This exploration is built around the "experience-a-foreign-culture" wrapper. The idea here is that they often read about different cultures. Maybe they even dress in the clothes of that culture or eat the food of the culture. But today they are going to try to think like the people in that culture.
However, as I tell them, the culture is imaginary. I will call it the culture of Rikitiki, though obviously the name could be anything.
This wrapper works great. If you just ask students to develop a way of representing numbers, they might develop some of these, but they would not develop all. If you work with actual ancient cultures, there becomes a right answer, not to mention you lose flexibility.
An imaginary culture is perfect. First, you can control exactly what happens. Second, because you are actually constructing the story of the culture as you go along, you can include the class in the construction. Third, because it is an imaginary culture, the students do become more involved in the construction of the story.
The idea then is to move through different methods of representing numbers. For each method, you ask questions that require number answers, such as "how many people are in your family?" If possible, you do addition and subtraction (and maybe even multiplication). You talk about whether the school children liked this method -- was it easy to learn or difficult.
But you also, to the extent you can, let them direct the flow of events. For example, if someone suggests a more advanced system, you wait until you get to that, then give them credit and incorporate their idea.
In terms of "surfin math", there is no right or wrong answer, they just supply ideas.
The original goal of this "exercise" is to understand the Base-10 method of representing numbers through the exploration of other ways of representing numbers. This leads to a sophisticated understanding of the base-10 system, including appreciating the advantages. But it also can be useful just for developing the basic concept of base-10, and it helps develop other concepts concerning the represention of numbers.
Because this is so basic, this should probably be one of the first things you do. Ideally, it would be a preliminary to learning base-10 operations (such as addition with carrying, or even just two-digit numbers).
As I discovered when I actually used this exercise, the early number systems teach a lot about the representation of numbers. These are mentioned in the discussion these early systems.
I know, this doesn't exactly fit the "insight method" of teaching math. There are four answers to that. First, while insight is in general the best way of teaching mental models in math, it is not the only way of teaching mentals. Second, there are problems involved. Explicitly, the students have addition problems and naming problems. Implicitly, they have the problem of trying to understand what the Rikitikian life was like and what the Rikitikian thinking was like. Third, it should be noted that the idea of surfin' math is to avoid the right-wrong answer perspective. This wrapper does that. So, the question could be, what did the Rikitikians invent next. The students actually get a chance to answer that, but only as volunteering an answer, as they think of it.
Technically, these are numeral systems, but number systems seems to be common terminology, and I tend to talk about representing numbers.
You might thing that "no representation" is unimportant. But I have found it to be very useful.
How many people are in your family/live in your house? How many arms do you have? How old are you? How many times have you been to Disneyworld? How many boys are in the class? How many buttons on your shirt. How many times have you been to Philadelphia?
In unique symbol method of representing numbers, every number is given its own name. Stick/finger and Unique Symbol are opposites, though both are conceptually simple.
There is a problem in that, because they are opposites, there is no logical progression of representing numbers that passes through both the stick method and the unique symbol method. The metaphor system leads naturally and logically to the finger/stick method, but it just as naturally and logically leads to the unique symbol method. One you have a Roman system, it would be a step backwards to unique symbol; once you had unique symbol, it would be step backwards to the finger/stick method.
I solve this problem by saying that the unique Symbol method was used in the southern half of Rikitiki.
The transition from metaphor to unique symbol is straightforward. At first, people would express two by number of arms, number of eyes, number of hands, or by a wide variety of methods. But eventually they would settle on a common method. And eventually that would become the number. (Or, at least, that's what happened in Rikitiki.) So, people would just say they had eyes pieces of gold.
The written representation then just corresponded to the spoken one -- a nose was written for one, a pair of eyes was written for two, etc. With time, a short-hand could be developed in both speaking and writing.
Because the Hopi Indians purportedly just had the numbers one, two, three, many, I think it is worthwhile to start with this system. You can explore all of the addition combinations. Subtraction would be interesting because you can't subtract well with "many". (And in that sense it is like infinite, but that's probably to deep for everyone, not to mention it is really irrelevant.)
Unique Symbol: Many Symbols
Of course, unique symbol gets "interesting" when there are different symbols for all of the numbers. Interesting in the sense that you really get to explore the properties (including advantages and disadvantages) of the system. At one point does it become too tedius to develop different symbols for all of the numbers? At one point does it become too difficult to remember all of the symbols?
I ask students to represent 237. Of course, this is simple. Then I say, "But you have to make up all the smaller numbers before you make up 237.
