Math Adventures at CROCUS

Here is my second post at CROCUS 2012.

Math can be fun. Puzzles, games and stories are some ways in which it can be made. Stories, especially work very well with children. And more so if they are stories of a mathematical cat!

A mathematical cat? What's that? Penrose is a mathematically gifted cat, who belongs to Theoni Pappas, the author of the stories.

The Adventures of Penrose
the mathematical cat        

The Further Adventures of Penrose
the mathematical cat

Fractals, Googols and other
mathematical tales
                         
Written by Theoni Pappas
Published by World Wide Publishing/Tetra
Ages: 8+

At the beginning of 'The Adventures...', she tells us the story of how she realised that the cat was interested in math. So she wrote down all of Penrose's stories for children- these stories tackle one concept at a time, and can be read as stand-alone stories, too.

The stories point out the association of math with the things we encounter in the real world: in nature and in man-made things.  Pappas is that rare mathematician, who breaks down the overwhelmingly complex into easily digestible bites, makes them palatable to all- children and adults alike. The books take into consideration the youngest possible reader, and the harried parent, who would much appreciate these handy stories to delve into the breathtakingly beautiful, yet sometimes alarming world of math.

pic courtesy domusweb.it

One day, Penrose finds his mistress playing with soap bubbles. He runs to play with them, and is amazed with the perfect sphere shapes, that magically change shapes as they coalesce. He discovers, (as did my daughter A, while playing with soap one day instead of washing her hands quickly and coming for dinner) that soap bubbles are always perfect spheres when discrete, as the sphere is the most stable form occurring in nature with the maximum volume for the smallest surface area - No matter what the shape of the bubble blower? Or that when they coalesce to form a cluster, they tessallate to form regular shapes once they stabilise? That if you draw a plane through this cluster, they will look like a cut-section of a honeycomb?

What is tessallation? Figures that fit together perfectly without any space in between. They could be the same figure or different figures. Nature creates many beautiful designs using this concept. Designs that are often copied by man in the arts.Quilts, tile designs, even the hexagons on a football are examples of tessallation.

pic courtesy scienceray.com

One day, Penrose comes across a snail in its shell. He is mesmerised by the shell- it is such a beautiful pattern. In conversation with the snail, Penrose discovers the secret of the golden rectangle- a very common concept that occurs in nature, creating breathtakingly beautiful designs. The ratio of length to breadth always tends to 1.618.

Start with a 1 by 1 square. Then, add another square of the same size. Subsequently, add squares whose sides are equal to the longest side of the existing rectangle. This can go on infinitely in both directions- increasing as well as decreasing. The rectangle that is formed every time is the golden rectangle. The numbers also form an interesting pattern-

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

These can be written as each consecutive number being the sum of the two numbers preceding it.

1, 0+1, 1+1, 1+2, 2+3, 3+5, 5+8, 8+13, 13+21, 21+34,...

This sequence is also called the Fibonacci sequence. How does this tie up with a snails shell? If you draw a curve inside each square in the figure, and join all the curves, they form what is called the golden spiral, that can extend infinitely inwards as well as outwards.

pic courtesy mathworld.wolfram.com
  

pic courtesy google images

More examples occurring in nature? Whorls of a flower, arrangement of seeds in spirals in flowers like the sunflower, branching of a tree, ...you may check this out at howstuffworks.

Art copying nature? Check this out. A very modern use of the golden rectangle- the credit card..

More of what occurs in nature? Fractals. What are those, and where do they come in daily life? Penrose was amazed when he found out how ubiquitous they were, as we are, too when we dig a little deeper. The definition says- duplication of a form inwards or outwards. So his, by definition, will incorporate and go beyond the golden spiral or Fibonacci numbers.

fractals from basic shapes
pic courtesy mathworld.wolfram.com

Ever seen a snowflake grow? The original nidus decides what design it will form- so that no two snowflakes are the same. Or just any crystals grow? Think clouds, galaxies, fern fronds, diffusion and movements of fluids, music, population explosion...the list could go on.

pic courtesy google images

Does Penrose encounter only these slightly unusual math concepts? Not at all. He comes across the very basics that our children might encounter on a daily basis too- numbers, the magic of prime numbers, number line, conventions in math, number processes, pi, irrational and rational numbers, graphs, estimates, probability, two-and-three-dimentional figures, x - the unknown in algebra, the concept of zero...and much more.

Each short story, illustrated with pictures of Penrose and simple figures for the math concept, ends with a puzzle which we are left to solve along with the cat. Pappas, of course, gives a very helpful solutions and explanations section at the end of each book that completely de-mystifies the concept at hand.    

Math Games and Activities from around the World

Here is my first post over at Saffron Tree for CROCUS 2012.

