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
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