Hierarchical inclusion, as explained in the following video, is the concept that a number contains all of the previous numbers. Imagine a number as a Russian nesting doll, if working with the number 4, imagine the largest doll is 4 and inside that doll are the smaller dolls, 3, 2, 1. Without this understanding a child will think that, when counting, the number he points to and names "3" is "3" in of itself without the other objects and when you ask him for "3" he will give you only that object. Hierarchical inclusion also helps students understand other number sense concepts, such as part/whole relationships and compensation because a child cannot understand that 4 can be broken up into the parts 3 and 1 without also understanding that the number "4" contains 3 and 1.
In order to complete the following common core standard, a child must understand hierarchical inclusion:
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
Magnitude is a child's ability to compare groups. Even children who cannot count have the ability to judge the relative size of groups of objects, but as a child's number sense develops, so should the sophistication of her understanding of magnitude. The following Common Core activities depend on a child's understanding of magnitude:
Understand that each successive number name refers to a quantity that is one larger.
Watch the following video. In it, groups of objects quickly flash on the screen. Can you tell how many objects there are in each grouping?
As explained in the following video, subitizing is the ability of a child to quickly recognize a number visually. Children do this by mentally grouping the objects they see (and you probably did this too, for example, seeing a group of 3 and 3 and 3 and knowing that there were 9 objects).
"Looking at a quantity for a short time and then being able to tell how many are in the group(s) without counting each object in the group begins to develop from small sets of two, three, four, and five objects, to parts of sets of six up to twenty. Generally, this development begins between ages 2 and 6. Later, the subitizer sees objects as groups of 10s and 1s and, combined with an understanding of place value, is able to see the numerosity of a large group of numbers quickly." (Copley 2010).
Subitizing strengthens a child's understanding of what numbers mean, and how they relate to one another. In fact, when children are taught to subtize, and their attention is drawn to the groups and patterns they see, it becomes a visual representation of addition and subtraction, as seen here. When strengthening a child's subitizing skills, we are teaching the following Core standards: