#18 Place Value as Groupings

Len made the observation that in school, we teach kids that place value is a grouping of counters into inclusive groups, not an algebraic operation. The common technique is to have kids make groups of beans on a bean stick or put Unifix cubes together, or use some other base ten manipulative materials. Isn’t that a simple operation of making inclusions much like classification that children easily master in the primary grades?

Place value is more complicated that simple groups of groups. Remember, it has mathematical properties but let’s look at these math properties from the point of view of making groups of groups.

We want to look at the place value operations themselves so we might think about a numeral as a representation of a system of quantified inclusions. This quantification of the classes must include the idea that 1 ten could only be considered as a class that includes 9+1 ones. I can't draw it here but we could show this as a hierarchy like a biological classification system in which the class of hundreds includes 9+1 subclasses of ten. Ten includes the ones. The numeral tells us in essence how much of a hierarchy is present and how many separate hierarchies there are. So 234 would tell us that there are 2 hierarchies of the hundreds classes both with their complete subclasses; there would also be 3 more hierarchies each of the tens classes though neither included yet in a hundred class but each having complete ones included in them; and finally there would be 4 individual classes of ones not included in any super class of ten.

Thus, place value operations could be represented as the creation and transformations in the hierarchies. For example with the hierachy I just described for 234, with the addition of more ones, the 9+1 ones could produce an inclusive class of a ten class. The same with the tens. The digit positions in 234 essentially name the highest classes and the number of hierarchies. in each hierarchy showing in the hierarchy. (so the 3rd position position of the 2 means there are hierarchies with three levels going up to the hundreds level and there are two of these hundreds hierarchies). Altogether, there would be 2 hierarchies up to hundreds level, 3 up to the tens level, and 4 up to the ones level.

But a mathematical requirement is that the size of any group at any level must be 9+1. In other words, the student must understand this constancy of the size of grouping that we call the "base." So to understand place value, the child has to quantify the groups, insure they are always exactly 9+1 in every group, and think in terms of how many hierarchies the numeral represents.

Making bean sticks doesn’t capture these mathematical properties very well unless somehow the child figures out that the base has to be constant for the place value operations to work. Grouping when he gets to 9+1 remains just an arbitrary rule whereas borrowing and carrying require the base concept for the operations to work.