Fingers/Sticks
Next, a Rikitikian invented the method of expresssing numbers by holding fingers. Again, I went around the class and most students wanted to answer a question using this method. (As it turns out, this method was only used in the Northern half of Rikitiki, but you can mention that later.)
The use of fingers is essentially metaphorical, and I am sympathetic to anyone who wants to argue that it should be allowable when there is no method of representing numbers. But in Rikitiki, it was not invented until later?
How do you write numbers using this method? As described above, Johnmichael had first used his fingers to represent numbers, so in my story Johnmichaelson from Rikitiki invented the finger method. Now the real Johnmichael wanted to show me on the board how to represent numbers. With great trepidation I let him -- and he wrote down the stick method. So, ||||| would be the representation of five.
Of course, any symbol would do. For my second grader, we used lollipops.
There is also an oral representation corresponding to this. This can be any sound. For my class, I used "wa" because it sounded like "one." So, five would be "wa-wa-wa-wa-wa". Again, students were happy to answer questions using wa. In fact, I asked how many people were in the United States, and they set out to tell me.
Name?
This system of representing numbers seems not to have a good name. I have seen it called the "unary" system and the "cardinal" system. In a sense it could be called a "tally" system, but to me, tally implies use of symbols to signal five or ten. For example, in the classic tally system, a line is put through a group of four tallies to indicate five. This more properly is a Roman-type system.
Addition and Subtraction
I told my class that the children of Rikitiki loved this system because it was so easy to learn. In fact, it is remarkably simple to add one or two to a large number. You don't have to figure out what the large number is -- you just add one (or two) lines to it. Subtraction is a simple matter of erasing. I gave them another problem corresponding to 6 + 8 and they could do that too.
It is also time to start talking about advantages and disadvantages. The advantage I already mentioned -- the children loved the ease of the system. The problem is representing large numbers. 1000 is impossible, but even 34 becomes tedious. Again, the class might lead in suggesting advantages and disadvantages, but if they don't, you just continue with the story.
Multiplication and Division
I did not ask my students to multiply or divide. I thought it was impossible, but instead it is remarkably easy. Essentially, if you are multipying ||| by ||||, you just write down ||| |||| times. The details are just in keeping track of how many times you have written down |||.
Similarly, to divide ||||||||| by |||, you would write down |||, but separate them, then starting putting sticks from ||||||||| underneath them. You would find that the answer is ||| with no remainder.
I am no fan of teaching procedures without understanding. But here, the procedures of multiplying and dividing (and addition and subtraction) are so close to the underlying meaning that I think you could teach just the procedures.
For these, I would emphasize that the goal is to think like a Rikitikian and solve the problem the way the Rikitikian would. So, the could translate ||| to 3 and |||| to 4, multiply 3 by 4 to get 12, then put 12 sticks on their paper. But the Rikitikians, in this phase of the represenation of numbers, didn't have 3, or 4, or times tables. So the goal is to solve the problem without using English numbers, and instead just to think in terms of sticks.
The same applies to some addition and subtraction problems. I think you can add or substract small numbers more easily in Rikitikian than by thinking in terms of English words. However, it might be easier to solve 6 + 8 by thinking in English. Again, the encouragement should be to think in terms of sticks.
Too Deep
It's a tough-to-get concept, and I didn't even try it, but there is no representation of zero in the stick system.
In this system, I think | and "wa" correspond to the number 1. Essentially, there is just one number in this system. There is a deeper sense, I think, in which | and "wa" still are not numbers. But that is real deep. This hence is a good precursor to learning about constructing the numbers (in which one plays a very important role). Or, not as deep, it is still interesting that you can express all of the counting numbers using just 1; no other counting number can function in this role.
Vestiges
The word "score" is a vestige of the tally system -- it means a cut or notch, and you can score a piece of word or score a goal in soccer. For that matter, tally means to cut and has the same origin as the word "tailor". A very famous tally is the gunslinger cutting notches in his gun.
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Next: Roman-type Representions
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