Geetha Iyer and Sundaram S

This is the centenary year of Charles Darwin, whose classical theory of the Origin of Life is still hotly debated. But to understand or debate the statement “Species originated by means of natural selection, or through the preservation of the favoured races in the struggle for life” in all its entirety, mere English language will not suffice. We would have to take the assistance of the language of mathematics. “Mathematics?” “Yes, mathematics”. It is this conviction that forms the body of this article.

Biology and mathematics share a complementary relationship which is not easily visible. In school, these two subjects are seen as mutually incompatible, categorised at the two extremes of the learning spectrum. However, mathematics originated from our biology and several aspects of biology are better understood if seen from a mathematical perspective. The scope of this integration is far too vast to be covered in a short article and hence we confine to highlighting this relationship through select examples.

Dr Geetha Iyer, is a consultant with several schools for curriculum design as well as Science & Environment education. She was previously a teacher at the Rishi Valley School and then Head of Sahyadri School (KFI) near Pune. She can be reached at scopsowl@gmail.com.

S Sundaram is the Principal of K D Ambani Vidyamandir in Jamnagar. He started his career in education with the Rishi Valley School. He is deeply interested in enabling students to learn Mathematics and in education for the 21st century. He can be reached at sundaram48@yahoo.com.

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The idea of using palm leaves as teaching material was a very new concept for teachers of Titan School. Although the idea was amusing initially we now have explored this material immensely in our classes in many ways.

Every year we have a Math Mela in our school and it was for this that the palm leaf kits that K-4 produces were first used and became an instant hit. These kits are extremely useful especially for primary school. The pre-primary students used the Design and Form kit to explore pattern making in multiple ways. A group of them working with shapes managed to create many interesting compositions. It was thrilling to see how these little kids seemed to be learning about different shapes without us having to intervene. Learning by exploring is the best way of learning. Whatever shape they came up with we helped turn them into beautiful mobiles that could be hung. The shells in the kit were used to play guessing games, for counting, pairing, making sets, etc.

The Number Boards have aided in the teaching of addition and subtraction. The shell box helps teach multiplication and division. The higher classes can use these same shells to learn factors, multiples and fractions. The materials being made of eco-friendly palm leaves have become a big hit with the pre-primary children as they use them in the learning of shapes and stringing. The Palm Mala with a variety of colours -threading them and making patterns has been an interesting learning material for kindergarten children. It was also used for counting and to learn other related skills. The stitching kits are helpful in learning to create new patterns.

The material being light, easy to handle and colourful makes it child-friendly. The palm kit has been such a wonderful addition to our teaching materials that we invited the craftswomen from K-4 to conduct a workshop in palm leaf weaving since weaving, knotting and patterns are all related closely to the concept of numbers and understanding them in other non verbal ways. Some of our senior students learnt the basic skills in weaving a palm leaf. We also used three dimensional palm leaf shapes like stars for decorating our school for the Math fest.

To know more about K-4 and their products visit www.k4arts.org.

The reviewer is Coordinator, Titan School, Hosur.

I am not a teacher of mathematics but have taught enough cousins in my school days the basics of percentages, time and distance, time and work, stocks and shares, etc. But what I am going to write about now is not about that kind of teaching, where concepts may or may not get cleared. Instead I am going to write about mathematics in everyday life; of the mathematics that we see on the streets yet do not bring into our classrooms, of the sense of numbers and space that people have all around us.I am going to write about two people who were a part of my life more than 35 years ago.

The first is an ‘illiterate’ milk man, by the very common name of Rangan; I have no clue whether he is still alive; chances are that he is no more. He was an inveterate drunkard and often, had to be ‘woken’ up from his stupor in the middle of the afternoon to relieve the ‘udder-full – cows’ of their milk. But this drunkenness notwithstanding, Rangan was amazing. He had never been to school, and did not know how to hold a pencil in his hand. Yet, that hand could cajole the most recalcitrant cow to yield milk. But this was not all! The hand that could milk cows but could not ‘count’ never ever made a mistake with calculations regarding the amount of milk that had to be supplied to people in about 10 houses. A piece of coal was all he used in those days to mark the milk he had given; a wall was his account book! The lady of the house would have to open her diary, add up all the halves, quarters, one and two litres to arrive at the total amount of milk that she had bought. But such ‘counting’ was not for Rangan; his eye would at one glance take in the long and short and not-so -long lines, the different strokes that only he could decipher, and out would come the total! And not once did he make a mistake; 15 minutes after he had ‘pronounced’ the amount, the ‘amma’ of the house would give him the figure and there was never a discrepancy!

