SecondSchool Smart Tuitions teachers are encouraged to give frequent, positive feedback to students as this bolsters their confidence in themselves. They also take care to give assignments to students of an appropriate level, not too easy nor too tough, so that each student gains confidence through small successes. We have also designed our Practice Books in such a way that problems are given in order of complexity. This enables the teacher to assign the right questions to a student based on his/ her current understanding level.

Our teachers are also urged to build an open and positive environment in class, where every student can speak freely without any embarrassment. And we want each student to feel that they are a valued part of a learning community.

Parents too have a critical role. Too often, we come across parents who openly put down their wards in front of others. They often unwittingly describe their children as weak, lazy, dull, etc. in their presence. Each time they do so, they reinforce the child’s own perception about these so called weaknesses. Instead, parents must take care not to make any disparaging remarks about the child in question. It is quite OK to say that Anil has scored only 52% in his last test, and even to say that he did not work enough , or that his concepts are weak. However, stay away from making any statements that imply that there is an inherent weakness in the child. Such statements are damaging.

Building motivation in a child can be a slow and gradual process. However, motivation can bring dramatic results. It is amazing how a motivated child can easily learn and comprehend something that he/she took hours to do, when pushed to learn. Building motivation and self-confidence is a part of the process at SecondSchool.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

SecondSchool Smart Tuitions follows a rigorous selection process to ensure the quality of its faculty. All faculty have to clear tests which evaluate their subject knowledge as well as teaching skills. Once they clear these gates, they go through training in the SecondSchool Smart Tuitions methodology, and are asked to demonstrate their understanding by taking mock class sessions. All of our faculty are required to complete this certification process before they can teach at our centres.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

A good preparation hinges  on the following actions:

1.  Understand what the key concepts are, and how they apply to real life situations.

2.  List down separately all those definitions, formulae, and rules that need to be remembered, and revise them frequently.

3.  Look out for the different kinds of problems that arise in any given chapter. Remember, there are only a certain number of variations in the kind of questions that can be given to you. Every time you come across a question which looks different from what you have solved before, check if it fits the pattern of an earlier question, or that its a new variation.

4.  Test yourself and check where you made mistakes. Focus on the kind of questions in which you went wrong, and make sure that you understand why that happened. It might be a lack of conceptual understanding , in which case, take care to correct your basics.

This cyclical process will invariably get you good results. It is not necessary to solve every problem in every reference book, and work all the time. Work smart. Focus on your areas of weakness and the gaps in your understanding.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

The other key advantage of the SecondSchool system of remediation is that the teacher is now supported with very specific information about each child. Our assessment system throws up specific information about each child’s individual performance. It provides topic wise break-up, type of question wise analysis, and even error wise analysis. If Anushka has just not understood the questions in the last test, and has simply tried fluke answers, the system gets to know. If Akash who has scored rather well, however  has not understood how to divide fractions, the analytical process leads to these revelations and helps the teacher focus on just those critical areas where each child is weak. Imagine the benefit of the efficiency that this information creates. Now, rather than revise the entire chapter, which is what Akash’s other teacher would have asked him to do, the SecondSchool tutor would get him to go over the specific problems of fraction division, which he has not understood, thereby making the process of remediation far more of a robust methodology than one of hit or miss.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

We have known for a while that ‘learning by doing’ is more effective than just reading about it or being taught it in a classroom.  Schools that emphasize this method of learning, have kids who are better adjusted and who perform better.  And it is easy to understand why that will work.  Imagine being taught to drive a car through a lecture or for that matter being taught how to drive a nail with a hammer.  The simplest tasks are better learned through experience.

Activities usually involve doing something physically and works best when more than one person is involved in the activity.  So rolling a ball and pushing a block down a slope in a physics class to understand the concept of friction works better when one student is applying the force and the other taking notes.  Discussion and having to accommodate another student’s learning approach usually enhances learning.

For younger students, activities result in a process of discovery.  Watching a plant grow, putting together a geography project with animals, all of this makes learning real.  By middle school, students need games and competition to remain fully engaged.  By senior school, many things cannot be learned through activities – specially subjects like Mathematics that tend to become more conceptual.  But even for these students an early grounding using activities helps them understand conceptual topics better.

So given the obvious benefits, why do children have to learn mostly from books and lectures?  Why not add activities in classrooms?  A few years ago, with this approach in mind, schools were required to have ‘Mathematics Labs’ – physical rooms designed to let children experience maths and equipped with kits to do many activities.  Schools now do have these labs – though the experience is not quite what was planned.

