Once you understand something, you tend to look at it in a completely different light. And you wipe out the memory of how it used to be before. I came across a curious case of this while teaching negative numbers.
The negative numbers carry sign of, - , in front. The sign's job is to suggest the value of the number. The value may be compared to zero, such as -30 or it could be a relative value such as -15 km from my house. The negative sign here is like an adjective.
Unfortunately, the exact same sign, -, is also there for the subtraction operation. Here it represents an operation between two numbers. Namely, the operation of taking difference between the values of two numbers. It is like a verb here. If you can't guess which role the, -, sign is playing then that leads to a great confusion.
Soon things get more complicated. The numbers whose difference is to be taken by subtraction could themselves be positive, +, or negative ,- . Many are unable to recognize the dual role played by, -. This may be one reason why so many children find concept of negative numbers baffling.
It is less confusing if its put in words, such as, "Subtract minus four from three". Here the two jobs played by our friend are clearer. Once you learn to recognize it as an operation or a value, then you see negative numbers in a more generic light.
PS: From another angle, subtraction operation is like finding the value of second number with respect to the first number. But that's even deeper to realize.
Showing posts with label maths. Show all posts
Showing posts with label maths. Show all posts
Sunday, August 1, 2010
Saturday, July 31, 2010
You can only learn it.
Recently a student of mine, who tried teaching maths to younger brother, narrated the frustrating experience to me - "its so obvious to me, but he doesn't get it. He just doesn't get it".
This is especially true of Maths. Maths is the ultimate compact way of saying things. Obviously a lot of thought is required to unpack all the thought that is hidden in a simple looking maths statement. A teacher can show what happens when different levers are pulled. But to understand how the Maths-machine works a student has to figure it out for himself/herself. Teacher may use various props to nudge a student to think about the inner working of maths. But for the student, think he must.
Maths demands a lot more "willingness to learn" from a student in that sense. The best kind of maths-teaching is when student does most of the thinking and teacher remains invisible.
Some things can not be taught, but can be only be learned.
This is especially true of Maths. Maths is the ultimate compact way of saying things. Obviously a lot of thought is required to unpack all the thought that is hidden in a simple looking maths statement. A teacher can show what happens when different levers are pulled. But to understand how the Maths-machine works a student has to figure it out for himself/herself. Teacher may use various props to nudge a student to think about the inner working of maths. But for the student, think he must.
Maths demands a lot more "willingness to learn" from a student in that sense. The best kind of maths-teaching is when student does most of the thinking and teacher remains invisible.
Some things can not be taught, but can be only be learned.
Tuesday, June 22, 2010
Maths as a story
It has amazed me to see the extent to which children equate maths with numbers and formulae. They have strong conviction that maths is numbers and rules. They hardly think about maths as a story, drama or an event. As a result simple word-problems baffle them.
Well before we get down to actually doing the maths, we must understand what happened in this story. Say, Meena goes to market to get a dozen bananas, a dozen times. We need to play this drama in our mind, only then can we figure-out what to do. Is this a multiplication or a division sum. A fraction or a square-root sum. Unfortunately, well before children are hooked to stories of maths, they are made to learn the maths of the story.
One way out is to start teaching elementary maths only through stories. It doesn't matter if they don't know the generalized rules, as long as they know how to resolve the situation in the story. In Meena's case she could have made 12 trips and buy 12 bananas each time (hence 12 times 12, which is 144 bananas). Or, she could have made 12 trips to buy one banana each (hence 12 times, which is 12 bananas). These possibilities are there. The maths (as symbols and rules) hardly enters here.
It would be easy to teach children generic rules once they understand many maths-stories. However, if they know the many generic-rules, its not easy to figure-out which of these would apply to a specific story-sum. So treat maths more like a story than a formula.
Well before we get down to actually doing the maths, we must understand what happened in this story. Say, Meena goes to market to get a dozen bananas, a dozen times. We need to play this drama in our mind, only then can we figure-out what to do. Is this a multiplication or a division sum. A fraction or a square-root sum. Unfortunately, well before children are hooked to stories of maths, they are made to learn the maths of the story.
One way out is to start teaching elementary maths only through stories. It doesn't matter if they don't know the generalized rules, as long as they know how to resolve the situation in the story. In Meena's case she could have made 12 trips and buy 12 bananas each time (hence 12 times 12, which is 144 bananas). Or, she could have made 12 trips to buy one banana each (hence 12 times, which is 12 bananas). These possibilities are there. The maths (as symbols and rules) hardly enters here.
It would be easy to teach children generic rules once they understand many maths-stories. However, if they know the many generic-rules, its not easy to figure-out which of these would apply to a specific story-sum. So treat maths more like a story than a formula.
Monday, January 25, 2010
When going gets tough...
I had to do a tight-rope walk in giving Maths sums in the class. Give simple problems and fast-learners get impatient. Give tough problems and slow-learners are lost. Soon the class would end in disarray. This went on for a while till I got the idea of - "Star" problem !
Start with simple problem that everyone can solve, and make it progressively difficult. At the end comes the Star-problem, which is challenging. Fast-learners look forward to the star-sum. Slow-learner don't need to do the Star-sum, instead get this additional time to catch-up with the back-log of sums. This made the class-room much quiet and organized.
A teacher-friend however suggested that I do the star-sum in the middle of the period. And give a problem that every one can solve at the end of the class. Why ? because it's important to leave children with positive feelings in the end. This seem to work even better.
On a recent open-day at school, I heard from parents that kids seem increasingly enjoying maths. Much thanks to my teacher-friend.
Start with simple problem that everyone can solve, and make it progressively difficult. At the end comes the Star-problem, which is challenging. Fast-learners look forward to the star-sum. Slow-learner don't need to do the Star-sum, instead get this additional time to catch-up with the back-log of sums. This made the class-room much quiet and organized.
A teacher-friend however suggested that I do the star-sum in the middle of the period. And give a problem that every one can solve at the end of the class. Why ? because it's important to leave children with positive feelings in the end. This seem to work even better.
On a recent open-day at school, I heard from parents that kids seem increasingly enjoying maths. Much thanks to my teacher-friend.
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