But Rangan had never been to school, did not know how to read or write, was branded illiterate and so, ‘could not have done’ any mathematics!

The second person is a lovely old lady who is no more, by the grand old name of ‘Pattu’. She too, did not know how to read or write and had in her own words, ‘never stepped across the threshold of a school!’ Yet, if you gave Pattu a canvas, (the front court yard of your house) you were in for a visual feast! First, she would ‘look’ at the spaces she had, and in the blink of an eye, the estimate would come “amma, grind two azhaakkus (that is approximately half a kilo) of rice!” This would of course happen one day before the festival. This ground rice, on festive occasions, and rice powder on ordinary days was her ‘medium’.

And then, the day of the kolam creations would dawn; Pattu would turn up either early in the morning, if it was a powder kolam, and the afternoon of the day before the festival if it was a ground rice kolam and then her work would begin. It was a fantastic treat to watch that lady at work; not schooled, never used a protractor or compass or scale, but her kolams always had perspective! Small, medium, or large spaces; narrow, broad, rectangular, square, odd shaped, none of this mattered! The eye would ‘measure’ the space and in accordance with availability, the spaces between the dots would increase or decrease; alternatively, ‘bigger’ and more appropriate kolams would get chosen!

This choice, of course, never came out of a book, but from inside her head! A chariot, a peacock, kolams that had dots and lines and kolams that had only intersecting lines; you name it and she had the designs in her head. The lady is no more, but yet, when I reflect, I can only look and wonder, at how that small ‘unlettered’, head could carry all that information; how that eye could judge space and size and do the needful.

Today, if we want to reduce or enlarge a diagram, we go to the photocopier, and painfully select the percentage, check and by trial and error, get it right! Pattu seemed to have an innate reducer/enlarger in her head! The eye glanced and the hand did! I have spent hours watching the speed and grace with which she worked her magic on courtyards and floors to create these amazing pieces of art; art that got washed out and wiped clean and not preserved, but dexterity, imagination, skill, and talent it definitely exhibited! And what was most important, was that her innate sense of geometry never let her down! She had a steady hand and an amazing eye! Never used a ruler, yet drew straight lines; did not know how to hold and use a compass but her points made perfect arcs! Her circles were picture perfect and her triangles and rectangles and rhombuses would have put any student of geometry to shame! A13 by 8 series of dots with interspacing dots in between, or a 9 by 9 square; … you name it and she could do it, and then, using those coordinates, weave amazing designs.

Yet, being ‘unlettered’ no school would have let her step in and would never have given her a certificate!

I am reminded of the grouch that the students of the Barbiana School have: …that they failed in their gymnastics examination because they could not play basketball. But any one of the children in that school (an unconventional one run by an Italian priest for school drop outs) could climb an oak tee, let go with their hands, and chop off a two hundred pound branch with a hatchet and then drag it through the snow to their mother’s door step! But these children will never get an ‘A’ in gymnastics just as Pattu and Rangan will never get an ‘A’ in mathematics.

The author is Reader, Department of evaluation, EFL Universal, Hyderabad. She can be reached at gdurairajan@gmail.com.

Traditionally mathematics is taught using a problem solving approach. Text books are filled with plenty of math exercises. Students are kept busy completing homework. Most students learn by the rote method and apply mathematical formulae in solving problems without any understanding. The rote learning method is mechanical and less inspiring. This leads to disinterest in the subject as children do not realise its incredible hidden beauty. As teachers of maths we have to devise ways to foster creativity and create interest in the subject amongst all our students. I constantly develop new methods to teach the subject and I’d like to share one such method here – that of paper folding to teach abstract math concepts. This technique can significantly benefit a teacher in the effective engagement of students and in developing their cognitive skills. The technique allows students to actually visualise the results and proofs of various concepts.

