Now, I do not mean to suggest that counting-numbers were invented at the dinner table, but I am not so sure that they could have been invented for keeping track of his flock by the proverbial shepherd ! Not unless he had a very large flock - say, 20 or more - so that he could not keep track of them by some distinguishing features. Why, he might even have a name for each one of them, just as in a human family each member has a name.

The point is : what pretext - or context - could we use to get children acquainted with this activity. Today, in a country like the USA where you can still count your pennies, we could tell the children to look at the grocery store receipt and count the change returned if the payment was by cash. In India, years ago, we had 100 "naye paise" coins making up a rupee. Then came 5 paise, 10 paise 25 paise and 50 paise coins. But these have all disappeared and one will have to work with 1 Rupee , 2 Rupee, 5 Rupee and 10 Rupee coins. But that is still not a small amount of money to let children play with. One can, of course, use nursery rhymes or jingles - one, t wo, buckle my shoe,... !

Perhaps, we should think of contexts where counting is not done just to find out how many individual entities - "ones"- there are in a collection. These could begin with pouring out juice into mugs or paper-cups, but going beyond that, perhaps measuring a piece of string with a measuring tape. ( I was about to write "measuring the length", but what is "length" for a beginner ? ) One could go on to measure height of the child, girth, height of a wall, length of a wall, depth of a ditch. One will soon discover that counting numbers do not suffice. ( Perhaps, we could make and use a tape with only centimeter or inch marks. ) So one would be led to meet "fractions", not through slicing of a bread loaf, cutting an apple, slicing a pizza. Can one ever slice a bread loaf into perfectly equal slices ? The measuring tape comes very close to a perfect cutting or division.

Arithmetic by children

Are there any situations - not related to school ! - where children might need, do, and perhaps even discover, arithmetic ? Let us assume that the child has acquired rudimentary number language and associated experience.  How rudimentary ? Perhaps a few number words, say, from "one" to "ten", the associated finger gestures, and the idea of a normal sequence.

Suppose you are going to sit down for your family dinner. You might ask your young ones whether they could help, may be get the dinner plates (unbreakable ones ! ) from the pantry closet or the dishwasher. Most probably, the first time they do it, they may not resort to counting and so bring too few or too many plates to the table. In that case, you could show them how counting, using the mumbo-jumbo words that they have heard from you before, could help. Perhaps, you could count out all the family members seated at the table and then count out the plates already brought out by the child. The next time, you could show how you would take them out yourself by counting aloud, even saying, this one is for you, then this one for your brother, then this one for Mom, and finally this one for Dad. And the next time count out aloud your family when they are not there, say, sister, one, brother , two, Dad, three, and Mom, four. There, that is all  of us. Then follow this out by taking out the plates one by one, saying, this is one, then saying one more - oops !, saying this is two, and so on.
After the plates have been laid out, point out a one-to-one correspondence between the plates and the seated family members. You could also introduce the difference between the cardinal and ordinal aspects of numbers - take out a plate and say "one" and also "first", then take out one more, saying "this is one more", and yes, "a second one", and "now we have two", and so on. You got the idea ? But doing it with the children will require a lot of your patience and perhaps self-control by the older ones who already know all this.

Yes, next go on to the knives, forks, spoons - soup spoons, dessert spoons. What if you require "two" dessert spoons for each member ? So you take out two, and two, and two, and two, so that is twos four times. But hey, they are eight in all. So twos four times make eight. And eight distributed into four - assigning two for each family member - result in two for each. Yes, eight divided by - better, divided among - four results in twos !

What about planning a birthday party ? Plenty of counting, multiplying, dividing to do, before, during, and after the party.

One could introduce the measurement aspect of numbers. Pouring out milk or juice or water into cups or glasses. How may cups can be filled with this carton of juice ? Look, it says "one litre" on the carton, and "100 cc" on the cup or glass. We need pieces of string for playing some game. How long each piece ? Oh, about this long ? But look, there is a "reference" piece of length here, a marked scale or ruler.

What about counting beyond the fingers of your two hands ? Why ? How ?

Why arithmetic for children

Did you note that I made a "silly" mistake in my last post ? In trying to work out an answer to my question "What is three times four ?", I started with the number three, and then went on to add three to the last result three more times so that I had four threes in all. But look, my "times" should have been three and I should have started with four ! Of course, we know the end result would be the same, thanks to the "commutative law of multiplication" - or better, the "associative law of addition". But how do the numbers know these laws ? Or, for that matter how do I know them ?

Well, why should children know this and do these activities ? Because they will "need" them later ? Or because they will help them in their daily lives ? Do these activities enter in their lives at all ? Children of age three, or even younger, might enjoy the "raising or folding fingers and making some sounds" activity in itself or because of the approval of adoring parents. ( Incidentally, is playing such games  AT HOME common all over the world, over all strata of society ? ) What situations we can find in the social lives of children which "invite " numbers and arithmetic ? PLEASE stop reading on and think over this. I will give you my answers in my next post !

How do I know ?

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.

In the beginning

Teaching and Learning with UNDERSTANDING :

When a newborn arrives in our family, we almost instinctively start TALKING to it.  Why do we do it  ? ( We humans are talking animals - homo loquens ! ) At first, we may just babble and not say any  words of our language. Eventually, the baby responds to our talk by making its own sounds, and finally learns to say OUR words, i.e., learns to imitate our words. Perhaps, the most common word successfully imitated - in the English language - is MAMA. Why does the baby do this ? Does it "understand" what the word means, or even that the word has a "meaning" ? Our talking is usually accompanied by some actions ; for example, we may touch the baby, we may smile at it , we may hold it up close, and so on. Most importantly, perhaps, it is Mama herself who talks. She may give feedback by smiling back at the baby after its successful imitation. By the time the baby has completed three years of its journey in OUR world, including our LANGUAGE, perhaps sooner, perhaps later, our language has become a part of its world and its language. It has learnt to use our common language correctly - most of thhe time, at least.  The child is now ready to go to school !

Now imagine that YOU were the child.  What words - and actions - did you learn? And how ? That is to say, what THEY were doing to you when you were learning and what you were doing ?