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Count With Me!

Activities that involve counting have been shown to be very effective for helping young children understand the concept of number. From evidence that children form many necessary language associations at a very early age (Fuson, Richard, & Brials, 1982) to findings that even at the age of three certain rational counting principles were in place (Aubrey, 1993; Gelman & Meck, 1983; Schaeffer, Eggleston, & Scott, 1974), we have determined that young children are prepared to engage and benefit from preschool exposure to counting. Though these examples are selective, they reflect an extensive corpus of research evidence. A common thread is that children can make effective use of guided experiences that help them build developmentally appropriate pre-formal mathematics understandings. Counting exercises provide an avenue for presenting these challenges and opportunities in a manner that can be scaled to meet a child's need where he/she is in understanding. To determine how best to use counting to reinforce and extend children's natural learning, it is helpful to explore how counting addresses the types of important precursors to formal mathematics understanding outlined in Early Childhood Numeracy. Widely cited research by Gelman and Gallistel (1978) resulted in a set of counting principles that bear a remarkable resemblance to those pre-formal conceptions, and found as well that counting exercises emphasizing these principles contributed greatly to children's pre-formal understanding and progress toward formal understanding.

Principles of counting

Gelman and Gallistel's principles are unique in that all are fully attainable by age five and some by age three, and that they do not refute but rather extend Piaget's findings. Many counting exercises that emphasize these principles also employ the types of logical activities recommended by Piaget—classification, seriation (ordering things by size), matching and comparison—for developing awareness of number properties as a foundation for understanding number concepts (Piaget & Szeminska, 1952), as follows.

When counting with your child...

To ensure that children get the highest benefit possible from counting exercises, and that the focus is on pre-formal development of number concept rather than rote or memorization, there are a number of considerations to keep in mind. These also underscore the importance of many findings throughout the years, including early work by Piaget and colleagues through those considerably more recent. In addition to the clear need to simply emphasize the things made explicit in the principles, in general, when counting with your child, make sure he/she:

- recites the sequence of counting words up to the required number and in the correct order. Research indicates that most three-year-old children can count in English to the numeral 5, and five-year-olds can count through 10 or more (Jedrysek, 2000);

- always assigns a number word for each object and avoids repeating or assigning the same number word for two or more objects. This should be addressed both verbally, and when appropriate in writing (note as well that the ability to "make the symbol" in writing will occur later than the ability to correctly say the word);

- learns to count carefully. Slow him/her down when necessary, and realize that it is common for the child to become impulsive and to rush, even more so as counting proceeds, which will result in "skipping" words or objects;

- slowly progresses to counting a set of objects without regard for which one is counted first, and to applying principles of counting to a variety of objects with different attributes;

- begins over time to establish the understanding that the final number word counted for a set of objects represents the total number of objects (cardinality), and typically later, that a certain number word in a series represents a certain object in the series, such as the third block in a line of five (ordinality);

- is provided developmentally appropriate opportunities to reason logically with objects being counted—to match, classify, order, and compare in a way that progressively extends overall understanding rather than hinders early counting progress;

- coordinates the assignment/recitation of each number word with the physical act of either moving, touching with a finger, or at least pointing at the object it represents. When this activity is carried out kinesthetically, children are implicitly exposed to the concept of one-to-one correspondence from the earliest possible age; and

- can easily see and physically deal with objects. Even very minor physical disabilities, or lack of eye-hand coordination or other fine motor control, can pose problems for young children that have otherwise indicated understanding of various counting principles. Where these situations exist, don't hesitate to make adjustments (e.g., making the objects bigger, easier to see, even holding or touching the objects with the child).

Finally, it is important that teachers and parents of preschool children be thoughtful when planning to incorporate logical skill development activities of the type recommended by Piaget—matching, classification, ordering, and comparing—with counting exercises. This relates to the general suggestion (see Early Childhood Numeracy) that addressing more than one understanding in tandem is often overwhelming. Cognitively speaking, the whole of these exercises is often far greater than the sum of its parts. For example, when children are in the early stages of understanding the principles of counting, consider equally spacing objects in a row prior to counting, and ensure that objects to be counted are the same or at least very similar to one another. If you wish a very young child to sort (early classification), count and compare the number of objects of different colors in a group, consider working on sorting first, then proceed to counting, and finally to comparing. To avoid disrupting a childÕs progress due to his/her lack of ability to conserve number—realization of equivalence regardless of configuration (see the Research Précis - Early Numeracy: Counting and Conservation)—configure the objects in each set identically, then count and compare. There is always time and opportunity to extend the complexity and required level of reasoning by combining and integrating understandings as the child progresses in number understanding.

Aubrey, C. (1993). An investigation of the mathematical competencies which young children bring into school. British Educational Research Journal, 19(1), 27-41.

Fuson, K., Richard, J., & Brials, D. (1982). The acquisition and elaboration of the number word sequence. In C. Brainerd (Ed.), Children's logical and mathematical cognition: Progress in cognitive development research. New York: Springer-Verlag.

Gelman, R., & Gallistel, C. (1978). The child's understanding of number. Cambridge, MA: Harvard University Press.

Gelman, R., & Meck, E. (1983). Preschoolers' counting: Principles before skill. Cognition, 13, 343Ð359.

Jedrysek, E. (2000). Number concept development in young children. In S. Vig, & R. Kaminer (Eds.), Early Intervention Training Institute Newsletter (pp. 1-3). Bronx, NY: Rose F. Kennedy Center.

Piaget, J., & Szeminska, A. (1952). Child's conception of number. London: Routledge & Kegan Paul.

Schaeffer, B., Eggleston, V., & Scott, J. (1974). Number development in young children. Cognitive Psychology, 6, 357-379.

Siegler, R. (2003). Implications of cognitive science research for mathematics education. In J. Kilpatrick, W. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 219-233). Reston, VA: National Council of Teachers of Mathematics.

Sophian, C. (1987). Early developments in children's use of counting to solve quantitative problems. Cognition and Instruction, 4, 61-96.

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