Bringing up children today is increasingly more complex. The degrees of freedom for children are increasing; with more screens, more options for recreational activities, more opportunities; parents are finding it hard to keep up.
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The new CBSE standards and processes are presently being implemented, and there are many questions that have yet to be answered. This article answers some of the most frequent questions through a set of FAQs for parents with children in middle school (classes VI to X) who would like to know implications of this for their children.
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In a worldwide poll sometime ago, about two-thirds of the people surveyed indicated a dislike for Maths. Their responses ranged from “I do not like it” to “I am scared of it” to “I hate it”. Read the rest of this page »
How do you reconcile this fact with the fact that Maths is all around us in almost anything we do in our daily lives. Almost every waking hour. Are you wondering where?
There has been considerable debate on the different ways of teaching Maths, and the last word on the subject has not been said, nor will be in the near future. One of the hot topics in recent times, is the advocacy of maths teaching based on first understanding how real life situations lead to the need for constructs in maths, before further expanding on that knowledge.
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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.
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As parents we spend enormous amounts of time nurturing our infants, providing answers patiently to their every “why?” or “how?”. They go to preschool one day and are suddenly one in a group of children being looked after by a couple of ‘teachers’. Of course this is usually for a few hours a day and your child is really only having fun and learning the basics of group dynamics. And then one day she is ready for primary school and a larger group of learners. Come high school and your child becomes a speck in a sea of children.
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.
While there does not appear to be a specific age or class when Maths Anxiety occurs, in India it does seem to come to the fore sometime between the 4th and 6th grades. This is when Maths steps out of the purview of arithmetic and elements like integers, geometry and algebra are introduced into the lexicon.
There is sometimes a single event; often a test, where a student who has been used to getting marks in the 90s, encounters an event where suddenly marks drop by 20% or more. In the typical Indian context with a high level of performance expectation, this is traumatic for the child and often for the parents as well. The single event hence is magnified, analysed and discussed to a very high level which ends up lowering the confidence level of a child.
The Negative Cycle
The negative cycle has three elements – Anxiety, Avoidance and Failure. Each of these three feed off the other. An anxious child avoids the subject to the extent possible which gives rise to Failure. A single failure can cause anxiety which can lead to avoidance. A child who missed a few classes, may end up with bad marks and hence lowered confidence. Each of these three elements feed off each other. This is amplified if the child is good at other subjects; Maths can then be avoided with the belief that the child is differently oriented.
The Positive Cycle
The positive cycle is the opposite – its three elements are Confidence, Pursuance and Success. A child who is diligent will end up doing well and hence will build up confidence. A confident child will score better and will be encouraged to pursue the subject as well. The social system will further propagate the belief – a child who is seen to be good at the subject will have proud parents supporting them and encouraging them.
As mentioned before, a child often moves from the positive to the negative cycle. This can happen at any time. A child who finds number systems easy may have a hard time with algebra because the concept of a variable is something that was confusing. This single element can start the entire negative cycle – with the corresponding lack of confidence.
From the Negative to the Positive
It is possible to also move from the Negative to the Positive cycle. The first is to build up confidence through positive reinforcement. Encouragement and self belief is vital along with working on baby steps – focusing on things that the child finds easy to do. It is an important place to start but it can only be effective for a short time; till another difficult math concept lowers confidence again.
Rakesh is a case in point. Till the 5th grade he found Maths easy and was able to score well with no difficulty. His doting parents were convinced he was a genius (as are most parents). In the 6th, he just could not grasp integers – specially applications involving negative numbers. His parents had long conversations with the class teacher and put Rakesh into Maths tuitions. One day Rakesh heard his mother discussing with her brother that she just did not understand what had happened to her brilliant son. Rakesh decided that he had to change this impression and put his heart into his studies. He solved hundreds of problems with his tutor and his grades improved. Not quite to where they were earlier, but certainly much better than his first class test.
He worked even harder for his next test and once again had marks in the 70s. While this seems reasonable, some of his friends continued to score in the high 90s. He began to believe that Maths was not his subject and reconciled himself to having to get though another few years before he could move onto non-Maths subjects.
Rakesh’s problems are not atypical and were a result of rote learning. With sufficient practice it is possible to get decent marks and even do well for a while – but it is always a struggle. Research on Maths Anxiety indicates that there is only one sustainable way to stay in the positive cycle – viz by building a strong understanding of the subject. The opposite – role learning, will certainly result in moving to the negative cycle – usually sooner rather than later.
Building Deep Understanding
There are four elements that contribute to building deep understanding
1. Understanding the ‘why’ of a topic. Maths can seem irrelevant to regular life though it is there and is used by us many times during a day. A discussion around the history of a particular topic along with examples that make sense to the child provide the context for interest
2. Understanding the ‘core’ elements of the topic through activities. The building blocks to Maths are not that many – specially in the early classes. For example understanding the purpose and the concept of a ‘variable’ is at the heart of algebra. Taking time to really ‘get’ the notion of a variable changes algebra from a monster to a friend
3. Regular Practice. Practice in Maths is like making a road in a forest. Driving a car through a forest will not set the foundation for a road – but a few hundred cars following the same path, will. Once a concept is clear, it is retained through discipline and extensive practice
4. Leveraging individual learning styles. We have known for a while that people learn differently, a kinaesthetic learner will understand differently from a visual or aural learner. Knowing and using learning styles helps a child understand a concept more deeply.
In summary then, Maths anxiety is a real issue that exists in society today. Unfortunately for those who have it, Maths is necessary for almost all forms of higher education – specially entrance tests in engineering, science even design.
And the only real solution is Deep Understanding – the sooner, the better.
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.
There is a great deal of interest nowadays in the book, Life of Pi, and a number of people must be awaiting the release of the movie by Ang Lee featuring Suraj Sharma, a student from Delhi. The protagonist of this story, seemingly quite by chance, and in quite an unrelated context, has taken on a name which has a rich history of its own. The concept of Pi has fascinated people through the ages, from unknown Egyptian and Babylonian (and perhaps Indian) architects who must first have encountered its challenge to modern mathematicians and scientists aided by supercomputers who have now gauged its value to a trillion decimal places.
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.
Many children have an easy time with Maths upto Class IV and then start to show signs of strain. Marks start to drop and things are no longer as easy. That is because till Class IV, the only subject taught is arithmetic. Other branches of mathematics start between Class IV and Class VI and include Algebra, Geometry, Mensuration, Number Systems and Probability & Statistics.
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.