Counting
“In the middle of the 20th century, Piaget’s research on number strongly influenced views of early mathematics. Among the many positive influences were an appreciation for children’s active role in learning, and the depth of the mathematical ideas they constructed. “ (Clements & Sarama, 2014) Counting is a complex learning process involving many skills that develop in sequence, students must master a prior skill before moving to the next. Counting involves subitizing, verbal counting, rote counting, cardinality, object counting, one-to-one correspondence, counting on, counting back, skip counting and estimation. Counting, while it sounds simple to adults and older students, is a complex process for young learners developing their number sense. Students must develop their ability to apply the five principles of counting to build mathematical fluency and flexibility with numbers.
“FIVE basic principles define our counting system1:
■ One-to-one principle: There must be one unique tag (i.e., number word or numeral) per item counted.
■ Stable order principle: The tags used must be ordered the same way consistently over time (i.e., we always count in the same sequence, “One, two, three, . . .”).
■ Cardinality principle: The last tag used in the stable order list uniquely represents the cardinal value of the set (i.e., the last number word used answers the question, how many?).
■ Abstraction principle: It does not matter what you are counting as long as you think of them as entities, real or imagined.
■ Order-irrelevance principle: As long as the first two principles are adhered to, it does not matter in what order you count the items (i.e., the tags must be in order, but the items tagged do not).”
(Krasa, Tzanetopoulos, & Maas, 2023)
Subitizing
“The ideas and skills of subitizing start developing very early, but they, as evert other area of mathematics, are not just “simple basic skills”. Subitizing introduces basic ideas of cardinality- “how many”, ideas of “more” and “less,” ideas of parts and wholes and their relationships, beginning arithmetic and, in general, ideas of quantity.” (Clements & Sarama, 2014) Subitizing is recognizing the numerosity or quantity of a group quickly, typically defined as within three seconds. When subitizing, the goal is for students to utilize visual patterns to determine precisely how many are in each group. Perceptual subitizing is the ability to see a specific pattern or group and instantly know how many there are such as playing games with dice and instantly knowing how many dots are present. In contrast conceptual subitizing requires understanding of patterns and being able to connect two quantities together quickly. Subitizing relies on patterns, visual, auditory, fingers and kinesthetic patterns. With practice subitizing becomes an unconscious action. A key element of subitizing is not having to individually count each object. “Which numbers are immediately recognized varies with students experience,” (Small, 2021) as students grow in their mathematical experience and understanding they begin to apply strategies such as grouping to subitize much larger numbers, using consistent patterns such as those seen on a dice or paring assists students in developing their subitizing skills. As students develop their understanding of patterns and ability to count on, they can begin to combine patterns and amounts to subitize larger numbers quickly.
Verbal Counting
Verbal counting comprises at least two separate skills counting requires both knowledge of number order and one to one correspondence. “This brief mathematical description suggests why we use the term “verbal counting” rather than “rote counting”—after 20, children need to use mathematical patterns and structures, not just “rote memory.” Further, children who can count starting at any number performed better on all numerical tasks, so fluent verbal is not rote, but, rather, is based on the recognition that the structure of numbers. There are other reasons. Without verbal counting, quantitative thinking does not develop.” (Clements & Sarama, 2014) . Verbal counting forms a foundation for later mathematical understanding both of number sense and operations.
Verbal counting reinforces the concept of stable order when counting (Krasa, Tzanetopoulos, & Maas, 2023) number words come in a specific order every time one, two, three, four and so on. Mastering these skills marks a big step in students’ ability to understand numbers and begin communicating mathematically. Children first master stable order counting much the same way they learn songs and nursery rhymes, though repetition and counting aloud with others. Learning to count aloud is a social activity for young children, focused work on counting often begins in Early Childhood Education programs typically from as young as two years old, students may count objects aloud with their teacher as well as reciting the number sequence or singing songs that include counting. Children learn the counting words by hearing older children and adults use them and apply them consistently to the child’s world. Children begin to perceive and identify the number words as different from other parts of language around eighteen months, long before the child can begin to say the words.
Object Counting
Object counting is the application of verbal counting to groups of objects. This includes counting with one-to-one correspondence. As students begin to count objects, they may make errors including skipping over objects or double counting them. Over time students begin to count objects with more accuracy and to relate the number they count to the number in the collection. Object counting requires students to apply all the principles of counting.
Cardinality
Cardinality is the understanding that the last number said when counting is the total number in the collection. Once a student develops cardinality they know to stop counting at the final item in the collection and that the final number they said is the total of the collection.
Abstraction
Over time students work on the idea of abstraction when counting. That the counting words do not apply to specific objects but to the number of objects. Students need to be able to apply the number words to a variety of contexts with real or imagined objects and images.
Counting On and Counting Back
Counting on is the ability to start anywhere in the number sequence and continue counting. The ability to count on shows the student has mastered stable order and has a solid understanding of what comes next at any point in the sequence. Counting back stretches a student further than counting on, still relying on stable order but with the added challenge of knowing the counting sequence backwards as well as keeping track of the change or subtraction required to count backwards. Counting on lays a foundation for students to be able to perform addition while counting back lays the groundwork for subtraction.
Skip Counting
Skip counting challenges students understanding of the counting principles, with students “skipping over” specific numbers in the number sequence upsetting the stable order principle. Students need to be taught the skip counting sequences as much as they are taught the regular counting sequence and encouraged to practice skip counting while relating their skip counting to groups of objects. Skip counting lays the foundation for multiplication and division as students move more into operations. “Skip counting is the switch from counting individual items when adding to counting a set as one whole group of a particular amount and adding that to the same amount.” (Tondevold) To skip count effectively students, need to see the whole set as an individual object yet also as containing a number, this requires a great deal of working memory. To effectively skip count students, need to be able to count groups of objects not just recite the counting sequence.