Developing Cardinality
Learning to count is an essential skill for daily life. For example, children count to make sure they have both shoes, communicate their age with their hands, and even make sure they get the same number of cookies as their siblings (Sadler, 2009). I used to think that children who can recite numbers 1-10 in order could count up to ten, but my clients taught me that sorting, counting, and finding the quantity of a set of items is a separate, complex cognitive process.
Counting requires procedural skills and conceptual understanding. Sadler (2009), a specialist in early childhood mathematics in special education, explains that counters must not only follow the procedure of saying number words in the correct order, but also demonstrate one-to-one correspondence by saying only one counting word as they point to each item. One-to-one correspondence is also known as “tagging” because it requires counters to assign one number to each item. Lastly, children must understand conceptually that when they count correctly, the final number is the total number of the set. This concept is called cardinality. Children who understand cardinality can count the number of items in a set and answer the question, “How many?”
How do children develop the skills needed for cardinality? Saddler (2009) synthesized research on the development of cardinality to construct a developmental trajectory. Both special learners and typically developing children usually acquire cardinality skills in the following order:
The above chart has greatly helped me when working on one-to-one correspondence or cardinality goals with young clients. This trajectory helps me assess where clients are in this developmental process, as well as create a task analysis for clients.
Music therapists can use counting songs to keep clients engaged and motivated while developing their cardinality skills. In her article, “The Who and What of Counting,” Plata (2017) writes, “For centuries, children’s rhymes, games, and songs have set the stage for the seemingly effortless acquisition of verbal counting. Young children happily sing songs in which ducks disappear and monkeys fall off beds.” Music therapists can adapt counting songs, such as “Who Stole the Cookies from the Cookie Jar,” “10 Little Racing Cars,“ “5 Little Ducks,” “10 Little Racing Cars,” “The Ants Go Marching,” “5 Little Apples/Cookies,” and “10 Little Monkeys Jumping on the Bed,” to help clients move through the stages of cardinality, or even write an original song.
The biggest tip I took away from Sadler’s (2009) article is the importance of understanding how counting errors can lead to an incorrect quantity. Sandler recommends asking children in stages 4 and 5 to check their answers by counting again. If children reach a different answer when checking their work, adults can ask questions such as, “I wonder which answer is right? How many are there for sure? How can we figure it out? Why do you think you got a different answer that time?” (Sadler, 2009). Asking children to reason their way to the correct answer prepares them for a lifetime of learning.
– Molly, Music Therapy Intern, who never thought she would teach math
References
Platas, L. M. (2017). The What and Why of Counting [Web page]. Development and Research in Early Math Education. Retrieved from: https://dreme.stanford.edu/news/why-and-what-counting
Sadler, F. H. (2009). Help! They Still Don’t Understand Counting. TEACHING Exceptional Children Plus, 6(1), n.p.