Rajan was one of the first children to use the spanking new Maths lab in their school.  He walked in with a sense of wonder.  Lovely colours, tables shaped as polygons, patterns on the floor, many charts on the walls.  It seemed a different world.  He and his other classmates seated themselves and waited expectantly for some fun.  Veena Ma’am his teacher came in and started explaining a Maths concept using the board.  Rajan waited for her to use the beautiful shapes and blocks to explain things – and let him play with them.  4 classes later, he is still waiting.  The Maths Lab turns out to be just another classroom.

The reasons for not using Activity Based Learning range from inadequate teacher training, insufficient numbers of kits, too many kids in a class and so on.  But the idea is right.  Children will learn better if they are able to do activities around concepts.

At SecondSchool, we use an enhanced approach to activity based learning.  Rather than create a set of tools that need to be used by the teacher and perhaps shared by a few students, we provide all our students with their very own Maths kit.  So when a teacher needs to do an activity, every student already has the required kit.

Not all activities can be physical.  For example, to understand that the angles of a triangle must be 180 degrees, providing multiple triangles and asking students to measure them, gets the point across to an extent but only proves that the specific triangles given total 180 degrees.  On the other hand a computer based activity allowing the teacher to increase or decrease any angle, and along with that showing the size of the other two angles, makes the point perfectly.  At SecondSchool, in addition to each student having an activity kit, we use computer-based activities.  Students engage here through games and through interaction.

For parents who want their children to learn using activities, the internet is a wonderful source of projects and assignments.  Do these activities with your children – in fact get them to invite their friends when you do some of them.  You will find that they look forward to these sessions.  And their understanding will improve dramatically, as will their school performance.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

Suddenly there is a bewildering array of terms and a completely new set of instructions with more getting added every day.  Bringing in a tutor temporarily solves these problems with extensive practice and shortcuts, but does not build the understanding required to enjoy and apply Maths.  These are a few tips of what you can do as a parent to improve your child’s liking for and performance in, mathematics

1. Discuss school chapters at home; asking both ‘why’ and ‘how’

The ‘why’ is the concept itself.   Why is it important to know about integers?  It is not important to necessarily tell the child the why, but to ask them to figure it out?  Hence integers allows for negative numbers and the child should be able to think of reasons to have negative numbers.

The ‘how’ is often called procedural understanding.  For example a child may know how to do long division or how to find the factors of a number.  It is possible to learn this mechanically, without understanding the function you are seeking to perform.

Interestingly in Maths, it is not always clear which comes first.  Both are important and feed off each other.  Hence knowing how to factorize and doing it often helps children understand easier the need for say the Highest Common Factor of two numbers.

Try alternating between the two – the why and the how.  Explain the concept then get your child to practice it.  Once that has been done for a while, explain the concept yet again with more examples.  Stay with a topic till your child understands both the how and the why.

2. Explain terms and their relevance

Maths uses language in a very precise manner.  For example, an irrational number is not a number which does not make sense.  It has a very specific definition.

Similarly in Maths conjunctions mean specific actions.  For example ‘divide by’ and ‘divide into’ have different operations.

Most children struggle with ‘Word Problems’ or applications.  In many cases it is because they do not understand what is required of them.  Spend time explaining the problem itself.  Once the problem is understood, the solution is usually easy

3. Do not let them get behind in class

Most children need substantial practice before each concept is firmly embedded in their mind.  If they miss a few classes, or if 2-3 chapters are not clear, future concepts are likely to become one big muddle.

Falling behind in class is almost certain to trigger a loss in confidence and hence understanding.  If for any reason, your child has missed a few sessions, immediately schedule time to catch up.  Once you wait everything becomes harder.

4. Make sure they spend time solving Maths problems regularly

Practice is key to Maths.  A child who is diligent and works regularly in solving problems will find it easier to understand and remember concepts.

In Maths, there is always a real risk of forgetting a concept – specially in younger children.  Someone who was perfect in one area of Maths a month back, may become hopelessly lost again.  It is only through regular repetition that the concepts separate out and their application becomes instinctive.

In an independent study, regularity and diligence have been the most important factors of success in Mathematics.  In fact another interesting factor is neatness.  A child doing their sums neatly and completely is likely to be better in maths later.

5. Give lots of encouragement for trying and performance

This advice applies to all subjects.  At the early stages, getting confidence is the key.  Celebrate small victories.  When your child comes with an average score, find sums he has done correctly – I can almost guarantee you will be surprised by how much your child actually knows.