Addition is also very quick. But how many number facts do you want to memorize? School children hated this system. (I was talking about the poor children who did not like this system because they had to memorize so many different addition facts. My second-grader did not seem to realize that she was in exactly the same situation.)
Alas, their impressions are accurate, though not perfectly accurate. With no experience, it is difficult to remember even 8 symbols. With practice, you could probably master 10 without difficulty. When you aren't even familiar with the symbols and haven't had any practice, you aren't going to memorize any addition facts. With work and practice, you could learn addition facts. (I do not know how high you could reasonably go -- eight for most people?).
To lighten their memory load, you can use the same symbols as we already use. Then of course, you are going to have to explain that 10 is not a symbol, so they will have to create the symbols beyond 9. (Orally, "ten" is a unique symbol, but "twenty-three" is not.) By the way, if it is useful, the written representation of 10 in computer programming is just the letter a, either capitalized or not. That gets you up to 15 (which is f), and the students will think of using the other letters to make it up to 36.
The point is, perhaps, that the investigation of unique symbol is going to be awkward, but I think you need to do it. They will experience the difficulties. The example of our current system might help them appreciate that in real life, the difficulties do not occur as soon as they seem to (in an unfamiliar system).
I don't know if it useful, but in writing, the English system has essentially no unique symbols for words, and the Chinese use all unique symbols for words.
The vestiges of unique symbol are still in our language, and captured in my exercise to name the numbers important enough to have names. We have a name for 12 (dozen) and a name for 144 (gross). We have old and rarely used names, like score, and names for irrational numbers (pi) and very large numbers (googol and Avogadro's Number).
Eventually there was a reunification of Rikitiki, so that Northern Rikitikians learned the unique symbol system. So they had one system -- 1, 2, 3, 4, 5, 6, 7, 8, 9 -- and another system |, X, C, M. It was natural for them to shorten some of there long strings using a mixed system.
So, instead of writing XXXXXXXX, they would write 8X; instead of XXXXX|||||| they would write 5X 6|. (Short-hand for this would be 5X 6. MMMCXXXX||||||| would be 4M C 4X 7|, which could also be written as 4M 1C 4X 7|. Of course, it could also be written as 4X 7| 1C 4M. But as you might guess, it became customary to write the numbers beginning with the largest.
Addition and Subtraction
Someone discovered that if you just memorized some 100 addition facts, you could add very quickly.
Once the numbers are lined up in order, this is equivalent to standard addition.
Multiplication and Division
Perhaps curiously, perhaps not, these now seem too complex.
There is not much of a step to now create the base-10 system.
One very common system is to give unique names to the numbers 1 to 9, distinctive names to the numbers 10 to 90 (counting by 10), distinctive names to the numbers 100 to 900, and so on.
Then a number could be expressed additively. This is exactly what we do in spoken english for the numbers less than 100 -- we call a number ninety seven, which means ninety plus seven.
The Greeks used each letter of their alphabet (with a few added in) to create 27 symbols, carrying them to 900. This apparently was in scientific use, not common use. This is the system apparently in use in China today.
I don't find this system interesting, and I can't think of much interesting to do with it. The only other interesting point, besides for finding it in spoken English, it is that is an additive system. In the spoken "3 hundred and nine", it is understood that hundred is multiplied by three. So this representation has multiplication. In the spoken "ninety seven", if 90 is taken as its own unique symbol, then there is just addition.
My speculation is that these additive systems are actually a hindrence to developing the base-10 system. In other words, they are an intellectual dead-end, but once you have driven down the dead-end, it is difficult to back up to the simple Roman system.
For all fo these reasons, I leave this type of representation out of my teaching.
So they used the "poetic" method. The idea is that class practices expressing numbers this way. I envision actually spending a session on this. The experience can include trying to add. (I am not sure quite what addition is in this culture.)
The rikimikians then go through all of the other methods of representing numbers. Again, you want to represent small numbers, represent large numbers, and add.
It would be nice to see if students eventually became frustrated with the rikimikians and suggested the base-10 system. I think the appropriate rejoinder, besides for congratulating the student, is to point out that the base-10 is difficult to invent, it took a while to invent, and that although the Rikimikians are imaginary, human history shows slow progress in developing a good system for representing numbers.
After you have experienced a Roman-type system, what comes next? It would be pointless to go on to the base-10. You could of course solicit other ideas, or improvements. Otherwise, you could stop with a Roman-type system. However, it would also be very interesting if the Rikimikians developed a base-8 system.