ing effectively (see research reviewed in Fuson, 1988, 1992a, 1992b, and in Clements and Sarama, 2007, 2008).

Operations: Addition and Subtraction

The 2- and 3-year-old children can solve change plus/change minus situations and put together/take apart situations with small numbers (totals ≤ 5) if the situation is presented with objects or if they are helped to use objects to model these situations (Clements and Sarama, 2007; Fuson, 1988). Children can have experience in learning how to do such adding and subtracting from family members, in child care centers, and from media such as television and CDs. Children may subitize groups of one and two or count these or somewhat larger numbers. To find the total, they may count or put together the subitized quantity into a pattern that is also just seen and not really counted (e.g., two and two make four).

Step 2 (Age 4 or Prekindergarten)

At this step, children learn to use conceptual subitizing and cardinal counting to solve situation, word, and oral number word problems with totals ≤ 8 and begin to count and to match to find out which set has more or less (see Box 5-10).

Cardinal counters at this age level can extend their understanding of relations and of all of the addition/subtraction situations and generalize them to a wider range of settings because their real-world knowledge is more extensive than it was at the previous level. Children can now also count out a specified number of objects, so they can carry out the count all and take away solution methods (Level 1 in Box 5-11) for numbers in their counting accuracy range. They also begin to use counting and matching as well as the earlier perceptual strategies to find which of two sets is more and begin to learn the meaning of the word less.

Relations

Children at this level continue to use the perceptual strategies they used earlier (general perceptual, length, density) but they can also begin to use matching and counting to find which is less and which is more (see research summarized in Clements and Sarama, 2007, 2008; Fuson, 1988, 1992a, 1992b; Sophian, 1988). However, they can also be easily misled by perceptual cues. For example, the classic tasks used by Piaget (1941/1965) involved two rows of objects in which the objects in one row were moved apart so one row was longer (or occasionally, moved together so one row was shorter). Many children ages 4 and 5 would say that the longer row has



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