When they get a sum right give lots of encouragement.  Even if it is wrong acknowledge the steps that are correct.

6. Play simple maths games like ‘find the mistake’ and ‘set a paper’

Left to their own devices, most children will prefer not to have anything to do with studies.  However, if some of their agreed study time can be spent on games, they would enjoy it tremendously.

‘Find a mistake’ is a common game.  Take a problem and write a wrong answer.  Make the mistake something that is linked to an important concept – eg 5 – (-3) is 8 but if you write it as 5 – (-3) is 2, and ask the child to find what is wrong, they will both enjoy the process and will understand the concept much better.

7. Encourage them to do puzzles from the daily newspaper

Mathematics is the study of patterns.   You may notice that puzzles like Sudoku are usually done by people who are comfortable with Maths.  The daily Times of India has several puzzles daily of which at least 3 are numeric puzzles requiring the understanding of patterns.

Get your children interested in these puzzles.  Soon they will be able to complete these quickly (TOI has the more simple range of puzzles) and with that will come confidence in numbers.

You may find that you are more comfortable with some of these methods than others.  We have seen good results by even following any 3 of these; as long as encouragement is one of the methods used.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

Pi simply is the constant, denoted ∏, which when multiplied with the diameter of a circle, will give its circumference. A simple approximation that many of us use is 3.142 but architects, engineers and scientists require a more precise definition and hence the search for a precise answer. Archimedes is known to have estimated the value of Pi by applying the common sense logic that a circle must be bounded internally and externally by a regular polygon of large enough number of sides. He is said to have calculated the dimension of a 96 sided polygon that bounded a circle and thereafter gave up attempting any more precise a definition of the elusive Pi.

We now know that Pi is an irrational number, which means that it cannot be expressed as a ratio of any two integers. There is archaeological evidence that the ancient Egyptians knew something about this fraction, and were taken up enough by its unfathomable nature, that they enshrined it in their monuments. The Great Pyramid at Giza constructed in 2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits which gives a ratio equivalent to 2 ∏. It cannot be pure chance that this great monument was built to this ratio. The equivalent value of ∏ in this ratio comes to be 3 + 1/7 or 22/7 which is what students in schools still commonly use in its place.

Around 1400 AD, an Indian, Madhava of Sangamagrama estimated Pi to eleven decimal places by equating Pi to an infinite series. The development of infinite series to estimate mathematical values was a great innovation that helped achieve greater precision in calculations. Srinivasa Ramanujam is known to have expressed Pi in the form of several such series, each of them giving another way of reaching closer to the value of Pi.

A German, Ludolph van Ceulen, devoted the greater part of his life to estimating the value of Pi, and he managed to solve value Pi to 35 decimal places using geometrical methods in the sixteenth century. He was so proud of his achievement, that he had these decimals inscribed on his tombstone. Ever since, the constant is often known as Ludolph’s Constant. It is also better known as Archimedes’ Constant.

It was only afterwards, in the 18th century that the nature of Pi was understood as irrational, and it took another century to understand that it is a transcendental number, which means that there is no polynomial with rational coefficients for which Pi is the root. That simply means that Pi is not constructible with compass and straight edge.

Teachers of mathematics have naturally tended to use Pi, as a symbol for mathematical enquiry. March 14 is celebrated as Pi Day in some parts of the world. The date relates closely to 3.14 which is an approximation of the value of Pi. Schools and colleges take the opportunity to hold competitions and displays related to maths on this day. A popular contest is often around how many digits of Pi can be memorised and retold accurately. People have managed to recite over 10000 digits from memory.

There are many websites dedicated to Pi, and its teaching. We have even come across verses and lyrics composed around the inscrutable Pi. Here is one from a website dedicated to the teaching of Pi:
“Oh, number Pi, Oh, number Pi
You’re truly transcendental.
Oh, number Pi, Oh, number Pi
You’re physical and mental.
You stretch the bounds…of all we know,
And tell our circles where to go
Oh, number Pi…..”

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

em>* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

Cooperating and competing in a group collaborative learning environment such as a school has some obvious and unique advantages. Some children really blossom in large groups. These are usually children who are wired for learning and performing in a group. Also, usually, these are also children who will grow up to become successful leaders of other people. These are the children who imbibe the value of competition early in life.

The Origin of Schools as We Know Them Today

Historically, at least in India, the most prevalent method of education was the gurukul system in which a small group of learners learnt from the ‘guru’ at their own pace. This was the original self-paced learning and it worked well. Faster learners moved on to newer things sooner while those who needed more practice had the time to hone their skills before moving to the next thing. The guru ensured that individual attention was available to each ‘shishya’ and was responsible for the successful development of each individual.

