Many of the problems that children have in subtraction especially in subtraction with renaming were attributable to a lack of complete understanding of place value. As Ma suggests, place value concepts must be taught over a lengthy period of time. When students begin to count they need to learn the difference between one and two digit numbers and two and three digit numbers, and so forth. They should understand what is meant by place value in math -- the name of the place and the relationship between the place like 1 ten equals 10 ones. The most important idea they form at the second and third grade level is that digits in different places have different values. In later addition of two or more digits they learn that they must line up the digits by place before they add - in order that they are adding like quantities - 10s to 10s and 1s to 1s..
The numerical system we use expresses all whole numbers as combinations of the symbols 0 through 9. Instead of continuing to make new symbols for each number it was probably inevitable that rules would be made that would allow for other numbers to be made with this finite number of symbols. Indeed if students look at a number line they can easily see the pattern of 0 -- 9 and then the returning back to have 1 combined with each number 0 -- 9, then 2 with each number and so on. Something unique happened after the number nine. The number 1 had already been used and so the 0 was put in as a way to fill in the place value. All numbers became combinations or ones, tens, hundreds, etc. As students learn addition with composing and subtraction with decomposing they learn the relationship between these procedures and their place value.e.
