Hierarchical Inclusion - A Building Block of Number Sense

I've been working with several districts on the building blocks of number sense.  This includes concepts such as, subitizing, unitizing, hierarchical inclusion, compensation, etc.  So what is hierarchical inclusion? 

Hierarchical Inclusion is a counting principle, sometimes referred to as the cardinal principle.  It is the understanding that numbers are nested inside of each other and that the number grows by one each count.  For example 19 is inside of 20 or 20 is the same as 19 + 1.  If you remove one, the number goes back to 19.

Labinowics stated, "Beyond labeling individual objects in a collection with a name, counting eventually involves a further mental act of relating the individual objects into wholes of increasing size."

"Relating the individual objects into wholes" is what Richards, Steffe, and Von Glasserfeld refer to as "constructing the number as a unit."  The child is able to "see" the number as a unit while at the same time "seeing" it made up of it's "parts." 

The number labeling how many objects in a group includes all the objects in the group.  For instance, "four" indicates the total number of cubes, rather than just the cube they labeled "four" (or the fourth curbe).   

I thought all students understood this concept.  I was shocked when I was working with a kindergarten student and had them count five teddy bears.  When I asked her to take away three bears, she picked up the bear she labeled "three" when she was counting.  Wow, this student did not understand that three bears included the bears she had named one and two as well.  The quantity of three was not secure for her or the concept of hierarchical inclusion.

Another thing to watch for with your students is the ability to count on when they can count a group of objects and continue from the total when an additional group of objects are added or subtracted, without needing to start at one again.

For more ideas to work with the concept of Hierarchical Inclusion, check out my weekly math idea. If you have other ideas, I would love to hear your ideas.