1+1 = 2 or 1+1 = 10?

I remember being taught 1+1 = 2. Simple. Sensible.

And then there came 1+1 = 10 (not 'ten', but 'one zero' at base 2). Complicated! Nonsense! One zero at base two! What the hell was that? Whatever for? Anybody any idea?

While 1x1 = 1 began to be taught as 1 of 1, therefore one kind or once (as in one parker pen or a dance to Grease Lightning one time)

So, 1x2 = 1+1, interpreted as twice of the same (in this case, that same 1) or two of the same kind (as in two parker pens with one in stock, or the dance to Grease Lightning two times on being encored by the audience)

Whereas 1÷1 = 1 began to be interpreted as one person enjoying that whole one apple (Oops! watermelon for Passu)

 

Which is actually the same as 10÷10 = 1, interpreted as ten people sharing ten watermelons and each enjoying one watermelon (Baapre!)

Which means 2÷1 = 2 is to be interpreted as one person enjoying two watermelons (How greedy!)

 

Which is actually the same as 20÷10 = 2, interpreted as twenty people sharing ten watermelons and each enjoying two watermelons (provided they are shared equally)

Which then means 1÷2 = ½ is to be interpreted as two people sharing one water melon and each enjoying half of it (hopefully equally shared)

Which is actually the same as 10÷20 = ½, interpreted as twenty people sharing ten watermelons and each enjoying half a watermelon  (quarter is actually quite a mouthful, isn't it?)

Yet,  ½ x ½ for some is equal to 1 !!! (Actually carelessly replacing ½ + ½ )

Why is ½ x ½ not equal to 1? Because it's equal to ¼.

How? Well, if 1x1 is 1 of 1, then ½ x ½ is ½ of ½, isn't it? So, if Passu who has half a water melon were to give half of it to his daughter, then he would have one fourth left because his daughter now has the other one fourth. I think the actual act of sharing is much easier than the paper and pen calculation of the sharing.

We do a lot of mathematics in real life and often so easily, but when it comes to doing it on paper it seems complicated. Can you imagine the task of the teacher then? Don't you think that's why Learning by doing makes sense? The teacher in a classroom might show the mathematical steps on the chalkboard or whiteboard, yet not be understood by the students. The teacher might then demonstrate the steps with a water melon or sweets and the students might see it clearer now. And if the students tried it out themselves with the water melon or sweets, don't you think the whole concept would actually sink in?

So, what Confucius of over 200 years ago said makes sense even today, doesn't it?

"I hear and I forget

I see and I remember

I do and I understand" 

Late Reverand Father William Mackey used to remark, "Who said Bhutanese do not know maths? Look at their dzongs. Look at their chortens. Amazing geometry! Jackpo!"

I watch little boys at the BOD (Bhutan OIL Distributors) trying to sell doma to people in the cars waiting in line for fuel. I see them doing a good job of mathematical calculations, although I'm at the same time sad that it's child labour. These same boys could actually be struggling with maths in school. Imagine!

And when I'm buying vegetables and fruit from farmers I find some of them, particularly in the east, struggling with how much money to give me back from a hundred note I've given them. Assuming this, I tried helping a Parop Aum at the Thimphu Famers' Market recently to calculate how much to return to me form a 500 note I'd given her for buying a kg of red dried chilly. She said, "Nga shey." Served me right for assuming! I then thought this Parop lady had probably become pretty used to calculations from years of such business. So, for some of the uneducated or illiterate Experience is a teacher. I guess that's where the theory of Experiential Learning came from. Is it a new theory? NO!!! Check it out!

(Oh! Remembered another incident with another Parop Aum - nothing to do with maths - at the same farmers' market. I asked if she had 'nam' to sell and she said, "Nam mee. Chhoe baynina?" My 'nam' was a sharchop word for don't know what you call it in English - served me right for using it with a Parop Aum - and hers was a dzongkha word for daughter-in-law. How hilarious!) 

Coming back to mathematics.....mathematics for sure can NEVER be understood from rote learning. I knew somebody who was good at rote memorizing questions and answers. At a maths test, she remembered the answer to a question, but had forgotten the steps. She wrote down some steps any how and proudly wrote the correct answer at the end. A lazy teacher would look at the answer and give full marks, whereas a conscientious teacher would go through the steps and give a zero if the steps were all wrong. What would she have learnt from the former? That it was ok to remember the correct answer even if she didn't understand head or tail of the whole thing. Well, this is quite similar to blind faith, isn't it? Whose fault, though? The teacher for encouraging rote memorization by setting questions from the textbook!

The whole thing about knowledge not being applied has to do mostly with NOT learning with understanding, besides some of it not being relevant to our lives (such as the 1+1 = 10 at base 2). Jigme might have learnt the same physics as Tika upto class twelve, but what makes Tika an engineer and not Jigme? Well, Tika went through intense practicals after that for four long years, while Jigme continued with the theory of physics or gave it up altogether. Both learnt about electrical circuit, but Tika can actually solve electrical problems while Jigme would have to call Tika each time he has an electrical problem at home. On the other hand, sixty year old Sonam, who studied only upto class two as a naughty little boy, with no education in any of the sciences, amazingly fixes radios and electrical problems. In his case, he took interest in learning it and learnt it on his own through years of practice. He had interest and passion and, therefore, was motivated to learn. Now, isn't it obvious then that motivation is necessary to take individual effort to learn? And what gives motivation? Interest! Passion!

So, what are some key things about learning?

• learning what is relevant
• learning with understanding, by doing, experiencing
• being motivated to learn, by developing interest and passion

This means a teacher who can ensure that his/her students are motivated to learn and is learning what is relevant and learning with understanding is basically a good teacher. Now, I leave it to you readers of my blog to think of whether a top performing graduate or a bottom performing graduate or both can make a good teacher.

(A view given on page 5 under Perspectives of Kuensel Issue dated January 16, 2010. An article on cover page of Kuensel Issue dated January 11, 2010)