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.

Just as a check, try and recall how very young children you have known, have learnt to count, which is commonly the first introduction to maths that most children have. There is the learning of number names by rote, which actually is about learning a set of abstractions. At the same time, children are taught to associate counts with numbers which gives these abstract number names a real meaning for the first time. With most children, it is a combination of both, and I am not sure if any one step precedes the other. That brings me the point I wish to make in this essay. It is to say that there is not one way of teaching or learning Maths. There are several ways, and a continuum of applications. The real question is which approach is most appropriate for different situations and concepts.

Keith Devlin, a mathematician who is a recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award among others, used to write a monthly column called Devlin’s Angle. In one of those columns, he discussed the pros and cons of two approaches to teaching maths, one commonly advocated in the US, and the other which had been adopted in Russia. The first approach starts the learning of maths with counting of numbers, much like we do in India. The other example that Devlin quotes is one inspired by a Russian educator, Davydov, which initiates children to measurements and relationships as the first step in maths. In the Russian system, children would first learn by comparing lengths and weights of different objects and so get an idea of dimensions and their relative sizes. Whereas American children started with a set of discrete numbers, the Russian children started with some idea of a continuous measurement scale. The Russians believed that this approach better prepared students for learning later concepts such as rational numbers and algebra.
The other major difference between the two approaches lie in their emphasis on learning maths from daily life examples. According to Devlin, the accepted wisdom in the US is that the spontaneous approach (learn to abstract from real life experiences) is the way to go at least all the way through K-8, and maybe all the way up to grade 12. However, this approach breaks down when it comes to subjects like calculus, when it becomes more expedient to adopt a rule based approach to teaching the topic.

On the other hand, the Davidov curriculum adopts the scientific-concepts approach (learn-it-by-the-rules approach) from day 1. Davydov believed that learning mathematics using a general-to-specific, “scientific” approach leads to better mathematical understanding and performance in the long run than does the spontaneous approach. His reasoning was that if very young children begin their mathematics learning with abstractions, they will be better prepared to use formal abstractions in later school years, and their thinking will develop in a way that can support the capacity to handle more complex mathematics.

However, Devlin hastens to add that the Davydov approach is grounded firmly in real-world experience, and lots of it. Indeed, students spend more time at the start doing nothing but real-world activities (before doing any explicit mathematics) than is the case in the US curriculum. But when the actual mathematical concepts are introduced, it is in a scientific fashion. The students have already been enabled to link the scientific concept to their real world experiences.
The fact is that both approaches have produced excellent results, and the benefit for us in India, is that we can refer to these different schools of thought and choose our own. The NCERT published a report in 2006 called the ‘Position Paper of the Focus Group on the Teaching of Mathematics’ which provides an incisive commentary on the state of maths teaching in India, and ideas on how we can improve. They studied the curriculum and methods adopted in the US, the UK, France, Hungary, Australia, New Zealand and several other countries across the world, and came up with a set of recommendations which is briefly summarised below.

The focus group saw that maths education in our schools suffers from various problems, viz. the sense of fear and failure regarding maths among a majority of students, a curriculum that disappoints the talented minority as well as the non-participating majority at the same time, crude methods of assessment that perceive maths as mechanical computation, and lack of teacher preparation and support.

The focus group recommended that the teaching of maths must result in the ‘mathematisation’ of the children’s thinking rather than packing children’s brains with mathematical content, which is the norm in most of our school systems. By ‘mathematisation’ the focus group meant the need to develop in children, clarity of thought and logic, ability to handle abstractions, and ability to understand and solve problems. They envisioned the teaching of maths in an environment in which children enjoy the process and feel engaged, see maths as part of their daily life, learn to use abstractions to perceive relationships and structure, and learn the value of maths through the solving of common and meaningful problems.
Much of the thinking of our policy makers on the approach to maths education is on the right lines. What we lag behind in, is implementation. If we can give effect to all the recommendations that have been spelt out in several policy documents, there will be enormous benefit to the school system, as well as to the entire community.

