Friday, July 6, 2012

Vedic maths. Facts + fiction = fantastic

It has been said that truth is stranger than fiction.  That is certainly true when it comes to the story of the origins of “Vedic” maths.  

Vedic mathematics first came to light in a book published in 1965 by the impressively named Bharati Krishna Tirthaji - an Indian mathematician ...and mystic. Tirathji claimed to have rediscovered Vedic mathematics after meditating on some neglected sacred scripts. This is where the story gets a tad bizarre - for there is no evidence in the Sutras of anything vaguely resembling the mathematics that Tirthaji proposed.  He is reported to have spent eight years meditating in a forest in order to discern the hidden mathematics.  This has been likened by Alex Bellos, author of “Alex’s Adventures in Numberland”, as  being akin to a “...a vicar announcing he had found a method for solving quadratic equations in the Bible”. When challenged by other Indian mathematicians who could read Sanskrit to identify the passages upon which he based his mathematics Tirthaji claimed that the messages were in his texts and in no other.  There was also mention of a 16 volume manuscript that Tirathji had written dealing with the subject - but which had somehow been lost prior to publication - indeed prior to him sharing it with a single other person. Such loss apparently did not disturb Tirathji who simply said he would re-write it from memory.  The first, and only book published (posthumously) was but a fraction of the mathematics that he claimed to have re-discovered in the sacred texts.   (A detailed discussion on the story of the “discovery” of Vedic mathematics can be found here.)  

In short, it appears unlikely that the mathematical techniques mentioned were “discovered” in any ancient Vedic text.  Intrigued by the power and simplicity of the mathematics Alex Bellos did some research to see if the techniques had been mentioned elsewhere, and it turns out that one of them, Vertically and Crosswise,  had been published in none other than Fibonnaci’s Liber Abaci (published in 1202 and credited with introducing Hindu-Arabic numerals and calculation techniques to Europe).  Another, All from 9 and the last from 10,  was apparently  a wide spread technique in Europe in the sixteenth-century - so wide spread that it may have been the origin of our current symbol for multiplication.

However, although many would find the story of the “re-discovery” of Vedic mathematics fanciful it is undeniable that the approaches  described in Tirathji’s text actually work.  In some cases they are no more efficient than contemporary approaches but in others they clearly are.  

Two of the techniques describes as “Vedic” maths that can be traced to European sources are All from 9 and the last from 10 and Vertically and Crosswise. Both are worth exploring.

All from 9 and the last from 10 is essentially a quick mental maths trick with limited application - the approach as depicted here can only be used when subtracting numbers from multiples of 10 (in other words 100, 1000, 10 000 etc.) 
but with minor modification can be used with any number ending in zero.  However it is useful in illustrating the nature of Vedic maths.  With this approach to subtraction you work from left to right and subtract all digits being subtracted from 9 except for the last digit which is subtracted from 10.

1000 - 578 becomes

-  578 (SUBTRACT the 5 and 7 from 9 and the 8 from 10)
= 422

Although limited in application it is easy to see how this might be a useful skill when using money for example.

A technique with wider application is  Vertically and Crosswise which allows rapid calculation by multiplication. Consider 63 x 28. The Vedic method would be;

Step 1. Write the digits being multiplied on top of each other.

6 3
2 8

Step 2. Multiply the numbers in the right hand column (24).  Write the 4 in the units column and carry the 2.

6 3
2 8
                                                                   2  4
Step 3.  Multiply diagonally opposite numbers (cross-wise) and then add the products (multiply 6 X 8 = 48, 2 X 3 = 6, 48 +6=54.  Use the carried 2 from the previous stage = 56. Write the 6 and carry the 5

6 3
2 8
                                                                5  6 4
Step 4. Multiply the left hand column (6 x 2 = 12)  Add the 5 being carried (12 + 5 = 17). Write the 17 to the left of the answer thus:

6 3
2 8
                                                               1 7 6 4

For those who prefer a more visual explanation this video gives a good indication of how the process works.

At first glance using two digits like this the process might easily be dismissed as a novelty and only marginally more efficient than  the “traditional” approach.  But when three or four digit numbers are being calculated (or indeed any  large number) the same method is employed and needs only one line of working.  It is much quicker than the standard approach to this problem - much more efficient.

Those wishing to explore some of the techniques might like to visit this site which provides opportunities to practice some of the techniques and this link to a free manual for teachers. This latter resource contains much that is not purely related to “vedic mathematics” and offers some sound techniques for teaching mental calculation techniques.There are more techniques than presented here to examine and explore - far more than could be presented in a post such as this.

I believe it would be unwise to teach this approach to young children. However, to students who have been introduced to and understand the principles and concepts of conventional mathematics this approach might add interest  and intrigue - and would promote mathematics as something worthy of exploration.    

There is much of worth in Vedic mathematics.  The murky story of its “re-discovery” should allow the story teller that is  in every effective teacher to create the setting for some effective exploration of mathematics.  At worst it still has to be better than providing yet another worksheet...

