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

1000
-  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...

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Credits;

• Image of Tirthaji: http://www.vedicmaths.org/Introduction/History/Tirthaji.jpg
• Links provided in text to original sources.
• Alex Bellos, 2010, Alex’s Adventures in Numberland”

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

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.

THE POOR SCHOLAR'S SOLILOQUY

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.

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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?

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

Image =  http://www.maebs.com/articles/Liz_Davies/TreeAcrossTheRoad.jpg

Educational reform; like building a plane - in flight!

I love this clip. To me it is a wonderful metaphor for educational reform.

One of the major issues with educational reform is that schools have to be operational while education is transformed. We can’t simply shut down the schools and learn new and more effective educational techniques. We need to test and trial our learning when and where it matters - in the classroom with our students. We need to learn new skills while we teach via established methods. There are those who suggest that teachers can learn new approaches when students are on vacation - and to some extent this is true. However, the important aspects of classroom teaching can only be learnt through implementation in the real world - and that means while the students are present. There is a time when a trainee pilot needs to land a plane in real life and not in a simulator.

There are obvious issues with this. Our students are not guinea pigs or lab rats. We should not experiment with them. On the other hand, we need to continually improve our practice - and that means we need to be constantly trialling new approaches or trying to refine and enhance established ones. This balance can be difficult to achieve - perhaps this explains the almost glacial pace of school reform.

Clearly there are risks involved in any educational reform - if any innovation is ineffective then students suffer in comparison to what they may have achieved with another approach. However, there are more risks involved in educational stagnation and ignoring new possibilities and potentials.

So how do we build an educational plane in flight? We start by being clear about what we want to do and why. We acknowledge that established techniques may not be appropriate in this operational environment. We try to anticipate any likely issues and develop responses in advance - or at least be observant and flexible to respond to issues as they arise. We clarify our roles - who does what, when and to what standard? Who checks and assists? How do we know when we have succeeded? We accept that the task may be difficult - but, if we are clear about our objectives, it should be worthwhile.


And then we take a deep breath and build our plane.

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Credits: EDS commercial via youtube @ http://www.youtube.com/watch?v=L2zqTYgcpfg

Original idea to link to education: Thanks to a long forgotten presenter at a SPERA conference in Darwin who used this clip in his session.

Ancient maths = modern learning

There are many issues of concern associated with mathematics teaching today.  One is the emphasis on procedural competence regardless of conceptual understanding. Another is the almost total lack of historical significance of mathematics. This is a pity as one can inform the other.

In our busy classrooms we tend to rush from one mathematical concept to another - often without any explanation given to the students. It is my experience that if we take the time to develop student interest in a topic then sound performance often follows  -  the “diversions” demonstrate that mathematics is more than learning algorithms  and makes the broad topic of mathematics more interesting - which engages the students - and engagement leads to understanding which in turn leads to higher performance.

My views on this are shaped not only by my experience as a teacher - but as a student. My primary years have been largely shrouded in the mist of time but I can still remember being stunned and amazed when I was briefly shown alternate historical methods of calculation. It was a revelation. Maths could be done more than one way? People once did maths differently - and accurately - and it worked … every time?  Ancient peoples were smart enough to do mathematics?  They discovered this stuff?

I’ve covered the same material myself in classrooms and  recognise the same lightbulb moment of understanding in students as they make the same discovery.  It is often the beginning of lasting interest in mathematics.

Today, thanks to web-based video we can show students “historical” mathematics as a way to scaffold and extend their mathematical knowledge.

One of the favourite “ancient” forms of mathematics taught in schools is often known as “Egyptian multiplication”.  It turns out that “Egyptian” part of the label may not be overly accurate - but the technique surely is.  This clip provides a good explanation of the process - together with a brief explanation as to why it works.

Ethiopian multiplication (also known as Egyptian)

From Egypt / Ethiopia / Elsewhere  we can make the short historical journey to the lattice method explained here.

Lattice Multiplication

The use of the diagonal columns will have parallels with the use of vertical columns in our current systems of calculations which will not escape the children and provides a useful opportunity to discuss the use of columns in modern calculations.
From there it is a short step to Napier’s Bones.  
Napier's Bones

                                        

 (This is a good clip despite it’s simplicity as it allows the students to study the structure of the “bones” and perhaps identify the pattern in the rods for themselves.) The clip below shows how the bones or rods were used.

The sense of amazement and wonder is enhanced if students actually complete some calculations themselves - in which case the pro forma here will be useful.  Once the students are capable of using the “bones” it is a useful extension to get them to use them to multiply two two digit numbers - and let them discover the technique themselves if they can.


The discussion around why devices such as Napier’s Bones were useful and necessary is a worthy way to spend some time.  Our students find it difficult to understand that it was once the norm for humans to perform all calculations, not silicon chips.  It is also worth raising awareness the universal education is a relatively new social phenomena - and discussing the impact of that.

Another avenue that can be useful is to look at numbers without using numerals at all (well, at first at least). This clip looks at the ancient Greek interest in square numbers and demonstrates the concept effectively using nothing more sophisticated than stones - which is all the Greeks needed to discover the concept.

Counting like an Ancient Greek

I’m not suggesting that the techniques presented here be “taught” in isolation - more that they can be explored and then, once students are familiar with the techniques, compare and contrast them to current methods.  It is often at this stage that students really develop an understanding of the strength of the place value system that we use today - after which their fluency with formal algorithms often improves significantly.  
If nothing else students will have enjoyed interacting with mathematics - and without a worksheet in sight.

