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word is said, and (2) the matching in space where the counting action points to an object once and only once. Children initially make errors in both of these kinds of correspondences (e.g., Fuson, 1988; Miller et al., 1995). They may violate the matching in time by pointing and not saying a word or by pointing and saying two or more words. They may also violate the matching in space by pointing at the same object more than once or skipping an object; these errors are often more frequent than the errors in time.

Four factors strongly affect counting correspondence accuracy: (1) amount of counting experience (more experience leads to fewer errors), (2) size of set (children become accurate on small sets first), (3) arrangement of objects (objects in a line make it easier to keep track of what has been counted and what has not), and (4) effort (see research reviewed in Clements and Sarama, 2007, and in Fuson, 1988). Small sets (initially up to three and later also four and five) can be counted in any arrangement, but larger sets are easier to count when they are arranged in a line. Children ages 2 and 3 who have been given opportunities to learn to count objects accurately can count objects in any arrangement up to 5 and count objects in linear arrangements up to 10 or more (Clements and Sarama, 2007; Fuson, 1988).

In groundbreaking research, Gelman and Gallistel (1978) identified five counting principles that stimulated a great deal of research about aspects of counting. Her three how-to-count principles are the three mathematical aspects we have just discussed: (1) the stable order principle says that the number word list must be used in its usual order, (2) the one-one principle says that each item in a set must be tagged by a unique count word, and (3) the cardinality principle says that the last number word in the count list represents the number of objects in the set. Her two what-to-count principles are mathematical aspects we have also discussed: (1) the abstraction principle states that any combination of discrete entities can be counted (e.g., heterogeneous versus homogeneous sets, abstract entities, such as the number of days in a week) and (2) the order irrelevance principle states that a set can be counted in any order and yield the same cardinal number (e.g., counting from right to left versus left to right).

Gelman took a strong position that children understand these counting principles very early in counting and use them in guiding their counting activity. Others have argued that at least some of these principles are understood only after accurate counting is in place (e.g., Briars and Siegler, 1984). Still others, taking a middle ground between the “principles before” view and the “principles after” view, suggest that there is a mutual (e.g., iterative) relation between understanding the count principles and counting skill (e.g., Baroody, 1992; Baroody and Ginsburg, 1986; Fuson, 1988; Miller, 1992; Rittle-Johnson and Siegler, 1998).

Each of these aspects of counting is complex and does not necessarily exist as a single principle that is understood at all levels of complexity at

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