Counting and Cardinality

 

The 2009 report of the Committee on Early Childhood Mathematics, National Research Council titled, Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, identifies four mathematical aspects of the number core below.  This “core” related to the content area of “number” is what we are focusing on in the early years.

THE NUMBER CORE
1.  Cardinality: Children’s knowledge of cardinality (how many are in a set) increases as they learn specific number words for sets of objects they see (I want two crackers).
2.  Number word list: Children begin to learn the ordered list of number words as a sort of chant separate from any use of that list in counting objects.
3.  1-to-1 counting correspondences: When children do begin counting, they must use one-to-one counting correspondences so that each object is paired with exactly one number word.
4.  Written number symbols: Children learn written number symbols through having such symbols around them named by their number word (Sees “2” and says “that is a two”).

Counting and Cardinality was identified in the Kindergarten Common Core State Standards as the foundation for learning about math. In earlier research, Number and Operations was identified as the most important content area to a young child’s mathematical learning in the early childhood years. Counting and Cardinality were an integral piece of this topic.  The TN-ELDS also referred to counting and cardinality under the component of Number and Operations. It is important to recognize the core of Number and Operations is Counting and Cardinality.  Without this core foundation, students will have difficulty meeting the standard for Number and Operations in Base Ten in kindergarten.  Keep this mind when reading earlier research that did not differentiate between counting and cardinality and number and operations.

Curriculum Focal Points


The National Council of Teachers of Mathematics (NCTM) developed Curriculum Focal Points in 2006 for all grade levels, including pre-kindergarten.  Curriculum focal points  identify the most Important mathematical topics for each grade level. They comprise related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning. They are the topics that should be considered as the basis for decisions about curriculum development. (www.nctm.org). It is important that these focal points be addressed in learning experiences that also promote the process skills: problem-solving, reasoning, communication, making connections, and designing and analyzing representations.

Pre-Kindergarten Curriculum Focal Points


Numbers and Operations: Developing an understanding of whole numbers, including concepts of correspondence, counting, cardinality and comparison.

Children develop an understanding of the meanings of whole numbers and recognize the number of objects in small groups without counting-----the first and most basic mathematical algorithm. They understand that the last word that they state in counting tells “how many”, they count to determine number amounts and compare quantities (using language such as “more than” and “less than”), and they order sets by the number of objects in them.

Teachers must possess the mathematical knowledge to understand how students learn mathematics as well as an understanding of the trajectory of mathematics content. Understanding the trajectory enables us to effectively plan and make instructional decisions to impact important mathematical learning. This continuum contains K-3 Number and Operations in Base Ten standards, Number and Operations – Fractions standards for grade 3, Counting and Cardinality standards for K, and Number and Operations and Algebraic Thinking standards K-3.

Tennessee’s Early Learning Developmental Standards


Tennessee’s Early Learning Developmental Standards (TN-ELDS) identified Learning Expectations and Performance Indicators aligned to the math area of Number and Operations as represented below.

4 to 5 years (49 through 60 months)
Component Learning Expectations
(examples)
Performance Indicators
(examples)
By the end of the age span
Correlations with K Common Core
Standards
Number and Operations Begins to identify and label objects using numbers Develops increased abilities to combine, separate and name “how many” concrete objects
  Develops understanding of numbers and their association with objects Begins to associate number concepts, vocabulary, quantities, and numerals in meaningful ways
    Develops increasing ability to count in sequence to 10 and beyond

Excerpted from TN Early Childhood Curriculum Standards http://tn.gov/education/ci/earlychildhood/sec3math.pdf 

Learning Paths and Teaching Strategies


Another resource for thinking about what and how you should teach number and operations to 4-5 year old children is the Learning Paths chart below that aligns knowledge with sample teaching strategies.

Content Area Examples of Typical Knowledge and Skills
From Age 3
Age 6 Sample Teaching Strategies
Number and Operations Counts a collection of one to four items and begins to understand that the last counting word tells how many.   
Counts and produces (counts out) collections up to 100 using groups of 10. Models counting of small collections and guides children’s counting in everyday situations, emphasizing that we use one counting word for each object:      

“One . . . two . . . three . . .”
Models counting by 10s while making groups of 10s (e.g., 10, 20, 30 . . . or 14, 24, 34 . . . ).
  Quickly “sees” and labels collections of one to three with a number.   
Quickly “sees” and labels with the correct number “patterned” collections (e.g., dominoes) and unpatterned collections of up to about six items.   
Gives children a brief glimpse (a couple of
seconds) of a small collection of items and asks how many there are.
  Adds and subtracts nonverbally when numbers are very low. For example, when one ball and then
another are put into the box, expects the box to contain two balls.
Adds or subtracts using counting-based strategies such as counting on (e.g.,
adding 3 to 5, says “Five . . . , six, seven, eight”), when numbers and totals do not
go beyond 10.
Tells real-life stories involving numbers and a problem. Asks how many questions (e.g., “How many are left?” “How many are there now?” “How many did they start with?” “How many were added?”).

Shows children the use of objects, fingers, counting on, guessing, and checking to solve problems.


Excerpted from Learning Paths and Teaching Strategies Chart in Promoting Good Beginnings:  Early Childhood Mathematics. NAEYC/NCTM Joint Position Statement, 2002, updated 2010.