. "5 The Teaching-Learning Paths for Number, Relations, and Operations." Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Washington, DC: The National Academies Press, 2009.
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Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity
4, 5, 6, 7, 8, 9, and 10. However, the 10 at this level means ten ones, the counted number ten that comes after nine. Not until the next level does it come to mean what the 10 symbols actually say: 1 ten and 0 ones. Children at this level can begin to write some numerals, often beginning with the easier numerals 1, 3, 4, and 7.
Counting Out “n” Things
Children at this level make one major conceptual advance. They move from knowing that the last number stated represents the amount in the group to knowing how to count out a given number of objects (Clements and Sarama, 2007; Fuson, 1988). Lots of counting of objects and saying the number word list enables their counting to become fluent enough that they can count out a specified number of things, for example, count out 6 things. Counting out n things requires a child to remember the number n while counting. This is more difficult for larger numbers because the child has to remember the number longer. So children may initially count past n because their counting is not fluent (overlearned) enough to count a long sequence of words, remember a number, and monitor with each count whether they have reached the number yet. Counting out a specified number is needed for solving addition and subtraction problems and for doing various real-life tasks, so this is an important milestone. Children can practice this conceptual task by counting out n things for various family and school purposes; such practice can also occur in game-like activities.
Counting out n things also requires a conceptual advance that is the reverse of learning that the last count word tells how many there are. To count out 6 things, a child is being told how many there are (a cardinal meaning) and must then shift to a count meaning of that 6 in order to monitor the count words as they are said (Have I said 6 yet?) so that they can stop when they say 6 as a counting word that corresponds to one object. They then have the set of 6 things they need.
Step 3 (Kindergarten)
At this step children work to integrate all of the core components of number. They are able to see that teen numbers are made up of tens and some ones. They also can come to understand that ten ones make one group of ten (see Box 5-8).
Kindergarten children can begin the process with seeing and making tens in teen numbers, and first graders can continue the process for tens and ones in numbers 20 to 100. At both grades this process helps children integrate the number components into a related web of cardinal, counting, and written number symbol knowledge. The first conceptual step is for chil-