The seminal premises around which the Position Paper is built are the two statement that all students can learn mathematics and that all students need to learn mathematics. Whether all children have the capability to learn maths is often one of the questions that pains parents and teachers who find that some children just do not seem to get it. The conclusive answer to that question has been given by Brian Butterworth in his well researched book: ‘What Counts: How Every Brain is Hardwired for Maths’. Dr. Butterworth, professor of cognitive neuropsychology at University College, London  draws upon a range of information from anthropology, psychology, neurophysiology, linguistics, psychometrics, and other ?elds to make the case that all of us are hardwired for mathematics. He very conclusively argues that barring serious disabilities, it is ‘zeal’ and ‘hard work’ that make all the difference.

According to Dr. Butterworth, the differences between mathematical ability of different individuals are due to acquiring the conceptual tools provided by our culture. That is a function of training. And practice. To become good at numbers, he avers that you must have plenty of practice in them.

Dr. Butterworth says that practice directed to understanding the basic principles of numbers, is critical, and lack of it is the reason some people get left behind by the educational system, and therefore end up both incompetent and anxious about their incompetence. Others manage to keep pace, grow their self confidence in the process, are motivated to work harder, and by virtue of this linked cycle of motivation and hard work, move decisively ahead.

In summary, some of the key points I would like to make are:

• Learning of maths building understanding on the basis of its relevance to real life situations and its practical applications is fundamental to deep understanding. If this approach is designed into the lesson plans especially in the earlier years, the capacity of students to understand complex concepts in practical terms will become ingrained.
• Knowhow of alternative ways of teaching maths are in the public domain. For educators, it is always useful to know what they are.
• The NCERT and the CBSE among other policy making bodies in India, have come out with approaches and curriculum which are scientifically drawn up. The challenge is to execute to the plan, and to create the resources which will provide the opportunities and experiences to students as has been envisioned.
• At the individual student level, the fact is that there is no substance in the oft held misconception that some people are inherently challenged in maths. Barring serious learning disability, it all boils down to motivated learning, and plenty of practice.

References

• Devlin’s Angle – a column by Keith Devlin, January 2009, on “Should Children Learn Math by Starting with Counting?’ in the Mathematical Association of America website
• Position Paper – National Focus Group on the Teaching of Mathematics, NCERT, New Delhi, published in 2006
  What Counts: How Every Brain is Hardwired for Maths by Dr. Brian Butterworth, University College, London, published by Free Press, 1999

* 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.

Cooking, Shopping

When you cook using a recipe book, you are putting ratios and proportion to work. Do you remember when the recipe for Paneer Butter Masala for 6 people said that you need 1 kg of Paneer, 200g of onions and 500g of tomatoes. The cooking time was 20 minutes. You were actually cooking for nine friends, so you figured out how to change the quantities of the ingredients. I recall the time I did a similar thing but also adjusted the time for cooking. But that’s a mistake I made because I knew less about cooking than about mathematics!

When you rush to that end-of-season discount sale and are busy trying to figure out which is the best deal from the various signs in each section, you are using a lot of the stuff you studied in high school. Signs like these are very familiar:

Buy 3 Get 3 FREE!!
Or
Buy ONE and get 50% off on the Second.

Exactly which deal is better? Did it take you long to figure out that the second offer is really a “25% off’ on your purchase?

How do you decide which petrol pump to buy your fuel from and which ones to avoid? What drives the decision of whether you should buy the large bottle of shampoo or two small ones? The large jar of your child’s favourite milk nutrient or two small refill packs?

Sports

You follow sport and are a keen Cricket enthusiast and get into heated exchanges with friends about whether Ricky Ponting can even be compared to Sachin. It usually boils down to statistics and the argument is usually won by the one who seems to know his numbers. Of course not all of us can be Duckworth or Lewis!