Paper folding strategy for teaching math
Try this simple activity using paper to explore a simple math idea and you will realise the many possibilities that exsist. When we fold a piece of paper, a crease is formed. The crease so obtained represents a line. When we unfold it, on that crease we can get a line segment of desired length. To get the mid point of the line segment students may perform the steps given on the facing page.

The author is a P.G.T. in Mathematics and teaches at the Kulachi Hansraj Model School, New Delhi. She can be reached at mathclass_khms@yahoo.co.in.

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The term ‘fraction’ in its literal sense means a part of the whole. That is how we’ve always learnt fractions in school and also as teachers have transferred the concept to children. At the centre for children where I work, we thought of several ways of making it easier for children to understand the idea of fractions. We procured and developed teaching-learning aids where we could demonstrate what was meant by ¼, ½, ¾, etc. There was an interesting variety of aids put together. However I was not very sure of the approach. While there was merit in what I was doing in class, there were some areas where I felt I was digressing from the principle of learning through practical experience and was going into an area of algorithmic expectations. Therefore, when we were approached by a team from Eklavya and the Homi Bhabha Science Centre to try out the model of conveying fractions with our group of children through the ‘sharing’ model, I was keen to do this in my class.

Difficulties with using only the conventional measure model in class
The measure model, by its definition, implies that a number is supposed to reflect a quantity; it is a measure of a specific quantity. It is not so challenging to teach unit fractions, as one-third, one-half, one-fourth because it can be introduced to children as a number which is less than a whole number and therefore answers the need to find a way of writing something that is less than one.

The author works for Muskaan, an NGO working with slum communities in Bhopal. A film has been made on the classroom processes described in this article. A copy could be made available on sending a request to muskaan_smiles@hotmail.com.

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Maths has evolved from Nature and therefore the connection between the two is but natural. Just as Nature is all around us so is the subject of mathematics. We can find maths in our culture, festivals and in day-to-day life. If we help students see this connection then the subject will acquire a new meaning for them. From a dreaded subject it could become one of their favourites. This article is an attempt to help math teachers develop activities related to math drawn from nature and life.

Numbers in day-to-day life
We may not notice them but numbers are everywhere. They are a part of our culture, environment and day-to-day life. Why not ask children to write down all the words they know that have a number in them? For instance, the number 5 appears in words like pentagon, panchatantra and panch pandav, the number 9 lends itself to words like navratri and navratna. Dasanana and dasavataar have the number 10 in them. Such an exercise is a fun way to not only teach children numbers but also to show that numbers are an integral part of our lives.

The author is Head, Training & Development, Calorx Foundation and Head, The Calorx School, Ahmedabad. He can be reached at sgupta@calorxmail.com.

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Mathematics is both an elegant tool and an exceedingly useful language. People often tend to equate mathematical ability with intelligence as they see it used brilliantly by some and poorly by others. Do we really believe that some people are born with a mathematical mind? Actually, the difference between high and low achievers lies not in the brain they were born with but in how they train and learn to use it.

Arithmetic is a powerful tool for day–to-day use. It enables us to handle our lives with greater efficiency. Beyond students and teachers, bank and billing clerks, accountants, business people and even housewives are constantly using numbers. The ability to increase speed and arithmetical agility definitely demands practice in figuring out the right way.

The author can be reached at saroja@gmail.com.

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The more serious issue of school maths is what is taught rather than how it is taught. In this article, I have picked up an example from fractions to illustrate the limitations of algorithmic learning and possible ways to overcome it.

Let us look at the idea of division of fractions and it’s implications to the process of teaching & learning mathematics.

Some teachers would be tempted to say “It is one of the easiest things to do and does not warrant a discussion”. To divide two fractions say 4/7 by 2/3 follow these steps.

The author works with Suvidya, an educational resource group, in Bangalore. He can be reached at sngananath@gmail.com.

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Several hundred years ago mathematicians realised that there were a lot of problems they could solve if they had a way to talk about numbers less than zero. So they decided to write 0-1=-1, which is read as “negative one” and means one less than zero. It’s easy to say what 12-11 is, so shouldn’t 11-12 have an answer too?

The first set of rules for dealing with negative numbers was stated in the 7^th century by the Indian mathematician Brahmagupta, (598 – 670). He used the ideas of ‘fortunes’ and ‘debts’ for positive and negative.

