THE PEDAGOGY OF DIVISION

Quotient = 121, Remainder = 9

This is taught to kids somewhere in 4th or 5th standard.  Some kids get it instantly while others struggle.  Eventually most achieve some proficiency in it.  The proficiency is retained and strengthens during school days because the technique is needed to get past higher hurdles in Maths.

The technique possibly fades away from within those adults who move away from Maths.  It is safe to say that most adults are fairly uncomfortable teaching it to their kids and leave the task to teachers.

There are variants of this problem which may flummox kids and adults alike.  Once flummoxed, most may not know how to get out of the mess.  Here is an example of a rat hole that one might venture down when dividing.







Most of us forget as we go to higher classes, what division is all about.  Or for that matter, what each basic thing is all about.  We proceed mechanically, applying formulae and methods, arriving at answers and crossing over into the next class.  When I work with my daughter, she is extremely worried about us "not doing things the teacher's way".  She firmly believes that Maths is about doing things the "teacher's way".  It takes a lot of effort to convince her - It is fine if you don't score a 100% because you did not write the answer exactly the way teacher taught you in school.

The chapter on DIVISION in school books all over the world will have "exercises" listing a number of practice problems such as the ones demonstrated above. Kids are expected to solve them using the "method" taught in class.  I am yet to see a text book which lists just one problem in an exercise and says "Solve it in 5 different ways".

Think about it.  If an adult is given a bag of chocolates (which by the way contains 1582 chocolates) and is told to pack them into 13 packets to be distributed equally among 13 kids, how would the adult do it?  Spread out all chocolates on a table.  Start making 13 heaps, adding maybe 5 to each heap - that takes care of 65.  Then the adult realizes this is too slow, I can do better. So the next round probably is 20 in each heap and so on and so forth.  Most adults would not count all chocolates, realize they are 1582, then use the "method", count out 121 in each heap and realize 9 remain, would they?

So I wanted to show my 12 year old daughter another, logical/intuitive way to divide 1582 by 13.  But before I could start she said - Wait!  Then she looked hard at the that I had written down, and then wrote the following 123 x 13 = 1599.  Then she looked back at the 1582, was puzzled for a while, and wrote 121 x 13 = 1573.  Then she needed some prodding to reach the conclusion.  The point is, the child is already thinking in a way different from what was taught by the teacher.  I didn't have time to ask her how she hit upon 123 in the first place.  It would be really interesting to know.  For the record, my daughter is generally tagged as a child uninterested in studies (I confirm it) and makes every effort possible not to study.  She typically scores 50% marks in school exams.  I mention this so that folks do not dismiss the above example as "Ahh must be a really smart kid, can't expect every kid to come up with such things."

I had another way in mind, which I then proceeded to show her...

1582 / 13 - REMEMBER, WE ARE DISTRIBUTING 1582 CHOCOLATES AMONG 13 KIDS
I split 1582 as 1000 + 500 + 82
I will always look for opportunities to create multiples of 13 i.e. 13, 130, 1300 so that I can quickly distribute them into 13 parts
So I come up with 1000 + 300 + 200 + 82 i.e. 1300 + 200 + 82
I distribute 1300 among 13 so that each gets 100.
I am left with 200 + 82 which is 130 + 70 + 82 which is 130 + 152
I distribute the 130 among 13, thereby each getting 10 - added to the earlier 100, each has 110
Now I am left with 152 = 130 + 22
I again distribute the 130 among 13, each now has the accumulated 110 and 10 more = 120
So now each has 120 and I am left with 22
I split 22 = 13 + 9, distribute the 13 among 13 so that each gets 1, that make the accumulation 121
So now each part of the 13 parts has 121 and I am left with 9
9 obviously cannot be divided among 13 so I leave it as it is
QUOTIENT = 121, REMAINDER = 9
Was it that difficult?  And wasn't it intuitive? If you actually try it out, it is a lot more fun that teacher's method and can actually get addictive - like Sudoku.

As kids go into higher classes, they are taught squares.  By 7th/8th standard most kids will know that 13 x 13 = 169.  So if the 1582 divided by 13 problem is revisited at that time, some kids are going to look for 169's within the 1582, realizing that each can dissolve into 13 in each part.  Millions of kids in schools, God knows how many distinct ways of arriving at the answer exist ... if they are left to play around.  Instead we send them home with "homework" to complete - 20 problems of the same type.  And the poor souls are left with no choice put to mechanically apply "teacher's method" to complete the assignment ... because there is a mountain of homework to complete including other subjects.

Courtesy Wikepedia...
When Ramanujan heard that Hardy had come in a taxi he asked him what the number of the taxi was. Hardy said that it was just a boring number: 1729. Ramanujan replied that 1729 was not a boring number at all: it was a very interesting one. He explained that it was the smallest number that could be expressed by the sum of two cubes in two different ways.
There are two ways to say that 1729 is the sum of two cubes. 1x1x1=1; 12x12x12=1728. So 1+1728=1729 But also: 9x9x9=729; 10x10x10=1000. So 729+1000=1729 There are other numbers that can be shown to be the sum of two cubes in more than one way, but 1729 is the smallest of them.
This story is very famous among mathematicians. 1729 is sometimes called the “Hardy-Ramanujan number”.
Ramanujan did not actually discover this fact. It was known in 1657 by a French mathematician Bernard Frénicle de Bessy.

We all know and acknowledge that Ramanujan was a genius.  But what made him a genius?  He definitely was in love with numbers, and his mind had probably swum in numbers since early childhood.

That's what we need to allow our kids to do - enjoy numbers.  It might not take much more for them to become say one tenth of a Ramanujan.