Saffron Tree prides itself on its multicultural identity. CROCUS is our acronym for Celebrating Reading of Culturally Unique Stories. All through CROCUS 2012 we shall have our contributors speak about various aspects about our theme for this year. How about focusing on Math across various cultures and through all of history in this post? Math Fun across time and geographies.

pic courtesy flipkart

Math Games and Activities
from around the World
Written by Claudia Zaslavsky
Published by Chicago Review Press
Ages 9+

Math games that originated in one part of the world went far and wide with travellers and traders, just as much as things like art & craft, language, scientific discoveries and inventions, and much else.

Do you know the story of the wise man from India who used the chessboard and some cool calculation to get rich when the king asked him to choose anything as a prize? He just placed one grain of rice on the first square of the chessboard, and asked the king to double the amount for each consecutive square, and total all the grains on all the squares to give him. Think how much that would be. This story, which can be found with many variations all over Asia and the Middle East, uses the math of exponential sequences, or 'raising a number to the power n' as it can be simply put, can be used as a puzzle.

There are similar stories about computing with numbers all over the world, dating back to ancient times. The story of the merchant's will dividing 7 camels between his 3 sons in the ratio of 1/2 : 1/4 : 1/8?

ancient 15x15 square

Magic Squares? Do you know that playing with numbers on a Magic Square was being done as far back as 4000 years ago. The Chinese knew it then as Lo Shu. From there, it spread to other places like India, Japan, the Middle East, Africa, and then to Europe, Americas and Australia. Today we know it as a great mind-bender game.

More games with numbers? Want to try secret codes using numbers for alphabets? Or the trick about 1089 that never ceases to amaze? I could go on...Of course, there is the number crunching involved in games involving any kind of gamble.

Now lets look at shapes. Can you draw a 5-point star without lifting your hand from the paper? Easy, isn't it? Now can you do that for a 7-point star? Think- it can be done with the help of a simple geometrical shape. Now, try this- can you draw a 6-point star like the Star of David without lifting your hand from the paper? No. Why?

pic courtesy google images

Patterns using stars? These are found all over the world- Indian rangoli and the rich tribal art. Native American art and craft. Quilting and embroidery, both from Europe and all the places colonised by them over history. These also use another geometrical concept- tessalation- repetitive figures fitting together snugly, that is also found in nature. Which can again be found in the art and craft of places as widely far-flung as Nigeria, Japan and Islamic nations.

Dots arranged in a certain manner? Add permutation-combinations, probability, and the possibilities for games are endless. 9 Men's Morris, Tic Tac Toe, Magic Squares, Su-do-ku...these are the ones that are widely known. They were played as far back as 3,300 years ago as found carved on the rooftops excavated in Egypt. Sounds interesting? Look up Shisima from Kenya, Tapatan from the Phillipines, Tsoro Yematatu from Zimbabwe, Picaria of the Native Americas, Nerenchi from Sri Lanka...they are found all over the world, and played with local variations.

pic courtesy indusladies.com

What about other kinds of fun using dot grids? We know rangolis and kolams, but there are similar uses elsewhere in he world. People from Angola, Africa, use the dot grid to draw and tell stories.

The book tells us about these and much more math fun, complete with historical and cultural tid-bits that add to the allure. Each chapter ends with a thought provoking section that has puzzles the reader can think on. The author has written many more Math Fun books, including a sequel- More Math Games and Activities from around the World.

There are so many more Math Fun books and activities out there, to suit each pocket. Do try them.

It's CROCUS time at Saffrontree

Once upon a time, my daughter was scared of Math. It was not that she could not comprehend it. It was just a matter of being overwhelmed. She has always been a words person, content with immersing herself in language and literature- not surprising since we've always been passionate about reading.

Until I decided to take things in hand. For, in my mind, the sheer symmetry and perfection of anything in nature, as well as the conscious or subconscious use of math in the arts- paintings, sculptures, music...had much to do with math. Arts here would include language and literature too- they are inextricably interwoven with math. Look at the beauty of poetry- the rhythm, metre, cadence, patterns- they all have to do with math. If the problem is couched in words, one reads the words, translates them into mathematical terms, does the math, and expresses the answer again in terms of words.

Sciences are close cousins to math. They all require the input of data, the use of logic to work through that data, and come up with a solution. These processes, and the conundrums they present, have been there since the beginning of time. Humans have been aware of them and trying to make sense of them, ever since evolution bestowed on them the power of reasoning. The pre-historic uses of available material for cutting and hunting, for protection as clothing or shelter; the procuring and cultivation of food, the use of fire for cooking, are all uses of math and science for survival.

The human body itself is an amazing feat of engineering- the perfect machine. Unlike in lesser animals, the superior intellect of humans is capable of analysing, reasoning and planning. This is the main defence, as well as offence humans have against the stronger forces of nature. Use of science and math enabled humans to provide for their survival, as well as have the time, energy and resources left over to put down cultural roots. All of culture, in its myriad and endless variety, is thus an offspring of the early use of math and science by humans. The fruits of leisure, after the basic necessities have been met.

