Let me ask you a very simple question - I hope you won't take it amiss.
What is three times four ?
I would answer : twelve. ( I suppose so would you. Also, I am assuming that you have figured out that my question is about "NUMBERS", and not about some words of the English language : "three", "times", "four" . Compare with the question "What is a noun ? ".)
But how do I know the answer ?
Well, I remember it. But how did I know it the first time in my life ? Perhaps I was told that it is so.
But could I figure it out NOW ? ( Could you ? ) Recalling the multiplication table will not do.
May be I can figure it out thus : I start out with "three" and add "three" to it. I get "six". ( How ? ) Then I add "three" to "six" and get "nine". And finally, I add "three" to "nine" and get "twelve". Whew, what a relief ! But look, how did I know when to stop adding "three" to a result obtained on the way ? Well, at the back of my mind, I was counting the "times" : start with "one", then "two", next "three", next "four", and stop.
But how did I know "three" added to "three" gives "six" ? I guess I have to figure that out afresh.
Let us imagine a three year old child trying to figure it out ( that could be me or you - when we were three year olds. ) What is the "background knowledge" and "context" ? I must "know" the words "three", "and", "six", and the context. What could be the CONTEXT ? Perhaps, some fingers of the hand raised or folded down. Perhaps, some pebbles ( in the good old days ) or chocolates spread out on the table, or perhaps, some tally marks "|" "|", "|", or some little circles, or blobs , on a piece of paper. And then doing something very strange : looking at or pointing to a pebble, chocolate, finger, tally mark, and so on, or picking up a pebble or a chocolate, and saying ( aloud at first, and then perhaps to myself !) "ONE". And then . . . . .
Was this all easy ? Why did I ( agree to ) do it ? Was it because I was promised that if I did it, I would get a chocolate ? It wasn't easy, and it isn't easy for today's three-year olds.
What is three times four ?
I would answer : twelve. ( I suppose so would you. Also, I am assuming that you have figured out that my question is about "NUMBERS", and not about some words of the English language : "three", "times", "four" . Compare with the question "What is a noun ? ".)
But how do I know the answer ?
Well, I remember it. But how did I know it the first time in my life ? Perhaps I was told that it is so.
But could I figure it out NOW ? ( Could you ? ) Recalling the multiplication table will not do.
May be I can figure it out thus : I start out with "three" and add "three" to it. I get "six". ( How ? ) Then I add "three" to "six" and get "nine". And finally, I add "three" to "nine" and get "twelve". Whew, what a relief ! But look, how did I know when to stop adding "three" to a result obtained on the way ? Well, at the back of my mind, I was counting the "times" : start with "one", then "two", next "three", next "four", and stop.
But how did I know "three" added to "three" gives "six" ? I guess I have to figure that out afresh.
Let us imagine a three year old child trying to figure it out ( that could be me or you - when we were three year olds. ) What is the "background knowledge" and "context" ? I must "know" the words "three", "and", "six", and the context. What could be the CONTEXT ? Perhaps, some fingers of the hand raised or folded down. Perhaps, some pebbles ( in the good old days ) or chocolates spread out on the table, or perhaps, some tally marks "|" "|", "|", or some little circles, or blobs , on a piece of paper. And then doing something very strange : looking at or pointing to a pebble, chocolate, finger, tally mark, and so on, or picking up a pebble or a chocolate, and saying ( aloud at first, and then perhaps to myself !) "ONE". And then . . . . .
Was this all easy ? Why did I ( agree to ) do it ? Was it because I was promised that if I did it, I would get a chocolate ? It wasn't easy, and it isn't easy for today's three-year olds.
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