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.

DON'T TEACH BODMAS IN SCHOOL

One of the topics taught in school that do a great job of mystifying Mathematics for young students and confuse the hell out of them is BODMAS.

So what is BODMAS?
It stands for BRACKET OPEN-DIVISION-MULTIPLICATION-ADDITION-SUBTRACTION.
What does it do?  It imposes rules for a potentially ambiguous arithmetical problem.
Take this example:
4+5x3 = ?
One could say 4+5x3 = (4+5)x3 = 9+3=27.
Another could say 4+5x3 = 4+(5x3)=4+15=19.
Thus the same problem throws up two answers.
If we deem that to be a problem, then we need a solution for it.
BODMAS is the known solution.  It loosely says, given an ambiguous arithmetical problem (such as the one above), you are commanded to first solve BRACKET OPEN, then division, followed by multiplication, addition and finally subtraction in that order.

This, by the BODMAS dikat, 4+5x3=4+(5x3)=4+15=19.
So the 19 answer is correct and the 23 answer is wrong.
Applause!  Bravo BODMAS!!

(For the purpose of this discussion, I will leave aside BRACKET OPEN because this is not meant to be a BODMAS eulogy.

Now let us look at 4+5x3 differently.

I was begging on the street.  A man gave me 4 rupees and after that 5 women were kind enough to give me 3 rupees each.  I thus had 4 then 5x3 so 4+5x3=19.  The way I have told the story, it is indisputable that I had 19 rupees.  The confusion is about the way I wrote it.
Well, firstly, if I go through the string of numbers and operations from left to right (as is the practice), my story and the sequence match.
So 4+5x3=19.

I was begging on the street.  A man gave me 4 rupees and then a woman gave me 5 rupees.  Coincidentally, this happened 3 times.  So I got 9 rupees thrice hence I got 27 rupees.
So 4+5x3=27.
Well you could say, but this is not OK as per BODMAS.  So this couldn't have possibly happened? :-)

The point is, BODMAS claims to remove ambiguity.  But why not place the onus of being unambiguous on the fella who writes the arithmetic?  Just say - look use brackets so that you are sure you convey the meaning you intend.  Else don't blame others for interpreting things in a way you did not intend.

So write (4+5)x3 and it is sure to be interpreted as 27 or write 4+(5x3) and be sure it is 19.

So basically BODMAS condones ambiguity and claims to resolve it.  But you know, Pythagoras believed that the universe is constructed out of numbers.  Some great Mathematician has said "It is evident that the Great Architect of the universe was a Mathematician".  So Mathematics reflects life and vice versa.  Given that we demonstrated how 4+5x3 can practically play out in two different ways, what gives anybody the divine right to lay down BODMAS as a law?

Not to mention that BODMAS confuses the hell out kids.  Forget 4+5x3.  Consider 12+6/9x14-4x8.

If I were a teacher I would just skip the section on BODMAS.  Neither teach it not test kids on it.

By the way, other than in a chapter or section on BODMAS, BODMAS is not needed.  All books, exercises and examinations are carefully presented using brackets to make the arithmetic unambiguous.  Such as ((12+6)/9)*((14-4)x8)=82.

BODMAS is a solution to a non-problem.  Ignore it.

Another such utterly useless topic I want to write about is SQUARE ROOT BY THE DIVISION METHOD.  As of 2016 it appears in the Maharashtra SSC Board's class VII text book. I have asked by daughter to ignore it and instead resort to trial and error to find square roots of large perfect squares. Saves the child a lot of anguish and replaces it with a most interesting exercise.