to use, the less common comparative terms such as less, shorter, smaller instead of only hearing or using more, taller, bigger. Initially some children think that less means more because almost all of their experience has been focused on selecting the set with more (e.g., Fuson, Carroll, and Landis, 1996). So children need to hear many examples of fewer and less, although it is not vital that they differentiate these from each other because that is difficult (fewer is used with things you can count, less is used with measured quantities and with numbers). Teachers can also use the comparative terms (for example, bigger and smaller rather than just big and small) so that children gain experience with them, although all children may not become fluent in their use at this level.

Operations

Problems expressed in words (word problems) can now be solved, although many children may need to act out some word problems in order to understand the meanings of the situation or of some of the words (see research summarized in Fuson 1992a, 1992b). Through such experiences relating actions and words, children gradually extend their vocabulary of words that mean to add—in all, put together, altogether, total—and of words that mean to subtract—are left, take away, eat, break. Discussing and sharing solutions to word problems and acting out addition/subtraction situations can provide extended experiences for language learning. Children can begin posing such word problems as well as solving them, although many will need help with asking the questions, the most difficult aspect of posing word problems. As with all language learning, it is very important for children to talk and to use the language themselves, so having them retell a word problem in their own words is a powerful general teaching strategy to extend their knowledge and give them practice speaking in English.

Drawing the solution actions using circles or other simple shapes instead of pictures of real objects can be helpful. The two addends can be separated just by space or encircled separately or separated by a vertical line segment. Some children can also begin to make mathematical drawings to show their solutions. Teacher and child drawings leave a visual record of the full solution that facilitates children’s reflecting on the solution, as well as discussing and explaining it. For children, making math drawings is also a creative activity in which they are somehow showing in space actions that occur over time. Children do this in various interesting ways that can lead to productive discussions.

Children also become able to use their fingers to add or to subtract using the direct modeling solution methods counting all or taking away (see Box 5-11, Level 1). When counting all, they will count out and raise fingers for the first addend, then for the second addend, and then count all



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