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.
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One-to-one Principle: When counting, only one number
word is assigned to each object. This principle is obviously
similar to the concept of one-to-one correspondence, only
as it specifically ties to both the verbal and mental act
of counting.
Stable Order Principle: When counting, number words
are always assigned in the same order. Though the tie of
number to language is important, exercises that employ stable
order are most useful when they simultaneously employ the
one-to-one principle, or the concept of one-to-one correspondence
at some level.
Cardinal Principle: The number of objects in the set
is the last number word counted. As with the one-to-one
principle, the cardinal principle is similar to the concept
of cardinality, of which children gain implicit (even nascent)
understandings long before they grasp the notion of numerical
quantity (Siegler, 2003; Sophian, 1987). See the Research
Précis - Early Numeracy: Initial Cardinality.
Order Irrelevance Principle: When counting the number
of objects in a set, the order in which they are counted
is not important, but rather simply that all objects are
counted. In other words, a set of objects may be properly
counted by starting with any object and going in any order.
Abstraction Principle: When counting any unique collection
or set of objects, the above principles all apply. When
counting a set of green dots, for example, the same principles
apply as when counting a set of dolls.
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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:
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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);
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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);
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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;
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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;
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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);
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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;
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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
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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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