ships can help children overcome the second kind of typical error in writing teen numbers, in which children write first what they say first. They hear eighteen and know that teens have a 1 in them (they may not yet think of this as one ten) and so they write 81.

Kindergarten children can also experience and learn all of the decade words in order from 20 to 100. Doing so while looking at a list of these number symbols grouped in tens can help to reinforce the pattern of the groups of ten.

Many states require that kindergarten children understand some aspects of money, but sometimes they have goals that are not sensible for this age group, even children who have had strong earlier mathematical experiences. The mathematical aspects of money that are most appropriate are the groups of ten pennies in dimes and the groups of five pennies in nickels. Children have been working with these cardinal groups of tens in this level and with 5-groups in the 4-year-old/prekindergarten level, so it is easy to build this understanding by extending this knowledge to coins by using any visual support that relates a 5-group of pennies to one nickel and one 10-group of pennies to one dime. Such supports were used successfully for first graders to construct the relationships for understanding two-digit numbers described next for first graders (Fuson, Smith, and Lo Cicero, 1997; Hiebert et al., 1997).

Learning the values of a dime and a nickel are of course particularly complicated because their values are not in the order of the sizes of the coins. In size, a dime < a penny < a nickel, but in value a penny < a nickel < a dime. For this reason, it is too difficult to work with these coins alone rather than with visual supports that show the values of these coins in pennies, as discussed above. Counting mixed collections of dimes, nickels, and pennies requires shifting counts from counting by tens when counting dimes to counting by fives when counting nickels to counting by ones when counting pennies. Such shifts are too complex for many children at this level, especially if they are looking at the coins rather than looking at their values as pennies. Practice just on the names of the coins and on their visual features, rather than on their value as ones, fives, or tens, is also not appropriate. It is the quantitative values that are mathematically important.

Step 4 (Grade 1)

At this step children see, say, count, and write tens and ones from 1 to 100 (see Box 5-9). To do this, they build on the integrations among cardinality, counting, and written number symbols that they have made in kindergarten. The major advance has two parts. First, children learn to count by two different units, units of ten and units of one. Second, they learn to shift from counting by units of ten to counting by units of one so that they can count cardinal sets up to 100. Children who have mastered



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