A few weeks ago, we shared some suggestions for strategies you can use to support students’ growing understanding of counting and cardinality (link here). This week, we’re going to focus on how these skills develop and how you can individualize your support for growing learners who are developing skills at different rates.
How Do Counting and Cardinality Skills Typically Develop?
Around age four, we see students’ skills developing in both counting and cardinality, and they’re working hand in hand. Preschoolers can typically count to 10, and can count up to 5 objects arranged in a line. And, when they’re finished counting and asked how many there are, they can say 5 without having to recount. Not only can four-year-olds count a group of given objects, but they can also produce a group of up to 5 objects. Producing the correct number of objects is a complex skill because it requires coordinating both counting and cardinality! In order to arrive at the correct amount, students must use one-to-one correspondence and understand the significance of the counting as it relates to producing the correct quantity of objects (cardinality).
As students reach age five and head into kindergarten, their counting and cardinality skills continue to grow. They can often count to twenty and can produce a group of up to 10 objects. Some students can do these things with as many as 30 objects! Most 5-year-olds have also developed some strategies to keep track of what objects have and have not been counted, like moving objects from one pile into another as they count.
As students move into first grade, they can count up to 200, and count backwards from 20. They can also “count on”, which means they can start counting at a number other than one. They can provide the “neighboring” number, which is the number that comes immediately before or after a given number (like knowing that fourteen comes before fifteen). And, they can count in groups rather than one-by-one (like counting a group of 10 objects by groups of 2 or by groups of 5).
Individualizing Supports for Students’ Counting and Cardinality Skills
The information shared above outlines the typical development of students’ counting and cardinality skills. However, not all students develop at the same rate and in the same ways. Here we focus on one of the counting skills students are developing: knowing number names and the counting sequence.
Students who are having difficulty learning the counting sequence may make several different types of errors including: saying the sequence out of order, skipping numbers, or using the same number more than once. Below we show examples of each of these types of errors along with strategies to support students’ understanding of the counting sequence.
|Type of Counting Error||Examples||Remedy|
|Saying the number sequence out of order, skipping numbers, or using the same number more than once.
Struggling with the count sequence past twelve
|“1, 2, 3, 6, 10”||Practice reciting (or singing) the single-digit sequence, first focusing on one to ten, then later moving on to numbers greater than ten.|
“1…13, 14, 16, 17, 18.”
|Highlight and practice exceptions, such as fif + teen. Fifteen and thirteen are commonly skipped because they are irregular.|
|Uses incorrect words:
“1…13, 14, fiveteen”
“1…18, 19, 10-teen” or
“1…29, 20-ten, 20-eleven”
|Recognize that a nine signals the end of a series and that a new one needs to begin (e.g., nineteen marks the end of the teens.)
Recognize that each new series (decade) involves combining a decade and the single-digit sequence, such as twenty, twenty plus one, twenty plus two, etc.
|Stops at a certain number:
“1…20” (starts from 1 again). Click here to see a video example of this type of counting error where a student stops at twenty-nine.
|Recognize the decade term that begins each new series (e.g., twenty follows nineteen, thirty follows twenty-nine, and so forth). This involves both memorizing terms such as ten, twenty, and thirty by rote and recognizing a pattern: “add –ty to the the single-digit sequence” (e.g., six + ty, seven + ty, eight + ty, nine + ty).|
|Table Reproduced from D. Frye, A. J. Baroody, M. Burchinal, S. M. Carver, N. C. Jordan and J. McDowell, Teaching Math to Young Children, A Practice Guide (NCEE 2014-4005) (Washington DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education, 2013).|
Click here for a more complete table that includes other common counting errors, and strategies for scaffolding learners who might need more support with the development of their counting and cardinality skills.
Knowing what students’ typical development of counting and cardinality skills looks like, and signs that a student might be struggling with their development of these skills, will help you provide appropriate supports for all learners.
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