Counting and Cardinality & Operations and Algebraic Thinking are about understanding and using numbers. Counting and Cardinality underlies Operations and Algebraic Thinking as well as Number and Operations in Base Ten. It begins with early counting and telling how many in one group of objects.
See the standards throughout the grades.
See this progression for Counting and Cardinality beginning in Kindergarten.
Overall Purpose: Establish early number sense development. This is the foundation for all the mathematics that students will be learning. This content helps future teachers diagnose and assess student understanding.
N1.a. Differentiate amongst digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), numerals (symbolic ways of representing quantities), and numbers (quantities).
N1.b: Recognize when students are displaying understanding of various components of the learning path for counting:
Ability to perceptually subitize regular patterns up to 5 and irregular patterns up to 4, transitioning to conceptual subitizing up to 10.
Ability to demonstrate principles of counting :
Stable Order Principle
One-to-one Correspondence
Cardinality
Order Irrelevance Principle
Abstraction
Conservation of number
Ability to apply conservation of number to counting on and identifying which number comes before or after
Ability to demonstrate understanding of the Hierarchical Principle of Numbers. (Understanding that all numbers preceding a number can be or are systematically included in the value of another selected number. For example, knowing that within a group of 5 items, there is also a group of 4 items within that group; 3 items within that group, etc.)
N1.c: Identify the relationship between counting and the concept of larger and smaller numbers.
Counting is Complex: Teacher candidates need to be able to understand the distinct differences between recognizing numerals, rote counting sequences, counting principles, and subitizing to support students as they build a solid foundation for early number sense.
Subitizing: Early childhood students come to quickly recognize the cardinalities of small groups without having to count the objects; this is called perceptual subitizing. 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”). Use of conceptual subitizing in adding and subtracting small numbers progresses to supporting steps of more advanced methods for adding, subtracting, multiplying, and dividing single-digit numbers (Counting and Cardinality Progressions, 2011).