If you enjoyed this post you may enjoy my other maths related posts available via the maths page or by clicking here.

Sunday, July 1, 2012

The Poor Scholar's Soliloquy

I came across this piece only recently - despite it first appearing in 1944. Despite its age it seems as modern as tomorrow.  I don’t think any commentary that I could add would improve it. The author, Stephen M. Corey, was Dean of Teachers College, Columbia University.

Stephen M. Corey "Childhood Education" - January 1944

No, I'm not very good in school. This is my second year in the seventh grade, and I'm bigger and taller than the other kids. They like me all right, though, even if I don't say much in the classroom, because outside I can tell them how to do a lot of things. They tag me around and that sort of makes up for what goes on in school.
I don't know why the teachers don't like me. They never have very much. Seems like they don't think you know anything unless you can name the books it comes out of. I've got a lot of books in my room at home-books like Popular Science Mechanical Encyclopedia, and the Sears & Wards catalogues--but I don't sit down and read them like they make us do in school. I use my books when I want to find something out, like whenever mom buys anything second-hand I look it up in Sears or Wards first and tell her if she's getting stung or not. I can use the index in a hurry.
In school, though, we've got to learn whatever is in the book and I just can't memorize the stuff. Last year I stayed after school every night for two weeks trying to learn the names of the presidents. Of course, I knew some of them--like Washington and Jefferson and Lincoln, but there must have been thirty altogether, and I never did get them straight. I'm not too sorry though, because the kids who learned the presidents had to turn right around and learn all the vice-presidents. I am taking the seventh grade over, but our teacher this year isn't so interested in the names of the presidents. She has us trying to learn the names of all the great American inventors.
I guess I just can't remember the names in history. Anyway, this year I've been trying to learn about trucks because my uncle owns three, and he says I can drive one when I'm sixteen. I already know the horsepower and number of forward and backward speeds of twenty-six American trucks, some of them Diesels, and I can spot each make a long way off. It's funny how that Diesel works. I started to tell my teacher about it last Wednesday in science class when the pump we were using to make a vacuum in a bell jar got hot, but she, didn't see what a Diesel engine had to do with our experiment on air pressure, so I just kept still. The kids seemed interested though. I took four of them around to my uncle's garage after school, and we saw the mechanic, Gus, tear a big truck Diesel down. Boy does he know his stuff!
I'm not very good in geography either. They call it economic geography this year. We've been studying the imports and exports of Chile all week, but I couldn't tell what they are. Maybe the reason is I had to miss school yesterday because my uncle took me and his big truck down and we brought almost 10 tons of livestock to the Chicago market.
He had told me where we were going, and I had to figure out the highways to take and also the mileage. He didn't do anything but drive and turn where I told him to, Was that fun. I sat with a map in my lap, and told him to turn south, or southeast, or some other direction. We made seven stops, and drove over 500 miles round trip. I'm figuring now what his oil cost, and also the wear and tear on the truck--he calls it depreciation--so we'll know how much we made.
I even write out all the bills and send letters to the farmers about what their pigs and beef cattle brought at the stockyards. I only made three mistakes in 17 letters last time, my aunt said, all commas. She's been through high school and reads them over. I wish I could write school themes that way. The last one I had to write was on, "What a Daffodil Thinks of Spring," and I just couldn't get going.
I don't do very well in school in arithmetic either. Seems I just can't keep my mind on the problems. We had one the other day like this:
If a 57 foot telephone pole falls across a cement highway so that 17 3/6 feet extended from one side and 14 9/17 feet from the other how wide is the highway?
That seemed to me like an awfully silly way to get the width of a highway. I didn't even try to answer it because it didn't say whether the pole had fallen straight across or not.
Even in shop I don't get very good grades. All of us kids made a broom holder and bookend this term, and mine were sloppy. I just couldn't get interested. Mom doesn't use a broom anymore with her vacuum cleaner, and all our books are in a bookcase with glass doors in the living room. Anyway, I wanted to make an end gate for my uncle's trailer, but the shop teacher said that meant using metal and wood both, and I'd have to learn how to work with wood first. I didn't see why, but I kept still and made a tie rack at school and the tail gate after school at my uncle's garage. He said I saved him ten dollars.
Civics is hard for me, too. I've been staying after school trying to learn the "Articles of Confederation" for almost a week, because the teacher said we couldn't be a good citizen unless we did. I really tried, though, because I want to be a good citizen. I did hate to stay after school because a bunch of boys from the south end of town have been cleaning up the old lot across from Taylor's Machine Shop to make a playground out of it for the little kids from the Methodist home. I made the jungle gym from old pipe. We raised enough money collecting scrap this month to build a wire fence clear around the lot.
Dad says I can quit school when I am sixteen, and I am sort of anxious because there are a lot of things I want to learn--and as my uncle says, I'm not getting any younger.

The piece speaks for itself and needs no augmentation from me.  However, I find it saddening that this piece could have been written today.  Has education changed so little over the years?


Text = Stephen M. Corey, "Childhood Education" - January 1944

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