January 10, 2019

Error in Mathematics - K . Bhanumoorthy

Analyzing the common errors made in Mathematics

Analyzing the common errors will lead us to identify the cause when it happens consistently. Here we discuss about some types of common errors.

Types of Errors: Careless Errors, Computational Errors, and Conceptual Errors.
Careless Errors

These errors occur on account of lack of attention or working too fast. Here are some examples: Copying the problem wrongly, posting a wrong number from the question, Sign, negative / positive. No clear handwriting, on some occasions even over writing.

Remedial ways: First and foremost is to slow down, pay attention what is given, what is asked. Either circling or under lining important information, what to do especially in word problems.

Computational Errors
Mistakes are committed in the process; mistakes are done on the incorrect operation (addition, subtraction, multiplication, division). Making one computational mistakes lead to further mistakes.

Remedial ways: A teacher has to tell the correct procedure. All required steps are to be written by the students. Conceptual learning is more important, small computational error may be avoided. Final solution is not important than understanding the Concept. One has to work carefully with given data. Check the answer step by step after solving, in order to have accuracy and correct solution.

Conceptual Errors
If students have not understood the concepts, these kinds of mistakes occur. We can say either incorrect logic or incorrect method. Preventing conceptual errors are not as easy as other errors.

Examples: Jim has a bag of 24 chocolates. He decided to share equally with 6 of his friends, how many each friend gets? Solution is  24 / 6=4 is the correct  answer, instead if they have multiplied 24 X 6 ,there we do understand ,the concept is not understood properly ,though the multiplication result may be correct . Student go wrong while calculating surface Area, Total Surface area, care to be taken when one side is closed, both sides are closed, hollow cylinder, especially in semi-sphere, etc. Same is applicable while calculating Volume. Area and volume to be defined clearly of 3 dimensional figures.

Remedial ways: Introduce concepts in hands-on, conceptual ways using teaching aids. Teach a concept more than one way (Open ended approach). Have Math talks. Use math journals.

- K. Bhanu Moorthy

December 18, 2018

How to get 100% in Mathematics - By K Bhanumoorthy

       How to get 100% pass in mathematics?
                                                                                       K Bhanumoorthy

 In continuation of “ How to get 100% pass in Mathematics , I suggested 5(five ) points in blog 2 ,  where teachers have to pay attention. Few more points are incorporated to attain the goals.

1. For board exam, students do practice many sample papers as per board pattern (Chapter wise weightage of marks, also Blue print to be discussed, for the general awareness.). We have to get to the point where the minimum is not enough.
2. Few students who wish to complete first thereby make mistakes, rechecking is to be insisted once over and follow up should be done.
3. Teachers have to identify certain chapters where students can easily score marks. In those chapters, teachers have to re-teach and give sufficient practice, to slow learners, instilling self confidence that they can clear the exam.
4. Within a period of 2 to 3months cordial relation has to be established so that students do not have fear to ask and clarify doubts.
5.Planned  home  work  for  feedback  and  assessing  work  carried out by students.
6. When teachers are paying attention to slow learners, gifted students should not be neglected .They are to be encouraged,  may assign  some  work  as  group leaders, taking them into confidence,
7. Last but not least in a test, if no one has attempted / not done correctly, any question, teacher has to think seriously and ponder over what went wrong, for self introspection. 
Parent’s role in a child’s life has far reaching impact .So teacher has to have very good rapport with parents so that they make a home a good place for learning. Teachers have to seek  parents  support  and   involvement .Parents are to be counseled, with a note, that they do  praise, encourage, and give positive influence so that their wards do better in examination.

When the teacher says, I am the one that makes learning possible in the classroom and I am committed to make it happen. And the student says, I will do everything that I can, to learn. I am ready to learn. That is when the magic of learning really happens. So student’s commitment depends on teacher’s commitment. One thing is, to study,  whom you are teaching, the other thing is, to study, the knowledge you are teaching. If you can interweave the two things together nicely, you will succeed in getting 100% result. Ultimately let students feel and give their hands in faith that “MY TEACHER SHALL LEAD ME IN THE RIGHT PATH”.
My sincere thanks to all my teachers, parents, students , friends ,  enabling me to express my views.

K Bhanumoorthy 

How to get 100% pass - Part 2 -

                                                         How to get 100 % pass
                                                  K Bhanumoorthy 

In continuation  to  blog  1 on the  topic “ How to get 100% pass in Mathematics”  I gave the details of various boards  and our commitments  to our profession, setting targets . Here  again   I have suggested  certain aspects where the teachers could pay attention to achieve the target .

Attention to Teachers:
1. To  be  aware  of  whole  syllabus  to be taught for the academic year.
2. In general a class is heterogeneous, so teaching has to be child centered cater to the needs all types of students (3 types of learners visual, auditory, kinesthetic).
3 .Case study of every student after 2 months or so, to be maintained .After conducting test(s), the teacher should be able to identify what types of errors/mistakes, does a student make. The errors may be in computation, comprehension, conceptual. The required remedial measures are to be carried out  then  and  there. Appropriate guide lines are to be given with all sorts of encouragement.
4. Group wise activities, now a day’s collaborative learning is suggested since the students feel ease, more comfortable to clarify doubts with friends circle and make up their mind for healthy competition. While forming groups care is to be taken that each group consists of students of different caliber.
5. Time management is the need of the hour, so students are to be trained / sufficient practice to complete the sum, correctly. Once the unit /chapter is over , for class assignments instead of 3 or 4 questions written on the black board, to solve, I would   suggest one question at a time  is written on the black board. This would help a teacher to find, who has completed on time, who have not completed, who and all need guidance. Then  to another question and so on.

To be continued -----

How to get 100% pass in Mathematics - Part 1 - K Bhanumoorthy

How to get 100% pass in Mathematics
 ---------- K  Bhanumoorthy

Introduction:   This title/topic has been chosen for the reason being no detention till Class VIII. Under the current provision of the RTE Act which provides for free and compulsory education  to  all  children  of the  age  6 to 14 years.  The idea may be to minimize the dropouts up to class VIII. Even otherwise, keeping 100% pass is a must in every class, from class I onwards by every teacher, sustaining our commitments to one of the noblest professions. “A TEACHER” .I wish to stress, the students go to the next higher class with at least minimum required abilities (MLL) there by no one blames our own  colleagues/ the other teachers.  Every child wants to learn. A teacher has to have some inert qualities, to my mind 3 C’s, Commitment, Concern for students, Character. It has been a myth that mathematics is a difficult subject and it is for who have high IQ. It is not so. It is quite  interesting and its applications are applied in all fields.

Let us discuss ways and means:
There are Different School Boards in India
·           Central board of Secondary education.(CBSE)
·           Council for the Indian School Certificate Examination (CISCE-ICSE/ISC)
·           International Baccalaureate (IB)
·          Cambridge Assessment International Education (Cambridge International).

As per Oxford dictionary the curriculum means “a course of study “. The curriculum gives guide lines what to teach in an academic year. How do we teach, learning outcomes, how the assessment is done, are matters of questions.

To achieve 100% pass in maths should be a target, to be fixed by the teachers in the beginning of the academic year. There is no compromise at all. Aim is to be followed by an action. The teachers have to ask students also to fix the target, in the year beginning, may be as initial set in goal. The students have aimed at, no other way; he/ she should work for it. Teachers have to guide, timely support and encouragement, remedial measures shall go a long way to attain. The target may be revised showing upward trend before the annual / board examination.

 To  be  continued  in the next blog---

K Bhanumoorthy

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

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.

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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.



So enjoy reading

Dr Rajesh Kumar Thakur