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

Mathematics education - as seen on the screen

The stats for this blog indicate that my previous collections of web-based videos have been popular. Hence this post...

Why use web based video in mathematics education? Don Tapscott, author of “Grown Up Digital”, uses the expression “screenagers” to describe the youth of today. Like it or loathe it, the fact remains that our students are conditioned to interact with screens. By using their medium of choice we are more likely to engage them - assuming that the content is worthy. Video - moving pictures and sound - is the language of our students. There is considerable research evidence that interest leads to engagement - which leads, on average, to better performance.

This classic clip of Abbott and Costello demonstrates how easily errors can flow from faulty understanding of procedures - procedural knowledge without conceptual understanding.
Abbot and Costello maths 7x13=28

Apart from the appeal of seeing adults make mistakes this clip can easily move from viewing to activity by investigating where is Abbot going wrong. How could students convince him that he is wrong - and help him from making the same mistake in the future?

This clip is one of many on Youtube featuring what has been called “Mayan multiplication”. (The name may be something of a misnomer as there is some evidence that it may have evolved in India as a part of the vedic tradition.)
Mayan multiplication

This clip is clear and “lo tech” which generates the impression that anyone could use this technique. Can they? Students could explore the technique - but then contrast it with the traditional method of multiplication to compare ease of use - especially with larger numbers.

So how does it work? The wondrous Vi Hart both demonstrates and explains here.



Younger students can also benefit from drawing lines - to investigate patterns in numbers.
Number patterns


Once this video has been viewed it is a small step to recreating it in real life - and then investigating the patterns created by other numbers.

Lines - of symmetry - also feature in this clip. In an earlier collection I included an amazing clip of paper placed in water which unfolded to create a complicated flower-like shape. This is similar - but much simpler and could also be used to prompt an examination of symmetry - and being simpler might be suitable for younger students.




This site features a reasonable number of educational videos with a special section on mathematics. Included is an “inspiration” section which links to photographs with an accompanying maths challenge.
It needs to be said that I see these clips as a means of promoting interest in mathematics rather than as an end in themselves - I see these as useful ways of introducing topics which can then be explored in a more traditional manner. Using visual images in mathematics classes can help bring the subject alive while still allowing teachers to address the requirements of the curriculum.
If you enjoyed this collection you may like to see other collections  here.

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Credits: all information available by following the relevant links.
Abbot and Costello maths 7x13=28
http://www.youtube.com/watch?v=KVn0aksCzNE&feature=related
Mayan multiplication
http://www.youtube.com/watch?feature=fvwp&NR=1&v=1bhOCJ8eijU  
Vi Hart
http://www.youtube.com/watch?v=a-e8fzqv3CE
Number patterns
http://www.youtube.com/watch?v=8TnEbL9vK5A
Floating flowers
http://www.youtube.com/user/dutchpapergirl#p/a/u/2/eHMPkLE0Qs0  

The most important sticky note you'll ever read?

I came across this video the other day -  a “feel good” effort with a twist at the end that I won’t spoil - all is revealed in the clip.

Video from KarmaTube


It occurs to me that this clip has implications beyond the relatively narrow focus of the target audience. So what relevance has this video to broader aspects of education?  There are several messages really - one is that  education requires open and genuine communication.  As teachers we need to listen to our students as much as we talk to them - if not more.

Another is that this relationship needed time to develop - the relationship could not blossom until there was a degree of trust.  Again, there is a message there for teachers.

But watching this reminded me of one of my favourite quotations which is attributed to Dr. James Comer;  “There is no significant learning without a significant relationship”.  Put simply, if we want our students to learn from us we need to spend the time to develop genuine relationships with them. How we do this is up to each teacher as an individual - for we too have our own personalities, interests and foibles. However,  unless we take the time to value our students they will not value us - or what we are trying to share with them.

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Dr. James Comer is more than a source of an inspirational quotation.  He has developed the Comer School Development program as a vehicle for school reform.  An interview with Dr. Comer can be accessed here.

A modern heresy?

I have a confession to make - one that will label me a heretic in this modern world, or worse, a luddite.

I turned my phone off.

Yes - my wonderful phone, miracle of modern technology, my link to the world and all the people I know - I turned it off.  Not to “in flight” mode.  Not to silent. Off.

I know, I know...my phone is really a powerful computer - I’m told that it has more computing power than NASA could muscle up in  the control room in 1969  used when they first put a man on the moon. It can find any information I think I might want to know  and much that I am sure I don’t. It can play my music or video. It can store and send email. I can read more news and current affairs than anyone outside of national security really wants to know.  I can search the data bases of the world. I can take photos, even video with it and upload it before I even leave the park.  It is my link to the world. And I turned it off.

At the time I committed my act of heresy I was at a family picnic. My loved ones were with me - I could talk to them in person. Work didn’t need to contact me - (and if it did what would that say about training, delegation and leadership density?). I did not need to access any of the social networks to which I belong. I did not need to tell the world I was having a great time.  But I needed to let my family know that they are important - much more important than the electronic distractions that masquerade  as essential elements of daily life. So I turned my phone off and “lived the moment”.

Technology might be important - but people are more so.

How often do you see people having a conversation stopping while one answers a phone? Even worse, to send a text that could be sent later when not talking to someone. Why is a person kilometres away more important than someone right there with you? Chances are they are not - so why put your face to face friend on hold?

Mobile technology is our servant - not our master.

Phone image:
http://cdn.cbsi.com.au/story_media/339302015/samsung-galaxy-s_4.jpg