Alexander,

Your January 2 page on counting reminds me of when I taught math for elementary school teachers. I wanted a way to help these future teachers get their kids to /experience/ the laws of arithmetic. I knew there was a challenge here because, historically, mathematicians /assumed/ the associative law of addition and multiplication and kids were taught these laws without any justification. The textbook used fussed about matching (1-1 correspondence) and made the concept of number look very abstract, and so much more esoteric than what ordinary people think.

Since children's blocks were available, I defined things using them. a + b is the number of things you get when you line up a things followed by b things. Hence a + b = b + a because b + a things is just a + b things flipped over. I defined a×b as the number of things you get when you take a copies of b things. If you arrange the a groups of b things as a rows of b things, you get an a by b rectangle. Rotating this rectangle 90° shows that a×b = b×a. Using this approach of building geometrical figures, one can also show the associative laws of addition and multiplication, of course limited to the natural numbers. In this way, kids can /experience/ the laws rather than just accept edicts from on high.

It is also interesting to note how multiplication evolved as repeated addition. In the days before electronic cash registers, MacDonald's used to have tables taped to the register giving the price of two, three, or more orders of fries. I wonder how long it took in the evolution of mathematics for people to invent multiplication to replace these tables.

And even more primitive, how was commerce conducted in societies where the only numbers were one, two, three, and many? We see some residue of those days at casinos. When the croupier pays off a bet involving many chips, he doesn't count chips, he /matches/ them with his payoff chips. He even pays off 3 to 2 bets the same way, by matching, without counting or calculating.

I heard that there was one primitive society that had no symbol for zero but they did have a number representation system as follows.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 I V N II IV IN VI VV VN NI NV NN III IIV IIN

My elementary education students had fun trying to decipher this system and do arithmetic with it. Afterwards the students had a much greater appreciation for the value of a notation for zero.

William

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