Have you ever wondered why each win in a football match counts for 3 points for the winner but in a draw 2 points are shared between the opponents? It wasn’t so till not so long ago. Winners used to be awarded 2 points. Why did it change? FIFA realized that enough teams were playing a different, defensive style of football because they knew they could progress in a tournament by just drawing matches. The sport was becoming boring. Imagine, basic Maths helped make the game more interesting.

Music, Photography

Music and photography and inextricably linked with Maths. Why do certain musical notes played together sound pleasant and certain others make the ear ache? I will not get into details here (there is a lot of stuff available on the internet which explains it better than I can), but the reason lies in the ratios of the frequencies of the notes played.
Most of us use the computer to touch up or improve the quality of the digital pictures we take, did you know that the algorithms are simply mathematical. That is how your favourite ageing hero still looks like he has no wrinkles on his face!

Investing for your Child’s future

Whether you are buying property or doing a systematic monthly investment you worry about the returns on your investment. There are so many differences between all the investment opportunities that you need to be able to figure out what is best for you.

If you have ever bought an apartment you would have found out the meaning of the terms carpet area, built-up area and super built-up area. These are mathematical constructs too. Why did you bother? All of us seem to be able to understand enough Maths when it is a question of making that vital decision on which our life’s savings depend. Should you buy at Rs 2500 per sq ft (super built-up area) or Rs 3000 per sq ft (carpet area)? Does this question give you enough information to arrive at the right answer? This is what the practicality of the application of Maths does to us at that moment when we need it.

The buying and selling of property requires knowledge of not just ratios and proportion but also of mensuration, profit and loss, decimals, percentages and fractions!

All Around us

How does the weatherman forecast the weather over the next few days? How did the ancient architect make their building so aesthetically appealing? The answer might lie in a mathematical concept called the “Golden Ratio”. What do certain flowers, honeycombs, pine cones and the pineapple have in common? The Fibonacci number.
One can go on and on. I cannot look around at people, things, places, tasks, careers, hobbies and pastimes without noticing what role Mathematics plays in it. Statistics, Algebra, Geomtery, Trigonometry and Arithmetic are lovely tools we all use, often without realising. Mathematics is indeed beautiful, and it is all around us. We can choose to ignore it and miss it or recognize it and enjoy it.

Answer to the Question

The answer to the question (at the beginning of this article) about why so many people believe they dislike Maths, probably lies in the way they are taught the subject in school. Someone needs to correct it. Someone will.

* 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.

Have the 10th board exams been scrapped? Is there a time limit for these?
The 10th board exams have not been scrapped, but have been made optional. There are three levels at which the option may be exercised

• At the state level. For example Bihar has retained the board exam at the 10th level. They want to retain the exam because Class X results may be linked to scholarships and jobs.
• At the school level. Schools may choose to adopt the CBSE guideline or may not. This is however likely to be a choice that the school can exercise for a year or two
• At the individual level. Individual students (specially those seeking to change their school) may want to take the board exam even if it is not compulsory.

There is presently no information available about whether the 10th paper will be available for years into the future.

If they have the option, who should consider doing the 10th board exam?
It is our opinion that once the paper is optional, it will decrease in importance and seriousness progressively. We believe that in 5 years, few students will opt for the exam.

The need to take the exam will exist for students who will not be continuing with their 11th and 12th in the same school, since it may facilitate admission into another education institution.

Another view is that the 10th exam can serve as a practice for the 12th boards, so that children can get used to the pressure.

What is the new grading system? What is it meant to achieve?

The new grading system will provide a grade for a range of marks. Specifically,

The purpose of moving to a grading system is to reduce stress caused by the current system, where a single question missed can make a large difference in ranking.

The guidelines for the use of grades removes promotion anxiety to a large extent. A student with an E1 rating in a subject will need to re-appear for the exam but can be promoted to Class XI. Only a person with an E2 rating will have to repeat the year. This effectively removes most of the pressure of promotion to the XIth standard.

