Most, if not all, elementary school teachers know that understanding of place value is the key to success in elementary school mathematics. But what does it mean to understand place value? How did we get stuck with such a complex notation system? What benefits are there that this particular system offers that other system didn't?
Probably the simplest number notation system is the tally system. It may have started with people carrying around some pebbles (or acorns or whatever), but eventually became a written system. If you see "|||||||||||||" you can actually "see" the number. Unfortunately, when the number gets large, it becomes difficult to distinguish numbers. Soon, people started to come up with simplified system, such as noting 5 as ||||. A further extension is a system like the Egyptian system where they used a new notation for 10, 100, etc. With those notations, it became easier to distinguish large numbers, but numbers themselves are no longer "visible."
One of the shortcomings of a system like the Egyptian system is that you need, in theory, infinite number of symbols in order to express very large numbers. Other people, like the Babylonians and the Mayans, came up with a system in which where a numeral is written also contributed in the way the number is written. The picture below shows how the Mayan system worked:
Each group of symbols actually represents a number between 0 and 19 inclusively. However, where it is written makes difference - so, "17" in the second from the bottom is in the 20's place, so it actually represents 340. Therefore, this system is like ours in that they used place values.
So, why didn't this system survive? It used only 3 symbols: 1 (dot), 5 (horizontal bar), and 0 (sea shell). Our system requires 10 numerals (0 through 9). One of the reason why this system did not survive is probably because of this economy of the symbols. When you have 5 in one place and another 5 in the adjacent place, it is difficult to distinguish that from 10 in one place.
So, it appears that our history can be characterized as the search for a balance between simplicity and complexity. Our current numeration system, typically called the Hindu-Arabic system, eliminated the confusion of the Mayan (and the Babylonian) system by using more symbols but a smaller exchange rate for adjacent places. So, here are the major "rules" of our number system:
1. Where a numeral is written matters, i.e., "1" in 31 and 15 represents different numbers.
2. Any pair of adjacent places have 10-to-1 relationship, i.e., you need 10 of the smaller place value to exchange with 1 of the larger value - therefore, the place values are all powers of 10.
3. The total value of a number is determined by multiplying the numeral by the place value and finding their sum.
4. There must be one and only one numeral in each place.
5. Because of (4), we must use "0" as a place holder - except for the leading 0's (and trailing 0's in decimal numbers).
(4) and (5) are often learned in the process of learning how to record addition/subtraction using written algorithms. Most teachers have seen children who tried to write "12" in the ones place when they add 35 + 17. Up till that point, (4) is not made explicit so children do what is most natural thing to do. In fact, (4) is the reason why we have to worry about re-grouping. And now we seemed to have introduced another complexity to our numeration system.
So, what are the merits of our number system? Probably the biggest merit of our number system is that calculation is simple. Wait a minute! we just said because of a rule of our system, we had to worry about re-grouping, a difficult idea for many children.
Yes, it is true that re-grouping is difficult, but with our system, if we know the basic addition and multiplication facts, we can do any calculation. Just think about this for a minute. If you know 100 addition facts (0+0 through 9+9) and 100 multiplication facts (0x0 through 9x9), you can do ANY calculation, no matter how large or how small numbers are. Just imagine how you would calculate 34x72 using the tally system, the Mayan system, the Roman numerals, etc.. So, one important, and often implicit, goal of teaching children computational algorithms is to help students understand this merit of our number system.