K.CC.B.4b
Understand the relationship between numbers and quantities; connect counting to
cardinality.
4.b Understand that the last number name said tells the number of objects counted.
The number of objects is the same regardless of their arrangement or the order in
which they were counted.
Unpacked
This standard calls for students to answer the question “How many are there?” by
counting objects in a set and understanding that the last number stated when counting a
set (...8, 9, 10) represents the total amount of objects: “There are 10 bears in this pile”
(cardinality). It also requires students to understand that the same set counted three
different times will end up being the same amount each time. Thus, a purpose of keeping
track of objects is developed. Therefore, a student who moves each object as it is counted
recognizes that there is a need to keep track in order to figure out the amount of objects
present. While it appears that this standard calls for students to have conservation of
number, (regardless of the arrangement of objects, the quantity remains the same),
conservation of number is a developmental milestone of which some Kindergarten
children will not have mastered. The goal of this standard is for students to be able to
count a set of objects; regardless of the formation those objects are placed.
Since an important goal for children is to count with meaning, it is important to have
children answer the question, “How many do you have?” after they count. Often times,
children who have not developed cardinality will count the amount again, not realizing
that the 10 they stated means 10 objects in all.
Young children believe what they see. Therefore, they may believe that a pile of cubes
that they counted may be more if spread apart in a line. As children move towards the
developmental milestone of conservation of number, they develop the understanding that
the number of objects does not change when the objects are moved, rearranged, or
hidden. Children need many different experiences with counting objects, as well as
maturation, before they can reach this developmental milestone.
Prior to reaching this understanding, a student who is asked “How many kittens?” may
regard the counting performance itself as the answer, instead of answering with the
cardinality of the set. Experience with counting allows students to discuss and come to
understand the second part of K.CC.4b—that the number of objects is more advanced
counting-on methods in which a counting word represents a group of objects that are
added or subtracted and addends become embedded within the total.
One idea to understand is the idea of subitizing. Students come to quickly recognize the
cardinalities of small groups without having to count the objects; this is called
perceptual subitizing, to recognize a small amount without any learned mathematical
knowledge. Perceptual subitizing develops into conceptual subitizing —recognizing that
a collection of objects is composed of two subcollections and quickly combining their
cardinalities to find the cardinality of the collection (e.g., seeing a set as two subsets of
cardinality 2 and saying “four”).
Subitizing is foundational to children’s number sense. “Children use counting and
patterning abilities to develop conceptual subitizing. This more advanced ability to group
and quantify sets quickly in turn supports their development of number sense and
arithmetic abilities”. (Clements, 1999, pg. 401)