Those who could not afford a ‘guru’, which was pretty much everyone else unless you were a prince or unless your name was Eklavya, were educated at home. Parents, essentially, farmers or craftsmen, involved children in everything they did so that the skills and the knowledge that they gained from their parents would get passed on to their children through induction.

With the arrival of the industrial age and the assembly line, however, parents had to go to work in factories and they had to find a way of keeping their children from falling into wells or playing with fire while they were away at work from dawn to dusk. (Children have not been allowed inside factories since time immemorial). The idea of a ‘community school’ was born to take care of precisely this problem. Education, as we expect from it today, was only supposed to be a by-product of such day-care places masquerading as schools.

As the population of the world increased rapidly and with it the competition for its resources, the importance of education became second to none. As the number of children who needed to go to school multiplied frantically, one thing that did not keep pace was the rate at which good teachers were born. As a result, we started packing more and more children into a class. Class sizes went from four or five in the gurukul to fifteen or twenty in the pre-school to 40-60 in high school. How did we decide that a class size of 50 or 60 was right for high school? We could have increased the class sizes further but for the emergence of two new problems on the school horizon:

1. The teacher was unable to cope with the load of evaluating more than that many homework assignments and answer scripts.
2. The teacher could no longer outshout the noise that a larger class made and the teacher ended up with a bad throat trying to be heard!

Individual Attention

But, seriously, how did we decide that a class size of 40 or 50 or 60 is good for a high school class? No one really knows. Fact is that every child, when given individual attention, will almost invariably perform better than otherwise. This is especially true when the child is an introvert and would rather keep quiet in front of a large bunch of children. This situation is a lot worse if he or she fears being laughed at or being ridiculed. Gentler, more sensitive children are often seen as meek by both the teacher as well as the other children in class and tend to get ignored.

Not understanding a concept which will be used again and again for learning other topics can have a disastrous, cascading effect. Before you know it, your child can start dreading Maths or Science, believing that she is inadequate in some way. This starts gnawing away at her confidence and eventually at her self-esteem.

All of us recognize that our child is different and has a unique combination of gifts. Yet, we, the children of the industrial age ourselves, believe that the assembly line production methodology of schools can help our child to realize her potential. All the way up to college, children go though just-in-case education placing pieces of a massive jigsaw exactly where they belong, making sense of what they are taught using what they already know as Lego building blocks. This is the process of wiring their brains in a manner that they can make the best use of it for the rest of their lives. This cannot and should not be a time-bound process. Certainly not time-bound on the basis of the ‘average’ student.

Of Averages and Means

That brings us to the fundamentally incorrect way in which we apply averages and means in education. We believe that as long as the average child can pass a test, the teacher has accomplished his job. Yet, if our own little one does not understand something, do we assume that it is beyond the child’s comprehension? Or do we try to explain the same thing in a hundred different ways till she gets it? So, exactly when does that change? Or, should it change at all? Isn’t the responsibility of the teacher to ensure that each child has learnt? If a child does not get it, is it the teacher or the student who has failed?

And what of the child who is particularly gifted in the subject? How does providing for the average learner help this child? It does not. The brightest learner needs as much individual attention to realize his potential.

Realizing that the success of education lies in measuring whether a child has learnt and not whether the teacher has taught, the American Congress passed an Act in 2002 which made schools more accountable for the learning of each child. The name of the Act, I believe, should serve as the motto of every school education system. The Act is called No Child Left Behind. And all that that would take is the right dose of individual attention. We owe it to our kids.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.

We may presume from notches found in sticks and bones, and from scratches on the walls of ancient caves, that our early ancestors had some inkling of numbers more than 30,000 years ago. Nobody can now claim to know as to when the first stirrings of mathematical thought emerged in human history, but it is not hard to imagine that it must have started with counting. They say that necessity is the mother of invention, and so it must have been with mathematics. The need to know if all the members of the tribe or hunting group have returned home, the need to count one’s precious possessions, and later when animal husbandry became prevalent, the need to tally the number of cattle would presumably have necessitated some form of counting, leading finally to the invention of number names.

Humankind has not looked back since, and now we have the capacity to carry out computations with millions of numbers in less than an infinitesimal part of second, and further progress is continuously being made.

The discovery and development of mathematical concepts is part of the history of many civilizations, most notably, the Egyptians, the Mesopotamians, the Greek, the Chinese and the Indian. India has made a rich contribution to the development of maths; some of the seminal concepts are said to have emerged here and then spread to the rest of the world.

Indus Valley Civilization

Excavations in Harappa and Mohenjodaro have revealed weights of 1/20, 1/10, 1/5,1/2, 1,2,5,10, 20, 50, 100, 200, 500 units of a standard weight of approximately 28 grams, indicating that they had a clear idea of weights and realised the value of precision. These weights have been found in various geometrical shapes including barrel, cylinders, hexahedrons and cones. Their bricks were cast in the proportions of 4:2:1, again indicating that they had some mastery over the dimensions of length. A Mohenjodaro ruler (scale) has been discovered (of length 3.4 centimeters) which has been divided into 10 equal parts, and the bricks used by them have been found to be dimensions which were multiples of this ruler unit.

The Vedas and the Sutras

The Vedas are an enormous compendium of knowledge which have come down to us from times immemorial, predating the start of writing. Through the ‘guru-shishya parampara’, knowledge was passed on from generation to generation verbally, before it was converted into written texts somewhere in the period 500-1000 BC. There are many instances of mathematical understanding which is reflected in different portions of the Vedas and allied texts such as the Samhitas, the Brahmanas and the Vedingas. For instance, very large numbers beyond million and billion are called out in a verse as part of a religious ritual. Astronomy and religious rituals were two areas which called for higher mathematical contributions, and there are treatises now available on how several aspects in these areas were to be treated. For instance, the Sulba Sutras deal with the construction of fire altars, where the challenge was to construct them of different shapes but having the same area. The Vedas provide evidence that the Pythogorean theorem, the value of Pi, and various number series were known to ancient Indians.

The Concept of Zero

The Jain sages are credited with the conception of shunya or zero, as well as that of infinity. They conceived infinity of various types: infinity in one direction, infinity in two directions, infinity in area, infinity in everywhere and infinity in perpetuity.

Some of the important Jain mathematical works include the Surya Prajnapti, the the Vaishali Ganit (c. 3rd century BC). Important Jain mathematicians include Bhadrabahu, Yativrisham Acharya, and Umaswati.

Aryabhata and Bhaskara – the Golden Age

Aryabhata in 499 AD, wrote a treatise on astronomy and mathematics called the Aryabhatiya. The mathematical portion of this text included 33 mathematical ‘sutras’ in the form of verse. Sutras were brief compositions often in verse, conveying some concept or mathematical result. Often they were made so brief as to appear to be in some codified form. Their explanation therefore had to be separate, and it can be assumed that the teachers of yore clarified the meaning of these sutras to their students separately. Another mathematician, Bhaskara, wrote a detailed commentary on the Aryabhatiya, where he went on to provide computations, and verification of the rules encapsulated in the sutras.

Aryabhata and Bhaskara are part of what is considered to be the golden period of Indian mathematics from 200 to 1200 AD. Other notable contributors to this era were Varamihira, Brahmagupta, Mahavira and Bhaskara II. During this period, the works of Indian mathematicians spread to Asia, Middle East and Europe.

The Middle Ages

India’s history is replete with a series of giants in mathematical thought who made great strides in understanding number systems, geometry, trigonometry, algebra and even the rudiments of calculus. Some of the more well known mathematicians are Sridhara of Bengal, Virasena of Karnataka and Madhava of Sangamagrama in Kerala (who founded the Kerala school of mathematics). It is also evident therefore that mathematical thought and learning was widespread throughout India, and that our scholars of the past came from different parts of the country.

Ramanujam

No historical note on mathematics in India can be complete without the inclusion of Srinivasa Ramanujam. He was born in 1887 and died at the early age of 32 in 1920. In his brief lifetime, he compiled over 3900 mathematical results (identities and equations) based on his research and intellectual exploration. He was a self taught mathematician, a mathematical prodigy who developed completely new and original ways to solve complex mathematical problems.

Ramanujam’s intuitive genius added to human knowledge in the fields of analytic number theory, continued fractions, elliptic integrals etc.

Why is history important

As you can see from the above narration, Indians have contributed in great measure to the development of mathematics all over the world. Mathematics is fundamental to modern science and is the cornerstone upon which most of  the modern world with its newest of technologies is built. As we enjoy this life, it helps to remember what our forebears have contributed and how their discoveries have helped shape the life we live today.

* This article has been written by Second School (www.secondschool.in) as a part of its endeavour to spread awareness in the areas of Maths, Curriculum and Standards. Second School is committed to providing school going children with whole brain learning through neighborhood tuition centers.