No Representation
The Rikitikians at first had no way of representing numbers. (In fact, they are famous for having an advanced culture with no way of representing numbers.)
So, when asked a question with a number as the answer, they just said "hard to say", or "can't explain it very well" or something like that.
Answering Questions
The first assignment for the class is that you ask them questions that have a number as the answer, and they answer as the Rikitikians would. I had a small collection of questions to ask. (I should have had more.) Some questions are below.
Surprisingly to me, most students in my fifth and sixth grade classes enjoyed doing this.
This is not as easy as you probably think. I told the students exactly how to answer the question. But then I gave them a real question to answer. I think some of them got wrapped up in figuring out the answer, and then they forgot about not having numbers and answered "two". And some just didn't get it in the first place. So this is a check that they are on the right page.
"Wrong" Answers
Some students apparently thought that I was just saying that the Rikitikians didn't have the English words "one", "two", etc. But they have their own system. So they suggested French or Spanish names. That is simply a wrong answer.
The more interesting "wrong" answer is a way that counts as representing numbers. Here, you are in control. Does holding up four fingers to resresent four count as a way of representing numbers? I said no. I said the Rikitikians would just see fingers, not a number.
However, while a "wrong" answer now, holding up four fingers to represent number is a great answer for later. Essentially, in this phase of the exercise, you are collecting "right" answers, which you can use later.
With a second grader, I pretty much suggested all of the improvements in the number system. But with my fifth graders, one way or another they made a comment or question presaging the advancements the Rikitikians would make in representing numbers.
There is a subtle teaching point here. I have tried exploring the representation of numbers by given students the task of constructing a number system. One problem with that is that you don't get to experience the variety of ways of representing numbers. But a second problem is that it generates the right-wrong milleau -- there is a right answer, and students are trying to find it.
With this structure, the students are not given the task of constructing a way of representing numbers. But, when they come up with answers to this, they are incorporated into the exercise. So this is a good way of implementing Surfing Math.)
The student who first showed me fingers (to represent four) was named Johnmichael. When I came to the finger method of representing numbers, I said that it was invented by the famous Rikitikian called Johnmichaelsahn.
And of course, once a student is rewarded for suggesting an advancement, the students will be motivated to suggest advancements whenever they think of them. That of course is exactly what you want.
A Culture without Numbers
This also teaches that we can have numbers without names. This is a useful concept -- most students think there are no numbers between one and two. My second-grader started the exercise with the claim that the numbers did not exist, then had the insight that there could be numbers that did not have names. I probably said that explicitly, but nonetheless she announced it cheerfully as if it were her own insight.
My sixth grade class spent an entire 40-minute session discussing this culture without numbers. One thing we explored is what it would be like to have a culture without numbers. This, as it turns out, gets very much into numbers being numbers versus numbers just being symbols.
For example, such a culture would have no problem with telephone numbers. They would just use something different. For example, they could use colors instead of numbers, and then a telephone number might be blue-green-green--orange-yellow-white-brown.
Someone in my sixth-grade class suggested using months on the telephone. The students then thought you would use the first ten months. I pointed out that it was odd we had only 10 buttons and that they would certainly use all twelve months.
We also discussed the numbers on a remote control, which I think would be like the telehpone. We did not get into house numbers, which is interesting. You could label houses with colors, but then there would no ordering, and you couldn't look at a wrong house number and know which way to go.
We did consider the fact that they could have months, but they couldn't know how long the months were.
Do the Rikitikians "Know" the Answer?
"I don't know," is an interesting answer to deal with. At one level, it is a "good enough" answer. If your class can stand the sophistication (or say the third student gave this answer), I think it is fair to say that the Rikitikians knew, in some sense, they just couldn't say. Of course, this in some sense is a massive concept. I don't think you could know exactly how many students are in a class without having numbers, but I think you could know how many arms you had without having any way of representing that number.
To teach this, you could stop for a moment and re-establish the goal of really understanding how the Rikitikians thought. You could ask they opinion -- did the Rikitikians really know large numbers? Did they really know small numbers?
What Counts as a Representation of a Number?
A primitive culture is almost certainly going to have names that correspond, at least roughly, to zero, one, and two. Whether or not these are numbers is debatable. The number "zero" is considered difficult for a culture to grasp. I think that grasping it as being a number is difficult, but the culture will have ways of expressing it, such as "none". Does "none" count as the representation of a number? "One" is obviously a number, but what about "a", as in "Go to the store and buy a loaf of breat." It means 'one' in that sentence. And in fact, the word "a" comes from the Old English word an meaning one.
Two is also tricky. The idea of "number" is that it is independent of what you are counting. Many cultures have words for two that depend on what you are counting. For example, a woman who gives birth to two children at the same time is said to have twins. We use the word "couple" to mean two, but it most obviously applies to people, and the origin of the word is to things bonded together. The word "brace", meaning two or couple, apparently is usually found when talking about pheasants. Pair I think it going to refer to two things that are alike. In that sense, it is the opposite of couple (in the sense of two things bonded together). Couplet, duet, duo, etc.
In English, plural is used to indicate more than one. Is that a representation of number?
What about words meaning more than one, as in some, many, and a lot?
If there is a right or wrong answer to these questions, I don't know what it is. I think it will depend on some rather precise definition of what we mean by "no representation of numbers", when the phrase was not well thought out or intended to be very precise. The point is, these are potentially interesting things to discuss. Having an informed opinion is important; deciding on a particular answer is not. I just had a class vote. Later, I let the students discuss this a little, but I did not have time for a lot of discussion, and only the better students were involved in the discussion.
Addition
My class voted that none and some were numbers. So, I wrote out the addition facts: none + none = none, none + some = some, some + some = some, and some + some + some + some + some = a lot. The Rikitikian children liked this method, because it took them five minutes to learn it in first grade.
Later, someone said that the Rikitikians were crazy. I think they were experiencing something different and calling it crazy. So I said they thought we were crazy. That's kind of true but trite -- as a quick reaction, if we think they are crazy because they are different, then they are going to think we are crazy because we are different. To give my claim some force, I added "They think we are crazy for having our system because their system only took five minutes to learn in first grade."
Next: Representing Numbers with Metaphor
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