such traditional knowledge should be passed to students. Appreciating the relevance of such mathematical values used by the villagers can enrich the child’s perception of mathematics and they can enjoy mathematics.

i) Fractions and Decimals: - Fractions and decimals constitute another problem area. I have seen students of secondary level sometimes fail to sum up or subtract the algebraic problem involving fractions. There is some evidence that the introduction of operations on fractions coincides with the beginning of fear of mathematics. It is the responsibility of primary teacher to clarify how to do fundamental operations in fractions. I am putting here some examples that need special care while teaching. If teacher takes the responsibility to make the fundamentals of why and how in such types of operations in fraction understandable to students then they will be able to apply the same in algebra.

Addition / Subtraction :- a) 2 ± 3/7 b) ½ + ¾ c) ¾ ± 5
Division a) 5/8 ÷ 10/12 b) 2/3 ÷ 4 c) 5 ÷3/7
In addition to this, students fail in multiply, add, subtract and divide the problem on decimals. Primary students generally face problem of placing decimals after multiplying or how to place different decimal numbers while doing addition or subtraction. The primary level courses designed by different text book publishing company don’t put much emphasis on the application of fractions. Such things have largely disappeared from the text book and that need special attention of teachers because the importance of fraction in conceptual structure of mathematics is undeniable.

j) Understanding of mathematical laws and formulas: - The syllabus of upper primary level is somehow dependent to that of primary level. In upper primary level, most of student encounters problem in algebra and applied geometry. In algebra the most confusing part is why---
i) + 5 – 7 = - 2
ii) – 2 x – 3 = + 6
iii) 6 x – 5 = -30
iv) 8 ÷ (- 2 ) = - 4

The root of the problem is that students are asked to learn the rule without giving any proper explanation. Mathematics is a reasoning based subject and until students are taught why and how in this subject, it can’t be made popular. I, therefore request all mathematical lover to be inquisitive and learn why and how of everything. Moreover, chapter like Percentage, Average, Exponent, Factorization, Ratio and Proportion, Profit and Loss, Mensuration, HCF and LCM, Square and Square root, Cube and Cube root should be dealt with rigor. You must learn the basic rules, definitions and formulas related to the above chapters. When I see students of class 10th unaware of the terms – cyclic factorization, componendo and dividendo, surds, rule of three etc, I feel sad about the level of mathematics being taught in our school. I am not blaming the students only the fault lies in our education system. The syllabus designing bodies should look into the matter. The NTSE, Math Olympiad asks question of high qualities and that require a sound knowledge of the subjects. If you compare the syllabus of the competitive examination with that of school curriculum you will find that there is a huge gap and that gap need to be bridged with standard textbook. Buy qualities text book and go through the chapters minutely. Consult other book for the same chapter and you will notice that your knowledge get updated every time you consult a new book.

Dr Rajesh Thakur
rkthakur1974@gmail.com