The practice of compartments and fail stand discontinued. The result of candidates is now declared in two categories: Eligible for qualifying certificate (QUAL) and Eligible for improvement of performance (EIOP). EIOP is awarded to candidates who have either E1 or E2 in their marksheets; or receive less than 33% marks in their external exams (the summative evaluations)

Handling of EIOP candidates will be different in classes IX and X.

In Class IX, students getting E1 or E2 in one or more subjects will be allowed one attempt to improve their grade through reappearing in the final summative exam. If they do not clear, they will not be eligible for admission to Class X. This does not apply to the ‘additional subject’ where they can progress with E1 or E2. The improvement exam will be set by the school following the guidelines earlier provided for the Summative Evaluation. It is recommended that this additional exam be given within one month of declaration of the results by the schools

In Class X, students getting E1 or E2 will get 5 attempts to clear each subject, and will hence be allowed to continue into Class XI without clearing every subject. It has not been decided yet, whether CBSE will provide the papers for subsequent attempts, or whether the school can set their own papers.

What is the CGPA? How is it calculated? How will it be used?

CGPA stands for Cumulative Grade Point Average, and is the approach used in many countries across the world and in a number of institutions of higher learning in India as well including the IITs and IIMs.

As the term indicates it the average grade points of all the subjects to be considered. To arrive at it, simply average the Grade Points (from the table above) for each grade received. For example a student with 3 A1 grades, 2 A2 grades and 1 C1 grade can compute their CGPA as

CGPA = ((3 x 10) + (2 x 9) + (1 x 6)) / 6 = 54/6 = 9.0

The CBSE has also provided a guideline to calculate a percentage by multiplying the CGPA by 9.5. This number was arrived at by analysing past student marks. Hence the percentage for a person with a CGPA of 9.0 is 85.5.

Where used in admission into college, the appropriate subjects will be used to calculate the CGPA. This is the same process as is followed today.

What is CCE?

CCE stands for Continuous and Comprehensive Evaluation. CCE is presently being mandated for the 9th and 10th standards.

The words in CCE have been carefully selected. ‘Continuous’ means that the evaluation should not depend solely on a single exam, but should involve ongoing processes where the teacher engages with each student and provides individualized support. ‘Comprehensive’ is meant to cover both scholastic and co-scholastic areas.

Scholastic Evaluation

The year will be divided into two terms. Each term will contain both formative and summative evaluation for each subject.
As the terms suggest, formative evaluation measures the continuing development of the child through projects and other developmental assessment. Summative evaluation measures whether or not learning has occurred. Summative evaluation is done through written tests.

Summative Assessment =20%

Formative Evaluation is meant to be both diagnostic and remedial. The actual implementation is left to the school. CBSE provides guidelines that recommend that Formative Evaluation should not only be about tests, but should include projects, quizzes, oral tests, class discussion, visual testing, practicals and assignments. The school can decide however, how exactly they would like to implement this guideline.

Summative Evaluation will be conducted through formal tests, one in each term. The specific chapters to be completed in each term are given by CBSE. The question forms are provided by the CBSE to ensure that there is consistency in implementation. CBSE will also periodically audit the process and sample some question papers to ensure continued consistency in evaluation.

Co-scholastic Evaluation

The idea of introducing co-scholastic evaluation was to ensure that schools provide a holistic environment for all-round development of children. Co-scholastic achievement in each area will be given as three or five grades (as compared to 9 for scholastic).

There will be comments provided to support the grades. There are no mark equivalents for co-scholastic grades.

What could be the positive and negative aspects of CCE and the new grading system?

In July 2004 the Executive Committee of the National Council for Educational Research and Training (NCERT) decided to revise the National Curriculum Framework. They came up with five guiding principles:

• To connect knowledge to life outside school;
• To ensure that learning shifts away from rote learning;
• To enrich the curriculum so that it goes beyond textbooks;
• To make examinations more flexible and integrate them with classroom life;
• To nurture an overriding identity informed by caring concerns within the democratic polity of the country.