The author is an educational consultant with Sparsh, a division of SEED infotech at Pune. She can be reached at shakun.jaisinghani@gmail.com .

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M – Mental
A – Ability
T – To
H – Handle Situations

Vocabulary enrichment is essential in the development of mathematical concepts in children. Most often we miss out on this, leading to a sense that math is difficult and complicated, and as a result children end up disliking the subject and developing a mental block towards it.

The teaching of math should happen with fun and laughter in class. The stereotypical math teacher is very serious and the subject is presented in a very strict, matter of fact manner, which robs it of the human element from it.

The following are some questions that as math teachers we often hear ourselves asking:
“Why can’t such a simple thing as this be understood by these children?”
“How can they misunderstand a basic fundamental thing as this?”
“I have taught this so systematically, so they have to understand it or they should have understood it, isn’t it?”

But can understanding be ensured? What we can do is to facilitate the child’s understanding by providing a variety of inputs.

Let me share with you my experience in class V.

Topic – perimeter, area and volume
The concepts were taught practically, i.e., children measured various things to determine these properties and learnt the formulae, then they also successfully completed exercises that suitably applied the various formulae.

The children also independently concluded that the perimeter is the measure of one dimension, i.e., linear; area is the measure of two dimensions; and volume is the measure of three dimensions.

Try this activity in your class.
Objects required: match boxes
Quantity: 6
What to do: Use three match boxes and form a cuboid by placing the match boxes vertically. Use another three match boxes to form a cuboid by placing the boxes horizontally. Ask the children to find the perimeter, area and the volume.

When I gave this activity to my class, the children measured the two cuboids and compared the measurements. They realised that although the number of boxes used to make the cuboids were the same, the measurements differed because of the way the match boxes were placed. This led to a discussion as to why there was this difference? They then arrived at the fact that horizontal and vertical placements altered the length, breadth, and height hence the measurements differed.

When they measured and calculated the perimeter, area and volume, some of them made mistakes in practically applying the formula although they could do it when given as a direct sum.

Here is when the question “I have taught this so systematically, so they have to understand it or they should have understood it, isn’t it?” surfaced and hit me hard.

They could calculate the measurements perfectly when given a sum and asked to do it on paper, but in this situation although they knew the formula, some of them were unable to apply it to the given situation.

Somewhere in the process of teaching there is a gap between theory and practice. We as math teachers should strive to bridge this gap to enable better understanding and application. This will foster confidence and logical thinking in children as that is what the subject represents.

We have to build a rich vocabulary in children from their formative years of schooling. This will strengthen their foundation of mathematics and will help them as they reach higher classes. They need to understand how to express mathematical ideas in a variety of ways that reflects the way numbers are used in daily life as well as in learning.

Enrichment of vocabulary is vital even when basic concepts are taught.
Take a simple example. 2 + 3 = 5. This should not end with a few more examples of other numerals.

The same concept can be taught with a variety of instructions such as

1. How much is this? (show them 3 objects)
   How much is this? (show them 2 objects)
   If you put together these, how many will it make?
2. If you take 3 objects and 2 more again, then how many objects will you have now?
3. I have 3 objects and if Raj gives me 2 more, then how many will they make?
4. We can use a variety of verbal examples with children within the class itself and organise an activity with other numbers to illustrate addition and subtraction.

We can use a variety of expressions like put together, count how many; how many do we have altogether; how many more do we need to make it a particular number; what is the result of taking away this number from the other; what kind of number will you get if you take away this number from that; will it be an odd number or an even number; two digit or a single digit number? These questions will get children to think, which is essentially what math is. Can we take away 12 from 5? If no what is the reason?

Standards III, IV and V are the formative stages where children are slowly switching from the concrete to the abstract stage and these strategies will ensure concrete understanding of concepts. Mathematical concepts can be successfully built with proper usage of vocabulary. Math and language are like fraternal twins. They appear non-related but they are interlinked. If math is the ladder on which we have to assist children to climb, then the rungs of this ladder are made up of words.

When verbal variety is used from the primary level, then the higher abstract concepts such as algebra will become easier for them to handle.

The author is with the AV Education Society in Bangalore. She can be reached at pitchin5@yahoo.com.