Developing understanding of place value
A student’s understanding of ten is central to the development of base ten place value. At the pre-place value level (Ten as a count), both ten and one are treated as simple numerical entities. Although the student may readily recite the sequence of multiples of ten, “Ten, twenty, thirty, forty…”, the names of these multiples are used as simple counting numbers in the same way that seven or thirty-three can be counting numbers. For students who are pre-place value level, reciting the names of the multiples of ten does not reflect a sense of increasing the size of the total (i.e. incrementing) by ten. Moreover, for these students ten is treated as something constructed of ten ones, but one ten and ten ones do not exist at the same time. Typically, students who are pre-place value reconstruct units of ten by counting them. That is, to add twenty they would typically count-on by ones.
Next, students come to understand ten as a unit. That is, ten becomes an abstract composite unit (both a unit of ten, and ten ones at the same time) where each act of counting by ten is a short cut to ten acts of counting by ones. However, at this level of place value understanding using ten as an abstract unit is in limited situations. These limited situations are typically where a student can access groups of tens and ones for only one of the numbers involved. Suppose a student is told that there are 46 mints in a sealed box and then shown 5 rolls of 10 mints and 7 individual mints. In this situation, when the student needs to find the total, the 46 mints representing the quantity 46 and the associated structure of units are not available.
Figure 1. Accessing the structure of tens and ones for only one number
That is, materials representing the tens and ones organisation for one quantity are hidden and the student must coordinate tens and ones to find the total. The student must have an abstract sense of 46 beyond being a number word before using the tens and ones support available to find the total. Students need to be able to simultaneously think about tens and ones to add or subtract tens and ones to a number. Adding ones to a multiple of ten, say twenty plus three, is easier than incrementing by tens off the decade. That is, 20 and 3 more is often easier than 3 and 20 more. Notice that when the student responds to “34 and another 10” the answer of “40” completes the ten. This could correspond to losing the structure of “tens and ones”. Some students have difficulty handling two lots of units (tens and ones) in numbers when operating with them. Without access to materials, keeping track of tens and ones as separate units can become so difficult that sometimes students do not increment the total but rather increment one of the units.