Our earliest stories, folktales and mythologies, too, have been an effort on the part of mankind to make sense of the amazing phenomena of nature. Sometimes the explanations thought of are in the realm of the fantastic; occasionally, they have come closer to what modern science has attempted to and managed to prove. For what are the stories of the Dashavatar anything other than an amazing attempt at explaining evolution as we now know of?

Subsequent advances in math and the sciences, especially those in the past 2-3 centuries, have done much to ease the lot of humankind, and improved our quality of life. In the process, though, we have been guilty of killing the goose that lays the golden egg. Indiscriminate exploitation of our natural resources have led to an untenable situation in terms of our ecology, If we are to bequeath a live-able world to future generations, we need to do something proactive about protecting the reserves that are still left. As well as work out eco-friendly methods of doing things that will sustain through many more centuries.

Children, and consequently parents, are often frightened of math and the sciences. To them, it is like an alien language. All one needs to do, however, 'is start at the very beginning', as Maria says in The Sound of Music- 'a very good place to start.' For musical notes too follow the logic of math, and that is what makes them so beautiful. Remember Irene from Enid Blyton's Malory Towers series- the music and math genius? That was a well-crafted character.

As for my daughter- as I said earlier, I decided to take things in hand. Clearly, there was something lacking in her mathematical and scientific experience up until then. The Montessori methods used in her early education had instilled a decided proficiency at the subject, yet she was faltering. As I have said, she was being overwhelmed by the enormity of it. Fortunately, math and science have a logic that is like building blocks of a building. Take hold of the basics, and work upwards, one step at a time. All I needed to do, was demonstrate to her that math and science could be fun, sit and work together with the logic and concepts, and viola...math and science are ogres no longer!   

So this is the theme for CROCUS 2012- MATH & ECO-SCIENCE.

And now, *DRUMROLL* unveiling our fabulous flyer- created by Saffrontree's very own in-house artist, Lavanya Karthik.

  
Come and join us in this celebration. And for all our readers - please share the news of CROCUS - blog about it, or share on Facebook, Twitter, or any other means you can think of. Participate, and have fun.

Cross-posted at Saffrontree .

Review: Saraswati's Way

Pic courtesy monikaschroeder.com

Saraswati's Way
Written by Monika Schroder
Published by Frances Foster Books
Ages 9+ years

"Nothing changes because of our doing. It's all in the hands of the gods." That is what Akash is told all his life. Gifted in Maths, all 12 yr old Akash would like to do is learn more, study and win a scholarship to a school in the city. Life has other things in store for him, though. His father dies, and Akash is sold off to the landlord to work off the family's debt.

Working at the landlord's quarry, breaking stones, Akash soon realises that there is no way out, except to run away. He escapes and lands up in the big city. At the New Delhi railway station, he is befriended by a gang of boys, who make a living by scavenging, stealing and drug-traffikking.

Akash does get tempted by the easy means of making money, to pay for a tutor in Maths, but his faith in Saraswati, the goddess of learning, brings him back on the right path. He is enouraged by Ramesh, the magazine stall owner on the platform who recognises that Akash is different from the other boys who work and live on the railway platform. He does not sit back waiting for the gods to act, but goes about finding his way proactively.

Is Akash's faith in himself vindicated? How does Saraswati show him the way? Does he realise his dream of going to a good school?

Monika Schroeder, the author, is of German origin, and has been a teacher in schools in many developing countries, and was a librarian at the American Embassy School in New Delhi at the time of writing this book. While reading the book, I was struck by the minute details about Indian customs and street life of the vagabond boys that the author seemed to be familiar with. Yet it was clearly an outsider's view. There was a clarity to the descriptions that is usually not found in the writing of someone who has grown up with the culture. It also had the unmistakeable stamp of making things understandable to a non-Indian. Like I found in this book. I was curious to know how she could manage that fine balance.

Searching online, I found a discussion between Uma Krishnaswami and the author. Ms Schroder speaks about her life in India, and the challenges in understanding and writing about a culture so different from the one in which she had grown up. She also speaks about her efforts to learn about the life of street children in India, so vividly potrayed in this book.

As she says in the author's note at the end of the book, "One of the sad aspects of living in India is seeing the effects of poverty on children. A boy like Akash has only a slim chance of fulfilling his dream in contemporary India. Yet I wanted to write a hopeful book about a child who, with determination, courage, and some luck, achieves his goal against all odds."

A Crystal Kite Award winner under Middle East/India/Asia for 2011.

A was fascinated by Akash's proficiency in maths, and we learnt a few tricks from vedic maths that Akash tries to teach the boys that befriend him. A system we will certainly want to explore more fully!

Click here to watch a trailer based on the book. Worth a watch.

Crossposted at Saffrontree.