once. Children may initially produce the first several number words and not even separate them into distinct words (Fuson, Richards, and Briars, 1982). They may think that they need to say the number word list in order as they count, but early on they cannot realize the implication that they need a unique last counted word, or they would not repeat words so frequently as they say the number word list.

The what-to-count principles also cover a range of different understandings. It takes some time for children to learn to count parts of a thing (Shipley and Shepperson, 1990; Sophian and Kailihiwa, 1998), a later use of the abstraction principle. And the order irrelevance principle (counting in any order will give the same result) seems to be subject to expectations about what is conventional “acceptable” counting (e.g., starting at one end of a row rather than in the middle) as well as involving, later on, a deeper understanding of what is really involved in 1-to-1 correspondence: Count-

BOX 5-5

Common Counting Errors

There are some common counting errors made by young children as they learn the various principles that underpin successful counting. Counting requires effort and continued attention, and it is normal for 4-year-olds to make some errors and for 5-year-olds to make occasional errors, especially on larger sets (of 15 or more for 4-year-olds and of 25 or more for 5-year-olds). Younger children may initially make quite a few errors. It is much more important for children to be enthusiastic counters who enjoy counting than for them to worry so much about errors that they are reluctant to count. If one looks at the proportion of objects that receive one word and one point, children’s counting often is pretty accurate. Letting errors go sometimes or even somewhat frequently if children are trying hard and just making the top four kinds of errors is fine as long as children understand that correct counting requires one point and one word for each object and are trying to do that. As with many physical activities, counting will improve with practice and does not need to be perfect each time. Teachers do not have to monitor children’s counting all of the time. It is much more important for all children to get frequent counting practice and watch and help each other, with occasional help and corrections from the teacher.

Very young children counting small rows with high effort make more errors in which their say-point actions do not correspond than errors in the matching of the points and objects. Thus, they may need more practice coordinating their actions of saying one word and pointing at an object. Energetic collective practice in which children rhythmically say the number word list and move down their hand with a finger pointed as each word is said can be helpful. To vary the practice, the words can sometimes be said loudly and sometimes softly, but always with emphasis (a regular beat). The points can involve a large motion of the whole arm or a smaller motion, but, again, in a regular beat with each word. Coordinating these actions



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