Metaphor
The Rikitikians went through a poetry phase. This helped them develop there first method of representing numbers -- metaphor.They could say, "I have as many coins in my pocket as there are ears on your head." Again, I let students answer my questions using this method. I also had coins in my hand and let a few students try to tell the class how many coins I had.
Answers
Again, the first exercise is answering questions. I also had a number of coins in my hand, showed them to a student, and had the student try to tell the class how many coins were in my hand.
It was beyond the sophistication of fifth-graders, but this method can be very poetical, and it can demonstrate great skill. I am not sure they thought of any good way of doing three -- someone came up with ears plus noses. I think clover is a better answer. Four is corners of a square. Most people can get five -- number of fingers on one hand.
Could the Ritikians Represent Large Numbers
In theory, 11 is the number of (American) football players a team has on the field. Therefore, 11 can be expressed as the number of football players on a football team.
In practice, as my second-grader noted to me, you cannot count with this system. So, if you had eleven pieces of gold, how would you know that it was the same number as the number of football players on a team?
It was, again, beyond the sophistication of my fifth-graders, at least for a beginning lecture, to try to get into this at all. (And, in fact, they couldn't go beyond 5 in their attempts to find metaphors for numbers.)
This leads into the important use of fingers in representing numbers.You cannot know that you have as many gold pieces as players on a football team. But you can know that you have as many gold pieces as you have fingers.
At this level, the exercise teaches the concept that you can know two sets have the same cardinality by putting them in one-to-one correspondence. In other words, if you pair up fingers with gold coins, and there are no fingers left over and no gold coins left over, you have as many gold coins as fingers.
If this does not come up here, it can be introduced in the discussion of the Roman-type systems of representing numbers.
Vestiges
One vestige of this system is saying that there are as many of something as their are stars in the sky. And we still use this system where appropriate. How many bacteria do you have in your stomach? More than there are people on the planet. And perhaps it helps makes sense of a very puzzling line from Jack and the Beanstalk. The man asks Jack, "I wonder if you know how many beans make five." Jack answers, as sharp as a needle, "Two in each hand and one in your mouth."
Too Deep
In some sense, we still use metaphor when we count. Suppose you count sheep and claim there are 8 sheep. You are, essentially, saying that there are as many sheep as their are numbers in the set {1,2,3,4,5,6,7,8}.
Next: Representing Numbers with Fingers
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