The new CCE and grading system conforms to these beliefs.
Hence the positives are:

• Holistic development of children
• Lower stress levels for children in the 9th and 10th
• Value development and performance beyond academic excellence
• Provide qualitative and quantitative inputs to parents
• Require teachers to play a more developmental role in child development
• Provide avenues for development of disadvantaged children

The negatives are largely (though not completely) in effective implementation

• Additional pressure on teachers who may choose shortcuts
• Lack of standardization. Grades depend upon implementation by schools. Even when papers are provided, schools will correct. It may not be possible to ensure standardization through audits alone
• Competence and Desire of faculty. The present schooling system does not have a uniformly high standard. The new system is highly dependent on competent and motivated faculty who are prepared to spend the additional time and effort in these areas.
• Extent of flexibility. There are only guidelines for many areas. As each school adapts to these broad guidelines, there will be pressure from many sources to manoeuvre these in a way to suit a certain section of people.
• Exceptionally bright children looking forward to the challenge of exams may not be as motivated
• The 10th exam was in a sense a precursor to the 12th board exams and provided children with a forum to get used to this method of studying. That will now be removed

What will happen in classes prior to the 10th? When will new guidelines come into effect?

The new guidelines came into effect from the middle of 2009. In this academic year the guidelines will be fully in effect for Classes IX and X.

Given the ambitious nature of the project, there are implementation challenges that will need to be worked through. For example, since the dates of the summative exams are left to the schools, even with multiple exam forms, children from schools having their exam later, can benefit from the papers that have been received by children doing their tests earlier. That was already a problem in the first IXth exam.

It is also CBSEs intention to move the guidelines into lower classes progressively, so that CCE is the approach to be followed consistently. Implementation will take longer to work through and expected dates have yet to be released.

Are the kinds of questions being asked changing? What are HOTS?

CBSE has stated that 20% of questions that will be asked will require the application of ‘Higher Order Thinking Skills’ (HOTS).

In 1956, Benjamin Bloom researched the method of evaluation at that time and found that 95% of questions simply tested the ability of students to recall information, which he defined as the lowest order of thinking skills. Bloom defined a taxonomy with 6 levels of cognitive ability. Since that time, levels 4,5 and 6 are often combined into one level.

Through HOTS, CBSE plans to introduce the ‘application’ layer of Blooms Taxonomy; which is the third layer and hence a higher order thinking skill than the base levels of Knowledge and Comprehension.

There are already questions in present CBSE papers that meet a Bloom Level 3 requirement. The idea however, is to continually increase this percentage and move away from role learning as far as possible.

What will schools now do differently?

At this time, most school teachers have been going through extensive training on the new system. Some schools have already been implementing comprehensive feedback for students, but for most, the workload for teachers will increase substantially.

Teachers will now have increased evaluation loads and will also have to engage students in co-scholastic activities. The evaluation of these activities will require a deep knowledge of each student which is difficult today given the workload in many schools.

At this time, the guidelines provided are open-ended and provide a fair amount of flexibility to each school and teacher. For example, the elements of formative evaluation can vary a great deal. Co-scholastic activities can range from teachers evaluating based on class participation and behaviour, to carefully structured sessions to elicit responses from children and encouraging them to develop to their full potential.

The exam system also has not settled in as yet. It is our expectation, that some schools will continue with live as usual with minimal changes. Others will view this as a transformational opportunity.

It is also our belief that the guidelines will become tighter with time, as CBSE starts to get feedback from initial implementations.

How should I prepare my child better for these new guidelines?

The new CBSE guidelines are far more in line with international standards. Success in life is less a function of academic performance than all round development.

This is a big change in mindset for all of us as parents, since the only measure of success during the school and college years seems to be academic performance. The level of competition between students will reduce over time, since it will no longer matter whether a student gets 99 or 91, and the difference between good students will start to involve co-scholastic measures and extra-curricular activities. This has already become evident in cases where students are going outside the country for studies.

The best thing for a parent to do is to review the complete report card that will now be available and understand all elements of the childs capabilities. Also, require the school to also provide this information, since it is now required.

* 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 centers.