Part I: Mathematics Everyday and Everywhere
Part II: Mathematics Content for Young Children

Counting and Cardinality

 "How many pennies do you have?" 

"One - two - three - four - five - six. Six!" 

Learning to count is a remarkable accomplishment for a young child. Counting, or producing sets of a given size, is not the concrete act we often assume it to be. There is nothing in a number word like “six” to suggest a specific sized collection. How can six pennies be the same quantity as six elephants? We can also count six jumping jacks and six drumbeats, but these six actions and sounds are gone the moment they occur. Connecting number names to quantities involves remembering number names in the correct order, connecting the count sequence in one-to-one correspondence to counting objects, understanding the principle of cardinality—that the last number word said tells “how many” objects there are (not just the name of the sixth object)—and realizing that quantity does not change when objects are moved around and recounted. On top of that, children are also expected to recognize that the numeral “6” and the written word “six” are also representations of the same quantity. While a preschooler might claim, “I can count to 20” or 50 or 100, they are likely referring only to one piece of the counting puzzle— knowing the number names in correct sequence while counting forwards. Early educators know that in order for children to claim they can count to 20 meaningfully, they need an understanding of the multiple connections between number names and their symbols and quantities.  

Understanding the research and theories behind counting and cardinality provides teachers with important insights into many of the taken-for-granted processes involved. In several of the articles, the counting principles and learning expectations are revealed through discussion and supporting activities (see Baroody and Benson 2001). In Griffin’s (2003) article, she identifies three important objectives in a preschool and kindergarten program along with practical examples, including (1) knowing the number names, (2) connecting those number names to a particular set size, and (3) realizing the inclusion principle that the set size increases by one for each successive number. Clement’s (1999) article on subitizing provides the history of its research and why it is an essential introduction to cardinality and conservation of number. Clements’ article here and Clements and Sarama (2008) in chapter 1 both point to the increasing levels of difficulty when subitizing objects arranged in a straight line or familiar pattern compared to a random arrangement. Cain and Faulkner (2012) address the key issue that “t-h-r-e-e is not three.” That is, the word “three” and the numeral “3” are abstract representations of the quantity three; they are verbal and written symbols assigned to a quantity to help us communicate about it. There is nothing inherent in either of these symbols that suggests the quantity three.   

A practical application of the principles of counting and cardinality (as well as the other aspects of early number) is found in the article of Moomaw et al. (2010). They describe an assessment tool that sidesteps the challenges of traditional assessments for preschoolers by situating the assessment in a motivating game context.